Approximation of explicit model predictive control using regular piecewise affine functions: an input-tostate stability approach

Piecewise affine (PWA) feedback control laws defined on general polytopic partitions, as for instance obtained by explicit model predictive control, will often be prohibitively complex for fast systems. In this work the authors study the problem of approximating these high-complexity controllers by low-complexity PWA control laws defined on more regular partitions, facilitating faster on-line evaluation. The approach is based on the concept of input-to-state stability (ISS). In particular, the existence of an ISS Lyapunov function (LF) is exploited to obtain a priori conditions that guarantee asymptotic stability and constraint satisfaction of the approximate low-complexity controller. These conditions can be expressed as local semidefinite programs or linear programs, in case of 2-norm or 1, ∞-norm-based ISS, respectively, and apply to PWA plants. In addition, as ISS is a prerequisite for our approximation method, the authors provide two tractable computational methods for deriving the necessary ISS inequalities from nominal stability. A numerical example is provided that illustrates the main results.