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A synthesis of Lyapunov's first and second methodsROXANNE MARY BYRNE a & EDWARD T. WALL aa Department of Electrical Engineering, University of Colorado, Denver, Colorado, 80202,U.S.A.Version of record first published: 08 Mar 2007.

To cite this article: ROXANNE MARY BYRNE & EDWARD T. WALL (1974): A synthesis of Lyapunov's first and second methods,International Journal of Systems Science, 5:12, 1179-1191

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INT. J. SYSTEMS SCI., 1074, VOL. 5, NO. 12, 1179-1191

A synthesis of Lyapunov's first and second methods

ROXANNE MARY BYRNE and EDWARD T. WALL Department of Electrical Engineering, University of Colorado, Denver, Colorado 80202, U.S.A.

[Received 28 November 19731

This paper considera the relation which exists between the first and second methods of Lyapunov. As a result of this study a useful nth order generating technique for Lyapunov functions is developed which applies to linear and non-linear systoms. The function obtained has a negativo definite time derivative so that regions of asymptotic stability or instability can be determined. The function is derived from the equations of the system and the curl relations. The final result requires the calculation of single and inverse coefficients using recursive formulae. The basic method for generating Lyepunov functions makes use of the first method of Lyapunov. I t is then extended to systems where the first method fails, to vs r ia tions which have proved useful, and to more general and, hence, more complex forms. Two theorems are proved and examples are given in each section to illustrate the method presented.

1. Introduction Two of the most powerful tools for determining system stability are

Lyapunov's first and second methods. Lyapunov's first method requires the calculation of the eigenvalues of

a linear approximation of the system. These eigenvalues are used to deter- mine whether the system is stable, asymptotically stable, or unstable. Under very general conditions, if the corresponding linear system is either asymp- totically stable or unstable, then so is the original system. However, nothing can be concluded if the corresponding linear system is only stable. Further- more, even when the first method determines asymptotic stability, there is no technique for finding the region in which the asymptotic stability applies.

To cover the case where the first method fails and also to prove the first method, Lyapunov developed the second or direct method. See for instance, Hsu and Meyer (1968), LaSalle and Lefschetz (1964), Hahn (1963), Kalman and Bertram (1960). He was able to show that the stability of a system can be established by investigating the signs of a scalar function and its time derivative evaluated along the system trajectories. The major advantage of the second method is that i t does not require the solutions to the differ- ential equations describing the system. However, charged against this advantage is the problem of determining a suitable scalar function. For each system there is a class of scalar functions which are suitable for deter- mining stability and another class without the desired properties and from which nothing can be concluded. Since Lyapunov's original investigations there have been many contributions made in the area of Lyapunov function generation. A very good summary is given by Gurel and Lapidus (1968 IL, b). Two of the more well-known techniques, the Variable Gradient Method, Schultz and Gibson (1962), and the Format Method, Peczkowski

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l 180 R. M . Byrne and E. 7'. Ilrall

and Liu (I 967), can be used on almost all systems. Unfortunately, however, although existing methods give some guidance in choosing scalar functions, thcy are so gencral as to bc difficult to use, because the solution of a system of algebraic or differential equations relating a set of arbitrary variables is involved. Evcn for system equations where the first method of Lyapunov applies, there is no guidc as to how to choose the arbitrary variables.

The process presented in this paper, Byrne (1972), mill generate a Lyapunov function with a negtztive definite time derivative for all systems in phase variablc form which satisfy the conditions of Lyapunov's first method and one extra continuity condition. For an nth order system, in order to deter- mine a set of n - 1 constants, the mcthod requires a calcnlation of one inverse of a matrix which is related to the well-known Routh-Hurmitz matrix. The rcmaindcr of the constants and the desired Lyapnnov function are found by using recursive formulae. Furthermore, except when the inverse above does not exist, the method generates a scalar function which will determine instl~bility by Chetakv's theorem for those systems where the first method applies. See Krasoviskii (1963). The process is then extended to the case where the first method fails.

Sincc it is desired to have V negative definite, i t is necessary to start with V' and detern~ine V . One way of obtaining V from 3 is to integrate tl V = V dt = V V dz from 0 to x. However, the value of the integral will dcpend upon the path of integration. In order to guarmtee a unique Y , V V must satisfy the curl relations. The difficulty then is to satisfy these curl relations a.nd to require V = V V . z to be negative definite.

There are two approaches to this problem to be considered. First, in calculating 1' from V, a series of products of the state variables and their time dcrivatives result. These system equations give a set of relations between z and x, and these relations are employed to force terms in V to be zero. In working with the system equations, a natural form for V develops, which is of the form v= - FTDF, where D is a positive definite matrix. In parti- cular, D is a diagonal matrix with positive diagonal terms. The case where 1) is a general positive definite matrix is considered in § 4. In the second approach, we assume that, after determining the form of V V , the curl rela- tions do not hold, in general, but are valid a t the origin and that V is obtained by intcgrsting d V along the 'elbow path '. The question then considered is : what effect does this have upon the actual V ? In particular, how will P be affected under the conditions of Lyapnnov's first theorem? The approach is applied to systems represented in phase variables. The theory employed applies to any system representation but the equations become too prohibitive unless a phase variable representation is used.

2. Analysis Any system described by an nth order differential equation

can bc rcpresentcd by a first order vector differential equation and linearized into tho form :

x= F ( z ) = Ax+ R(5) (1)

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A synthesis of L?japunou's first and second methods 1181

using the substitutions

where

and

ai,= Si-,, ,, for i 2 2

If q(x) has a Taylor series expansion about the origin then llR(x)11/11x11+0 as Ilxll-tO.

From the relations xi/xj =dxi/dxj and the equations describing the system, the following identities can be derived if R(z) =O.

Simplifying the above and multiplying by arbitrary constants results in

K,, l ( - x n - , d ~ , + ~ n 4 dx,,-I)=O K3, 1( - xn-, axn + xn-, = 0

K3, 2( - xn4 d ~ , - ~ +x,-, = 0 ...

Kn, l{g(x) dxn + xn-1 ax11 = 0 Kn, 2Mx) axn-,+ 2,-2 dx1) = 0

...

Kn, ,-I'&(x) dxg+zl ~ x I ) = O

In matrix form this becomes FTK dx= 0 for any skew symmetric matrix K Let D=diag. (dl, d,, d,, .. ., d,) be any diagonal matrix. Then set

d V = F T ( K - D)dx = - {d,g(x) ax1 + d p , dx, + . . . + d,xn-, axn}

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R. M . Byrne and E . T . Wall 1182

Thus,

If dl of the di's are non-negative, is negative semi-definite. Also, if g,# 0 and all of the dils are chosen positive, v then is negative definite.

Using eqn. ( 2 ) , V can be obtained by integrating along any path from the origin to an arbitrary point x if the generalized curl relations hold. The general form for these n(n- 1) /2 equations becomes

where K,, i=di and K,,,,, j = O , where the j th column elements are given in terms of the elements from the preceding colun~n and the first column. Therefore, if the elements in the first column are known, eqn. ( 4 ) can be used to determine the remaining elements by substituting known quantities. r 7 lhen, to determine the first column elements, we make successive substitu- tions into (4) . Letting go= 1 and KO, ,=O, Kij becomes

Equation ( 5 ) gives another recursive relation between K,, and the elements from the first column. Letting i = j = 1, 2, . .., n, eqn. ( 5 ) becomes, successively,

If now we let

and

Then

where

If the even rows and columns of H are multiplied by - I , the results are the first n- 1 rows and columns of the Routh-Hurwitz Stability matrix which is used to determine if the linear system is stable.

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A synthesis of lnjapunou's first and second methods 1183

If dl > 0, i = I , 2, ..., n, and K is chosen according to eqns. (5) and (6) or eqns. (4) and ( 6 ) , then V can be determined from eqn. (2) by integrating along the 'elbow path ' from 0 to (0, 0, ..., x,) to (0, 0, ..., xn-,, xn) to ... to ( x 1 , , . x x ) . By so doing, V becomes, after grouping like terms

The matrix representation of the foregoing is

where P = { p i j } is a symmetric matrix with pij given by

With V and 3 as defined in eqns. (7) and (3), i t is possible to prove the following theorem.

Theorem 1 Let a linear system be given by (1) with R=O and gn# 0. If (HI # 0,

d,> 0 for i = I , 2, ..., n and K is determined by (5) and (6) or (4) and (6), then V(x) as given in (7) will determine the stability or instability of the system.

Proof I H I # 0 =. 3 V(x) whose time derivative is given by (3). di > 0, i = 1 , . . . , n

and g, f 0+ 3(x) is negative definite. Then V(x) can be

(i) positive definite3origin is globally asymptotically stable,

(ii) indefiniteaorigin is unstable, or

(iii) positive semidefinite => 3%" # 03 V(xo) = 0. V(zo) < 0 =>for x(l) the solu- tion to the system with x(0)=xO, V[x(t)] is a decreasing function- 3 to > 03V[x(to)] < 0 which contradicts V(x) > 0. Thus, case (iii) can- not occur.

The theorem guarantees that for linear systems which are asymptotically stable, i.e. that satisfy the Routh-Hurwitz criteria, a Lyapunov function which will prove this stability will be generated. Determining a function from eqns. (4), (6) and (7) is easier than solving the n(n+ 1)/2 equations involved in the standard linear method. If I H 1 2 0 and g,, f 0, instability can also be established using the same function. For a linear system, 1H ( = 0 implies that there is a pair of eigenvalues which are negatives of each other. A zero eigenvalue results when g,=O. Therefore, all stable systems which are not asymptotically stable fall into one of the above two cases.

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11 84 R. M . Byrne and E. T. Wall

Example 1 (i) xl = -axl - bx,

x2 = xl.

(ii) Hk =d becomes [a:I[K2, ,] = [a2+ b dl]. Chooscd,=d,=a. ThenK,, ,=l+b.

(iii) Prom eqn. (4)-(G) : 2 V = xTPx where

From eqn. (3) : v = - a(axl + b ~ , ) ~ - ax12,

(iv) (1) If a, b >O, then V is positive definite and P' is negative definite. (2) If a > 0 and b < 0, then V is indefinite and P is negative definite.

(3) If a < 0 and b < 0, then - V is indefinite and - Y is negative definite.

(4) If a = 0 and b > 0, then V is positive definite and V = 0. (5) If a = 0 and b < 0, then V is indefinite and v = 0.

(6) If a+O and b= 0, then Y = [ - 2(a2+ 1)/a] V and V is positive semi-definite.

I n the above cases, ( I ) is globally asymptotically stable, (2) and (3) are unstable, (4) is stable, and with a < 0, ( G ) is unstable. For (5) and with a > 0, (6), no conclusion can be made using the given scalar function. Now if b < 0 and a=O, a new scalar function can be chosen. Let dl= 1, d,= - b> 0, and K,, l = 0. Then 2 V =xTPx,

P=[: 11 and v = - ( b x , ) ~ + bx12

V is indefinite and Tr is negative definite. Therefore, the origin is unstable.

Ezmnple 2

(i) x,= -axl-bx2-cx,-dx, xz = x1 x, = x, x, = x,.

(ii) gl =a, g, = b, g, = c and g,=d. Hk=d becomes

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A synthesis of hyapunov's first and second methods 1185

If b and d are positive, choose dl = abc - a2d - c2, d2 = bd,, d3 = dd, and d,= 2bddl. Then

and [K4, 3 ] = [2ab3d - 2b2cd].

(iii) From eqn. ( 8 ) :

and

(abc - a2d - c2)a + 2b2c

ab2c + a2bd + bc2

(abc - a2d - c2)c + 2ab2d

(abc - a2d - c2)d

abc2 + a2cd + c3 + 2a3bd + 2b2(bc -ad)

2a2b2d + a2d2 + c2d + 2b2c2 - abcd ,

2b2cd 1 Thus 2 V =xl 'Px, with P as given by the above, and

3 = - (abc - a2d - c2)[(axl + bx, + cx, + + bx12 + dxZ2 + 2bdx32].

(iv) By Theorem 1, if (abc-a2d-c2), b and d are all positive, then the above scalar function will prove stability or instability, which ever the case may be. If either b or d or both are negative, then the system is unstable. To generate a scalar function that will prove instability, choose d2 lbd, 1, d3 = Idd, ( and d, = 2 lbdd,).

3. Non-linear systems In the non-linear case, the curl relations are not satisfied if the Ktj's are

restricted to be constants. However, if they are allowed to vary, the diffi- culties of calculation increase. Hence, the K matrix is calculatecl as in the

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11 86 R . M . Byrne and E. T . Wall

linear case, using the linearized representation of the system. Then integrat- ing 3 along the same elbow path as before, it becomes

where P i s the same as in the linear case. Under the conditions of Lyapunov's first method, i.e. IIR(x)[l/llx11+0 as 11x11--+0, if 1' is definite or indefinite, then so is V, about the origin. The remainder term, r ( x ) , provides higher power terms to the function V.

Since the curl relations did not hold except a t the origin, must be deter- mined by differentiation. Doing this and rearranging terms results in

If the system satisfies the condition of Lyapunov's First Theorem, then the terms -d ,g2(x) -d2xI2 - ... - d , , ~ , - , ~ dominate 3. In view of this, a theorem similar to that stated in the linear case can be proved.

Theorem 2 Let a system be given by ( 1 ) with IIR(x)ll/llxll-tO as IIx11-tO, Vg continuous

a t 0 , l ~ n d gn # 0 . If I H I # 0, di > 0 for i = 1, 2, . . . , n, and K is determined by (5) and (6) or (4) and (5), then V ( x ) as given by ( 8 ) will determine the stability or instability of the system.

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A synthesis of Lyapunov's first and second methods 1187

(iii) If 8 is chosen so that all of the inequalities in (i) and (ii) hold, then

and

where p and + are both quadratic forms in the vector

and represent the upper bounds of the terms involving r in V and V respectively. Therefore, 6 can be made small enough so that if P is positive definite or indefinite, then V will be likewise, a t least in some neighbourhood about the origin. Furthermore, if di > 0 and g,#O, V will be negative definite about the origin. Now let V,= xTPx. By Theorem 1, this determines the stability or instability of the linear system x=Az. Therefore, the V given above will deter- mine the stability or instability of the non-linear system.

If the system satisfies the condition of Lyapunov's First Theorem, then the terms - dlh2(z) - dp,2 - . . . - dnxn-12 dominate P. Using this assumption, a theorem similar to that stated in the linear case can be proved.

Theorem 2 Let a system be given by (1) with 11 R(x)ll/llxll+O as 11x11-+0, Vg continuous

atO,andg,#O. If IH(#0 ,d i>Ofo r i=1 ,2 ,..., a , a n d Kisdeterminedby (5) and (6) or (4) and ( 5 ) , then V(x) as given by (8) will determine the stability or instability of the system.

The consequence of Theorems 1 and 2 is that for all systems, linear or non-linear, in phase variable form for which IHl #O, Vg is continuous a t 0, and Lyapunov's first method applies, eqns. (4) (6) and (8) generate a valid scalar function from which, if the system is stable, a region of asymptotic stability can be determined.

I t should be noted that if a negative semi-definite V function is acceptable, then not all of the d,'s need be positive. A study of the remainder terms in (9) can help in deciding which of the di can be chosen to be zero. For example, if r(x) is not a function of xn, then dl can be chosen zero. Also, if the remainder term is only a function of a few of the variables, the effect of the remainder term on 3 can be calculated using eqn. (9) and the di can be chosen accord- ingly.

Example 3

(i)

( i i ) Let g1=u1(O) and g,=w'(O). Hk =d becomes [gl][K2, ,] = [d, +g,d,]. Choosed,=d,=g,. Then K2,,=1+g2.

S.S. 4 1,

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1188 R. N . Byrne and E. 1'. Wall

Differentiating V to obtain J' yields

( iv) ( I ) If u(xl) /x , 2 a > 0 and tu(x,)/x, 2 b > 0, then V and - J' are positive dcfinite and the origin is asymptotically stable.

( 2 ) If gl = 0, u(xl)/xl 2 0, and w(x2)/x, 2 b > 0, then V is positive definite and J' is negative semi-definite. Therefore, the origin is stable.

(3) If u(x,)/xl 4 a < 0 and w(x,)/x2 $ b < 0 or w(x,)/x, 2 6 > 0, then V is indefinite and 3 or - v, respectively, is negative definite. There- fore, the origin is unstable.

(4) If g, = 0, u(x,)/xl 2 a > 0, and w(x2)/x2 < b < 0,

and the origin is unstable.

(ii) g, = ru(O), g2 = h2(0), g3 = h3(0) and g4 = h,(O). Then Ilk = d becomes

Let d, = g , g g 3 - g12g, - g,2, d, =g,dl, d3 = g4dl and d, = 2gg4d,, then the .K and I-' matrices arc the same as those of Example 2, with a=g, = 10(0), b =g,=h,(O), c =IJ, = h3(0) and d =g4 = h4(0).

( i i i ) From eqn. (a), with

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A synthesis of fijapunov's first and second methods

the remaining term, RT, is

Then 2 V=xTPx+ RT. Using eqns. (4)-(a),

(iv) By Theorem 2, if the system is asymptotically stable, hi(%) and wi(x) are continuous a t the origin and the first method of Lyapunov applies, then the above Lyapunov function will prove stability and could be used to establish a region of asymptotic stability.

4. Non-linear systems where Lyapunov's first method fails I n the previous two sections, if I H I # 0 and g, p 0, then a valid Lyapunov

function could be found by choosing all of the di positive (or a subset of them in some cases). Suppose now tha t IH( =O. Immediately, H-I does not exist. As an immediate consequence, if (6) has a solution, i t is not unique.

If all of the d i are chosen to be zero, then there will be infinitely many solutions. For example, let d, =a, ( H I , i = 1, 2 , . .. , n, and ai is constant. Then a solution would be k = adj (H)a where aT = [a, + g,al, a, - g4al, . . . , a,]. If the rank of H is n- 2 then there will be only one independent solution for k (with d , = O ) and choosing any value for a that does not result in a zero k will give a basis for the solutions of k. However, not all of the di need be chosen as zero. If rank H =S, then there will be S independent choices for

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1190 R. N . Byrne and E. T . Wall

d for which thcre will be infinitely many solutions. I n fact, for each value of cl for which there is a solution, there will be n- 1-8 independent solu- tions. Once k has been determined, K , V and v are calculated using the same set of equations as before in both the linear and non-linear cases.

This case is not as straightforward as when IHI # 0 since there are many choices for d and k. However, through eqns. ( 8 ) and (9), the form for V and for 3 is already known in terms of the constants K i , j. Therefore, the effect of a particular choice can be easily calculated. In fact, the effect on v can be determined from just d and k so that these vectors can be chosen to make 3 a t least negative semi-definite and V can then be calculated.

Another usable approach is to let some of the elements in K - D be func- tions which, when x = 0 , are the same as those in the original K - D. In this case, V and v must be determined by integration and differentiation since eqns. ( 8 ) and (9) are not valid. One circumstance where this would be advantageous is when g,=O. If g,(x) is not identically zero, then i t may be substituted for g,. Also, if J H 1 = 0 , let di = ai IH(x) I , where ai is a positive constant and H ( x ) is the matrix formed similar to H but with all or some of the gi replaced by hi(x) . Another useful substitution permits Ki+, . to vary with xi. Then, when V is determined from (1 V, a term of the form

is gencrated instead of K,,,, ixi2/2. This method can be used to generate the Lyapunov function obtained by Leighton (1968) and Skidmore (1966) .

Exanzple 5 (i) A simpler Lyapunov function for the system in Example 4 can be

found if is allowed to be negative semi-definite. Choose d l = & = d4 = 0 and d , = glg,g3 - g12g4 - g a s . Then

and [K4, 31 = k71gd4-93941.

K , , , is chosen to vary with ~ ( x ) .

Z

(i i i) V = j [ - Z U ( X ) X ~ - h (x ) , x l , x Z ] ( K - D) dx 0

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A synthesis oJ Lyapunov's Jirsl and second methods 1191

This is the same Lyapunov function tha t was obtained by Skidmore (1960). Differentiating V to obtain V and simplifying results in

5. Conclusion

This paper presents a straightforward method of obtaining Lyapunov functions which will prove stability or instability for systems in phase vari- able form where the conditions of the first method of Lyapunov apply. The scalar function obtained will have a negative definite time derivative so tha t other applications of Lyapunov functions, in addition to stability can be studied. The method is generalized t o include cases where the first method of Lyapunov gives no information.

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Indus. and Eng. Chem., 80, 13. HAHN, W., 1963, Theonj and Application of Liapunov's Direct Method (N.J.,

Englewood Cliffs : Prentice-Hall). Hsu, J. C., and MEYER, A. U., 1968, Modern Control Principles and Applications

(New York : McGraw-Hill Book Co.). KALMAN, R. E., and BERTRAM, J. E., 1960, J . Bwic Eng., Trans. ASHE, 00, 371. K ~ a s o v s ~ n , N. N., 1963, Stability of Motion (Standford, California : Standford

University Press). LASWE, J., and LEFSCHETZ, S., 1964, Stability by Liapuizov's Direct Method with

Applicatiow (New York : Academic Press). LEIOHTON, W., 1968, Contr. to Diff. Eqns., 2, 367. PECZKOWSKI, J. L., and LIU, R. W., 1967, J. Basic Eng., Trans. ASME, 89, 433. SCHVLTZ, D. G., and GIBSON, J. E., 1962, A.I.E.E., 81, 203. SKIDIMORE, S., 1966, J . London Math. SOC., 41, 649.

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