STABILITY AND STABILIZATION OF NONLINEAR DYNAMICAL SYSTEMS

Exponential and asymptotic stability for a class of nonlinear dynamical systems with uncertainties is investigated. Based on the stability of the nominal system, a class of bounded continuous feedback controllers is constructed. By such a class of controllers, the results guarantee exponential and asymptotic stability of uncertain nonlinear dynamical system. A numerical example is also given to demonstrate the use of the main result.


INTRODUCTION
In recent decades, the stability problem of nonlinear systems have been extensively studied ([1]- [3] and [4]).It is well known that the study of stability theory of nonlinear dynamical systems is carried out by one of two Lyapunov methods, one is the Lyapunov's linearization method, and the other is the Lyapunov's direct method which concerns with construction of the Lyapunov function.The stability problem has motivated the study of Lyapunov function in both finite ( [3], [5] and [6]) and infinite dimensional ( [1] and [2]) spaces.Here, the Lyapunov's direct method is used.It is the purpose of this paper to investigate the exponential and asymptotic stabilization for nonlinear dynamical systems with control constraint.This paper is organized as follows.In section II, a theorem which is a criterion for the exponential and asymptotic stability is proposed.Furthermore, based on this theorem, a bounded and continuous state feedback control is proposed to guarantee the exponential and asymptotic stability.In section III, a numerical example is given to illustrate the use of our main result.Finally, the conclusion follows in section IV.

PROBLEM FORMULATION AND MAIN RESULT
Consider a class of uncertain nonlinear dynamical systems described by the following state equations: ẋ(t) f (t, x) F(t, x) .(t, x,u), t t 0 0 where t R is time, x(t) R n is the state vector, u(t) R m is the control vector, and (t, x,u)represents the system uncertainties.The function, ( , , ): R n m , and f ( , ):[0, ) R n R n , are assumed to be continuous.
The corresponding system of (1) without uncertainties, called the nominal system, is described by ẋ(t) f (t, x), t t 0 0 x(t 0 ) x 0 . ( We assume further that the equation (2) has a unique solution corresponding to each initial condition and the origin is the unique equilibrium point.The state feedback controller can be represented by a nonlinear function in the form Now, the question is how to synthesize a state feedback controller u(t) that can guarantee the asymptotic and exponential stability of nonlinear dynamical system (1) in the presence of uncertainties (t,x,u).
Before giving our synthesis approach, we give some definitions and prove sufficient conditions for the asymptotic and exponential stability of system (2).
Definition 1.The equilibrium zero of (2) is stable if, for each > 0 and each, t o R , there exists a ( ,t 0 ) such that x 0 ( ,t 0 ) implies x(t, x 0 ) , t t 0 0.
Definition 2. The equilibrium zero of (2) is attractive if, for each, t 0 R ,there is an (t 0 ) 0 such that x 0 (t 0 ) implies that the solution x(t, x 0 ) approaches zero as t ap- proaches infinity.
Definition 3. The equilibrium zero of ( 2) is asymptotically stable if it is stable and attractive.
Definition 4. The equilibrium zero of ( 2) is exponentially stable if there exist positive constants,, , k and such that The following theorem provides sufficient conditions for the asymptotic and exponential stability of system (2).
Theorem 1. Assume there exist a sufficiently smooth function V(t,x), positive constants 1 , 2 , 3 , p and q such that, for all and for all t t 0 0 and for all x(t) R n and the derivative of V along the solution of (2) satisfies Then the equilibrium point of the system (2) is asymptotically stable.Moreover, it is exponentially stable if p = q. Proof.Let Then, from ( 5), ( 4) and (3), we have Integrate both sides of (6), we have, for all t t 0 0 Hence, it follows from (3), (5), and (7), we get where k = .Let > 0 be given and ( ,t 0 ) k p / q then whenever x 0 ( ,t 0 ) we have , t t 0 0.
Therefore, the equilibrium zero of (2) is stable.Moreover, one can easily see that the righthand side of (8) approaches zero when t approaches infinity.Hence, the equilibrium zero of ( 2) is attractive and therefore asymptotically stable.In particular, when p = q the inequality (8) becomes x(t) k x(t 0 ) e (t t 0 ) , t t 0 0.
that is the equilibrium zero of ( 2) is exponentially stable.
We shall use Theorem 1 to fine the condition on u(t) that can guarantee the asymptotic and exponential stability of nonlinear dynamical system (1).Let us introduce for system (1) the following assumptions: (B1) The components of the control vector are physically limited by with c i 0, i 1,2,..., m.
(B2) There exist a sufficiently smooth function W(t,x), positive constants 1 , 2 , p and q such that for all x R n , for all t t 0 0, we have and the derivative of W along the solution of the nominal system ẋ(t) f (t, x) satisfies Remark : The nominal system ẋ(t) f (t, x)is stable with (B2) (See [3] pp. 53-54).
(B3) There exist positive continuous functions (t, x), f 1 (t, x), f 2 (t, x), f 3 (t, x)and posi- tive constants 3 and such that where Theorem 2. The system (1) satisfying the assumptions (B1)-(B3) is asymptotically stable and if p = q it is exponentially stable under the control Here and . Proof.

CONCLUSION
In this paper, the exponential and asymptotic stabilization of nonlinear dynamical systems with control constraint has been considered.A bounded and continuous state feedback control for the exponential and asymptotic stability for the closed-loop system is proposed.Finally, a numerical example has also been given to demonstrate the use of our main result.

Figure 1 :
Figure 1 : The state trajectories of the feedback-controlled system for (25).