Stabilization on Null Controllable Region — Continuous-Time Systems

In this chapter, we will study the problem of stablizing a linear system with saturating actuators. The key issue involved here is the size of the result­ing domain of attraction of the equilibrium. Indeed, local stabilization, for which the size of the domain of attraction is not a design specification, is trivial. It is straightforward to see that any linear feedback that stabilizes the system in the absence of actuator saturation would also locally sta­bilize the system in the presence of actuator saturation. In fact, with the given stabilizing linear feedback law, actuator saturation can be completely avoided by restricting the initial states to a small neighborhood of the equi­librium. Our focus in this chapter is on the construction of feedback laws that would lead to a domain of attraction that contains any a priori given bounded subset of the asymptotically null controllable region in its interior. We refer to such a problem as semi-global stabilization on the asymptoti­cally null controllable region, or simply, semi-global stabilization. We recall from Chapter 2 that the null controllable region of a linear system subject to actuator saturation is the set of all the states that can be steered to the origin in a finite time by an admissible control, and the asymptotically null controllable region is the set of all the states that can be driven to the origin asymptotically by an admissible control