Discrete-time Piecewise Affine Systems Research Articles

Stabilizing low complexity feedback control of constrained piecewise affine systems

Piecewise affine (PWA) systems are powerful models for describing both non-linear and hybrid systems. One of the key problems in controlling these systems is the inherent computational complexity of controller synthesis and analysis, especially if constraints on states and inputs are present. In addition, few results are available which address the issue of computing stabilizing controllers for PWA systems without placing constraints on the location of the origin. This paper first introduces a method to obtain stability guarantees for receding horizon control of discrete-time PWA systems. Based on this result, two algorithms which provide low complexity state feedback controllers are introduced. Specifically, we demonstrate how multi-parametric programming can be used to obtain minimum-time controllers, i.e., controllers which drive the state into a pre-specified target set in minimum time. In a second segment, we show how controllers of even lower complexity can be obtained by separately dealing with constraint satisfaction and stability properties. To this end, we introduce a method to compute PWA Lyapunov functions for discrete-time PWA systems via linear programming. Finally, we report results of an extensive case study which justify our claims of complexity reduction.

A SURVEY ON STABILITY ANALYSIS OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

Piecewise affine (PWA) systems are useful models for describing nonlinear and hybrid systems. They also result from LTI systems subject to constrained optimal control. Recently stability analysis of PWA systems has received increased interest since it can help to obtain explicit feedback control laws of low complexity (Grieder et al., 2003b; Grieder and Morari, 2003; Grieder et al., 2004). A wide range of methods with varying degrees of conservativeness are available, for analyzing stability of PWA systems. This survey introduces the most promising existing methods and adapts them to discrete-time PWA systems. Specifically, we investigate the computation of common quadratic, common quartic, piecewise affine, piecewise quadratic and piecewise quartic Lyapunov functions using linear programming, semi-definite programming and sum-of-squares techniques. Subsequently, the different methods are compared regarding the likelihood of finding a certificate of stability and computation run time. The objective is to provide the reader with a practical ‘recipe' for analyzing PWA systems. To this end, an extensive case study is performed.

H∞ STATE FEEDBACK CONTROL OF DISCRETE-TIME PIECEWISE AFFINE SYSTEMS

This paper addresses the H∞ state feedback control problem for discrete-time piecewise affine systems. Our main objective is to derive design methods that take into account the partition information of the system so as to alleviate the design conservativeness embedded in conventional approaches where local Lyapunov functions are required to possess some global characteristics. To overcome the difficulty arising from non-positive definiteness of Lyapunov matrices, a transformation is introduced which converts the state feedback control problem into a bilinear matrix inequality (BMI) problem. We propose iterative linear matrix inequality (LMI) approaches to solve the BMI problem. Numerical examples demonstrate the effectiveness of the proposed design.

Lagrange stability and performance analysis of discrete-time piecewise affine systems with logic states

In this paper we consider discrete-time piecewise affine systems with boolean inputs, outputs and states and show that they can be represented in a logic canonical form where the logic variables influence the switching between different submodels but not the continuous-valued dynamics. We exploit this representation for studying Lagrange stability and developing performance analysis procedures based on linear matrix inequalities. Moreover, by using arguments from dissipativity theory for non-linear systems, we generalize our approach to solve the H ∞ analysis problem.

Analysis of discrete-time piecewise affine and hybrid systems

In this paper, we present various algorithms both for stability and performance analysis of discrete-time piece-wise affine (PWA) systems. For stability, different classes of Lyapunov functions are considered and it is shown how to compute them through linear matrix inequalities that take into account the switching structure of the systems. We also show that the continuity of the Lyapunov function is not required in discrete time. Moreover, the tradeoff between the degree of conservativeness and computational requirements is discussed. Finally, by using arguments from the dissipativity theory for nonlinear systems, we generalize our approach to analyze the l 2-gain of PWA systems.

An LMI approach for H ∞ analysis and control of discrete-time piecewise affine systems

In this paper we investigate some analysis and control problems for discrete-time hybrid systems in the piece-wise affine form. By using arguments from the dissipativity theory for non-linear systems, we show that H X analysis and synthesis problems can be formulated and solved via linear matrix inequalities by taking into account the switching structure of the considered system. In this paper we address the generalized problem of controlling hybrid systems whose switching structure does not depend only on the state but also on the control input.