As in Chap. 8, we consider in this chapter constraints on state as well as input variables. As discussed in detail in Chap. 8, if the given system has at least one of the constraint invariant zeros in the open right-half plane (continuous time) or outside the unit disc (discrete time), that is, if it has non-minimum-phase constraints, then neither semi-global nor global stabilization in the admissible set is possible. Thus, whenever we have non-minimum-phase constraints, the semi-global stabilization is possible only in a certain proper subset of the admissible set. This gives rise to the notion of a recoverable region (set), sometimes also called the domain of null controllability or null controllable region. Generally speaking, for a system with constraints, an initial state is said to be recoverable if it can be driven to zero by some control without violating the constraints on the state and input. We can appropriately term the set of all recoverable initial conditions as the recoverable region. The recoverable region is thus indeed the maximum achievable domain of attraction in stabilizing linear systems subject to non-minimum-phase constraints. The goal of stabilization is to design a feedback, say u = f(x), such that the constraints are not violated and moreover the region of attraction of the equilibrium point of the closed-loop system is equal to the recoverable region or an arbitrarily large subset contained within the recoverable region. Such a stabilization is termed as the semi-global stabilization in the recoverable region, and this is what we pursue in this chapter.
State Feedback Controller Input Constraint Measurement Feedback Invariant Zero Static State Feedback
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