Disturbance Rejection with Stability

Abstract

In this chapter, we will study the following linear systems subject to actu­ator saturation and persistent disturbances

= Ax + B sat(u) + Ew,(10.1.1)

and

x(k + 1) = Ax(k) + Bsat(u(k)) + Ew(k),(10.1.2)

where x E TV is the state, u E Rt is the control and w E BY is the dis­turbance. Also, sat : Rm is the standard saturation function that represents the constraints imposed by the actuators. Since the terms Ew and Ew(k) are outside of the saturation function, a trajectory might go unbounded no matter where it starts and whatever control we apply. Our primary concern is the boundedness of the trajectories in the presence of disturbances. We are interested in knowing if there exists a bounded set such that all the trajectories starting from inside of it can he kept within it. If there is such a bounded set, we would further like to synthesize feed­back laws that have the ability to reject the disturbance. Here disturbance rejection is in the sense that, there is a small (as small as possible) neigh­borhood of the origin, such that all the trajectories starting from inside of it (in particular, the origin) will remain in it. This performance is analyzed,for example, for the class of disturbances with finite energy in [32]. In this chapter, we will deal with persistent disturbances