A relation between the output regulation and the observer design for nonlinear systems

K e y w o r d s O u t p u t regulation, Observer design, Exponential observers, Discrete-time systems. 1. I N T R O D U C T I O N O u t p u t regula t ion problem is the problem of controlling the ou tpu t of a control system so as to achieve asympto t i c t racking of prescribed reference t ra jec tor ies a n d / o r asympot ic rejection of undesired d is turbance signals. I t is a central problem in control theory. This problem has been solved for nonlinear control systems with a Poisson s table exosystem by Isidori and Byrnes [1]. The problem of designing observers for nonlinear systems was in t roduced by Thau [3]. Over the past three decades, considerable a t ten t ion has been paid in the l i te ra ture to the construct ion of observers for nonlinear systems. In [4], a Lyapunov-l ike method was presented for exponent ia l observer design. In [5], Xia and Gao derived a necessary condit ion for the existence of an exponent ia l observer for nonlinear systems. Explicit ly, in [5], Xia and Gao showed tha t a local exponent ia l observer exists for a nonlinear sys tem only if the l inearizat ion of the nonlinear syst em is detectable . In [2,6,7], the exponent ia l observer design problem was complete ly solved for Lyapunov s table nonlinear systems using classical and geometrical methods. In this paper , we present a new method for solving the exponent ia l observer design problem for a special t ype of Lyapunov s table nonlinear systems, namely neut ra l ly s table systems. Our method is to set up the observer design problem as an ou tpu t regulat ion problem and solve the new problem using the ou tpu t regulat ion techniques of Isidori and Byrnes [1]. 0893-9659/03/$ see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. Typeset by .4A/eS-2~X PII: S0893-9659 (02)00185-4 236 V. SUNDARAPANDIAN This paper is organized as follows. In Section 2, we discuss the state-feedback regulator problem for nonlinear systems. In Section 3, we discuss the exponential observer design problem for nonlinear systems. In Section 4, we demonstrate how the exponential observer design problem can be solved using the output regulation techniques of [1] for neutrally stable nonlinear plants. Next, we present the corresponding results for the discrete-time case. In Section 5, we discuss the state-feedback regulator problem for discrete-time nonlinear systems. In Section 6, we discuss the exponential observer design problem for discrete-time nonlinear systems. In Section 7, we demonstrate how the exponential observer design problem can be solved using the output regulation techniques of [1] for neutrally stable discrete-time nonlinear plants. 2. O U T P U T R E G U L A T I O N OF N O N L I N E A R S Y S T E M S In this section, we consider a nonlinear system of the form = f (z , ~) + g(x)u, = s ( o ~ ) , e = h(x) + q(w), (i) where x E R n is the state, u E R m the input, and e E R p the error between the plant output h(x) and the reference signal -q (w) . The state x is defined in a neighborhood X of the origin of R '~. The dynamics ~b = s(w) is the exosystem for plant (1). The state w of the exosystem is defined in a neighborhood W of the origin of R ~. We assume that the functions f , g, s, h, and q a r e C 1 and also thgt f(0, 0) = 0, g(0) = 0, s(0) = 0, h(0) = 0, and q(0) = 0. Next, we define a basic output regulation problem for plant (1). DEFINITION 1. The state feedback regulator problem for plant (1) is to find, i f possible, a state feedback law of the form u = ~(~, ~), where a(x , w) is a C 1 mapping defined on X x W such that the following two conditions axe satisfied. 1. The equilibrium x = 0 of : f (x , o) + g(x) ~(~, o) is locally exponentially stable. 2. There exists a neighborhood U C X x W of (0, O) such that, for each initial condition (x(O), w(O)) E U, the solution of the composite system = f (x , w) + g(x)c~(x, ~), = S(~), satisfies IIh(x(t)) + q(w(t))ll <_ Mexp( -a t ) I Ih (x (O) ) + q(w(O))ll, for some positive constants M and a. Next, we define a neutrally stable nonlinear system. DEFINITION 2. A C 1 nonlinear system defined by the equation