Continuous Piecewise Linear Control for Nonlinear

Selected Papers from the WSEAS Conferences in Istanbul, Turkey, May 27-30, 2008

Continuous Piecewise Linear Control for Nonlinear Systems: The Parallel Model Technique Andr´es G. Garc´ıa Universidad Nacional del Sur Dto. de Ing. El´ectrica y de Computadoras Instituto de Investigaci´on en Ing. El´ectrica (IIIE, UNS-CONICET) Av. Alem 1253, B. Blanca Argentina

Osvaldo E. Agamennoni Universidad Nacional del Sur Dto. de Ing. El´ectrica y de Computadoras Instituto de Investigaci´on en Ing. El´ectrica (IIIE, UNS-CONICET) Av. Alem 1253, B. Blanca Argentina

Abstract: This paper describes a systematic technique for obtaining controllers for Nonlinear Systems using a Continuous Piecewise Linear Approximation (CPWL) of the given Nonlinear vector field. The method proposes the use of a CPWL approximation of the Nonlinear System and then a theory is developed to show that the stabilization of the CPWL aproximation ODE yields stability for the Nonlinear ODE. An example is presented in order to show the capabilities of this idea but also the practical applicability. Finally some conclusions and future work are depicted. Key–Words: Nonlinear ODE’s, Continuous Piecewise Linear ODE’s, Control Systems.

1

Introduction

First item was pioneered studied and defined by Chua in the early’s 700 s (see [16] and [17]). This work gave origin to a Canonical Representation for a CPWL Basis able to represent any CPWL vector field developed by Juli´an (see [9]). Using this basis a reliably toolbox written in Matlab code were developed in order to approximate a given vector field with some desired degree of error (see [10]), on the other hand with a different procedure than a CPWL basis the Phd thesis of Johansson also leads a Matlab toolbox (see [11]).

Nonlinear Controller design is a very involved issue into the control community. While for linear systems there exists a wide range of methodologies from analysis to readily implementable strategies (see [1], [2] and [4]), for Nonlinear dynamics only a few methodologies are known to be effective and usually encounter in practice hard problems regarding time consuming or demanding tremendous amounts of capabilities from computer resources (see [6],[7] and [8]). Since Control problems are not more than a parameterized Initial Value Problem (IVP) for Ordinary Differential Equations (ODE’s), for designing control strategies, is natural to resort primarily to techniques able to solve -or approximate- such a problems. One methodology which has shown to be effective either from numerical issues or theoretical analysis is the use of Continuous Piecewise Linear Approximations (CPWL), this idea was early studied by Sacks in a qualitative fashion (see [14]) and later extended to approximate solutions by Girard, De Feo, Storace and Johansson (see [13], [19] and [11]). Is worth noticing that two main research streams are related to Continuous Piecewise Linear (CPWL) ODE’s:

Second item was conducted by Storace and De Feo who made several numerical experiments to investigate topological properties of Nonlinear ODE’s of low dimensions. However, this work is not proving in rigor that the properties of the given the Nonlinear ODE and the CPWL approximation are shared by both systems, they only present extensive simulations and Continuation numerical packages to show this idea (see [15]). One attempt to overcome this inconvenient providing a way to decide if both (Nonlinear ODe and its CPWL approximation) share properties, are the works in [13] and [18] where Dynamic Error Bounds are derived. The former paper proved that the trajectories of the Nonlinear ODE and the trajectories of the approximation CPWL differs in norm around the order of the grid size -see [9] for a precise definition of gride size, while the second paper proved that the dynamics of

◦ Dynamic Systems which are written as CPWL ODE’s. ◦ As approximation to Nonlinear Continuous ODE’s. ISBN: 978-960-6766-91-6

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ISSN 1790-2769

Selected Papers from the WSEAS Conferences in Istanbul, Turkey, May 27-30, 2008

the error bound introduced in that paper is the same as the CPWL approximation ODE. As is evident, the result in [18] when applied to a system with a stable CPWL approximation will lead the conclusion that the trajectories of both systems are stable and this precisely the topic of the present paper, the concept of ”Parallel Model”, that means, ˙ = given a Nonlinear control system to the type x(t f (x) + B · u and a CPWL approximation of f (x), fCP W L (x), then a CPWL controller uCP W L (x) stabilizing xCP W˙ L (t) = fCP W L (xCP W L )+B ·uCP W L ˙ = f (x) + B · u. is also stabilizing x(t A remarkable point here is the fast implementation of CPWL functions with microelectronics suggests that the future implementation of electronic controller could be appealing via CPWL vector fields which approximate any nonlinear dynamical system with some degree of accuracy, see [20] This paper is organized as follows: Section 2 introduces precise definitions for the kind of Nonlinear systems considered in this paper and the goal addressed ,Section 2.1 presents the formulation of an error bound able to ensure stability for both systems, Section 3 provides an analysis of the asymptotic properties of the approximate and real systems, Section 4 shows how the developed theory works in practice applicability in a real case and finally Section 5 depict some conclusions and future directions for research.

[11]. In this paper we aim to develop a controller u(t) for the following class of Nonlinear Systems: ˙ = f (X) + B · u(t), X(t)

where X ∈