Mode-Conscious Stabilization of Switched Linear Control Systems against Adversarial Switchings

The stabilization problem of a discrete-time switched linear control system using continuous control input against adversarial switchings is studied. This problem is formulated as a two-player dynamic game, and it is assumed that the continuous controller has access to the adversary’s switching mode at each time and can be of the form of an ensemble of mode-dependent state feedback controllers. Under this information structure, the (fastest) stabilizing rate is proposed as a quantitative metric of the system’s stabilizability. Conditions are derived on when the stabilizing rate can be exactly achieved by an admissible control policy and a counter example is given to show that the stabilizing rate may not always be attained by a mode-dependent linear state feedback control policy. Bounds of the stabilizing rate are derived using (semi)norms. When such bounds are tight, the corresponding extremal norms are characterized geometrically. Numerical algorithms based on ellipsoid and polytope norms are developed for computing bounds of the stabilizing rate and illustrated through examples.