Linear Fractional Representations and L₂-Stability Analysis of Continuous Piecewise Affine Systems

This letter addresses L
₂
-stability analysis of discrete-time continuous piecewise affine systems described in input-output form by linear combinations of basis piecewise affine functions. The proposed approach exploits an equivalent representation of these systems as the feedback interconnection of a linear system and a diagonal static block with repeated scalar nonlinearity. This representation enables the use of analysis results for systems with repeated nonlinearities based on integral quadratic constraints. This leads to a sufficient condition for L
₂
-stability that can be checked via the solution of a single linear matrix inequality, whose dimension grows linearly with the number of basis piecewise affine functions defining the system. Numerical examples corroborate the proposed approach by providing a comparison with an alternative approach based on the computation of piecewise polynomial storage functions.