Lyapunov theorems with semidefinite functions

LY APUNOV THEOREMS WITH SEMTDEFTNTTE FUNCTIONS ...

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Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China

LYAPUNOV THEOREMS WITH SEMIDEFINITE FUNCTIONS

Abderrahman Iggidr",l Boris Kalitine** Gauthier Sallet'" .. CONGE Project, INRIA Lorraine fj UPRES A 7035 Dept. of Mathematics, University of J.\Jetz I.S, G,M,P. Bat, A, !le du Saulcy 57045 Metz cedex 01, France, phone +33 3 87 54 72 80, Fax: +33 3 87 54 72 77 e-mail: [email protected]@loria.fr

** Belarusian State University Faculty of Applied Mathematics & Computer Science Chair of Optimal Control Methods 220050, Republic Belarus, Minsk, Fr. Skariny au., 4 e-mail: [email protected]

Abstract: This paper gives a new generalization of Lyapunov theorems. It shows how to explore the stability properties of a given system by using a Lyapunov function which is nonnegative semidefinite, rather than positive definite. Several examples and an application to the algebraic Riccati Equation are given to illustrate the new approach. Copyright © 1999 IFAC . Keywords: Lyapunov functions, nonlinear systems, stability, semidefinite functions, Riccati equations.

1. INTRODUCTION

In this paper, we are interested in the stability properties of an autonomous ordinary differential equation defined on an open subset oflR", with an equilibrium point. We suppose, without restricting the generality of our results that the origin is the equilibrium point:

{ x(t) = X{x(t» , X(O) = 0

(1)

X is assumed to be locally Lipschitz continuous. The most efficient tool for the study of the stability of a given nonlinear system is provided by Lyapunov theory. This theory is based on the use of positive definite functions that are nonicreasing along the solutions of the considered system. It

can be summarized as follows : If there exists a positive definite function V whose derivative, V, along the trajectories of system (1) is negative semidefinite then the syst.em is stable and it is asymptotically stable if V is negative definite.But finding an appropriate positive definite Lyapunov function is in general a difficult task. Thanks to LaSalle's invariance principle (LaSalle and Lefschetz, 1961; Krasovski, 1959), the assumption on V in the asymptotic stability theorem !;tas been considerably relaxed: the definiteness of V is more required. It is sufficient to have V :5 0 and the largest positively invariant set contained in the locus V = 0 must be reduced to the equilibrium point. This is very useful in practice, for it is easier to find Lyapunov fundions satisfying these assumptions than it is to find Lyapunov functions which satisfy the assumptions of the original Lyapunov Theorem. In (Iggidr et al., 1996), a new

1 corresponding author

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generalization of Lyapunov's theorems has been derived: the authors has relaxed the definiteness requirement not only on V ::; 0 but also on the Lyapunov function used in the stability theorem as well as in the asymptotic stability theorem. Roughly speaking, it has proved in (Iggidr et al., 1996) ~hat if there exists a function V 2: 0 such that V ::; and the set {x: V{x) = O} does not contain any complete negative orbit except the trivial one x == 0 then system (1) is Lyapunov stable and it is asymptotically stable if no solution of (1). can stay for all negative time in the set where V vanishes, other than the trivial one x == O. This result has been proved by using the properties of backward solutions of (1).

°

The first goal of this pa.per is to formulate the results of (Iggidr et al., 1996) without using the negative trajectories because we think that the concept of negative orbits (or negative trajectories or backward solutions) of a system is not a familiar concepct for the community of Automatic Control and is, may be, not very useful in practice. Our result can be formulated as follows: The null solution of system (1) is stable if it is asymptotically stable with respect to perturbations belonging to the set where the Lyapunov function V vanishes. It is asymptotically stable if it is asymptotically stable with respect to perturb at ions belonging to the largest invariant set contained in the set where the difference V vanishes. We also give a proof of this result that avoid the use of backward solution and so this proof can be used to derive the analogous results for difference equations Xk+l = f{Xk) even when it is not possible to define negative solutions. The second goal is to give some examples and applications in order to show ho\,.,. this tool is easy to use for exploring the stability of a given nonlinear system: it is often much easier to find a nonnegative Lyapunov function satisfying the conditions of our results than it is to find a positive definite one which satisfies the assumptions of the classical Lyapunov's theorems. Our result allows also to give simpler proofs for some classical results as it is done in this article (Section 3) for the algebraic Riccati Equation and as it has done in (Sepulchre et al., 1997) in connection with passivity.

14th World Congress ofIFAC

L+(x) is the w-limit set of x.

A = {x E U : L+ (x) :::: {O}} = {x E U 1imt-Hoo Xt(x) = O} is the domain of attractivity. For a given set E, XdE) = {Xt(x), x E E}

E is positively invariant if Xt(E) C E, for all t 2:: O. E is negatively invariant if Xt(E) C E, for all t ~

o.

For any positively invariant set E, AE will denote the relative domain of attractivity i.e, AE :::: An

E. Definition 1. Let E be a closed positively invariant set such that the origin belongs to E. The system 1 (or the vector field X) is said to be : Ca) stable on E if

E) C BE for all t

:1/ x - y 11< E}, B,(y) = {x E :11 x-y lis; E}, S€(y) = {x E IR,n :11 x-v 11= t}. m+ :-::: {t E 1R. : t ~ O} is the set of nonnegative =

{x E IRn

IR

n

real numbers and IR.- = {t E IR. : t S; O} the set of nonpositive real numbers.

Xt(X) is the solution of (1) starting at x, i.e., ftXt(x) = X(Xt(x)) and Xo(x) = x.

"If > 0 38 > 0 : Xt(B/i n O.

(b) asymptotically stable on E if it is stable on E and there exists J > 0 such that lim Xt(x):;;: 0, V x E EnB,j.

t-++oo

2. STABILITY THEOREMS Theorem 1. If on a neighborhood n of the origin there exists a function V E Cl (n, IR) such that

(i) I:'(x) 2: 0 for all x E n and V(O) = O. (ii) V"(x) = XY(x) = (~V"(x), X(x» ::; 0 for all xE

n.

(iii) On the positively invariant set M

{x E is

Vex) = O} The restriction of X asymptotically stable.

n :

Then the origin is a Lyapunov stable equilibrium point for system (1).

PROOF. We shall use the following two lemmas whose proofs are given in the appendix. Lemma 1. Suppose that the origin is not stable. Then there exist ( > 0, a sequence (Xn)nEIN CBE, lim x n = 0 and a strictly increasing sequence 72-+00

(tn)nElN C lR+ in such a way that

{

o::;tI!Xt(Xn) 11<£, I! Xtn(xn) 11= t: "In EN, lim tn = +00

(2)

n~oo

Throughout this paper, we shall use the following notations : B,{y)

~

lim Xtn (xn) = X E S<

n---+-oo

Lemma 2. Let E be a clolled positively invariant set for system (1), with 0 E E. If the vector field X is asymptotically stable on E then, for any compact neighborhood K of the origin satisfying En K c AB and for any ( > 0, there exists a time TE for which one has Xt(x) E B
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Let x be the point defined by Lemma 1. We claim that if is in M and that the whole trajectory {Xt(x), t E IR} of X starting at x is contained in M;

The sequence (Xn}nEN tends to the origin as n tends to +00, so there exists no E IN such that IIxnll < 8 for all n ~ no. Thus, by (4), one has

'Ve have, by continuity of V,

So, by Lemma 1, this implies that T" < tn for all n 2:: no· Therefore, we have 0 < tn - TE < tn for all n ~ no. Taking into account Lemma 1, we get

and, thanks to (ii), V(Xt,,(x n )) ~ V(x n ). So, V(x) ~ 17(xn).

= 0, it follows 17(x) = O. So x EM n-+= and therefore Xt(x) E M for all t 2:: 0 because

Since lim

Xn

M is positively invariant. It remains to show that X-t(:i:) E Ai for all t 2:: O. As previously V(X_t(x» = lim V(X_tHn(xn)).ByLemmal,

11 Xt(x n )

o ~ V(z)

2nd Proof. VVe take c sufficiently small in order

to have B,

n AI CAM .

The origin is asymptotically stable on M, so, by Lemma 2, there exists T. E m+ such that the solutions of (1) satisfy

11 Xt(Y)

11< ~,

"It

2:: T.

and Vy E Rn Ai.(3)

The continuity of the solutions with respect to initial conditions ensures the existence of § > 0 such that

11 x c 11 Xt(x) - Xt(Y) 11< 2'

V(X, y) E B.

X

BE,

y

11< 5 =:::}

'it ~ T • .

(4)

Vn ~ no·

and

=

lim V(uq,(n))

n-++=

= n--++oo lim V{X t 1>.( n. )-T (x.I.(n))

tn tends to += so for n sufficiently large, -t + tn > 0, therefore, V(X-t+tn (xn)) ~ V(x n ). We, then, deduce as before that V(X_t(x) = O.

Remark 1. The above proof is intuitive and it is based on geometric considerations. However, it uses the properties of the solution starting at x and evolving backward in time. So, it can not be used, for instance, to derive analogous result for discrete-time systems for which the backward solutions can not be well defined. To this end, we give an alternative proof that uses only the properties of solutions evolving forward in time.

'it E {O, . . . ,TE }

Therefore, the sequence (u" )n>no defined by Un = Xt~-T.(Xn) has a convergent subsequence say (u.p(n)) rl_tlO > . Let z = tl-r+CO lim u
n-+oo

Now, we shall use Lemma 2 with E = Ai and we take f: sufficiently small in order to have B,(O) n M c AB. Therefore, x E M n B,(O). Let T, be the time defined by this lemma. On the one hand, X_T.(X) E M. on the other hand X_T.(i) = lim X-T.+tn(X n), For n sufficiently n~= large, 0 < tn - T, < tn so, by the definition of tn and Xn (Lemma 1), X-T.+tn(X n ) E B , (O). Hence by taking limits we get 11 X -T. (x) II~ t", thus X-T. (x) E M n B,(O). Therefore, by Lemma 2, x = X T • (X-TJX)) E B E/ 2 (O) which is a contradiction to 11 x 11= c. This yields the theorem. 0

II<~,

~

::;

hm V(x.p(n))

n-++=

= O.

'I-"

Hence, z belongs to B. n Af and then (3) yields t

11 XT.(Z) 11< 2' Since z =

(5)

hm Xt,p( n )-T.(X(n»), there exists

1'1""";'+00

p ~ no such that

11

z - Xtp-T«x p )

11<

J. So, by

(4), we get

Finally, the combination of (5) and (6) leads to

11

Xtp(xp)

11< f.

which is a contradiction to (2). This ends the proof of Theorem 1. 0

V being a nonnegative Lyapunov function for system (1), we denote by L the largest positively invariant set contained in {x En: V(x) = a}. Clearly 1vJ = {x En: V(x) = O} c L. Theorem 2. If on a neighborhood n of the origin there exists a function V E Cl (0, IR) such that (i) ~(x) 2: 0 for all x E nand V(O) = O. (ii) V(x) = X.V(x) = ('VV(x) , X(x) ::; 0 for all x E n. (iii) On the positively invariant set L, the restriction of X is asymptotically stable.

Then the origin is is an asymptotically stable equilibrium point for system (1). PROOF. the set M is contained in L so, by Theorem 1, the origin is Lyapunov stable. Therefore, for any J > 0 there exists a positive number '1 such that any solution of (1) which starts in B"( (0) remains in B6(0) for all positive time t. 1804

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L Y APUNOV THEOREMS WITH SEMTDEFTNTTE FUNCTIONS ...

Let AL be the domain of attractivity relative to L. We choose 0 > 0, such that Bb (0) n L c AL· To show the attractivity of the origin, we shall prove that B'") (0) is contained in the domain of attractivity, i.e,

\;/x E 8,.,.(0):

lim Xt(x) = O. t-++oo

(7)

Let x E B-/(O), " be any positive real number and let L+(x) be the w-limit set of x. First of all, L+(x) i 0 because the whole trajectory {Xt(x), t 2 O} of X starting at x is contained in Eo (0). Thanks to the stability of the origin, it is possible to find TJ > 0 in such a way :

X t (B'1(O)) c B.(O), Vt:?: o. Since Bo(O) n L that

c

AL, there exists T

11 Xt(y) 11<"2'1 Vt 2: T, Vy

(8)

2: 0 such

--

E Bel (0)

11

x- Y

~ T.

Proposition 1. If there exists a function V C1(JRn , JR+) satisfying

E

=

(1) V(x) > 0 for all x E }Rn and V(O) 0, (2) 11(x) ~ 0 for all x E lR'" ( 3) On the posi ti vely in variant set L, the restriction of X is globally asymptotically stable., (4) All the solutions of system (1) are bounded, then the origin is globally asymptotically stable.

Proposition 2. If the assumptions (I), (2) and (3) of Theorem 1 are satisfied and the function V is radially unbounded ( lim V(x) = +00) then 11",11",,+00 the origin is globally asymptotically stable.

n L. (9)

The continuity of the solutions ensures the existence of a > 0 such that V(x,y) E Bs(O) X BJ(O) , 11 Xdx) - Xdy) 11< ~ Vt

in example (2) below. Nevertheless, we have the following global results that are consequences of LaSalle Invariance Principle and Theorem 2.

3. EXAMPLES Example 1. : Consider a system described by

11< 0:" ===?

{

(10)

. 3 x=y·-x, if = _y3 ,

(14)

(x,y) Em}. Now, let y be an element of L + (x). According to LaSalle Invariance Principle, y belongs to Bo (0) n L, so, by (9), we have

11 Xt(Y) II<~,

\;/t

2: T.

(11)

On the other hand yE L+(x), hence

3p E IN : 11 Xtp{x)""" y

11< CI'.

(12)

Using (12), (10) and (11) we get

(13)

TJ·

And from (8), it follows E,

+ y(t)2) == -X(t)4 -- y(t)4 + x(t)y(t) Ixl > 1 and Iyl > l.

(X(t)2

11 XT+tp(X) 11< ~ + ~ =

11 Xt(XT+tp(X)) 11<

One can remark that the linearized system around the equilibrium (0,0) is unstable so the linearization techniques do not allow to conclude about the stability of system (1). However, all the assumptions of Theorem 2 are satisfied with the semi definite Lyapunov function Vex, y) = y2. Hence, the equilibrium (x, y) == (0,0) is asymptotically stable. Moreover, all the solutions of (1) are bounded. This follows from

Vt;::

o.

if

::; 0

Therefore, all the hypotheses of Proposition 1 are fulfilled which implies that the zero solution of (1) is globally asympt.otically stable. Example 2. : Consider a system described by

This proves that

lim Xt(x) = O.

t--t+oo

0

X = -x

+ x 2 y)

{ y= -y, Now suppose that the system (1) is defined on ]Rn and there exists a nonnegative function V E Cl (m. n 1 m.+) which is a Lyapunov function for (1) that is (T(X) = X.V(x) = {VV(x),X(x) ::; o for all x E JRn . As above, L denotes the largest positively invariant set contained in {x E ]Rn : V(X) = O}. One can ask the following : Does the global asymptotic stability of the system restricted to the positively invariant set Limply the global asymptotic stability of system (1) ? The answer is unfortunately no as it can be shown

(15)

(X,y) E IR2. If we take V(x, y) = y2 th'on V(x, y) = _2y2 ::; 0,

Here, L::= {(.E, y) E lR? : y = O}. The origin is globally asymptotically stable on L but the system is not globally asymptotically stable. Indeed, one can see that the set :

{(X,Y) E R2: xy

== 2}

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is invariant, and so global asymptotic stability can not be expected.

that detectability implies uniqueness of the positive semidefinite solution of the ARE (16). Remark 3. Usually, in the literature, one ca.n find proofs of this result with the pair (C, A) observable and the use of LaSalle's Invariance Principle (see (Anderson and Moore, 1971»). Our result avoid the use of positive definite solution of the ARE.

4. APPLICATION TO ALGEBRAIC RICCATI EQUATION It is well known that if (A, B) is stabilizable and

(C, A) is detectable then there exists a unique solution P, positive semi-definite matrix, to the celebrated Algebraic Riccati Equation (ARE) : PA

+ ATp -

PBR-1BTp

+ cTe =

Remark 4. Since We do not have any hypothesis on the matrix B, our result (Proposition 3) contains the celebrated Lyapllnov criterion :

0 (16)

If P 2: 0, (C, A) detectable and P A CTC = 0 then A is stable.

with R a symmetric positive definite matrix. Furthermore it is shown that the matrix A BR- 1 BT Pis Hurwitz (see (Wonham, 1985), Theorem 12.2). The remark that this result is true with (C, A) detectable is due to Kucera (Kikera, 1972). We prove here, in an elementary manner, the following result :

Once more we do not need P > 0 and the proof is elementary (compare with (Wonham, 1985)).

5. APPENDIX

Proposition 3. If P is a symmetric positive semidefinite solutuion of (16), with (C, A) detectable, then A - BR-lBT P is a HUrwitz matrix.

Proof of lemma 1. By using the definition of unstable equilibrium, we can find f > 0, which can be chosen a.s small as we want, such that, for every neighborhood B 1 / n (0) of the origin, at least one solution of (1), starting at x,.. E Bl/n (0) does not lie entirely in the ball B. (0). Thus, there exists a point of the trajectory in the boundary Sf (0) of B.(O). \Ve define tn to be the first time for which Xt(x,.,,) intersects 5.(0). S,), we have

PROOF. "We consider the following function V(x) = xTpx. V is a semidefinite positive Lyapunov function for :i: = (A - BR- 1 BT P)x.

(17)

< tn =:;,-I! Xt(x n ) 11< c , 11 Xtn (x n ) 11= [ \/n E IN.

{ 0 :::; t

Indeed, Vex) = xT (PA+ATp_2PBR-l ET P)x. ~sing the fact that P is a solution of (16), we get

V(x) =

+ AT P +

-IICxIl 2 - (R- 1 BTpx, BT Px).

By extracting sequences ~ince Se(O) is compact, we have limn-++oo Xtn (Xn~ = X with x E S.(O).

Since R is positive definite, we have V(x) S O. It remains to show that the matrix A - BR- 1 BT P is asymptotically stable on L the largest invariant set by A - BR- 1 BTp contained in M = {x : V(x) = a}. But M = {x: ex = 0 and BTp)x = O}. Hence system (17) is governed on AI by x = Ax. Thus it is clear that the largest invariant set contained in M is contained in the unobservable subspace .!V = {or E lRn : Cx = CAx == ... = CAn-Ix = O}.

Morevover, limn-t+oo tn = +00. To prove this, suppose tha.t in is bounded. By extracting sequences i nk1 since the extracted sequence tnk has a limit T, the continuity of the flow implies

x"'" k-++oo lim Xt" ~ (X"'h)

= XT(O} =

0

W'e reach a contradiction. Since tn --+ +00, if needed by extracting sequence, we can suppose that tn is inceasing.

Detectability implies that A is asymptotically stable on N. Since L C N, this implies that A is asymptotically stable on L which ends the proof. 0

Proof of lemIrla 2. Th-~ vector field X being asymptotically stable on E:, there exists 6 > 0 such that

It is worthwhile to compare this proof with the classical proof in the literature (Sontag, 1990; Wonham, 1985).

EnB6 CAB = {x E E: lim Xt(x) = O}. t-++oo

Let then, K be any compa.ct neighborhood of t.he origin satisfying E n K C AB.

Remark 2. It is known (see (Brockett, 1970)) that there is only one positive semidefinite solution of the ARE (16) that yields a stable closed loop system. Hence our proposition proves immeditely

By stability on E, we have a ball B",(O) for which: Z E Bc, (0) :I E ==> Xt(z) E B
(18)

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Since E n K is in the basin of attraction of of the origin, for any x E E n K, we have hm X.(x) = O. Therefore, there exists Ta: 2: 0 (-++00

such that XT",(X) E BO!(O). By continuity of the flow, we have a ball B"", (x) such that:

We have En Kc

U

BT/r (x). Since En]{ is

xEEroK

compact, we can extract from this open covering of EnK a finite covering {B"'''i (x;), i == 1, ... , k}. That is En Kc

U

B,.,x, (Xi).

ifJ,EEnK

Let T~ = maxi T Xi ' Then, for any y E En K, we have for an index i, y E B"", (x;), thus X Tx , (y) E BO! (0). Therefore, by (18), .

X. (XTXi (y)) E Be/z(O), Vs

> o.

Or

Thus, Lemma 2 is proved.

6. REFERENCES Aderson, B.D.O. and J.B. Moore (1971). Linear Optimal Control. Prentice-Hall. Brockett, R. VV.(1970). Finite Dimensional Linear Syste·ms. Wiley. Iggidr, A., B. Kalitine and R. Outbib (1996). Semidefinite Lyapunov Functions : Stability and Stabilization. In MeSS, 9, 95-106. Krasovski, N. N.(1959). Problems of the theory of stability of motion. Stanford Univ. Press, Stanford, California 1963 ; translation of the Russian edition, Moskow, 1959. Kiicera, k. (1972). A contibution to matrix quadratic equations. IEEE Trans. Aut. Control, 17(3), 344-347. LaSalle, J. and S. Lefschetz (1961). Stability by Liapunov's direct method with applications. Academic Press, New-York. Sepulchre, R., M. Jankovic and P. Kokotovic (1997). Constructive Nonlinear Control. Springer Verlag, 1997. Sontag. E.D. (1990) Mathematical Control Theory - Deterministic finite dimensional systems. Texts in Applied Mathematics 6, Springer Verlag. \Vonham, W.M. (1985). Linear Multivariable Control - A Geometric Approach. Third Edition (Applications of mathematics; 10), Springer Verlag~New York.

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