Optimal stabilizing rates of switched linear control systems under arbitrary known switchings

Abstract

The problem of stabilizing discrete-time switched linear control systems using continuous control input under arbitrary mode switchings is studied. It is assumed that at each time instance the switching mode can be arbitrarily chosen but is always known by the controller designing the continuous control input; thus the continuous controller is of the general form of an ensemble of mode-dependent state feedback controllers. Under this setting, the fastest (worst-case) stabilizing rate is proposed as a quantitative metric of the systems’ stabilizability. Conditions are derived on when this 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 on 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.