Explicit fragility margins for PWA control laws of discrete-time linear systems

The paper considers a given discrete-time linear dynamic in closed loop with a piecewise affine control law defined over a bounded polyhedral partition of the state space χ. The objective is to describe the largest error affecting the control law coefficients over each region in the above partition, for which the positive invariance of the set χ is guaranteed with respect to the closed loop dynamics. This subset describes in fact a robust invariance margin with respect to a subset of closed-loop parameters and subsequently represents the explicit fragility margin of the given piecewise affine controller. The fragility characterizes the impact of an error in the implementation of a given controller. It is shown that the largest subset of errors in the state feedback gains is represented by a polyhedral set explicitly constructed with operations over convex domains. A series of remarks on the structure of the numerical problems are discussed and an example is provided for illustration.