2154 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. zyxwv IO. OCTOBER IWJ zyx Structure Invariance for Uncertain Nonlinear Systems R. Castro-Linares and C. H. Moog work of [ 11 where the so-called generalized matching condition for a class of nonlinear systems is introduced. More recently, the robust output tracking problem was addressed in [I81 for a class of single-input single-output (SISO) nonlinear systems, of which the zyx Absfract-A unified study of three control problems associated to multi- input multi-output nonlinear systems with uncertainties is presented, namely, input-state linearization by static state feedback, input-output decoupling by static state feedback and input-output decoupling by dynamic compensation. For each of these problems, geometric conditions which describe intrinsic structural invariance properties are given. I. INTRODUCTION During the last two decades, nonlinear control theory was very actively developed (see e.g., [14], [16], [20] and the references therein). In particular, geometric and algebraic methods have received considerable attention in the literature. Exact control problems such as linearization and decoupling of the closed-loop system have been studied in various ways, e.g., linearization of the closed-loop state equations [ 131, [ 151, linearization of the closed-loop input-output map [ 141, and input-output decoupling of the closed-loop system [9], [ 121. A major question arises when the model of the system contains uncertain elements such as constant or varying parameters that are not known or imperfectly known. Under such imperfect knowledge of the model, one tries to design a control such that the model can still achieve the desired closed-loop behavior. The matter of robustness of such control methods is often argued in some case studies. The idea for studying some classes of uncertainties in the system model is growing in the current literature since it aims to develop a general theory for robustness. To deal with uncertain nonlinear systems, two main approaches have been proposed in the literature: adaptive control and Lyapunov-based control. The first one is applied to systems with parameterized uncertainties (see e.g., [22]), while the second allows nonparameterized uncertainties. The Lyapunov-based approach relies on an explicit construction of a Lyapunov function from which a state feedback control is synthesized assuming bounds on the uncertainties. To obtain either stabilization or tracking, however, some assumptions were introduced regarding the structure of the uncertainties. That is to say that the uncertainties have to enter into the state equation in a certain way; such conditions are often referred to as matching conditions. Some studies have been carried out on the stability analysis of uncertain dynamical systems not satisfying matching conditions. In [21], it is pointed out that mismatched uncertainty may affect robust stability when the feedback linearizing method is applied. Such systems which contain mismatched uncertainties are not considered in the rest of this paper. In addition, several authors have contributed to the robust tracking problem for nonlinear systems with uncertainties. A multivariable tracking problem is studied in [IO], using a measurement of the tracking error which is a general function of the system's state and input; the resulting controller is robust in the sense that the tracking error is ultimately bounded in the presence of modeling errors which satisfy the matching conditions. A similar result is obtained in the Manuscript received August 9, 1993; revised December 13, 1993. The authors are with the Laboratoire d'Automatique de Nantes, Unit6 AssociCe au C.N.R.S., Ecole Centrale de Nantes, UniversitC de Nantes, 1 rue de la No&, 44072 Nantes Cedex 03, France. R. Castro-Linares is on sabbatical leave from CINVESTAV-IPN, Dept. of Electrical Eng., Apdo. Postal 14-740, 07000 Mexico, D.F., Mexico. IEEE Log Number 9403957. uncertainties may not satisfy the conventional matching condition. using a Lyapunov-based approach. In this note, we consider a multi-input multi-output (MIMO) nonlinear system in the presence of uncertainties, referred to as the uncertain system, described by zyxwv (1) where zyxwvut s E W" is the state vector, zyxwv 11 E zyxwvu RP is the input vector, y E R" is the output vector, f(s) and thep columns zyxw gl(s) zyxw :...g,( r) of the matrix g(x) are meromorphic vector fields of s. and thep components hl (s). . . . . hp(r) of the vector h(s) are meromorphic functions of s. Af(.r) and the p columns Ag1 (.r). . . . . AgP(.r) of the matrix Ag(s) are also meromorphic vector fields of s which represent the disturbance and model uncertainties. The corresponding nonlinear system without uncertainty, called the nominal system, is then defined as zyxwv s~: .i. = f(s) + Af(.r) + (g(.r) + Ag(n))i19 { y = h(s) j. = f(r) + g(r)7r. y = h(r) c: { i.e., Af(a) F 0 and Ag(s) 0 in (1). In the rest of the paper, dim D denotes the generic dimension of a meromorphic distribution D. i.e., its constant dimension on a suitable dense submanifold of R" . The goal of the note is to give a unified geometric study for three control problems associated to the uncertain nonlinear system F": input-state linearization by static state feedback, static state feedback decoupling, and decoupling under dynamic compensation. In particular, some geometric intrinsic conditions on the uncertainties are given for each problem. These conditions are innovative in the sense that they describe intrinsic structural invariance properties from the nominal system to the uncertain system. Under these conditions, robust trajectory tracking is studied in [3] when dynamic decoupling is considered. Equivalent results for input-state linearization and input-output decoupling by static state feedback can be found in the literature (see, for example, [l], [16], [181, [231). The note is organized as follows. In Section 11, we recall the input- state linearization problem by static state feedback and give geometric conditions so that the uncertain system in closed-loop form has a particular representation in a new set of coordinates. The same is done in Sections I11 and IV for the input-output decoupling problem by static state feedback and by dynamic compensation, respectively. Finally, a conclusion is offered in Section V. 11. INPUT-STATE LINEARIZATION BY STATIC-STATE FEEDBACK Through this section we consider an uncertain system zyx Xp and a nominal system without outputs, i.e., each system is just defined by the corresponding state equation. Given the vector fields f and the matrix g. the input-state lineariza- tion problem we address here consists in supposing the existence of a state-space coordinate transformation and a regular static state feedback defined on W". such that the nominal nonlinear system Y is equivalent to a linear controllable system ( = A< + Bil (3) with A and B constant matrices of dimension n x 71 and n x m, respectively, and 11 = col( 111.. . . .vP) a new input. It is well known 0018-9286/94$04.00 0 1994 IEEE