An implicit representation for the analysis of piecewise affine discrete-time systems

• Published in: Automatica (Oxford) Vol. 147; p. 110730
• Main Authors: Broering Groff, Leonardo, Valmorbida, Giorgio, Gomes da Silva, João Manoel
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-01-2023; Elsevier
• Subjects: Automatic Control Engineering; Computer Science; Discrete-time systems; Engineering Sciences; Piecewise affine systems; Piecewise quadratic Lyapunov functions; Stability; Discrete-time systems; Piecewise affine systems; Stability; Piecewise quadratic Lyapunov functions; discrete-time systems; piecewise quadratic Lyapunov functions; piecewise affine systems; stability
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2022.110730

This paper presents a new framework for stability assessment of discrete-time piecewise affine systems. An implicit representation for piecewise functions based on ramp nonlinearities is proposed. Instead of the usual sector inequalities adopted in the study of Lurie systems, we directly exploit properties of the ramp function, which are given by a particular set of identities and inequalities. These properties are then used to derive stability conditions associated to piecewise quadratic Lyapunov functions. These functions are implicitly defined from quadratic forms involving ramp functions. These conditions can be cast as LMIs and their numerical solution is illustrated by examples highlighting the advantages of the proposed method over existing stability analysis methods for PWA systems.