Dissipative Design of Observers for Multivalued Nonlinear Systems Marisol Osorio and Jaime A. Moreno Abstract— In this article, a new dissipativity based method- ology to design observers for nonlinear systems is extended to cover systems with discontinuous and non monotone nonlinear- ities. A generalization and improvement of previous results for this class of systems is obtained. The class of systems covered are those in Lur’e form, with a LTI system in the feedforward and a discontinuous or multivalued nonlinearity in the feedback path. I. INTRODUCTION Although discontinuous systems are present in control since the early days, classical control theory is mainly developed for systems with smooth nonlinearities. This is probably related to the mathematical difficulties involved in a rigorous treatment of discontinuous systems, as for example, the lack of existence and uniqueness of solutions for these kinds of systems in a classical sense. However, in the last years a renewed interest in discontinuous systems has appeared, and a rigorous treatment is now made pos- sible by the advances in multivalued differential equations and differential inclusions. Moreover, besides the intrinsic interest in mechanical systems, where friction and backslash nonlinearities, for example, are naturally discontinuous or multivalued, the increasing interest in switching and hybrid systems calls for the necessity of studying more deeply the class of discontinuous dynamical systems. Although observation issues for dynamic systems is a basic topic in control theory relatively few work has been done for general classes of discontinuous or multivalued systems. There are many publications dealing with the prob- lem of inserting discontinuities in the observer [14], [9], [26], in order to obtain a improved estimate. For switching nonlinearities there is also a big amount of literature, in particular for the observation of hybrid and switching linear systems (see for example [18], [8], [4], and the references therein). However, only few observation results are known for general nonlinear, discontinuous or multivalued systems. An exception is the work of Juloski [18], [17], where an observer design for nonlinear systems of the Lur’e type, with a linear time invariant system in the forward path and a discontinuous or multivalued nonlinearity in the feedback path, is proposed. The basic idea of the method is borrowed from the circle criterion observers proposed by [1], [2] for This work has been done with the financial support of DGAPA-UNAM under project PAPIIT IN111905-2, UPB and Colciencias. M. Osorio is with Escuela de Ingenier´ ıa, Grupo de Investigci´ on en Autom´ atica y Dise˜ no, Universidad Pontificia Bolivariana, Cir. 1 Num. 70- 01, Medell´ ın, Colombia mosorio@upb.edu.co J.A. Moreno is with the Coordinaci´ on de Automatizaci´ on, Instituto de In- genier´ ıa, Universidad Nacional Aut´ onoma de M´ exico, Ciudad Universitaria, 04510 M´ exico, D.F., Mexico JMorenoP@ii.unam.mx Lure systems with smooth nonlinearities. Similarly to [1], [2], hard restrictions are imposed on the systems that can be treated by the method proposed in [18], [17], heavily restricting its applicability. The most important requirements of this method are: (i) The nonlinearities are restricted to be square, i.e. the number of inputs and outputs has to be equal. (ii) The nonlinearities are restricted to be (maximal) monotone. (iii) The designed observer is required to have unique solutions, despite of the fact that the observer is a discontinuous/multivalued nonlinear system. Recently, one of the authors [19] has proposed a method- ology to design nonlinear observers for smooth nonlinear systems that can be transformed to the Lur’e form, based on the dissipativity theory, the so called Dissipative Design Technique. This method generalizes the circle criterion de- sign [1], [2], eliminating both restrictions (i) and (ii). More- over, many well-known observer design strategies can be treated and improved in a unifying manner by the dissipative method. So, for example, the High-Gain [13], the Thau [21], the circle criterion [1], [2] and the Lipschitz [20] observer design methods are special cases of the method proposed in [19]. The objective of this work is to extend the dissipa- tive design of [19] to the case of nonlinear discontinu- ous/multivalued systems. It will be shown that this is pos- sible and advantageous, so that the restrictions (i), (ii) and (iii) of the method proposed in [18], [17] are eliminated. Requirement (iii), in particular, although attractive for the numerical realization and from a simulation point of view, is very restrictive. It is well known that for systems with discontinuous or multivalued nonlinearities the uniqueness of solutions is more the exception than the rule [7]. So, a general observation theory for this class of systems has to consider observers with multiple solutions. Due to this fact a big amount of effort has been done in [18], [17] to assure the well-posedness, i.e. the existence and uniqueness of solutions, of the method. Since well-posedness is not required in this paper, only the existence of solutions has to be assured, that is a much easier and good understood issue for differential inclusions. Of course, the price to be paid is the difficulty in establishing numerical algorithms that converge to the solutions of the observer, an issue shared by the whole field of differential inclusions. A further advantage of the proposed method is that for the observer design in most cases Linear Matrix Inequalities (LMIs) can be used, for which excellent and efficient numerical algorithms exist Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006 FrB05.5 1-4244-0171-2/06/$20.00 ©2006 IEEE. 5400