IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 5, MAY 2002 609 On Simplifying and Classifying Piecewise-Linear Systems Victoriano Carmona, Emilio Freire, Enrique Ponce, and Francisco Torres Abstract—A basic methodology to understand the dynamical behavior of a system relies on its decomposition into simple enough functional blocks. In this work, following that idea, we consider a family of piecewise-linear systems that can be written as a feed- back structure. By using some results related to control systems theory, a simplifying procedure is given. In particular, we pay at- tention to obtain equivalent state equations containing both a min- imum number of nonzero coefficients and a minimum number of nonlinear dynamical equations (canonical forms). Two new canon- ical forms are obtained, allowing to classify the members of the family in different classes. Some consequences derived from the above simplified equations are given. The state equations of dif- ferent electronic oscillators with two or three state variables and two or three linear regions are studied, illustrating the proposed methodology. Index Terms—Circuit analysis, nonlinear oscillators, piecewise linear approximation. I. INTRODUCTION AND MAIN RESULTS P IECEWISE-LINEAR systems are the most natural exten- sions to linear systems in order to capture nonlinear phe- nomena observed in practice. In fact, the richness of dynamical behavior found in piecewise-linear systems seems to be almost the same of general nonlinear systems: limit cycles, homoclinic and heteroclinic orbits, strange attractors The consideration of this class as an alternative to general nonlinear system is gaining popularity due to the fact that one can integrate in closed form the solutions when they are re- stricted to a region of the phase space where the system becomes linear. Nevertheless, the analysis of the corresponding dynamics is far from being trivial. As far as we know, the pioneering investigation of piece- wise-linear systems in a rigorous way is due to Andronov and coworkers [1]. Their ‘Theory of Oscillations’ remains nowadays an obligated reference, being a source of ideas for recent works [2], [3]. The analysis of piecewise-linear systems has received some growing attention after the work of Chua and coworkers on chaotic systems, see for instance [4]. Attempting to do a systematic study of piecewise-linear sys- tems, some canonical forms are tackled with in several works, see [5]–[10] and [11]. On the other hand, most of nonlinear cir- cuits found in practice do not need such rather general canon- ical forms, since they can be adequately modeled with only two Manuscript received June 19, 2001; revised December 5, 2001. This work was supported in part by the spanish Ministerio de Ciencia y Tecnología under Grant PB98-1152 and Grant DPI2000-1218-C04-04. This paper was recommended by Associate Editor M. J. Ogorzalek. The authors are with Universidad de Sevilla, Departamento de Matemática Aplicada II, Escuela Superior de Ingenieros, Seville 41092, Spain. Publisher Item Identifier S 1057-7122(02)04727-X. or three linear regions separated by parallel boundaries hyper- planes, see [12]. Moreover, for elementary circuits, the number of state variables is typically two or three. Even for those simple piecewise-linear circuits, the knowl- edge of the essential structures required to get rich nonlinear behavior is not completely well understood. Recently, several works have appeared about decomposition of circuits in func- tional blocks ([13], [14]), looking for a systematic procedure suitable for designing oscillators with prescribed properties. Our approach has some points in common with the quoted works, but we are rather interested in the analysis of the dynamics of the systems and so we search for simpler equations, i.e., canon- ical forms, as a starting point to describe the corresponding dy- namics and its eventual bifurcations. Thus, in this paper, as a previous step to facilitate their math- ematical analysis, we will study canonical forms for -dimen- sional continuous piecewise-linear circuits ( , for short) taking as reference common oscillators whose nonlinearity is a scalar piecewise-linear function with three linear pieces (the two-pieces case is obviously included by assuming that two of the three pieces glue not only continuously but also with con- tinuous derivative). To begin with, we formally introduce the family of systems to be studied through the paper. Definition 1: A differential equation with is said to be a system if there exist three vectors and one vector in , two scalars and three matrices in so that if if if (1) and is continuous, that is, we have for all such that One relevant subclass of systems can also be distin- guished, namely the case when the linear part of the outer re- gions coincide. Definition 2: A system with is called a quasi-symmetrical system. Of course, if the linear part of two adjacent regions are equal, then the continuity assumption enforces that the corresponding nonhomogeneous terms are also equal, so that the hyperplane 1057-7122/02$17.00 © 2002 IEEE