Any Domain of Attraction for a Linear Constrained System is a Tracking Domain of Attraction

Abstract

In the stabilization problem for systems with control and state constraints a domain of attraction is a set of initial states that can be driven to the origin by a feedback control without incurring constraints violations. If the problem is that of tracking a reference signal, that converges to a constant constraint-admissible value, a tracking domain of attraction is a set of initial states from which the reference signal can be asymptotically approached without constraints violation during the transient. Clearly, since the zero signal is an admissible reference signal, any tracking domain of attraction is a domain of attraction. We show that the opposite is also true. For constant reference signals we establish a connection between the convergence speed of the stabilization problem and tracking convergence which turns out to be independent of the reference signal. We also show that the tracking controller can be inferred from the stabilizing (possibly nonlinear) controller associated with the domain of attraction.