A New Converse Lyapunov Result on Exponential Stability

Copyright (l:J IFAC Design Methods of Control Systems. Zurich. Switzerland. 1991

A NEW CONVERSE L YAPUNOV RESULT ON EXPONENTIAL STABILITY M. Corless* and L. Glielmo** *School of AeronauJics and AstronauJics, PUTd~ University , West Lafayelle,lndUlna 47907, USA **Dipartimento di Informatica e Sistemistica, Universita degli StuLIi di Napoli "Feduico 1/" , VUl Claudio 21 , 1-80125 Napoli, Italy

Abstract. In this paper we treat the problem of constructing Lyapunov functions for systems which are, by assumption, exponentially stable. The construction we present results in a larger set of functions than those obtainable by previously known methods. A useful property of the suggested Lyapunov functions is that they preserve information on the rate of exponential convergence of the system. Some possible applications are presented. Keywords. Lyapunov methods, stability, linear systems, linearization techniques, nonlinear systems.

1

ble systems. Even though the existence of Lyapunov functions for such systems is known, our construction results in a larger set of functions . Moreover, we claim that , if the system under consideration is exponentially stable with a "rate of exponential convergence" 1 Cl! and some further technical conditions are satisfied, one can choose any positive (3, less than Cl! but arbitrarily close to it, and construct a Lyapunov function which guarantees exponential stability of the system with rate (3. This was not possible with the previously known constructions. In the next section, we present this converse result. Section 3 illustrates the results for linear systems. Then, in Section 4, we illustrate some of its possible applications. Proofs of all the results can be found in [3].

Introd uction

It is well known that Lyapunov stability theory provides sufficient conditions which ensure various types of stability for dynamical systems described by ordinary differential equations . Over the years, this powerful class of results has been used extensively in system theory research, for both analysis and design purposes . The interested reader is referred to such classical sources as [5,6] for the basic results, and, for example, to [4] for a review of research where the results are utilized in controller design.

A less well known class of results is that of converse Lyapunov theorems . Whereas the more well-known results deduce a particular stability property of a system by the existence of a suitable (Lyapunov) function, the converse results ensure that, given a system with certain stability properties, there exists a Lyapunov function which guarantees those stability properties . It follows that , for certain classes of systems and types of stability, the existence of an appropriate Lyapunov function is not only a sufficient, but also a necessary condition . A thorough description of converse Lyapunov results can be found in [5,7].

1.1

Notation

The following notation will be employed the sequel.

R

the set of real numbers;

R+

the set of nonnegative real numbers;

B(8)

the open ball {x E Rn :

IIxll

10

<

8};

In this paper we present a new converse Lyapunov result for exponentially sta-

1 A precise definition of this individual is given in Section 2.

341

B[o]

the closed ball { x ERn Ilxll ~ O}j

a(A)

the set of eigenvalues of the square matrix Aj

Some regularity conditions are needed. Assumption 2.2 For each 8 E e, D2f( ., ·,8) exists and is continuous on R x Rn. Moreover, there exists a positive constant 1 such that

the real part of the complex number Aj

Di f

2

IID2I(t, x, 8) 11

the "block" partial derivative of the function f w .r.t. its i-th argument [1, p . 360].

for all (t,x,8) ER x Rn

(3)

e.

Our main result is stated in the following theorem.

The main result

Theorem 1 Consider the parameterized dynamical system (1) and suppose Assumptions 2.1, 2.2 are satisfied. Then, for any positive /3 < a, there exists a parameterized Lyapunov function V : R x Rn X e ~ R+ with the following properties:

We consider systems described by the differential equation x(t) = f(t,x(t),8),

X

~ I,

(1)

where t E R is the "time," x(t) E Rn is the state vector, 8 E e is a parameter vector, and f is a continuous function. A solution, corresponding to an initial condition x( to) = Xo, will be denoted by 4>( . ; to, Xo, 8) or, when no confusion is likely to arise, simply by x( . ). We need the following definitions.

1. There exist positive real numbers Wl,W2, such that

e, the function V( ., . ,8) is continuously differentiable. Moreover, for all (t,x,8) ER X Rn X S,

2. For each 8 E

Definition 1 The parameterized dynamical system (1) is (locally) uniformly exponentially stable (u. e .s') if there exist positive real numbers 0, c and a such that, for all to E R, Xo E B[o], 8 E e and t 2 to,

114>(t; to, Xo, 8)11 ~

cIlxoll e-a(t-to).

6-

=

D 1V(t,x,8)

+D2V(t, x, ())f(t, x, 8)

(2)

< -2/3V(t,x,()),

Definition 2 The parameterized dynamical system (1) is globally uniformly exponentially stable (g.u.e.s') if there exist positive scalars c and a such that inequality (2) holds for all to E R, Xo E Rn, 8 E e and t 2 to.

where

W3

(5)

is a suitable positive number.

Furthermore, if system (1) is timeinvariant, the Lyapunov function can be chosen to be time-invariant.

The scalars c and a are called a gain and a rate of exponential convergence, respectively. Note that, in the above definition, they are independent of 8. Moreover, for the case of g.u .e.s., the supremum a of all the rates of exponential convergence will be called the supremal rate of exponential convergencej this mayor may not be an actual rate of convergence . We assume the following.

Remark 1 Note that, using (4) and (5), one can demonstrate that system (1) is g.u.e.s. with rate f3 and gain (w2/wd1/2, i.e., Ilx(t)11 ~ (w2/wt)1/2I\xoll e-.8(t-to). As a consequence of this, we say that the Lyapunov functions of the above theorem preserve information on the rate of convergence of the system. 0 Remark 2 In [3] it is shown that, if e ~ R n 8 and f is continuous, then the function V of Theorem 1 can be chosen to be

Assumption 2.1 System (1) is g.u.e.s . with rate of convergence a and gain c. 342

o

continuous.

Remark 4 Sometimes the supremal rate of convergence a is an actual rate of convergence; when this is the case Theorem 1 just assures that it is possible to construct a Lya.punov function satisfying (5) with (3 < a. However, in the next section we will see that, for linear time-invariant systems, a is an actual rate of convergence iff there exists a Lyapunov function satisfying (5) with (3 = a; see Theorem 3. 0

If some additional hypotheses are satisfied, and e ~ R n 8 , it is possible to guarantee smoothness of V w.r.t. (). To make this precise, we replace Assumption 2.2 with the following two assumptions.

Assumption 2.3 Dd exists and is continuous on R X Rn X e. Moreover, there exists a positive constant / s.t.

IIDd(t, x, ())II ::; /, for all (t,x,()) ER

X

Rn

X

(7)

Remark 5 More general versions of Theorems 1 and 2, applicable to the case of local uniform exponential stability, can be found

e.

Assumption 2.4 D3f exists and is continuous on R X Rn X e. Moreover, there exists a function k : R+ X e ----> R+, continuous and nondecreasing w.r.t . its first argument, such that

IID3f(t, x, ())II ::; k(llxll, ()), for all (t,x,()) ER X Rn X e .

in[~.

In the next section we illustrate the results of Theorem 1 for linear time-invariant systems.

(8)

3

We can now state the following theorem.

Linear time-invariant systems

Theorem 1 can be readily illustrated for linear time-invariant systems. Consider a system described by

Theorem 2 Consider the parameterized dynamical system (1) with e ~ Rn0, and suppose Assumptions 2.1, 2.9, 2.4 are satisfied. Then, for any positive (3 < Q, there exists a parameterized Lyapunov function V : R X Rn X e ----> R+ with the following properties : 1. There exist positive real numbers such that inequality (4) holds.

0

x=

(10)

Ax,

with x E Rn, A E Rn xn and define /:;.

a = - max lR('x).

Wl, W2,

(11)

>'Eo-(A)

First we have the following lemma.

2. V is continuously differentiable. Moreover, inequality (S) is satisfied, and there exist positive real numbers W3, W4 such that inequality (6) holds and

Lemma 1 System (10) is g.u.e.s. iff a > O. Moreover, if (10) is g.u.e.s., then supremal rate of convergence.

II D 3V (t, x, ())II ::; W4 k(c Ilxll ,()) Ilxll (9) for all (t,x,()) ER X Rn X e.

a

is the

Consider now any (3 E (O,a) . A Lyapunov function V satisfying (4), (5) and (6) can readily be constructed as follows. Choose any symmetric positive definite Q E Rnxn and let

Furthermore, if system (1) is timeinvariant, the Lyapunov function can be chosen to be time-invariant.

V(x) ~ x T Px, where P is the unique symmetric positive definite solution of the modified Lyapunov equation

Remark 3 It should be clear that Theorems 1 and 2 apply also for any positive f3 < a, where a is the supremal rate of convergence for (1). In this case one can select any actual rate of convergence Q E ((3, a) and apply Theorem 1 or 2. The constant c appearing in inequality (9) will then be a gain associated with the rate of convergence Q . 0

P(A+ (3I)

+ (A+ (3Ifp+ Q =

0;

such a solution exists since the eigenvalues of A + f3I have negative real parts; see [61. Then, it is readily seen that V satisfies the above-mentioned inequalities. We also have the following more general 343

Assumption 4.3 The convergence of

result. Denote the eigenvalues of A by .Ai, i = 1, ... ,rn, where .Ai =I=- .Aj for i =I=- j . Let rii denote the multiplicity of .Ai, i = 1, ... ,rn, in the the minimal polynomial of A [2]. The following theorem holds.

lim z-+O

_ f::,

max lR(.A) > 0.

= -

i) The supremal rate of convergence a

1S

= -a then rii = I, i = 1, . . . , rn .

2

iii) There exists a Lyapunov function V Rn ---+ R+ for system (10) such that

•

1. V ( . , . ,0) is continuously differentiable for each E 8;

°

2 ::; w211 xl1 ,

2. inequalities (4), (6) (t,x,O) ER x Rn X 8;

f::,

V(lO)(X) = DV(x)Ax ::; -2aV(x) .

4

Linearization about equilibrium state

an

all

Corollary 1 Assume that the hypotheses of Theorem 4 hold. Then, for any real positive {3 < 0(, system (1) is tt .e.s. with rate of convergence {3.

f(t,O,O):=o

Remark 6 If e ~ R n9 and A satisfies a suitable smoothness condition w.r .t. 0, it is possible to obtain results using Theorem 2. 0

A(t,O) ~ D2 f(t,0,0)

exists for each (t,O) E R x e. Hence, the linearization of system (1) around zero is defined and is given by

A(t,O)x.

for

The following corollary readily follows .

°

x=

hold

9. inequality (5) holds for all (t, x, 0) ER x B(8) x e.

Suppose x = is an equilibrium state for system (1) , i.e.,

and

(13)

Theorem 4 Consider system (l). Suppose its linearization (12) satisfies Assumptions 4.1-{9. Then, for any real positive f3 < 0(, there exist real positive numbers 8, Wi, i ·= 1, 2, 3, and a function V : R x Rn X 8 ---+ R+, such that

an actual rate of convergence.

2 Wl IIxl1 ::; V(x)

°

Utilizing Theorem 1, one can readily obtain the following result .

>'EC1(A)

Then, the following statements are equwalent:

ii) /flR(.Ai)

=

is uniform w.r.t. (t,O) E R x e, i.e., for any f > 0, there exists '1 (f) such that Ilf(t, x, 0) - A(t, O)xll ::; f Ilxll for all x E B['1], (t,O) ER x 8.

Theorem 3 Consider system (10) and suppose that 0(

Ilf(t, x, 0) - A(t, O)xll Ilxll

It is known that, if a nonlinear system is asymptotically stable, then its linearization cannot be unstable (e.g., see [8, Corollary 4.8.8, p. 179]). In the case of a nonlinear exponentially stable system, we can make a stronger statement by utilizing Theorem 1. This is illustrated by the following theorem.

(12)

Consider the following assumptions. Assumption 4.1 System (12) is g.u.e.s. with rate of convergence 0( and gain c.

°

Assumption 4.2 For each E e, the matrix function A(·, 0) is continuous. Moreover, there exists / > such that 11 A( t, 0) 11 ::;

°

Theorem 5 Consider system (1) and suppose Assumptions 2.1 and 2.2 hold. Then its linearization (12) exists and is g.u.e.8. with 8upremal rate of convergence greater than or equal to 0( .

/. 2This is equivalent to nondefectiveness of the eigenvalues of A whose real part is -a.

344

5

of Lyapunov's second method. Symp. intern. Ecuac. dilf. ordin., 158-163, Mexico.

Conclusions

Even though exponential stability is a rather strong kind of asymptotic stability, it is of interest because, for linear systems, it is equivalent to asymptotic stability; for nonlinear systems, it is related to the stability of the linearization. The converse Lyapunov results which are presented provides a qualitative and quantitative method for the analysis of exponential stability and related properties. Their usefulness have been pointed out by presenting several new results.

6

[8] Sontag, E. D. (1990). Mathematical Control Theory. Springer-Verlag, New York.

Acknowledgments

M. Corless is supported by the US National Science Foundation under Grants MSM-8706927 and MSS-90-57079. L. Glielmo performed this work during a stay at the School of Aeronautics and Astronautics, Purdue University, and was supported by Consiglio Nazionale delle Ricerche, Italy, under Grant 203.07 .17.

References [1] Bartle, R. G. (1976). The Elements of Real Analysis. J. Wiley, New York, 2nd edition. [2] Chen, C. T. (1970). Introduction to Linear System Theory. Holt, Rinehart and Winston, New York. [3] Corless, M., and L. Glielmo. A new converse Lyapunov result on exponential stability. In preparation. [4] Corless, M., and G. Leitmann (1988). Controller design for uncertain systems via Lyapunov functions. Proc. Amer. Contr. ConE., Atlanta. [5] Hahn, W. (1967). Stability of Motion. Springer-Verlag, Berlin. [6] Kalman, R. E., and J. E . Bertram (1960). Control system analysis and design via the "second method" of Lyapunov, I: continuous-time systems. ASME J. Basic Engineering, 82, 371-393. [7] Massera, J. L. (1961). Converse theorems 345