High gain sampled data systems: deadbeat strategy, disturbance decoupling and cheap control

In this paper, it is shown that a sampled data system under a high gain state feedback displays a two time-scale behaviour. The two time-scale structure of the system is then used to study three seemingly different classes of problems. First, the pole placement problem is considered and a two-step design based on the theory of singular perturbation is presented. Secondly, the asymptotic disturbance decoupling problem is solved using two different methods. The first method reduces the problem to one of an exact disturbance decoupling problem for the reduced-order model, while the second method shows that the problem can be solved by means of the output deadbeat response. The third problem considered in this paper is the cheap regulator problem where a near optimal composite controller based on the optimal controllers of the slow and fast subsystems is constructed. For all three problems, it is shown that by exploiting the two time-scale property of the system and solving the problem in terms of two lower dimensional subsystems, the ill-conditioning and high dimensionality associated with the high gain feedback system can be alleviated.