Nonlinear Systems 2nd- Hassan K. Khalil

Chapter 4

Advanced Stability Analysis

In Chapter 3, 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 Chapter 3 how to use linearization to study stability of equilibrium points of an autonomous system. We saw also that linearization fails when the Jacobian matrix, evaluated at the equilibrium point, has some eigenvalues with zero real parts and no eigenvalues with positive real parts. In Section 4.1, we introduce the center manifold theorem and use it to study stability of the origin of an autonomous system in the critical case when linearization fails.

The concept of the region of attraction of an asymptotically stable equilibrium point was introduced in Section 3.1. In Section 4.2, we elaborate further on this 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 was presented in Theorem 3.4. There are, however, theorems which capture some features of the invariance principle. Two such theorems are given in Section 3.6. The first theorem shows convergence of the trajectory to a set, while the second one shows uniform asymptotic stability of the origin.

4.1 T h e Center Manifold Theorem Consider the autonomous system

x = f (x) (4.1) where f : D + Rn is continuously differentiable and D is a neighborhood of the origin x = 0. Suppose that the origin is an equilibrium point of (4.1). Theorem 3.7

168 CHAPTER 4. ADVANCED STABILITY THEORY

states that if the linearization o f f a t 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 (4.1) in order to determine stability of the origin. The interesting finding that we shall present in the next few pages is that stability properties of the origin can actually be determined by analyzing a lower-order nonlinear system, whose 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 definition.' For our purpose here, it is sufficient to think of a k-dimensional manifold as the solution of the equation

v(x) = 0 where q : 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 {q(z) = 0) is said to be an invariant manifold for (4.1) if

q(x(0)) = 0 + q(z(t)) 0, V t E [O, t l) C R where [0, t l ) is any time interval over which the solution x(t) is defined.

Suppose now that f (x) is twice continuously differentiable. Equation (4.1) can be represented as

'The center manifold theory has several applications to dynamicalsysterns. It is presented here only insofar as it relates to determining the stability of the origin. For a broader viewpoint of the theory, the reader may consult [28].

'See, for example, [60].

4.1. T H E CENTER MANIFOLD THEOREM

where a f P(") = f(") - az(o) "

is twice continuously differentiable and

Since our interest is in the case when linearization fails, assume that A has R eigenvalues with zero real 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, Al is k x k and A2 is m x m. The change of variables

transforms (4.1) into the form

where gl and gz inherit properties off . In particular, they are twice continuously differentiable and

for i = 1,2. If z = h ( y ) is an invariant manifold for (4.2)-(4.3) and h is smooth, then it is called a center manifold if

Theorem 4.1 If g, and g2 are twice continuously differentiable and satisfy (4.4), all eigenvalues of A1 have zero real parts, and all eigenvalues of A2 haue negative real pads, then there exist 6 > 0 and a continuovsly differentiable function h ( y ) , defined for all [ly[[ < 6 , such that z = h ( y ) is a center manifold for (4.2)-(4.3). 0 Proof: Appendix A.7


If the initial state of the system (4.2)-(4.3) 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; 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

which we shall refer to as the reduced system. If z (0 ) # h(y(O)), then the difference z ( t ) - h ( y ( t ) ) represents the deviation of the trajectory from the center manifold a t any time t . The change of variables

transforms (4.2)-(4.3) into

In the new coordinates, the center manifold is w = 0. The motion in the manifold is characterized by

w ( t ) z 0 ~ ( t ) 0 Substitution of these identities in (4.7) results in

Since this equation must he satisfied by any solution which lies in the center manifold, we conclude that the function h ( y ) must satisfy the partial differential equation (4.8). Adding and subtracting g ~ ( y , h ( y ) ) to the right-hand side of (4.6), and suh- tracting (4.8) from (4.7), we can rewrite the equation in the transformed coordinates as

Y = A I Y + ~ I ( Y , ~ ( Y ) ) + N I ( Y , ~ ) (4.9) w = Azw + Nz(y , w ) (4.10)

where N I ( Y , ~ ) = ~ I ( Y , w + W Y ) ) - ~ I ( Y , h ( y ) )

and d h

Nz(Y ,w) = ~ z ( Y > w + h ( y ) ) - g 2 ( ~ , h ( y ) ) - -(y) N I ( Y , w ) 8 ~

4.1. THE CENTER MANIFOLD THEOREM 171

It is not difficult to verify that N1 and N2 are twice continuously differentiable, and

for i = 1,2. Consequently, in the domain

N1 and N2 satisfy N ( Y ) 5 I i = 192

where the positive constants k1 and k2 can be made arbitrarily small by choosing p small enough. These inequalities, together with the fact that A2 is Hurwitz, suggest that the stability properties of the origin are determined by the reduced system (4.5). The following theorem, known as the reduction principle, confirms this conjecture. Theorem 4.2 Under the assumptions of Theorem 4.1, if the origin y = 0 of the reduced system (4.5) is asymptotically stable (respectively, unstable) then the origin of the full system (4.2)-(4.3) is also asymptotically stable (respectively, unstable).

0

Proof: The change of coordinates from (y, z) to (y, w) does not change the stability properties of the origin (Exercise 3.27); therefore, we can work with the system (4.9)-(4.10). If the origin of the reduced system (4.5) is unstable, then by invariance, the origin of (4.9)-(4.10) is unstable. In particular, for any solution y(t) of (4.5) there is a corresponding solution (y(t),O) of (4.9)-(4.10). Therefore, stability of the origin of (4.9)-(4.10) implies stability of the origin of (4.5). This is equivalent t o saying that instability of the origin of (4.5) implies instability of the origin of (4.9)-(4.10). Suppose now that the origin of the reduced system (4.5) is asymptotically stable. By (the converse Lyapunov) Theorem 3.14, there is a continuously differentiable function V(y) which is positive definite and satisfies the following inequalities in a neighborhood of the origin

where a3 and a4 are class X functions. On the other hand, since A2 is Hurwitz, the Lyapunov equation

PA2 + ATP = -I


has a unique positive definite solution P. Consider

u ( y , w ) = V ( y ) + L z G as a Lyapunov function candidate3 for the full system (4.9)-(4.10). The derivativr of u along the trajectories of the system iq given by

av i . (y, w ) = - [ A I Y + ~I(Y, h ( y ) ) + N I ( Y , w ) ] a~

1 + 2-

[ w T ( p ~ z + A T P ) ~ + 2wTPN2(y , w ) ]

Since kl and kz can be made arbitrarily small by restricting the domain around the origin to be sufficiently small, we can choose them small enough to ensure that

Hence,

which shows that u(y , w ) is negative definite. Consequently, the origin of the full system (4.9)-(4.10) is asymptotically stable. 0

We leave it to the reader (Exercises 4.1 and 4.2) to extend the proof of Theo- rem 4.2 to prove the following two corollaries.

Corollary 4.1 Under the assumptions of Theorem 4.1, if the origin y = 0 of the reduced system (4 .5) i s stable and there is a continuously differentiable Lyapunov

3The function v(y,w) is continuously differentiable everywhere around the origin, except on the manifold w = 0. Both v ( y , w ) and "(9, w ) are defined and continuous around the origin. It can be essily seen that the statement of Theorem 3.1 is still valid.


function V ( y ) such that

in some neighborhood of y = 0 , then the origin of the full system (4.2)-(4.3) is stable. 0

Corollary 4.2 Under the assumptions of Theorem 4.1, the origin of the reduced system (4.5) is asymptotically stable if and only if the origin of the full system (4.2)-(4.3) is asymptotically stable. 0

To use Theorem 4.2, we need to find the center manifold r = h(y ) . The function h is a solution of the partial differential equation

d e f a h N ( h ( y ) ) = -(y) [AIV + ~ I ( Y , h(y))l - Azh(y ) - ~ J Z ( Y , h ( ~ ) ) = 0 (4.11) a y

with boundary conditions a h

h(0) = 0; -(0) = 0 OY

This equation for h cannot be solved exactly in most cases (to do so would imply that a solution of the full system (4.2)-(4.3) has been found), but its solution can be approximated arbitrarily closely as a Taylor series in y.

Theorem 4.3 If a continuously differentiable function 4 ( y ) wath 4(O) = 0 and [a$/ay](O) = 0 can be found such that N ( # ( y ) ) = O(IIyIIP) for some p > 1, then for suficiently small llyll

MY) - Q ( Y ) = ~ ( I I Y I I ~ ) and the reduced system can be represented as

Proof: Appendix A.7

'The existence of the Lyapunov function V(y) cannot be inferred from a converse Lyapunov theorem. The converse Lyspunov theorem for atabjlity [62, 961 guarantees the existence of a Lyapunov function V(t, y) whose derivative satisfies V(t, y) 5 0. In general, this function cannot be made independent o f t [62, page 2281. Even though we can choose V(t,y) to be continuously differentiable in its arguments, it cannot be guaranteed that the partial derivatives aV/ayi, aV/at will be uniformly bounded in a neighborhood of the origin for all t > 0 [96, page 531.

174 CHAPTER 4 . ADVANCED STABILITY THEORY

The order of magnitude notation O ( . ) will be formally introduced in Chapter 8 (Definition 8.1) . For our purpose here, it is enough to think of f ( y ) = O(llyllP) as a shorthand notation for ( I f ( y ) ( J 5 lCllyllP for sufficiently small ilyll. Let US now illustrate the application of the center manifold theorem by examples. In the first two examples, we shall make use of the observation that for a scalar state equation of the form

Y = a y + 0 (IylP+l) where p is a positive integer, the origin is asymptotically stable if p is odd and a < 0 . It is unstable if p is odd and a > 0 , or p is even and a # 0.5 Example 4.1 Consider the system

X I = 2 2

xz = - 2 2 + a z f + bzlz2 where a # 0 . The system has a unique equilibrium point at the origin. The linearization at the origin results in the matrix

which has eigenvalues at 0 and -1 . Let M he a matrix whose columns are the eigenvectors of A; that is,

M = [ ' 0 -1 ' 1 and take T = M-' . Then,

The change of variables

puts the system into the form

The center manifold equation (4.11)-(4.12) becomes N ( ~ ( Y ) ) = ~ ' ( Y ) [ Q ( Y + h ( ~ ) ) ~ - b(yh(y) + h 2 ( y ) ) ] + h ( y )

+ "(Y + h ( ~ ) ) ' - b(yh(y) + h 2 ( y ) ) = 0 , h (0 ) = h f ( 0 ) = 0 "ee Exercise 3.2.


We set h ( y ) = hzy2 + h3?/3 + . . . and snbstitute this series in the center manifold equation to find the unknown coefficients h z , hs , . . . by matching coefficients of like powers in y (since the equation holds as an identity in y) . We do not know in advance how many terms of the series we need. We start with the simplest approximation h ( y ) k 0 . We snbstitute h ( y ) = O ( l y ( 2 ) in the reduced system and study stability of its origin. If the stability properties of the origin can be determined, we are done. Otherwise, we calculate the coefficient h z , substitute h ( y ) = hzy2 + O(JyI3) , and study stability of the origin. If it cannot he resolved, we proceed to the approximation h ( y ) = hzy2 + hsy3, and so on. Let us start with the approximation h ( y ) w 0 . The reduced system is

Notice that an O ( l y l z ) 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 g l ( y , r ) (which appears on the right-band side of the reduced system (4.5) as g ~ ( y , h ( y ) ) ) has a partial derivative with respect to z which vanishes at the origin. Clearly, this observation is also valid for higher-order approximations; that is, an error of order O ( l y l k ) in h ( y ) results in an error of order O(lylk+') in g ~ ( y , h ( y ) ) , for k 2 2. The term ay2 is the dominant term on the right-hand side of the reduced system. For a # 0 , the origin of the reduced system is unstable. Consequently, by Theorem 4.2, the origin of the full system is unstable. A

Example 4.2 Consider the system

which is already represented in the ( y , r ) coordinates. The center manifoldequation (4.11)-(4.12) is

hl ( y ) [ yh (y ) ] + h ( y ) - ay2 = 0, h(O) = h'(0) = 0 We start by trying 0 ( y ) = 0 . The reduced system is

Clearly, we cannot reach any conclusion about the stability of the origin. Therefore, we substitute h ( y ) = hzyZ + O(lv13) in the center manifold equation and calculate h z , by matching coefficients of y2 , to obtain hz = a . The reduced system is6

Y = all3 + O(Iyl4) "he error on the right-hand aide of the reduced system is actually O(ly15) since if we write

h(y) = hzy2 + h3y3 + . . ., we will find that ha = 0.


Therefore, the origin is asymptotically stable if a < 0 and unstable if a > 0. Con- sequently, by Theorem 4.2, we conclude that the origin of the full system is asymptotically stable if a < 0 and unstable if a > 0. A

Example 4.3 Consider the system (4.2)-(4.3) with

It can be verified that d(y) = 0 results in N ( ~ ( Y ) ) = 0 ( I ~ Y I I Z ) and

Using V(y) = $(y: + y;) as a Lyapunov function candidate, we obtain

in some neighborhood of the origin where k > 0. Hence,

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

We have to find a Lyapunov function that confirms asymptotic stability of the origin for all perturbations of the order 0 ( I I Y I I ; ) . The importance of this observation is illustrated by the following example.

Example 4.4 Consider the previous example, but change A1 to

With 4(y) = 0, the reduced system can be represented as

4.2. REGION OF ATTRACTION 177

Estimate of the Region of attraction

Figure 4.1: Critical clearance time.

Without the perturbation term 0 (Ilyll;), the origin of this system is asymptotically stable.' If you try to find a Lyapunov 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 (4.11)-(4.12) has the exact solution h(y) = $, so that the reduced system is given by the equation

whose origin is unstable.'

4.2 Region of Attraction Quite often, it is not sufficient to determine that a given system has an asymptotically stable equilibrium point. Rather, it is important to find the region of attraction 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 happen in the operation of a nonlinear system. Suppose that a nonlinear system has an asymptotically stable equilibrium point, which is denoted by Zp, in Figure 4.1. Suppose the system is operating at steady state at Z p , Then, at time t o a fault that changes the structure of the system takes place, for example, a short circuit in an electrical network. Suppose the faulted system does not have equilibrium at Zp, or in its neighborhood. The trajectory of the system will be driven away from

'See Exercise 5.24. 'See Exercise 3.13.


Zp,. Suppose further that the fault is cleared a t time tl and the postfault system has an &ymptotically stable equilibrium point at Zp,, where either spa = zpr or Spa is sufficiently close to xp, SO that steady-state operation a t i!,, is still accept- able. At time t~ the state of the system, say, z(tt), could be far from the postfault equilibrium ZP,. Whether or not the system will return to steady-state operation at Zps depends on whether x(t1) belongs to the region of attraction of Zp,, as determined by the post-fault system equation. A crucial factor in determining how far z(t1) could be from Zp, is the time it takes the operators of the system to remove the fault, that is, the time difference ( t ~ -to). If ( t l - t o ) 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 Zp.. 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, ( t l - to) must be less than t,. If we know the region of attraction of x p 3 , we can find t, by integrating the faulted system equation starting from the prefault equilibrium Zp, until it hits the houndary of the region of attraction. The time it takes the trajectory t o reach the boundary can he taken as the critical clearance time because if the fault is cleared before that time the state z(t1) will be within the region of attraction. Of course, we are assuming that .tpr belongs to the region of attraction of z p a , which is reasonable. If the actual region of attraction is not known, and an estimate i, o f t , is obtained using an estimate of the region of attraction, then i, < t, since the houndary of the estimate of the region of attraction will be inside the actual boundary of the region; see Figure 4.1. This 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 which are not too conservative. A very conservative estimate of the region of attraction would result in i, 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 he an asymptotically stable equilibrium point for the nonlinear system

x = f(x) (4.13) where f : D - Rn is locally L'ipschitz and D C Rn is a domain containing the origin. Let 4( t ;z ) be the solution of (4.13) that starts a t initial state z at time t = 0. The region of attraction of the origin, denoted by Ra, is defined by

Ra = {x E D ( $(t; x) i 0 as t --t co) Some properties of the region of attraction are stated in the following lemma, whose proof is given in Appendix A.8.

*see [I311 for an introduction to the transient stability problemin power systems.

4.2. REGION O F ATTRACTION 179

Figure 4.2: Phase portrait for Example 4.5

Lemma 4.1 If z = 0 is an asymptolically stable equilibrium point for (4.13), then its region of attraction RA is an open, connected, invariant set. Moreover, the boundary of RA is formed by trajectories. 0

This lemma suggests that one way to determine the region of attraction is to characterize those trajectories which lie on the boundary of Ra. There are some methods which approach the problem from this viewpoint, hut they use geometric notions from the theory of dynamical systems which are not introduced in this book. Therefore, we will not describe this class of methods."' We may, however, get a flavor of these geometric methods in the case of second-order systems (n = 2) by employing phase portraits. Examples 4.5 and 4.6 show typical cases in the state plane. In the first example, the boundary of the region of attraction is a limit cycle, while in the second one the boundary is formed of stable trajectories of saddle points. Example 4.7 shows a rather pathological case where the boundary is a closed curve of equilibrium points.

Example 4.5 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 4.2. The phase portrait shows that the origin is a stable focus; hence, it is asymptotically stabIe. This can be confirmed by

"'Examples of these methods csn he found in [30] and [191].


Figure 4.3: Phase portrait for Example 4.6.

has eigenvalues a t -112 * j&/2. Clearly, the region of attraction is bounded because trajectories starting outside the limit cycle cannot cross it to reach the origin. Since 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 4.6 Consider the second-order system

This system has three isolated equilibrium points a t (0, 0), (fi, O), and (-&, 0). The phase portrait of the system is shown in Figure 4.3. The phase portrait shows that the origin is a stable focus, and the other two equilibria are saddle points Hence, the origin is asymptotically stable and the other equilibria are unstable; a fact which can be confirmed by linearization From the phase portrait, we can also see that the stable trajectories of the saddle points form two separatrices which are the boundaries of the region of attraction. The region is unbounded. A

Example 4.7 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 this system are the radii of the unit circle. This can be seen by transforming the system into polar coordinates. The change of variables

yields I = -p(l - p2) , 4 = 0

All trajectories starting with p


and V(X) = - ;.:(I - f .f) - +$

Defining a domain D by

it can he easily seen that V(z) > 0 and ~ ( z ) < 0 in D - {O). Inspection of the phase portrait in Figure 4.3 shows that D is not a subset of RA. A

In view of this example, i t is not difficult to see why D of Theorem 3.1 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) = cz, with cz < el , there is no guarantee that the trajectory,will remain forever in D. Once the trajectory leaves D , there is no guarantee that V(x) will be negative. Hence, the whole argument about V(x) decreasing to zero falls apart. This problem does not arise in the estimates of R a , which we are going to present, since they are positively invariant sets; that is, a trajectory starting in the set remains for all future time in the set. The simplest estimate is provided by the set

when Cl, is bounded and contained in D. This follows as a corollary of Theorem 3.4. The simplicity of obtaining n, has increased significance in view of the linearization results of Section 3.3. There, we saw that if the Jacobian matrix

is a stability matrix, then we can alwaysfind a quadratic Lyapunovfunction V ( r ) = zTPz by solving the Lyapunov equation (3.12) for any positive definite matrix Q. Putting the pieces together, we see that whenever A is a stability matrix, we can estimate the region of attraction of the origin. This is illustrated by the following two examples.

Example 4.9 The second-order system

was treated in Example 4.5. There, we saw that the origin is asymptotically stable since


is a stability matrix. A Lyapunov function for the system can be found by taking Q = I and solving the Lyapunov equation

for P. The unique solution is the positive definite matrix

The quadratic function V(z) = x T P z is a Lyapunov function for the system in a certain neighborhood of the origin. Since our interest here is in estimating, the region of attraction, we need to determine a domain D about the origin where V(z) is negative definite and a set R, C D, which is bounded. We are also interested in the largest set R, that we can determine, that is, the largest value for the constant c, because 0, will be our estimate of Ra. Notice that we do not have to worry about checking positive definiteness of V(x) in D because V(x) is positive definite for all z . The derivative of V(z) along the trajectories of the system is given by

The right-hand side of V(X) is written as the sum of two terms. The first term, -IIzII;, is the contribution of the linear part Ax while the second term is the contribution of the nonlinear term g(z) = f ( z ) - A s , which we may refer to as the perturbation term. Since

we know that there is an open ball D = {z E R2 I llollz < r} such that ~ ( z ) is negative definite in D. Once we find such a ball, we can find R, c D by choosing

c < min V(x) = X,i.(P)r2 11~11.=1-

Thus, to enlarge the estimate of the region of attraction we need to find the largest ball on which V(X) is negative definite. We have

where we used 1x11 < 112112, 1x1221 < ;IIxll$, and 1x1 - 2z21 < &11+112. Thus, ~ ( z ) is negative definite on a ball D of radius given by r2 = 2/& = 0.8944. In this second-order example, a less conservative estimate of R, can be found by searching for the ball D in polar coordinates. Taking


we have

v = -p2 + p4 cos2 9 sin 9(2 sin 9 - cos 9) < - -p2+p41cos29sin91.12sinB-cos91 < -p2 + p4 x 0.3849 x 2.2361

1 < -p2 + 0.861p4 < 0, for p2 < - 0.861 Using this, together with A,i,(P) 2 0.69, we choose

The set 0, with c = 0.8 is an estimate of the region of attraction.


There are two equilibrium points a t (0,O) and (1,2). The equilibrium point a t (1,2) is unstable since

9 1 a z ,,=,,,2=2 = [ ; :I

has eigenvalues a t f fi it is a saddle point. The origin is asymptotically stable ! since

is a stability matrix. A Lyapunov function for the system can be found by taking Q = I and solving the Lyapunov equation

for P. The unique solution is the positive definite matrix

The derivative of V(z) = z T p z along the trajectories of the system is given by V ( Z ) = -(x: + 2;) + ( + z ? z ~ + 212;)

v'3 5 -11.11; + ;lz:xzl 1.1 +2.21 < - 1 1 ~ 1 1 ; + qll.ll;


Thus, V(X) is negative definite in a ball of radius r = 4/&. Since Ami.(P) = 1/4, we choose

The set R, with c = 0.79 is an estimate of the region of attraction. A

Estimating the region of attraction by means of the sets R, is simple, hut is usually conservative. We shall present two ideas to obtain better estimates of the region of attraction. The first idea is based on LaSalle's theorem (Theorem 3.4) while the second one is the trajectory-reversing method, which uses computer simulation. LaSalle's theorem provides an estimate of Ra by the set 0, which is a compact positively invariant set. Working with the set R,, whose boundary is a Lyapunov surface, is a special case where the invariance of the set follows from the negative definiteness of V(x). We can work with more general sets, but then we need to establish that the set is a positively invariant one. This typically requires investigating the vector field at the boundary of the set to ensure that trajectories starting in the set cannot leave it. The following example illustrates this idea.


where h : R -r R satisfies

Consider the quadratic function

as a Lyapunov function candidate." The derivative V(X) is given by

"This Lyapunov function candidate can be derived using the variable gradient method. See Exercise 4.14.

CHAPTER 4. ADVANCED STABILITY THEORY

Figure 4.4: Estimates of the region of attraction for Example 4.11.

Therefore, V ( Z ) is negative definite in the set

and we can conclude that the origin is asymptotically stable. To estimate RA, let us start by an estimate of the form R,. We need to find the largest c > 0 such that R, C G. It can be easily seen that this c is given by

c = min V ( z ) = mint min V ( x ) , min V ( z ) ) In+z,l=l zl+c,=l 21+2.=-1

The first minimization yields

Similarly, min V ( z ) = 1

21+z,=-1

Hence, R, with c = 1 is an estimate of RA; see Figure 4.4. In this example, we can obtain a better estimate of RA by not restricting ourselves to estimates of the form R,. A key point in the following development is to observe that trajectories inside G cannot leave through certain segments of the boundary Ixl + x21 = 1 . This can be seen by plotting the phase portrait of the system, or by the following analysis. Let

such that the boundary of G is given by o = 1 and o = -1. The derivative of u2 along the trajectories of the system is given by


On the boundary u = 1,

This implies that when the trajectory is at any point on the segment of the houndary o = 1 for which xz 5 4, it cannot move outside the set G because at such point a2 is uonincreasing. Similarly, on the houndary u = -1,

Hence, the trajectory cannot leave the set G through the segment of the boundary o = -1 for which x2 2 -4. This information can be used to form a closed, hounded, positively invariant set R that satisfies the conditions of Theorem 3.4. Using the two segments of tbe boundary of G identified above to define the boundary 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 he such that the Lyapunov surface V(x) = cl intersects the houndary zl + x2 = 1 at x2 = 4, that is, at the point (-3,4); see Figure 4.4. Let cx he such that the Lyapunov surface V(z) = c2 intersects the boundary XI + x2 = -1 at xz = -4, that is, at the point (3, -4). The required Lyapunov surface is defined by V(x) = min{cl, cz). The constant cl is given by

Similarly, cz = 10. Therefore, we take c = 10 and define the set R by

Q = {z E R2 I V(x) < 10 and 1x1 + z21 5 1) This set is closed, hounded, and positively invariant. Moreover, V ( Z ) is negative definite in 12 since R c G. Thus, all the conditions of Theorem 3.4 are satisfied and we can conclude that all trajectories starting in R approach the origin as t -+ m; that is, C Ra. A

The trajectory-reversing method uses computer simulation to enlarge an initial estimate of the region of attraction. It draws its name from reversing the direction of the trajectories of (4.13) via backward integration. This is equivalent to forward integration of the system

x = -f(x) (4.14) which is obtained from (4.13) via replacing t by -t. System (4.14) has the same trajectory configuration in the state-space as system (4.13), but with reversed arrow heads on the trajectories. Let Wo C D he a compact set that contains the origin in


its interior. Suppose Wo is positively invariant with respect to (4.13) and there is a positive definite, continuously differentiable function V : Wo -+ R such that

V(Z) < 0 v z E Wo - {0) According to Theorems 3.1 and 3.4, the origin is asymptotically stable equilibrium for (4.13) and Wo C Rn. Consequently, Wo is connected.I2 For z E Wo, let $(t, z) be the solution of (4.14) that starts at z a t time t = 0. This is the backward integration of (4.13) starting at z . Since f ( x ) is locally Lipschitz and Wo is compact, there is T > 0 such that $(t ,z) is defined on [0, T) for all x E Wo. Notice that backward integration starting at z E Wo may have a finite escape time even though forward integration is defined for all t since z E Ra. A simple example that illustrates this point is the scalar system

i = -23 , z(O) = xo which, in reverse time, becomes

Forward integration is defined for all t and z(t) approaches the origin, while back. ward integration results in

which has a finite escape time at t = 1/22;. Let t l E (O,T), and consider the backward mapping

( 2 ) = 1 2 ) 2: E Wo Due to uniqueness and continuity of solutions in both forward and backward integration, this mapping is continuous and one-to-one on Wo, and its inverse F-' is continuous on F(Wo). Let

WI = F(Wo) Then, WI is compact and connected.13 Moreover,

The fact that Wo is a subset of WI follows from the fact that WO is a positively invariant set in forward integration. Suppose there is a point Z E Wo such that 2: # ljl(tl, x) for any z E Wo. Then, the forward trajectory starting at 2: must leave

12See Exercise 4.15. 13This follows from the fact that compactness andcoonectedngsof a set remain invariant under

a topological mapping or a homeomorphism [?I. A mapping F ( . ) is a topological mapping on a set S if it is continuous, one-to-one, and its inverse F-' is continuous on F(S ) . The backward mappins F(z) = rL(t1, r) is a topological mapping on Wo .


the set Wo, which contradicts the forward invariance of Wo. Actually, Wo is a proper subset14 of Wl. To see this, note that no point on the boundary of Wo may be taken by backward integration into the interior of Wo since, again, this would contradict the fact that Wo is a positively invariant set in forward integration. Moreover, since every forward trajectory starting on the boundary of Wo must eventually approach the origin, there must be some points on the boundary where the vector fields of forward trajectories point into the set Wo. Backward integration must take these points outside Wo. The fact that Wl C RA follows from the fact that forward integration of any point in Wl would take it into Wo, which is a subset of RA. Thus, Wl provides a better estimate of the region of attraction than Wo. Now, let us increase t l to t2,ts, . . . , t i , . . ., where ti < ti+, < T for all i. Repetition of the argument shows that the backward mappings $(t i ,x) define nested compact sets W; R U C ~ that

Wi=F;(Wo), W;C W ; + I C R A where Fi(z) = $(ti, 2). Therefore, the sequence W I , W2, . . . provides increasingly better estimates of RA. The utility of the trajectory-reversing method is not in generating the sets W; analytically, but in using computer simulations to approx- imate these sets.15 Let ri be the boundary of Wi. It is not difficult to see that the backward mapping F;(x) = $(ti, I) transforms ro into ri . Discretize I'o using an adequate number of points and carry out backward integration of these points over the interval [0, t l] to obtain a discretized image of rl . To obtain a better estimate, choose tz > t l and repeat the backward integration. It is enough, however, to integrate the points on rl over the interval [tl, tz] since, due t o uniqueness of solution, this would be equivalent to integrating the points on To over the interval [to, t z ] . The process can be repeated to obtain r3, r4,. . .. The method will function particularly well when Ra is bounded since backward integration cannot escape to infinity. Example 4.12 Consider the second-order system

2 1 = - 1 2

i.2 = X I + (x? - 1)xz We found in Example 4.5 that the region of attraction RA is the interior of the limit cycle shown in Figure 4.2. In Example 4.9, we used Lyapunov's method t o estimate RA by the set

Wo = { x 6 R~ I xTpx _< 0.81, where P = [!:5


Figure 4.5: Estimates of the region of attraction for Example 4.12

Let us use the trajectory-reversing method to improve the estimate Wo. We have discretized the surface xTPz = 0.8 using 16 points which are equally-spaced angle- wise, that is, the points are given in polar coordinates by (pk, Bk), where

2 ~ ~ 0 8 / [ c o s @ i ] T p [ c ~ s @ k ] @k = - 16 > Pk - . sin Bk sin Ok

for k = 0 ,1 ,2 , . . ., 15. Figure 4.5 shows the surfaces obtained by backward iutegra- tion of these points over the interval [0, ti] for ti = l , 2, 3. It is clear that if we continue this process, the estimate Wi will approach RA. A

If RA is unbounded, the method will still enlarge the initial estimate of the region of attraction but we might not. be able to asymptotically recover RA. One prohlenl is that backward integration may have a finite escape time. Even without facing this problem, we may still experience numerical difficulties due to the fact that backward trajectories approaching infinity may have different speeds. While the backward integration of some points on ro might still be close to Po, the backward integration of other points might produce solutions large enough to induce numerical difficulties in the computer simulation routine. The following example illustrates this point.


4.3. INVARIANCE THEOREMS 191 I I

Figure 4.6: Estimates of the region of attraction for Example 4.13

In Example 4.10, we estimated RA by the set

0.25 0 WO = {x E R' I xTpx 5 0.79). where P = [ 0,5 ] Discretizing the surface zTPz = 0.79 using the scheme from Example 4.12, we obtain the surfaces Ti for t i = 0.2, 0.5, and 0.55, which are shown in Figure 4.6. The wedge-like shape of the surfaces is due to backward integration of two points on f o which approach infinity rapidly. At ti = 0.6, the simulation program suffers from singularities and the calculations are no longer reliable.16 A

4.3 lnvariance Theorems In the case of autonomous systems, LaSalle's invariance theorem (Theorem 3.4) shows that the trajectory of the system approaches the largest invariant set in E, where E is the set of all points in 12 where ~ ( z ) = 0. In the case of nonautonomous systems, it may not even he clear how to define a set E since V(t, x) is a function of both t and r. The situation will he simpler if it can be shown that

I6The integration is carried out using a classical fourth-order Rung-Kutta algorithm with au- tomatic step adjustment. Of course, using another algorithm we might be able to exceed the 0.6 limit, but the problem will happen again at a higher "due oft,.


for, then, a set E may he defined as the set of points where W ( x ) = 0. We may expect that the trajectory of the system approaches E as t tends t o m. This is, basically, the statement of the next theorem. Before we state the theorem, we state a lemma that will he used in the proof of the theorem. The lemma is interesting in its own sake and is known as Barbalat's lemma. Lemma 4.2 Let 4 : R -+ R be a uniformly continuous function on [O,oo). Suppose that lirnt,- Ji 4(r) d r exists and is finite. Then,

Proof: If i t is not true, then there is a positive constant kl such that for every T > 0 we can find TI 2 T with Id(T1)I 2 k l . Since 4(t) is uniformly continuous, there is a positive constant kz such that IQ(t + r ) - O(t)l < k1/2 for all t > 0 and a11 0 5 r 5 k2. Hence,

Therefore.

where the equality holds since 4 ( t ) retains the same sign for Tl 5 t 5 TI + k2. Thus, the integral 5; 4 ( ~ ) d r cannot converge t o a finite limit as t -+ oo, a contradiction.

0

Theorem 4.4 Let D = { z E R" 1 11x11 < T } and suppose that f ( t , x ) is piecewise continuous in t and locally Lipschitz in z , uniformly in t , on [O,m) x D . Let V : [0, m) x D --t R be a continuonsly differentiable function such that

w, (2 ) _< V ( t , 2 ) 5 W 2 ( z )

V t 2 0 , V z E D, where W l ( z ) and W 2 ( z ) are continuous positive definite functions and W ( x ) is a continnous po~itive semidefinite function on D. Let p < rninllzll,, W i ( z ) . Then, all solutions of x = f ( t , x ) with x(to) E { z E B, 1 W z ( x ) 5 p } are bounded and satisfy

4.3. INVARIANCE THEOREMS 193

Moreover, if all the assumptions hold globally and W l ( z ) as radially unbounded, the statement is true for all x(to) E R". 0 Proof: Similar to the proof of Theorem 3.8, it can be shown that

since ~ ( t , z ) < 0. Hence, (Iz(t)(l < r for all t 2 to. Since V ( t , z( t ) ) is monotonically nonincreasing and bounded from below by zero, it converges as t i m. Now,

Therefore, limt,, J: W ( z ( r ) ) d~ exists and is finite. Since JJx(t)Jl < r for all t > to and f( t , z ) is locally Lipschitz in z , uniformly in t , we conclude that z ( t ) is uniformly continuous in t on [to, m). Consequently, W ( z ( t ) ) is uniformly continuous in t on [to, m) since W ( z ) is uniformly continuous in z on the compact set B,. Hence, by Lemma 4.2, we conclude that W ( z ( t ) ) -+ 0 as t i oo. If all the assumptions hold globally and W l ( z ) is radially unbounded, then for any z(ta) we can choose p so large that z(t0) E { z E Rn I Wz(z) 5 p } . o

The limit W(x( t ) ) + 0 implies that z ( t ) approaches E as t i m, where E = { z E D I W ( z ) = 0)

Therefore, the positive limit set of z ( t ) is a subset of E. The mere knowledge that z ( t ) approaches E is much weaker than the invariance principle for autonomous systems which states that z ( 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 3.1, namely the positive limit set is an invariant set. There are some special classes of nonautonomous systems where positive limit sets have some sort of an invariance property.17 However, for a general nonautonomous system, the positive limit sets are not invariant. The fact that, in the case of autonomous systems, z ( t ) approaches the largest invariant set in E allowed us to arrive at Corollary 3.1, where asymptotic stability of the origin 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 nonautonomous system, there is no exten- sion of Corollary 3.1 that would show uniform asymptotic stability. However, the following theorem,shows that it is possible to ~onclude uniform asymptotic stability if, in addition to V( t , I ) 5 0 , the integral of V(t , z ) satisfies a certain inequality.

"Examples are periodic systems, almost-periodic systems, and asymptotically autonomom sye tern. See (137, Chapter 81 for invariance thcorems for these classes of systems. See, also, [I191 for a different generalisation of the invariance principle.


Theorem 4.5 Let D = { x E Rn I )lxJI < r) and suppose that f ( l , x ) is piecewise conlinuons in t and locally Lrpschitz in x on [O, oo) x D. Let x = 0 be an equilibrium point for& = f ( t , x ) at t = 0. Let V : [O, m)XD i R be a continuously differentiable function such that

W I ( X ) 5 V ( t > x ) 5 W z ( x )

tt6 , v ( T , * ( T , ~ , x ) ) . i ~ 5 - A v ( t . ~ ) , o c A <

V t 2 0, V x E D, for some 6 > 0, where W I ( X ) and W z ( x ) are conlinuous positive definite functions on D and $ ( ~ , t , x ) is the solution of the system that starls a t (1 , x ) . Then, the origin is uniformly asymptotically stable. If all the assumptions hold globally and W t ( x ) is radially unbounded, then the origin is globally uni fomly asymptotically stable. If

then the origin is exponentially stable. 0

Proof: Similar to the proof of Theorem 3.8, it can be shown that

where p < minllrll=v W l ( x ) because i / ( t , x ) < 0. Now, for all t 2 to, we have

Moreover, since ~ ( t , x ) 5 0 ,

For any t 2 to, let N be the smallest positive integer such that t _< to + N6. Divide the interval [to, to+(N - 1)6] into ( N - I ) equal subintervals of length 6 each. Then,

"There is no loss of generality in assuming that X < 1, for if the inequality is satisfied with A 1 2 1, then it is satisfied for any positive X < 1 since -XIV < -XV. Notice, however, that this inequality could not be satisfied with X > 1 since V(t,s) > 0, V s # 0.

4.3. INVARIANCE THEOREMS 195

where 1 1 b=-In- 6 ( I - A ) Taking

- h a U ( T , S) = - ( 1 -A)e it can he easily seen that o(r, s) is a class KC function and V ( t , x ( t ) ) satisfies

V( t , x ( t ) ) 5 u(V(to,x(to)), t - to), v V(to,z(to)) E [O,pI From this point on, the rest of the proof is identical to that of Theorem 3.8. The proof of the statements on global uniform asymptotic stability and exponential stability are the same as the proof of Corollaries 3.3 and 3.4.

Example 4.14 Consider the linear time-varying system

2 = A(t)x where A(t) is continuous for all t 2 0. Suppose there is a continuously differentiable, symmetric matrix P(t) which satisfies

0 < ell < P(t) 5 c z l , v t 2 0 as well as the matrix differential equation

-P(t ) = ~ ( t ) ~ ( t ) + ~ ~ ( t ) ~ ( t ) + cT ( t )C( t ) where C ( t ) is continuous in t . The derivative of the quadratic function

V ( t , 2 ) = xTp( t )x along the trajectories of the system is

The solution of the linear system is given by

4 ( ~ , t , X ) = @(T, t ) ~


where @(T, t ) is the state transition matrix. Therefore,

where i t 6

W ( t , t + 6 ) = 1 aT(r , t ) c T ( ~ ) c ( ~ ) @ ( ~ , t ) d~ Suppose there is a positive constant k < cz such that

then tt6 , k

V T ~ ) d 5 k - - v ( t , ~ ) c2

Thus, all the assumptions of Theorem 4.5 are satisfied globally with

and we conclude that the origin is globally exponentially stable. Readers familiar with linear system theory will recognize that the matrix W ( t , t + 6 ) is the observability Gramian of the pair (A( t ) ,C( t ) ) and that the inequality W ( t , t + 6 ) > kI is implied by uniform observability of (A( t ) ,C( t ) ) . Comparison of this example with Example 3.21 shows that Theorem 4.5 allows us to replace the positive definiteness requirement on the matrix Q(t) of (3.21) by the weaker requirement. Q(t) = CT(t)C(t) , where the pair (A( t ) ,C( t ) ) is uniformly observable. A

4.4 Exercises Exercise 4.1 Prove Corollary 4.1

Exercise 4.2 Prove Corollary 4.2.

Exercise 4.3 Suppose the conditions of Theorem 4.1 are satisfied in a case where g ~ ( y , O ) = 0, g*(y,O) = 0 , and A l = 0. Show that the origin of the full system is stable.

Exercise 4.4 Reconsider Example 4.1 with a = 0 , and Example 4.2 with a = 0 In each case, apply Corollary 4.1 to show that the origin is stable.

4.4. EXERCISES 197

Exercise 4.5 ( [ S O ] ) Consider the system

where dim(%) = nl, dim(zs) = nz, Aa is a Hurwitz matrix, j, and j b are continuously differentiable, [afb/azb](O, 0) = 0, and fb(z., 0) = 0 in a neighborhood of 2, = 0.

(a) Show that if the origin z, = O is an exponentially stable equilibrium point of X. = f,(z., 0), then the origin (I., za) = (0,O) is an exponentially stable equilibrium point of the full system.

(b) Using the center manifold theorem, show that if the origin 2, = 0 is an asymptotically (but not exponentially) stable equilibrium point of x, = f.(z,,O), then the origin (z., zb) = (0,O) is an asymptotically stable equilibrium point of the full system.

Exercise 4.6 ([59]) For each of the following systems, investigate the stability of the origin using the center manifold theorem.

Exercise 4.7 For each of the following systems, investigate the stability of the origin using the center manifold theorem.

Exercise 4.8 ((281) Consider the system

Investigate the stability of the origin using the center manifold theorem for each of the following cases.


Exercise 4.9 ([28]) Consider the system

Investigate the stability of the origin using the center manifold theorem for all possible values of the real parameter a.

Exercise 4.10 ([SO]) Consider the system

Investigate the stability of the origin using the center manifold theorem for all possible values of the real constants a, b , c.

Exercise 4.11 (Zubov's Theo rem) Consider the system (3.1) and let G C Rn be a domain containing the origin. Suppose there exist two functions V : G + R and h : R" - R with the following properties:

. V is continuously differentiable and positive definite in G and satisfies the inequality

0 < V(z) < 1, v z E G - { 0 ) As z approaches the boundary of G, or in case of unbounded G as llzll + m, limV(z) = 1. . h i s continuous and positive definite on Rn

For z E G, V(z) satisfies the partial differential equation

Show that z = O is asymptotically stable and G is the region of attraction.

Exercise 4.12 ([62]) Consider the second-order system

4.4. EXERCISES

where

for some positive constants a,, b; (ai = cu or bi = oo is allowed). Apply Zubov's theorem to show that the region of attraction is {z E R2 I - a; < xi < bi]. Hint: Take h(z) = gl(zl)hl(zl) and seek a solution of the partial differential equation (4.15) in the form V(z) = 1 - Wl(zl)W2(z2). Note that, with this choice of h, V(Z) is only negative semidefinite; apply LaSalle's invariance principle.

Exercise 4.13 Find the region of attraction of the system

Hint: Use the previous exercise.

Exercise 4.14 Consider the system of Example 4.11. Derive the Lyapunov function V(z) = 22; + 221x2 +I: using the variable gradient method.

Exercise 4.15 Let Cl be an open, positively invariant set containing the origin. Suppose every trajectory in Cl approaches the origin as t - w. Show that R is connected.

Exercise 4.16 Consider a second-order system 2 = f(z) with asymptotically stable origin. Let V(z) = zf +zz, and D = {z E R2 I lzZl < 1, lzl-z21 < 1). Suppose that [aV/az] f(z) is negative definite in D. Estimate the region of attraction.

Exercise 4.17 Consider the system

(a) Using V(z) = 52: + 2z1r2 + 2 4 , show that the origin is asymptotically stable. (b) Let

S = {Z E R2 ( V(I) 5 5) n {Z E R2 1 1z21 5 1) Show that S is an estimate of the region of attraction

CHAPTER 4. ADVANCED STABILITY THEORY

SX2

Figure 4.7: Exercise 4.19.

Exercise 4.18 Show that the origin of

is asymptotically stable and estimate the region of attraction.

Exercise 4.19 Consider a second-or+ system 2 = f (z ) , together with a Lya- punov function V(x). Suppose that V(x) < 0 for all xf + zg 2 a2. The sketch, given in Figure 4.7, shows four different directions of the vector field at a point on the circle xf + x i = a2. Which of these directions are possible and which are not? Justify your answer.

I Exercise 4.20 Consider the system

where sat(.) is the saturation function. (a) Show that the origin is the unique equilibrium point.

(b) Show, using linearization, that the origin is asymptotically stable (c) Let u = X I + 1 2 . Show that

u u u , for 1 0 1 2 1

4.4. EXERCISES 201

(d) Let V(z) = z: + 0 . 5 4 + 1 - coszl, and M, be the set defined, for any c > 0, by

M, = {I E R2 1 v(2) 5 C} n {Z E R2 I ( 0 1 5 1) Show that M, is positively invariant and that every trajectory inside M, must approach the origin as t -+ co.

(e) Show that the origin is globally asymptotically stable

Exercise 4.21 Consider the tunnel diode circuit of Example 1.2. The circuit has three equilibrium points.

( a ) Investigate the stability of each point using linearization. (b) For each asymptotically stable equilibrium point, find a quadratic Lyapunov

function V(z) = z T p z and use it to estimate the region of attraction. (c) Continuing part (h), extend the region of attraction using the trajectory re-

versing method.

Exercise 4.22 Consider the third-order synchronous generator model described in Exercise 1.7. Take the state variables and parameters as in parts (a) and (b) of the exercise. Moreover, take r = 6.6 sec, M = 0.0147 (per unit power) x sec2/rad, and DIM = 4 sec-'.

(a) Find all equilibrium points in the region -s < z l < n, and determine the stability properties of each equilibrium using linearization.

(b) For each asymptotically stable equilibrium point, estimate the region of attraction using a quadratic Lyapunov function.

Exercise 4.23 ((1001) Consider the system

where g(t) is continuously differentiable and 0 < kl < g(t) < k2 for all t > 0 (a) Show that the origin is exponentially stable.

(b) Would (a) be true if g ( t ) were not hounded? Consider g(t) = 2+exp(t) in your answer.

Hint : Part (a) uses observability properties of linear systems. In particular, for any bounded matrix IC(t), uniform observability of the pair (A(t),C(t)) is equivalent to uniform observability of the pair (A(t) - K(t)C(t), C(t)).



where g(t) is continuously differentiable and 0 < k1 < g(t) < k2 for all t > 0. Show that the origin is exponentially stable. Hint: Use the previous exercise.


where a ( t ) = sint. Show that the origin is exponentially stable. Hint: Try V(z) = zTz and apply the result of Example 4.14.

Chapter 5

Stability of Perturbed Systems

Consider the system i: = f(t , x) + g(t, 2) (5.1)

where f : [O, co) x D + Rn and g : [O, co) x D + Rn are piecewise continuous in t and locally Lipschitz in x on [O, co) x D, and D C Rn is a domain that contains the origin z = 0. We think of this system as a perturbation of the nominal system

The perturbation term g(t, z) could result from modeling errors, aging, or uncertainties and disturbances which exist in any realistic problem. In a typical situation, we do not know g(t, z) but we know some information about it, like knowing an upper bound on llg(t, z)ll. Here, we represent the perturbation as an additive term on the right-hand side of the state equat,ion. Uncertainties which do not change the system order can always be represented in this form. For if the perturbed rigbt- hand side is some function f(t , z), then by adding and subtracting f ( t , x) we can rewrite the right-hand side as

and define g(t ,z) = f ( t , x ) - f ( t , z )

Suppose the nominal system (5.2) has a uniformly asymptotically stable equilibrium point a t the origin, what can we say about the stability behavior of the perturbed system (5.1)? 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 approach in Sections 3.3 and 3.5. The new element here is that the perturbation term could be

204 CHAPTER 5. STABILITY OF PERTURBED SYSTEMS

more general than the perturbation term in the case of linearization. The conclu- sions we can arrive at depend critically on whether the perturbation term vanishes a t the origin. If g(t,O) = 0, the perturbed system (5.2) has an equilibrium point a t the origin. In this case, we analyze the stability behavior of the origin as an equilibrium point of the perturbed system. If g ( t , 0) # 0, the origin will not be an equilibrium point of the perturbed system. Therefore, we can no longer study the problem as a question of stability of equilibria. Some new concepts will have to be introduced.

The cases of vanishing and nonvanishingperturbations are treated in Sections 5.1 and 5.2, respectively. In Section 5.3, we use the results on nonvanishing perturbations to introduce the concept of input-to-sta1.e stability. In Section 5.4, we restrict our attention to the case when the nominalsystem has an exponentially stable equilibrium point a t the origin, and use the comparison lemma to derive some sharper results on the asymptotic behavior of the solution of the perturbed system. In Section 5.5, 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 systems, respectively. In both cases, stability analysis is simplified by viewing the 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 subsystems, while in the case of slowly varying systems, a nonautonomous system with slowly varying inputs is approximated by an autonomous system where the slowly varying inputs are treated as constant parameters.

5.1 Vanishing Perturbation Let us start with the case g(t, 0) = 0. Suppose z = 0 is an exponentially stable equilibrium point of the nominal system (5.2), and let V(t, x) be a Lyapunov function that satisfies

c111x1I2 5 V ( t , z ) 5 czllxllZ (5.3)

for all ( t , x) E [0, m) x D for some positive constants c l , c2, CQ, and c4. The existence of a Lyapunov function satisfying (5.3)-(5.5) is guaranteed by Theorem 3.12, under some additional assumptions. Suppose the perturbation term g(t, z ) satisfies the linear growth bound

5.1. V A N I S H I N G PERTURBATION 205

where y is a nonnegative constant. This bound is natural in view of the assumptions on g ( t , x ) . In fact, any function g ( t , z ) that vanishes at the origin and is locally Lipschitz in z , uniformly in t for all t _> 0, in a bounded neighborhood of the origin satisfies (5.6) over that neighborhood.' We use V as a Lyapunov function candidate to investigate the stability of the origin as an equilibrium point for the perturbed system (5.1). The derivative of V along the trajectories of (5.1) 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 (5.4) . The third term, [ a V / a z ] 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 , z ) negative definite. With the growth bound (5.6) as our only information on g, the best we can do is worst-case analysis where [ a V / a x ] g is majorized by a nonnegative term. Using (5.4)-(5.6), we obtain

If y is small enough to satisfy the bound

then ~ ( t , 2 ) 5 -(c3 - 7~4)llzIl~, (c3 - 7 ~ 4 ) > 0

Therefore, by Corollary 3.4, we conclude the following lemma.

Lemma 5.1 Let z = 0 be an ezponentially stable equilibrium point of the nominal system (5 .2) . Let V ( t , x ) be a Lyapunov function o f t he nominal system that satisfies (5.3)-(5.5) i n [O, oo) x D. Suppose the perturbation t e r m g ( t , z ) satisfies (5.6)-(5.7). Then , the origin i s an exponentially stable equilibrium point of the perturbed system 5 . 1 ) Moreover, if all the assumptions hold globally, then the origin i s globally exponentially stable. 0

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 (5.6)- (5.7). To assert this robustness property, we do not have to know V ( t , z ) 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

'Note, however, that the linear growth bound (5.6) becomes restrictive when required to hold globally, because that would require g to be globally Lipschite in s.


exponentially stable without actually finding a Lyapunov function that satisfies (5.3)-(5.5).2 Irrespective of the method we use to show exponential stability of the origin, we can assert the existence of V(t, x) satisfying (5.3)-(5.5) by application of Theorem 3.12 (provided the Jacobian matrix [af/ax] is bounded). However, if we do not know the Lyapunov function V(t ,x) we cannot calculate the bound (5.7). Consequently, our robustness conclusion becomes a qualitative one where we say that the origin is exponentially stable for all perturbations satisfying

with sufficiently small y. On the other hand, if we know V(t, x) we can calculate the bound (5.7), which is an additional piece of information. We should be careful not t o overemphasize such bounds because they could be conservative for a given perturbation g(t, I). The conservatism is a consequence of the worst-case analysis we have adopted from the beginning.


where A is Hurwitz and llg(t, z)l12 < 711x1/2 for all t 2 0 and all x E R". Let Q = QT > 0 and solve the Lyapunov equation

for P. From Theorem 3.6, we know that there is a unique solution P = PT > 0. The quadratic Lyapunov function V(x) = xTPx satisfies (5.3)-(5.5). In particular,

~mim(p ) I l~ I I~ < V(X) 5 ~ r n ~ ~ ( p ) I I ~ l l ~

The derivative of V(x) along the trajectories of the perturbed system satisfies

Hence, the origin is globally exponentially stable if y < Xmi,(Q)/2A,,,(P). Since this bound depends on the choice of Q , one may wonder how to choose Q to maxi- mize the ratio hrni,(Q)/Xrn,,(P). I t turns out that this ratio is maximized with the choice Q = I (Exercise 5.1). A

ZThis is the case, for example, when exponential stability of the origin is shown using Theo- rem 4.5.

5.1. VANISHING PERTURBATION 207


where the constant p 2 0 is unknown. We view the system as a perturbed system of the form (5.1) with

The eigenvalues of A are -1 + jd. Hence, A is Hurwitz. The solution of the Lyapunov equation

P A + A ~ P = -I is given by

As we saw in Example 5.1, the Lyapunov function V(x) = zTPx satisfies inequalities (5.3)-(5.5) with cs = 1 and

The perturbation term g(x) satisfies

for all lz21 < k2. At this point in the analysis, we do not know a hound on x2(t), although we know that x2(t) will be bounded whenever the trajectory x(t) is confined to a compact set. We keep k2 undetermined and proceed with the analysis. Using V(x) as a Lyapunov function candidate for the perturbed system, we obtain

Hence, V(X) will he negative definite if

To estimate the bound kz, let 0, = { z E R2 ( V(z) < c ) . For any positive constant c, the set 0, is closed and bounded. The boundary of 0, is the Lyapunov surface


The largest value of 1x21 on the surface V(z) = c can he determined by differentiating the surface equation partially with respect to z l . This results in

Therefore, the extreme values of 1 2 are obtained at the intersection of the line ZI = -z2/12 with the Lyapunov surface. Simple calculations show that the largest value of zz on the Lyapunov surface is 96~129. Thus, all points inside 0, satisfy the honnd

96c )z2) 5 k2, where k i = Therefore, if

29 0.1 C -

< 3.026 x 96c c ~ ( z ) will be negative definit,e in R, and we can conclude that the origin z = 0 is exponentially stable with R, as an estimate of the region of attraction. The inequality P < O.l/cshows a tradeoff between the estimate of the region of attraction and the estimate of the upper hound on P. The smaller the upper hound on P , 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 4.5 to have a bounded region of attraction surrounded by an unstable limit cycle. When transformed into the z-coordinates, the region of attraction will expand with decreasing P and shrink with increasing P. Finally, let us use this example to illustrate our remarks on the conservative nature of the honnd (5.7). Using this bound, we came up with the inequality P < l/3.026kz. This inequality allows the perturbation term g(1, z) to be any second-order vector that satisfies llg(t, z)112 5 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

5.1. VANISHING PERTURBATION 209

an unslructured perturbotion 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 the analysis, this time taking into consideration the structure of the perturbation. Instead of using the general bound (5 .7 ) , we calculate the derivative of V ( t , z ) along the trajectories of the perturbed system to obtain

Hence, V ( Z ) is negative definite for P < 4/3k;. Using, again, the fact that for all z E R,, 1221' < 822 = 96c/29, we arrive at the bound /3 < 0.4/c, which is four times the bound we obtained using (5.7). A

When the origin of the nominal system (5.2) is uniformly 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 function V ( t , z ) that satisfies

for all ( t , z ) [0, co) x D, where W3(2) is positive definite and continuous. The derivative of V along the trajectories of (5.1) is given by

Our task now is to show that

for all ( 2 , z ) E [0, co) x D, a task which cannot he done by a putting a simple order of magnitude bound on [lg(t, z)l( as we have done in the exponentially stable case. The growth bound on ilg(t, z)11 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 is the case when V ( t , z ) is positive definite, decrescent, and satisfies


for all (t, x) E [O, m) x D for some positive constants CQ and c4, where 4 : Rn -t R is positive definite and continuous. A Lyapnov function satisfying (5.8)-(5.9) is usually called a quadratic-type Lyapunov function. It is clear that a Lyapunov function satisfying (5.3)-(5.5) is quadratic type, but a quadratic-type Lyapunov function may exist even when the origin is not exponentially stable. We shall illustrate this point shortly by an example. If the nominal system (5.2) has a quadratic-type Lyapunov function V(t,z), then its derivative along the trajectories of (5.1) satisfies

i.(t, 2) 5 - ~ 3 4 ~ ( z ) + ~44(z)llg(t>z)Il Suppose now that the perturbation term satisfies the bound

Then, v ( t , 2) 5 -(c3 - c47)4=(z)

which shows that ~ ( t , z) is negative definite. Example 5.3 Consider the scalar system

The nominal system i = -z3

has a globally asymptotically stable equilibrium point at the origin but, as we saw in Example 3.23, the origin is not exponentially stable. Thus, there is no Lyapunov function that satisfies (5.3)-(5.5). The Lyapunov function V(x) = x4 satisfies (5.8)- (5.9), with +(z) = 1zI3, c3 = 4, and c4 = 4. Suppose the perturbation term g(t, z) satisfies the bound lg(t, z)1 5 y(x(3 for all r, with y < 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 notice that a nominal system with uniformly asymptotically stable, but not exponentially stable, origin is not robust to smooth perturbations with arbitrarily small linear growth bounds of the form (5.6). This point is illustrated by the following example.3

-

3See, also, Exercise 5.8.

5.2. NONVANISHING PERTURBATION 21 1

Exaniple 5.4 Consider the scalar system of the previous example with perturbation g = yx where y > 0 ; that is,

It can be easily seen, via linearization, that for any y > 0 , no matter how small y, the origin is unstable. A

5.2 Nonvanishing Perturbation Let us turn now to the more general case when we do not know that g(t , 0 ) = 0. The origin x = 0 may not be an equilibrium point of the perturbed system (5.1). We can no longer study stability of the origin as an equilibrium point, nor should we expect the solution of the perturbed system to approach the origin as t i m . The best we can hope for is that if the perturbation term g ( t , x ) is small in some sense, then x ( t ) will be ultimately bounded by a small bound; that is, IIx(t)ll will be small for sufficiently large t . This brings in the concept of ultimate boundedness.

Definition 5.1 The solutions of x = f ( t , x ) are said to be uniformly ultimately bounded if there exist positive constants b and c, and for every or E (0 , c ) there is a positive constant T = T ( a ) such that

[Iz(to)ll < a * Ilx(t)ll 5 b, Q t > to + T (5.10) They are said to be globally uniformly ultimately bounded if (5.10) holds for arbitrarily large a.

We shall refer to the constant b in (5.10) as the ultimate bound. In the case of autonomous systems, we may drop the word "uniform" since the solution depends only on t - to. The following Lyapunov-like theorem is very useful in showing uniform ultimate boundedness.

T h e o r e m 5.1 Let D C R" be a domain containing the origin and f : [ O , m) x D - R" be piecewise continuous an t and locally Lipschitz in x . Let V : [O, m) x D - R be a continuously differentiable function such that

W I ( X ) 5 V ( t , x ) 5 Cl/z(x) (5.11) av av

+ z f ( t 3 ~ ) 5 - W ~ ( X ) , Q llzll 2 P > 0 (5.12) V t 2 0, V x E D where W I ( X ) , W z ( x ) , and W3(t) are continvous posiiive definite functions on D. Take r > 0 such that B, C D and suppose that p is small enough that

rnax W z ( x ) < min W t ( x ) 1111


Let q = maxll,ll

5.2. NONVANISHING PERTURBATION 213

Corollary 5.2 Under the assumptions of Corollary 5.1, the solutions of x = f ( t , X ) satisfy

II4t)ll < P(IIx(to)ll, t - to ) + a ; l ( a z ( ~ ) ) > v t l t o (5.16) Corollary 5.3 Suppose the assumptions of Theorem 5.1 are salisfied with

for some positive constants kj and c. Suppose p < r ( k l / k z ) ' / c and (Ix(to)(( < r ( k l / k 2 ) ' l c . Then, (5.14)-(5.15) take the form

where k = (kz/kl) ' / ' and y = (R3/kac). It is significant that the ultimate bound shown in Corollary 5.1 is a class K:

function of p, because the smaller the value of p the smaller the ultimate bound. As p --t 0, the ultimate bound approaches zero.

Let us illustrate how Theorem 5.1 is used in the analysis of the perturbed system (5.1) when the origin of the nominal system is exponentially stable. Lemma 5.2 Let z = 0 be an exponentially stable equilibrium point of the nominal system (5.2). Let V ( t , z ) be a Lyapunov function of the nominal system that satisfies (5.3)-(5.5) in [O, m) x D, where D = { z E R" ( ( lx( l< P). Suppose the perturbation term g ( t , z ) satisfies

(5.19)

for all t 2 0, all z E D, and some positive constant 8 < 1. Then, for all (1z(to)ll < a?, the solution z ( t ) of the perizrbed system (5.1) satisfies and

Ilz(t)ll l b, V' t 2 t l for some finite time t l , where


Proof: We use V ( t , z ) as a Lyapunov function candidate for the perturbed system (5.1). The derivative of V ( t , z ) along the trajectories of (5.1) satisfies

Application of Corollary 5.3 completes the proof.

Note that the ultimate bound 6 in Lemma 5.2 is proportional to the upper bound on the perturbation 6. Once again, this result can be viewed as a robustness property of nominal systems having exponentially stable equilibria a t the origin because it shows that arbitrarily small (uniformly bounded) perturbations will not result in large steady-state deviations from the origin.


where p > 0 is unknown and d(t) is a uniformly hounded disturbance that satisfies Id(t)l 5 6 for all t > 0. This is the same system we studied in Example 5.2 except for the additional perturbation term d(t). Again, the system can be viewed a9 a perturbation of a nominallinear system that has a Lyapunov function V(2) = z T P z where

We use V(z) as a Lyapunov function candidate for the perturbed system, but we treat the two perturbation terms 42; and d(t) differently since the first term vanishes at the origin while the second one does not. Calculating the derivative of V(z) along the trajectories of the perturbed system, we obtain

where we have used the inequality

(221 + 5221 C J I x I l z ~

5.2. NONVANISHING PERTURBATION 215

and icz is an upper bound on 1x2(. Suppose P 5 4(1 - {)/3k;, where 0 < { < 1. Then,

where 0 < 9 < 1. As we saw in Example 5.2, is bounded on R, by 96~129. Thus, if 0 5 0.4(1- { ) / c and 6 is so small that p2X,,(P) < c, then B,, C Rc and all trajectories starting inside R, remain for all future time in R,. Furthermore, the conditions of Theorem 5.1 are satisfied in R,. Therefore, the solutions of the perturbed system are uniformly ultimately bounded with an ultimate bound

In the more general case when the origin x = 0 is a uniformly asymptotically stable equilibrium point of the nominal system (5.2), rather than exponentially stable, the analysis of the perturbed system proceeds in a similar manner.

L e m m a 5.3. Let x = 0 be a uniformly asymptotically stable equilibrium point of the nominal system (5.2). Let V ( t , x ) be a Lyapunov function of the nominal system that satisfies the inequalities4

in [0, m) x D, where D = {I E Rn 1 11x11 < r } and ai(.) , i = 1 ,2 ,3 ,4 , are class X functions. Suppose the perturbation term g ( t , x ) satisfies the uniform bound

'The existence of a Lyapunov function satisfying these inequalities (on a bounded domain) is guaranteed by Theorem 3.14 under some additional assumptiom.


for all t > 0, all x E D, and some positive constant 8 < 1. Then , for all llx(to)ll < a ; ' ( a l ( r ) ) , the solution x ( t ) of the perturbed sys tem (5.1) satisfies

I < l l ( 0 I l - 0 v to < t < t l and

I 5 P v t 2 t l f o r s o m e class XL function P(., .) and some finite t i m e t l , where p(6) is a class K funct ion o f 6 defined by

a4(r) ~ ( 6 ) = a;' (a. (a;' (+)))

Proof: We use V ( t , x ) as a Lyapunov function candidate for the perturbed system (5.1). The derivative of V ( t , x ) along the trajectories of (5.1) satisfies

,, ,.

< -a3(llxll) + ~a4(11~11) < -(I - S)a3(11~II) - ~ ~ 3 ( 1 1 ~ 1 1 ) + 6a4(r ) , 0 < 8 < 1 5 -(I - ~)a3(11xII), v 11x11 > 013' (Y)

Application of Theorem 5.1 and Corollary 5.1 completes the proof.

This lemmais 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, 6 is required to satisfy (5.19). The right-hand side of (5.19) approaches m as r + co. Therefore, if all the assumptions hold globally, we can conclude that for all uni formly bounded disturbances, the solution of the perturbed sys tem will be uni formly bounded. This is the case because, for any 6 , we can choose r large enough to satisfy (5.19). In the case of uniform asymptotic stability, 6 is required to satisfy (5.23). Inspection of the right-hand side of (5.23) shows that, without further information about the class K functions, we cannot say anything about the limit of the right-hand side as r - m. Thus, we cannot conclude that uniformly hounded perturbations of a nominal system with 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 statement is not true. It is possible to construct examples (Exercise 5.19) where the origin is globally uniformly asymptotically stable, but a bounded perturbation could drive the solution of the perturbed system to infinity.

5.3. INPUT-TO-STATE STABILITY 217

5.3 Input-to-State Stability Consider the system

i = f ( t , x , u ) (5.24) where f : [O, m) x D x D, + R" is piecewise continuous in t and locally Lipschitz in x and u , D C Rn is a domain that contains x = 0, and D. C Rm is a domain that contains u = 0. The input u ( t ) is a piecewise continuous, bounded function of t for all t 2 0. Suppose that the unforced system

has a uniformly asymptotically stable equilibrium point a t the origin x = 0. By viewing the system (5.24) as a perturbation of the unforced system (5.25), we can apply the techniques of the preceding section to analyze the input-to-state behavior of (5.24). For example, if the unforced system satisfies the assumptions of Lemma 5.3 and the perturbation term satisfies the bound

for all t 2 0 and all ( I , u) in some bounded neighborhood of ( x = 0, u = 0 ) , then the conclusion of Lemma5.3 shows that for sufficiently small 11x(to)ll and supttto IIu(t)ll, the solution of (5.24) satisfies

This inequality motivates the following definition of input-to-state stability

Definition 5.2 The system (5.24) is said to be locally input-to-state stable if there exist a class KL: function P, a class K: function y, and positive constants k1 and k z such that for any initial state x ( t o ) with llx(to)ll < k , and any input u ( t ) with sup,>,o IIu(t)ll < k z , the solution x ( t ) exists and satisfies

-

for all t 2 to 2 0. It is said to be input-to-state stable if D = Rn, D, = Rm, and inequality (5.27) is satisfied for any initial state x ( t o ) and any bounded input u ( t ) . Inequality (5.27) guarantees that for a bounded input u ( t ) , the state x ( t ) will he bounded. Furthermore, as t increases, the state x ( t ) will be ultimately bounded with an ultimate bound that is a class K function of sup,>,, llu(t)ll. We leave it to

-


the reader (Exercise 5.12) to use inequality (5.27) to show that if u ( t ) converges to zero as t + m, so does ~ ( t ) . ~ Since, with u ( t ) E 0 , (5.27) reduces to

Ilz(t)ll < 8(1 lz ( to) l l~ t - to ) local input-to-state stability implies that the origin of the unforced system (5.25) is uniformly asymptotically stable, while input-to-state stability implies that it is globally uniformly asymptotically stable.

The following Lyapnnov-like theorem gives a sufficient condition for input-to- state ~ t a b i l i t y . ~ Theorem 5.2 Let D = {x E R" 1 11211 < T}, DU = {U E Rm I llull < r u ) , and f : [ O , cu) x D x D, + Rn be piecewise continuous i n t and locally Lipschitz i n z and u . Let V : [ O , m) x D + R be a continuously differentiable function such that

V (1, z , u ) E [0 , m) x D x D, where al, a2, a3 and p are class K functions. Then , ihe 1

system (5.24) is locally input-lo-state stable with y = a;' o a z o p , kl = a; ( a l ( r ) ) , and k2 = p-l ( m i n { k l , p ( r , ) } ) . Moreover, if D = Rn, D, = Rm, and a1 i s a class K, function, then the system (5.24) i s input-to-state stable with y = a;' o a2 o p.

0

Proof: Application of Theorem 5.1 (with Corollaries 5.1 and 5.2) shows that for any z ( t o ) and any input u ( t ) such that

the solution z ( t ) exists and satisfies

Since the solution z ( t ) depends only on u(r) for to < T 5 t , the supremum on the right-hand side of (5.31) can be taken over [to, t] which yields (5.27).7 In the

5Another interesting use of inequality (5.27) will be given shortly in Lemma 5.6. 6For autonomous systems, it has been shown that the conditions of Theorem 5.2 are also

necessary; see [164]. 71n particular, repeat the above argument over the period [O,T] to show that

llz(u)ll 5 O,llz,to)ll.. - to) + Y ( WP llu(T)ll) . V to 5 0 5 T tO


global case, the function a;' o a, belongs to class IC,. Hence, for any initial state z(to) and any bounded input u(t), we can choose r and r, large enough that the inequalities (5.30) are satisfied.

The next two lemmas are immediate consequences of converse Lyapunov theorems.

Lemma 5.4 Suppose that, in some neighborhood of ( z = 0,u = 0), the function f ( t , z , u) is continuously differentiable and the Jacobian matrices [af/az] and [aflau] are bounded, uniformly in t. If the unforced system (5.25) has a uniformly asymptotically stable equilibrium point at the origin z = 0, then the system (5.24) is locally input-to-state stable. 0

Proof: (The converse Lyapunov) Theorem 3.14 shows that the unforced system (5.25) has a Lyapunov function V( t , z ) that satisfies (5.20)-(5.22) over some bounded neighborhood of x = 0. Since [af lau] is bounded, the perturbation term satisfies (5.26) for all t > to and all (x ,u) in some bounded neighborhood of (z = 0, u = 0). I t can be verified that V(t,x) satisfies the conditions of Theo- rem 5.2 in some neighborhood of ( z = 0, u = 0).

In the case of autonomous systems, the boundedness assumptions of Lemma 5.4 follow from continuous differentiability of f (x , n). Therefore, for autonomous systems the lemma says that ef f ( z , u) is continuously differentiable and the origin of (5.25) is asymptotically stable, then (5.24) is locally input-to-stale stable. Lemma 5.5 Suppose that f ( t , x , u ) is continuously differentiable and globally Lip- schitz in (z,u), uniformly in t . If the unforced system (5.25) has a globally exponentially stable equilibrium point at the origin x = 0, then the system (5.24) is input-to-state stable. 0

Proof: (The converse Lyapunov) Theorem 3.12 shows that the unforced system (5.25) has a Lyapunov function V( t , z ) that satisfies (5.3)-(5.5) globally. Due to the uniform global Lipschitz property of f , the perturbation term satisfies (5.26) for all t 2 to and all (x, u). It can be verified that V(t, x) satisfies the conditions of Theorem 5.2 globally.

Recalling the last paragraph of the preceding section and Exercise 5.19, it is clear that if the origin of the unforced system (5.25) is globally uniformly asymptotically stable but not globally exponentially stable, then the system (5.24) is not necessarily input-to-state stable even when f is globally Lipschitz in (2, u). We illustrate the use of Theorem 5.2 by three first-order examples.

Example 5.6 The system x = - x 3 + u

220 CHAPTER 5. STABEITYOFPERTURBED SYSTEMS

has a globally asymptotically stable origin when u = 0. Taking V = i z 2 , the derivative of V along the trajectories of the system is given by

where 0 is any constant such that 0 < 8 < 1. Hence, the system is input-to-state stable with y(a) = (a/8)'I3. A

Example 5.7 The system

has a globally exponentially stable origin when u = 0, hut Lemma 5.5 does not apply since f is not globally Lipschitz. Taking V = i z 2 , we obtain

Hence, the system is input-testate stable with y(a) = aZ.


Similar to the previous example, when u = 0 the origin is globally exponentially stable but Lemma 5.5 does not apply since f is not globally Lipschitz in z . This time, however, the system is not input-to-state stable as can be seen by taking u(t) G 1. The solution of the resulting system

that starts at z(0) = 0 diverges to co; note that X 2 2 . According to Lemma 5.4, the system is locally input-to-state stable. Theorem 5.2 can be used to estimate the bounds on the initial state and input (the constants El and E2 in Definition 5.2). Let D = {JzJ < r) and D, = R. With V(z) = $z2, we have

where 0 < 8 < 1. Thus, the system is locally input-to-state stable with El = r, E2 = r8/(1 + r2), and y(a) = a( l + r2)/8. A


An interesting application of the concept of input-to-state stability arises in the stability analysis of the interconnected system

where fl : [O,m) x Dl x Dz + Rnl and fz : [O,m) x D2 + Rn2 are piecewise continuous in t and locally Lipschitz in z ;

The set Di is a domain in Rn' that contains the origin zi = 0; in the global case, we take Di = Rn=. Suppose that

has a uniformly asymptotically stable equilibrium point at z l = 0, and (5.33) has a uniformly asymptotically stable equilibrium point a t 2 2 = 0. It is intuitively clear that if the input z2 into the 21-equation is "well behaved" in some sense, then the interconnected system (5.32)-(5.33) will have a uniformly asymptotically stable equilibrium point at z = 0. The following lemma shows that this will he the case if (5.32), with 2 2 viewed as input, is input-to-state stable.

Lemma 5.6 Under the stated assumptions,

If the sys tem (5.32), with zz as input , i s locally input-to-state stable and the origin of (5.33) i s uni formly asymptotically stable, then the origin of the interconnected sys tem (5.32)-(5.33) i

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Nonlinear Systems 2nd- Hassan K. Khalil