AJSTD Vol. 20 Issue AJSTD 1 pp Vol. 61-70 20 Issue (2003) 1
STABILITY AND STABILIZATION OF NONLINEAR DYNAMICAL SYSTEMS P. Sattayatham*, R. Saelim School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 30000. Thailand S. Sujitjorn School of Electrical Engineering, Suranaree University of Technology, Nakhon Ratchasima, 30000 Thailand Received 12 July 2002, Accepted 15 January 2003
ABSTRACT 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. Index Terms : Control constraint, feedback control, stability, stabilization, uncertainty, uncertain systems
Nomenclature Rn R n= m AT A
x ¢ x V(t, x) a
Bl (0)
n-dimensional real space Set of all real n by m matrices Transpose of matrix A Induced Euclidean norm of matrix A n Euclidean norm of x D R Gradient of smooth scalar function V(t, x) Absolute value of a real number a n Ball in R of radius l 0 and center at the origin
Parly supported by Suranaree University of Technology, Thailand (2000). *Corresponding author-e-mail:
[email protected]
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1.
Stability and Stabilization of Nonlinear Dynamical Systems
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.
2.
PROBLEM FORMULATION AND MAIN RESULT
Consider a class of uncertain nonlinear dynamical systems described by the following state equations:
˙ f (t, x) F(t, x). q (t, x,u), t * t0 * 0 x(t)
(1)
x(t0 ) x0 n
m
where t DR is time, x(t) DR is the state vector, u(t) DR is the control vector, and q (t, x,u) represents the system uncertainties. The function, q (u,u,u):[0, ') = R n = R m A R m , n n F(u,u):[0, ') = R n A R n= m , and f (u,u):[0, ') = R A R , are assumed to be continuous. The corresponding system of (1) without uncertainties, called the nominal system, is described by
˙ f (t, x), t * t0 * 0 x(t)
(2)
x(t0 ) x0 .
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
u(t) < a (t, x)K T (t, x). 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 q (t,x,u).
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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, to DR , there exists a b b ( ¡ ,t0 ) such that x0 ) b ( ¡ ,t0 ) implies x(t, x0 ) ) ¡ , t * t0 * 0. Definition 2. The equilibrium zero of (2) is attractive if, for each, t0 DR , there is an d(t0 ) 0 such that x0 ) d(t0 ) implies that the solution x(t, x0 ) approaches zero as t approaches 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,, l, k and a such that
x(t, x0 ) ) k x0 e < a (1
