The l1 sampled-data problem for controllers with optimal samplers and optimal hold functions

Considers several versions of the l¹ sampled-data control design problem. Given a continuous-time plant, with continuous-time performance objectives, expressed in terms of the L^∞-induced norm, the author considers two possible controller configurations. The first is a controller with sampled measurements and continuous-time control signals, and the second is a controller with continuous-time measurements but with fixed hold functions for the controls. These problems lead to optimal “hold functions”, and optimal sampling operations respectively. The author shows that these two problems are in some sense dual problems. These problems differs from standard discrete-time methods in that it takes into consideration the inter-sample 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. These problems further differ from standard so-called sampled-data problems in that the sampler and hold operations are not both fixed, but are allowed to be part of the design process. The author addresses two main issues, namely the structure and parametrization of such types of controllers, and a computational procedure for designing such l¹ optimal controllers. The controllers in question represent systems with discrete-time inputs and continuous-time outputs (or vice versa), and the author shows that such systems are most conveniently parametrized in terms of their lifted representations. Furthermore, the author shows that for linear time-invariant (LTI) plants, the optimal performance is achieved by controllers that have LTI lifted representations. This settles the issue of controller structure, which is then exploited to obtain a convergent approximate procedure for reducing the problem to a standard l¹ discrete-time problem. The structure of the resulting optimal controllers, and the convergence properties of the approximation procedure are closely analysed