Proceedings of the 2OQ1 IEEE International Conference on Control Applications September 5-7,2001 Mexico City, Mexico Passification by Output Injection of Nonlinear Systems Edmundo Rocha-C6zatl and Jaime Moreno Coordinacibn de Automatizacibn Instituto de Ingenieria, U.N.A.M. Ap. Postal 70-472 Ciudad Uniyersitaria 04510 COYOACAN D.F., MEXICO eddrochaQautomatizacion.iingen.unarn.mx morenoQpumas.iingen.unam.mx Abstmct- Passivity is an important and useful property for analysis and design of nonlinear control systems. The problem of when a system can be rendered passive by state or output feedback has being very well studied in the last years. For linear time invariant systems it follows by duality that if a system is state feedback passive, then it can also be made passive by output injection. Since such a duality concept is (in general) not true for nonlinear systems in this paper sufficient conditions for rendering a nonlinear system passive by output injection are given. We concentrate on global conditions for (strict) passivity, but similar results can be obtained along the same lines for local or semiglobal conditions. Keywords: Dissipative and Energy-based Design, Geometry and Structure of Non-linear Systems, Lyapunov Design, Non-linear Observers. I. INTRODUCTION Passivity is a very basic concept in control theory. Its physical motivation, as an energy balance concept, makes it very appealing. Furthermore, it has shown to play a key role in analysis and design of linear and nonlinear control systems, and there is a vast literature about this concept and its possible uses (see [7] for a recent summary of the subject). Powerful constructive algorithms for the control of nonlinear systems have been developed in recent years taking advantadge of passivity concepts [SI. Although the basic concepts and characterizations of passivity were in- troduced and developed in the 1960's and the 19707s, a fundamental question was posed and solved recently: when can a nonlinear system be rendered passive via state feed- back? This question is of utmost importance in the control context, since it extends the class of systems that can be given the nice properties that passive systems share. This question was solved in [l] (see also [2] for further results), and, under mild regularity assumptions, it basically states that a system can be rendered passive by state feedback if and only if it has relative degree 1 and is minimum phase. It is interesting to note that for linear time invariant sys- tems, by duality, a system can be rendered passive by state feedback (and a change of input) if and only if it can be made passive by output injection (and a change of output). Although trivial this seems not to have been noticed be- fore. Passification by output injection is the natural con- cept for the observer design, and it can be used for this porpose in a similar manner as the state feedback one is used for stabilization. Since the duality concept is not valid in the nonlinear context, it is of interest, both theoretically and for observer design objectives, to pose the question: When can a finite-dimensional (quadratic) nonlinear sys- tem be rendered passive by output injection? The aim of this paper is to give a partial answer to this question and to study the relationship between the output injection and the state (and output) feedback passification problems. For nonlinear systems all these properties can be local, global or semiglobal, and passivity can be strict or not. Instead of giving all possible versions we will concentrate in the global case. Other versions can be stated likewise. We will not consider applications for design of these results, which are going to be explored somewhere else. The rest of the paper is organized in the following form. In section I1 some preliminary concepts will be recalled and the problem will be precisely posed. Section I11 presents the main result, i.e. sufficient conditions for a nonlinear system to be output injection passifficable. Section IV dis- cusses the relationship between state and output feedback, and output injection passive systems. In section V an ex- ample will be presented, to illustrate the possibility of ren- dering a concrete system passive by these three ways. Fi- nally, in section VI some conclusions are stated. 11. PRELIMINARIES Consider the nonlinear control system [4] (1) j.=f(z)+g(x)u, z(O)=zo,' Y=h(x) , where x E R" is the state, y E R" the output and U E R" the input vectors. Let f(x) and the m columns of g(x) be smooth vector fields and h (x) a smooth mapping. Sup pose that (1) has an equilibrium point at the origin, i.e. f (0) = 0, and h (0) = 0. The set of admissible inputs is as- sumed to be all locally square integrable Lze. It is assumed that outputs are also in L2,. Denote as x (t, 20, U (t)) the solution of (1) that start at xo when t = 0 and is excited by ~(t), and as y(t,xo,u.(t)) = h(x(t,xo,u(t))) its corre- sponding output. If no confusion arises these functions will 0-7803-6733-2/01/$10.00 0 2001 IEEE 1141