H K Khalil Nonlinear Systems 3rd Edition 2002

Nonlinear Systems Third Edition

HASSAN K. KHALIL Department of Electrical and Computer Engineering

Michigan State University

PRENTICE HALL Upper Saddle River, New Jersey 07458

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Contents

< , .

Preface xiii

1 Introduct ion ' 1 1.1 Nonlinear Models and Nonlinear Phenomena . . . . . . . . . . . . . . . 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples 5 . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Pendulum Equation 5

1.2.2 Tunnel-Diode Circuit . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Mass-Spring System . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.4 Negative-Resistance Oscillator . . . . . . . . . . . . . . . . . . 11 1.2.5 Artificial Neural Network . . . . . . . . . . . . . . . . . . . . . 14 1.2.6 Adaptive Control . . . . . . . . : . . . . . . . . . . . . . . . . 16 1.2.7 Common Nonlinearities . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Second-Order Systeins 3 5 . . .

. . . . . . . . . . . . . . . . . . 2.1 Qualitative Behavior o f Linear Systems 37 2.2 Multiple Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ~ ' . . 2.3 Qualitative Behavior Near Equilibrium Points . . . . . . . . . . . . . . . 51 2.4 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

. . . . . . . . . . . . . . . . 2.5 Numerical Construction o f Phase Portraits 59 . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Existence o f Periodic Orbits 61

2.7 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3 F'lindamental Properties 87 3.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.2 Continuous Dependence on Initial Conditions

and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3 Differentiability o f Solutions and Sensitivity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations 99 3.4 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

vii

continuous, square-integrable functions; that is, Sr uT(t)u(t) dt < m. TO 1 1 1 ~ -

sure the size of a signal, we introduce the norm function ( ( ~ ( 1 , which satisfies tllrce properties:

The norm of a signal is zero if and only if the signal is identically zero and is strictly positive otherwise.

Scaling a signal results in a corresponding scaling of the norm; that is, llnull = aJlu\l for any positive constant a and every signal u.

The norm satisfies the triangle inequality ((ul + u2(( I ((ulll + Ilu2ll for any signals ul and uz.

For the space of piecewise continuous, bounded functions, the norm is defined as

and the space is denoted by L g . For thc space of piecewise continuous, square- integrable functions, the norm is defined by

and the space is denoted by LF. More generally, the space L r for 1 < p < m is defined as the set of all piecewise continuous functions u : (0, bo) -r Rm such that

The subscript p in L r refers to the type of pnorm used to define the space, while the superscript m is the dimension of the signal u. If they are clear from the context, we may drop one or both of them and refer to the space simply as Lp, Lm, or L. To distinguish the norm of u as a vector in the space t from the norm of u(t) as a vector in Rm, we write the first norm as 1 1 l l ~ . ~

If we think of u E Lm as a "well-behaved" input, the question to ask is whether the output y will be "well behaved" in the sense that y E Lq, where LQ is the same space as Lm, except that the number of output variables q is, in general, different from the number of input variables m. A system having the property that any "well-behaved" input will generate a "well-behaved" output will be defined as a stable system. However, we cannot define H as a mapping from Lm to LQ, because we have to deal with systems which are unstable, in that an input u E Lm may

2Note that the norm I(. 1) used in tho definition of 11. \IC,, for any p E (1, oo], can be any pnorm in fin'; thc number p is not necessarily thc samc in the two norms. For example, we may define the LW spwc with Ilullc, =SUPDO Ilu(t)l/l, II~llc, = SUPr>o Ilu(t)llz~ Or IIullc, = " P P ~ ~ O II4t)ll- Ijowevcr, it is common to define the Lz space with the 2-norm in Rm.

gcl~crate all output y that does not bclo~ig to LQ. Therefore, H is usually defined as a mapping from an extended space tz to an extended space Lj , where L r is defined by

L: = { u 1 u, E L ~ : V T E [O,m)}

and u, is a truncation of u defined by

The extended space L r is a linear space that contains the unextended space Lm as a subset. It allows us to deal with unbounded "ever-growing" signals. For example, the signal u(t) = t does not belong to the space L,, but its truncation

belongs to L, for every finite T. Hence, u(t) = t belongs to the extended space .c,e

A mapping H : L F -$ L j is said to be causal if the value of the output (Hu)(t) at any time t depends only on the Values of the input up to time t. This is equivalent

to ., (Hu)r = ( H U T ) ,

Causality is an intrillsic property of dynalnical systems represented by state models. With the space of input and output signals defined, we can now define input-

output stability.

Definition 5.1 A mapping H : Lp --, L j is L stable i f there exist a class K: function a, defined on (0, m ) , and a nonnegative constant P such that

II(Hu)r llc 5 c-k (llurllc) + P (5.1) for all u E LT and T E [0, m ) . It is finite-gain L stable i f there &t nonnegative constants y and ,O such that

for all u E L r and T E (0, m ) .

The constant ,f3 in (5.1) or (5.2) is called the bias term: It is included in the definition to allow for systems where Hu does not vanish a t 'u = 0.3 When inequality (5.2) is satisfied, we are usually interested in the smallest possible y for which there is P such that (5.2) is satisfied. When this value of y is well defined, we will call it the gain of the system. When inequality (5.2) is satisfied with some y 2 0, we say that the system has an L gain less than or equal to y.

3See Exercise 5.3 for a diKerent role of thc bias term.

198 CHAPTER 5. INPUT-OUTPUT STABILITY

For causal, L stable systems, it can be shown by a simple argument that

and IIHullc 5 a (llullc) + PI V E Lm

For causal, finite-gain L stable systems, the foregoing inequality takes the form

IlHullc I rlluTllc + P, v u E Lm

The definition of L, stability is the familiar notion of bounded-input-bounded- output stability; namely, if the system is L, stable, then for every bounded input u(t), the output Hu(t) is bounded.

Example 5.1 A memoryless, possibly time-varying, function h : [0, m) x R -, R can be viewed as an operator H that assigns to every input signal u(t) the output signal y(t) = h(t,u(t)). We use this simple operator to illustrate the definition of L stability. Let

for some nonnegative constants a, b, and c. Using the fact

we have Ih(u)l I a + bclul, V u E R

Hence, H is finite-gain L, stable with 7 = bc and /3 = a. F'urt,hermore, if a = 0. then for each p E [1 , oo) ,

I 1 Thus, for each p E 11, oo], the operator H is finite-gain Lp stable with zero bias and 7 = bc. Let h be a time-varying function that satisfies

for some positive constant a. For each p E [l, oo], the operator H is finite-gain L, stable with zero bias and 7 = a. Finally, let

Since 2

t>o

H is L, stable with zero bias and a ( r ) = r2. It is not finite-gain L, stable because the function h(u) = u2 cannot be bounded by a straight line of the form (h(u)( 5 r(u( + P, for fill u E R. A

C

5.1. 1: STABILITY 199

Example 5.2 Consider a single-input-singleoutput system defined by the causal convolution operator

y(t) = I t o h(t - u)u(u) du

where h(t) = 0 for t < 0. Suppose h E Lie; that is, for every T E [0, m),

I l h r l l ~ . = 1- lhr(u)l da = 1; l h ( a )~ da <

If u E Lxe and T 1 t , then

Consequently, Il~rllc, 5 Ilhrllcl ll~TlIc,r V E 1 0 , ~ )

C This inequality resembles (5.2), hut it is not the same as (5.2) because the constant

c -y in (5.2) is required to be indcpentlcnt uf T . While IlhTllcl is finitc for every finite T ,

C it inay not be bounded uniformly in r. For example, h(t) = et has l)h,)\c, = (eT -I), C which is finitc for all T E [O, oo) but not uniformly bounded in r. Inequality (5.2) will be satisfied if h E L1; that is.

Then. the inequality '1 !& I -

;:& sho~vs that the system is finite-gain L, stable. The condition Ilhllc, < actually

guarantees finite-gain Lp stability for rnrll p E [l,co]. Consider first. the case p = 1. : c:

For t 5 T < m, we have 1.d i _

Reversing the order of integration yields I. # I L..-

Thus,

. / . . . _ ' _ .._

200 CIIAPTER 5 . INPUT-OUTPUT STABILITY

Consider now the case p E (1, co) and let q E ( 1 , ~ ) be defined by l l p + l / q = 1. For t 5 T < co, we have

Jo l/P

5 (l lh(t - 41 d o ) llq (/f o lh(t - 41 iu(o)lp d o )

wherc the second inequality is ol)taiiletl I y npplying Holder's inequality.4 Thus,

By reversing the order of integration, we obtain

Hence, ~ ~ ~ ~ I L ~ ~ 5 ~ ~ h ~ ~ ~ l ~ ~ u r ~ l c J ~ 1

In summary, if h l l L , < co,,t.helhfor each p E [ I , co], the causal convolution operator is finite-gain Lp stable and (5.2) is satisfied with 7 = llhllLl and P = 0. A

One drawback of ~efini i ion 5.1 is the implicit requirement that inequality (5.1) or inequality (5.2) be satiified for all signals in the input space Lm. This excludes systems where the input-out.put relation may be defined only for a subset of the i~lput slx~(:(:. The next cxairiplc c~xplorcs t.hc point and motivates the definition of small-signal L stability that follows tlie example.

.'H61<ler1s inequality states that if f E C,,,. and g E Cp,, where p E ( 1 , ~ ) and l / p + l /q = 1, tllcll

for every r E [O,m). (See [14].)

,. - . ... . . - , < ; . . . - -- . . , . I ...- -.----*- "' . . . .lt, . : < . - . . _ . ._"

8 . . . . \ ' h a . . . . . .

2 5.2. L STABILITY OF STATE MODELS 201

Example 5.3 Colisider ti sii1gle-i1i~)11t-si1ig1~out~)ut systelii dcli~icd by 1110 lionlill- earity

y = t anu

The output y ( t ) is defined only when the input signal is restricted to

Thus, the system is not L, stable in the sense of Definition 5.1. However, if we restrict u ( t ) to the set

A ( u J 5 r < -

2 then

l ~ l 5 (F) I4 and the system will satisfy the inequality

for every u E Lp such that Iu(t)l 5 r for all t 2 0 , where p could be any number in [1, oo]. Ig the space L,, the requirement Iu(t)l 5 r implies that ~ J U ( ( ~ , 5 r , showing that the foregoing inequality holds only for input signals of small norm. However, for other Lp spaces with p < oo the instantaneoua bound on (u(t)l does not necessarily restrict the norm of the input signal. For example, the signal

u ( t ) = 4 > 0 -. ' ,-> .

which belongs to Lp for each p E [l,oo], satisfi'& the-instantaneous bound lu(t)1 5 r ' '

while its L, norm

can be arbitrarily large. A (

Definition 5.2 A mapping H : L r -, Lz is small-signal L stable (respectively, small-signal finite-gain L stable) if there is a positive constant r such that inequality (5.1) [respectively, inequality (5.2)] is satisfied for all u E L r with sup,,^,^, ((u(t)(l 5 r . 1

f

5.2 f, Stability of State Models

The notion of input-output stability is intuitively appealing. This is probably why most of us were iiltroduced to dynamical systein stability in tlic framework of bounded-input-bounded-output stability. Since, in Lyapunov stability, we put a

I


lot of emphasis 011 studying the stability of equilibriu111 poilits and blie asymptotic behavior of state variables, one may wonder: Whnt can we see about input-outpl~t stability starting from the formalism of Lyapunov stability? In this section. we show how Lyapunov stability tools can be used to establish L stability of nonlinear systems represented by state models.

Consider the system

where x E Rn, u E Rm, y E Rq, f : [0, co) x D x D, 4 Rn is piecewise continuous in t and locally Lipschitz in (2 , u), h : [O, m) x D x D, 4 Rq is piecewise cont.inuous in t and continuous in (I, u), D c Rn is a domain that contains x = 0, and D, C R'" is a domain that contains u = 0. For each fixed xo E D, the state model given by (5.3) and (5.4) defines all operator H that assigns to each input signal ~ ( t ) the corresponding output signal y(t). Suppose x = 0 is an equilihritim point of the unforced system

i = f(t,x,O) (5.5)

The theme of this section is that if the origin of (5.5) is uniformly asymptotir~lly stable (or exponentially stable), then, under some assumpt.ions on f and h. the system (5.3) and (6.4) will be L 3able or small-signal L stable for a certain sigl~ill space L. We pursue this idea first in the case of exponent.ially stability, and then for the more general case of uniform asymptotic stability.

Theorem 5.1 Consider the system (5.3)-(5.4) and take r > 0 and r, > 0 smh that (11x11 i: r r c D and (I(uJ( I rU) c Du. Suppose that

x = 0 iB an eezponentially stable equilibrium point of (6.5), and therc is u Lyapunov hnction V(t,x) that satisfies

1 for all ( t , x ) E [ O , c u ) x D for some positive constank cl , cz, CJ, an.d c4.

f and h satisfy the inequalities

I l f ( 4 X I U) - f ( t , x,O)Il I LII'LLII (5.9)

Ilh(t, 21 411 < 771 11x11 + 772 llull (5.10)

for all ( t , x , u ) 6 [O, m) x D x D, for some nonnegative constants L. 71, and 112

5.2. L STABILITY OF STATE MODELS 203

Then, for each xo with lxol\ 5 r m , the system (6.3)--(6.4) is snrcrll-siglrul finite-gain L, stable for each p E [ I , co]. In particular, for each u E Lp, with supo,,,, ((u(t)ll 5 mill{r,, c lc3r / ( c~c~L)} , the output ~ ( t ) satisfies

I I Y ~ I I L , I YJI'LLTIIL, + P (5.11)

for all 7 E [O, m), with

f 1, if p = m

Furthermore, if the origin is globally exponentially stable and all the assumptions hold globally (with D = Rn and D, = Rm), then, for each x0 E Rn, the system (5.3)-(5.4) is finite-gain Lp stable for each p E. [I, co].

0

Proof: The derivative of V along the trajectories of (5.3) satisfies

I -c311xl12 + c4LIlxl1 lI'1~1l

Take \V(t) = d m . Lirhen V ( t , r ( t ) ) # 0, use w = ~ / ( 2 f l ) and (5.6) to obtain

When V ( t , x ( t ) ) = 0, it can be verified5 that

Hence,

for all values of V( t , x ( t ) ) . By (the comparison) Lemma 3.4, I.V(t) satisfies the inequality

Using (5.6), we obtain

" ~ e e Exercise 5.6.

. ,' .. . . -: - - 2 .


It, (:ail I,(! casily verified that

ellsure that ( ( x ( t ) ( ( 5 r; hence, z ( t ) stays within the domain of validity of the n.sr~mpt~ioris. Using (5.10), we 1 1 i ~

where c3

= *, k3 = m,' a = - 2c1 2c2

Set

Suppose now that u E C$ for some p E [l , oo]. Using the results of Example 5.2, it can be easily verified that

k2 ( ( M ~ L T ( ~ c , ~ 5 a ( ( u T ( ( ~ p

It is also straightforward to see rhat

As for the first term, y l ( t ) , it car1 be verified that

Thus, by the triangle in equal it.^, (5.11) is satisfied with

When all the assumptions hold globally, there is no need to restrict l(xoll or the illst,ant,ancous values of Ilu(t)ll. Therefore, (5.11) is satisfied for each xo E Rn and 21 E Lpe.

The ~ w c of (the converse Lynpiinov) Throrrm 4.14 shows t.he existence of a Lya- 1111nt1v fiirlc*t,ion satisfying (5.6) tllrc111g11 (5.8). Comcc~uc~~tly, we haw t l ~ e following corol1al.y.

5.2. C STABILITY OF STATE AlODELS 208

Corollary 5.1 Suppose that, i n sonre neighborhood of ( x = 0 , u = 0) , the junction f ( t , x , u) is continuously diflerentiable, the Jacobian matrices [ 8 j /Bx] and [a f /au] are bounded, uniformly in t , and h ( t , x, u) satisfies (5.10). If the origin x = 0 is an exponentially stable equilibrium point of (5.5), then there i s a conatant ro > 0 such that for each xo with llxoll < ro, the system (5.3)-(5.4) is small-signal finite-gain tp stable for each p E [ I , oo]. Furthermore, i j all the assumptions hold globally and the origin x = 0 i s a globally exponentially stable equilibrium point of (5.5), then for each t o E Rn, the system (5.3)-(5.4) i s finite-gain Lp stable for each p E [l ,oo]. 0

For the linear time-invariant system

the global exponential stability condition of Theorem 5.1 is equivalent to the condition that A is Hurwitz. Thus, we have the following result for linear systems:

Corollary 5.2 The linear time-invariant system (5.14)-(5.15) is finite-gain Lp stable for each p E [ l , oo] i f A i s Hurwitz. Moreower, (5.11) i s satisfied with

where

f i f p = o o

and P is the solution of the Lyapunov equation PA + A ~ P = - I . 0

We leave it for the reader to derive the foregoing expressions for 7 and P.

Example 5.4 Consider the single-input-single-output, first-order system

The origin of j. = -I - x3 is globally exponentially stable, as can be seen by the Lyapunov function V ( x ) = x2/2 . The function V satisfies (5.6) through (5.8) globally with cl = c2 = 1/2 , cs = cd = 1. The functions f and h satisfy (5.9) and (5.10) globally with L = 711 = q2 = 1. Hence, for each xo E R and each p E 11, oo], the system is finite-gain L,, stable. A


Example 5.5 Consider the single-input-singleoutput second-order system

$/ = 21

where a is a nonnegative constant. Use .". U V(x) = xTpx = p11x: f 2~1251~2 +p22x; &"

as a Lyapunov function candidate for the ullforced system:

li = -2plz(x:+ax1 tanhx1)+2(~ll - P I P - P ~ ~ ) X I X P - ~ ~ P P ~ X P tanh XI - 2 ( p ~ ~ - ~ ~ ~ ) x i

Choose pll = pl2 + p22 to cancel the cross-product term 21x2. Then, taking p22 = 2p12 = 1 makes P positive definite and results in

v = -2: - x; - ax1 tanhxl - 2ax2 t,anhxl

Using the fact that XI tanhxl 2 0 for all X I E R, we obtain

Thus, for all a < 1, V satisfies (6.6) through (5.8) globally with cl = Xmi,(P), CP = Am,(P), ca = 1 - a, and c4 = 21(P((2 = 2Am,(P). The functions f and h satisfy (5.9) and (5.10) globally with L = ql = 1, 112 = 0. Hence, for each $0 E R and each p E [1, oo], the system is finite-gain L, stable. A

We turn now to the more general case when the origin of (5.5) is uniforlnlp asymptotically stable and restrict our attention to the study of L, stability. The next two theorems give conditions for small-signal L, stability and L, stability, respectively

Theorem 5.2 Consider the systen~ (5.3)-(5.4) and take r > 0 such that {(Jx(I 5 r ) C D. Suppose that

x = 0 is a uniformly asymptotically stable equilibrium point of (5.5). nnd there is a Lyapunov function V(t,x) that satisfies

for all ( t . ~ ) E [O. c ~ ) x D fov some class K functions crl to a4. 2,

5.2. L STABILITY OF STATE MODELS 207

f and h satisfy the inequalities

I l h ( t j ~ l ~ ) \ l 5 a~(11~11) + a7(llull) + q (5.20)

for all (t, x, u) E [O, CO) x D x D,, for some class K functions as to a7, and a nonnegative constant q.

Then, for each xo with llxoll 5 cr;'(al(r)), th'< system (5.3)-(5.4) is small-signal L, stable. 0

Proof: The derivative of V along the trajectories of (5.3) satisfies

where 0 < 0 < 1. Set

ad?.)as ( s u ~ o ~ t ~ , Ilu(t) 11) =a,' 6

and choose r, > 0 small enough that {lluII I r,) C DU and 1-1 < a i l ( a l ( r ) ) for SUP,^,^, llu(t)ll 5 ru. 'rhen,

By app1yillg Theorem 4.18, we conclude from (4.42) and (4.43) that ((x(t)(( satisfies the inequality

(5.21)

for all 0 I t I s, whcre f i and -, arc class KC and class K filnctions, respectively. Using (5.20), we obtain

where .we used t.he general property of class K functions

a (a + b) 5 a(2a) + a(2b)

=See Exercise 4.35.

:> ; Thus, ' j I I Y T I ~ L , 5 YO ( I IUT~IL~) + PO (5.22)

where yo = a s 0 27 + a7 and Po = a~(2P(Il~oll , 0)) +

.., The use of (the converse Lyapunov) Theorem 4.16 shows the existence of a - Lyapunov function satisfying (5.16) hhrough (5.18). Consequently, we have the

-\ following corollary: .-I

Corollary 5.3 Suppose that, i n some neighborhood of (x = 0, u = O), the function ./ f ( t , x, U) is continuous1y differentiable, the Jacobian matrices [a f /ax] and [a f /Bu] . ~,) are bounded, uniformly in t , and h(t, x, u) satisfies (5.20). If the unforced system -. . . .i (5.5) has a uniformly asymptotically, stable equilibrium point at the origin x = 0, - then the system (5.3)-(5.4) is small-signal L, stable. 0 .-.'\

To extend the proof of Theorem 5.2 to show Lm stability, we need to demon- .J - strate that (5.21) holds for any initial state s o E Rn and any bounded input. As we .-' discussed in Section 4.9, such inequality will not automatically hold when the con- . I - clitions of Theorem 5.2 are satisfied globally, even when the origin of (5.5) is globally

uniformly asymptotically stable. However, it will follow from input-testate stability ,/' of the system (5.3), which can be checked using Theorem 4.19. , , ,'

Theorem 5.3 Consider the system (5.3)-(5.4) with D = Rn and D, = Rm. Sup- .~,, .. pose that

The system (5.3) is input-to-state stable.

h satisfies the inequality

for all ( t , x, u) E [0, oo) x R" x Rm for some class K: functions a l , 122, and a nonnegative constant q .

Then, for each xo E Rn, the system (5.3)-(6.4) is L, stable. 0

Proof: Input-testate stability shows that an inequality similar to (5.21) holds for any xo E Rn and any u E Lmf Tlie rest of the proof is the same as that of Theorem 6.2.

Example 5.6 Consider the single-input-singlooutput first-order system

" $ . . .. 4, . -.. I . .... . . i-. .. .. ' . d a , . . ' _ . _ .-.-..- -! .,-.,,,----. - ..a

5.3. L2 GAIN 209

We saw in Example 4.26 tliut the s t~i tc otluntio~l is input-t o-stat>e st.111)lc. T l ~ c ootput function h satisfies (5.23) globally with a l ( r ) = r2, a2(r) = r , and q = 0. Thus, the system is L, stable. A

Example 5.7 Consider the singleinputsingleoutput second-order system

where g(t) is continuous and bounded for all t 2 0. Taking V = (x: +xi ) , we have

Using

yields

Thus, V satisfies inequalities (4.48) and (4.49) of Theorem 4.19 globally, with al (r) = az(r) = r2, W3(x) = -(1 - 0) llxllf, and p(r) = (2r/B)ll3. Hence, the state equation is input-testate stable. Furthermore, the function h = x l + 2 2 satisfies (5.23) globally with a1 (r) = f i r , a 2 = 0, and q = 0. Thus, the system ia L, stable. A

5.3 L2 Gain

L2 stability plays a special role in systems analysis. It is natural to work with square- integrable signals, which can be viewed as finiteenergy signals.' .In many control

the system is represented as an input-output map, from a disturbance input u to a controlled output y, which is required to be small. .With L2 input signals, the control system is designed to make the input-output map finitegain L2 stable and to minimize the L2 gain. In such problems, it is important not only to be able to find out that the system is finitegain L2 stable, but also to calculate the L2 gain or an upper bound on it. In this section, we show how to calculate the L2 gain for a special class of timeinvariant systems. We start with linear systems.

'If you think of u(t) as current or voltage, then uT(t)u(t) is proportional to the instantaneoue power of the signal, and its integral over all time is a measure of the energy content of the signal.

the literature on H , control; for example, (201, [54], [61], [go], (1991, or (2191.


Theorem 5.4 Consider the linear time-invariant system

2 = A x + B u (6.24) y = C x + D u (5.25)

where A is ~ u k t z . Let G ( s ) = C ( s I - A)- 'B + D . Then, the Lz gain oj the system is SUPwE~ l l G ( j w ) l l ~ . ~ 0

Proof: Due to lincnrity, we set x(0) = 0. From Fourier trnnsforln tlleory,'" wc know that for a causal signal y E L2 , the Fourier transform Y ( j d ) is given by

and Y ( j w ) = G ( j w ) U ( j w )

Using Parseval's theorem," we can write

I which shows that the Lz gain is less than or cqual to SUPwE~ (IG(jw)1J2. Showing that the Lz gain is equal to S\IPwE~ 1JG(jw)II2 is dolie by a contrudictioii arguilieilt that is given in Appelidis C.10.

The case of linear time-inwriant systems is cxccptional ill that we can actually find the exact Lz gain. In more general cases. like the case of the next theorem. we can only find an upper bound on the Lz gain.

BThis is the induced 2-norm of the complex matrix G ( j w ) , which is equal to Am,[GT(-jw)G(jw)] = umsx[G(jw)]. Thls quantity is known as the H, norm of G ( j w ) ,

$en G ( j w ) is viewed r an element of the Hardy apace H,. (See [GI].) 'Osee (53). 'LPamval's theorem (531 states that for a caus~l signal y E Ls,

5.3. L2 GAIN

Theorem 5.5 Consider the time-invariant nonlinear system

where f ( r c ) is locally Lipschitz, and G ( x ) , h ( x ) are continuous over Rn . The matrix G is n x m and h : Rn -t Rq. Tllc fi~nctions f and h vanish at the origin; that is, f ( 0 ) = 0 and h(0) = 0 . Let y be a positive number and suppose there is a continuowly diffei~ntiable, positil~e senridefinite function V ( x ) that satisfies the inequality

def DV 1 av 1 ( V , f G , h 7 ) = f ( x ) - G ( x ) G T ( ) ( ) +5hT(x )h (x ) 5 0 (5.28)

27 a x

for all x E Rn. Then, for each xo E Rn, the system (5.26)-(5.27) is finite-gain L2 stable and its L2 gain is less than or equal to 7 . 0

Proof: By completing the squares. we have

Substituting (5.28) yields

Hence,

Sote that. tlie left-liand side of (5.29) is the derivative of V along the trajectories of the system (5.2G). Integrating (5.20) yields

where x( t ) is the solution of (5.26) for a given u E LZe. Using V ( x ) > 0, we obtain

Taking the square roots and using the inequality d m 5 a + b for nonnegative! numbers a and b, we obtain

(6.30) ll~l'llc? 5 711~~rllc2 + d m

which completes the proof. •

Inequality (5.28) is knawn as the Hamilton-Jacobi inequality (or the Hamilton- Jacobi equation when < is replaced by =). The search for a function V(x) that satisfies (5.28) requires basically the solution of a partial differential equation, which might be difficult to solve. If we succeed in finding V(x), we obtain a finite-gain L2 stability result, which, unlike Theorem 5.1, does not require the origin of the unforced system to be exponent,ially stable. This point is illustrated by the next example. . .

Example 5.8 Consider the single-input-single-output system

where a and k are positive constants. The unforced system is a special case of the class of syst.ems treatad in Esnlnplc 4.9. In that example, ivc used the energy- like Lyapunov function V(x) = ar f /4 + xi12 to show that the origin is globally asymptotically stable. Using V(x) = a(axfl4 + x;/2) with a > 0 as a candidate for the solution of the Hamilton-Jacohi inequality (5.28), it can be shown that

The idea of t.he preceding example is generalized in the next one.

Example 6.9 Consider the nonlinear system (5.26)-(5.27), with m = q, and suppose there is a continuously differentiable positive semidefinite function W(x) that satisfies12

aw ( X ) < -khT&)h(x), k > O (5.32)

for all x E Rn. Using V(x) = aW(x) with a > 0 as a candidate for the solution of the Hamilton-Jacobi inequality (5.28), it can be shown that

To satisfy (5.28), we need to choose a > 0 and -y > 0 such that

This inequality is the same as inequality (5.31) of Example 5.8. By repeating the argument used there, it can be shown that the system is finite-gain L2 stable and its Lz gain is legs than or equal to l lk . A

a2 1 Z(V, f., G, h, y) = -ak + - ( 272 + 1) xi ,

Example 5.10 Consider the nonlinear system (5.26)-(5.27), with m = q, and suppose there is a continuously differentiable positive semidefinite function W(x) that

To satisfy (5.28), we need to choose a > 0 and 7 > 0 such that satisfies13 aw

d2 1 - f ( ~ ) ax 5 0 (5.34) - a b + - + - < O

27' 2 - (5.31) aw -G(x) = hT(x) ax (5.35)

By simple algebraic manipulati~n, we can rewrite this inequality as ' for all x E Rn. The output feedback control \ a2

7 2- u = - k y + v , k > O 2ak - 1 results in the closed-loop system Since we are interested in the smallest possible 7, we choose a to minimize the

right-hand side of the preceding i~aquality. The minimum value l/k2 is achieved = aw

at a = 1/k. Thus, choosing 7 = l /k , we conclude that the system is finite-gain f(x) - ~G(X)GT(X) (z) + G(X)V dl( fC(x) + G(X)V L2 stable and its Lz gain is less than or equal to I lk . We note that. the conditions of Theorem 5.1 are not satisfied in t.his example because the origin of the unforced y = h(x) = GT(x) - system 'is not exponentially stable. Linearization at the origin yields the matrix (z)

I

''A system satisfying (5.32) and (5.33) will be defined in the next chapter as an output strictly passive system.

13A system satisfying (5.34) and (5.35) will be defined in the next chapter as a passive system. We will come back to this example in Section 6.5 and look at it as a feedback connection of two

which is not Hurwitz. A passive systems.


It can bc easily verified that. for tlic closed-loop systcni, I.V(x) sntisfics (5.32) and (5.33) of the previous example. Hence, the input-output map from v to y is finite- gain L2 stable and its L2 gain is less than or equal to ilk. This shows, in essence, that the L2 gain can be made arbitrarily small by choosing the feedback gaiu k sufficiently large. A

Example 5.11 Consider the linear time-invariant syst.em

5 = A x + B u

y = C x

Suppose there is a positive semidefinite solution P of the Riccati equation

for some 7 > 0. Taking V ( x ) = ( 1 / 2 ) x T P x and using the expression [ 8 V / a z ] = x T P , it can be easily seen that V ( x ) satisfies the Hamilton-Jacobi equation

Hence, the system is finite-gain L2 stable and it,s L2 gain is less than or equd to 7. This result gives an alternative method for computing an upper bound on the L2 gain, as opposed to the frequency-domain calculation of Theorem 5.4. I t is interesting to note that the existence of a positive semidefinite solution of (5.3G) is a necessary and sufficient co~idition for the L2 gain to be less than or equd to ?.I4

A

In Theorem 5.5, we assumed that the assumptions hold globally. It is clear from the proof of the theorem that if the assumptions hold only on a finite domain D, we will still arrive a t inequality (5.30) as long as the solution of (5.26) stays ill D.

Corollary 5.4 Suppose the assumptions of Theorem 5.5 are s a t w e d on n domnin. D C Rn that contains the origin. Then, for any x0 E D and any u E Cle for which the solution of (5.26) satisfies x ( t ) E D for all t E 10, T ] , we have

I I ~ r l l ~ z I7l l~rl l~Z + @TJ

Ensuring that the solutioil x ( t ) of (5.26) remains in some neighborhood of the origin, when both ))xo(( and supoltlrl)u(t)ll are sufficiently small, follows from asymp totic stability of the origin of k = f ( x ) . This fact is used to show small-signal L2 stability in the next lemma.

'"SW 1541 for tlrc proof of ~ r ~ c ~ i t y . C

5.3. Lz GAIN

Lemma 5.1 Suppose tlre assuttlplio7rs of Tlreorem 5.5 an: sulis/icrl on u rlo71rr~i7r D c R" that contains the origin, f ( x ) is continuously diferentiable, and x = 0 is an asymptotically stable equilibrium point of x = f ( x ) . Then, there is kl > 0 such that for each xo with Ilxol) <- kl, the sy8tem (5.26)-(5.27) is small-signal finite-gain L2 stable with L2 gain less than or equal to 7. 0

Proof: Take r > 0 such that {llxll 5 r ) c D: By (the converse Lyapunov) The- orem 4.16, thcrc exist ro > 0 and a coiitinuously differentiable Lyapunov function W ( x ) that sat.isfics

a1(lIxll) 5 I.V(x) 5 a2(11x11)

for all J J x ) ) < ro: for some class K functions a1 to as . The derivative of W along the trajectories of (5.26) satisfies

c. 4

where k is an upper bound on )/6L1'/8x(J. L is a Lipschitz constant of f with respect I &

to u , and 0 < 0 < 1. Similar to the proof of Theorem 5.2, we can apply Theorem 4.18 j < to show that there exist a class KC function 0, a class K function 70, and positive constants kl and k2 such that, for any initial state xo with llxoll 5 kl and any input u ( t ) with supost,, JJu(t)ll 5 k2, the solution x ( t ) satisfies 'C

I I X ( ~ ) I I 5 P ( I I X O I I . ~ ) + 70 (ozwT I I ~ ( ~ ) I I )

for all 0 I t 5 T . Thus, by choosing k1 a id k2 small enough, we can be sure that ' L<

((x(t)(I 5 I. for all 0 5 t 5 7. The lemnia follows then from Corollary 5.4. : a: 3

; d: To apply Lemma 5.1, we need to check asymptotic stability of the origin of x = f ( x ) . This task can be done by using linearization or searching for a Lyapunov function. The next lemma shows that, under certain conditions, we can use the same function V that satisfies the Hamilton-Jacobi illequality (5.28) as a Lyapunov function for showing asyniptotic stability.

Lemma 5.2 Suppose the assumptions of Theorem 5.5 are satisfied on a domain D C Rn that contains the origin, f ( x ) is continuously diflerentiable, and no solution

' L

of x = f ( x ) can stay identically in S = { x E D I h ( x ) = 0) other than the trivial I

solution x ( t ) r 0. Then, the ortgin of x = f ( x ) is asymptotically stable and there is kl > 0 such that for each xo with ))xO1l 5 kl: the system (5.26)-(5.27) is small-signal finite-gain L2 stable u~t th L2 gain less than or equal to 7.

- - - . . / . .............. ..... ....... .- . . -- . - . . .-

k.

. - - - --.-. .. . . - .. - . / , . . , , . . . .I..-.

216 CHAPTER 5 . INPUT-OUTPUT STABILITY

Proof: Take u(t) 3 0. By (5.28), we have

Take T > 0 such that B, = {JJxJJ 5 T ) c D. We will show that V(x) is positive definite in B,. Toward that end, let +(t; x) be the solution of i = f (x) that starts at d(0;x) = x E B,. By existence and uniqueness of solutions (Theorem 3.1) and continuous dependence of the solution on initial states (Theorem 3.4), there exists 6 > 0 such that for each x E B, the solution d(t; x) stays in D for all t E [O, 61. Integrating (5.37) over [0, r] for r < 6, we obtain

Using V(+(r; x)) 2 0, we obtain 07

Suppose now that there i s 5 + 0 su~cll that V(E) = 0. The foregoing inequality implies that

1' Ilh(d(t; s))ll; dt = 0, V r E [O, 61 + h(+(t; 2)) r 0. V t E (0.4

Since during this interval the solr~tion stays in S and, by assumption, the only solution that can stay identically in S is the trivial solution, we conclude that $(t;x) r 0 + x = 0. Thus, V(x) is positive definite in B,. Using V(s) as a Lyapunov function candidate for x = f (x), we conclude from (5.37) and LaSalle's invariance principle (Corollary 4.1) that the origin of x = f (x) is asymptotically stable. Application of Lemma 5.1 completes the proof. 0

Example 5.12 As a variation on tke theme of Examples 5.8 and 5.9, consider the system

where a, k > 0. The function V ( x ) = a [a (2712 - xf/12) + xi/2], with cr > 0, is positive semidefinite in the set {Ixll 5 6). Using V(x) as a candidate for the solrrtion of t,he Hamilton-Jncobi inrtl~~alit,y (5.28), it can be shown that .

5.1. FEEDBACK SI'STEAfS: TIIE SAfALL-GAIN THEOREAl 217

Repeating the argument used in Example 5.8, it can be easily seen that by choosing a = 7 = 1/k, inequality (5.28) is satisfied for all x E R2. Since the conditions of Theorem 5.5 are not satisfied globally, we investigate small-signal finite-gain stabiiity by using Lemma 5.1. We need to show that the origin of the unforced system is asymptotically stable. This can be shown by linearization at the origin, which results in a Hurwitz matrix. Alternatively, we can apply Lemma 5.2, whose . conditions are satisfied in the domain D = {(xl( < d), because

Thus, we conclude that the system is small-signal finite-gain L2 stable and its Lz gain is less than or equal to 1/k. A

5.4 Feedback Systems: The Small-Gain Theorem

The formalism of input-output stability is particularly useful in studying stabiiity of interconnected systems, since the gain of a system allows us t o track how the norm of a signal increases or decreases as it passes through the system. This is particularly so for the feedback connection of Figure 5.1. Heri, we have two systems H1 : C r C,9 tnd Hz : Lz -* L,". Suppose both systems are finite-gain L stable;15 that is,

Suppose further that the feedback system is well defined in the sense that for everv .

pair of inputs u1 E L r and u2 E Lz, there exist unique outputs el, y2 E C r and e2, yl E Cf.16 Define

The question of interest is whether the feedback connection, when viewed m n .- - - mapping from the input u to the output e or a mapping from input u to the output y, is finite-gain C stable.17 It is not hard to see (Exercise 5.21) that the mapping from u to e is finite-gain C stable if and only if the mapping from u to y is finite-gain L stable. Therefore, we can simply say that the feedback connection is finite-gain

'=In this section, we present a version of the cla~sical small-gain theorem that applies to finite- gain t stability. For more general versions which apply to C stability, eee [93] and (1231.

'"ufficient conditions for existence and uniqueness of solutions are available in the iiterature. The most common approach uses the contraction mapping principle. (See, for example, (53, Theorem IIL3.11.) A more recent approach that makes use of existence and uniquenw of the solution of state equations can be found in (931.

"See Exercise 5.20 for an explanation of why we have to consider both Inputs and outputs in studying the stabiiity of the f e e d b d connection.

CHAPTER 5. INPUT-OUTPUT STABILITY

Figure 5.1: Feedback connection.

L stable if either mapping is finite-gain L stable. The following theorem, known as the small-gain theorem, gives a sufficient condition for finite-gain L stability of tlie feedback connection.

Theorem 5.6 Under the preceding assumptions, the feedback connection is finite- gain L stable ifrlr2 < 1- 0

Proof: Assuming existence of tlie solution, we can write

elr = ulr - ( H ~ Q ) ~ , e 2 ~ = U ~ T + (Hledr

Then,

'* Since 7172 < 1,

1 llelrll~ 5 - (Ilulrllc + r z l l~2~ l l c + P2 + r ~ P 1 ) (5.40)

1 - 7172

for all r E [0, oo). Similarly, ,

1 l(e2~llc I - (11wtTllc + n ~ ~ ~ l r ~ ~ c +PI + ~ 3 2 ) (5.41)

1 - 7172

for all r E [O,oo). The proof is competed by noting that Ilellc 5 Ilelllc + llezllc, which follows from the triangle inequality.

5.4. FEEDBACK SYSTEAIS: THE SMALL-GAIN TIfEOREM 219

The feedback connection of Figure 5.1 provides a convenient setup for studying robustness issues in dynamical systcms. Quite often, dynamical systems suhjcct to model uncertainties can be represented in the form of a feedback connection with HI, say, as a stable nominal system and Hz as a stable perturbation. Then, the requirement 7172 < 1 is satisfied whenever 7 2 is small enough. Therefore, the small-gain thcorem provides a conceptual framework for understanding many of tlie robustness results that arise in the study of dynamical systems, especially when feedback is used. Many of the robustness results that we can derive by using Lyapunov stability techniques can be interpreted as special cases of the small-gain theorem..

Example 5.13 Consider the feedback connection of Figure 5.1. Let H1 be a linear time-invariant system with a Hurwitz square transfer function matrix G(s) = C(sI- A)-'B. Let Hz be a memoryless function e2 = $(t, y2) that satisfies

fiom Theorem 5.4, we know that HI is finite-gain L2 stable and its L2 gain is given by

71 = SUP llG(j4112 W E R

We have seen'in Example 5.1 that Hz is finitegain L2 stable and its L2 gain is less than or equal to 72. Assuming the feedback connection is well defined, we conclude by the small-gain theorem that it will be finite-gain L2 stable if 7172 < 1. A

Example 5.14 Consider the system

whcre f is a smooth function of its argumcnts, A is a Hurwitz matrix, -CA-'B = I , -. is a small positive parameter, and dl, dz arc disturbaiice signals. The liiiear part of this modrl represents actuator dynamics that are, typically, much faster t,haii thc plant dynamics represented here by the nonlinear equation S = f . The disturbance signals dl and d2 enter the system at the input of the plant and the input of the actuator, respectively. Suppose the disturbance signals dl and d2 belong to a signal space L, where L could be any L, space, and the control goal is to attenuate the effect of this disturbance on the state x. This goal can be met if feedback control can be designed so that the closed-loop input-output map from (dl, d2) to x is finite- gain L stable and the L gain is less than some given tolerance 6 > 0. To simplify the design problem, it is common to neglect the actuator dynamics by setting E = 0 and substituting v = -CA-I B(u + d2) = u + d2 in the plant equation to obtain the reduced-order model

x = f(t,rr,u f d)

....... - .... ....... .. .. ., - . I '. . , .. *!, ' . + .

. & * ..\_ . .

... . . .... . . . . . ..... . . - -.- - . . .-.. . '. . .-:. .- ,.- -.. r ...- ... b + I .

CIqAPTER 5. INPUT-OUTPUT STABILITY 5.4. FEEDBACK SYSTEMS: THE SAlALL-GAIN THEOREAl 221

where d = dl +d2 . Assuming that the state variables are available for measurement, opens the loop and the overall closed-loop system reduces to the nominal one. ~~t we use this model to design a state feedback control law u = y ( t , x ) to meet the us assume that the feedback function 7 ( t , x ) satisfies the inequality design objective. Suppose we have succeeded in designing a smooth state feedback control u = ~ ( t , x ) such that 11: + $ f ( t l x j 7 ( t j x ) + e l ) i c i l l~ l l +c2llelll I1 (5.43)

I I ~ L 5 7 ~ ~ d ~ ~ c + f i (5.42) for all ( t ~ X, e l ) , where ci and c2 are nonnegative constants. Using (5.42) and (5.43),

for some y < 6. Will the control meet the design objective when applied to the it can be shown that

actual system with the actuator dynamics included? This is a question of robustness IlYlll~ 5 ~ l l e l l l ~ + PI of the controller with respect to the unmodeled actuator dynamics.18 When the where control is applied to the actual systeln, the closed-loop equation is given by 71 = c17 + C2, Pi = C I P

Since H2 is a linear time-invariant system and A is Hurwitz, we apply Corollary 5.2 to show that H2 is finitegain tp stable for any p E [ I , w ] and

Let us assume that d2( t ) is differentiable and d2 E L. The change of variables I ~ Y ~ I L 6 ̂ ~zlkzllr. + P 2 U~yf l l e z l l r . + , ~ 2

where q = z + A-'B(?(t , X ) + d?( t ) ]

brings the closed-loop system into the form 7 f = 2~k~=(Q)IIA-1Bl1211~l12, P2 p , l c l 1 2 1 1 ~ ~ o ~ l l /- L , = ( Q )

Amin(&) xmin(Q) j. = f ( t . x , $ 4 5 ) + d ( t ) + C q )

1, i f p = w €4 = AV + E A - ~ B [ ? + d 2 ( t ) ]

where 7 or = - + - f ( t , x , y ( t , x ) + d ( t ) + C q ) a t a x and Q is the solution of the Lyapunov equation QA+ATQ = -I . lg Thw, ssuming

It is not difficult to see that the closed-loop system can be represented in the form the feedback connection is well defined, we conclude from the small-gain theorem of Figure 5.1 with Hl defined by that the input-output map from u to e is f, stable. &om (5.40), we have

S = f ( t , x , r ( t , 5 ) + e l ) ar a7

Yi .= ? = ~ + ~ f ( t , s , ~ ( t , x ) + e ~ )

H2 defined by

and ~1 = dl + d? = d , u2 = d2

In this representation, the system H I is t,he nominal reduced-order closed-loop system, while Hz represents the effect of the u~lmodeled dynamics. Setting E = 0

'"n Example 11.14, we investigate a sinrilar robustness problem, in the context of stabilization, using singular perturbation theory.

Using

llxllr. 5 ~l le l l l r . + P which follows from (5.42), and the definition of u1 and u2, we obtain

I t is interesting to note that the righehand side of (5.44) approaches

as E -+ 0 , which shows that for sufficiently small E the upper bound on the L gain of the map from d to x , under the actual closed-loop system, will be close to the corresponding quantity under the nominal closed-loop system. A

''P of Corollary 5.2 is taken as EQ SO that (fQ)(A/f) + (A/f)=(fQ) = - I .


5.5 Exercises P. 5.1 Show that the series connection of two L stable (respectively, finite-gain L

stable) systems is L stable (respectively, finite-gain L stable).

5.2 Show that the parallel connection of two L stable (respectively, finitegain L stable) systems is L stable (respectively, finite-gain L stable).

5.3 Consider a system defined by the memoryless function y = u'l3.

(a) Show that the system is L, stable with zero bias.

(b) For any positive constant a, show that the system is finite-gain L, stable with 7 = a and 0 = (l/a)'I2.

(c) Compare the two statements.

5.4 Consider a system defined by the memoryless function by y = h(u), where h : Rm + R* is globally Lipschitz. Investigate L, stability for each p E 11, co] when

5.5. EXERCISES

(a) on-off with hysteresis

94

(1) h(0) = 0. (2) h(0) # 0.

6.5 For each of the relay charactcristices sllown in Figure 5.2, investigatc L, aid "

L2 stability.

5.6 Verify that D+W(t) satisfies (5.12) when V(t, x(t)) = 0.

. ?- (c) Ideal on-off

Hint: Using Exercise 3.24, show that V(t + h,x(t + h)) 5 ~ 4 h * L ~ 1 / ~ ) ) ~ / 2 + h ~ ( h ) , ~t

where o(h)/h + 0 as h + 0. Then, use the fact that q 2 2cl.

5.7 Suppose the assumptions of Theorem 5.1 are satisfied, except (5.10), which is Figure 5.2:

replaced by Ilh(t,x,~)II < 7 1 ~ 1 1 ~ 1 1 + v2llull+ 1731 713 > 0 5.11 For each of the following systen

Show that the system is small-signal finite-gain L, stable (or finite-gain L, stable, (1) i l = -XI + x:x2, if the assumptions hold globally) and find the constants 7 and 0 in (5.11). ,

5.8 Suppose the nss~lmptions of Theorem 5.1 are satisfied, except (5.10), which i (2) ~1 = -XI + 2 2 ,

replaced by (5.20). Sliow that the systelrl is small-signal L, stable (or L, stab if the assumptions hold globally). (3) 51 = (xl + u)(llx11; - I),

5.9 Derive a result similar t,o Corollary 5.2 for linear time-varying systems. (4) 5 1 = - ~ 1 - ~ ~ + ~ ~ ,

5.10 For each of tlic following syst.clns, investigate L, and finitegain L, stability: (5) ~1 = -XI +x:x2,

jr = - ( I + U ) X ~ jr = -(l + u)x3 - x5 y = x (2) y = X + U (6) 51 = x2,

x = -x/(l +x2) + u .+ = -x - 53 +x2u (7) 51 = -XI - 12, (3) = r / ( l + s 2 ) (4) 9 = xsinu

. , , , . , . - : : . . -.r, .I-..I. . . s .

. a , .

," , , - , , . . . - ". , :. . ..--- . .. . ' . ..--

(b) On-off with dead zone and hysteresis

I ((1) 011-off with dead zone

Relay characteristics

as. illvestigate L, and finite-gain L, stability:

x2 = -x; -x2 + u I y = x 1

x2 = -x; - x2 +u , Y = $2

2 = 2 - 1 y = X l

f 2 = X I -x;+u2, y=x1(x2+u1)

x2 = 21 - 2 2 + u, y = x 1 + u

i 2 = -2; - 2 2 + U, y = x*

~2 = XI - xg + u, ~ ( t ) = xi (t - 2')

3 I., -.

1' - I)

> -1

I

. --- . . . - . .. -. . . -- -- - . . . . . .. .. . .. . ... . ... . . . - 7 :" " $ . . ..:. . ... . - . .%. , -,. . ; L. .4.., . . d ,

, . . , . .. ' ... . .-. -,L, . ...

CIIAPTER 5. INPUT-OUTPUT STABILITY 5.5. EXEItCISES 226 224

5.12 Consider the system

where h is continuously differentiable, h(0) = 0, and zh(t) > az2 for all z E R, for some a > 0. Show that the system is finite-gain Lp stable for each p E [ l ,m].

5.13 ([192]) Consider the time-invariant system

j. = f(x, u), y = h(x, 21)

whcrc f is locally Lipschitz, h is continuous, f (0,O) = 0, and h(0,O) = 0. Suppose there is a continuously differentiable, positive definite, radially unbounded function V(x) such that

Ef a x (2, u) 5 -IV(x) + $(,)I (x,u)

where W(x) is continuous, positive definite, and radially unbounded, $(u) is continuous, and $(O) = 0. Show that the system is Lm stable.

. .

5.14 Let H(s) be a Hunvitz strictly proper transfer function, and h(t) = L-'{H(s)} be the corresponding impulse response function. Show that

5.15 For each of the following systems, show that the system is finite-gain (or small-signal finite-gain) C2 stable and find an upper bound on the L2 gain:


where a is locally Lipschitz, u(0) = 0, and zu(z) 2 0 for all z E R.

(a) Is the system finite-gain L, stable?

(b) Is it finite-gain L2 stable? If ycs, find an upper bound on the L2 gain.

5.17 (1771) Consider tlie systcln

2 = f (x) + G(x)u, y = h(x) + J(x)u

where f , G, h, and J are smooth functions of x. Suppose there is a positive constant 7 such that 721 - JT(x)J(x) > 0 and

V x. Show that the system is finite-gain L2 stable with L2 gain less than or equal to 7. Hint: Set

and show that the following inequality holds V u

5.18 ([199])a Consider the system

where u is a control input and w is a disturbance input. The functions f , G, K, and h are smooth, and f(0) = 0, h(0) = 0. Let 7 > 0. Suppose there is a smooth posit.ive semidefinite function V(x) that sat.isfies

V x. Show that with the feedback control u = -GT(x)(dV/ax)T the closed-loop

map from u to [ : ] is finite-gain L2 stable with 4 gain less than or equd to 7.

5.19 ([200]) The purpose of this exercise is to show that the 122 gain of a linear time-invariant system of the form (5.24)-(5.25), with a Hurwitz matrix A, is the same, whether the space of functions is defined on R+ = [0, m ) or on the whole real line R = (-m, m). Let L2 be the space of square integrable functions on R+ with the norm ( I ~ l l : ~ = Jr uT(t)u(t) dt and L ~ R be the space of square integrable .

m functions on R with the norm llu((inR = J-m uT(t)u(t) dt. Let 72 and 72R be the L2 gains on L2 and LZR, respectively. Since L2 is a subset of LZR, it is clear that % 5 %R. We will show that 72 = YZR by showing that, for every E > 0, there is a signal u E LZ such that y E L2 and JJyllr, 2 (1 - E ) Y ~ R ~ ~ u ~ ~ L ~ .

226 CIfAPTER 8. INPUT-OUTPUT STABlLl'l'Y

(a) Given E > 0 show that we can always choose 0 < 6 < 1 such that

(b) Show that we can always select u E tzR and time tl < oo such that

(c ) Let u(t) = ~ ( t ) +uz(t), where u1 vanishes for t < tl and u2 vanishes for t > tl. Let pl(t) be the output corresponding to the input ul(t). Show that

(d) For d ' t > 0, define u(t) and p(t) by u(t) = ul(t + tl) and y(t) = yl(t + tl). Show that both u and y belong to L2, y(t) is the output corresponding to

and I ~ u I I c ~ 2 (1 - ~ ) " 1 2 ~ ~ 1 ~ 1 1 ~ ~ .

5.20 Consider the feedback connection of Figure 5.1, where Hl and Hz are linear timeinvariant systems represented by the transfer functions Hl (s) = ( s - l ) / ( s + 1) and Hz(s) = 1/(3 - 1). Find the closed-loop transfer functions from (u1,u2) t~ (pi, p2) and from ( ~ 1 , uz) to (el, el). Use these transfer functions to discuss why we have to considcr both inputs (u1,uz) and both outputs (el, e2) (or (ply B)) in studying the stability of the feedback connection.

6.21 Consider the feedback connection of Figure 5.1. Show that the mapping from ( ~ 1 , uz) to (pl,yz) is finitegain C stable if and only if the mapping from (ul, a2) to (el, ez) is h i t ega in t stable.

6.22 Let dz(t) = asinwt in Example 5.14, where a and w are positive constants.

(a) Show that, for sufficiently small E , the state of the closed-loop system is uniformly bounded.

(b) Investigate the effect of increcrsi~lg w.

5.23 Consider the feedback connection of Figure 5.1, where H1 and Hz are given by

il = -XI + 2.2 HI : { i2 = -xt - x2 + el and Hz : { :: = -x: e2

Yl = 2 2 = (1/2)x;

Let uz = 0, u = u1 be the input, and y = yl be the output.

(a) Using x = 1x1, xz, x3IT as the state vector; find the state model of the systsem.

(b) Is the syst.cm t2 st,able?

Chapter 6

Passivity

passivity provides us with a useful t.001 for the analysis of nonlinear syst.eins! whic2.i relates nicely to Lyapunov and C2 stal~ility. We start in Scctioll6.l I)y dcfiliing passivity of memorylcss ~lo~llil~caritics. \V(! cxtcnd the definition t . ~ tlynamiwl systc~ns, represented by state models, in Scctio~i 6.2. In both cases, we use clcctrical 1iet.works to motivate the definitions. In Section 6.3, we study positive real and stxictly positive real t.ra11sfer fullctions ant1 sho~v that they represent passive and strictly passi~~e systems, respectively. The coil~~ectioi~ bct~veen passivity ant1 both Lyapuno~~ and C2 stability is estahlislled in Section G.4. These four sections prepare us to address the main results of the c11al)tcr. 11~11101y. t,l~e passivity theorems of Sect.io11 6.5. The main passivity theorem stat.es that the (negative) feedback connection. of two passive systems is passive. Under additiollal obser~.ability conditions, t.lie feedback connection is also asymptotically stable. The passivity theorems of Section 6.5 and the small-gain theom111 of Sect.io11 5.4 l)rovidc a co~~ccptually importiult gc~lcri~li./.i~t.i~ of the fact that the feedback connectioll of two stable linear systems will be stable if t.he loop gain is less than 'one or the loop phase is less than 180 degrees. The connection between passivity a ~ ~ d thc phase of a transfer function comes from the frequency-domain cl~aractcrizfitio~l of positive real transfer funct,ions, give11 i11 Scc- tion 6.3. There we kl~ow that. tllc pllt~sc of u positive real transfer funct-io11 c~1111ot. exceed 90 degree. Hence, t.he lool) phase cannot exceed 180 degrees. If one of tile two transfer functions is strictly positive real, the loop phase will be strictly less than 180 degrees. Section 6.5 discusses also loop transformations, which allow us, in certain cases, to transform the feetlback connection of two s y s t e ~ ~ s that may l~ot, be passive into t.he feedback connection of t.wa passive systems! hence ext,endil~g t , l ~ utility of tlic passivity thcorc~lis.

. -

228 CHAPTER 6. PASSIVITY

Figure 6.1: (a) A passive resistor; (b) u-y characteristic lies in the first-third quadrant.

6.1 Memoryless Functions

Our goal in this section is to define passivity of the memoryless function y = h(t , u), where h : [O, co) x RP + RP. We uso electric networks to motivate the definition. Figure 6.l(a) shows a one-port resistive element with voltage u and current y. We view this element as a syst*em with input, u and output y. The resistive element is passive if the inflow of power is always n~nnegat~ive; that is, if uy 2 0 for all points (u, y) on its u-y characteristic. Geometrically, this means that the u-y curve must lie in the first and third,quadrants, as shown in Figure 6.l(b). The simplest such resistive element is a linear resistor that obeys Ohm's law u = Ry or y = Gu, where R is the resistance and G = l /R is the conductance. For positive resistallce, the u-y characteristic is a straight line of slope G and the prod~lct uy = Gy2 is always nonnegative. In fact, it is always positive except at the origi'n point (0,O). Nonlinear passive resistive elements have nonlinear u-y curves lying in the first and third quadrants; examples are shown in Figures 6.2(a) and (b). Notice that the tunnel-diode characteristic of Figure 6.2(b) is still passive even though the curve has negative slope in some region. As all example of an element that is not passive, Figure 6.2(c) shows the. u-y characteristic of a negative resistance that was used in Section 1.2.4 to construct the. negative resistance oscillator. Such characteristic can be only realized using active devices such as the twin tunnel-diode circuit of Figure 1.7. For a multipart network where u and y are vectors, the power flow into the network is the inner product uTy = Cf,,uiyi = Cy=luihi(u). The network is passive if uTy 2 0 for all u. ' This concept of passivity is now abstracted and nssigned to any function y = h(t, u) irrespective of its physical origin. We think of uTy as the power flow into the system and say that the system is passive if uTy 2 0 for all u. For the scalar case, the graph of the input-output relation must lie in the first and third quadrants. We also say that the graph belongs to the sector [O, a], where zero and infinity are the slopes of the boundaries of the first-third quadrant region. The graphical representation is valid even whea h is time varying. In this case, the u-y curve will be changing with time, but will always belong to the sector 10, oo]. For a vector function, we can give a graphical representation in the special case when h(t, u) is decoupled in t.11~ sc~~sc: t.h;~t. hi ( t , u) dc~)c~lcls ollly on ui; that is,

G. 1. MEA,IORYLESS FUNCTIONS 9'19 --

(a) (b) I

(c)

Figure 6.2: (a) and (b) are examples of nonlinear passive resistor characterirtiu; (c) is an example of a nonpassive resistor.

L hp(ttu,) 1 In this case, the graph of each component belongs to the sector [ O , 4 In the general case, such graphical representation is not possible, but we will continue to use the sector terminology by saying that h belongs to the sector [O, cm] if uTh(t, u) 1 0 for all (t,u).

An extreme case of passivity happens when uTy = 0. In this case, we say that the system is lossless: An example of a Iossless system is the ideal transformer shown in Figure 6.3. Here y - Su, where

The matrix S is skew-symmetric; that is, S + ST = 0. Hence, uTy = uTSu = (1/2)uT(S + ST)u =o.

Figure 6.3: Ideal transformer

Consider now a function h satisfying uTy 2 uTv(u) for some fuoetion ~ ( u ) . Wllcn uTp(U) > 0 for all u # 0, h is called input strictly passive because passivity is strict in the sense that uTY = 0 only if u = 0. Equivalently, in the malar case,


Figure 6.4: A graphical representation of uTy 2 euTu for (a) E > 0 (excess of passivity); (b) E < 0 (shortage of passivity): (c) removal of excess or shortage of passivity by input-feedforward operation.

the u y graph does not touch the u-axis, except at the origin. The term uTP(u) represents the LLexcess" of passivity. On the other hand, if uTp(u) is negative for some values of u, then the function h is not necessarily passive. The term uTp(u) represents the "shortage" of passivity. Excess and shortage of passivity are more transparent when h is scalar and ~ ( u ) = EU. In this case, h belongs to the sector [E, m), shown in Figure 6.4, with excess of passivity when E > 0 and shortage of passivity when E < 0. Excess or shortage of passivity can be removed by the input- feedforward operation shown in Figure 6.4(c). With the new output defined as 8 = y - p(u), we have

Thus, any function satisfying uTy > uTQ(u) can be transformed into a function that belongs to the sector [0, m] via input feedforward. Such a function is called input- feedforwad passive. On the other hand, suppose uTy > yTp(y) for some function p(y). Similar to the foregoing case, there is excess of passivity when yTp(y) > 0 for all y # 0, and shortage of passivity when yTp(y) is negative for some values of y. A graphical representation of the scalar case with p(y) = dy is shown in Figure 6.5. There is "excess" of passivity when 6 > 0 and shortage of passivity when 6 < 0. Excess or shortage of passivity can be removed by the output-feedback operation shown in Figure 6.5(c). With the new input defined as ii = u - p(y), we have

Hence, any function satisfying uTy 2 yTp(y) can be transformed into a function that belongs to the sector [0, m] via output feedback. Such a function is called output-feedback passive. When yTp(y) > 0 for all y # 0, the function is called output strictly passive because passivity is strict in the sense that uTy = 0 only if y = 0. Equivalently, in the scalar case, the u y graph does not touch the y-axis, except at the origin. For convenience, we summnrize the various notions of passivit.~ in tlic nest dchition.

C

6.1. IIIEI\IORYLESS FUNCTIONS 231

Figure 6.8: A graphical representation of r r T y 2 6yTy for (a) 6 > 0 (excess of passivity); (b) 6 < 0 (shortage of passivity); (c) removal of excess or shortage of passivity by output-feedback operation.

Definition 6.1 The system y = h(t:u) is

passive if uTy 2 0.

lossless if uTy = 0.

input-feedforward passive if uTy 2 uTrp(u) for some function rp.

input strictly passive if uTY 2 uTrp(u) und uTp(u) > 0, V u # 0.

output-feedback passive if uTy 2 yTp(y) for some function p.

output strictly passive i juTy 2 yTp(y) and yTp(y) > 0, V y # 0.

In all cases, the inequality sl~ould hold for ull (t, u).

Consider next a scalar function y = ll(t, u), which satisfies the in equal it,^

for all (t, u), where a and are real num1)crs with p 2 a. The graph of this function belongs to a sector whose boundaries are the lines y = a u and y = pu. We say that h belongs to the sector [a,P]. Figure 6.6 shows the sector [a ,P] for /3 > 0 and different signs of a. If strict inequality is satisfied on either side of (6.2), we say that h belongs to a sector (a,P], [a,P). or (a,P), with obvious implications. Comparing the sectors of Figure 6.6 with thosc of Figures 6.4 alid 6.8 shows that a function in the sector (a, P] combines input-feedforward passivity with output strict passivity since the sector [a, P] is the intersection of the sectors [a? m] and [O,p]. Indeed, we can show that such a function can be transformed into a function that belongs to the sector [O! x] by a sequence of input-feedforward and output-feedback operations. Before we do that. we extend the sector definition to the vector case. Toward that end, note that (6.2) is equivalent to

CHAPTER 6. PASSIVITY

. , __ __ _ _.-- __ . . T " - " , -- . ' < . . . . . ' .;; ,. ," ... ~ 3 . ,a

, ,.. . . . . . ;: 9, ., , b 4 . . . ~ . . . . .. ..- *-,:.,,- . . .n&a-.-"""-- -,

6.2. STATE AlODELS 233

Figure 6.6: The sector [alp] for ,f3 > 0 and (a) a > 0; (b) a < 0.

for all (t,u). For the vector case. let us consider first a function h(t,u) that is decoupled as in (6.1). Suppose each component h, satisfies the sector condition (6.2) with constants ai and 0, > a,. Taking

it can be easily seen that . 4

for all ( t ,u) . Note that EC; = K2 - K1 is a positive definite symmetric (diagonal) matrix. Inequality (6.4) may hold for more general vector functions. For example, suppose h(t, u) satisfies the inequality

for all (t, u). Taking K1 = L - TI and K2 = L + 71, we can write 4

Once. again, I< = K2 - KI is a positive definite symmetric (diagonal) matrix. We 4 use inequality (6.4) with a positive definite symmetric matrix K = K2 - K1 as a definition of the sector [K1, K2] in the vector case. The next definition summaries the sector tcrminology.

Definition 6.2 A memoyless ji~nction h : [O,m) x RP 4 fl is said to belong to the sector

[O, K2] with Kz = KT > 0 ij hT(t, u)[h(t, u) - K ~ u ] 5 0.

Figure 6.7: A function in the sector [Kl, Kz], where K = K2 - K1 = K T > 0, can be transformed into a function in the sector 10, w] by input feedforward followed by output feedback.

In all cases, the inequality should hold for all (t,u). I' in any case the inequality is strict, we write the sector as (0, w ) , (Kl , m) , (0, K2), or (K1, K 2 ) . In the scalar case, we mite (a , 01, [a, P), or (a , P) to indicate .that one or both sides of (6.2) is satisfied as a strict inequality.

The sector 10, m] corresponds to passivity. The sector IKl, w] corresponds to input- feedforward passivity with p(u) = Klu. The sector [0, K2] with K2 = (l/d)I > 0 corresponds to output strict passivity with p(y) = 6y. We leave it to the reader (Exercise 6.1) to verify that a function in the sector IK1, Kz] can be transformed into a function in the sector (0, m] by input feedforward followed by output feedback, as shown in Figure 6.7.

6.2 State Models

Let us now define passivity for a dynamical system represented by the state model

where j : Rn x RP --t Rn is locally Lipschitz, h : Rn x RP --t RP is continuous, j (0,O) = 0, and h(0,O) = 0. The system has the same number of inputs and outputs. The following RLC circuit motivates the definition.

Example 6.1 The RLC circuit of Figure 6.8 features a voltage source connected to an RLC network wit,h linear inductor and capacitor and nonlinear resistors. The nonlinear resistors 1 and 3 are represented by their v-i characteristics il = h l ( y ) and i3 = h3(v3), while resistor 2 is represented by its i-v characteristic v2 = hz(i2).

62. STATE I\IODEL.S CIIAPTER (i. PASSIVITY

We take the voltage u as the input and the current y as the output. The product uy is tbe power Row i1lt.o the netww.ork. Takillg the c~~rrellt 51 t l~ r~ug l l tllc ili(ltl(:t.or and the voltage xz across the capacitor as the state variables, we call write the state model as

The new feature of an RLC network over a resistive network is the presence of the energy-storing elements L and C. The system is passive if the energy absorbed by the network over any period of time [0, t ] is greater than or equal to the increase in the energy stored in the network over the same period; that is,

(6.8)

where V(x) = (1/2)Lx: + (1/2)Cxg is the energy stored in network. If (6.8) holds with strict inequality, then the difference between the absorbed energy and the increase in the stored energy must be the energy dissipated in the resistors. Since (6.8) must hold for every t 1 0, the illstantaneous power inequality

must hold for all t; that is, the power Row into the network must be greater than or equal to the rate of change of the energy stored in the network. We can investigate inequality (6.9) by calculating the derivative of V along the trajectories of the system. We have

Thus, try = l'~ + t~hl(tl) + .rlhz(.~l) t .r2h3(.r?)

Figure 6.8: RLC circuit of Example 6.1.

Tlle term uhl (u) could represent excess or shortage of passivity. If uhl ( u ) > 0 for all u + 0. there is excess of passivity since the energy absorbed over [0, t] will be greater than the increase ill the stored energy, unless the input u(t) is identically zero. This is a case of input strict passivity. On the other hand, if uhl(u) is negative for some values of u, there is shortage of passivity. As we saw with memoryless fullctions. this type of excess or shortage of passivity can be removed by input fectlforwartl shown in Figure 6.4(c).

Case 3: If hl = 0 and h3 E [ O ? x ] . Ic

Excess or shortage of passivity of h2 results in the same pr0pcrt.y for the network. Once again, as with ~nomoryless functions, this type of excess or shortage of passivity call be rr.1~1ovrtl by output feedback, as ill Figure G.5(c). When yh2(y) > 0 for all y # 0. wvc liave output strict passivity because the ' I

energy absorbed over (0: t] wvill be greater than the increase ill the stored energy, unless the output y ( t ) is identically zero.

Case 4: If hl E (0, m]? h2 E (0: x) . and h3 E (0, m) ,

Lf hi. hz. and ha me pi~.ssi\~. uu 2 i' nnd the system is passive. Other possibilities (UC i l l ls t ratd by follr different spcrinl rnscs of the network.

k s e 1: If hl = h: = h3 = 0. tiy = i- m d there is no energ dissipation iu the lwt\\vrk: lh~r is. tht- s p t t ~ n i s l l ~ s l c s .

whcrc x1h2(x1) + ~ ~ 1 1 : ~ ( x ~ ) is ;I j)osil.iv~ d~fillitc 1111i(:t,io11 01 .r. ?'his is ;I c:ilS(! of state strict passivity because thc crlcrgy absorbed over [O: t] wvill bc grcatcr than the increase in the stored cilcrgy, unless the state x(t) is identically zero. A system having this property is called state strictly passive or. simply strictly passice. Clearly there is 110 counterpart for state strict pasivity ill memoryless functions since there is no state.

1

' . . -- -. .. .. . . - .- - - . . . .

. . -:. . . . . . CHAPTER 6. PASSIVITY

Definition 6.3 The system (6.6)-(6.7) is said to be passive if there exists a continuously differentiable positive semidefinite function V ( x ) (called the storage function) such that av

uTY > v = - f ( x , u ) , V ( I , U ) E Rn x RP (6.10) a x

Moreover, it i s said to be

input-feedforward passive if u T y 2 v + uTcp(u) for some function cp. . i n p t strictly passive i f u T y _> v + uTcp(u) and u T q ( u ) > 0, v # 0

ontput-feedback passive if uT?j _> v + yTP(y) for some function p.

output strictly passive i f u T y > v + yTp(y) and yTp(y) > 0 , V y # 0.

strictly passive if u T y 2 v + $ ( x ) for some positive definite function $.

In all cases, the inequalitu sho?clrl liold l o r all ( I , u ) .

Definition 6.3 reads almost the same as Definition 6.1 for memoryless functions, except for the presence of a storage function V ( x ) . If we adopt the convention that V ( x ) = 0 for a memoryless function, Definition 6.3 can be used for both state models arid memoryless functioils.

Example 6.2 The integrator of Figure 6.9(a), represented by '

is a lossless system since, with V ( z ) = (1 /2)x2 as the storage function, we have u y = V . When a memoryless function is connected in parallel with the integrator, as shown in Figure 6.9(b), the system is represented b y

'\

Clearly, the system is input-feedforward passive since the parallel path h ( u ) can be cancelled by feedforward from the input. With V ( x ) = (1 /2 )x2 as the storage function, we have u y = ~ + u h ( u ) . If h E [O, m], the system is passive. If u h ( u ) > 0 for all u # 0 , the system is input strictly passive. When the loop is closed around an integrator with a memoryless funct.ion, as in Figure 6.9(c), the system is represented by

x = - h ( z ) + u , y = x

Plainly, the system is output-feedback passive since the feedback path can be call- celled by a feedback from the output. With V ( x ) = (1 /2 )x2 as the storage function, we have u y = v + yh(y) . If h E [O, oo], tllc systen~ is passive. If yh(y) > 0 for all y # 0, the system is output strictly passive. A

6.3. POSITIVE REAL TRANSFER FUNCTIONS 237

Figure 6.9: Example 6.2 I

Figure 6.10: Example 6.3 I Example 6.3 The cascade connection of an integrator and a passive memoryless function, s h o h jn Figure 6.10(a), is represented by

Passivity of h guarantees that h ( u ) du 2 0 for all x. With V ( x ) = h ( o ) du as the storage function, we have v = h ( x ) x = yu. Hence, the system is lossleap. Suppose now the integrator is replaced by the transfer hnction l / ( a s + 1) with a > 0, as shown in Figure 6.lO(b). The system can be represented by the state model

a x = - x + u Y = h ( x ) With V ( x ) = a l ; h ( u ) du as the storage function; we have

Hence, the system is passive. When x h ( x ) > 0 for all x # 0, the system is strictly passive.

A

6.3 Positive Real Transfer Functions B Definition 6.4 A p x p proper mtional tmnsfer function m a t h G ( s ) is called positive real if

poles of all elements of G ( s ) are i n Re[s] 5 0,


for all real w for which jw is not a pole of any element of G(s) , the matrix G(jw) + GT(-jw) is positive semidefinite, and

any pure imaginaly pole jw of any element of G(s) is a simple pole and the residue matrix lim,,jw(s - jw)G(s) is positive semidefinite Hermitian.

The transfer function G(s) is called strictly positive real ' if G(s - E ) is positiz~e real for some E > 0.

When p = 1, the second condition of Definition 6.4 reduces to Re[G(jw:)] 2 0, V w E R, which holds when the Nyquist plot of of G(jw) lies in the closed right- half complex plane. This is a condition that can be satisfied only if the relative degree of the transfer function is zero or one.2

The next lemma gives an equivalent characterization of strictly positive real transfer functions.

Lemma 6.1 Let G(s) be a p x p proper rational transfer function matrix, and suppose det [G(s) + GT(-s)j is not identically zero.3 Then, G(s) is strictly positive real if and only if

G(s) is Hurwitz; that is, poles of all elements of G(s) have negative real parts,

G(jw) + GT(-jw) is positive definite for all w E R, and

either G(w) + G T ( w ) is positive definite or it is positive semidefinite and lirn,,, w2 M T [ G ( j w ) + G T ( - j w ) ] ~ is positive definite for any px (p-q) full-

I rank mat* M such that h f T [ G ( ~ ) + G T ( ~ ) ] M = 0, where q = rank[G(w)+ GT(m)l .

Proof: See Appendix C.11.

> If G ( w ) + G T ( w ) = 0, we can take ll1 = I. In the scalar case (p = 1). the frequency-domain conditioii of the lemma reduces to Re[G(jw)] > 0 for all w E R and either G ( m ) > 0 or G ( w ) = 0 and lirn,,, w2Re[G(jw)] > 0.

Example 6.4 The transfer function G(s) = l / s is positive real since it has no poles in Reis] > 0, has n simple pole at s = 0 whose residue is 1, and .

'The definition of strictly poaitive real transfer functions is not uniform in the literature. (See [206] for various definitions nnd the relntionsl~ip between them.)

h he relatiw degree of a rational transfer function G(a) = n(a)/d(a) is deg d - deg n. For a proper trnnrfcr ft~nction. thc relnti\.e dcgrce is a nonnegntive integer.

;'Kqt~i\nln~fl~. C(r) + (:I.(-$) IIW n nnr~~~nl r n ~ ~ k 11 clr.cVr t ltc lic.l<I of rnf ioiinl f~iiirt ioiis of 8.

6.3. POSITIC'E REAL TR;\NSFER FUh1C,'T1O~Y.S 23!)

It is not strictly positivc real siilcc: l / (s - E ) lias a pole in Re[s] > 0 ror iuly E > 0. Thc t.ra~~sfcr h111ctio11 G(s ) = l / ( s + n ) wit.11 (1 > 0 is positive rcal, si11c:c il, I~tls 11o poles in Re(s] 2 0

Since this is so for cvrry a > 0. \\T SCP tli;rt ror ally E E (0, a) tlic transfer f~~nctioii G(s - E ) = 1 / ( s + a - E ) will I J ~ positivc rcal. Hence, G(s) = l / ( s + a) is strictly positive real. The sallic conclusioli (.;11i I ) ( \ clra~vii from Lemma 6.1 by noting that

w2a liin w2Re[G(jw)] = lim - = a > 0

w-32 d-z ~2 +a2

The transfer functiori C ( s ) =

1 s Z + s + l

is not positive real because it,s relativc tlcgrce is two. We can see it also I J ~ calculatirig

Consider t.he 2 x 2 transfer functioii iilatrix

\\lc cannot apply Leinma 6.1 because tlat[G(s) + GT(-s)] G 0 V s. I-Iowcva, G(s) is strictly posit.ive real. as call be seen by checking the conditions of Definition 6.4. Note that.! for E < 1? the poles of the elcinents of G(s - E ) are in Re[s] < 0 and

is positive semidefillite for all w E R. Si~nilarly. it can be seen that tlie 2 x 2 transfer

is strictly posit,ive ~ e a l . This tinic, ho~vcver, det[G(s) + G ~ ( - s ) ] is not idelltically zero, and we can apply Leinlna G . 1 to arrive at the same conclusioii by noting that G ( w ) + G T ( w ) is positive dcfiiiitc arid

is positive definite for all w E R. Finally. the 2 x 2 transfer function matrix

- - - - - - . . . -


has

-I

- 1 It can be verified that

.d

,i G(jw) + ~ ~ ( - j d ) =

-

is positive definite for all w E R. Taking AIT = [ 0 1 1, it can be verified that -

lim w2~'[G'( jw) + ~ ~ ( - j w ) ] ~ l = 4 w d m

.> Consequently, by Lemma 6.1, we collclude that G(s) is strictly positive real. A

Passivity properties of positivc real transfer functions can be shown by using the next two lcmmas, which are known, rcspcctivcly, as thc positive real lemma and the Kalman-Yakubovich-Popov lemma. The lemmas give algebraic characterization of positive real and strictly positive real transfer functions.

Lemma 6.2 (Positive Real) Let G(s) = C(sI - A)-'B + D be a p x p transfer function matrix where (A , B ) is controllabh and ( A , C ) is obsennble. Then. G(s) is positive real if and only i f thew exist mntrins P = PT > 0, L, m d IV S U ~ thnt


Lemma 6.3 (Kalman-Yakubovich-Popov) Let G(s) = C(sl-A)-lB+D be a pxp transferhnction mat*, where ( A , B ) is controllable and (A, C ) is observable. Then, G(s) is strictly positive real if and only i j there exist matrices P = PT > 0, L , and IY, arid a positive constartt E sricli tllat

6.4. Lz AND LYAPUNOV STABILITY 231

Proof: Suppose there exist P = pT > 0, L, 1Y, and E > 0 that satisfy (6.14) through (6.16). Set p = ~ / 2 and recall that G(s - 11) = C(sI - pI - A)-IB + D. From (6.14), we have

It follows from Lemma 6.2 that G(s - p) is positive real. Hence, G(s) is strictly positive real. On the other hand, suppose G(s) is strictly positive real. There exists p > 0 such that G(s - p) is positive real. It follows from Lemma 6.2 that there are matrices P = PT > 0, L, and W, which satisfy (6.15) through (6.17). Setting E = 2p shows that P, L, IY, and E satisfy (6.14) through (6.16).

0 Lemma 6.4 The linear time-iwvariant minimal realization

x = Ax+Bu

y = Cx+Du with G(s) = C ( s l - A)-'B + D is

passive if G(s) i s positive real;

strictly passive if G(s) is strictly positive real.

". 0 Proof: Apply Lemmas 6.2 and 6.3, respectively, and use V ( s ) = ( l /2)xTPx 8~ the storage function.

av uTg - z(Ax + Bu) = uT(Cx + Du) - ~ P ( A X + Bu)

= U ~ C X + ; u ~ ( D + D ~ ) U - ; X ~ ( P A + A ~ P ) X - X ~ P B U

= u T ( B T p + W T L ) x + $ u T 1 l r T ~ ~ u

+ $ X ~ L ~ L X + ; E X ~ P X - X ~ P B U

= ;(Lx + W U ) ~ ( L X + W u ) + ;ExTPx 2 ;txTPz

In the case of Lemma 6.2, E = 0, and we conclude that the system is passive, while in the case of Lemma 6.3, E > 0, and we conclude that the system is strictly pwsive.

6.4 L2 and Lyapunov Stability

In this section, we study Lp and Lyapunov stability of passive systems of the form

ri. = f (21 u ) (6.18)

. - ~ , wl~ere f : R" x RJ' - Rn is locally Lipschitz, h : Rn x RJ' - RJ' is continuous, f (0,O) = 0, and h(0,O) = 0.


Lemma 8.5 If the system (6.18)-(6.19) is output strictly passive with u T y > v + 6yTy, for some 6 > 0 , then it is finite-gain Lz stable and its L2 gain is less than or equal to 1/6.

ProoE The derivative of the storage function V ( x ) satisfies

integrating both sides over [0, T ] yields f-

Thus, 1

I I V T I I L ~ 5 J ~ ~ u T l l ~ 2 + /- where we used the facts that V ( x ) > 0 and d m 5 a + b for nonl~egative numbers a and b.

Lemma 6.8 If the system (6.18)-(6.19) is passive with a positive definite storage . . function V ( x ) , then the origin of x = f ( x , 0 ) is stable.

. . , .. . ,

Proof: Take V as a Lyapunov function candidate for x = f ( x , 0) . Then v 5 0. $,>?;

T? show asymptotic stability of the origin of j. = f ( x , 0 ) , we nccd to either sl~ow that V is negative definite or apply the invariance principle. In the next lemma, we apply the invariance principle by considering a case where V = 0 when v = 0 and then require the additional property that

for all solutions of (6.18) when u = 0. Equivalently, no solutions of x = f ( x , 0 ) can stay identically in S = { x E Rn I h ( x , 0 ) = 0 } , other than the trivial solution x ( t ) I 0. The property (6.20) can be interpreted as an observability condition.

.Recall that for the linear systenl

x = A x , y = C x

observability is equivalent to C.

For eRsy reference. we define (6.20) ns m obsrrvnhility property of the system. C

6.4. L2 AA'D LY.4PUSOi' STABILIT\' 243

Lemma 6.7 Considei the systcn~ (6.18)-(6.19). The origin of x = f (x ,O) is asymptotically stuble if the systeirl is

Definition 6.5 The systenr (6.18) (6.13) is said to be zeiu-state olr.sci.~~c~blc if iro . ~ o l ~ ~ t i o n of x = f (1. 0) can. stay idci~.Linlll!l in S = { x E Rn I h ( x , 0 ) = O ) , olhei, thnn. the trivial solrltioit. x ( t ) = 0. .

strirtly pnrrsirlc or. IC

C 6. C,

Proof: Suppose thr.system is st.rictly passive and let V ( x ) he its storage function. Then. with u = 0. I/ satisfies thc inrqr~ality V < --$J(x), where * ( x ) is positive definite. \Ve can use this incquality to show that V ( x ) is positive definite. In particular, for any x E Rn. the equation i = f ( x , 0 ) has a solution o ( t : x ) . starting ffom x at t = O arid defined on sotne ir~terval [0,6]. Integrating the inequality V 5 -v(x) yields

output strictly passive and zeiv-state observable. .

Furtheirnore, if the storage function is i,adially unbounded, the origin will be globally asymptotically sfable. 0

t

8 .. i L

which c0ntradict.s t,he claim that f # 0. Thus! V ( x ) > 0 for all x # 0. This qualifies V ( x ) as a Lyapunov function candidate. and since V ( X ) 5 -$(I) , we conclude that the origin is asymptotically stable.

Suppose now the system.is output strictly passive and let V ( x ) be its storage function. Then, with u = 0 , V satisfies the inequality V 5 -yTp(y) , where yTp(y) > 0 for all y # 0. By repeating the precetlii~g argument, we can use the inequality to show that V ( x ) is posit.ive definite. In part.icular, for any x E Rn: we have

V(.(T. x ) ) - V ( x ) < - lr $ ( 4 ( t ; x ) ) dt , V T E [ O dl

Using V ( Q ~ ( T . x ) ) > 0. we obtain

~ ( r ) 2 1' il(m(t; z)) dt ' 1

Suppose now that there is f # O such that V ( Z ) = 0. The foregoing inequa1it.y implies

"T

V ( x ) > 1' l ~ ~ ( b ( f : r ) , ~ ) p ( h ( m ( t ; x ) , 0 ) ) dt

L L i, k

b & '!' 1,

. 4~ @: '

j:. lfi:

- . - . . - - - . . -. . . . - - - - . . . . .. .- . -. . .. .. . -- ' / . - :. .- ....


Suppose now that there is r # 0 s~ich that V( i ) = 0. The foregoing inequality implies

LT hT(m(t; l),O)p(h(#(t; I ) , 0)) dt = 0, V 7 E lO,6] * h(#(t; 5) . 0) 0

which, due to zero-state observability, implies

Hence, V(x) > 0 for all x # 0. This qualifies V(x) as a Lyapunov function candidate, and since V(x) < -yTp(y) and y(t) = 0 + x(t) = 0, we conclude by the invariance principle that the "rigill is asymptotically stahle. Finally, if V(x) is radially unbounded, we can infer global asymptotic stability from Theorem 4.2 and Corollary 4.2, respectively.

Example 6.5 Consider the pinput-poutput system4

where f is locally Lipschitz, G and h are continuous, f(0) = 0, and h(0) = 0. Suppose there is a continuously differentiable positive semidefinite function V(t) such that av av

%f (x) < 0, %G(x) = hT(x)

Then,

BV av T BV uTY - -[f (x) + G(x)u] = uTh(x) - - f (x) - h (x)u = - % f (2) 2 0

Ox Bx

which shows that the system is passive. If V(x) is positive definite, we can conclude that the origin of x = f (2) is stable. If we have the stronger condition

av av - f (x) < -khT(x)h(x), ,-jj$x) = hT(x) ax

for some k > 0, then

and the system is output strictly passive with p ( ~ ) = ky. It follows from Lemma 6.5 that the system is finitegain L2 stable and its L2 gain is less than or equal to Ilk. If, in addition, the system is zero-state observable, then the origin of x = f(x) is asymptotically stable. Furthermore, if V(x) is radially unbounded, the origin will he globally asymptotically stable. A

"Lz stability OF this system was studied in Examples 5.9 and 5.10.

. ... . .. -.--..-.----.- .--..---.a 'T.". - -- . a " $ . . .

-.- .. . . . . , ;:L .$..,' A d * -.-,.,, >,-, ~.~"..;,-&~&2.,L.A..

6.5. PASSIVITY THEOREblS 245

where a and k are positive constants. Consider also the positive definite, radially unbounded fi~nction V(x) = (1/4)ax: + (1/2)xi as a storage function candidate. The derivative v is given by

Therefore. the system is output strict.1~ pnssive wit.11 p ( y ) = by. It follows froin Lemrna 6.5 that the systeln is firiite-gain L2 stable with L2 gain less than or equal to I lk. h,loreover, when u = 0,

Hence, the system is zero-state observable. It follows from Lemma 6.7 that the origin of the unforced system is globally asymptotically stable. A

6.5 Feed back Systems: Passivity Theorems

Consider the feedback connection of Figure 6.11 where each of the feedback components HI and H2 is either a timeinvariant dynamical system represented by the state model

or a (possibly time-varying) memoryless function represented by

We are interested in using passivity properties of the feedback components HI and Hz to analyze stability of the feedback connection. We will study both L2 and Lyapunov stability. We require the feedback connection to have a well-defined state model. When both components HI and Hz are dynamical systems, the closed-loop state model takes the form

j. = f (x, U) (6.24) y = h(x ,u) (6.25)

'La and Lynpunov stability of this system were studied in Examples 5.8 and 4.9.


Figure 6.11: Feedback connection.

We assume that f is locally Lipschitz, h is continuous, f (0,O) = 0, and h(0, 0) = 0. It can be easily verified that the feedback connectioli will have a well-defined state model if the equations

have a unique solution (el, en) for every (21, x2, ~1,212). The properties f (0,O) = 0 and h(0,O) = 0 follow from fi(O,O) = 0 and hi(0,O) = 0. It is also easy to see that (6.26) and (6.27) will always have a unique solution if hl is illdependent of el or h2 is independent.of e2. In this case, the functions f and h of tlic closetf-loop state model inherit smoothnessproperties of the functions fi and hi of the feedback components. In particular, if fi and hi are locally Lipschitz, so are f and h. For linear systems, requiring hi to be independent of ei is equivalent to requiring t,he transfer function of Hi to be strictly proper.G

When one component, Hl say, is a dynamical system, while the other one is a memoryless function, the closed-loop stat,e model takes the form

where

We assume that f is piecewise continuous in t and locally Lipschitz in (x, u), h is piecewise continuous in t and cont.inuous in (x, u), f (t, 0,O) = 0, and h(t, 0,O) = 0.

5,::

:. , "Tl rslstrnr~ of soll~lions for (6.26) n~lcl (6.27) is ~rors~~cul fr~rtl~rr in Esrrrisr 6.12. I .,,

"

, , j. : 1, k *

I

6.5. PASSIVITY THEOREbfS

The feedback coi~nection will haw a \rcll-defined state model if the cquatiorls

have a unique solution (el, ez) for every (xl, t , ~1,212). This will be always the casc when hl is independent of el. The case when both components are memoryless functions is less important and follows trivially as a special case when the statc x does not exist. In this case. the feedback connection is represented by y = h(t,u).

The starting point of our analysis is the following fundamental property:

Theorem 6.1 The feedback connection of two passive systems is passive.

Proof: Let Vl(xl) and f i (x2) be t.he storage functions for H1 and Hz, respectively. If either component is a memoryless function, take = 0. Then,

e: yi 2 v i

From t.he feedback connection of Figme 6.11, we see that

Hence:

Taking V(x) = 6 ( x l ) + Vz(x2) as the storage function for the feedback connection, na obtain

.uTy 2 v

Using Theorem 6.1 and the results of the previous section on stability properties of passive systems, we can arrive at some straightforward conclusions on stability of the feedback connection. We start with t2 stability. The next lemma is an immediate consequence of Lemr~ia 6.5.

Lemma 6.8 The feedback connection of two output strictly pass'ive sy.~tems with

is finite-gain tz stable and its Lz gain is less than or equal to 1/ min{gl, 62).

Proof: With V = Vl + l5 and b =nlin{bl, 621, we have

. / . -:. . . - 21 8 CHAPTER 6. PASSIVITY

Reading t,he proof of Lemma 6.5 shows that we use the inequality

to arrive at the inequality 1 6 v 5 -uTu 26 - pTY (6.33)

which is then used to show finite-gain C2 stability. In Lemma 6.8, we establish (6.32) for the feedback connection, which then leads to (6.33). However, even if (6.32) does liot hold for the feedl~i~ck connection, we may still be able to show an inequality of the form (6.33). This idea is used in the next theorem to prove a more general result that includes Lemina 6.8 as a special case.

Theorem 6.2 Consider the feedback connection of Figure.6.11 and suppose each feedback component satisfies the in,equulity

eTyi 2 fi + +;eTei + biYTyi, for i = 1,2 (6.34)

for some storage function K(xi). Then, the closed-loop map from u to y is finite gain C2 stable if

€1 + 62 > 0 and €2 + 61 > 0 (6.35)

0

Proof: Adding inequalities (6.34) for i = 1,2 and using

we obtain

where

and V(r) = VI(XI) + Vz(x2). Let a = inin{e2 + bl, +d2) > 0, b = ((Nllz 2 0, and c = JJn'1))z 1 0. Then

_ . . . ..... _ _ _ ......I - -<- ---. -,.. ,I-.- , - . . - . , w-- .. .$, " $ . ' , . . . ,.. - . -. .-L-

-..-, .. ' . . . . . . -.. . . . . . . . . : . . . -- d .

6.5. PASSIVITY THEORE!vlS 210

where k2 = b2 + 2ac. Iiltegratiilg over (0, T ] , using V(r) 2 0, and taking tlle square roots, we arrive at

k 11~7I l~ l 5 ;ll~711Ll + JmmE

which completes the proof of the theorem. 0

Theorem 6.2 reduces to Lemma 6.8 when (6.34) is satisfied with €1 = €2 = 0, 61 > 0, and 62 > 0. However, condition (6.35) is satisfied in several other cases. For example, it is satisfied when both H1 and Hz are input strictly passive with eTyi 2 fi+~iu'ui for some ei > 0. It is also satisfied when one component (HI say) is passive, while the other component satisfies (6.34) with positive €2 and 6 ~ . What is more interesting is that (6.35) can be satisfied even when some of the constants ~i and 6i are negative. For example, a negative €1 can be compensated for by a

6 ~ . This is a case where shortage of passivity (at the input side) of HI is compensated for by excess of passivity (at the output side) of Hz. Similarly, a negative J2 can be compensated for by a positive €1. This is a case where shortage of passivity (at the output side) of H2 is compensated for by excess of passivity (at the input side) of HI.

Example 6.7 Consider the feedback connection of

where k > 0 and ei, yi E RP. Suppose there is a positive definite function VI fx) such that

av1 ah zf (x),< 01 -G(x) ax = hT(x), V E R n

Both components are passive. Moreover, Hz satisfies

Thus, (6.34) is satisfied with €1 = 61 = 0, €2 = yk, and 62 = (1 - y)/k. This shows that (6.35) is satisfied, and we conclude that the closed-loop map from u to y is finite-gain C2 stable. A Example 6.8 Consider the feedback connection of

31 = xz ~2 = -ax? -U(XZ) + el and H2 : yz = kez Y1 = 5 2

where o E [-a, co], a > 0, a > 0, and k > 0. If o was in the sector (0, co], we could have shown that HI is passive with the storage function Vi (x) = (a/4)xf + (1/2)x;. For u E [-a, co], we have


Hence, (6.34) is satisfied for H1 with €1 = 0 ~ i i d 61 = -a. Since

(6.34) is satisfied for HZ with €2 = yk and 62 = (1 - y)/k. If k > a, we can clloose y such that yk > a. Then, €1 + 62 > 0 and 9 + 61 > 0. We conclude that the closed-loop map from u to y is finite-gain t2 stable. A

Let us turn now to studying Lyapunov stability of thc feedback conllectioll. \Ire are interested in studying stability and asymptotic stability of the origin of the closed-loop system when the input u = 0. Stability of the origin follows trivially from Theorem 6.1 and Lemma 6.6. Therefore, we focus our attention on st,udying asymptotic stability. The next theorem is an im1nediat.e consequence of Theorem 6.1 and Lemma 6.7.

Theorem 6.3 Consider the feedback connection of ttrro time-in11ntian.t dyna~n.ica1 systems of the f o m (6.21)-(6.22). The origin of the closed-loop system (6.24) (when u = 0) is asymptotically stable if

both feedback com.ponents are strictly passiz~e,

both feedback cony)onents are output strictly passive and zero-state obsenmble, or

one component is strictly passiire and the other one is oi~tput strictly pa~si~re and zero-state observable.

firthemore, if the storage bnction for each componen,t is radially unbounded. the origin is globally asymptotically stable. 0

Proof: Let Vl(x1) and Vz(x2) be the storage functions for H1 and Hz, respectively. As in the proof of Lemma 6.7, we can show that Vl(x1) and Vz(x2) are positive definite functions. Take V(z) = Vl(z1) + V2(x2) as a Lyapupov function candidate for the closed-loop system. In the first case, the derivative V satisfies

since u = 0. Hence, the origin is asymptotically stable. 111 the second case,

where y'~i(vi) > 0 for all yi # 0. Here v is only negative semidefinite and v = 0 + y = 0. To apply the invariance principle, we need to show that y(t) n 0 + x(t) E 0. Note that y2(t) = 0 + el(t) m 0. Then, zero-state observability of HI sho~vs that yl(t) E 0 =+ xl(t) E 0. Similarly, yl(t) E 0 + ez(t) = 0 and zero- state observability of H2 shows that y2(t) E 0 =+ x2(t) r 0. Thus. the origin is asymptotically stable. 111 the third case (with H1 as the strictly passive component).

6.5. PASSIVITY THEOREMS 251

and v = 0 implies $1 = 0 and yz = 0. Note that y2(t) a 0 + el(t) = 0, which together with xl( t ) = 0 imply that !jl(t) = 0. Hence, e2(t) = 0 and zero-state observability of Hz shows that y2(t) 0 + xz(t) G 0. Thus. the origin is asymptotically stable. Finally. if Vl(xl) arid 15(z2) are radially unbounded, so is V(x), and we can conclude global asymptotic stabilify.

The proof uses a simple idea, namely, that the sum of the storage functions for the feedback components is usctl as a Lyapunov function candidate for the feedback connection. Beyond this sirnple itlea, the rest of the proof is straightfor- yard Lyapunov analysis. In fact. tlie analysis is restrjctive because to show that V = Vl + Vz 5 0, we insist that both Vl 5 0 and V2 < 0. Clearly, this is not necessary. One term. vl say, could be positive over some region as long as the sum v 5 0 over the same region. This is again a manifestation of the idea that shortage of passivity of one component can be compensated for by excess of passivity of the other component. This idea is exploitrd in Examples 6.10 and 6.11, whilc Exarnplc 6.9 is a straightforward application of Theorem 6.3.


where a, b, and li are positivc coilstiuit.~. Using Vl = (a/4)x: + (1/2)2: as the storage function for H1, we obt.ain

Hence, H1 is output strictly passive. Besides, with el = 0, we have

which shows that HI is zero-state observable. Using V2 = (b/2)x: + (1/2)x: as the storage function for Hz, we obtain

4 % = bx3x4 - b 5 3 ~ 4 - .r: + x4e2 = -y2 + yzez

Therefore, Hz is output strictly passivc. Rloreover, with ez = 0: we have

y2(t) = 0 H x4(t) 0 .=. x3(t) I 0

which shows that Hz is zero-state observable. Thus, by the second case of Theo- rem 6.3 and the fact that Vl and 14 arc radially unbounded, we conclude that the origin is globally asymptotically stahlr. A

. . . . . . -- --.-. -- . . ' . . . . . - .. - ...,


Example 6.10 Reconsider the feedback coililection of the previous example, hut change the output of Hl to yl = x2 + el. F+om the expression

we can conclude that H1 is passive, but we cannot conclude strict passivity or output strict passivity. Therefore, we cannot apply Theorem 6.3. Using

as a Lyapunov function candidate for the closed-loop system, we obtain

Moreover, v = 0 implies that 1 2 = xa = 0 and

Thus, by the invariance principle and the Fact that V is radially unbounded, we conclude that the origin isglobally asymptotically stable.

Example 6.11 Reconsider the system

from Examples 4.8 and 4.9, where hl and h2 are locally Lipschitz and belong to the sector (0, oo). The system can be viewed as the state model of the feedback connection of Figure 6.12, where H1 consists of a negative feedback loop around the integrator x2 with hz in the feedback path, and Hz consists of a cascade connection of the integrator xl with hl. We saw in Example 6.2 that H1 is output strictly passive with the storage function 13 = (1/2)x: and, in Example 6.3, that H2 is lossless with the storage function V2 = S:' hl (u) du. We cannot apply Theorem 6.3 because Hz is neither strictly passive nor output strictly passive. However, using V = Vl -I- IT2 = stoZ' hl(u) du + (1'/2)xi as a Lyapunov function candidate, we can proceed to investigate asymptotic stability of the origin. This is already done in Examples 4.8 and 4.9, where it is shown that the origin is asymptotically stable and will be globally asymptotically stable if S l hl(z) dz + m as (yl + m. We will not repeat the analysis of these two examples here, but let us note that if hl(y) and h2(y) belong to the sector (0, m) only for y E (-a, a), then the Lyapunov analysis can be limited to some region around t.he origin, leading to a local asymptotic stability conclusion, as in Example 4.8. This shows that passivity remains useful as a tool for Lyapunov analysis evcii wlic!li it l~oltls o~ily on a finite region, rather than the whole space. ' A

<--. -. " T I

. '. . :',* *;',A <sic /,: ;?$>>:g?q

- . ' . , . -. . .-.--.I="L; : .--A -.. .! ,... > : .*,A, . 3 , I 6.5. PASSIVITY THEOREAlS 253 i

L - - - - - - - J

Figure 6.12: Example 6.11.

When the feedback connection has a dynamical system as one component and a memoryless function as the other component, we can perform Lyapunov analysis by using the'i;torage function of the dynamical system as a Lyapunov function candidate. It is important, however, to distinguish between time-invariant and time-tarying memoryless functions, for in the latter case the closed-loop system will he nonaritonomo~ls and we cannot apply the invariance principle as we did in the proof of Theorem 6.3. We treat these two cases separately in the next two theorems.

Theorem 6.4 Consider the feedback connection of a strictly passive, time-invariant, dynamical system of the form (6.21)-(6.22) with a passive (possibly time-varying) memotyless function of the form (6.23). Then, the origin of the closed-loop system (6.28) (when ri = 0) is uniformly asymptotically stable. firthemnore, the storage function for the dynamical system is mdially unbounded, the origin will be globailg uniformly asymptotically stable. 0

Proof: As in the proof of Lemma 6.7, it can be shown that Vl(zl) is positive definite. Its derivative is given by

8% fi = - f lk i , e i ) 2 eTyl- h ( z 1 ) = -eTy2 - qjl(zl) 5 -qj1(zl) &1

The conclusion follows from Theorem 4.9.

Theorem 6.5 Consider the feedback connection of a time-invariant dylamieal system H1 of the form (6.21)-(6.22) with a time-invariant memoryless function Hz of

CHAPTER 6. PASSIVITY 6.5. PASSIVITY THEOREAIS

theform (6.23). Suppose that Hl is rem-state obsertrable and has a positive definite stomge function, which satisfies

eTyi L li, + yTpl(y1)

and that HZ satisfies

eTyz 1 eTvz(ez) (G.37) Then, the o ~ g i n of the closed-loop system (6.28) (when u = 0) is asymptotically stable if

vT[Pi(v) + vz(v)] > 0, V v # 0 (6.38) Furthermom, if Vl is mdially unbounded, the odgin will be globally asymptotically stable.

0 Proof: Use K(x1) as a Lyapunov function candidate, t.o obtain

= -e?yz - YTPI(YI) 5 -[YTv2(~1) + Y ? P ~ ( Y ~ ) ]

Inequality (6.38) shows that I$ < 0 and f i = 0 9 yl = 0. Noting that yl(t) 0 * ez(t) 0 * el(t) s 0, we see that zeroatate observability of HI implies that xi (t) 0. The conclusion follows from the invariance principle. 0

Example 6.12 Consider the feedback connection of a strictly positive real transfer function and a passive time-varying memoryless function. From Lemina 6.4. wc know that the dynamical system is strictly passive with a positive definite storage function of the form V(x) = (1/2)xTPx. J+om Theorem 6.4, we conclude that the origin of the closed-loop system is globally uniformly asymptotically stable. This is a version of the circle criterion of Section 7.1. A Example 6.13 Consider the feedback connection of

where a E (0,m) and ei, yi E R*. Suppose there is a radially unbounded positive definite function K(x) such that

av1 av1 - f (x) 5 0 -G(x) = hT(x), V x E Rn ax ax ' and HI is zero-state observable. Both components are passive. bIoreover. Hz

satisfies T e2 YZ = era(e2)

Thus, (6.36) is snt,isfirtl wit,ll pi = 0, ailtl (6.37) is sat.isfictl wit,h cpz = a. Sillce a E (O,oo), (6.39) is sntisfictl. It follows from Tlieorcin 6.5 that the origin of tlie closed-loop system is globnlly asymptoticnlly stnblr. A

We conclude this scction by prcsci~tiiig loop transformations, which extend the utility of passivity theorems. Starting with a feedback connection in which one of the two feedback components is not passivc or does not satisfy a condition that is needed in one of the theorems. we may be alde to reconfigure the feedback connection into an equivalent connection that has thc desired properties. We illustrate the process first for loop transformations that use constant gains. Suppose H I is a time-invariant dyliamical system, while Hz is a (possibly timevarying) memoryless functioii that belongs to the sector [IC1, K2], where K = Kz - Ki is a positive definite symmetric matrix. We saw in Section 6.1 that a function in the sector [K1. K2] can he transformed into a function inthe sector [O, CO] by input feedforward followed by output feedback, as shown in Figure 6.7. Input feedforward on Hz can be nullified by output feedback on Hi. a~ shown in Figure 6.13(b), resulting in an equivalent feedback connection, as far as asymptotic stability of the origin is concerned. Similarly. premultiplying the modified Hz by K-' can be nullified by postmultiplying the modified Hi by I<, as shown in Figure 6.13(c). Finally, output feedback oil the component in the feedback path can be nullified by input feedforward on the component in the forward path, as shoyn in Fi_gure 6.13(d]. The reconfigured feedback connection has two components H1 and HZ, where Hz is a memoryless function that belongs to the sector [O,CO]. We can now apply Theorem 6.4 or 6.5 if H1 satisfies the conditions of the respective theorem.

~ x a m ~ l e ' 6 . 1 4 Consider the feedback coiiiicction of

f 1 = x2 j.2 = 4 x 1 ) + bx2 + e l and Hz : yz = a(ez) Y1 = 5 2

where a E [a,P], 12 E [al, CO], b > O! 01 > 0! and k = P - a > 0. Applying the loop trnnsfoririatioii of Figure G.13(d) (witli IC1 = a aud K2 = P) results ill tlie feedback connection of

= 1 2

f i : [ = h - 2 + and & : h = b ( h )

where 5 E 10, CO] and a = a - b. If a > b. it can he shown (Exercise 6.4) that f i l is strictly passive with a storage funct.iori of the form Vl = k St' h(s) ds + xTPx, where P = P' > 0. Thus, we conclude from Theorem 6.4 that thc origiii of trhc feedback connectiou is globally asymptotically stable. A

Next. we consider loop transformations with dynamic multipliers, as shown in Figure 6.14. Premultiplying Hz by a transfer function W(s) can by nullified by postmultiplying HI by W-'(s), providctl thc inverse exists. For example, wlicrl Hz is a passive, time-invariant, memoryless function h, we saw in Example 6.3 that premultiplying h by the transfer fu~~ct ion l / (as + 1) results in a strictly passive

,. d,. -

- /

.- -- . - . -?

" 7


Figure 6.13: Loop transformation with constant gains. A memorylew function Hz in the sector [K1, 1c2] is transformed into a memoryless function H2 in the sector (0, CO].

. - -. " ..-.-.------ + .. .. \r- - - . , < . . .

- ..- . . . . . . -.:..., >,L.*..... . d . . ..&- - .. ... n -.-' ' _.. .-

6.5. PASSIVITY THEOREMS 257

Figure 6.14: Loop transformation with dynamic multipliers.

dynamical system. If postmultiplying H1 by (as + 1 ) results in a strictly passive system or an output strictly passive system that is zero-state observable, we can employ Theorem 6.3 to conclude asymptotic stability of the origin. This idea is illustrated in the,,next two examples for cases where H1 is linear and nonlinear, respectively.

Example 6.15 Let H1 be a linear time-invariant system represented by the state model

x=Ax+Bel, y , = C x

Its transfer function 1/(s2 + s + 1 ) has relative _degree two; hence, it is not positive real. Postmultiplying HI by (as+ 1 ) results in HI, which can be represented by the state model

x = A x + B e l , , $1 = e x where C = C + aCA = [ 1 a 1. Its transfer function (as + l ) / ( s 2 + s + 1 ) satisfies the condition

it a 2 1. Thus, choosing a 2 1, we can apply Lemmas 6.3 and 6.4 to conclude that Hl is strictly passive with the storage function ( l /2 )xTPx where P satisfies the equations

PA + A ~ P = - L ~ L - EP, PB = CT

258 (7IIAPTER 6. PASSIVITY 6.6. EXERCISES

for some L and E > 0. On the otlier lialid. let H2 be given by y2 = A(e2). where h E [0, w]. We saw in Example 6.3 tliat prelni~ltiplyilig h by tlie tralisfer fiinctioli l / ( as+ l ) results in a strictly passive system with the storage function a Joe' h(s) ds. Application of Theorem 6.3 shows that the origin of the transformed feedback toll-

nection of Figure 6.14(b) (with zero input) is asymptot.ically stable with tlie Lye- punov function V = (1/2)xTPx + a Jbe2 h(s) ds. Notice, hotvevrr. tliat tllr transformed feedback connectioll of Figure 6.14(b) has a state model of dimension three, while the original feedback connection has a state model of dimellsioli two; so more work is needed to establish asymptotic stability of the origin of the original fecclback connection. The extra work can be alleviatecl if n7e use the trarisformed feedback connection only to come up with the Lyapunov function V and tlien proceed to calculate the derivative of V with respect to tlie original feedback collnectioll. Sricli derivative is given by

which shows that the origin is asymptotically stable. In fact, silice I' is radially unbounded, we conclude tllnt the origin is globally asymptotically stable. A


where b > 0,- k > 0, and h E [0, w]. Postmultiplying H1 by (as + 1) resu1t.s in a system H1 represented by the same state equation but with a new output fil =-XI + ax2. Using Vl = (1/4)bxi + (1/2)xTpx as a storage functioli caiididntc for HI, we obtain

Taking pll = k, pl2 = p22 = l9 a = 1, and assuming that k > 1, we obtain

wliicli shows that H] is strictly passive. On the other hand. prcniultiplying h by thr trnrisfer fiilictioli l / (s + 1) results ill n strictly passive systrin with tlie stnr-

r ~ ~ g r IIIIN.~ io11 /I(x) (I,*. l'si~ig IIiv stonigv hnic-tion (or tlis 1 r o t ~ s i o ~ ~ ~ i ~ ~ c l fi~>tll):ttsk

connect ion) e2

1' = (1/4)b x j + ( l /2)xTpx + h(s) ds

as a Lyapuliov function caiididate for tlic original feedback connection (whcn u = 0) yields I

which is negative definite. Since V is positive definite and radially unbounded, we conclude tliat tlie origin is glol~ally nsympt,otically stable. A

6.6 Exercises

6.1 Verify thaea function in the sector [K1, K2] can be transformed into a function in the sector [O! x ] by input feedforward followed by output feedback, as shown in Figure 6.7.

I! 6.2 Consider the system

where a a11d I; are positive colistallts iuld h E [O, k]. Show that the system is pacsive with V(x) = a St h(u) du as the storage fuliction.

\vhrre 0 < a < a and h E (0. m]. Sliotv that the system is strictly passive. Hint: Usc \'(I) of Example 4.5 as thc storage function.

i 6.4 Consider t,he system

where a > 0: k > O? h E [al, and a, > 0. Let V(x) = k S t ' h(s) ds + xTPx,

where pi1 = uplz, p22 = k/2. and 0 < pi2 < min{2al, ak/2). Using V(x) as a storage function, show that the systein is strictly passive.

6.5 Consider t.he system represented by the block diagram of Figure 6.15, where u? y E RJ', A.1 and K are positive definite symmetric matrices, h E [O, K], and st hT(u)N do 2 0 for all x. Sho~v that the system is output strictly passim.

. - . CHAPTER 6. PASSI\'ITY

Figure (3.15: Exercise 6.5 t 6.6 Show that the parallel connection of two passive (respectively, input strictly

passive, output strictly passive, strictly passive) dynamical systems is passive ( r e spectively, input strictly passive, output strictly passive, strictly passive).

6.7 Show that the transfer function (bos + bl)/(s2 + a l s + a2) is strictly positive real if and only if all coefficients are positive and bl < ulbo.

6.8 Consider equations (6.14) through (6.16) and suppose that (D + DT) is nonsingular. Show that P satisfies the Riccati equation

where A. = - (~ /2 )1 - A + B(D + DT)-'C, Bo = B(D + DT)-IBT, and Co = -cT(D + D ~ ) - ~ c .

6.9 Show that if a system is input strictly passive, with cp(u) = eu, and finite-gain C2 stable, then there is a storage function V and positive constants e l and 61 such ,

that

6.10 Consider the equations of motion of an m-link robot, described in Exer- cise 1.4. Assume that P(q) i s a positive definite function of q and g(q) = 0 has an 4 isolated root at q = 0. if (a) Using the total energy V = 1 ' TAl (q )~ + P(q) as a storage function, show that z q

the map from u to q is passive.

(b) With u = -Kdq + v, where Kd is a positive diagonal constant matrix, show t,hat the map from v to Q is 04tput strictly passive.

\

( c ) Show that u = -Kdq, where Kd is a positive diagonal constant matrix, makes the origin (q = 0, q = 0) asyn~ptotically stable. Under what additional conditions will it be globally asyfnptotically stable?

6.11 ([151]) Euler equations for a rotating rigid spacecraft are given by 2

where wl to w3 are the components of the angnlar velocity vector along the principal axes, ul to ug are the torque inputs ilppliccl about the principal axes, and J1 to J3

are the principal moments of inertia.

. ,- . .... ---* , - . . I . - . ..

' ' * . * , ..._....... .--- C ..; ' .> ..:

, - . . : ., . , ,. ... _ ? _ - 6.6. EXERCISES 261

(a) Sliow tliat tlie Jnltp from u = [ul, uz,u3IT to w = [dl, ~ ~ , u ~ ] ~ is lossless.

(b) Let u = -Kw + v, where K is a positive definite symmetric matrix. Show that the map from v to w is finite-gain t 2 stable.

(c) Show that, when v = 0, the origin w = 0 is globally asymptotically stable.

6.12 Consider the feedback system of Figure 6.11 where HI and Hz have the state models

xi = fi(xi) + Gi(~i )e i , Pi = hi(zi) + Ji(Xi)ei for i = 1,2. Show that the feedback system has a well-defined state model if the matrix I + J2(x2) JI (XI) is nonsingular for all xl and x2.

6.13 Consider equations (6.26)-(6.27) and (6.30)-(6.31), and suppose hl =. hl (xl), independent of el. Show, in each case, that the equations have a unique solution (e1,ez).

6.14 Consider the feedback connection of Figure 6.11 with

351 = x2 -21 - h r ( n ) + e l and H2 : { = -23 + e2

QI1 = 2 2 Y2 = h2 (~3 )

where hl and h2 are locally Lipschitz functions, which satisfy hl E ( 0 , 4 , h2 E (0, oo], and (hz(z)( 2 (z ( / ( l + z2) for aI1 Z.

(a) Show that the feedback connection is passive.

(b) Show that the origin of the unforced system is globally asymptotically stable.

6.15 Repeat the previous exercise for

6.16 ([78]) Consider the feedback system of Figure 6.11, where H1 and H2 are passive dynamical systems of the form (6.21)-(6.22). Suppose the feedback connection has a well-defined state model and the series connection HI (-Hz), with input e2 and output yl, is zero-state observable. Show that the origin is asymptotically stable if Hz is input strictly passive or HI is output strictly passive.

6.17 ([78]) Consider the feedback system of Figure 6.11, where Hl and Hz are passive dynamical systems of the form (6.21)-(6.22). Suppose the feedback can- nection has a well-defined state model and the series connection H2H1, with input el and output y2, is zero-state observable. Show that the origin is asymptotically stable if HI is input strictly passive or Hz is output strictly passive.

CHAPTER ti. PASSIVITY

6.18 ([78]) As a generalization of the concept of passivity, a dynalnical system of the form (6.6)-(6.7) is said to be dissipative with respect to a supply rate U J ( U , y) if there is a positive definite storage function l f ( x ) such that v 5 w. Consitler the feedback system of Figure 6.11 where H1 and H2 are zero-state observable, dynalnical systems of the form (6.21)-(6.22). Suppose each of HI and H2 is dissipative with storage function K( . r ; ) nlid supply rate I U ~ ( U ; , p i ) = $Qiyi + + I I T R ~ u ~ , where Qi and Ri are real symmetric matrices and Si is a real matrix. Show tliat the origin is stable (respectively, asymptotically stable) if the matrix

is negative semidefinite (respectively, negative definite) for some a > 0.

6.19 Consider the feedback connection of two time-invariant dynamical systems of the form (6.21)-(6.22). Suppose both feedback cornpolleiits are zero-state observable and there exist positive definite storage functions which satisfv

eTyi 1 ~ + eTcpi(ei) + YrPi(31i), for i = 1,2 F Show that the origin of the closed-loop system (6.24) wheu u = 0 is asymptotically stable if 1

vT[p1(v) + ~ 2 ( v ) ] > 0 and vTb2(v) - pi(-v)] > 0, V v # 0 1 Under what additional conditions will the origin be globally asymptotically stable?

Chapter 7

Frequency Domain Analysis of Feedback Systems

Many nonlinear physical systems can be represented as a feedback connection of a linear dynamical system and a nonlinear element, as shown in Figure 7.1. The process of representing a system in this form depends on the particular system involved. For instance, in the case in which a control system's only nonlinearity is in the form of a relay or actuator/sensor nonlinearity, there is no difficulty in representing the system in the feedback form of Figurc 7.1. In other cases, the representation may be less obvious. We assume that the csteriial input r = 0 and study the behavior of the unforced system. What is unique about this chapter is the use of the frequency response of the linear system, which builds on classical control tools like the Nyquist plot and the Nyquist criterion. In Section 7.1, we study absolute stability. The system is said to absolutely stable if it has a globally uniformly asymptotically stable equilibrium point at the origin for all nonlinearities in a given sector. The circle and Popov criteria give frequency-domain sufficient conditions for absolute stability in

Figure 7.1: Feedback connection. t

. . . . . . . . CIMPTER 7. I*'EEDBACh' SYSTEAJS

the forin of strict positive realness of certain transfer functions. In the single-input- single-output case, both criteria can be applied graphically. In Section 7.2, we use tlic tlt!st:ril)irig function method to study the existence of periodic solutions for a single-input-single-output system. We derive frequency-domain conditions, which can be applied graphically, to predict. the existence or absence of oscillations and estimate the frequency and arnplit.ude of oscillation when there is one.

Consider the feedback connection of Figure 7.1. We assume that the external input r = 0 ant1 study the behavior of tlie unforced system, represented by

whcre .r E R", U, y E RP, (A, B) is coiit,rollablc, (A, C) is observable, and $J : [O, w) x R" --I RP is a mel~~oryless, possibly time-varyillg, i~oi~lincurity, wliich is piecewise coiitinuous in t and locally Lipschitz in y. We assunie that the feedback connection has a well-defined state inodel, which is the case when

has a unique solution u for every (t, x) in the domain of interest. THis is always the case when D = 0. The transfer funct,ion matrix of the linear system

is square and proper. The controllability and observability assumptions ensure that {A, B, C, D) is a minimal realization of G(s). From linear system theory, we know that for any rational proper G(s), a minimal realization always exists. The nonlinearity 1/, is required to satisfy a sector condition per Definition 6.2. The sector condition may be satisfied globally, that is, for all y E RP, or satisfied only for y € Y , a subset of Rp, whose interior is connected and contains the origin.

For all nonlinearities satisfying the sector condition, the origin r = 0 is an equilibrium point of the system (7.1)-(7.3). The problem of interest. here is to study the stability of the origin, not for a given nonlinearity, but rather for a class of nonlinearities that satisfy a given sector condition. If we succeed in showing that t.lle origin is uniformly asymptot,ically stable for all nonlinearities in the sector, the system is said to be absolutely stahlc. The problem was originally formulated by Lure and is sometimes called Lure's.proble7n. Tkaditionally, absolute stability has been definer1 for the case when the origin is globally uniformly asymptotically stable. To keep up t.his tradition, we will use the phrase "absolute stability" when the sector coritlition is satisfied globally anrl t11c origiii is globally uniformly asymptotically stable. Ot,hermise, we will use the phrase '.ahsolute stability with a finite domain."

- - -------A- - --- ---- . r $7 -- . , ' 4 . . ... . . . . . .

0 . 4 . - -. +. -...- . . . .

7.1. ABSOLllTE STABILITY 266

Definition 7.1 Consider the system (7.1)-(7.3), ,where I I , satisfies a sector condition per Definition 6.2. The system is absobtelg stable if the origin is globally uniformly asymptotically stable for any nonlinearity in the given sector. It is absolutely stable with a finite domain if the origin is uniformly asymptotically stable.

We will investigate asymptotic stability of the origin by using Lyapunov analysis. A Lyapunov function candidate can be chosen by using the passivity tools of the. previous chapter. In particular, if the closed-loop system can be represented as a feedback connection of two passive systems, then tlie sum of the two storage functions can be used as a Lyapunov function candidate for the closed-loop system. The llse of loop transformations allows us to rover various sectors and Lyapunov function candidates, leading to the circle and Popov criteria.

7.1.1 Circle Criterion

Theorem 7.1 The system (7.1)-(7.3) is absolutely stable if

$J E [Icl, m] and G(s)[I + KlG(s)]-' is strictly positive real, or

$ E [K1,K2], with K = Kz - K1 = K~ > 0, and [ I + ~ z G ( s ) ] [ l + KIG(S)]-' is st~z'ctly positive real.

If the sector condition is satisfied only on a set Y C RP, then the foregoing conditions ensure that the system is absolutely stable with a finite domain. 0

We refer to this theoremas the multivariable cimle criterion, although the reason for using this name will not be clear until we specialize it to the scalar case. A necessary condition for equation (7.4) to have a unique solution u for every 11, E [Kl, m] or $J E [Kl, Kz] is the nonsingularity of the matrix ( I + K1 D). This can be seen by taking $J = Kly in (7.4). Therefore, the transfer function [I+ KlG(s)]-I is proper;

Proof of Theorem 7.1: We prove the theorem first for the sector [0, m] and re- cover the other cases by loop transformations. If 11, € (0, m] and G(s) is strictly positive real, we have a feedback connection of two passive systems. From Lemma 6.4, we know that the storage function for the linear dynarnical system is V(x) = (1/2)xTPx, where P = PT > 0 satisfies the Kalman-Yakubovich-Popov equations

and E > 0. Using V(x) as a Lyapunov function candidat,e, we obtain

CIfAPTER 7. FEEDBACK S)-STEAIS

? - - - - - - - - - -

Figure 7.2: 11 E [I<l, oo] is transformed to 4 E [0, oo] via a loop transformation.

Using (7.6) and (7.7) yields

Using (7.8) and the fact that uTDu = ;uT(D + DT)u, we obttlill

Since yTIL(t, y) 1 0, we have v 5 - ;€xTpx

which shows that the origin is globnlly cxpo~icntinlly stablc. If $J sntisfics thc sccutor condition only for y E Y , the foregoing analysis will be valid in some neighborhood of the origin, showing that the origin is exponentially stable. Tlle case JI E [ICl, m] can be transformed to a casc where the nonlinearity beloligs to (0, oo] via tlie loop transformation of Figure 7.2. Hence, the systeln is absolutely stable if G(s)[I + I<lG(s)]-' is strictly positive real. The case II, E [I<l, I<2] call be transforlncd to a case where the 1ionlinearit.y beloilgs to [O,oo] via the loop t,rai~sformatio~l of Figure 7.3. Hence, the system is absolutely stable if

I + I<G(s)[I + I<iG(s)]-' = [I + I<~G(s)] [I + IC1G(s)]-l

is strictly positive real. 0

Example 7.1 Consider the system (7.1)-(7.3) and suppose G(s) is Hurwitz and strictly proper. Let

% = SUP gm,[G(jw)l= sup IlG(jw) 112 UF R wER

7.1. ABSOLUTE STABILITY 267 r - - - - - - - - - - - - - - -

I ,

Figure 7.3: 11, E [I<1, Kz] is transformed to 4 E [0, oo] via a loop transformation.

where urn,[.] denotes the maximum singular value of a complex matrix. The constant 71 is finite, since G(s) is Hurwit.~. Suppose ?I, satisfies the inequality

then it belongs to the sector [I<1, If2] with K1 = - 721 and K2 = y21. To apply Theorem 7.1, we need to show that

is strictlv positive real. LVc not(> t.l~;l.t tlrt.[Z(s) + ZT(-s)] is not itlcnt.ically zcro because Z(m) = I. LVe apply Lclllina G.1. Since G(s) is Hurwitz, Z(s) will be Hurwitz if [I - -y2G(s)]-I is Hurwitz. Noting that1

we see that if 71-72 < 1, the plot of rlet[I --y2G(jw)] will not go t,hrough nor encircle the origin. Hence, by the multivariablc Nyquist criterion,' [I-y2G(s)]-1 is Hurwitz; consequently, Z(s) is Hurwitz. Next. we show that

'The rollott4ng properties of singular values of a complex matrix are used:

det G # 0 o o,I~[G] > 0

amaxIG- ' ] = l l a m i n [ q , if am~. [G l > 0 a m i n [ l + G ] 2 1 - amax[Gl

o m a x [ G ~ G 2 ] < ~ r n a x [ G l ] ~ r n a x [ G ~ ]

2See (33. pp. 160-161) lor a statement of tlie nulltivariable Nyquist criterion.

:&, . L id-

1 , - !lp

ii

f :A - , .v:'d$&

,

Tlle left.-hand side of this inequality is give11 by

Hence, Z(jw) + ZT(-jw) is positive definite for all w if and only if

Now, for y1y2 < 1, we h a w

Finally, Z(w) + ZT(co) = 21. Thus, all the conditions of Lemma 6.1 are satisfied and we co~lclude that Z(s) is strictly positive real and the system is absolutely stable if 7172 < 1. This is a robustness rcsult, which shows that closi~~g the loop around a Hurwitz transfer function with a nonlinearity satisfying (7.9). with a sufficiently small y2, does not destroy t.he stabilit,y of the system.3 A

In the scalar case p = 1, the collditions of Theorem 7.1 call be verified graphically by exainil~ing the Nyquist plot of G(jsr). For ?I, E [a , PI, with > a, the system is

absolutely stable if the scalar transfrr function

is strictly positive real. To verify that Z(s) is strictly positive real, we can use Lemma 6.1 which states that Z(s) is strictly posit,ive real if it is Hurwitz and

i To relate condition (7.10) to the Nyquist plot of G(jw), we have to distinguish between three different cases, depending on the sign of a. Consider first the case when p > a > 0. In this case, condition (7.10) can be rewritten as

For a point q on the Nyquist plot nf G(jw). tlir two col~iplrx nr~mbers (l/P)+G(jw) ant1 (l/tu) + G(jw) can bc rcprc~sciitc~tl by tllc lir~cs conncrting q to -(l/P) + j0 ant1

J'rhe inequality 7172 < 1 can he clcrivcrl also lro~n the srr~all-gain theorem. (See Example 5.13.) 6

. . _ ,.. . ____ - .. . ' , . I .- . . < * ,.. * . -_ - . " . .

-.. ..., - 2 '!.. * . . . . -: , .- . ,.. . .-

7.1. ABSOLUTE STABlI,lTY 260

Figure 7.4: Graphical representation of the circle criterion.

- ( l /a) + j O , respectively, as shown in Figure 7.4. The real part of the ratio of two complex numbers is positive when the angle difference between the two numbers is less than n/2; that is, the angle (81 - 82) in Figure 7.4 is less than n/2. If we define D(a,P) to be the closed disk in the coinplex plane whose diameter is the line segment'connecting the points -(l/a) + j O and -(l/P) + j0, then it is simple to see that the angle (81 - 82) is less than n/2 when q is outside the disk D(a, P). Since (7.11) is required to hold for all w, all points on the Nyquist plot of G(jw) must be strictly outside the disk D(a,P) . On the other hand, Z(s) is Hurwitz if G(s)/[l -i- aG(s)] is Hurwitz. The Nyquist criterion states that G(s)/[l + rrG(s)] is Hurwitz if and only if the Nyquist plot of G(jw) does not intersect the point -(l/a)+jO and encircles it exactly m times in the counterclockwise direction, where m is the number of poles of G(s) in the open right-half complex plane.4 Therefore, the conditions of Theorem 7.1 are satisfied if the Nyquist plot of G(jw) does not enter the disk D(a ,P) and encircles it m times in the counterclockwise direction. Consider, next, the case when 0 > 0 and a = 0. For this case, Theorem 7.1 requires 1 + flG(s) to be strictly positive real. This is the case if G(s) is Hurwitz and

Re[l + PG(jw)] > 0, V w E [-co, w]

The latter condition can be rewritten as

which is equivalent to the graphical condition that t.he Nyquist plot of G(jw) lies to the right of the vertical line defined by Re[s] = -1/P. Finally, consider the case

'when G(s) has poles on the imaginary axis, the Nyquist path is indented in the right-half plane, as usual.

270 C W P T E R 7. FEEDBACK SI'STEAIS

when a < 0 < 0. In this case, conditioii (7.10) is cquivalcnt to

where the inequality sign is reversed because. as we go from (7.10) to (7.12), ;ve multiply by alp. which is iiow iiegative. Repeati~ig previous arguments, it can be easily seen that for (7.12) to hold, the Nyquist plot of G(jw) must lie inside the disk D(a,D). Consequently, the Nyquist plot cannot encircle the point - ( l / a ) + jO. Therefore, from the Nyquist criterion, we see that G(s) must be Hurwvitz for G(s)/[l + aG(s)] to be so. The stability criteria for the three cases are summarized in the following theorem, which is known as the circle criterion.

Theorem 7.2 Consider a scalar system of the form (7.1)-(7.3), where { A : B:C: D} is a minimal realization of G(s) and $J E [a, 01. Then, the svstem is nbsoltrtelg stable if one of the following conditions is satisfied, as appropriate:

I. If 0 < a < p, the Nyquist plot of G(jw) does not enter the disk D(a, 3) and encircles it m times in the counterclockwise direction, where rn is the i~.ulnber of poles of G(s) with positive real parts.

2. I f 0 = a < p, G(s) is Hunuitz and the ~ ~ ~ u i s i plot of G(jw) lies to the right of the vertical line defined by Re[s] = -lip.

9. If a < 0 < p , G(s) is Htrnuitz and the Nyquist plot of G(jw) lies in the interior of the disk D(a,P).

If the sector condition is satisfied on$ on an interval [a, b], then the f0regoin.g conditions ensure that the system is absolutely stable with a finite domaiii. 0

The circle criterion allows us to investigate absolute stability by using only the Nyquist plot of G(jw). This is important because the Nyquist plot can be determined directly from experimental data. Given the Nyquist plot of G(j&), we can determine permissible sectors for which the system is absolutely stable. T l ~ c next two exrllnples il11ist.rnte tlic usc of tlic circlc criterion.

Example 7.2 Let

The Nyquist plot of G(jw) is show11 ill Figure 7.5. Since G(s) is Hurwitz. we can allow a to be negative and apply the third case of the circle criterion. So, we need to determine a disk D(a ,P) that encloses the Nyquist plot. Clearly, the choice of the disk is not uniqm. Suppose wc decide to locatme tile center of tlie disk at t l ~ c origin of the complex plaiic. This means that wwte will work with a disk D(-72,32). R ~ C ~ C tllc rndills ( l / ~ ? ) > 0 is to br cliosrii. Tllr Nyquist plot wwpill br i ~ ~ s i d r this

7.1. ABSOLUTE STABILITY

t Figure 7.5: Nyquist plot for Example 7.2.

disk if IG(jw)l < 1/72. In particular, if we set 71 = SUP,,=R IG(jw)I, then 7 2 must be chosen to satisfy 71% < 1. This is the same condition we found in Example 7.1. It is not hard to see that IG(jw)( is maximum at w = 0 and 71 = 4. Thus, 7 2 must be less than 0.25. Hence, we can concludc that the system is absolutely stable for all nonlinearities in the sector [-0.25+~,0.25-€1, where E > 0 can be arbitrarily small. Inspectioh of the Nyquist plot and the disk D(-0.25,0.25) in Figure 7.5 suggests that the choice to locate the center a t the origin may not be the best one. By locating the center a t another point, we might be able to obtain a disk that encloses the Nyquist plot more tightly. For example, let us locate the center at the point 1.5 + jO. The maximum distance from this point to the Nyquist plot is 2.834. Hence, choosing the radius of the disk to be 2.9 ensures that the Nyquist plot is inside the disk D(-1/4.4,1/1.1), and we can conclude that the system is absolutcly stable for all nonlinearities in the sector [-0.227,0.714]. Comparing this sector with the previous one (see Figure 7.6) shows that by giving in a little bit on the lower bound of the sector, we achieve a significant improvement in the upper bound. Clearly, there is still room for optimizing the choice of the center of the disk, but we will not pursue it. The point we wanted to show is that the graphical representation used in the circle criterion gives us a closer look at tlie problem, compared wit11 tlie use of norm inequalities as in Example 7.1, which allows us to obtain less conservative estimates of the sector. Another direction we can pursue in applying the circle criterion is to restrict a to zero and apply the second case of the circle criterion. The Nyquist plot lies to the right of the vertical line Re[s] = -0.857. Hence, we can conclude that the system is absolutely stable for all nonlinearities in the sector [O, 1.1661. This sector is sketched in Figure 7,6, together with the previous two sectors. It gives the best estimate of p , which is achieved at the expense of limiting the nonlinearity to be a first-quadrant-third-quadrant nonlinearity. To appreciate how this flexibility in usiiig tlie circle criterion could be useful in applications, let us suppose that we are interested in studying the stability of the system of Figure 7.7, which includes a liiniter or saturation noillinearity (a typical nonlinearity in feedback co~itrol systems

. _. . ,_ "__I___.._.___ "-- ..-- -"--- ti-- . 8 " . : . -, . . . .a>. . . . . \ ' b - ' . . ..- . . .- , ... " , ,

. - . . . - CHAPTER 7. FEEDBACK SYSTEMS 7.1. ABSOLUTE STABILITY 273

Figure 7.6: Sectors for Example 7.2.

due to constraints on physical variables). The saturation nonlinearity belongs to a sector [O,1]. Therefore, it is included in the sector [O, 1.1661, but not in the sector (-0.25,0.25) or [-0.227,0.714]. Thus, based on the application of the second case of the circle criterion, we can conclude that the feedback system of Figure 7.7 has a globally asymptotically stable equilibrium point at the origin.

A

Figure 7.7:. Feedback connection with saturation nonlinearity.

', Example 7.3 Let a

This transfer function is not Hurwitz, since it has a pole in the open right-half plane. So, we must restrict a to be positive and apply the first case of the circle criterion. The Nyquist plot of G(jw) is shown in Figure 7.8. From the circle criterion, we know that the Nyquist plot must encircle the disk D(a,P) once in the counterclockwise direction. Inspection of the Nyquist plot shows that a disk can be encircled by the Nyquist plot only if it is totally inside one of the two lobes formed by the Nyquist plot in the left-half plane. A disk inside the right lobe is encircled once in the clockwise direction. Hence, it does not satisfy the circle criterion. A disk inside the left lobe is encircled once in the counterclockwise direction. Thus, we need to

Figure 7.8: Nyquist plot for Example 7.3.

choose a and p to locate the disk D(a,P) inside the left lobe. Let us locate the center of the disk a t the point -3.2 + jO, about halfway between the two ends of the lobe on the real axis. The minimum distance from this center to the Nyquist plot is 0.1688. Hence, choosing the radius to be 0.168, we conclude that the system is absolutely stable for all nonlinearitiea in the sector [0.2969,0.3298]. A

In ~xa rn~ l$A 7.1 through 7.3, we have considered cases where the sector condition is satisfied globally. In the next example, the sector condition is satisfied only on a finite interval.

Example 7.4 Consider the feedback connection of Figure 7.1, where the linear system is represented by the transfer function

G(s) = s + 2 (8 + 1)(8 - 1)

and the nonlinear element is $(y) = sat(y). The nonlinearity belongs globally to the sector (0, I]. However, since G(s) is not Hurwitz, we must apply the first case of the circle criterion, which requires the sector condition to hold with a positive a. Thus, we cannot conclude absolute stability by using the circle ~ r i t e r i on .~ The best we can hope for is to show absolute stability with a finite domain. Figure 7.9 shows that on the interval 1-a,a], the nonlinearity @ belongs to the sector [alp] with a = l / a and p = 1. Since G(s) has a pole with positive real part, the Nyquist plot of G(jw), shown in Figure 7.10, must encircle the disk D(a , l ) once in the counterclockwise direction. It can be verified, analyticauy, that condition (7.10) is satisfied for a > 0.5359. Thus, choosing a = 0.55, the sector condition is satisfied on the interval [-1.818,1.818] and the disk D(0.55.1) is encircled once by t h e ' N y q ~ t plot in the counterclockwise direction. From the first case of the circle criterion, we

=1n fact, the origin is not globally asymptotically stable because the system has three equilib rium points.

CHAPTER 7. FEEDBACK SYSTEMS

Figure 7.9: Sector for Example 7.4.


conclude that the system is absolutely stable with a finite domain. We can also use a quadratic Lyapunov function V(x) = xTPx to estimate the region of attraction. Consider the state model

'The loop transformation of Figure 7.3 is given by

Thus, the transformed linear system is given by I

7.1. ABSOLUTE STABILITY

Figure 7.11: Region of attraction for Example 7.4.

where

A = [ - i 5 5 ] , B = [ 1 , C = [ 0.9 0.45 1 , and D = l

The matrix P is the solution of equations (7.6) through (7.8). It can be verified that6

E = 0.02, . P = [ 0'4946 0'4834 ] , L = [ 0.i946 -0.4436 ] , and bV = & 0.4834 1.0774

satisfy (7.6) through (7.8). Thus, V(x) = xTPx is a Lyapunov function for the system. We estimate the region of attraction by

where c < min~lyl=1.818) V(x) = 0.3445 to ensure that 0, is contained in the set {Jyl < 1.818). Taking c = 0.34 gives the estimate shown in Figure 7.11. A

7.1.2 Popov Criterion

Consider a special case of the system (7.1)-(7.3), given by

where x E Rn, u, y E RP, (A, B) is controllable, ( A , C) is observable, and $J~ : R * R is a locally Lipschitz memoryless nonlinearity that belongs to the sector [0, ki]. In this special case, the transfer function G(s) = C ( s I - A)-'B is strictly proper

6The value of E is chosen such that C(s - ~ / 2 ) is positive real and [(~/2)1 +A] is Hurwitz, where G(s) = C(sI - A)-'B + D. Then, P is calculated by solving a Riccati equation, as described in Exercise 6.8.

. - 1. . ...., CHAPTER 7. FEEDBACK SYSTEBIS '

Figure 7.12: Loop transformation. f and $J is time invariant and decoupled; that is, $Ji(y) = $Ji(yi) . Since D = 0, the feedback connection has a well-defined state model. The following theorem, known as the multivariable Popov criterion, is proved .by using a (Lure-type) Lyapunov function of the form V = (1/2)xTPx + C y i J ' l(li(u) du, which is motivated by the application of a loop transformation that transforms the system (7.13)-(7.15) into the feedback connection of two passive dynamical systems.

Theorem 7.3 The system (7.13)-(7.15) is absolutely stable ifl jor 1 5 a 5 p, qi E [O, ki], O < ki 5 CO, and the~e edsts a constant ~i 2 0, urith (1 + h y i ) # 0 for every eigenualue Xk of A, such that M + ( I + sr)G(s) is stn'ctly positive real, where r = diag(yl,. . , -yp) and M = diag(l/kl,. .., Ilkp). If the sector conditiorc qi E [O, ki] is satisfied only on a'set Y c RP, then the foregoing conditions ensure that the system is absolutely stable w.th a finite domain.

0

Proof: The loop tr_ansformation of Figure 7.12 results in a feedback connection of H~ and &, where HI is a linear system whose transfer function is

. .

Thus, M + ( I + sr)G(s) :can be realized by the state model {A, B,C, 271, when: A = A, B = B, C = C + rCA, nl~d D = hf + r C B . Let Xk be an eigenvalue of A ant1 ur; be the associated eigenvector. Then

(C + rCA)vh = (C + rCXk)Uk = (I + Xpr)Cvk

7.1. ABSOLUTE STABILITY 277

The condition (1 +Xl;y,) # 0 inlplies that (A,C) is observable; hence, tlle realizatioll {A, B,C, D) is minimal. If A4 + ( I + sr)G(s) is strictly positive real, we c m apply the ICalman-Yakubovich-Popov lemma to collclude that there are matrices P = PT > 0, L, and Wl and a positive constant E that satisfy

and V = (1/2)xTPx is a-storage function for f i l . One the other hand, it can be verified (Exercise 6.2) that H2 is passive with the storage function xf=l yi $" A(@) do. Thus, the storage function for the transformed feedback connection of Figure 7.12 is i

We use V as a Lyapunov function candidate for the original feedback connection i ., (7.13)-(7.15). The derivative V is given by 3

i

Using (7.16) and (7.17) yields

Substituting u = -$J(v) and using (7.18), we obtain

v = - I E X ~ P X - ~ ( L x + W U ) ~ ( L X + Wu) - ( ~ ( y ) ~ [ y - M$(y)] 5 - $CX~PX

which shows that the origin is globally asymptotically stable. If $J satisfies the sector i condition only for y E Y , the foregoing analysis will be valid in some neighborhood of the origin, showing that the origin is asymptotically stable. i

For M + (I + sr)G(s) to be strictly positive real, G(s) must be Hurwitz. As we have done in the circle criterion, this restriction on G(r) may be removed by performing a loop transformation that replaw C(s) by G(s)P + K,G(s)J-l. We will not repeat this idea in general, but will illustrate it by an example. In the scalar case p = 1, we can test the strict positive realneaa of Z(s) = (I/*) + (1 + s?)G(s) graphically. By Lemma 6.1, Z(s) is strictly positive real if G(8) is Hunsitz and

CHAPTER 7. FEEDBACK SYSTEhlS 7.1. ABSOLUTE STABILITY 279

Figure 7.13: Popov plot.

where G(jw) = Re[G(jw)] + jIm[G(jw)]. If we plot Re[G(jw)] versus wIm[G(jw)] with w as a parameter, then condition (7.19) is satisfied if the plot lies to the right of the l i e that intercepts the point -(l/k) + j O with a slope 117. (See Figure 7.13.) Such a plot is known as a Popov plot, in contrast to a Nyquist plot, which is a plot of Re[G(jw)] versus Im[G(jw)j. If condition (7.19) is satisfied only for w E (-co, w ) , while the left-hand side approaches zero as w tends to oc, then we need to analytically verify that

This case arises when k = w and G(s) has relative degree two. With 7 = 0, condition (7.19) reduces to the circle criterion condition Re[G(jw)] >

-1/k, which shows that, for the system (7.13)-(7.15), the conditions of the Popov criterion are weaker than those of the circle criterion. In other words, with 7 > 0, absolute stability can be established under less stringent conditions.

Example 7.5 Consider the second-order system

' This system would fit the form (7.13)-(7.15) if we took $J = h, but the m a t r ~ A would not be Hurwitz. Adding and subtracting the term ay to the right-hand side of the second state equation, where a > 0, and defining $J(y) = h(y) - ay, the system takes the form (7.13)-(7.15), with

A = [ - : B = [ : ] , and C = [ 1 O ]

Figure 7.14: Popov plot for Example 7.5.

Assume that h belongs to a sector [a,P], where P > a. Then, $J belongs to the sector [0, k], where k = /3 - a. Condition (7.19) takes the form

1 a - ~ ' + ~ ~ - + > 0, V w E 1-03, co] k ( a - w ~ ) ~ +w2

For all 6iiite positive values of a and k, this inequality is satisfied by choosing 7 > 1. Even at k = m, the foregoing inequality is satisfied for all w E (-co, oo) and

lim w2(a-J+7w2) > O

u-00 ( a - w2)2 + w2

Hence, the system is absolutely stable for all nonlinearities h in the sector [a, co], where a can be arbitrarily small. Figure 7.14 shows the Popov plot of G(jw) for a = 1. The plot is drawn only for w 2 0, since Re[G(jw)] and wIm[G(jw)] are even functions of w. The Popov plot asymptotically approaches the line through the origin of unity slope from the right side. Therefore, it lies to the right of any line of slope less than one that intersects the real axis at the origin and approaches it asymptotically as w tends to co. To see the advantage of having 7 > 0, let us take 7 = 0 and apply the circle critcriol~. Fkom the second case of Theorem 7.2, the system is absolutely stable if the Nyquist plot of G(jw) lies to the right of the vertical line defined by Re[s] = -l/k. Since a portion of the Nyquist plot lies in the left-half plane, k cannot be arbitrarily large. The maximum permissible value of k can be determined analytically from the condition

which yields k < l f 2 f i . Thus, using the circle criterion, we can only conclude that the system is absolutely stable for all nonlinearities h in the sector [a, l+a+2fi-E], where a > 0 and E > 0 can be arbitrarily small. A

. / . . - . . - . - 280 CHAPTER 7. FEEDBACK SYSTEMS

7.2 The Describing Function Method

Consider a single-input-singleoutput nonlinear system represeoted by the feedback wnnection of Figure 7.1, where G(s) is a strictly proper, rational t r d r function and 4 is a time-invariant, memoryless nonlinearity. We assume that the externd input T = 0 and study the existence of periodic solutions. A periodic solution satisfies y(t + 27r/w) = y(t) for all t, where w is the frequency of oscillation. We will use a general method for finding periodic solutions, known as the method of hanonic balance. The idea of the method is to represent a periodic solution by a hurier series and seek a frequency w and a set of Fourier coefficients that satish, the system's equation. Suppose y(t) is periodic and let

be its Fourier series, where a t are complex coefficients: a* = I - k and j = fl. Since $(.) is a time-invariant nonlinearity, $(g(t)) is periodic with the same f r e quency w and can be written as

m

Il(y(t)) = ckex~(jkwt) k=-m

where cadi complex co&cient is a function of all ails. For y(t) to be a solution

of the feedback system, it must satisfy the differential equation

d(~)ll(t) + n(p)$(y(t)) = 0

where p is the differential operator P(.) = d(.)/dt and n(s) and d(s) are the numer- ator and denominator polynomials of G(s). Because

we have M M

d(p) ak exp(jk*t) = d(jkw)ai exp(jkut) k=-w k = - m

and

- - k=-w k=-m

Substituting these expressions back into the differential equation yields w

[d(jkw)ab + n(jkw)ck] exp(jkwt) = 0 k=-m

'A bar over a complex variable denbtes its complex conjugate.

_ , , . ._. _._. _ ..... _- r t 7- .- '

' . . . a < .

. . . - .- .$. , . . 98. . . . . ,: ,-,&.A . -.. . .. ' . d "

7.2. THE DESCRIBING FUNCT~ON METHOD 281

Using tlle ortllogondity of t l ~ e fu~~ctions exp(jl;wt) for different values of I*., we find that the Fourier coefficients must satisfy

G(jkw)ck + ak = 0 (7.20) for all integers k. Because G(jkw) = G(-jksr), ar, = &,k, and ck = 5-k, we need only look at (7.20) for k 2 0. Equation (7.20) is an infinitedimensional equation, which we can hardly solve. We need to find a finite-dimensional approximation of (7.20). Noting that the transfer function G(s) is strictly proper, that is, G(jw) -, 0 as w -, oo, it is reasonable to assume that there is an integer q > 0 such that for all k > q, (G(jkw)( is small enough to replace G(jltw) (and consequently ak) by 0. This approximation reduces (7.20) to a finite-dimensional problem

G(jkw)&+Bk=O, k = 0 , 1 , 2 ,..., q (7.21) where the Fourier coefficients are written with a hat accent to emphasize that a solution of (7.21) is only an approximation to the solution of (7.20). In essence, we can proceed to solve (7.21). However, the complexity of the problem will grow with q and, for a large q, the finite-dimensional problem (7.21) might still be difficult to solve. The simplest problem results if we can choose q = 1. This, of course, requires the transfer function G(s) to have sharp "low-pass filtering'' characteristics to allow us to approximate G(jkw) by 0 for all k > 1. Even though we know G(s), we cannot judge whether this is a good approximation, since we do not know the frequency of oscillation w. Nevertheless, the classical describing function method makes this approximation and sets Bk = 0 for k > 1 to reduce the problem to one of solving the two equations

G(0)?o(Borci,)+cio = 0 (7.22) G(jw)E1(cio,B1) + B1 = 0 (7.23)

Notice that (7.22) and (7.23) define one real equation (7.22) and one complex equation (7.23) in two real unknowns, w and Bo, and a complex unknown B1. When expressed as real quantities, they define three equations in four unknowns. This is expected because the time origin is arbitrary for an autonomous system, so if (B0,dl) satisfies the equation, then (Bo, BleJe) will give another solution for arbitrary real 6'. To take care of this nonuniqueness, we take the first harmonic of y(t) to be asinwt, with a 2 0; that is, we choose the time origin such that the phase of the first harmonic is zero. Using

a a a sin wt = 3 [exp(jwt) - exp(-jut)] + bl = - 2 j

we rewrite (7.22) and (7.23) as

i re r g ."** s:. J,% B '*I

.? ' ?& .>> ,?< .,<.: ,*.

38 r ; i

.$,*,

:"P '*>'A, ;* ., . .. ., ;&<

'*>

989 CHAPTER 7. FEEDBACK SYSTEMS

Since (7.24) does not depend on w, it may be solved for 60 as a function of a. Note that if $(a) is an odd function, that is,

then 60 = & = 0 is a solution of (7.24) because

For convenience, let us restrict our attention to nonlinearities with odd symmetry and take do = & = 0. Then, we can rewrite (7.25) es

The coefficient &(0, a/2j) is the complex Fourier coefficient of the first harmonic at the output of the nonlinearity when its input is the sinusoidal signal asinwt. It is given by

2 a / w

PI (0, a/2j) = & & $(a sin wt) eip(- jwt) dt

2n /w

= E 1 W(a sin wt) cos wt - j$(a sin wt) sin wt] dt

The first term under the integral sign is an odd function, while the second term is an even function. Therefore, the integration of the first term over one complete cycle is zero, and the integral simplifies to

61 (0, a/'&) = - J ; $(a sin wt) sin wt dt 6"" Define a function @(a) by

so that (7.26) can be rewritten as

Since we are not interested in a solution with a = 0, we can solve (7.28) completely by finding all solutions of

G(jw)@(a) + 1 = 0 (7.29) Equation (7.29) is known as the first-order harmonic balance equation, or simply

the harmonic balance equation. The function @(a) defined by (7.27) is called the

7.2. THE DESCRIBING FUNCTION METHOD 283

describing function of the nonlinearity 4. It is obtained by applying a sinusoidal signal asinwt at the input of the nonlinearity and by calculating the ratio of the Fourier coefficient of the first harmonic at the output to a. It can be thought of as an "equivalent gain" of a linear time-invariant element whose response to a sinwt is @(a)asinwt. This equivalent gain concept (sometimes called equivalent linearization) can be applied to more general time-varying nonlinearities or nonlinearities with memory, like hysteresis and back la~h .~ In that general context, the describing function might be complex and dependent on both a and w. We will only deal with describing fuilctions of odd, time-invariant, memoryless nonlinearities for which @(a) is real, dependent only on a, and given by the expression

@(a) = A & $(a i n 0) s ins do (7.30)

which is obtained from (7.27) by changing the integration variable from t to 0 = wt. The describing function method states that if (7.29) has a solution (a,, w,), then

there is "probably" a periodic solution of the system with frequency and amplitude (at the input of the nonlinearity) close to w, and a,. Conversely, if (7.29) has no solutions, then the system "probably" does not have a periodic solution. More analysis is needed to replace the word "probably" with "certainly" and to quantify the phrase "close to w, and a," when there is a periodic solution. We will postpone these investigations until a later point in the section. For now, we would like to look more closely at the calculation of the describing function and the question of solving the harmonic balance equation (7.29). The next three examples illustrate the calculation of the describing function for odd nonlinearities.

Example 7.6 Consider the signum nonlinearity $(y) = sgn(y). The describing function is giver1 by

Example 7.7 Considcr the piecewise-linear function of Figurc 7.15,'If s sinusoidal input to this nonlinearity has amplitude a < 6, the nonlinearity will act as a linear gain. The output will be a sirlusoid with amplitude s la . Hence, the describing function is Q(a) = s l , independent of a. When a > 6, we divide the integral on the right-hand side of (7.30) into pieces, with each piece corresponding to a linear portion of $(.). Furthermore, using tlie odd symmetry of the output waveform, we simplify the integration to

@(a) = 2 /' $(a sin 0) sin 0 do = na 0

~ ( a sine)

Figure 7.15: Piecewise-linear function.

Thus,

l ( a ) = - lin-l (:) + + s2 R

4 q' A sketch of the describing function is shown in Figure 7.16. By selecting specific

values for 6 and the slopes sl and sz, we can obtain the describing function of severd common nonlinearities. For example, the saturation nonlinearity is a special case of the piecewise-linear function of Figure 7.15 with 6 = 1, sl = 1, and s 2 = 0. Hence,

6 , K'~

* * 4. , , - . , . - - A . , - . r - L _ I. --I

7.2. T H E DESCRIBING FUNCTION M E T H O D 285

Figure 7.16: Describing function for the piecewise-linear function of Figure 7.15.

its describing function is given by

Example 7.8 Consider an odd nonlinearity that satisfies the sector condition

all2 < Y*(Y) I Py2

for all y E R. The describing function l ( a ) satisfies the lower bound

~ ( a ) = $1' *(a sin 0) sin 0 do 2 - L r s i n 2 0 do = or

and the upper bound

a ) = $ " s i n ) s i n 0

Therefore, or 5 l ( a ) 5 P, V a 2 0

A

Since the describing function l ( a ) is real, (7.29) can be rewritten as

{Re[G(jw)] + jIm[G(jw)]) @(a) + 1 = 0

This equation is equivalent to the two real equations

Because (7.32) is independent of a , we can solve it first for w to determine the poc sible frequencies of oscillation. For each solution w, we solve (7.31) for a. Note that the possible frequencies of oscillation are determined solely by the transfer function G(s); they are independent of the nonlinearity $I(.). The nonlinearity determines the corresponding value of a, that is, the possible amplitude of oscillation., This prc- cedure can be carried out analytically for low-order transfer functions, as illustrated by the next examples.

Example 7.9 Let

and consider two nonlinearities: the signum nonlinearity and the saturation nonlinearity. By simple manipulation, we can write G(jw) as

Equation (7.32) takes the form

which has one positive root w = ,fi. Note that for each positive root of (7.32), there is a negative root ofequal magnitude. We only consider the positive roots. Note also that a root a t w = 0 would be of no interest because it would not give rise to a nontrivial periodic solution. Evaluating Re[G(jw)] at w = & and substituting it in (7.31), we obtain Q(a) = 6. All this information has.been gathered without specifying the nonlinearity $(.). Consider now the signum nonlinearity. We found in Example 7.6 that Q(a) = 4/na. Therefore, Q(a) = 6 has a unique solution a = 2/3?r. Now we can say that the nonlinear system formed of G(s) and the signum nonlinearity will "probably" oscillate with frequency close to fi and amplitude (at the input of the nonlinearity) close to 2/3?r. Consider next the saturation nonlinearity. We found in Example 7.7 that *(a) < 1 for d l a. Therefore, Q(a) = 6 has no solutions. Therefore, we expect that the nonlinear system formed of G(s) and the saturation nonlinearity will not have a sustained oscillation. A

Example 7.10 Let

and consider two nonliiearities: the saturation nonlinearity and a dead-zone nonlinearity that is a special case of the piecewise-linear function of Example 7.7 with sl = 0, s2 = 0.5, and 6 = 1. We can write G(jw) as

7.2. THE DESCRIBING FUNCTION IIIETHOD 287

Equation (7.32) has a unique positive root w = 2f i . Evaluating Re[G(jw)] at w = 2 f i and substituting it in (7.31), we obtain Q(a) = 0.8. For the saturation nonlinearity, the describing fulictioll is give11 in Example 7.7, arid lP(a) = 0.8 has the unique solution a = 1.455. Therefore, we expect that the nonlinear system formed of G(s) and the saturation nonlinearity will oscillate with frequency close to 2 f i and amplitude (at the input of the nonlinearity) close to 1.455. For the dead-zone nonlinearity, the describing function q ( a ) is less than 0.8 for all a. Thus, Q(a) = 0.8 has no solutions, and we expect that the nonlinear system formed of G(s) and the dead-zone nonlinearity will not have sustained oscillations. In this particular example, we can confirm the no oscillation conjecture by showing that the system is absolutely stable for a class of sector nonlinearities, which includes the given dead-zone nonlinearity. It can be easily checked that

Therefore, from the circle criterion (Theorem 7.2), we know that the system is absolutely stable for a sector [O,P] with 0 < 0.8. The given dead-zone nonlinearity belongs to this sector. Consequently, the origin of the state space is globally asymptotically stable and the system cannot have a sustained oscillation., A

Example 7.11 Consider Raleigh's equation

where E is a positive constant. To study existence of periodic solutions, we represent the equation in the feedback form of Figure 7.1. Let u = -i3/3 and rewrite the system's equation as

The first equation defines a linear systeni. Taking y = i to be its output, its transfer function is ES

G(s) = s2 - ES + 1

The second equation defines a nonlinearity $(Y) = y3/3. The two equations together represent the system in the feedback form of Figure 7.1. The describing function of $(y) = y3/3 is given by

@(a) = & sin o ) ~ sin 0 dB = f a 2

The function G(jw) can be written as

. -. . . .- . . . - /

288 CHAPTER 7. FEEDBACK &TE~?S " <@ d The equation Im[G(jw)] = 0 yields ~ ( 1 - w 2 ) = 0; hence, there is a unique positive solution w = 1. Then,

1 + @(a)Re[G(j)] = 0 =+ a = 2

Therefore, we expect that Raleigh's equation has a periodic solution of frequency near 1 rad/sec and that the amplitude of oscillation in t is near 2. n

For higher-order transfer functions, solving the harmonic balance equation (7.29) analytically might be very complicated. Of course, we can always resort to numerical methods for solving (7.29). However, the power of the describing function method is not in solving (7.29) analytically or numerically; rather, it is the graphical solution of (7.20) that made the method popular. Equation (7.20) can be rewritten as

Equation (7.33) suggests that we call solve (7.29) by plotting the Nyquist plot of G(jw) for w > 0 and the locus of -1/P(a) for a 2 0. Intersections of these loci give the solutions of (7.29). Since @(a) is real for odd nonlinearities, the locus of -1/8(a) in the complex plane will be confined to the real axis. Equation (7.34) suggests a similar procedure by plotting the inverse Nyquist plot of G(jw) (that is, the locus in the complex plane of l/G(jw) as w varies) and the locus of -@(a). The important role of Nyquist plots in classical control theory made this graphical implementation of the describing function method a popular tool with control engineers as they faced nonlinearities.

Example 7.12 Consider again the transfer function G(s) of Example 7.9. The Nyquist plot of G(jw) is shown in Figure 7.17. It intersects the real awis a t (-1/6,0). For odd nonlinearities, (7.29) will have a solution if the locus of -l/Q(a) on the real axis includes this point of intersection. A

Let us turn now to the question of justifying the describing function method Being an approximate method for solving the infinitedimensional equation (7.20), the describing function method .can be justified by providing estimates of the error caused by the approximation. In the interest of simplicity, u7e will pursue this analysis only for nonlinearities with the following two feature^:^

Odd nonlinearity, that is, $(y) = -+(-y), V y ;f. 0.

Single-valued nonlinearity with a slope between a and P; that is,

for all real numbers yl and 32 > yl. %ee [24], [129], and [I891 for descriljirrg function theory for more general nonlinearities


A nonlinearity $(-) with these features belongs to a sector [a,A. Hence, from Example 7.8, its describing function satisfies a 5 Q(a) < P for all a 2 0. It should i

d be noted, however, that the slope restriction is not the same as the sector condition. A nonlinearity may satisfy the foregoing slope restriction with bounds a and 0, and #'I could belong to a sector [a, f l with different bounds B and P.1° We emphmize that 1

.I( in the forth$ominp analysis, we should use the slope bounds a and P, not the sector I

boundaries 6 and j.

Example 7.13 Consider the piecewiselinear odd nonlinearity

2'1 f 0 1 0 l y l 2 2 ' = 4 -Yl { y-2 ,

for2_<2 '_<3 for y 1 3

shown in Figure 7.18. The nonlinearity satisfies the slope restriction

as well as the sector condition

In the forthcoming analysis, we should take a = -1 and P = 1. A i

We will restrict our attention to the question of the exk.tenm of half-wave symmetric periodic solutions;ll that is, periodic solutions that only have odd harmonics. This is a reasonable restriction in view of the odd symmetry of .31. The Fourier coefficients

loverify that [ti, C [a, 61. "This restriction is made only for convenience. See I1281 for a more general analysis that d o e

not make this assumption.


kr: Figure 7.18: Nonlinearity of Example 7.13. f~ 2'

of the odd harmonics of a periodic solution y(t) satkfy (7.20) for k = 1,3,5,. ... %? The basic idea of the error analysis is to split the periodic solution y(t) into a first

harmonic yl(t) and higher harmonics yh(t). We choose the time origin such that - r

T s b ~ the phase of the first harmonic is zero; that is, yl(t) = asinwt. Thus, &% U" l b " l

,y ~ y(t) = a sin wt + yh(t)

Using this representation, the Fourier coefficients of the first harmonic of y(t) and *(v(t)) are

a a1 = -

2 j

cl = flrh $(asinut + yh(t)) exp(-jut) dt i T

Fkom (7.20), with k = 1, we have

G(jw)cl + a1 = 0

Introducing the functiol;

@*(a, yh) = - = j- 2w J"'~ $(aSinwt + yh(t))exp(-jut) dt a1 0

we can rewrite the equation as

1

G(jw) - + **(a, yh) = 0 (7.35)

Adding Q(a) to both sides of (7.35), we can rewrite it as

1 - C(jw)

+@(a) = 6Q (7.36)

7.2. THE DESCRIBING FUNCTION METHOD

Figure 7.19: Finding p(w).

where 6Q = @(a) - @*(a, yh)

When yh = 0, @*(a,O) = ,@(a). Thus, 6Q = 0 and (7.36) reduces to the harmonic balance equation

Therefore, the harmonic balance equation (7.37) is an approximate version of the exact equation (7.36). The error term 6Q cannot be found exactly, but its size can often be estimated. Our next step is to find an upper bound on 6%~. To that end, let us define two functions p(w) and a(w). Start by drawing the locus of l/G(jw) in the complex plane. On the same graph paper, draw a (critical) circle with the interval [-P, -a] on the real axis as a diameter. Notice that the locus of -@(a) lies inside this circle on the real axis, since a 5 Q(a) < 0. Now consider an w such that the points on the locus 1/G corresponding to k u (k > 1 and odd) lie outside the critical circle, as shown in Figure 7.19. The distance from any one of these points to the center of the critical circle is

Define 1

in, lT+-l p(w) = k>l ;k odd G(jkw)

Note that we have defined p(w) only for w at which l/G(jkw) lies outside the critical . circle for all k = 3,5,. .; that is, for w in the set

On any connected subset R' of R, define

. ... . . . . -- - .. - -. . --" "",% $- - . ..

. .-.. . .. -.. . . , ,


The positive quantity u(w) is an upper bound on the error term 69, as stated in the next lemma.

Lemma 7.1 Under the stated assumptions,


The proof of Lemma 7.1 is based on writing an equation for yh(t) in the form yh = T(gh) and showing that T(.) is a contraction mapping. This allows us to calculate the upper bound of (7.40), which is then used to calculate the upper bound of (7.41) on the error term. The slope restrictions on the nonlinearity $J are used in showing that T(.) is a contraction mapping.

Using the bound of (7.41) in (7.36), we see that a necessary condition for the existence of a half-wave symmetric periodic solution with w E R' is

Geometricdly, this condition states that the point -@(a) must be contained in a circle with a center at l/G(jw) and radius o(w). For each w E R' C a, we can draw such an error circle. The envelope of all error circles over the connected set fl' forms an uncertainty band. The reason for choosing a subset of R is that, as w approaches the boundary of R, the error circles become arbitrarily large and cease to give any useful information. The subset R' should be chosen with the objective of drawing a narrow band. If G(jw) llas sharp low-pass filtering characteristics, the uncertainty band can be quite narrow over 0'. Note that p(w) is a measure of the low-pass filtering characteristics of G(jw); for the smaller (G(jkw)l for k > 1, the larger p(w), as seen from (7.38). A large p(w) results in a small radius u(w) for the error circle, as seen from (7.39).

We are going to look a t intersections of the uncertainty band with the locus of -@(a). If no part of the band intersects the -@(a) locus, then clearly (7.36) has no solution with w E 0'. If t11c barlcl intersects the locus completely, as in Figure 7.20, then we expect that there is a solution. This is indeed true, provided

i " _. , .

' , ' I - , . . . . , , , . 2. .,. , . , 4 5 .. . . .e- . . .-.- -.. '. . c .

7.2. TIIE DESCRIBING FUNCTION AfETIIOD

Figure 7.20: A complete intersection.

we exclude some degenerate cases. Actually, we can find error bounds by examining the intersection. Let a1 and a2 be the amplitudes corresponding to the intersections of the boundary of the uncertainty band with the -@(a) locus. Let wl and w2 be the frequencies corresponding to the error circles of radii u(wl) and u(wz), which are tangent to the -@(a) locus on either side of it. Define a rectangle r in the (w, a ) plane by

... ~ = { ( w , a ) ( w 1 < w < w z , a l < a < a 2 )

The rectangle r contains the point (w,,a,) for which the loci of 1/G and -!? intersect, that is, the solution of the harmonic balance equation (7.37). It turns out that if certain regularity conditions hold, then it is possible to show that (7.36) has a solution in the closure of J?. These regularity conditions are

A complete intersection between the uncertainty band and the -!?(a) locus can now be precisely defined as taking place when the l/G(jw) locus intersects the -rk(a) locus and a finite set r can be defined, as shown, such that (w,, a,) is the unique intersection point in r and the regularity conditions hold.

Finally, notice that a t high frequencies for which all harmonics (including the first) have the corresponding l/G(jw) points outside the critical circle, we do not need to draw the uncertainty band. Therefore, we defme a set

and take the smallest frequency in fi as the largest. frequency in R', then decrease w until the error circles become uncomfortably large.

The next theorem on the justification of the describing function method is the main result of this section.


Theorem 7.4 Consider the feedback connection of Figure 7.1, where the nonlin- eority *(a) is memoryless, time invariant, odd, and single valued with slopes between a and 0. Dmw the loci of l/G(jw) and -9(a) in the complez plane and construct

$:$* 1.*

the critical circle and the band of uncertainty as described earlier. Then, $&4,

sv %y<T 1-+ the system has no half-wave symmetric periodic solutions with fundamental

frequency w E 6. i

t

$t * > the system has no half-wave symmetric periodic solutions with bndamental &6$ frequency w E R' if the comsponding error circle does not intersect the -9(a) *&' locua. k ?& for each complete intersection defining a set in the (w, a) plane, there is at

least one half-wave symmetric periodic solution Y

y(t) = a sin wt + yh(t)

with (w, a) in f and yh(t) satisfies the bound of (7.40). 0

>*<, Proof: See Appendix C.14.

,- Note that the theorem gives a sufficient condition for oscillation and a sufficient condition for nonoscillation. Between the two conditions, there is an area of ambiguity where we cannot reach conclusions of oscillation or nonoscillation.

+ , Example 7.14 Consider again *\,,

* together with the saturation nonlinearity. We have seen in Example 7.10 that the

&* harmonic balance equation has a unique solution w. = 2 f i = 2.83 and a, = 1.455.

T . The saturation nonlinearity satisfies the slope restrictions with a = 0 and ,8 = 1. 5 Therefore, the critical circle is centered at -0.5 and its radius is 0.5. The function

.* l/G(jw) is given by

p 9;; Hence, the locus of l/G(jw) lies on the lime Rels] = -0.8, as shown in Figure 7.21.

Q The radius of the error circle u(w) has been calculated for eight frequencies starting + * with w = 2.65 and ending with w = 3.0, with uniform increments of 0.05. The * , - centers of the error circles are spread on the line Re[s] = -0.8 inside the critical

circle. The d u e of u(w) st w = 2.65 is 0.0388 and at w = 3.0 is 0.0321, with t - monotonic change between the two extremes. In all cases, the closest harmonic to *\ the critical circle is the third harmonic, so that the infimum in (7.38) is achieved at

k = 3. The boundaries of the uncertainty band are almost vertical. The intersection :" - of the uncertainty b m d with the -@(a) 1oc11s correspond to the points a1 = 1.377

4 .

i v

7.2. THE DESCRIBING FUNCTION METHOD 290

, Figure 7.21: Uncertainty band for Example 7.14.

and a2 = 1.539. The error circle corresponding to w = 2.85 is almost tangent to the real axis from the lower side, so we take w2 = 2.85. The error circle corresponding to w = 2.8 is the closest circle to be tangent to the real axis from the upper side. This means that wl > 2.8. Trying w = 2.81, we have obtained a circle that is almost tangent to the real axis. Therefore, we define..the set r as

r = {(w, a) 1 2.81 < w < 2.85, 1.377 < a < 1.539)

There is only one intersection point in I?. U'e need to check the regularity conditions. The derivative

is different from zero at a = 1.455, and

Thus, by Theorem 7.4, we conclude that the system indeed has a periodic solution. Moreover, we conclude that the frequency of oscillation w belongs to the interval

Figure 7.22: Inverse Nyquist plot and critical circle for Example 7.15.

[2.81,2.85], and the amplitude of the first harmonic at the input of the nonlinearity belongs to the interval [1.377,1.539]. F'roin the bound of (7.40), we know also.that the higher harmonic component yh(t) satisfies

!! ~ " y 2 ( t ) dt 5 0.0123, Q (w, a) E r X 0 .

which shows that the waveform of the oscillating signal at the nonlinearity input is fairly close to its first harmonic asin wt. A

Example 7.15 Reconsider Example 7.9 with

and the saturation nonlinearity. The noillinearity satisfies the slope restriction with a = 0 and p = 1. The inverse Nyq~ ist plot of G(jw), shown in Figure 7.22, lies outside the critical circle for all w > Hence, fl = (0, m), and we conclude that there is no oscillation. 3 A

7.3 Exercises

7.1 Using the circle criterion, study absolute stability for each of the scalar transfer functions given next. In each case, find a sector [a,P] for which the system is absolutely stable.

(1) G(s).= S

I s 2 - S + 1

1 (2) G(s) = (S +2)(s +3)

7.3. EXERCISES 297

7.2 Consider the feedback connection of Figure 7.1 with G(s) = 2s/(s2 + s + 1).

(a) Show that the system is absolutely stable for nonllnearities in the sector [O, 11.

(b) Show that the system has no limit cycles when $(y) = sat(y).


where h is a smooth function satisfying

J h ( y ) l l c, for a1 c v c a2 and - a2 c y c -a1

(a) Show that the origin is the unique equilibrium point.

(b) Show, using the circle criterion, that the origin is globally asymptotically stable.

7.4 ([201]j Consider the system

where p, a, q, and w are positive constants. Represent the system in the form of Figure 7.1 with $(t, y) = qy coswt and use the circle criterion to derive conditions on p, a, q, and w, which ensure that the origin is exponentially stable.

7.5 Consider the linear time-varying system x = [A + BE( t )qx , where A ia Hurwitz, JIE(t)lJz I 1, Q t 2 0, and s ~ p , , ~ u , , ~ ( j w I - A)-'B] c 1. Show that the origin is uniformly asymptotically stable.

298 9" CHAPTER 7. FEEDBACK SYSTEMS

ki 7.6 Consider the system x = Ax + Bu and let u = -Fx be a stabilizing state

feedback control; that is, the matrix (A - BF) is Hurwitz. Suppose that, due to physical limitations, we have to use a limiter to limit the value of ui to (ui(t)( <_ L. The closed-loop system can be represented by x = Ax - BL sat(Fx/L), where sat(v) is a vector whose ith component is the saturation function.

(a) Show that the system can be represented in the form of Figure 7.1 with G(s) = F(sI - A + BF)-lB and $(y) = L sat(y/L) - y. i

(b) Derive a condition for asymptotic stability of the origin using the multiwiable circle criterion. 6

u* z

(c) Apply the result to the case

and estimate the region of attraction.

7.7 Repeat Exercise 7.1 using the Popov criterion.

7.8 In this exercise, we derive a version of thepopov criterion for a scalar transfer function G(s) with all poles in the open left-half plane, except for a simple pole on &. the imaginary axis having a positive residue. The system is represented by

E = Az - B$(y), 6 = -$(y), and y = Cz + dv k where d > 0, A is Hurwitz, (A, B) is controllable, (A,C) is observable, and $ belongs to a sector (0, k]. Let V(z , v) = zTPz + a(y - C Z ) ~ + b J: $(u) du, whrre ~ = ~ ~ > 0 , a > o , a n d b > 0 .

(a) Show that V is positive definite and radially unbounded.

(b) Show that v satisfies the inequality

v 5 Z ~ ( P A + A ~ P ) Z - 2zT(PB - w)$(y) - r$2(Y) .r

where w = adCT + (1/2)bATCT and 7 = (2adlk) + b(d + CB). Assume b is chosen such that 7 2 0.

(c) Show that the system is absolutely stable if t 1 b - + Re[(l + jwg)G(jw)] > 0, V w E R, where g = k f

7.9 ([85]) The feedback system of Figure 7.23 represents a control system where H(s) is the (scalar) transfer function of the plant and the inner loop models the actuator. Let H(s) = ( s + 6)/(s + 2)(s + 3) and suppose k 2 0 and $ belongs to a sector (O,@], where p could be arbitrarily large, but finite.

7.3. EXERCISES

(a) Show that the system can be represented as the feedback connection of Fig- ure 7.1 with G(6) = [H(s) + li]/s.

(b) Using the version of the Popov criterion in Exercise 7.8, find a lower bound kc such that the system is absolutely stable for all k > kc.

Figure 7.23: Exercise 7.9.

7.10 For each odd nonlinearity $(y) on the following list, verify the given expree sion of the describing function Q(a):

(I) $(y) = y5; *(a) = 5a4/8

(2) $(y) = y3(yl; *(a) = 32a3/15a

where

4A (3) $(y) : Figure 7.2.L(a); *(a) = f - a a

(4) $(y) : Figure 7.24(b) 0, for a I A

= { ( ~ S / r a ) [ l - ( ~ / a ) ~ ] ' / ~ , for a 2 A

(5) $(y) : Figure 7.24(c) 0, for a 5 A

*(a) = k[l - N(a/A)I, for A i a I B k[N(a/B) - N(a/A)], for a 2 B

7.11 Using the describing function method, investigate the existence of periodic solutions and the possible frequency and amplitude of oscillation in the feedback connection of Figure 7.1 for each of the following cases:

jfp''= i(y)l 7: 7.14 Coasiii~r tlic fixdbnrk colulcct i o ~ ~ of I:igl~ns 7.1 wit11 i

1 - 1 S b y S 1 G(s) =

7 A B Y (8 + 1 ) y s + 2)2 ' *(Y) = sgn(y) blyl > 1 { "

where b > 0.

(a) (b) (c) (a) Using the circle criterion, find the largest b for which we can confirm that the origin of the closed-loop system is globally asymptotically stable.

Figure 7.24: Exercise 7.10. a

(b ) Using the Popov criterion, find the largest b for which we can confirm that the origin of the closed-loop system is glibally asymptotically stable.

(1) G(S) = (1 - s)/s(s + 1) alld rl/(u) = y5. (c) Using the describing function method, find the smallest b for which the system .

(2) G ( ~ ) = (1 - s)/s(s + 1) and $J is the nonlinearity of Exercise 7.10, part (51, will oscillate and estimate the frequency of oscillation. A = 1, B = 312, and k = 2.

(d) For b = 10, study the existence of periodic solutions by using Theorem 7.4. For : (3) ~ ( s ) = l / ( s + 1)6 and $(Y) = sgn(y). each oscillation, if any, i

(4) G(s) = (s + 6)/s(s + 2)(s + 3) and $J(y) = sgn(y). a ( 5 ) G(S) = s/(s2 - s + 1) ant1 $(y) = y5. .I (6) G(s) = 5(s + 0.25)/s2(s + 2)2 and $ is the nonlinearity of Exercise 7.10, part

(3), with A = 1 and k = 2. 1 0

(7) G(s) = 5(s + 0.25)/s2(s + 2)2 and $ is the nonlinearity of Exercise 7.10, part (4), with A = 1 and B = 1.

(8) G(s) = 5(9 + 0.25)/s2(s + 2)2 and $ is the nonlinearity of Exercise 7.10, part (5), with A = 1, B = 312, and k = 2. a

1 (9) G(s) = l / ( s + 1)3 and $(Y) = sgn(y).

. i (10) G(s) = l / ( s + and $(y) = sat(y).

7.12 Apply the describing function method to study the existence of periodic solutions in the negative resistance oscillator of Section 1.2.4 with h(v) = -u+v3 - v5/5 and E = 1. For each possibleperiodic solution, estimate the frequency and amplitude of oscillation. Using computer simulation, determine how accurate the describing function results are.

7.13 Consider the feedback connection of Figure 7.1 with G(s) = 2bs/(s2 - bs+ 1) and $(y) = sat(y). Using the describing function method, show that for sufficiently small b > 0 the system has a periodic solution. Confirm your conclusion by applying Tl~eorem 7.4 and estimate the frequency and amplitude of oscillation.

i. find the frequency interval [wl , wz] and the amplitude interval [ a ~ , az];

ii. use Lemma 7.1 to find an upper bound on the energy content of the higher- , order har~nonics and express it as a percentage of the energy content of

::$ the first harmonic; and

iii. simulate the system and compare the simulation results with the foregoing analytical results.

(e) Repeat part (dJ for b = 30.

., , 7.15 Repeat parts (a) to (c) of the previous exercise for G(s) = 1 0 / ( ~ + 1 ) ~ ( 8 + 2 ) . * . 7.16 Consider the feedback connection of Figure 7.1, where G(s) = l / (s +

and $(y) is the piecewise-linear function of Figure 7.15 with d = l/k, sl = k, and s 2 = 0.

(a) Using the describing function method, investigate the existence of periodic *, ~ solutions and the possible frequency and amplitude of oscillation when k = 10.

(b) Continuing with k = 10, apply Theorem 7.4. For each oscillation, if any, find the frequency interval [wl, w2] and the amplitude interval [al, a2].

(c) What is the largest slope k > 0 for which Theorem 7.4 ensures that there is no oscillation?

7.17 For each of the following cases, apply Theorem 7.4 to study the existence of periodic solutions in the feedback connection of Figure 7.1. For each oscillation, if any, find the frequency interval [wl,w2] and the amplitude interval [all az].

302 CHAPTER 7. FEEDBACK SYSTEMS

(1) G(a) = 2(s - l)/s3(s + 1) and $(y) = sat(y).

(2) G(s) = -s/(s2 + 0.85 + 8) and $(y) = (112) sin y.

(3) G(s) = -a/($ + 0.85 + 8) and $(y) is the nonlinearity of Example 7.13.

(4) G(5) = -24/s2(s + 1)' and $(y) is an odd nonlinearity defined by

y3 + 1112, f o r O < y l l $(I/) =

2~ - 112, for y 2 1

Chapter 8

Advanced St ability Analysis

In Chapter 4, we gave the basic concepts and tools of Lyapunov stability. In this chapter, we examine some of these concepts more closely and present a number of extensions and refinements.

We saw in Chapt.er 4 how to use linearization to study stability of equilibrium points of autonomous systems. We saw also that linearization fails when the Ja- cobian matrix, etaluated at the equilibrium point, has some eigenvalues with zero real parts and no eigenvalues with posit.ive real parts. In Section 8.1, we introduce the center manifold theorem and use it to study stability of equilibrium points of autonomous systems in the critical case when linearization fails.

The concept of the region of attraction of an asymptotically stable equilibrium point was introduced in Section 4.1. In Section 8.2, we elaborate further on that concept and present some ideas for providing estimates of this region.

LaSalle's invariance principle for autonomous systems is very useful in applications. For a general nonautonomous system, there is no invariance principle in the same form that tvas presented in Theorem 4.4. There are, however, theorems which capture some features of the invariance principle. Two such theorems are given in Section 8.3. The first theorem shows convergence of the trajectory to a set, while the second one shows uniform asymptotic stability of the origin.

Finally, in Section 8.4, we introduce 11oLio11s of stability of periodic solutions and invariant sets.

8.1 The Center Manifold Theorem

Consider the autonomous system 15 x = f ( x ) (8.1)

where f : D 4 Rn is continuously differentiable and D c Rn is a domain that contains the origin x = 0. Suppose that the origin is an equilibrium point of (8.1).

. . . . . . - .- .. .. , , - . - - -. . *. .- . -. - . -. - . - :/ . -:. - . L )

304 CHAPTER 8. ADVANCED STABILITY ANALYSIS

Theorem 4.7 states that if the linearization of f at the origin, that is, the matrix

has all eigenvalues with negative real parts, then the origin is asymptotically stable; if it has some eigenvalues with positive real parts, then the origin is unstable. If A has some eigenvalues with zero real parts with the rest of the eigenvalues having negative real parts, then linearization fails to determine the stability properties of the origin. In this section, we take a closer look into the case when linearization fails. Failure of linearization leaves us with the task of analyzing the nth-order nonlinear system (8.1) in order to determine stability of the origin. The interesting finding that we are going to present in the next few pages is that stability properties of the origin can be determined by airalyzing a lower order nonlinear system - a system whone order is exactly equal to the number of eigenvalues of A with zero real parts. This will follow as an application of the center manifold theory.'

A k-dimensional manifold in Rn (1 5 k < n) has a rigorous mathematical defini t i~n.~ For our purpose here, it is sufficient to think of a k-dimensional manifold as the solution of the equation

V(X) = 0

where 7 : Rn -, Rn-k is sufficiently smooth (that is, sufficiently many times continuously differentiable). For example, the unit circle

is a one-dimensional manifold in R2. Similarly, the unit sphere

is an (n - 1)-dimensional manifold in Rn. A manifold {g(x) = 0) is said to be an invariant manifold for (8.1) i f ,

q(x(0)) = 0 & v(x(t)) = 0, V t E [0, t l ) C R

where (0, tl) is any time interval over which the solution x ( t ) is defined. Suppose now that f(x) is twice cont,inuously differentiable. Equation (8.1) can

be represented as

'The center manifold theory has several applications to dynamical systems. It is presented here only insofar as it relates to determining the stability of the origin. For a broader viewpoint of the ti~&ry, the reader may consult 1341.

2See, for example, (711.

8.1. THE CENTER MANIFOLD THEOREM 305

is twice continuously differentiable and 1

Since our interest is in the case when linearization fails, assume that A has k eigenvalues with zero r ed parts and m = n - k eigenvalues with negative real parts. We can always find a similarity transformation T that transforms A into a block diagonal matrix, that is,

where all eigenvalues of A1 have zero real parts and all eigenvalues of A2 have negative real parts. Clearly, A1 i6 k x k and A2 is m x m. The change of variables

transforms (8.1) into the form I

where gl and g2 inherit properties of j. In particular, they are twice continuously differentiable and

for i = 1,2. If z = h(y) is an inmiant manifold for (8.2)-(8.3) and h is smooth, then it is called a center manifold if

Theorem 8.1 If gl and g2 are twice continuously dgerentiable and satCb (8.4), a l l eigenvalue8 of A1 have zero realparts, and all eigenwaluea.o&A2. have negative ml; parts, then there exist a constant 6 > 0 and a wntinuowly diflerentiable functio; h(y), defined for all llyll < 6, n u h that z = h(y) C a center manifad for (8.2)-(8.3).

0 Proof: See Appendix C.15.

If the initial state of the system (8.2)-(8.3) lies in the center manifold; that is, z(0) = h(y(O)), then the solution ( y ( t ) , z ( t ) ) will lie in the manifold for all t 2 0;

i ?

CHAPTER 8. ADVANCED STABILITY ANALYSIS

$&' P. $3

that is, z(t) = h(y(t)). In this case, the motion of the system in the center manifold is described by the kth-order differential equation

. .

9 = A ~ Y + gl(y, h(y)) (8.51 . ,

which we refer to as the reduced system. If z(0) # h(y(O)), then the difference ~ ( t ) - h(y(t)) represents the deviation of the trajectory from the center manifold at any time t. The change of variables

transforms (8.2)-(8.3) into

In the new coordinates, the center manifold is w = 0. The motion in the manifold is characterized by

w ( t ) s O =+ w ( t ) e O

Substituting these identities into (8.7) results in

Since the equation must be satisfied by any solution that lies in the center manifold, we conclude that the function h(y) must satisfy the partial differential equation (8.8). Adding and subtracting gl(y, h(y)) to the right-hand side of (8.6), and sub-

I tracting (8.8) from (8.7), we can rewrite the equation in the transformed coordinates as

-,

+ = Alar + gl(y, h ( d ) + Nib, w) (8.9) = A2w+ N2(y,w) (8.10)

where

Nl(~1.w) = gl(y,w + h b ) ) - g1(y, h(y))

and a h

N2(2/1 w) = g2(v, w + h(y)) - g2(y, h(y)) - -(y) N i b , w) ay

, ,

It is not difficult to verify that Nl and N2 are twice continuously differentiable, and

8.1. THE CENTER AIANIFOLD THEOREAI

for i = 1,2. Consequently, in the domain

Nl and N2 satisfy IINi(ylw)l12 I killwll, i = 192

where the positive constants kl and k2 can be made arbitrarily small by choosing p small enough. These inequalities, together with the fact that A2 is Hurwitz,

C

suggest that the stability properties of the origin are determined by the reduced C

system (8.5). The next theorem, known as the reduction principle, confirms this C conjecture. c- Theorem 8.2 Under the &;&itions of Theorem 8.1, if the origin y = 0 of the reduced system (8.5) is asymptotically stable (respectively, unstable) then the origin

6

ofthe full system (8.2) and (8.3) is also asymptotically stable (respectively, unstable). ;c

0 ' Q Proof: The change of coordinates from (y, I ) to (y, w) does not change the stability ,C properties of the origin (Exercise 4.26); therefore, we can work with the system (8.9) /&- and (8.10). If the origin of the reduced system (8.5) is unstable, then, by invariance, 1 k. the origin of (8.9) and (8.10) is unstable. This is so because for any solution y(t) of (8.5), there is a corresponding solution (y(t),O) of (8.9) and (8.10). Suppose / (. now that the origin of the reduced system (8.5) is asymptotically stable. By (the c converse Lyapunov) Theorem 4.16. there is a continuously differentiable function V(y) that is positive definite and satisfies the inequalities

L 1

in a neighborhood of the origin, wlicre a3 and a4 are class K functions. On the c other hand, since A2 is Hurwitz, tlic Lyapunov equation

PA2 + A ~ P = -I & QP C

has a unique positive definite solution P. Consider b "(Y, w) = V(y) + b

as a Lyapunov function candidatevor the full system (8.9) and (8.10). The dcriva- & tive of v along the trajectories of the system is given by bc

sWB 3The function u(y ,w) is continuously differentiable everywhere around the origin, except on

. the manifold w = 0. Both u ( y , w ) and ir (y,w) are defined and continuous around the origin. It fb can be easily seen that the statement of Theorem 4.1 is still valid. I ie

.. . . . . - -- - .. -. - .- . . . -. .. -- .- .. -. . . . . . , - , . . .. . . . ..

. / . . . . - . . - . .


max :. . : ,

Since k1 and kz can be made arbitrarily small by restricthg the domain around the origin to be sufficiently small, we can choose them small enough to ensure that

Hence, 1

4 % ~ ) 5 -a3(llvl12) - 4 & J F l l w 1 1 2

which shows that L(y, w ) is negative definite. Consequently, the origin of the full system (8.9)-(8.10) is asymptotically stable.

We leave it to the reader (Exercises 8.1 and 8.2) to extend the proof of Theo- rem 8.2 to prove the next two corollaries.

Corollary 8.1 Under the assumptions of Theorem 8.1, if the origin y = 0 of the reduced system (8.5) is stable and there is a continuously differentiable Lyapunov function V ( y ) such that

in some nezghborhood of y = 0, then the origin of the full system (8.2)-(8.3) is stable. 0

\

Corollary 8.2 Under the assumptions of Theorem 8.1, the origin of the reduced system (8.5) is asymptotically stable i f and only i f the origin of the full system (8.2)-(8.3) is asymptotically sttzble. 0 .

4The existence of the Lyapunov function V ( y ) cannot be inferred from a converse Lyapunov theorem. The converse Lyapunov theorem for stability [72, 107) guarantees the existence of a Lyapunov function V( t , y ) whose derivative satisfies ~ ( t , y ) < 0. In general, this function cannot be mule indepcn[l(!nt o f t. (See (72, page 2281.) I<vc:r~ t l~ol~gb we can choose V ( t , y) to be continuously differentiable in its arguments, it cannot be gllarantccd that the partial derivatives aV/Byi, 8V/8t will be uniformly bounded in a neighborhood of the origin for all t > 0. (See (107, page 531.)

. . ., . - -. -T; - . . ... . .. .. . r . .- . . *r, " < . . . --. - -.- , . . . . - . . . . ' b d . @.. . . .-, ,? . .-..

8.1. THE CENTER AIANIFOLD THEOREAI 309

To use Theorem 8.2, we 11c4 to find the ccllter ~llw~ilold : = h(y). The f u ~ ~ c t i o ~ ~ h is a solution of the partial differential equation

def Bh N ( h b ) ) = ~ ( 9 ) [ A l p + gi(& h ( ~ ) ) l - A2h(y) - h ( p , h ( y ) ) = 0 (8.11)

with boundary conditions Bh

h (0 ) = 0; - ( 0 ) = 0 BY 1

This equation for h cannot be solved exactly in most cases (to do so would imply that a solution of the full system (8.2)-(8.3) has been found), but its solution can be approximated arbitrarily closely as a Taylor series in y.

Theorem 8.3 If a continuously diflerentiable function d ( y ) with 4(O) = 0 and [O@/ay](O) = 0 mn be found such that N ( 4 ( y ) ) = O(IIyIJP) for some p > 1, then for suficiently small J ( y (J

h ( v ) - 4 ( v ) = 0(lll/llP) and the reduced system can be represented as

B = Ala, + g i b , 4 ( y ) ) + O ( I I Y I I ~ ~ ' 1


The order of magnitude notation O(.) will be formally introduced in Chapter 10. (Definition 10.1). For our purpose here, it is enough to think of f ( y ) = O ( J J y p ) as a shorthand notation for ( I f (y)l( 5 kJJy(lP for sufficiently small I JyJJ . Let us now:; illustrate the application of the center manifold theorem by examples. In the first. two examples, we will make use of the observation that for a scalar state equation of the form

I

where p is a positive integer, the origin is asymptotically itable if p L - o d d b d a < 0. It is unstable if p is odd and a > 0, or p is even and a # 0.5


x1 = 2 2

x2 = -22 + ax: + bx1x2

where o # 0. The system has a unique equilibrium point a t the origin. The linearization a t the origin results in the matrix


"! which has .eigenvalues a t 0,and -1. Let M be.a,&atrix +hose col-s are the " ei&nvectors .of A; jthat,is; : .i

c<7.e,z: ..?. ~..? I.! -:'.... : ....

and3:t,+ii.,T =.M;~? Then,

I The change of variables

3 - "

j , . puts the system into the form

The center manifold equation (8.11) with the bo&dary condition (8.12) becomes

N(h(v ) ) = h'(ar)[a(e + - b(yh(y) + h2(y))l + h(y) + a(y + h ( ~ ) ) ~ * b(yh(y) + h2(y)) = 0 , , h(0) = hr(0) = 0

. . . . . . . . . . . . . . . . . , . : - . .-,.,

"'At h ( y ) = hi# + hay3..+. and substitute' this series in the center manifold ustionto find t h e u n k n o p coefficients h2, ks, ... by matching coefficients of like

powers in y (since the equation holds as an identity in y). We do not know in a d W c e howmany t e r m of the series we need. Westart'with the simplest np-.

' ' ' pdmat ion h(y)"= O. :We substitute h(y) = O(JyI2) into the reduced system and study stability of its origin. If the stability properties of the origin can be determined, we & don$. Otherwise, we calculate the coefficient h2, substitute h(y) =

O(bP), and study st,ability of the origin. If it c t g o t be resolved, we pro- ceka to the approximation h(y) %. h2y2 + hi$, and so on. Let us start with the approximation h(y) * 0. The reduced system is

d J ..: ?,:. ai = all2 + O ( I V I ~ )

Notice that an O(IY12) error in h(y) results in an O(lyI3) error in the right-hand side of the reduced system. This is a consequence of the fact that the function gl(y, z ) , which appears on the right-hand side of the reduced system (8.5) as gl(y, h(y) ) ,

,': has a partial derivative with respect to z that vanishes at the origin. Clearly, this

>'A .?, observation is rrlso valid for higher order approximations; that is, an error of order .&, ,, : pp.:i.,:, O(ly(k) in h(y) results in an error of order O(JylkC1) in gl(y, h(y)) , for k 2 2. The p$*;:;. &A .X term ay2 is the dominant term on the right-hand side of the reduced system. For

p+' , a # 0, the origin of t b reduced system isunstable. Consequently, by ~ i e o r e m 8.2, .$ tile origin of the full system is unstable. A t $

8.1. THE CENTER AlANIFOLD THEOREhl


which is already represented in the ( y , z ) coordinates. The center manifold equation (8.11) with the boundary condition (8.12) is

a) We start by trying 4(y) = 0. The reduced system is

'7): Y = 0 ( l y 1 3 )

Clearly, we cannot reach any conclusion about the stability of the origin. Therefore, we substitute h(y) =h& +O(JyI3) into the center manifold equation and calculate hz, by matching coefficients of y2, t o obtain h2 = a. The reduced system is"

# ~ k 3 y = ay3 + O((yI4) /

Therefore, the origin is asymptotically stable if a < 0 and unstable if a > 0. Conse- quently, by Theorem 8.2, we conclude that the origin of the full system is asymptotically stable if a < 0 and unstable if a > 0. If a = 0, the center manifold equation (8.11) with the boundary condition (8.12) reduces to

which has the exact solution h(y) = 0. The reduced system y = 0 has a stable origin with V ( y ) = y2 as a Lyapunov functio~~. Tlicrefore, by Corollary 8.1, we conclude that the origin of the full system is stnhle if a = 0 A

f Example 8.3 Consider the system (8.2)-(8.3) with

6 ) It can be verified that ?(y) = results in N ( ( ( y ) ) = 0 ( [ ~ Y I I ; ) and 4

1 Using V ( y ) = (y i + y;)/2 as a Lyapunov function candidate, we obtain

v = -Y: - Y: + y T o ( I l ~ l l : ) r -Il~lli + kllvll;

=The error on the right-hand side of the reduced system is actually 0((y15) since, if we wrlte .... h(y) = hzy2 + hay3 + we will find that 113 = 0.

- ...... - ... - .-..... - ......... .- . . : ...


ill sonie ileighborhood of the origin where I; > 0. Hence,

1 I - ;llYJI;, for l I ~ J l 2 <

which shows that the origin of the reduced system is asymptotically stable. Conse- quently, the origin of the full system is asymptotically stable. A

Notice that in the preceding example it is not enough to study the system

tVe have to find a Lyapunov function that confirms asymptotic stability of the origin for all perturbations of the order 0 (lly11;). The importance of this observation is illustrated by the next example.

Example 8.4 Consider the previous example, but change A1 to

With d(y) = 0, the reduced system can be represented as

Without the perturbation term 0 (lly114), the origin of this system is asymptotically stable.' If you try to find a L ~ R ~ U ~ O V function V(y) to show asymptotic stability in the presence of the perturbation term, you will not succeed. In fact, it can be verified that the center manifold equation (8.11) with the boundary condition (8.12) has the exact solution h(y),= y;, so that the reduced system is given by the equation -

whose origin is unstable.' A

8.2 Region of Attraction , D-k sf o * ~ a c ~ ~ @*f~hh Quite often, it is not sufficient to determine that a given system has an asymp totically stable equilibrium point, Rather, it is important to find the region of att,raction of that point, or at least an estimate of it. To appreciate the importance of determining the region of attraction, let us run a scenario of events that could

?See Exercise 4.56. BSee Exercise 4.13.

......................... . . . '".':'---- -" "'

. , . . sf,

" . : . .... - I . - , . . . . . . ....- . 4 , . . ^ , . .

. . 8.2. REGION OF ATTRACTION 313

Estimate anra@ of the region toe tc

Region of attraction

Figure 8.1: Critical clearance time.

1 happen in the operation of a nonlinear system. Suppose that a nonlinear system has an asymptotically stable equilibrium point, which is denoted by x,, in Suppose the system is operating at steady state a t xpr. Then, at time to a fault' that ch*iges the structure of the system takes place, for example, a short circuit in an electrical network. Syppose the faulted system does not have equilibrium points a t xpr or in its neighborhood. The trajectory of the system will be driven away from xpr. Suppose further that the fault is cleared at time tl and the postfault system has an asymptotically stable equilibrium point at xps, where either xpa = xp, or xps is sufficiently close to xpr so that steady-state operation at x,, is still acceptable. At time tl the state of the system, say, x(tl), could be far from the postfault equilibrium xp8. Whether or not the system will return to steady-state operation at x,, depends on whether x(tl) belongs to the region of attraction of xps, as determined by the postfault system equation. A crucial factor in determining how far x(tl) could be from xp8 is the time it takes the operators of the system to remove the fault, that is, the time difference (tl - to). If ( t l - to) is very short, then, by continuity of the solution with respect to t , it is very likely that x(tl) will be in the region of attraction of xp,. However, operators need time to detect the fault and fix it. How much time they have is a critical question. In planning such a system, it is valuable to give operators a "critical clearance time," say t,, such that they have to clear the fault within this time; that is, (tl - to) must be less than t,. If we know

+ the region of attraction of xps, we can find t, by Figure 8.1. integrating the faulted system equation starting from the prefault equilibrium xpr until the trajectory hits the boundary of the region of attraction. The time it takes the trajectory to reach the boundary can be taken as the critical clearance time because if the fault is cleared before that time the state x(tl) will be within the region of attraction. Of course, we are assuming that xpr belongs to the region of attraction of xp,, which is reasonable. If the actual region of attraction is not known, and an estimate t,, oft, is obtained by using an estimate of the region of attraction, then t , < t,, since the

. - * 5 &$#< * *"%- 314 CHAPTER 8. ADVANCED STABILITY ANALYSIS

' boundary of the estimate of the region of attraction will be inside the actual bound- : ary of the region. (See Figure 8.1.) Thii scenario shows an example where finding

the region of attraction is needed in planning the operation of a nonlinear system. It also shows the importance of finding estimates of the region of attraction that are not too conservative. A very conservative estimate of the region of attraction would result in t,, that is too small to be useful. Let us conclude this motivating discussion by saying that the scenario of events described here is not hypothetical. It is the essence of the transient stability problem in power system^.^

Let the origin z = 0 be an asymptotically stable equilibrium point for the nonlinear system . ,

x = f (z) (8.13) : where f : D +-Ryis locally Lipschit2 and D c Rn is a domain containing the

origin.) Let $(t;z) be the solution of (8.13) that starts at initial state z at time t = 0. he region of attraction of the origin, denoted by RA, is defined by,

;- s;;j c

R;={ZE D 1 q5(t;z) isdefined'dt 2 0 a n d q5(t ;z)+0ast+oo)

?x .%a - Some properties of the region of attraction are stated in the next lemma, whose % proof is given in Appendix C.16.

"- - - . k

~e lhma 8.1 If z = 0 an asynptotically stable equilibrium point for (8.13), then itg region of attmction RA i s an open, connected, invariant set. Moreover, the boundary of RA b foned by tmjectories. 0

Lemma 8.1 suggests that one way to determine the region of attraction is to # A ' ,

e characterize those trajectories that lie on the boundary of RA. There are some methods that approach the problem from this viewpoint, but they use geometric

I . notions from the theory of dynamical systems that are not introduced in this book. Therefore, we will not describe this class of methods.1° We may, however, get a

$be ?+-Yt . flavor of these geometric methods in the case of second-order systems (n = 2) by &'>> .' <2

employing phase portraits. Examples 8.5 and 8.6 show typical cases in the state s* ('mt ,

plane. In the first example, the boundary of the region of attraction is a limit cycle, P'c ' ew r 4

while in the second one the boundary is formed of stable trajectories of saddle points. Example 8.7 shows a rather pathological case where the boundary is a closed curve of equilibrium points.

Example 8.5 The second-order system

is a Van der Pol equation in reverse time, that is, with t replaced by -t. The system has one equilibrium point at the origin and one unstable limit cycle, as determined from the phase portrait shown in Figure 8.2. The phase portrait shows that the

Osee (1701 for an Introduction to the transient stability'problem in power systems. l0Exnrnplcs of these ~ncthocis cnn bo founcl in (361 ant1 (216).

8.2. REGION OF ATTRACTION 315

Figure 8.2: Phase portrait for Example 8.5.

origin is a stable focus; hence, it is asymptotically stable. This can be confirmed by linearization, since

has eigenvalues at -112 f j f i /2. Clearly, the region of attraction is bounded because trajectories starting outside the limit cycle cannot cross it to reach the origin. Because there are no other equilibrium points, the boundary of RA must be the limit cycle. Inspection of the phase portrait shows that indeed all trajectories starting inside the limit cycle spiral toward the origin.

A

Example 8.6 Consider the second-ordcr system

This system has three isolated equilibrium points at (0,0), (fi, o), and (-8.0). The phase portrait of the system is shown in F i g m 8.3. The phase portrait shows that the origin is a stable focus, and the other two equilibria are saddle points. Thus, the origin is asymptotically stable and the other equilibria are unstable; a fact that can be confirmed by linearization. From the phase portrait, we can also

. see, that the stable trajectories of the saddle points form two separatrices that are . . the boundaries of the region of attraction. The region is unbounded.

A

'? -. ................ - . . . . . . . . . y- . . . . . . . . . . . . . . . . . . . . ... , .-.. .- - / . . . . . . . . .

> 7. :$l(j CIIAP'l'ER 8. ADVANCED STABILITY ANALYSIS

Figure 8.3: Phase portrait for Example 8.6.

I-, ,

Example 8.7 The system

has an isolated equilibrium point at. the origin and a continuum of equilibrium points on the unit circle; that is, every point on the unit circle is an equilibrium point. Clearly, RA must be confined to the interior of the unit circle. The trajectories of the system are the radii of the unit circle. This can be seen by transforming the system into polar coordinates. The change of variables

yields p = -p(l - p2), 0 = 0

L All trajectories starting with p < 1 approach the origin as t -+ co. Therefore, RA

c, is the interior of the unit circle. A fi

C Lyapunov's method can be used to find the region of attraction RA or an es-

timate of it. The basic tool for finding the boundary of RA is Zubov's theorem, I which is given in Exercise 8.10. The theorem, however, has the character of an

6 existence theorem and requires the solution of a partial differential equation. Via much simpler procedures, we can find estimates of RA by using Lyapunov's method.

I By an estimate of RA, we mean a set R C RA such that every trajectory starting

a:.. in R approaches the origin as t + co. For the rest of this section, we will discuss some aspects of estimating RA. Let 11s start by showing that the domain D of The-

( I f l y 0 c ~ & orem 4.1 (or Corollary 4.1) is not an estimate of R A We have seen in Theorem 4.1 B \ and Corollary 4.1 that if D is a domain that contains the origin, and if we can

I 8.2. REGION OF ATrRAC'170N 317

find a Lyapunov function V(x) that is positive definite in D and G(x) is negative definite in D or negative eemidefinite, but no solution can stay-identically in the t

set {V(x) = 0) except for the zero solution x = 0, then the origin is asymptotically stable. One may jump to the conclusionthat- of R A . This c o n j e c t u r e j s n o t as illustrated by the next example.

Example 8.8 Consider again the system of Example 8.6:

x1 = x2

x2 = -21 + i x f - 22 2 t

This system is a special case of that of Example 4.5 with !

h(xl) = xl - 1x3 and a = l

Therefore, a Lyapunov function is given by

= ix; - ~~4 12 1 + iX 2 12+Bx$

and ~ ( z ) = - 1x2(1- 1x2) - ix;

2 1 3 1

Defining a domain D by

it can be easily seen that V(x) > 0 and V(x) < 0 in D - (0). Inspecting the phase portrait in Figure 8.3 shows that D is not a subset of RA. A

In view of this example, it is not difficult to see why D of Theorem 4.1 or Corollary 4.1 is not an estimate of RA. Even though a trajectory starting in D will move from one Lyapunov surface V(x) = cl to an inner Lyapunov surface V(x) = q, with c2 < c1, there is no guarantee that the trajectory will remain forever in D. Once the trajectory leaves D, there is no guarantee that ~ ( x ) will be negative. Consequently, the whole argument about V(x) decreasing to zero f& a s This problem does not arise when RA is estimated by a compact positively invariant subset of D; that is, a compact set R c Dxsuch that every trajectory starting in R stays in R for all future time. Theorem 4.4 shows that R is a subset of RA. The simplest such estimate is the setu

R c = { x E Rn I V(X) 1 C) "The set {V(z) 5 c} may have more than one component, but there can be only one bounded

component in D, and that is the component we work with. For exam~le. if VIrl = xa/(l + . -, -- ,-, z4) and D = {CI < I ) , the set {V(x) S 1/41 hm two mmponentu {hI 5 \/z--J i d

{[*I 2 m}. We work with {lzl 5 -1.


..~~... :.. .-. 7..i_.--..l . . when,^^ is b ~ d e d and containtid iid. For a quadratic Lyapunov function V ( x ) = xTp,and D {l /xl/z < r } , we can ensure that c D by choosing . .. . .

c < min x T p x = X ~ ~ , , ( P ) ~ ' CL-r

For D = {(bTx( < T ) , where b E Rn,12

r min xTPx = (bTzI=i

Therefore, {xTPx < c) will be a subset of D = {IbTxl < ri, i = 1, ...PI. if we choose

ri" c < min - I < ~ _ < P bTP-1 bi

The simplicity of estimating the region of attraction by 0, = { x T p x _< c) has increased significance .in.rie.y;of.the linearization results of Section 4.3. There, we saw that if the J&obihmatrix i

A = 31 a x .=o

is Hurwitz, then we can always find a quadratic Lyapunov function V ( x ) = xTPx by solving the Lyapunov equation p A + A T P = -Q for any positive definite matrix Q. Putting the pieces together, we see that wheneverA is Hunvitz, we can atimate the region of attmctta of the ora'gin. This is illustrated by the next example.

Example 8.9 The second-order system (4 k1 = - 2 2

k2 = 21 + (2: - 1)x2 was treated in Example 8.5. There, we saw that the origin is asymptotically stable

8 f ~=bl,~=[! I:]

' is Hurwitz. A Lyapunov function for the system can be found by tsking Q = I and solving the Lyapunov equation

PA+ A ~ P = -I

12~ollowing 1122, Section 10.31, the Lagrangian arrsociated with the conatrained optimization problem is L(z, A) xTPx + X[(bTz)2 - r2). The first-order necessary conditions are 2Px + 2X(bTz)b = 0 and (bTr)l - ? = 0. It cnn he vcrlficd tl~nt the solr~t io~~s X = -1/(bTP-'b) and z = *r~-'b/(b~'P-'b) yicld the rt~lnlmd mluc ~ ~ / ( b ' ~ P - l b ) .

.. , . '-'

,, , *; . .,.-,-. - . ' " . ..' .+- " m- - - . , . .

I

. . . 2 .

> -. - - . . - . . - .


for P. The unique solutiol~ is the positive definite matrix

The quadratic function V ( x ) = xTPx is a Lyapunov function for the system in a certain nrighhorhood of the origin. Because our interest here is in estimating the region of attraction, we need to dcter~nine a domain D about the origin where V ( x ) is negative definite and a constant c > 0 such that R, = { V ( x ) 5 c ) is a subset of D. \Ire are interested in the largest set R, that we can determine, that is, the largest value for the constant c. Notice that we do not have to worry about dieckiiig positive definiteiless of V ( 2 ) ill D hecatrse V ( x ) is positive definite for all x. hloreovcr, V ( x ) is radially unbounded: hence R, is bounded for any c > 0. The derivative of V ( x ) along the trajectories of the system is given by

The right-hand side of V ( X ) is writtcn as the sum of two terms. The first term, -IIxII$, is the contribution of the linear part Ax, while the second term is the contribution of the nonlinear tern1 g(x) = f ( x ) - Ax. Since

we know that there is an open hall D = {z E RZ I llxl12 < r ) such that V ( x ) is negative definite in D. Once we find such a hall, we can find 52, C D by choosing

Thus, to cnlar~e the estimate of the rrgion of tit,trrrction, we need to find t.he largest ball on which V(z) is negative definite. \Ve have

where we used J x l J 5 llxllz, lrlxzl 5 11x11;/2, and )xl - 2x21 5 fillxllz. Thus, V ( X ) is negative definite on a ball D of radius give11 by rZ = 2/& = 0.8944. In this second-order example, a less conservative estimate of fi, can be found by searching for the ball D in polar coordinates. Taking

we get

v = -p2 + p4 cos2 6 sin e(2 sin 8 - cos 8 ) < -p2 + p4 1 cos2 9 sin 9 ) .I2 sin 6' - cos 81 - <_ -p2 + p4 x 0.3849 x 2.2361

1 5 -p2 + 0.8~1~" 0, for pZ < - 0.861

. / , - . . -

:i20 C:IIA PTER R. ADVANCED STAB~L~TY ANALYSIS

Figure 8.4: (a) Contours of V(X) = 0 (dashed), V(x) = 0.8 (dash-dot), and V(x) = 2.25 (solid) for Example 8.9; (b) comparison of the region of attraction with its estimate.

Using this last equation, together with Xmi,(P) 2 0.69, we choose

The set R, with c = 0.8 is an estimate of the region of attraction. A less conservative (that is, larger) estimate can be obtained by plotting contours 'of V(x) = 0 and V(x) = c for increying values of c until we determine the largest c for which V(x) = c will be in {V(x) < 0). This is shown in Figure 8.4(a) where c is determined to be c = 2.25. Figure 8.4(b) compares this estimate with the region of attraction whose boundary is a limit cycle.

n

Estimating the region of attraction by R, = {V(x) 5 c) is simple, but usudly conservative. According to LaSa e's theorem (Theorem 4.4), we can work with any compact set R c D provided 'I we ca;n show that R is positively invariant. It typically requires investigating the vector field at the boundary of 0 to ensure that trajectories starting in fi cannot leave it. The next example illustrates this idea.


where h : R -, R is a locally Lipschitz function that satisfies

Considcr the quadratic function

. . . . . . . . . . . . . . . . . - - 1 , ' i . q . . . . . . . . . . .r 4 . , , . < :. _ . . _ . _ . . . . ..-


-5 0 5

Figure 8.5: Estimates of the region of attraction for Example 8.10.

as a Lyapunov function candidate.13 The derivative V ( X ) is given by

Therefore, V(X) is negative definite in the set

and we can conclude that the origin is asymptotically stable. ib estimate RA, let us start by an estimate of,the form fl, = (V(x) 5 c) . The largest c > 0 for which R, C G is given by i

1 c = min xTPx = - 121+221=1 bTp-lb = I

where bT = [l 11. Hence, R, with c = 1 is an estimate of RA. (See Figure 8.5.) *Ipthis,example, wegan obtain a better estimate of RA by not restricting ourselves to estimates of the form R,. A key point in the development is to observe that trajectories inside G cannot leave through certain segments of the boundary Ixl + 221 = 1. This can be seen by examining the vector field at the boundary or by the following analysis: Let

u = x 1 +x2

1 3 ~ h i s Lyapunov function candidate can be derived by using the variable gradient method or by applying the circle criterion and the Kal~nan-Yakubovich-Popov lemma.

22 CHAPTER 8. ADVANCED STABlLlTY ANALYSlS

such that the boundary of G is given by a = 1 and u = -1. The derivative of u2 along the trajectories of the system is given by

d ;7ja2 = 2u(.i1+ .i2) = 20x2 - 8a2 - 2ah(a) 5 20x2 - 8u2, V In( 5 1

On the boundary a = 1, f

d - a 2 < 2 x 2 - 8 5 0 , ~ 2 2 1 4 dt

This implies that when the trajectory is at any point on the segment of the boundary u = 1 for which xz 5 4, it cannot move outside the set G, because at such point u2 is nonincreasing. Similarly, on the boundary a = -1,

Hence, the trajectory cannot leave the set G through the segment of the boundary a = -1 for wbkh x2 2 -4. This information can'be used to form a closed, bounded, positively invariant set R that satisfies the conditions of Theorem 4.4. Using the two segments of the boundary of G just identified to define the boundtuy of R, we now need two other segments to close the set. These segments should have the property that trajectories cannot leave the set through them. We can take them as segments of a Lyapunov surface. Let cl be such that the Lyapunov surface V(x) = c1 intersects the boundary XI + x2 = 1 at 2 2 = 4, that is, a t the point (-3,4). (See Figure 8.5.) Let cz be such that the Lyapunov surface V(x) = c2 intersects the boundary XI +xz = -1 a t 3 2 = -4, that is, a t the point (3, -4). The required Lyapunov surface is defined by V(x) = min{cl, c2). The constants cl and cz are given by

C1 = v(x)/,1=-3,2,=4 = 10, c2 = V(X)I,~=~ = 10

Therefore, we take c = 10 and d e h e the set R by

= {x E R2 1 V(x) 5 10 and )xl + xzl < 1)

' This set is closed, bounded, and positively invariant. Moreover, V(X) is negative 4" definite in 52, since R C G. Thus, all the conditions of Theorem 4.4 are satisfied and

we can conclude that all trajectories starting in R approach the origin as t -+ m; % , that is, R C RA. A k

8.3 ' Invariance-like Theorems EnR E B4mLRT f- In the case of autonomous systems, LaSalle's invariance theorem (Theorem 4.4)

sl~ows thnt tho t.rnjcct,onr of t.lie system approaches the largest invariant set iii E,

. . , . . , . * . . ,, ...., ) I : . . ' - - . .-l ;...-- ... . I . . . . . . -_.. -- .1 .--.-A. . ..-2-..---.- . .

where E is the set of all points in 0 where V(X) = 0. In the case of lionautoi~ol~iouv systems, it. may not even be clcar how to define a set E , since V ( t , x ) is a fu~lctiol~ of both t and x. The situatio~l will bc siinplcr if it can be shown t h d

for, then, a set E may be dcfincd as tlir set of points where W ( x ) = 0. We may expect that the trajectory of the systeiri appyoaclies E as t tends to w . This is, basically, the statement of the next theoreln. Before we state the theorem, we state a lemma that will be used in the proof of the t.heorem. The lemma is interesting in its own sake and is known as Barhalat 's 1em.m.a.

Lemma 8.2 Let d : R -+ R be a unijomly continuow function on [0, w ) . Suppose that limt,, Si 4(7) d~ exists and is finite. Then,

Proof: If it is not true. then there is a positive constant kl surh that for every T > 0. we can find Tl > T with 14(&)J 2 kl. Since $( t ) is uniformly continuous, there is i positive constant k2 such tllrit I@(t + T) - qh(t)J < k1/2 for all t 2 0 and all 0 5 T 5 k2. Hence,

Therefore?

d(t1 d t I = Jl'lrk2 id(t)\ dt > fr,b TI

where the equality holds, since 4(t) rctains t,he same sign for TI 5 t 5 TI + kz. Thus, fi I#J(T) d7 cannot convergc 10 it fil~itc limit as 1 -+ ca7 a contradiction. Q

Theorem 8.4 Let D c Rn he a dornain containing x = 0 and suppose f (t, x) is piecewise continuous in t and locally Lipscllitz in x, uniformly irl t , on [0, m ) x D. Furthermore, suppose f(t, 0) is 1~r~ifu17r~ly bounded for all t 2 0. Let S : (0, oo) x r ) -, R be a continuously differantiable function such that

11'1(x) 5 V(t,x) 5 I.Y?(x)

. - - ' / . - . ..

: r2 4 ('11~\1W"1 H. ADVANCED S'1'AOILj'j'Y ANALI'SIS

V t 2 0, V x E D , where lV1(x) nnd 1lJ2(x) are contin.uol~s positive definite functions and IY(x) is a continuous positive semidefinite function on D. Choose r > 0 such that Br c D and let p < minll,ll=r 1.K (x ) . Then, all solutions of x = f ( t , X ) vlth x(to) E { x E Br ( W2(x) 5 p ) are bounded and satisfy

Moreover, if all the assumptions hold globally and Wl ( x ) is radially unbounded, the statement is tme for all x(to) E R n .

0

Proof: Similar to the proof of Theoreln 4.8, it can be shown that

since ~ ( t , x ) 5 0. Hence, (Ix(t)(( < T for all t 2 to. Because V ( t , x ( t ) ) is m o m tonically o on increasing and boilndrtl from below by zcro, it converges as t -+ M .

Now,

Therefore, limt,, J:~ W ( X ( T ) ) dr exists and is finite. Since x( t ) is bounded, i ( t ) = f ( t , x ( t ) ) is bounded, uniformly in t. for all t 2 to. Hence, x(t) is uniformly continuous in t on [to, co). Consequently, 1V(x(t)) is uniformly continuous in t on [to, M )

because 1V(x) is uniformly continuous in x on the compact set,Br. Therefore, by Lemma 8.2, we conclude that lV(x( t ) ) -, 0 as t -, M. If all the assumptio~ls 110ld globally and W l ( x ) is radially uribounded, then for any x(to), we can choose p so large that x(t0) E { x E Rn ( I Y ~ ( x ) <_ p } .

The limit W ( x ( t ) ) -+ 0 iml)lics that x(t) approaches E as t -+ co, whcrc

Therefore, the positive limit set of z ( t ) is a subset of E. The mere knowledge that x(t) approaches E is much weakcr tl1a11 thc invariance principle for autonomous syst,ems, which states that x ( t ) approaches the largest invariant set in E. The stronger conclusion in the case of autonomous systems is a consequence of the property of autonomous systems stated in Lemma 4.1, namely the positive limit set is ,an invariant set. There are somc special classes of nonautonomous systems where positive limit sets have some sort of an invaria~ice property. I.' However, for a general nonautonomous system, the positive limit sets are not invariant. The fact t,liat, in the case of autonomous systems. x ( t ) approrrches the largest invariant S C ~

l.li.'xir~~l~~ias arc pcriotlic systcms, irlrrlcrsL-~rc!rioclic syslcrs, and asymptotically a~~tonomolls fiy* lclos. Scc (154, Chapter Bj for invariance Lhcorcn>s for tiles@ classes of systems. See, a\=, (1361 for a tlilicrent.gcneralization bf the invariance principle.

in E allowed us to arrive at Corolla~y 4.1, where asyniptotic stability of the origill is established by showing that the set E does not contain an entire trajectory of the system, other than the trivial solution. For a general nonautono~nous system, there is no extension of Corollary 4.1 that would show uniform asymptotic stability. However, the next theorem shows'that it is possible to conclude uniform asymptotic stability if, in addition to V ( t , x ) I 0, we can show that V decreases over the interval [t,t + 6].15 t

Theorem 8.5 Let D C Rn be a domain containing x = 0 and suppose f ( t , x ) b piecewise continuous in t and locally Lipschitt in x for all t 2 0 and x E D. Let x = 0 be an equilibrium point for x = f ( t ,x ) at t = 0. Let V : 10, oo) x D -+ R be a I

: continuously differentiable function such that

bVl(.) I. V ( t , x ) 5 LV2(x)

V ( t + 6, q5(t + 6; t , 2 ) ) - V ( t , x ) 5 -XV(t, x ) , 0 < X < 1'' V t 2 0, V x E D , for some d > 0, where W l ( x ) and IV2(x) are continuous positive definite functions on D and d ( ~ ; t , x ) is the solution of the system that starts at ( t ,x ) . Then,. the origin is uniformly asymptotically stable. 0 all the assumptions hold globally and W l ( x ) is radially unbounded, then the origin is globally uniformly asymptotically stable. If

then the origin is exponentially stable. 0 Proof: Choose r > 0 such that B, E D. Similar to the proof of Theorem 4.8, it can be shown that

where p < minll,ll=, ~ V I ( X ) , because ~ ( t , x ) 1 0. Now, for all t 2 to, we have i

V ( t + 6,x(t + 6) ) 5 V ( t , 4 t ) ) - XV(t,x(t)) = (1 - X)V(t ,x(t))

hloreover, since ~ ( t , x ) < 0,

V(T , x ( T ) ) < V ( t , x ( t ) ) , V f E [t, t + 61 151t L sllown in [I] that the condition v _< 0 can be dropped and uniform aqymptotic stability

can be shown if

V(t + 6, b(t + 6; t,z)) - ~'(L,z) 5 -r(Ilz(J) for some class K function Y I ......

t lGTherc is no loss of generality in assu~ning that X < 1, for if the inequality is satisfied with X I > 1, then i L is satisfied for any positive < 1, since -A1 V 5 -XV. Notice, however, that this inequality co~lld not be satisfied with A > 1, since V(t,z) > 0, V z # 0.

For any t 2 the interval


to, let N be the smallest positive integer such that t 5 to + N6. Divide [to, to+ ( N - 1)6] into ( N - 1) equal subintervals of length 6 each. Tllen,

V(t ,x( t ) ) ' < V(to + ( N - l ) b , ~ ( t o + ( N - 1)6))

5 (1 - X)V(to + ( N - 2)6,x(t0 + (N - 2)6))

5 (1 - X)(N- ' )~ ( to , x(t0)) 1 5 -(I-- ~ ) (~- '0 ) '~v ( to , x(to))

(1 - 4

where 1 1 b = -In- 6 (1 - A)

Taking r

u(r, 9 ) = m)e-b8 it can be easily seen that u(r, s) is a class K& function and V ( t , x(t)) satisfies

V ( t l ~ ( t ) ) 5 4V(to1 x(to))1 t - to), v V(to1 x(t0)) E [O,p]

&om this point on, the rest of the proof is identical to that of Theorem 4.9. The proof of the statements on global uniform asymptotic stability and exponential stability are the same as the proofs of Theorems 4.9 and 4.10.

Example 8.11 Consider the linear time-"arying system

where A(t) is continuous for all t > 0. Suppose there is a continuously differentiable, symmetric matrix P(t) that satisfies

0 < c1I I P(t) < czI, v t > 0 as well as the matrix differential equation

-P(t) = ~ ( t ) ~ ( t ) + ~ ~ ( t ) ~ ( t ) + c T ( t ) c ( t )

where C(t) is continuous in t. The derivative of the quadratic function

V ( t , x) = xTp(t )x

. . along the trajectories of the system is

3 P(t,x) = -x=cT(t)c(t)x 5 0

8.3. INVARlANCELlKE THEOREAIS 327

The solution of the linear system is given by 4(r; t , x ) = @(T, t )x , whcrc ~ I ) ( T , t ) is the state transition matrix. Therefore,

where

Suppose there is a positive constarit k < cz such that

then k

V( t +6,d(t + 6; t , x ) ) - V ( t : x ) 5 - k l ( ~ ( ( $ < - -V( t ,x) c2

Thus, all the assumptions of Theorem 8.5 are satisfied globally with

and we conclude that the origin is globally exponentially stable. Readers familiar with li~iear system theory will recogiliiic that the matrix IV(t, t + 6 ) is the observability Gramian of the pair (A( t ) , C( t ) ) and that the inequality LV(t,t + 6 ) 2 kI is implied by uniform observabi1it.y of (A( t ) , C( t ) ) . Comparing this example with Example 4.21 shows that Theorem 8.5 allows us to replace the positive definite- ness requirement on the matrix Q(t) of (4.28) by the weaker requirement Q(t) = CT(t)C(t) , where the pair (A( t ) , Cbt)) is uniformly observable. A

Theorems 8.4 and 8.5 and their application to linear systems, as in Example 8.11, are cstcrisively used in the analysis of nrlaptivc control syst,ems.17 As an example, we analyze the adaptive control system of Section 1.2.6.

Example 8.12 In Section 1.2.6, we saw that the closed-loop equation of a model reference adaptive control system, with plant 9p = apyp + kpu and reference model aim = amym + k,r, is given I)y

dl = -reor(t) $2 = -woko + ym(t)l

':See, for cxan~ple, 1871 and 11681.

. + . . . ., , , -. . . . . -- , ... .. . . - - . . .. . . . ., - . / . . - . - ..-,


where > 0 is the adaptation gain, e, = yp - ym is the output error, and $1 and 4 2

are the parameter errors. It was assumed that kp > 0 and, of course, the reference model must have a, < 0. Furthermore, we assume that ~ ( t ) is piecewise continuous and bounded. Using

as a Lyapunov function candidate, we obtain

By applying Theorem 8.4, we conclude that for any c > 0 and for. all initial states in the set {V 5 c}, all state variables are bounded for all t 2 to and limtdm eo(t) = 0. This shows that the output of the plant yp tracks the desired output y,, but says nothi~ig aboi~t the convergence of the parameter errors 41 and 42 to zero. In fact, they may not converge to zero. For example, if T and ym are nonzero constant signals, the closed-loop system will have an equilibrium subspace {e, = 0, d2 = (am/km)41)l which shows clearly that, in general, dl and $2 do not converge to zero. -Toderive conditions under which and will converge to zero, we apply Theorem 8.5. This will yield conditions under which the origin (c, = 0, dl = 0, d2 = 0) is uniformly asymptotically stable. Since we have already shown that all state variables are bounded, we can represent the closed-loop system as the linear time-varying system

am kPr(t) kPyp(t) I = [ -,.(ti 0 0 ] x., whe r~ x = [ 2 ]

-7yp(t) 0 0

Suppose the reference signal r( t ) has a steady-state value r,,(t); that is, limt,,[T(t)- 1-,,(t)] = 0. Then, lirnt,,[y,(t) - y,,(t)] = 0, where y,,(t) is the steady-state response of the reference model. These limits, together with limt,,, e,(t) = 0, show that the linear system can'be represented by

where

am kp~ss(t) hyhS(t) , and lim B(t) = 0

0 O I t-m

If we can show that the origin of x = A(t)x is uniformly asymptotically stable, we cikn IISC t,hc property limr,, B(t) = 0 to show that t,he origin of x = [A(t) + B(t)]x is uniformly asymptotically stable.18 Therefore, we concentrate our attention on

LWSoo Example 9.6.

, " -. " < .

%-:- -- . a . -.. , . " .li... , , . . .,. .

8.4. STABILITY OF PERIODIC SOLUTIONS 329

I tlie systmii i = rl(t).r. Olicc again, nsing \ ' LU 11 L ~ ~ I ~ I I I ~ O Y I111ictio11 ct~iitlid~ke, \\v obtain

P=a" e: = -xTCTCx, where C = ,/? [ 1 0 0 ] kP

From Example 8.12, we see that the origin will be uiliformly asymptotically stable if ~

the pair (A(t), C ) is uniformly observable. Since uniform observability of (A(t),C) is equivalent to uniform observability of (A(t) - I((t)C, C) for any piecewise continuous, bounded matrix K(t),lg we take

~ ( t ) = [ am - ~ a s ( t ) - ~ ~ a ( t ) 1 -am

1. to simplify the pair to

By investigating observability of this pair for a given reference signal, we can determine whether the conditions of Theore111 8.5 are sntisfied. For example, if r is a nonzero cdnstant signal, it can be easily SCCII t l ~ t ~ t the pair is not observable. This is not surprising, since we have already seen that in this case the origin is not uniformly asymptotically stable. On the other hnnd. if ~ ( t ) = asinwt with positive a and w, we have ~, , ( t ) = ~ ( t ) and ya,(t) = ad4sin(wt + 6), where M and 6 are determined by the transfer function of the reference model. It can be verified t.hat

, the pair is uniformly observable; hence the origin (e, = 0, $1 = 0, $2 = 0) is uniformly asymptotically stable and the parameter errors &(t) and &(t) converge to zero as t tends to infinity.20 A

8.4 Stability of Periodic Solutions

In Chapter 4, we developed an extensive theory for the stability of equilibrium points. In this section, we consider the corresponding problem for periodic solutions. If u(t) is a periodic solution of the system

what can we say about other solutions that start arbitrarily close to u(t)? Will they remain in some neighborhood of u(t) for all t? Will they eventually approach

lgSee (87, Lemma 4.8.11. 2oFor this example, the reference r(t) = asinwt is said Lo be persistently exciting, while a

constant reference is not persistently exciting. 'Ib read more about persistence of excitation, see 151, (151, (871, [139], [lea], or Section 13.4 of the second edition of this book.

CHAPTER 8. ADVANCED

? Such stability properties of the periodic solution u(t) investigated in the sense of Lyapunov. Let

y = s - u ( t )

at the origin y = 0 becomes an equilibrium point for

STABILITY ANALYSIS

can be characterized

the nonautonomous

e +$ P k ; 6 = f ( t , ~ + u(t)) - f (t,u(t)) (8.15) ,

:$4% The behavior of solutions of (8.14) near u(t) is equivalent to the behavior of solu- &%$.- tions of (8.15) near y = 0. Therefore, we can characterize stability properties of $& qd-2

u(t) from those of the equilibrium y = 0. In particular, we say that the periodic

$ j s solution u(t) is uniformly asymptotically stable if y = 0 is a uniformly asymptot-

p<fd> ically stable equilibrium point for the system (8.15). Similar statements can be made for other stability properties, like uniform stability. Thus, investigating the

&>e2 stability of u(t) has been reduced to studying the stability of an equilibrium point of 2 g a nonautonomous system, which we studied in Chapter 4. We shall find this notion z -LGG

' of uniform asymptotic stability of periodic solutions in the sense of Lyapunov to be %<: p9 - $ i .

useful when we study nonautonomous systems dependent on small parameters in Chapter 10. The notion, however, is too restrictive when we analyze periodic solu-

y"T 5 3 , tions of autonomous systems. The next example illustrates the restrictive nature of Z . $ z s *J+ this notion.

le 8.13 Consider the second-order system

1-2: -28) . L 1 = R [ ( Z: + X; 3 ] - z 2 1 + 1 - x ~ - 2 8 ) I ( 'I

1-2: -2i) * z = ~ ~ [ ( X: + X; 3]+21b+(1-x;-x;) 'I

represented in the polar coordinates

21=rcosB, x 2 = r s i n e

, )=1+(1-,.')' r

The solution starting at (ro, 00) L given by

8.4. STABILITY OF PERIODIC SOLUTIOXS 331

h o m these espressions, we see that the systcm has a periodic solution

Sl (t) = cost? ' S2(t) = sin t

The corresponding periodic orbit is the unit. circle r = 1. All nearby solutioils spiral t.oward this periodic orbit as t - cc. This spiralling is clearly the kind of "asymptotically stable" behavior we expect. to see with a periodic orbit. In fact, the periodic orbit has been known classically as a stable limit cycle. However, tlie periodic solution f ( t ) is not uniformly asymptotically stable in the sense of Lyapunov. Recall that for the solutioll to be ulliformly asymptotically stable, we must have

[r(t) cos e(t) - cost]' + [ ~ ( t ) sin B(t) - sin t]' -+ 0 as t --, SC)

for sufficiently small [ro cosBo - 11' + [ro sin^^]'. Because r ( t ) -, 1 as t -, m, we nlust have

11-cos(B(t)-t)l-,O as t - t c a

which clearly is not satisfied when ro # 1: since (B(t) - t) is an ever-gro\ving mono- tonically increasing function of t. A

The point illustrated by this example is true in general. In particular, a nontrivial periodic solution of an autonomous system can never be asymptotically stable in the sense of LYapunov.''

The stability-like propcrties of the periodic orbit of Example 8.13 can be cap- tured by extending the notion of stability ill the sense of Lyapunov from stability of an equilibrium point to stability of all illvuriallt set. Co~lsidcr the autonomous system

j. = f ( r )

where f : D -t R" is a continuously diffcrerltiable map from a doinain D C Rn into R". Let A l C D be a closed invariant set of (8.1G). Define an E-neigliborllood of M by

U, = {x E Rn I tlist(r, A1) < E )

where dist(2,ll.i) is the minimum distance from z to a point in ill; tliat is,

dist(x, AI) = i l l f 11. - 1/11 YEAI

1 Definition 8.1 The closed invariant set 111 of (8.16) is

1-r$ I "' stable if, for each E > 0, there is 6 > 0 such that. =

1 + 4t (1 r;) x ( 0 ) ~ U ~ ~ x ( t ) ~ U , , V t 2 0

6(t) = 6'0 + t + 4 ln [l + 4t ( l - r;)'] 2'See (72, Theorem 81.11 for a proof of this statement.

t _ I ~ . . . . . ......, ', - . . -.., . ,-. - _ . . . . - ,,--;-** -.. .\.- .*- .-- . . . ,,

. a ; . 3.. .. .. 1 ' .

. "-..- . . . ."

.J . . ...... ................................... ............ - . " q . ' . 2 ,

'7 . . . .* '(. . . . . . . . . - I . -.: ..-- a , ' / . - I . .., . .

L

.) ( ~ ~ J , \ P ' ~ E R 8. ADVANCED STABILITY ANALYSIS 8.4. STABILIT'I' 01.' PBRIOI)IC S O L L ~ ~ I O N S :rq 332 J

aslJmptotically stable if it is stable and 6 can be chosen such that

x(0) E Ua + t-m lim dist(x(t), M) = 0

This definition reduces to Definition 4.1 when M is an equilibrium point. Lyapunov stability theory for equilibrium points, as presented in Chapter 4, can be extended to invariant sets.22 For example, by repeating the proof of Theorem 4.1, it is not hard to see that if there is a function V(x), which is zero on A4 and positive in some neighborhood D of MI excluding M itself, and if !he derivative V(X) = [OV/ax] f ( x ) 5 0 in D, then M is stable. Furthermore, if V(x) is negative in D, excluding hi, then M is asymptotically st,able.

Stability and asymptotic stability of invariant sets are interesting concepts in their own sake. We will apply them here to the specific case when the invariant set A.I is the closed orbit associated with a periodic solution. Let u(t) be a nontrivial periodic solution of the autonomous syst.em (8.16) with pcriod T I and let y be the closed orbit defined by

The periodic orbit y is the image of u(t) in the state space. It is an invariant set whose stability properties are characterized by Definition 8.1. It is common, especially for second-order systems, to refer to asymptotically stable periodic orbits as stable limit cycles

Example 8.14 The harmonic oscillator

has a continuum of periodic orbits, which are concentric circles with a center at the origin. Any one of these perio ic orbits is stable. Consider, for example, the periodic orbit yc defined by 4 \

y c = { x ~ R 1 I r = c > O } , wherer= Js ,

The U, neighborhood of y, is,defined by the annular region

This annular region itself is an invariant set. Thus, given E > 0, we can take 6 = E

and see that any solution starting in the U6 neighborhood at t = 0 will remain in the U, neighborhood for all t 2 0.' Hence, the periodic orbit yc is stable. However, it is not asymptotically stable, because a solution starting in a Ua neighborhood

%ee, for example, I2131 and [221] for comprehensive coverage and (1181 for some interesting results on converse Lyapunov theorems.

of docs not approach 7, as t -, m, 110 lrlatter IIOIV sn~all d is. Stability of the periodic orbit { r = c) can be also shown by the Lyapunov function

V(x) = (r2 - c ~ ) ~ = (x: + x i - c ~ ) ~

whose derivative along the trajectories of the system is

A Example 8.15 Consider the system of Example 8.13. It has an isolated periodic orbit

Y = { x E R ~ l r = l ) , wherer= X:+X

For x $ y, we have m

dist(x, 7) = YET inf Ilx - vll2 = YET illf J(xl - y1)2 + (x2 - 2/2)2 = I r - 11 Recalling that

- it can be easily seen that the E-6 requirement for stability is satisfied and

Hence, the periodic orbit is asymptotically stable. The same conclusion can be arrived at using the Lyapunov function

whose derivative along the trajectories of the system is

~ ( x ) = 4(r2 - 1 ) r i = -4(r2 - I)' < 0, for r # 1

Having defined the stability properties of periodic orbits, we can now define the stability properties of periodic solutions.

Definition 8.2 A nontrivial periodic solution u(t) of (8.16) is

orbitally stable if the closed orbit y generated by u(t) is stable.

asymptotically orbitally stable if the closed orbit y generated by u(t) ia asymptotically stable.

Notice that different terminology is used depending on whether we are talking about t.he periodic solution or the corresponding periodic orbit. In Example 8.14 we say that the unit circle is an asymptotically stable periodic orbit, but we my that the periodic solution (cos t, sin t) is orbitally asymptotically stable.

CHAPTER 8. ADVANCED STABILITY ANALYSIS 8.5. EXERCISES 335

Investigate stability of the origin by rlsilig the center manifold theorem for each of the following cases:

( 1 ) a+c>O. ( 2 ) a + c < 0. ( 3 ) a + c = 0 and cd + bc2 < 0. ( 4 ) a + c = 0 and cd + bc2 > 0. ( 5 ) a + c = cd + bc2 = 0.

.1 are satisfied in a case where gl(y, 0 ) = 0, g2(1,0) = 0, and A1 = 0. Show that the origin of the full system is stable. 8.8 ([34]) Consider the system

8.1 with a = 0. Show that the origin is stable. = ax? + r:x2, .t2 = -.r2 + X: +,xlx2 - I ;

1nvestigat.e stability of t,he origin by using the center manifold theorem for all possible values of the real parameter a.

50 =' fa(&, xb) kb = Abxb + f b ( ~ a , ~ b ) 8.9 ( [88]) Consider the system

where dim(x,) = n l , dim(xb) = nz, Ab is Hurwitz, fa and fb are cont,inuously k l = ax1x2 - x;, 52 = -12 + bxlx2 + cx: dserentiable, (8 f b / & b ] ( ~ , 0) = 0, and fb(x,, 0 ) = 0 in a neighborhood of x, = 0.

Investigate stability of the origili by using the center manifold theorem for all p a - (a) Show that if x, = 0 is an exponentialljr stable equilibrium point of 5, = sible values of the real coilstants a, b, and c.

fa(Xa9 O ) , then (x,, xb) = (0,O) is an exponentially stable equilibrium point of the full system. 8.10 (Zubov's Theorem) Consider the system (8.13) and let G C Rn be a do-

totically (but not exponentially) stable equilib main containi'ng the origin. Suppose there exist two functions V : G -, R and h : Rn 4 R with the following properties:

,XI,) = (0,O) is an asymptotically stable t V is continuously differentiable and positive definite in G and satisfies

ms, investigate stability of the origin by 0 < V ( x ) < 1 , V x E G - {O)

k l = ax2 1 - x& a # o t As x approaches the boundary of G. 01. in case of unbounded G 1 1 ~ 1 1 -+ 00,

( 1 i2 = -xz+x:+ 21x2 lim V ( X ) = I .

h is continuous and positive definite on Rn. 51 = -12 + X l X 3

x , = x;x2 (3 ) 52 = 2 1 + 12x3 (4) C = -x! - 1 2

t For x E G, V ( x ) satisfies the partial differential equation x3 = -53 - (x: + 2'2) + x i av

- f ( x ) = -h(z)[ l - V ( X ) ] (8.17) (6) dl = -xi + xf(x1 + x2 - 1) ax

5, = xg(xl+ 1 2 - 1 ) Show that x = 0 is asymptotically stable and G is the region of attract.ion.

- ..

, . 51 = xz xl = -2x1-3a2+x3+xz *: I, .

<,.. . 8.11 ( [ 72 ] ) Consider the second-order system .; ? (7) x~ = -12 + a x f / ( l + x;) ( 8 ) x~ = X I + x: + 1 2

, ::F*: a#O x3 = X: X I = - h 1 ( ~ 1 ) + g 2 ( ~ 2 ) ? .t2 = -gi(xl)

8.7 ( [34]) Consider the system where

xl = xlxz +ax: + bxlxi, x2 = -12 + CX? + dxtx2 hl(0) = 0 , rh l ( r ) > 0 V -a1 < z <bl

B . 'U... - .- - . . . ,

," . . .. .. . . . -. ,. ,..,., ,,.- .,-... * - . .,. - - - . , . . 1 . . * : . - - -. . , . - - . . . .- . - . . \ -.--I-p-. "..:I-"-. --

J O

for some positive constants ai, bi (ai ,= w or bi = w is allowed). Apply Zubov's theorem to show that the region of attraction is {X E R2 1 - ai < xi < bi). Hint: Take h(x) = gl(xl)hl(xl) and seek a solution of the partial differential equation (8.17) in the form V(x) = 1 - IVl(xl)W2(x2). Note that, with this choice of h, V(x) is only negative semidefinite; apply LaSallels invariance principle.

8.12 Find the region of attraction of the system

Hint: Use the previous exercise.

8 . 1:) 1 , 1 8 1 . $ 2 \I(! 1111 o11(!11, l)osil.i\-c!ly i~~viuirrl~b sot cont.nining the origin. Suppose c!vtbry t.ri{ic!ct.ory ill R approi~clics t.11~ origin as t -+ m. Show that R is connected.

8.14 Consider a second-order systeln x = f (x) with asymptotically stable origin. Let V(x) = x: + x;, and D = {x E R2 I lxZl < 1, lxl - 221 < 1). SIII)I)OW' [8V/a.-c] f (x) is negative definite in D. Estimate the region of attraction.


(a) Using V(x) = 51: + 2 x 1 ~ ~ + 2x22. show that the origin is asymptotically stable.

(b) Lct, S = {X E n2 1 V(X) I 5 ) n {X E R2 ( 1x21 5 1)

Show that S is an estimate of the region of attraction.

8.16 Show that the origin of

is asymptotically stable and estimate the region of attraction.

8.17 Consider a second-order system x = f (x), together wit.h a Lyapunov function V(x). Suppose that V (x) < 0 for all x: +xi 2 a2. The sketch, given in Figure 8.6, shows four different directions of the vector field at a point on the circle x f + x i = aZ. Which of these directions are possible and which are not? Justify your answer.


x l = x2, .i.? = - . ~ 2 - sin X I - 2 sat(xl + x2)

Figure 8.6: Exercise 8.17.

(a) Show that the origin is the unique equilibrium point.

(b) Show, using linearization, that the origin is asymptotically stable.

(c) L c ~ U = 21 + 2 2 . S ~ ~ O W tllilt U b < -1Ul (u( 2 1.

(d) Lct V(x) = x: + 0.52; + 1 - cosxl. Sliow tliat,

is positively invariant and trajectories in Ad, approach the origin as t 4 oo.

(e) Show t,hat the origin is globally asymptotically stable.

8.19 Consider the synchronous generator model described in Exercise 1.8. Take the state variables and parameters as in parts (a) and (b) of the exercise. Moreover, take T = 6.6 sec, A.I = 0.0147 (per unit power) x sec2/rad, and D/Ai = 4 sec-I.

(a) Find all equilibrium points in the region -n 5 x l < n , and determine the stability properties of each equilibrium by using linearization.

(b) Estimate the region of attraction of each asymptotically stable equilibrium.

8.20 ([113]) Consider the system

x l = 52, x2 = -51 - g(t)z2

where g(t) is continuously differentiable and 0 < kl 5 g(t) 5 k2 for all t 2 0.

(a) Show that the origin is exponentially stable.

(b) LVould (a) be hue if g(t) were not bounded? Consider g(t) = 2 + exp(t).


' ;; >8.21 Consider the system

x l = x2, x2 = - sinxi - g(t)x2

where g(t) is continuously differentiable and 0 < kl 5 g(t) 5 k2 for all t 2 0. Show 9 ' that the origin is exponentially stable.

Hint: Use the previous exercise.


*s

$5 , k.,' where a( t ) = sin t + sin 2t. Show that the origin is exponentially stable.

&& 8.23 Consider the single-input-singleoutput nonlinear system

' where fo, fl, and go are known smooth functions of x, defined for all x E Rn, while 8' E RP is a vector of unknown constant parameters.. The function go(x) is bounded away from zero; that is, Jgo(x)l 2 ko > 0, for all x E Rn. We assume that all state variables can be measured. It is desired to design a state feedback adaptive controller such that x l asymptotically tracks a desired reference signal ~ ( t ) , where T and its derivatives up to T(") are continuous and bounded for all t 2 0.

(a) '@king ei = x1-r('-l) and e = [el, . . . , enIT, show that e satisfies the equation

d = Ae + B [ fo(x) + (19')~ fl (x) + go(x)u - T(")]

By? , where (A, B) is a controllable pair. p

: (b) Design K such that A - B K is Hurwitz and let P be the positive definite solution of the Lyapunov equation P(A - B K ) + (A - B K ) ~ P = - I . Using the Lyapunov function candidate V = eTpe + &TI'-1&, where & = 9 - O* and

'r I' is a symmetric positive definite matrix, show that the adaptive controller

@ H $9

ensures that all state variables are bounded and limt-+ao e(t) = 0. 1: .

(4 Let

"t) = [ -Bf'(R) ] , c = [ I n Onrp 1 OPXP

where R = [T, . . . , T ( ~ - ' ) ] ~ . Show that if (A(~),c) is uniformly observable,

@d

then the pnrruneter error & converges to zero as t -+ 00. rCo*r a:

Chapter 9 : ~ b ( h M \ f IC&:~N ( M ~ I

Stability of Perturbed Systems

Consider the system x = f( t ,x) +g(t.x) (9.1)

where f : [0, x) x D -+ Rn and g : [0, m) x D -+ Rn are piecewise continuous in t aiid locally Lipschitz in x on [0, oo) x D, aiid D c Rn is a domain that contains the origin x = 0. \Ire think of this system as a perturbation of the nominal system

Tlic perturbatioi~ term g(t, x) could result from inodcling errors, aging, or uncertainties and disturbances, which exist in any realistic problem. In a typical situation, nrc do not know g(f.x). hut we know some information about it, like knowing an upper bound on Ilg(t.x)I/. Here, we represent the perturbation as an additive term on the right-hand side of the state equation. Uncertainties that do not change the system's order can always be represented in this form. For if the perturbed right- hand side is some function f (t, x), then by adding and subtracting f (t, x), we can rewrite the right-liaiid side as

and define g(t:x) = J(t, 5) - f (t, X)

Suppose the nominal system (9.2) has a'uniformly asymptotically stable equilibrium poilit a t the origin, what can we say about the stability behavior of the pertutbed system (9.1)? A natural approach to address this question is to use a Lyapunov function for the nominal system as a Lyapunov function candidate for the perturbed system. This is what we have done in the analysis of the linearization appioach in Sections 4.3 and 4.6. The new element here is that the perturbation term could be more general than the perturbation term in the case of linearization. The conclusions we can arrive a t depend critically on ivliether the perturbation term vanishes

) I

'1 at the origin. If g(t, 0) = 0, tlie perturbed system (9.2) has an equilibrium point at the origin. In this case, we analyze the stability behavior of the origin as an

" i ! equilibrium point of the perturbed system. If g(t,O) # 0, the origin will not be 5 . c 1

f i an equilibrium point of the perturbed system. In this case, we study ultimate

j A 5 ] boundedness of the solutions of the perturbed system.

5' The cases of vanishing and nonvanishing perturbations are treated in Sections 9.1 and 9.2, respectively. In Section 9.3, we restrict our attention to the case when tlie

5 nominal system has an exponentially stable equilibrium point at the origin and use

3 the comparison lemma to derive some sharper results on the asymptotic behavior of the solution of the perturbed system. In Section 9.4, we give a result that establishes continuity of the solution of the state equation on the infinite-time interval.

The last two sections deal with interconnected systems and slowly varying sys- % , tems, respectively. In both cases, stability analysis is simplified by viewing the

to i i i~~st igate tlie stability of tlic origill IIS all equilibrium poillt for tlie perturbed system (9.1). The derivative of V along the trajectories of (9.1) is given by

The first two terms on the right-hand side constitute the derivative of V(t,x) along the trajectories of the nominal system, which is negative definite and satisfies (9.4). The third term, [bV/bx]g, is the effect of the perturbation. Since we do not have complete knowledge of g, we cannot judge whether this term helps or hurts the cause of making V(t,x) negative definite. With the growth bound (9.6) 8s our only information on g, the best we can do is worst case analysis where [bV/8x]g is bounded by a nonnegative term. Using (9.4) through (9.6), we obtain

system as a perturbation of a simpler system. In the case of interconnected systems, the analysis is simplified by decomposing the system into smaller isolated ( ) - + -c311xl12 + q.rllx~12

~ubsystems, while in the case of slowly varying systems, a nonautonornous system with slowly varying inputs is approximated by an autonomous system where the If 'Y is small enough to satisfy the bound slowly varying inputs are treated as constant parameters. --

c *uuz;J * 0 9.1 vakshing Perturbation ( c k d ~ o s ~ b r a\ &>qcwi H ,GI) then

~~t us with the case g(t, 0) = Oi Suppose X = 0 is an exponentially equi- w, x) I -(c3 - .r~4)11~II~, (c3 - 7 ~ 4 ) > 0 librium point of the nominal system (9.2), and let V(t,x) be a,Lyapunov Therefore, by Theorem 4.10, we conclude the next lemma. that satisfies

~111x11~ 5 v(tt1.31 < ~211x11~ (9.3) Lemma 9.1 Let x = 0 be an exponentially stable equilibrium point of the nominal

av + av system (9.2). Let V(t, X) be a Lyapunovfunction of the nominal system that eat&&

- f - 3 er6b;l&J (9.4) (9.3) thm~gh (9.5) in [O, m) x D. Suppose the perturbation te rn g(tlx) 8&&fi@ a t

(9.6) and (9.7). Then, the origin b an clponentially stable equilibrium point of BV the perturbed system (9.1). Moreover, if all the assumptions hold globally,. then the 11 11 5 ~ ~ 1 1 ~ 1 1 $ origin is globally exponentially stable. 0

for all (t ,x) E [O, m) x D for some positive constants cl, c2, c3, and cd. The existence of a Lyapunov function satisfying (9.3) through (9.5) is guaranteed by :

Theorem 4.14, under some. additional assumptions. Suppose the perturbation term g(t, x) satisfies the linear growth bound

where 7 is a nonnegative constant,. This bound is natural in view of thq assumptions - ,

on g(t,x). In fact, any function g(t,x) that vanishes at the origin and is locally Lipschitz in x, uniformly in t for all t 2 0, in a bounded neighborhood ofthe origin :

satisfies (9.6) over that neighborhood.' IVe use V as a Lyapunov function candidate

'Note, liowever, that the linear growl11 bo l~~ ld (9.6) beco~nes restrictive when required to hold globally, because that would require g to be globally Lipschitz in I.

This lemma is conceptually important because it shows that exponential stability of the origin is robust with respect to a class of perturbations that satisfy (9.6) and (9.7). TO assert this robustness property, we do not have to know V(t, x) explicitly. It is just enough to know that the origin is an exponentially stable equilibrium of the nominal system. Sometimes, we may be able to show that the origin is exponentially stable without actually finding a Lyapunov function that satisfies (9.3) through (9.5).' Irrespective of the method we use to show exponential stability of the origin, we can assert the existence of V(t,x) satisfying (9.3) through (9.5) by application of Theorem 4.14 (provided the Jacobian matrix [af/ax] is bounded). However, if we do not know the Lyapunov function V(t, x) we cannot calculate the

2Tl~is is the case, for example, when exponential stability of the origin is shown using T h e rem 8.5.

342 CHAPTER 9. STABILITY OF PERTURBED S17STEh4S

bound of (9.7). Consequently, our robustness conclusion becomes a qualitative one where we say that the origin is exponentially stable for all perturbatiolls satisfying

wit.h sufficiently small 7. On t.he other hand? if we know V(t, x)? we can calculate the bound of 19.71. which is an additional ~ i e c e of informntio~i. We ssl~oultl be , ,. caroful not t.o o\~erernphasize the usefulness of such bounds because they could bc conservative for a given perturbation g(t, x). The conservatism is a consequence of the worst case analysis we have adopted from the beginning.


where A is Hurwitz and 11g(t,~)11~ I r11x112 for a11 t 2 0 and a11 x E R". Let Q = QT > 0 and solve the Lyapunov equation

for P. From Theorem 4.6, we know that there is a unique solution P = PT > 0. The quadratic Lyapunov function V(x) = xTPx satisfies (9.3) through (9.5). In particular,

An~ilP+l/~/l; -(WY &nnx(P)llxII~

The derivative of V(x) along the trajectories of t.he perturbed system satisfies

~ ( x ) < -Xrn~n(Q)Ilxllf + 2~rnex(P)7l l~l l~

Hence, the origin is globally exponeritially stable if 7 < Xrnin(Q)/2Xrn,,(P). Since this bound depends on the choice of Q, one may u-onder how to choose Q to maxi- mize the ratio XmIn(Q)/Xrn,(P). 'It turns out bhat this ratio is maximized with the, choice Q = I (Exercise 9.1). A


where the constant /3 2 0 is unknown. We view the system as a perturbed system of the form (9.1) with

9.1. VANISHII\'G PERTURBATION 343

The eigenvalues of A are -1 f j L 6 . Hence, A is Hurwitz. The solution of the Lyapunov equatior~

PA + A'"P = -1

is given by

As we saw in Example 9.1? the Lyapurlov function V(x) = xTPx satisfies inequalities (9.3) through (9.5) with ca = 1 and

The perturbation tcrin g(x) satisfies

for all 1x2 1 < k2. At this point ill the analysis, we do not know a bound on x2(t), alt.hough we know that x2(t) will hc bounded whenever the trajectory x(t) is confined to a compact set. We keep kz undetermilled and proceed with the analysis. Using V(x) as a Lyapunov funct.ioi1 cantlidate for the perturbed syst.em, we obtain

Hence, ~ ( x ) will LC negative tlcfiliitc! if . . .

1 . . . \ . . . . . . . :.. . . . . . . . : . . . . . . . . . . . "-:* ! ' : ;. <;..::*~'. f l < - . .

3.02Gk;

To estimate the bound k2, let 52, = {x E R2 I V(x) 5 c). For any positive constant c: the set 52, is closed and bounded. The boundary of 52, is the Lyapunov surface

The largest value of Ix21 on t.he surface V(r ) = c can be determined by differentiating the surface equat.ioxl partially with respcct to xl. This results in

Therefore, the ext,reme values of 1 2 are obtained at the interscctioxl of tllc lilic xl = -22112 with the Lyapunov surface. Simple calculations show that the largest value of xif on the Lyapunov surface is 9Gc/29. Thus, all points inside 52, satisfy t.he bound

96c 1x2( < kZ, wllere kg = - 29

Therefore, if

i- t:: ,. 3

k"' i; CJ

b-

ir(r) will be ~legatise definite in R, and wc call coliclude that the origin x = 0 is exponentially stable with R, as an estimate of the region of attraction. The inequality /3 < O.l/c shows a tradeoff between the estimate of the region of attraction and the estimate of the upper bound on P. The smaller the upper bound on 0 , the larger the estimate of the region of attraction. This tradeoff is not artificial; it does exist in this example. The change of variables

transforms the state equation into

which was shown in Example 8.5 to have a bounded region of attraction surrounded by an instable limit cycle. When transformed into the x-coordinates, the region of attraction will expand with decreasing and shrink with increasing P. Finally, let us use this example to illustrate our remarks on the conservative nature of the bound of (9.7). Using this bound, we came up with the inequality0 < 1/3.026k;, which allows the perturbation term g(t, x) to be any second-order vector that satisfies 11g(t,~)11~ < Pk2211~11~. This class of perturbations is more general than the perturbation we have in this specific problem. We have a structured perturbation in the sense that the first component of g is always zero, while our analysis allowed for an unstructured perturbation where the vector g could change in all directions. Such disregard of the structure of the perturbation will, in general, lead to conservative bounds. Suppose we repeat t e analysis, this time taking into consideration 'a the structure of the perturbation. In tead of using the general bound of (9.7), we calculate the derivative of V(t, x ) along the trajectories of the perturbed system to obtain

I

V(X) = -11x11; + 2xTPg(x) = -11x11; + 2Px5 ( Q x ~ x ~ + Ax;) 5 -llxll; + 2Px; ( ~ l l x l l ; + &llxll;) 5 - 1 1 ~ 1 1 2 2 + ~ ~ ~ ; 1 1 ~ 1 1 2 2

Hence, V(X) is negative definit.e for B < 4/3k;. Using, again, the fact that for all x E R,, (x2I2 5 ki = 96~129, we arrive at the bound < 0.4/c, which is four times the bound we obtained by using (9.7). A

. . . .___l_ll._...^_. --- .- - ' ' 5 1

I f

, ,.. .,.. :L .!!A,, * , . _ - . . " . ... 9.1. VANISHING PERTURBATlON 345

When the origin of the iloluinal systeiu (9.2) is uiliforluly asymptotically stable, but not exponentially stable, the stability analysis of the perturbed system is more involved. Suppose the nominal system has a positive definite, decrescent Lyapunov I

function V(t, x) that satisfies

for all (t, x) E [0, oo) x D, where W3(x) is positive definite and continuous. The derivative of V along the trajectories of (9.1) is given by

Our task now is to show that

for all (t,x) E (0,oo) x D, a task that cannot be done by putting a simple order of magnitude bound on IJg(t,x)JI, as we have done in the exponential stability case. The growth bound on ((g(t, x)JJ will depend on the nature of the Lyapunov function of the nominal system. One class of Lyapunov functions for which the analysis is almost as simple as in exponential stability ip the case when V(t,x) is positive definite, decrescent, and satisfies

. 2 . - av sv . . - + -f (t , 2) < - ~ 3 4 ~ ( 2 ) a t . BX

for all (t, x) E (0, oo) x D for some positive constants c3 and c4, where 4 : Rn -r R is positive definite and continuous. A Lyapunov function satisfying (9.8) and (9.9) is usually called a quadratic-type Lyapunov function. It is clear that a Lyapunov function satisfying (9.3) through (9.5) is quadratic type, but a quadratic-type Lya- punov function may exist even when the origin is not exponentially stable. We will illustrate this point shortly by an example. If the nominal system (9.2) has a quadratic-type Lyapunov function V(t, x), then its derivative along the trajectories of (9.1) satisfies

i.(t, 2) I - ~ 3 4 ~ ( 2 ) + c44(x)llg(t, x)ll Suppose now that the perturbation term satisfies the bound

346 CHAPTER 9. STADILITY OF PERTURBED SlSTEAIS

Then, ~ ( t , x ) 5 -(c3 - c47)d2(x)

which slio\vs that ~ ( t , .r) is negative dcfiiiite.

Example 9.3 Consider the scalar system

The nominal system 1 ) = -z3

has a globally asymptotically stable equilibriuin point at tlie origin. but., as we s a w in Example 4.23, the origin is riot exponeiit,i~lly stable. Tlius, there is no Lyapuiiov function that satisfies (9.3) tlirough (9.5). The Lyapunov function V(x) = z4 satisfies (9.8) and (9.9), wit11 4 ( x ) = lrI3! c3 = 4, and cd = 4. Suppose tllc perturbation term g ( t , x ) satisfies the bound Jg( t , x)l 5 y1xI3 for all x, with 7 < 1. Then, the derivative of V along the trajectories of the perturbed system satisfies

Hence, the origin is a globally uniformly asymptotically stable equilibrium point of the perturbed system. A

In contrast to the case of exponential stability, it is important to iiotice tllot. a nominal system with uniformly asyinptotically stable, but not expolientially stable, origin is not robust to smooth perturbations with arbitrarily small lii~car gro\vtll bounds of tlie form of (9.G). This point is illustrated by t.he next e ~ a m p l e . ~

Example 9.4 Consider the scn l~r system of tlie previous example with perrurba- tion g = yx where 7 > 0; that is,

It can be easily seen, via linearization, that for any 7 > 0 the origin is unstable, no matter how small r is. A

19.2 Nonvanishing Perturbation

Let us turn now to the more general case wheu we do not know that g(t, 0 ) = 0. The origin x = 0 may not be an equilibrium point of the perturbed system (9.1). We can no longer study stability of the origin as an equilibrium point, nor should we expect the solution of the perturbed system to approacli the origin as t -i oc. The best we can hope for is that r(t) will be ulti~nat~ely bounded by a sillall bound. if the perturbation term g( t , x ) is small in some sense. We start with the case when the origin of the nominal system (9,2) is exponentially stable.

3 ~ w , also, Excrciar 9.7.

9.2. NONVANlSHlNG PERTURBATIOK 347

,-

Lemma 9.2 Let s = 0 be an exponen.tially stlblble equilibrium point of the nominal system (9.2). Let V ( t , x ) be a L?japl~n,olr fim.ction of the nonrinal system that sata.$es (9.3) through (9.5) in (0, oo) x D, where D:= { x E R" 1 ))I)) < r } . Suppose the perturbation t e r n g( t , x) satisfies /-----. .

for all t > 0 , all x E D. and some positive constant 0 < 1. Then, for all Ilx(to)ll < GT, the solution x ( t ) of the pertui.bed system (9.1) satisJcs

and

6 for some finite T , where

. - 1 0

I B Proof: X'e use V(t . t) as a Lyapunov function candidate for the perturbed system

1 (9.1). The derivative of V ( t , x ) along the trajectories of (9.1) satisfies

Applying Theorem 4.18 and Exercise 4.51 completes the proof.

N0t.e that the ultimate bound b in Lemma 9.2 is proportional to t.he upper bound on the perturbation 6. Once again, tliis result can be viewed as a robustllcss property of iiominal systems having espol~efially stable equilibria at the origin, because it shows that arbitrarily small (uniformly bounded) pert.url)i~tions will not result in large steady-state deviations from the origin.


where 0 2 0 is unkno~vn and d( t ) is a uniformly bounded disturbance that satisfies Id(t)l 5 6 for all t 2 0. This is the same system we studied in Example 9.2, except

.- . - - -- - - - - - . / . : - ..-.

348 CHAPTER 9. STABILITY OF PERTURBED SYSTEAIS

for the additional perturbatiol~ terin d(t). Again, the system can be viewed as a ~erturbation of a nominal linear system that has a Lyapunov function V(x) = X?PX, where

We use If(x) as a Lyapunov function candidate for the perturbed system, but we treat the two perturbation terms pxi and d(t) differently, since the first term vanishes at the origin while the second one does not. Calculating the derivative of V(x) along the trajectories of the perturbed system, we obtain

where we have used the inequality

12x1 + 5x21 5 11x112-

and k2 is an upper bound on 1x21. Suppose 0 I 4(1 - ~)/3k;, where 0 < C < 1. Then,

m 6 8 3 6 v t x I X I + I I I I I - 1 - I l l v lIxII2 2 P = -

where 0 < 0 < 1. As we saw ill Example 9.2, 1x2I2 is bounded on Cl, by 96~129. Thus, if 0 5 0.4(1- C)/c and d is so small that p2X,,(p) < c, then B, c Cl, and all trajectories starting inside Cl, remain for all future time in R,. Furthermore, the conditions of Theorem 4.18 are satisfied in Cl,. Therefore, the solutions of the perturbed system are uniformly ulti~nately bounded by

In the more general case when the origin x = 0 is a uniformly asymptotically stable equilibrium point of &the nominal system (9.2), rather than exponentially stable, the analysis of the perturbed system proceeds in a similar manner.

Lemma 9.3' Let x = 0 be a unifo~mly asymptotically stable equilibrium point of the nominal system (9.2). Let V(t,x) be a Lyapunov function of the nominal system that satisfies the inequalities4

al(llxll) I v( t !x) 5 az(llxll) (9.11)

"he existence of a Lyapunov function satisfying these inequalities (on a bounded domain) is guaranteed by Theorem 4.16 under somc additional assumptions.

r-. - .. . -. " . ' " $ .

. ., -- . . ., ; .-q . . , < , . . - ' .--. . , , ,.. ....., ; L..!... , . * + . . . _. , . . I . . . . *\._: :- ..-- :"&- -. "..," ..,< 11:. *a-

9.2. NONVANISIIING PERTURBATION 3119 I

1 in [O,oo) x D, where D = {x E Rn ( 11x11 < r) and ai(.), i = 1,2,3,4, are class K functions. Suppose the perturbation term g(t, x) satisfies the uniform bound i

for all t 2 0 , all x E D, and some positive constant 0 < 1. Then, for all 11x(to)ll < a;'(al(r)), the solution x(t) of the perturbed system (9.1) satisfies

and Ilx(t)ll I P@), v t 2 to + T

for some class KL function p and some finite T , where p is a class K function of d defined by

p(6) = a;' (a2 (a;' (y ))) 0

Proof: We use V(t, x) as a Lyapunov function candidate for the perturbed system (9.1). The derivative of V(t, x) along the trajectories of (9.1) satisfies

Applying Theorem 4.18 completes the proof.

This lemma is similar to the one we arrived at in the special case of exponential stability. However, there is an important feature of our analysis in the case of exponential stability, which has no counterpart in the more general case of uniform asymptotic stability. In the case of exponential stability, d is required to satisfy (9.10). The right-hand side of (9.10) approaches oo as r -+ oo. Therefore, if the assumptions hold globally, we can conclude that for all uniformly bounded disturbances, the solution of the perturbed system will be uniformly bounded. This is the case because, for any 6, we can choose r large enough to satisfy (9.10). In the case

350 CHAPTER 9. STXBILITI' OF PERTURBED SI'TEAlS E 9.3. COAIPARISON METHOD @ I Using (9.3), we can find an upper boulltl on v as of uniform asymptotic stability, 6 is required to satisfy (9.14). Iilspcctio~l of (9.14)

shows that, without further informati011 about the c l ~ s s K functions, we cannot say anything about the limit of the right-hand side as r -+ oo. Thus, we cannot conclude that uniformly bounded pertorbations of n nominal system 1vit.h a uniformly asymptotically stable equilibrium at the origin will have bounded solutions irrespective of the size of the perturbation. Of course the fact that we cannot show it, does not mean it is not true. It turns out., however, that such a st.ntenient is not true. It is possible to construct examples (Exercise 9.13) where the origin is globally uniformly asymptoticellp stable, but a bounded perturbation could drive the solution of the perturbed system to infinity.

w

t To obtain a linear differential inequality. we take W(t) = and use the

dt fact lif = ~ / 2 f l . when V # 0. to obtain

I When \/ = 0. it can be showllS that D+IV(t) 5 c46(t)/2fi. Hence, D+\V(t) satisfies (9.17) for all values of V. By the comparison lemrna, W(t) satisfies tllc b.3 Comparison ~e thod ] inequality

\.\f(t) < 4(t.to)lV(to) + & 1; 4(t17)6(7) dr Consider the pert,urbed system (0.1). Let V(t, x) be a Lyapunov filnct.ion for t.he nominnl system (9.2) and suppose the tlerivative of V along the t,rajectories of (9.1) satisfies the differential inequality

! where the transitioll function d(t t to) is given by

By (the comparison) Lemma 3.4,

1 Using (9.3) in (9.18), we obtain where y(t) is the solution of t.he differential equat.ion

This approacli is particularly useful when the differential ir~cqualit~y is linear, that is, when h(t, V) = a(t)V + b(t). for then we can write down a closed-form expression for the solution of the first-order linear differential equation of y. Arriving at. a linear differential inequality is possible when the origin of the nominal system (9.2) is exponentially stable.

Let V(t,x) be a Lyapullor function of the nornilla1 system (0.2) that satisfies (9.3) through (9.5) for all (t..r) E [O,x) x D, whcrc D = {T E Rn 1 llrll < r ) . Suppose the perturbation term g(t, x) satisfies the bound

I Suppose now that 7(t) satisfies t,lie contlit.ion t

1 7 ( 7 ) d r 5 ~ ( t - to) + n (9.20)

I for some nonnegati\re const.ants E anrl q. where

I Defining the constants a and p by where y : R -+ R is ilorlneg~t,ive ant1 contiriuous for nll t 2 O? and 6 : R -+ R is nonnegative, continuous, and bounded for all t 2 0. The derivative of V along the trajectories of (9.1) satisfies

t and using (9.20) and (9.21) in (9.19). we obtain . - -av + av av- V(t,x) = - -f ( t , ~ ) + zg(t,x)i a t ax

"~ee Exercise 9.14.

7 +---

--- . - - - - 7 - . . / 2 352 CHAM'ER 9. STABILITY OF PERTURBED SYSTEAlS

7 . 3

For this bound to be valid, we must ensure that IJx(t)lJ < r for all t > to. Noting thatG

3 we see that the condition I(x(t)ll < r will be satisfied if I

and 2clar

sup 6 ( t ) < -- ,>to c4 P

For easy refel.ence, we summarize our findings in the next lemma.

Lemma 9.4 Let x = 0 be an ezporientially stable equilibrium point of the nominal system (9.3). Let V ( t , x ) be a Lynp~lnor~frmction of the nominal svstem that satisfies (9.3) through (9.5) i n [0, ao) x D , where D = { x E Rn I llxl12 < T ) . Suppose the perturbation term g ( t , x ) satisfies (9.15), where y ( t ) satisfies (9.20) and (9.21). Then, provided x ( t o ) satisfies (9.24) and sup,,,,, 6 ( t ) satisfies (9.25), the solution of the perturbed system (9.1) satisfies (9.23). bh>hermore, if all the assumptions hold globally, then (9.23) is satisfied for any .r(to) and any bounded 6 ( t ) . 0

Specializing the foregoing lemma to the casc of vanishing perturbations; that is, when 6 ( t ) E-0, we obtain the following result:

Corollary 9.1 Let x = 0 be an ezponentially stable equilibrium point of the nominal system (9.2). Let V ( t , x ) be a Lyapunolrfunction of the nominal system that satisfies (9.3) through (9.5) i n [O, ao) x D . Suppose the perturbation term g( t , x ) satisfies

11g(t, 411 1 r(t)llxll

where y ( t ) sntisfies (9.20) and (9.21). Then, the origin is an ezponentially stable equilibrium point of the perturbed system (9.1). Moreover, if all the assumptions Fold globally, then the origin is globnlly ezponentially stable. 0

If y ( t ) = y = constant, then Corollary 9.1 requires y to satisfy the bound y < c1c3/c2c4, which has no advnntage over the bound y < ca/c4 required by Lemma 9.1, since (c l /c2) < 1. In fact, whenever ( c l / c2 ) < 1, the current bound will be more conservative (that is, smaller) than the bound required by Lemma 9.1. The advantage of Corollary 9.1 is seen ill the case when the integral of y ( t ) satisfies conditions (9.20) and (9.21), even when sup,,,,, y ( t ) is not small enough to satisfy sup,,,,, - y ( t ) < c3/c4. Three such cases arc giGn in the next lemma.

w e use the fact that the function ue-''I + b( l - e - I " ) , witli positive a, b, and a, relaxes monoton~cally from its initial value a to its filial value b. Hence, it is bounded by the maximum of the two ~ ~ l i n ~ l ~ c r s .

9.3. COhlPARISON METHOD

Lemma 9.5

1. If

then (9.20) is satisfied with E = 0 and q = k .

then for any E > 0 , there is q = q ( ~ ) > 0 such that (9.20) is satisfied.

3. If there are constants A > 0 , T 2 0 , and €1 > 0 such that I

then (9.20) is satisfied with E = E I and q = E I A + JoTy(t) dt. 0 Proof: The first case is obvious. To prove the second case, note that, because limt+rn y ( t ) = 0 , for any E > 0, there is TI = TI(€) > 0 such that y ( t ) < E for all t 2 TI. Let q = J,T' y ( t ) dt. If to > TI , then

' 6 . t

l o t y ( T ) d7 < lo E d7 = ~ ( t - to)

If t < TI , then

If to <TI < t , then

In the last case, if t 5 T , then

r t rT

For t 1 tl 2 T , let N be the integer for which ( N - l ) A 5 t - tl 5 N A . Then,

354 CHAPTER 9. STABILITY OF PERTURBED SYSTEAB

This inequality is used next with t i = to when t 2 t o 2 T ? and with t l = T when to _< T I t . If t _> to 2 T. then

while if to 5 T j t , then

1 In the first case of the foregoing lemma, condition (9.20) is satisfied wit11 E = 0 , !

while in the second case. it is satisfied with arbitrarily small E. Therefore. in both cases, condition (9.21) is always satisfied and the origin of the pertllrhed system (9.2) is exponentially stable. The third case of the lemma sets a bound on a moving average of 7 ( t ) as t becomes sufficiently Inrge. The o4gin of the perturb~d system ! (9.2) will be exponentially stable if this bound is sufficiently small. r Example 9.6 Consider the'linear system I

n l ~ c w r l ( t ) nnd B ( t ) are c~olitinuous and A ( t ) is bo~ultlctl on (0, cc). Suppose thc origin is all exponelltially stable equilibrium point of the nominal system

I i

and

From Theorem 4.12, we know that there is a quadratic Lyapunov function V ( t . x ) = x T P ( t ) x that satisfies (9.3) through (9.5) globally. The perturbation ten11 S ( t ) r satisfies the inequality

IlB(t)xll s IlB(t)ll Ilxll Since ( ( B ( t ) ( ( 4 0 as t -, w, we conclude from Corollary 9.1 and the second case of Lemma 9.5 that the origill is a glohnllp expol~cliti~lly stablc eqllilibriun point of the perturbed system. A

Similar conclusiorn can be drawn when S: IIB(t)ll dt < M (Exercise 9.16) and Sr 11B(t)l12 dt < w (Exercise 9.16). ! I

In the case of nonvanishing perturbations, that is, when b ( t ) $ 0. the next lemma states a number of conclusions concerning the asymptotic behavior of x ( t )

t w t f --., I

9.4. CONTINUITY OF SOLUTIONS 355

Lemma 9.6 Suppose the conditions of Lemma 9.4 are satisfied, and let ~ ( t ) denote the solution of the perturbed system (9.1).

1. If e-4 ' - ' )6(r) d~ 5 P, V t 2 to

for soirlr positive coilstal~t a, tho^ .r(t) is uniformly ultintutely bounded wit11 the ~rltimate bound

c4pP b = - - 2~~ e

where 0 E ( O ? 1 ) is an arbiti'ary constant.

then x ( t ) is uniformly ultimately bounded with the ultimate bound

c4pbm , b e - 2nc1e :

where 8 E (0.1) is an arbitrary conatnnt.

lim 6(1) = 0 , ~ ~ I ( ' I I lim ~ ( t ) = 0 i-u t-m

V the conditions of Lemma 9.4 are s~rtiqficd then the foregoing statente?its

hold for any initial state x( t0) . 0

Proof: All three cases follow easily from inequality (9.23). In the 'first two cases, we use the property t,hat if u ( t ) = ~ ( t ) + a with a > 0 and liml,, w ( t ) = 0 , then u ( l ) is ullill~atcly bounded by.a/8 for auy pusitiw 0 < 1. This is su because there is a finite t i~nc T sucl~ that (w( t ) ( 5 a(1 - B)/Q for all t 2 T. In tho last tw! cases, w , tho property that if u ( t ) = ee\p(-a(t - T ) ) w ( T ) d~ where ul(t) is bounded

aiid li111~,~ ~ ( 1 ) = wrn: thcn limt,, u ( t ) - v!;/n.7 0

9.4 Continuity of Solutions on the Infinite Interval

In Section 3.2, we studied continuous dependence of the solution of the state equation on init.ia1 states and parameters. h particular, in Theorem 3.1, we examined the nominal system

j. = f ( t , x ) (9.26)

7Sec (33. Theorem 3.3.2.333.

I '. - -., ..A

L-, J

r',, i t.,

and t.he pert.urbed system xi.= f ( t , x ) + g ( t , x ) (9.27)

under the assumption that 11g(t,x)ll 5 b in the domain of interest. Using the Gronwall-Bellman inequality, we found that if y(t) and z ( t ) are well-defined solutions of the nominal and perturbed systems, respectively, then

where L is a Lipschitz constant for f . This bound is valid only on compact time intervals, since the exponential term exp[L( t - to)] grows unbounded as t -+ cm. In fact, the bound is useful only on an interval [to, t l ] where t l is reasonably small, for if t l is large, the bound will be too large to be of any use. This is not surprising, because in Section 3.2, we did not impose any stability conditions on the system. In this section, we use Lemma 9.4 to calculate a bound on the error between the solutions of (9.26) and (9.27) that is valid uniformly in t for all t 2 to.

Theorem 9.1 Let D 1 c Rn be domain that contains the origin and suppose

f ( t , x ) and its first partial det-ivntives with respect to x are continuous, bounded, and Lipschitz in x , uniformly in t , for all ( t , x ) E [0, cm) x Do, for every compact set Do C D;

g(t, x ) is piecewise continuous in t , locally Lipschitz in x , and

.the origin x = 0 is an exponentially stable equilibrium point of the nominal sl~stem (9.26);

there is a Lyapunov fut~ction 17(t,x) that satisfies the conditions of Theo- rem 4.9 for the nominal system (9.26) for ( t , x ) E [O, oc) x D and {lVl ( x ) 5 c} is a compact subset of D. \

Let y( t ) and z ( t ) denote solutions of the nominal system (9.26) and the perturbed system (9.27), respectively. Tlien, for each compact set R C {lVz(x) 5 pc, 0 < p < 11, there emkt positive constants p, 7, q, p, and k , independent of 6, such that i f y(to) E R, 6 < q, and (Iz(to) - y(to)(l < p , the solutions y( t ) and z ( t ) will be uniformly bounded for all t .? to 2 0 and

While the origin is exponentially st,able, the Lyapunov function V is required to satisfy the conditions of uniform tlsyrnptotic stability, rather than (the more stringent) co~lclitions of exponential stsbi1it.y. This provides less conservative estimates

. . . -. --- f ' ' 2 .

. . , -.. . . A .d*-., d *

9.4. CONTINUITY OF SOLUTIONS 357 i

of the set 0. When tlie ~lonli~lal systeln (9.26) is autonomous, the function V is provided by (the converse Lyapunov) Theorem 4.17 and the set S-2 can be any com- !

! pact subset of the region of attraction. Exponential stability is used only locdy when the error z ( t ) - y(t) is sufficiently small. !

i

Proof of Theorem 9.1: The derivative of If along the trajectories of the per- i f

turbed system (9.27) satisfies 1

for aU, x , ~ {Wl(x).-s,.~}., .where kl is an upper bound on BV/ax over { W l ( x ) < c}. Let k2 > 0 be the minimum of W3(x) omr the compact set A = { W l ( x ) 5 c and W2 ( x ) 2 c). Then

This shows that v is negative on V ( t , x ) = c; hence the set { V ( t , x ) 5 c} is positively invariant. Therefore, for all t ( t 0 ) E { W ~ ( X ) < c) , the solution z ( t ) of (9.27) is uniformly bounded. Since S-2 is, in the interior of { W 2 ( x ) < c), there ie > 0 such that at01 E {Wz(x) < c) +enever y(t0) E 0 and (Iz(to) - y(to))) 5 pl. I t is also clear that for y(t0) E a, y(t) is uniformly bounded and y(t) - 0 as t - oo, uniformly in to. The error e(t) = z ( t ) - y(t) satisfies the equation

k = 5 - Y = f ( t! 2 ) + g(t, z ) .- f (t , y) = f (t , e ) + A ( t , e) + g(t, z ) (9.31)

where A ( t , e ) = f ( t , YO) + e ) - f ( 4 y ( t ) ) - f ( t , e )

We analyze the error equation (9.31) over the ball {Ilell I r ) c D. Equation (9.31) can be viewed as a perturbation of the system

' : . i

whose origin is exponentially stable. By Theorem 4.14, there exists a Lyapunov . ' I

function V ( t , e) that satisfies (9.3) through (9.5) for llell < ro < r. By the mean . value theorem, the error term Ai can be written as ,.. :

where 0 < A, < 1. Since the Jacobian matrix [ a f / a x ] is Lipschitz in x, uniformly in t , the perturbation term ( A + g) satisfies

IlA(t1e) + g(t,t)ll 5 L1llelI2 + L2llell l l~( t) l l+ 6

358 CHAPTER 9. STABILITY OF PERTURBED Sl'STEAlS

- where y(t) -+ 0 as t -+ x. uniformly in to. Consequently.

IlA(t, e) + g(t, 2)ll I {Llr l + Lzll~(t)ll) llell + 6

for all llell I r~ < ro. This inequality takes the forin (9.15) with

~ ( t ) = {LIT] + L2Ily(t)(l} and 6(t) r 6

Given any €1 > 0, there is Tl > 0 such that Ily(t)ll 5 c1 for a11 t 2 to+Tl. Therefore, (9.20) is satisfied with

1 lo '(7) d~ 5 (€1 + L I ~ I ) ( ~ - to) + T I maxLzllu(t)Il t a l l

By taking €1 and rl small enough, we can satisfy (9.21). Thus, all the assumpt.ions of Lemma 9.4 are satisfied and (9.30) follows from (9.23).

9.5 Interconnected Systems ~1st- b p b d * When we analyze the stability of a nonlinear dynainical system, the complexity of the analysis grows rapidly as the order of the system increases. This situat.ion motivates us to look for ways to simplify the analysis. If the system can be modeled as an interconnection of lower order subsystems, then we ]nay pursue the st.abi1it.y analysis in two steps. 111 the first step, we tlccornl~ose t l ~ e syst.cin illto srrlnllcr isolnted siibsystenls by igl~oril~g interconnections, and analyze the st.ability'of each subsystem. In the second step, we combine our co~~clusions from the first step with information about the int.erconnections to draw conclusions about the st.abi1it.y of the interconnected system. In this section, we illustrate how this idea can he utilized in searching for Lyapunov functions for interconnected systems.

Consider the interconnected system

T where xi E Rn', nl + ... + n, = n, nntl x = [r:?. .. ,I:] . Suppose fi aild gi are smooth enough to ensure local existence ant1 uniqueness of the solution for all initial conditions in a domain of interest, and that

fi(t,O) = 0, gi(t,O) = 0, V i

so that the origin x = 0 is an equilibrium point of the system. Ignoring the interconnection terms gi, the system decomposes into m isolated subsystems:

with ench one having an equilibrium point a t its origin xi = 0. i r e start. by searching for Lynp~lllov fu~lctions t Ililt cstnl)lisl~ u11ifo1.111 i1syi111)totic stnbility of tllr origiii for

9.5. INTERCONNECTED Sl'STEAIS

each isolat.ed subsystem. Suppose this scnrcl~ has' been successful and tliat, for cacli subsystem, we have a positivc dcti~iitc! tlccrcsccnt Lyapunov function V,(t, xi) whosc derivative along the trajectories of the isolated subsystem (9.33) is negative definite. The function m

\ / ( t , r ) = x d i ~ , ( t , x i ) , i= I di > 0

is a composite Lyapzrnov function for the collection of the m isolated subsystems for all values of the positive constants d,. Viewing the interconncctcd system (9.32) as a perturbation of the isolated subsystems (9.33), it is reasonable to try V(t,x) as a Lyapunov function candidate for (9.32). The derivative of V(t,x) along the trajectories of (9.32) is given by

for all t 2 0 and 11x11 < r for so~nc positive constants ai and Pi, whcre di : Rni + R are positive definite and continuous. Furthermore, suppose that tlie interconncctiorl terms gi(t, x) satisfy the hound

m

JIgi(t,x)II < x T i j d j ( x j ) j= 1 (9.3G)

for all t 2 0 and JJxJJ < r for some nonnegative constants yij. Then? the derivative of V(t,x) = CEl di&(t,xi) along t . 1 ~ t.rajectories of the interconnected system (9.32) satisfies the inequality

The first term on the right-hand side is negative definite by virtue of the fact that & is a Lyapunov function for the ith isolated subsystem, but the second term is, in general, indefinite. The situation is si~nilar to our earlier investigation of perturbed systems in Section 9.1. Therefore, 1ve inay approach the problem by performing

C I L

L

worst cask analysis where the term [al/,/a.r,Ig, is bounded by a nonnegative upper bound. Let us illustrate the idea hy using quadratic-type Lyapunov functions, introduced in Section 9.1. Suppose that. for i = 1,2,. . . , m , &(t, r,) satisfies

L

c: C

-- . . . . . . . . . . . . . .... .. .... -. .. - . ' . . . . . . . . .- :. ... r>r

3GO CHAPTER 9. STABILITY OF PERTURBED SYSTEhfS

The right-hand side is a quadratic form in d l , . .. ,dm, which we rewrite as

~ ( t , x ) 5 - ~$(Ds+ STD)d

where d = [ ~ l l . . . , ~ m ] T l D = d i a g (d l , . . . ,dm)

and S is an m x m matrix whoseelements are defined by

If there is a positive diagonal matrix D such that

then ~ ( t , x) is negative definite. since d(x) = 0 if and only if x = 0; recall that di(xi) is a ~~ositive definite fuiictioii of xi. Tlius, a si~ficient colldition for u~iiforiil asymptotic stability of the origin as an equilibrium point of the interconllected system is the existence of a positive diagonal matrix D such that DS+STD is positive definite. The matrix S is spccial in that its off-diagonal elements are nonpositive. The next lemma applies to this class of matrices.

Lemma 9.7 There exists a positive diagonal matriz D such that D S + STD is positive definite if and only if S is an M-matrix; that is, the leading principal minors of S are positive:

Proof: See [57].

The.Af-matrix condition can be interpreted as a requirement that the diagonal elements of S be "larger as a whole" than the off-diagonal elements. It can be seen (Exercise 9.22) that diagonally dominant matrices with nonpositive off-diagonal elements are M-matrices. The diagonal elements of S are measures of the "degree of stability" for the isolated subsystems in the sense that the constant cti gives a lower bound on the rate of decrease of the Lyapunov function with respect to &(xi). The off-diagonal elements of S reprtxent the "strength of the interconnections" in the sense that they give an upper hound on gi ( t , x ) with respect to $j ( z j ) for j = 1,. ... m. Thus, the M-matrix coiidit~ion says that if the degrees of stability for the isolatc(l subsystems are larger ns n ~rrllole than the strength of the interconnections, then the interconnected system has a uniformly asymptotically stable equilibrium at the o7Qin. We summarize our conclusion in the next theorem.

....... - I I . : $. , -----

. . . . . . . . . ....,, .;L..!*,.,., d . . .. 9.6. INTERCONNECTED SYSTEhIS 361

Theorem 9.2 Coi~sider. the systeiu (9.3'2) urid si~ppose Uheia a~ positiue definite decrescent Lyapunovfvnctions K ( t , x i ) that satisfy (9.34) and (9.35) and that gi( t lz)

3

satisfies (9.36) for all t 2 0 and llxll < r . Suppose the matrix S defined by (9.37) is an kl-matrix. Then, the origin is uniformly asymptotically stable. Moreover, if all the assumptions hold globally and K ( t , x i ) are radially unbounded, it will be globally uniformly asymptotically stable. 0

I Example 9.7 Consider the second-order system

i The system can be represented in the form (9.32) with

t f i (x1) = - 2 1 , g 1 ( ~ ) = -1.5~:x,3, fz(x2) = -x i , and g2(x) = 0 . 5 ~ : ~ ;

The first isolated subsystem X I = -XI has a Lyapunov function K ( x l ) = x:/2, which satisfies i I

sv1 .A -f1(x1) = -x: = -ald;(xl) ax1

where ct1 = 1 and dl(x1) = 1x11. The second isolated subsystem x2 = -xi has a Lyapunov function h ( x 2 ) = ~ 4 1 4 , which satisfies 1

1

where cr2 = 1 and h ( x 2 ) = )x2I3. The Lyapunov functions satisfy (9.35) with P1 = = 1. The interconnection term gl (x ) satisfies the inequality

for all 1x11 5 cl. The interconnection term g ~ ( x ) satisfies the inequality

Igz(X) I = 0 . 5 ~ : ~ ; < O . ~ C I & I ( X I ) : for all 1x11 < cl and 1x21 5 ~ 2 . Thus, if we restrict our attention to the set

s

we can conclude that the interconnection terms satisfy (9.36) with

711 = OI 712 = 1 . 5 ~ : ~ 721 = 0 . 5 ~ ~ ~ 2 2 ~ and rz2 = 0

The matrix

362 C H A P T E R 9. STABILIT't' OF P E R T U R B E D S Y S T E M S

is an A{-matrix if 0.75~:~; < 1. This will bc the case, for csamplc, when cl = cz = 1. Thus, the origin is asymptotically stable. If we are interested in est.imnting the region of attraction? we need t.0 k n o ~ ~ ~ the col~lposite Lyapu~lov fu~lction I/ = dlV; + dzV;; that is. nre need to ~ I I O I V a positive diagonal matrix D such t,hat DS + STD > 0. Taking cl = c2 = 1: we have

which is positive definite for 1 < d2/dl < 9. Siiice there is 110 loss of generality in multiplying a Lyapunov function by a positive const.ant., we take dl = 1 and write the composite Lyapunov function as

An estimate of the region of attraction is given by

where c < min {1/2, d2/4} to ensure that R, is il~siclc the rectangle ( . E ~ ( 5 1. Noting that the surface V(x) = c iiltcrsccts the rl-asis and the x2-axis at 6 and (4c/d2)'I4, respectively. we max~mize these distances by choosing d2 = 2 and c = 0.5. A

Example 9.8 The lnr~tl~eillatical nlotlol of NI nrtificinl neural rletwork was pro sentcd in Section 1.2.5, ~ I I I ~ its sbability properties were analyzed in Example 4.11 by using LaSalle's invariance principle. A key assumpt,ion in Example 4.11 is the symmetry requirement Tij = T j i , which allows us t,o represent. the right,-hand side of the state equation as tihe gradient of n scnlnr function. Let us relax this rrquire- ment and allow Tij # Tji. We will analyze the stability properties of the net\vork by viewing it as an intercoliiiect~ion of subsystems; each subsystem corresponds to

.. one neuron. We find it convenient here to work with t,he voltages.at t,l~e amplifier inputs ui. The equations of motion are

for i = 1,2,. . . ,n , where gi(.) are sigmoid functions, Ii are constant current inputs, Ri > 0, and Ci > 0. We assume that the system has a finite number of isolated equilibrium point,s. Each equilibrium point u' satisfies the equation

9.5. I N T E R C O N N E C T E D S Y S T E h l S 363

Figure 9.1: The sector nonlinearity qi(xi) of Example 9.8.

To analyze the stability properties of a given equilibrium point u*, wesl~ift it to the origin. Let xi = ui - ul. Then:

where vi(ri) = gi(ai + u;) - g i (~ ; )

Assume that qi(.) satisfies the sector condition

u2kil _< uqj(u) 5 u2ki2, for u E I-~i , r i ]

where k,] and ki2 are positive co~s ta~l t s . Figure 9.1 shows that such co~ldit~io~l is indeed satisfied when gi(ui) .= (2l/n,/T) tan-' (X~ui/2Vj\,), X > 0. tVe can recast this system in'the form (9.32) with

Using

, .. _ . . - . . . . . . -. , . .- - . . ~

. ' . . , . . . . - . . .- _ , L C .

364 CIII\PTE~~. !I. STABILITY OF PERTURBED SYSTEMS

as a Lyapunov function candidate for the ith isolated subsystem, we obtain

If T,i 5 0, then

which is negative definite. If Tii > 0, then

and

In this casc, we assume that Tj~l;?~ < l /Ri , so that thc dcrivativc of V, is llcgativc definite. To simplify the notation, let

Then, K(xi) satisfies (9.34) and (9.35) on the interval [-ri:ri] with

where cri is positive by assumption. The interconnection term gi(x) satisfies the ineaualitv

Thus, gi(r) satisfies (9.36) with yii = 0 and yij = kj21Tijl/Ci for i # j. Now wc can form the matrix S as

6i + l /R , , for i = j

-ITii152 for i # j

The equilibrium point u* is asyinptotically stable if S is an Al-matrix. We may estimate the region of attraction by t-he set.

9.6. SLO\VI,Y VARYING SI'STEhlS 365 ? -

where c 5 0.5mini{diCir?) to ensure that R, is inside the set lxil -< ri. This analysis is repeated for each asymptotically stable equilibrium point. The conclusions we could arrive at in this example are more conservative compared with the conclusions we arrived a t using LaSalle's invariance principle. First, the interconnection coefficients Tij must be restricted to satisfy the M-matrix condition. Second, we I obtain only local estimates of the regions of attractions for the isolated equilibrium points. The union of these estimates does not cover the whole domain of inte'rest. On the other hand, we do not have to assume that Tij = Tji. A

9.6 Slowly Varying Systems

The system s = f XI^)

where x E Rn and u(t) E r C Rm for all t > 0 is considered to be slowly varying if u(t) is continuously differentiable and Ilu(t) (1 is "sufficiently" small. The components i of u(t) could be input variables or time-varying parameters. In the analysis of (9.38), one usually treats u as a "frozen" parameter and assumes that for each ,

fixed u = a E r , the .frozen system has an isolated equilibrium point defined by r x = h(cr):.. If a property of a: = h(a) is uniform in h, then it i s reasonable tb 9

I expect that the slowly varying system (9.38) will possess a similar property. The underlying characteristic of such systems is that the motion caused by changes I of initial conditions is much faster than that caused by inputs or timevarying parameters. In this section, we will see how Lyapunov stability can be used to

i analyze slowly varying systems.

1 Suppose f (2, u) is locally Lipschitz on Rn x r, and for every u E r the equation i

O=f (x ,u )

has a continuously differentiable isolated root x = h(u); that is,

0 = f ( h ( ~ ) , U)

To analyze the stability properties of the frozen equilibrium point x = h(a), we ! shift it to the origin via the change of variables z = x - h(a) to obtain the equation

Now we search for a Lyapunov function to show that z = 0 is asymptotically stable. Since g(z, a ) depends on the parameter a , a Lyapunov function for the system may

StiU CHAPTER 9. STABILITY OF PERTURBED SI'STEAlS

depend, in general. on a. Suppose we cnn fincl a Lyapullov fuilction V(n , a ) that satisfies the conditions

dl' z g ( z 7 a ) 5 - ~ 3 1 1 ~ 1 1 ~ (9.42)

for all z E D = { z RR 1 IJzIJ < r ) and a E r. where c,, i = 1,2.. .. , 5 are positive constants independent of a. Inequalities (9.41) and (9.42) state the usual requirements that V be positive definite and dccrescent nnd has a negative definite derivative along the trajectories of the system (9.40). Furthermore. they show that the origin z = 0 is exponentially stable. The special requirement here is that these inequalities hold uniformly in a. Inequalities (9.43) and (9.44) are needed to handle the perturbations of (9.40). which will result from the fact that u ( t ) is not constant, but a time-varying function. \Vit,h V ( z , u ) as a Lyapunov function candidate, the analysis of (9.38) proceeds as follows: Tlie change of variables z = z - h ( u ) transforms (9.38) into the form

wlicre the effect, of the time variat.ion of u appears as a perturbation of the frozen system (9.40). The derivative of V ( Z , U ) a1011g the trajectories (9.45) is given by

Setting

7( t ) = ~ l l u ( t ) l l and d( t ) = LIld(t)ll

we can rewrite the last in equal it..^ as

which takes the form of inequality (9.16) of Section 9.3. Therefore, by applying the comparison lemma, as in Sect,ion 9.3. it can be shown that, if 4 ( t ) satisfies

in which a1 and pl are defii~ed I)!.

then z ( t ) sat.isfies the inequality

Depending upoil tlie assuinpt ioiis for ( 1 i 1 1 , several conclusiol~s call be drawn from the foregoing inequality. Sonle of these coilclusions are stated in the next theorem.

Theorem 9.3 Consider the system (9.45). Suppose that [ah/8u] satisfies (9.39), ( lu( t ) ([ 5 E for all t 2 0. and there is a Lyapunou function V ( z , u ) that satisfies (9.41) through (9.44). If c i c ~ r

E < - X czc; r + cq L/c5

then for all J ( t ( 0 ) I ( < r a , the sohriiails of (9.45) are unifolnlly 6ol~n.dcd for 011 t 2 0 and unifoinlly ultiinately Do~~nded 6y

where 6' E (0 , 1) is an arbitrary constnnt. If, in addition, u ( t ) -+ 0 as t + m, then t ( t ) -t 0 as t -, m. Finally, if h ( u ) = 0 for all u E r and E < c3/c5. then z = 0 is an ezponentially stable equili6r.ilrn1 point of (9.45). Eguiualen.tly, x = 0 is an exponentially stable egtrilibri~tm point of (9.38). 0

Proof: Since Ilu(t)ll 5 E < cIc3/c;c5. ineq~lality (9.46) is sat,isfied wit.h E I = E and

Using the give11 upper bound on E , we have

. ........... .......... ................. ---- ......... ...- ..... . . . . . . . , , ,.

. . -- . . , . . . . . . . .

.- " T .

. . . . . . ,.. ,....,, . .A.. .A,. . A ..... --- ..% :, :, ._ -.-.-.* . . . . . ..... . . ' . . -

CHAPTER 9. STABILITY OF PERTURBED W S ~ E M s 9.6. SLOWLY VARYING S1'Sl'/?hIS 368 369

Hence, the inequality suptyo l l~( t) l l < 2 c i a l ~ / c d L is satisfied and (9'47), we where T = h1 (2k2)/27. Silllilar to the prod 01 Tlicorelu 4.14, it can be &own.

obtain that V ( z , a ) satisfia (9.41) through (9.43) with cl = [l - exp ( - ~ L ~ T ) ] / ~ L ~ , c2 = k2[l - exp(-27T)V27, CJ = 112, and CA = 2k{l - exp [-(7 - L l ) T ] } / ( 7 - L ~ ) . To ahow that V ( z l a ) satisfies (9.44), note that the sensitivity fvnction &(ti z ,a) satisfies the sensitivity equation

After a finite time, the exponentially decaying term will be less than (1 -%)b, which shows that z( t ) will be ultimately bounded by b. I f , in addition, C(t) --+ 0 as t -+ oo, then it is clear from (9.47) that z ( t ) -+ 0 as t -+ oo.. I f h (u) = 0 for all u E r, we can take L = 0. Consequently, the upper bound on V simplifies to

which shows that z = 0 will be exponentially stable i f E < c3/c5.

Theorem 9.3 requires the existence o f a Lyapunov function V ( z , a ) for the frozen system (9.40), which satisfies inequalities (9.41) through (9.44). Lemma 9.8 shows that such Lyapunov function will exist, under some mild smoothness requirements, i f the equilibrium point z = 0 o f the frozen system is exponentially stable uniformly in a . This is done by deriving a converse Lyapunov function for the system, as in the converse Lyapunov theorems o f Section 4.7.

Lemma 9.8 Consider the system (9.40) and suppose g(z ,a) is continuously differentiable and the Jacobian matrices [8g/Bz] and [Bg/bo] satisfy

B 47 4 = 4 ; 4 + ( 4 ) 4.(0; z , a ) = o

from which we obtain

1 1

I I 5 1 Llll4.(r;~,a)li2 dr + / ~2l l4(r ; z ,a ) l12 dr 0

1

5 1 Lil14a(~; 'z , a1112 dr + L2ke-7T dr~lzIl2 I' L2k < 1' ~ l l l + ( r i 2, a1112 dr + -11~112 7

Use o f the Gronwall-Bellman inequality yields

. 114.(t; 2 , 4 1 1 2 5 *11z112e~l t 7

Hence,

for all ( z , a ) E D x I?, where D = { z E Rn 1 1 1 ~ 1 1 < r } . Let k , 7 , and ro be positive constants with ro < r / k , and define'go = { z E Rn I llzll < ro}. Assume that the trajectories of the system satisfy , \

which completes the proof of the lemma. IIz(t)ll I k l l z (~) l l e -~ ' , Q 4 0 ) E Do, E r, t 2 0 o

Then, there 2s a function V : Do x I' -, R that satisfies (9.41) through (9.44). when the frozen System (9.40) is linear, a Lyapunov function s a t k h g (9.41) Moreover, if all the assumptions hold globally (in z), then v ( z , a ) is defined and through (9.44) can be explicitly determined by solving a parameterized Lyapunov

0 equation. This fact is stated in the next lemma. satisfies (9.41) through (9.44) on Rn x r. Proof: Owing to the equivalence of norms, it is sufficient to Prove the lemma Lemma 9.9 Consider the system i = A(a)z , whe,hen o E I. and A ( ~ ) is for the norm. Let d ( t ; z , a ) be the solution of (9.40) that starts at (092); that O u s l ~ diffe~ntiable. ~uPPose the elements of A and their jrst padial derivcrthes is, 4(0; z , a ) = *. The notation emphasizes the dependence o f the Solution on the with respect to a are uniformly bounded; that is, parameter a. Let

T

V ( Z , Q ) = l mT(t; 2, a)m(t; 2, a ) dt IIA(a)II2 5 c, A ) < bi, Q a E I', Q 1 < i < rn 2

370 CHAPTER 9. STABILITY OF PERTURBED SYSTEMS

Suppose further that A(a) is Hurwitr uniformly in a; that is,

Rc[X(A(a))] 5 -0 < 0, V a E r Then, the Lyapunov equation

has a unique positive definite solution P (a ) for every a E I?. Moreo.ver, P ( a ) is continuously differentiable and satisfies

for all ( z , a ) E Rn x I?, where cl, c2, and pi are positive constants independent of a. Consequently, V(z,a) = (9.42) through (9.44) in the 2-nonn with cs = 1, cg = 2c2, and 0

Proof: The uniform Hurwitz property of A(a) implies that the exponential matrix exp [tA(a)] satisfies

11 exp [tA(a)]ll ,< k ( ~ ) e - O ~ , V t L 0, V a E

where p > 0 is independent of a, but k(A) > 0 depends on a. For the exponentially decaying bound to hold uniformly in a, we need to use the property that IIA(a)ll is bounded. The set of matrices satisfying Re[X(A(a))] _< -a and JJA(a)JJ ,< c is a compact set, which we denote by S. Let A and B be any two elements of S. Consider8

exp[t(A + B)] = exp[tA] + exp[(t - r)A]Bexp[r(A + B)] d r 1' Using the exponentially decaying bound on exp[tA], we get

Multiply through by ePt,

'This matrix identity follows by writing i = (A + B)z a s i = Az f Bz and viewing Bz as an input term. Substituting x(t) = exp[t(A $ B)]zo into the input term yields

Slnm this c'xprmion Ilolds for nil $0 E R". we nrrive nt the nlntrix identity.

Applying the Gro~~\vall-Bcll~r~a~~ inrc111;llity yicltls

11 cxp\f ( A + B)]\\ 5 k ( . ~ \ ) r ~ - ( ~ ' - k ~ " ) ~ ' B ~ ~ ) ' , V t 2 0

Hcrice. tllcre csists n positive coi~stal~t 7 < +3 n~rtl a neighborhood N(A) of A sucl~ that if C E N(A). t h w

Siurr S is rolnpact. it is roi.crec1 by a fil~itr 1111n1l,cr of these neighborlloods. There- fore. cnll fincl a positivc cons~iuit k i~~( lv l ) (*~~( le l~ t of n S U C ~ I that.

Considcr no\\- the Lyapunov cquat io~~ (9.48). Existence of a unique positdive definite soh~tion for every a E I? follows from Theorein 4.6, Moreover, the proof of that theorem shows that

Shce A(a) is continuously differentia1,le. su is P(o). LVe have

1" k2 k2 zTp(n)z 5 k2e-27'1(;11i dt = -));)I; 27 * cz = - 27

Let y(t) = el"(")z. Then. y = A(a)gl

-yT(t)3i(t) = -yT(t)~(a)y(t) I I A ( ~ ) I I ~ Y ~ ( ~ ) u ( ~ ) I cyT(t)y(t)

and

Z ~ P ( ) Z = bx yT(t)y(t) dt 2 &- ;vT(t)i(t) dt

Diffcrerltiatc P(a)A(a)i-AT(a)P(a) = -I p;1ltially wit.hrespect to iu~y co~nponent ai of a, and denote the derivative of P ( a ) by PJ(cr). Then,

P1(a)A(a) + A ~ ( ~ ) P ' ( O ) = -{P(a)A1(n) + [ ~ ' ( a ) ] ~ ~ ( a ) }

Thus, P1(a) is given by

. .. . , . . . - ,. . . ... - . , . . ..- . , . : < . \ 1 .

. ' .,. . . . . ,. . .:L .... . . ; . ~ . 3 .. . , . r , .. .. . .., >.*..

9, STABILITY OF PERTURBED SYSTEAIS a 9.7. EXERCISES 373 372

t ~t follows that 9.2 Co~isider the system S = Ax + Bu ulid Ict 11 = -Fr be a stl\bilizilg stntc

k2 b.k4 b, k4 feedback control; that is, the matrix (A - B F ) is Hurwitz. Suppose that, due to IIP1(n)l\2 5 lm k2e-2Tt2-b, dt = - * r = - t physical limitations, we have to use a limiter to limit the value of ui to Iui(t)l 5 L. .

27 2y2 2-Y2 The closed-loop system can be represented by x = Ax - B L sat(Fx/L), where

4 sat(v) is a vector whose ith component is the saturation function. By adding and

which completes the proof of the lemma. subtracting the term BFx, we can rewrite the closed-loop state equation as j. =

I+, be noted that the set r in emm ma 9.9 is not n e c e s s ~ i l ~ compact. When ( A - BF)X - ~ h ( F x ) , where h(v) = L sat(u/L) - v. Thus, the effect of the limiter . can be viewed as a perturbation of the nominal system without the limiter. . r is the boundedness of A(a) and its partial derivatives follow from the 3".

assumption that A(a) is continuously differentiable. ( a ) Show that 6 .y'

Iht(v)I I - Ivil, V lvil < L(l + 6) .* Example 9.9 Consider the system (1 + 6)

x = A ( ~ t ) x where 6 > 0.

where E > 0. When e is s~fficient~ly small, we can treat this system as a slowly varying system. It is in the form of (9.38) with u = et and = [O,m). For all u E r, the origin x = 0 is an equilibrium point. Hence, this is a special case where h(u) = 0. Suppose Re[A(A(a))] < -0 < 0, and A(a) and A1(a) are uniformly bounded for all a E I?. Then, the solution of the Lyapunov equation (0.48) holds the properties stated in Lemnla 9.9. Using V ( x , u ) = xTP(u)x as a Lyapunov function candidate for x = A(u)z , we obtain

where cs is an upper bound on / / P ' ( ( Y ) ( ( ~ . Therefore, for all E < l/c5, the origin x = 0 is an exponentially stable equilibrium point of 5 = A ( ~ t ) x . A

.J (b) Let P be the solution of

-1 P(A - BF) + ( A - B F ) ~ P = -I

Show that the derivative of V ( x ) = xTPx along the trajectories of the closed- loop system will be negative definite over the region ( ( F X ) ~ ~ < L(1+ 6) , V i, provided 6/(1+ 6) < 1/(211PBllz IIFIIz).

(I

(c) Show, that the origin is asymptotically stable and discuss how you would estimate the region of attraction.

e (d) Apply the result obtained in part (c) to the case

A = [ , & i ] , B = [ ; ] , ' F = [ ~ 2 1 , and 1 - 1

1 8 and estimate the region of attraction.

a 9.7 Exercises 9.3 Consider the system /*

9.1 ( [ l so] ) Consider the Lyapunov equation pA+ATP = -8, where Q = QT > 5 = f ( t , X ) + Bu, y = Cx, and u = -g(t, y) 4

0 and A is Hurwitz. Let p(Q) = ~ r n i n ( ~ ) / ~ m a x ( P ) . t

where f ( t , 0) = 0, g(t, 0) = 0, and I(g(t, y) 1 1 I 711y(I for all t L 0. Suppose that the I

(a) show that p ( k ~ ) = p(Q) for any positive constant k. origin of 5 = f ( t , x ) is globally exponentially stable and let V ( t , x ) be a Lyapunov function that satisfies (9.3) through (9.5) globally. Find a bound 7' on 7 such that

(b) Let Q = QT > 0 have x,~,,(Q) = 1. Show that ~ ( 1 ) 2 P(Q). the origin of the given system is globally exponentially stable for 7 < 7'.

(c) Show that p ( I ) > p(Q), v Q = QT > 0. b k 9.4 Consider the perturbed system I

Hint: In (b), let PI and 9 be the solutions of the Lyapunov equation for Q = I x = Ax+ B[u +g( t ,x )] and Q = Q, respectively. Show that '

where g( t , x ) is continuously differentiable and satisfies Ilg(t,x)(12 I k11~11~, v t 2 m 0, V x E B, for some r > 0. Let P = PT > 0 be the solution of the Riccati equation

p, - p2 = exp(ATt)(I - Q ) e x p ( ~ t ) dt 5 0 P A + A ~ P + Q - P B B ~ P + ~ ( Y P = O

374 CHAPTER 9. STABILITY OF PERTURBED SYSTEMS

where Q > k?1 and a > 0. Show' that u = -BTPx stabilizes the origin of the perturbed system.

I 9.5 ([ lol l ) Consider the perturbed system

I x = A x + B u + D g ( t , y ) , y = C x 3

where g(t, y) is continuously differentiable and satisfies Ilg(t, y)(J2 5 k(lyl12, V t > 0, V ( ( ~ ( ( 2 < 7 for Some T > 0. Suppcise the equ,@tion

where Q = QT > 0, E > 0, and 0 < y < I l k has a positive definite solution P = pT > 0. Show that u = - ( 1 / 2 ~ ) B ~ P x stabilizes the origin of the perturbed system.


where a > 0, 0, y, and w > 0 are constants.

(a) By viewing this system as a perturbation of the linear system

show that the origin of the perturbed system is exponentially stable with (11x112 5 r ) included in the region of attraction, provided 1/31 and (y( are sufficiently small. Find upper bounds on 1/31 and 171 in terms of r.

(b) Using V(X) = sf + x i as a Lyapunov function candidate for the perturbed system, show that the origin is globally exponentially stable when /3 5 0 and exponentially stable with {JJxJJz < m) included in the region of attraction when 0.

(c) Compare the results of (a) and (b) and comment on the conservative nature of the result of (a).

9.7 Consider the perturbed system

s = f ( 4 + 9(x)

Suppose the origin of the nominal system x = f(x) is asymptotically (but not exponentially) stable. Show that, for any y > 0, there is a function g(x) satisfying ()g(x)ll < rllxll in some neighborhood of the origin such that the origin of the perturbed system is unstable.

9.7. EXERCISES 375

wlicrc j (x) and g(x) arc colitinuously differentiable and Jlg(x)JJ < yJIxJl for all llxll < r . Suppose the origin of the llolninal system j. = f(x) is asymptotically stable and t.l,ere is a Lyapunov functioll V(x) that satisfies inequalit.ies (9.11) t.hrough (9.13) for all IJ.rll < 1'. Lct = {If(.) 5 c)! with c < crl(r).

(a) Sl~ow that there is a positive col~si ant. y' such that, for y < y*. the solutions of tlie perturl>ed syste111 startillg ill stay in for all t 2 0 and are ultilnately bounded by a class K fullct ioil of y. .

(b) Suppose the no~ninal systcin liils t11c additional property that A = [6j/6x](0) is Hurwitz. Show that tllere is 7; such that, for y < y;. the solutiolls of the perturbed system starting in fl converge to the origin as t - w.

(c) \\'auld (b) hold if A was i ~ o t Hurwitz? Consider

Hint: For the exiumplc of part (c). ~lsc!

If (x) = .rf + i . r i + ixg + 11x3

to show that the origin of x = jS() is nspmpt.otically stable and t.hcn apply Thco- rem 4.16 t.o obtain a Lyapunov function that satisfies (9.11) through (9.13).

9.9 Collsidcr t,llc system

(a) With y = 0, sl~ow that the origill is globally asymptotically stable. Is it exponentially stable?

(b) With 0 < y 5 112: show that. the origin is unstable and the solutions of the system are globally ultimately bounded by an ultimate bound that is a class K function of y.

9.10 ([19]) Consider tlie system :

where a? b > a, c! and y arc positive consbants and q(t) is a continuous function.

(a) With q(t) = 0, use

V ( x ) = (b + ic2) x i + ~ ~ 1 x 2 +xi + 2 4 1 - cosxl)

to show that the origin is globally exponentially stable.

(b) Study the stability of the system when q( t ) # 0 and (q(t)l 5 k for all t 2 0.

9.11 Consider the system f

( a ) With b = 0 , show that the origin is exponentially stable and globally asymp totically stable.

(b) With b # 0, show that the origin is exponentially stable for sufficiently small lbl, but not globally asymptotically stable, no matter how small Jbl is.

(c) Discuss the results of parts (a) and (b) in view of the robustness results of 1 Section 9.1, and show that when 6 = 0 the origin is not globally exponentially stable.

9.12 ([S]) Consider the system 1 ( a ) Let b = 0. Show that the origin is globally asymptotically stable. Is it expo-

nentially stable? ! (b) Let b > 0. Show that the origin is exponentially stable for b < min{l,az). :i (c) Show that the origin is not .globally asymptotically stable for any b > 0.

( d ) Discuss the results of parts '(a) through (c) in view of the robustness results of Section 9.1, and show that when b = 0 the origin is not globally exponentially stable.

Hint: In part (d), note that the Jacobian matrix of the nominal system is not globally bounded.

9.13 Consider the scalar system x = - x / ( l + x Z ) and V ( x ) = x4. 1 (a) Show that inequalities (9.11) through (9.13) are satisfied globally with f

(b) Verify that these functions belong to class K,.

(c) Sliow t.lit\t tlic riglit-llcui(1 sick ol' (!l.lil) sppronc.11rs wro ns 1% + CQ. a

( d ) Consider the perturbed system i = - x / ( l + x 2 ) + 6, where 6 is a positive i 1

constant. Show that whenever 6 > 1/2, the solution x ( t ) escapes to w for any initial state x(0). 4

b 9.14 Verify that D+W(t ) satisfies (9.17) when V = 0. Hint: Show that V ( t + h , x ( t + h ) ) 5 0.5c4h2(lg(t, 0)(12 + h o(h), where o ( h ) / h 0 as h + 0. Then, use the fact that 2 1.

9.15 Consider the linear system of Example 9.6, but change the assumption on B ( t ) to jp IIB(t)ll dt < w. Show that the origin is exponentially stable.

I

9.16 Consider the linear system of Example 9.6, but change the assumption on B ( t ) to llB(t)112 dt < w. Show that the origin is exponentially stable. i

Hint: Use the inequality

I ) v ( t ) dt 5 \ l (b - a) [ G ( t ) dt, V u( t ) > 0

which follows from the Cauchy-Schwartz illcquality. .

9.17 Consider the system j. = A( t )x where A ( t ) is continuous. Suppose limt,, A(t) = A exists and A is Hurwitz. Show that the origin is exponentially stable.

9.18 Repeat part(b) of Exercise 9.10 when q(t) is bounded and q( t ) + 0 as t -, w.

9.19 Consider the system x = f ( t , x ) , where 11 f ( t , x ) - f (0, x)112 5 -y(t)llxll2 for all t 2 0 , x E R 2 , ~ ( t ) + 0 as t + 00,

and a, P, w, R are positive constants. Show that the origin is globally exponentially stable.

9.20 Consider the system x = f ( x ) + G(x)u + w(t ) , where I(w(t))lz 5 a + c e". Suppose there exist a symmetric positive definite matrix P , a positive semidefinite function 1Y(x), and positive constants 7 arid u such that

2 x T p f ( x ) + -yxTpx + W ( x ) - 2uxTPG(x)GT ( X ) P X 5 0, V x E Rn

Show that with u = -uGT(x)Px, the trajectories of the closed-loop system are uniformly ultimately bounded by 2akX,,(P)/.yXm1,(P), for some k > 1.

378 CHAPTER 9. STABILI??' OF PERTURBED SYSTEMS

9.21 Consider the perturbed system (9.1). Suppose there is a Lyapunov function V ( t , x ) that satisfies (9.11) through (9.13), and the perturbation term satisfies lJg(t,x)l( < 6( t ) , V t 2 0, V x E D. Show that for any E > 0 and A > 0 , thcrc exist q > 0 and p > 0 such that whenever ( l / A ) J,'+* 6 ( ~ ) d~ < q, every solution of the perturbed system with I(x(to))l < p will satisfy Ilx(t)ll < E , V t > to. (This result is known as total stability in the presence of perturbation that is

the time interval with sampling points at to + i A for i = 0,1 ,2 , . . ., and show that W(to + iA ) satisfies the difference inequality

9.22 Let A be an n x n matrix with ai j < 0 for all i # j and aii > CjZi (aij!, i = 1,2, . . . , n. Show that A is an M-matrix. Hint: Show that Cj",, ai j > 0 for i = 1 , . . . , n, and use mathematical induction to show that all the leading principal minors are positive.

9.23 Suppose the conditions of Theorem 9.3 are satisfied with

Show that the origin is exponentially stable.

9.24 ([132]) Study the stability of the origin of the system

by using composite Lyapunov analysis.

9.25 Study the stability of the origin of the system

by using composite Lyapunov analysis.

9.26 Consider the linear interconnected system

where, for each i, xi is an ni-dimensional vector and Aii is a Hurwitz matrix. Study the stability of the origin by usilig composite Lyapunov analysis.

9.27 ( [175]) Complex interconnected systems could be subject to structural perturbations that cause groups of subsystems to be connected or disconnected from each other during operation. Such structural perturbations can be represented a s

9.7. EXERCISES 379

\\,liere e i j is a I)ii~;~ry variilblc tli;~t t:~kcs the value 1 when t,he jth s~ll)syst,c!in tu:t.s on the ith subsystcin aiid the valuc 1) otherwise. The origin of tlic iiitcrco~i~rnct.c!tl systcln is said to I)c coniicctivcly ;rsyli11)tol.ic;~lly stable if it is ;~s.vliil,I.ol,i(:i~lly sl,id)l(: for a11 iiitercuiiiicctioli pattcriis. tlii~t is, for all possible valucs of the binary varia1)les ei j. Supposc that. all the assuinptioiis of Theorem 9.2 are satisfied, with (9.36) t- J i i ig the foiln m

Sllow that the origin is ronncctively asyniptot ically stable.

9.28 ((491) The output y(t) of the liliear system

is required to track a reference iiiput I . . Consider the integral controller

where have assurned that the state x can be measured and the matrices Fi and F2 can be designed suc11 that the inatrix

is Hurwitz.

( a ) Show that. if r = constant, then y(f) --t r as t -+ oo.

( b ) St.ucly the trackil~g properties olt.11~ syst,e111 when r ( t ) is a slowly varying input.

9.29 ( [86 ] ) The output y(t) of the nonlinear system

is required to track a reference inp11t r. Consider the int.egra1 colltroller

where we have assumed that the state x can be measured, tlw f\mction y can be designed such that t l ~ e closcd-lool) syst.c!111

has an exponentially stable equilibrium point (2, z), and the funct,ions f: h? and y are twice continuously differentiable in tlleir i argument.^.

( a ) Show that. if r = constant. a i d tlie initial state ( ~ ( 0 ) ~ ~ ( 0 ) ) is sufficiently close to ( f , f ) . then y(t) --t r as t --t x.

/ . . . . :. .. ,'?

CHAPTER 9. STABILITY OF PERTURBED SYSTEhfS

(b) Study the tracking properties of tlib systelll wllen ~ ( t ) is a slowly varying input.

9.30 ([86]) Consider the tracking problem of Exercise 9.29, but assume that we can only measure y = h(x). Consider the observer-based integral controller

i 1 = f ( n 1 u ) + G ( r ) b - h ( a ) ] , & = T - y , and u = ~ ( z ~ , z z , ~ )

Suppose 7 and G can be designed such that the closed-loop system has an exponentially stable equilibrium point (2, i l , i2). Study the tracking properties of the system when

(1) T = constant. (2) ~ ( t ) is slowly varying.

9.31 Consider the linear system x = A(t)x where I(A(t)\\ 2 k and the eigendues of A(t) satisfy Re[A(t)] 5 -o for all t 2 0. Suppose that so / ( ~ ( t ) l l ~ dt < p. Show that the origin of x = A(t)x is exponentially stable.

Chapter 10

Perturbation Theory and Averaging

Exact closed-form analytic solutions of nonlinear differential equations are possible only for a limited number of special classes of differential equations. In general, we have to resort to approximate solutions. There are two distinct categories of approximation methods that engineers and scientists should have a t their disposal as they analyze nonlinear systems: (1) numerical solution methods and (2) asymptotic methods. In this and the next chapter, we introduce the reader to some asynlpt,otic methods for'the analysis of nonlinear differential equations.'

Suppose we are given the state equation

where E is a "small" scalar parameter, and, under certain conditions, the equation has an exact solution d(t, E ) . Equations of this type are encountered in many applications. The goal of an asymptotic method is to obtain an approximate solution Z(t, E) such that the approximation error x ( t , E ) - Z(t, E) is small, in some norm, for small I E I and the approximate solution 5(t, E) is expressed in terms of equations simpler than the original equation. The practical significance of asymptotic methods is in revealing underlying structural properties possessed by the original state equation for small I E ~ . We will see, in Section 10.1, examples where asymptotic methods reveal a weak coupling structure among isolated subsystems or the structure of a weakly nonlinear system. More important, asymptotic methods reveal multiple-t.imoscale structures inherent in many practical problems. Quite often, the solution of the state equation exhibits the phenolnenon that some variables move in time faster than other variables, leading ta the classification of variables as "slow" and "fast." Both the averaging method of this chapter and the singular perturbation method of the next chapter deal with the interaction of slow and fast variables.

'N~~rnericsl solution methods are not stutlied in illis textbook on !Ile premise lhat most studcnts arc introdllcctl Lo t1le111 in elel~lentary diflerential equation courses and they get their in-depth study of the subject in numerical analysis courses.

382 CHAPTER 10. PERTURBATION AND AVERAGING "?p",

Section 10.1 presents the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The asymptotic .,

validity of the approximation is established in Section 10.1 on finite time intervals and in Section 10.2 on .the infinite-time interval. Section 10.3 examines an autonomous system under the iduence of a weak periodic perturbation. While the results of the f i s t three sections are interesting in their own sake, they provide the technical basis for the averaging method. In Section 10.4, we introducc the averaging method in its simplest form, which is sometimes called "periodic averaging" since the right-hand side function is periodic in time. Section 10.5 gives an application of the averaging method to the study of periodic solutions of weakly nonlinear second-order systems. Finally, we present a more general form of the averaging $? method in Section 10.6. ,\@>

10.1 The Perturbation Method e Q

Consider the system k = f ( t , X , € )

where f : [to, t l ] x D x [ - E ~ , €01 -' Rn is "sufficiintly smooth" in its arguments over ik a domain D C Rn. The required smoothness conditions will be spelled out as we proceed. Suppose we want to solve the state equation (10.1) for a given initial state

where, for more generality, we allow the initial state to depend "smoothly" on f E . The solution of (10.1) and (10.2) depends on the parameter E , a point that we I' emphasize by writing the solution as x( t , E) . The goal of the perturbation method is to exploit the "smallness" of the perturbation parameter E to construct approximate solutions that are valid for sufficiently small lal. The simplest approximation results + by setting E = 0 in (10.1) and (10.2) to obtain the nominal or unperturbed problem 5

where qo = q(0). Suppose this problem has a unique solution xo(t ) defined on [ to,t l ] and xO( t ) E D for all t E [ tO l t l ] . Suppose further that f is continuous in ( t , x , ~ ) and locally Lipschitz in (x , E ) , uniformly in t , and q is locally Lipschitz in E

for ( t , x , E ) in [to, t l ] x D x [-Eo, E O ] . The closeness of the solutions of the perturbrd and unperturbed problems follows from continuity of solutions with respect to initial states and parameters. In particular, Theorem 3.5 shows that there is a positive constant €1 5 EO such that for all J E I 5 ~ 1 , the problem of (10.1) and (10.2) has a unique solution x( t , E ) defined on [to, t l ] . Furthermore, Theorem 3.4 shows that there is a positive constant k such that

10.1. THE PERTURBATION AIETHOD 353

When the approxilnatioll crror s;ltislics the bound of (10.4), we say that the error is of order O ( E ) and write

. K ( / . E ) - .Ucr(t) = O ( E )

This crder of rnngilitude notatiou will I,e used frequently in this chapter and the next one. It is defined nest.

Definition 10.1 61 ( E ) = O ( & ( E ) ) if there exist positive constan.ts k and c such that

Example 10.1

E" = O ( E ~ ) for all n 1 m? since

I E J " = J E ~ " ' J E ( ~ - ~ < / E l r n , v I € ( < 1

. ~ ~ l ( 0 . 5 + E ) = O ( E ~ ) : since

. i + 2~ = O(1) . since

11 + 2 ~ 1 < 1 + 2a, V J E ( < a

exp(-U/E) \vith positive a all11 5 is O ( c n ) for ally positivc irltcger n, since

\\:hat can we say about the ~lulllrrical value of the approsilnatiun error z( t , E ) - ~ ( t ) for a given iiurnerical value OI.E wheli the error is O(E) ' ! Uiifortullately! we callnot trailslate tile O ( E ) ortlcr u l ~l l i l~l~i tudc stillc~~lcllt ill1 o ;I 11111ilc:ri~;l1 1)01111(1

on the prror. Knowing that the error is O ( E ) means t,hat it.s norin is less than ~ ( E J for some positive constalit k tliat. is independent of E. However, we do not know the value of k? which rniglit>be 1: 20, or any positive The fact that k is independent of E guar.antees that the bound k l ~ l decreases rnoiiotoliically a. J E ~ decreases. Therefore, for sufficiellt.ly small IE(, the error will be small. hlore precisely, given any tolerance 6: we kuow that the norm of the rrror will be less - -

21t should be noted. however, that in a well-formulated perturbation problen~ where variables are normalized to have dimensionless slntc variables, time, and perturbatio~~ parameter, one should expect the numerical value of k not t o bc nll~ch larger than one. See Example 10.4 for further discussion of normalization, or consult 198) and [141] for more examples.

than 6 for all ( E I < 6lk. If this ralige is too sillall to cover the numerical values of Q

interest for E, we then need to extend the range of validity by obtaining a higher 4 order approximation. An O(.c2) approximation will meet the same 6 tolerance for all J E ~ < $&, an O ( E ~ ) approximation will do it for all ~ E I < (6/k3)'13, and so on. Although the constants k, k2, k3,. . . are not necessarily equal, these intervals are increasing in length, since the tolerance 6 is typically much smaller than one. Another way to look at higher order approximations is to see that, for a given "sufficiently small1' value of E, an O(E") error will be smaller than an O ( E ~ ) error for n > m, since

l/(n-m)

Higher-order approximations for sol~~tions of (10.1) and (10.2) can be obtained in a straightforward manner, provided the functions f and q are sufficiently smooth. Suppose f and q have continuous partial derivatives with respect to (x, E) up to order N for ( t , x , ~ ) E [to, tl] x D x [ - E ~ , E ~ ) TO obtain a higher order approximation of x(t, E), we construct a finite Taylor series

N-1

x ( t , ~ ) = C x k ( t ) ~ ~ + E ~ & ( ~ , E ) (10.5) k=O

Two things need to be done here. First, we need to calculate the terms XO, XI, . . . , XN-~; in the process of doing that, it will be shown that these terms are well defined. Second, we need to show that the remainder term R, is well defined and bounded on [to, tl], which will establish that x k ( t ) ~ ~ is an O(cN) (Nth-order) approximation of x(t, E). By Taylor's theoremI3 the smoothness requirement on the initial state q ( ~ ) guarantees the existence of a finite Taylor series for q ( ~ ) ; that is,

Therefore, Xk(tO) = qk, k = 0,1,2,. . . , N - 1

Substituting (10.5) into (10.1) yields 4

where the coefficients of the Taylor series of h(t, E) are functions of the c0efficient.s of the Taylor series of x(t, E). Since (10.6) holds for all sufficiently small E, it must hold

3See [lo, Theorem 5-14].

as an identity in E. Helice, coetficicnts of likc po\\.vrs of E ~ilust be equal. hlatching those coefficients, we can derive the equations that must be satisfied by xo, xl, and so on. Before we do that, we have to generate the coefficients of the Taylor series of h(t, E). The zeroth-order term h,-,(t) is given by

ho(t) = f(tI~0(t)lO)

Consequently, matching coefficients of EO in (10.6), we determine that xo(t) satisfies

which, not surprisingly, is the unperturbed problem (10.3). The first-order term hl(t) is given by

Matching coefficients of E in (10.6), we find that xl(t) satisfies

af ~1 = - P I xo(t), 0) X l + $(tI xo(t), o), , x1(t0) = q1 ax

Define

ax a f A ) = t I 0 0 ) g1(t, xo(t)) = -(t, xo(t), 0) a& and rewrite the equation for zl as

This linear equation has a unique solution defined on [to, tl]. The process can be continued to derive the equations satisfied by x2, xa, and

so on. This, however, will involve higher order differentials o f f with respect to x, which makes the notation cumbersome. There is no point in writing the equations in a general form. Once the idea is clear, we can generate the equations for the specific problem of interest. Nevertheless, to set the pattern that these equations take, we will, a t the risk of boring some readers, derive the equation for x2. The second-order coefficient in the Taylor series of h(t, E) is given by

386 CHAPTER 10. PERTURBATION AND AVERAGING

Now,

To simplify the notation, let

and continue to calculate the second derivative of h with respect to E:

Thus, h2(t) = A ( t ) x ~ ( t ) + g2(t, xo(t)r XI(^))

where

Matching coefficients of E' in (10.6) yields

In summary, the Taylor series coefficients so, X I , . . . , X N - 1 are obtained by solving the equations

ko = f ( t , xo, O ) , xo(to) = 770 (10.7)

5k = A(t)zk + gk(t,x0(t)1 xk-l(t)), xk(t0) = 771; (10.8)

for k = 1,2, ... ,N - 1, where A(t) is the Jacobian [a f lax] evaluated at x = xo(t) and E = 0, and the term gk(tr xO(t)! xl( t) , . . . , xk-l(t)) is a polynomial in X I , . . . ,xk-1 with coefficients depending continuously on t and xo(t). The assump tion that xO(t) is defined on [to, t l ] implies that A(t) is defined on the same interval; hence, the linear equations (10.8) have unique solutions defined on [to,tl]. Let us now illustrate the calculation of the Taylor series coefficients by a second-order csmplc.

10.1. THE PERTURBATION i\fE?'HOL)

Example 10.2 Consider the Van drr Pol state equation

.fl = n . ~ . xl(0) = 7 1 ( ~ )

~2 = -Xi f ~ ( 1 - . T ~ ) x z , ~ ~ ( 0 ) = 7 p 2 ( E )

Suppose lie want to construct a finite Taylor series with N = 3. Let

X i = X,IJ + E X 1 ] + E2x,2 E3&,, i = 1,2

and

qi = 7i0 + E l l i l + E27i2 + E ~ R ? ~ , i = 2

.f- Substituting the series for xl alirl 2-2 into the state equation results in

x (a20 + E X Z ~ + ~ ~ ~ 2 2 + E ~ R ~ ~ )

hlatching coefficic~its of EO: ~ v e obt,ai~i

which is the unperturbed probleln at z = 0. hlatching coefficicnts of E. we obtain

while matching coefficients of E' results ill

x22 = -xl2 f (1 - x:o)x2l - 2 ~ 1 0 ~ 1 1 ~ 2 0 ~ ~22(0 ) = 722

The lat.ter two sets of equations are in t.he form of (10.8) for k = 1: 2. A

Having calculated t.he terms xo, X I : . . . , X N - 1 , our task now is to show t,hat ~ r s ~ x k ( t ) ~ % indeed an O ( E ~ ) approximation of x(t , E ) . Consider the approximation error

N-1

.! &. ; &.

'id- ! &

;c. . .

r t..

. -. -. -. - . - .-. -- . /

388 CIIAPTER 10. PERTURBATION A N D ' A M C I N G '

Differentiating both sides of (10.9) with respect to t and substituting for the derivatives of x and xk from (10.1), (10.7), and (10.8), it can be shown that e satisfies the equation

c = A(t)e+ pl(t,e,€) + pz(t,E), e(to) = E ~ R ? ( & ) (10.10) i

where 9

By assumption, xo(t) is bounded and belongs to D for all t E [to, tl]. Hence, there exist X 1 0 and el > 0 such that for all llell <- X and \&I1. <- €1, the functions xo(t), z;=o1 xk(t)rk, and e + ~ ~ ( t ) & ~ belong to a compact subset of D. It can be

I - . .

easily verified that

( l ~ l ( t , e z , ~ ) - PI(^, el,E)II 5 k~llez - ell1 (10.12)

Ilpz(t,&)I\ -< h!€IN (10.13)

for all t E [to, tl], el,e2 E BA, E E [ - ~ l , & l ] , for some positive coytant. kl and k2. Equation (10.10) can be viewed as a perturbation of

Co = A(t)eo + pl(t, eo, E), eo(t0) = 0 (10.14)

which has the unique solution eo(t,s) = 0 for t E [to, tl]. Applying Theorem 3.5 shows that (10.10) has a unique solution defined on [to, tl] for sufficiently small 161. Applying Theorem 3.4 shows that

We summarize our conclusion in the next theorem.

Theorem 10.1 Suppose .$

f and its partial derivatives vnth respect to (x, E ) up to order N are continuow in (t, 2, E) for (t, x, E) E [to, tll x D x [-€0, €01; T,- and its derivatives up to order N are continuorn for E E [-EO,EO];

the nominal problem given in (10.3) has a unique solution xo(t) defined on [to, tl] and xo(t) E D for all t E [to, ti].

Tlren, tlrei-e exists E* > 0 stir11 thrrt V It) < e*, tltt, 1)tnblctti !)ituS11 I I ~ (10.1) ui~ti (10.2) lras a ui~iqzre sol~rtioil x(t,e), defiiied on [to, tl], wlrich satisfies

When we approximate x(t, E ) by xo(t), we need not know the value of the parameter E, which could be an unknown parameter that represents deviations of the system's parameters from their nominal values. When we use a higher order a p proximation z,"=ol x k ( t ) ~ ~ for N 2 2, we need to know the value of E to construct the series, even though we do not need it to calculate the terms X I , x2, and so on. If we have to know E to construct the Taylor series approximation, we must then compare the computational effort needed to approximate the solution via a Taylor series with the effort needed to calculate the exact solution. The exact solution x(t, E ) can be obtained by solving the nonlinear state equation (10.1), while the approximate solution is obtained by solving the nonlinear state equation (10.7) and a number of linear state equations (10.8), depending on the order of the approximation. Since, in both cases, we have to solve a nonlinear state equation of order n, we must ask ourselves, What do we gain by solving (10.7) instead of (10.1)? One situation where the Taylor series approximation will be clearly preferable is the case when t i e solution is sought for several values of E. In the Taylor series approximation, equations (10.7) and (10.8) will be solved only once; then, different Taylor expansions will be constructed for different values of E. Aside from this special (repeated values of E )

case, we find the Taylor series approximation to be effective when

r the unperturbed state equation (10.7) is consitlerilbly simpler than the E-dependent state equation (10.1),, and

E is reasonably small that an "acceptable" approximation can be achieved with a few terms in the series.

In most engineering applications of the perturbation method, adequate approximac tions are achieved with N = 2 or 3, and the process of setting 6 = 0 simplifies the state equation considerably. In the next two examples, we look at two typical cases where setting E = 0 reduces the complexity of the state equation. In the first example, we consider again the Van der Pol equation of Example 10.2, which represents a wide class of "weakly nonlinear systems" that become linear at E = 0. To construct a Taylor series approximation, we only solve linear equations. In the second example, we look at a system formed of interconnected subsystems with ''weak" or E-coupling. At E = 0, the system decomposes into lower order decoupled subsystems. To construct a Taylor series approximation, we always solve lower order decoupled equations as opposed to solving the original higher order equation (10.1).

CHAPTER 10. PERTURBATION AND AVERAGING

Figure 10.1: Example 10.3 at E = 0.i: (a) xl(t, E) (solid) and xlo(t) (dashed); (b) xl(t, E) - xlO(t) (solid) and xl(t, E) - xlo(t) - EXII ( t ) (dashed).

Example 10.3 Suppose we want to solve the Van der Pol equation

over the time interval [0, T ] . We start by setting E = 0 t,o obtain the linear unperturbed equation

xl0 '= 5201 510(0) ' 1

k2" = -510, x20(0) = 0

whose solution is xlo(t) = cost, xZ0(t) = - sint

Clearly, all the assumptions of Theorem 10.1 are satisfied, and we conclude that the approximation error x(t, E ) - xo(t) is O(E). Calculating x(t, E) numerically a t three different values of E and using

. .

Eo = max Ilx(t, €1 - xo(t)ll2 a<t<_rr

' as a measure of the approximation error, we find that Eo = 0.0112, 0.0589, and 0.1192 for E = 0.01, 0.05, and 0.1, respectively. These numbers show that the error is bounded by 1.26 for E < 0.1. Figure lO.l(a) shows the exact and approximate trajectories of the first component of the state vector when E = 0.1. Suppose we want to improve the approximation a t E = 0.1. Rom Example 10.2, we know t,hat x l l and xzl satisfy the equation

10.1. THE PERTURUAI'ION i\IEl'HULJ oJ1

Figure 10.2: Electric circuit of Example 10.4.

whose, solution is

By Theorem 10.1, the second-order approximation xo(t) + &xl(t) is O($) close to the exact solution for sufficiently sniall E . To compaie the approximate solution wit.h thc cract one at E = 0.1, ~ve cnlculatc

1 which shows a reduction in the approximation error by almost an order of magnitude. Figure lO.l(b) shows the approxilnation errors in the first coinpollent of thc state vector for the first-order approximation x0 and the second-order approximation s o + €21 at E = 0.1. A

Example 10.4 The circuit shown in Figure 10.2 contains nonlinear resistors whose I-V characteristics are given by i = @(v). The differential equations for the voltages across the capacitors are

The circuit has two similar RC sections connected through' the resistor R,. When R, is "relatively large," the colinection between the two sections becomes "weak." In particular, when R, = oo, the connection is open circuit and the two sections are decoupled from each other. This circuit lends itself to an E-coupling representation where the coupling between the t,wo sections may be parameterized by a small

p;traiiic!t.cr E. At. first glance, it appcars t.hat. a reasonable choice of E is E = l/Rc. Indeed, with this choice, the couplil~g terins in the foregoing equations will be multiplied by E. However, such achoice makes E dependent on the absolute value of a physical parameter whose value, no rnatter how small or large, has no significance by itself without considering the values of other physical parameters in the system. In a well-formulated perturbation problem, t.he parameter E would be chosen as a ratio betwcci~ physical parameters that rcflects the "true smallness" of E in a relative sense. To choose E this way, we usually start by choosing the state variables or the time variable (or both) as dimensionless quantities. In our circuit, the clear choice of st,at.e variables is vl and v2. Instead of working with vl and v2, we scale them in such a way that the typical extreme values of the scaled variables would be close to ztl. Duo to thc weak coupling betwrnn tlie two identical sections, it is reasonable to use the same scaling factor a for both state variables. Define the state variables as xl = v l / a and x2 = v2/a. Taking a dimensionless time T = t /RC and writing dxldr = i-, we obtain the state equation

It appears now that a reasonable choice of E is RIR,. Suppose that R = 1.5 x lo3 0, E = 1.2 V, and the nonlinear resistors are tunnel diodcs with

Take a = 1 and rewrite the state equation as

where h(v) = 1.5 x lo3 x $(TI). Suppose we want to solve this equation for the initial state

x1 (0) As0.15; ~ ~ ( 0 ) = 0.6

X Setting E = 0, we obtain the decoupl d equations

kl =. 1.2 -z l - h(r l ) , ~ ~ ( 0 ) = 0.15

x2 = .1 .2-x2-h(x2)? x~(O) = 0.6

which are solved independdntly of each other. Let xlo(t) and xzo(t) be the soh- tions. According to Theorem 10.1, tliey provide an O(E) approximation of the exact solution for sufficiently small E. To obtain an O ( E ~ ) approximation, we set up the equations for 211 and x21 as

. - - ---.- --"-. - --.- . I

5 a - ',---- - 6, ' . - . - 20.2. P E R ? ~ B A T ~ O N 6 ~ i ' l r ~ INIZINI'I'E INTERVAL 393

0.05 I 0 0.5 1 1.5 2

Time

Figure 10.3: Exact solution (solid), first-order approximation (dashed), and second- order approximation (dash-dot) for Example 10.4 at E = 0.3.

where hl( . ) is the derivative of h(.). Figure 10.3 shows the exact solution as well as the first-order and second-order approximations for E = 0.3. A

A serious limitation of Theorem 10.1 is that the O ( E ~ ) error bound is valid only on finite (order O(1)) time intervals [to, tl]. It does not hold on intervals like [ to,T/~] nor on the infinite-time interval [to,oo). The reason is that the constant k in the bound kJ€JN depends on tl in such a way that it grows unbounded as t l increases. In particular, since the constant k results from application of Theorem 3.4, it has a component of the form exp(Lt1). In the next section, we will see how to employ stability conditions to extend Theorem 10.1 to the infinite interval. In the lack of such stability canditions, the approximation may not be valid for large t, even though it is valid on O(1) time intervals. Figure 10.4 shows the exact m d approximate solutions for the Van der Pol equation of Example 10.3, at E = 0.1, over a large time interval. For large t, the error xl(t, E ) - xlo(t) is no longer O(E). bo re seriously, the error x l (t, E) - xlo(t) - ~ x 1 1 (t) grows unbounded, which is a consequence of the term t cos t in 311 (t).

10.2 Perturbation on the Infinite Interval

The perturbation result of Theorem 10.1 can be extended to the infinite time interval [to, oo) under some additional stability conditions. In the next theorem, we require the nominal system (10.7) to have an exponentially stable equilibrium point at the origin and use a Lyapunov function to estimate its region of attraction. There is no loss of generality in taking the equilibrium point a t the origin, since my equilibrium point can be shifted to the origin by a change of variables.

Theorem 10.2 Let D C Rn be a domain that contains the origin and suppose

CHAPTER 10. PERTURBATION AND AVERAGING 10.2. PERTURBATION ON THE INFINITE IN W R V A L 395

Figure 10.4: Exact solution (solid), first-order approximation (dash-dot), and second- order approximation (dashed) for the Van der Pol equation over a large time interval.

f and its partial derivatives with respect to (2 , E ) up to order N are continzlous and bounded for ( t , x , ~ ) E [O,oo) x Do x [ - E O , E O ] , for every compact set Do c D; if N = 1, [a f /ax](t , x , E ) is Lipschit2 in (1, E ) , unifonnly in t;

17 and its derivatives up to order N are continuous for E E [-.so, €01;

the origin is an exponentially stable equilibiizrm point of the nominal system (10.7);

there is a Lyapunov function V ( t , x ) that satisfies the conditions of Theo- rem 4.9 for the nominal system (10.7) for ( t , x ) E [O, oo) x D and { lVl (x) 5 c ) is a compact subset of D.

Then, for each compact set R c {1.V2(x) 5 pc, 0 < p < I ) , there is a positizie constant E* such that for all to 2 0, 770 E R, and ( € 1 < E * , eqlrations (10.1) and (10.2) have a unique solution x(t , E ) , unifonnly bounded on [to, oo), an,d

where O ( E ~ ) holds uniformly in t for all t 2 to. 0

If the nominal system (10.7) is autonomous, the set R in Theorem 10.2 can be any compnct subset of the regio~i of at.tract,ion of the origin. This is a co~ise- quence of (the converse Lyapunov) Theorem 4.17. since the Lyapunov function V ( x ) provided by the theorem has the property that any compact subset of the region of attraction can be included in the interior of a compact set. of the form { V ( x ) 5 c).

Proof of Theorem 10.2: Application of Tlirorriii 9.1 sllows that tlirrr is 51 > 0

such that for all [el < .ell x ( t , ~ ) is uniformly bounded and x ( t , ~ ) - xo(t) is O(E) , uniformly in t , for all t 2 to. It is also clear that for qo E R, xo(t) is uniforn~ly bounded and limr,, xo(t) = 0. Col~sitlcr the linear equations (10.8). Wc know from bounded-input-bounded-output. stability (Theorem 5.1) that the solution of (10.8) will be uniformly bounded if the origin of i.= A(t)z is exponentially stable and the input term gk is bounded. The input gk is a polynomial in X I , . . . , xk-1 with coefficients depending on t arid xo(t) . The dependence on t comes through the partial derivatives of f , which are hounded on compact subsets of D. Since xo(t) is bounded, the polynon~ial coefficients are'bounded for all t 2 to. Hence, boundedness of gk will follow from boundedness of xi, . . . , xk-1. The matrix A(t ) is given by

where xo(t) is the solut,ion of the nominal system (10.7). It turns out that exponential st.ability of the origin as an equilibrium for (10.7) ensures that the origin of i = A(t)z will be exponentially stable for every solution xo(t) that starts in the set R. To see this point, let

a f Ao(t) = z ( t , O,0)

and write , A(t) = Ao(t) + [A( t ) - Ao(t)] ef Ao(t) + B( t )

so that the linear system i = A(t)z can be viewed as a linear perturbation of 3i = Ao(t)y. Since (af/ax](t ,x ,O) is Lipschitz in x, uniformly in t ,

On the other hand. by exponential stnbility of the origin of (10.7) and Theorem 4.15, we know that the origin of the linear system a = Ao(t)y is exponentially stable. Therefore, similar to Example 9.6. we can use limt,, xo(t) = 0 to show that the origin of the linear system i = A(t)z is exponentially stable.

Since Ilxo(t)() is boulided alid gl(t, t o ( t ) ) = [Df/aE](t, xo(t), O), wc sac that yl is bounded for all t 2 to. Hence, by Theorem 5.1, we conclude that x l ( t ) is bounded. By a simple induction argument, we can see that x2(t), . . . , ~ ~ - ~ ( t ) are bounded.

So far, we have verified that the exact solution x(t , E ) and the approximate solution ~ f ! : xk(t)Ek are uniformly bounded on [to, oo) for sufficiently small I E I . All that remains now is to analyze the approximation error e = x - C~G' x k ( t ) ~ ~ . The error analysis is quite similar to what we have done in Sectioli 10.1. The error satisfies (10.10), where pl and p l satisfy (10.11), (10.13) and

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H K Khalil Nonlinear Systems 3rd Edition 2002