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Stability analysis of piecewise affine discrete-time systems*

This paper presents an implicit model for piecewise affine functions as the interconnection of a linear function and ramp nonlinearities. We show how to verify positive-definiteness of piecewise quadratic functions by exploiting sector properties of the ramp function. These properties are characterized by a set of identities instead of the usual sector inequalities adopted in the study of Lurie systems. Based on this setup, we then formulate conditions for stability of discrete-time piecewise systems using piecewise quadratic Lyapunov functions in terms of linear matrix inequalities. The results are illustrated in numerical examples.