Minimization of the L∞-Induced Norm for Sampled-Data Systems

In this paper, a complete solution for the l¹ sampled-data problem is furnished for arbitrary plants. The l¹ sampled-data problem is described as follows: Given a continuous-time plant, with continuous-time performance objectives, design a digital controller that delivers this performance. This problem differs from the standard discrete-time methods in that it takes into consideration the inter-sampling behavior of the closed loop system. The resulting closed loop system dynamics consist of both continuous-time and discrete-time dynamics and thus such systems are known as hybrid systems. It is shown that given any degree of accuracy, there exists a standard discrete-time l¹ problem, which can be determined apriori, such that for any controller that achieves a level of performance for the discrete-time problem, the same controller achieves the same performance within the prescribed level of accuracy if implemented as a sampled-data controller. This is accomplished by first converting the the hybrid system into an equivalent infinite dimensional discrete-time system using the lifting technique in continuous time, then the infinite dimensional parts of the system which model the inter-sample dynamics are approximated. This approximation is done independently of the controller, and explicit bounds are obtained for the degree of approximation. It is shown that the convergence of this approximation is at least as 1/n.