H² and $H^\infty$ Design of Sampled-Data Systems Using Lifting. Part I: General Framework and Solutions

This paper presents a complete solution to the H2 and $H^\infty$ problems for sampled-data systems. As opposed to previous works in the area, it is assumed here that all or some of the sampling function, the discrete-time controller, and the hold function are available for design. The solution is obtained by transforming the problem to discrete time using the well-known lifting technique. It is then shown that the desired components of the sampled-data controller can be "peeled-off" from the inherently infinite-dimensional description in the lifted-domain. The procedure for doing this last step is central to the approach in this paper. Both new and revised solutions are presented in this paper. The solution to the H2 problems is completely new. The solution to the $H^\infty$ problems is presented in a unifying framework and is more transparent than the previous existing solutions in the literature. Transparency pays in the form of clearer results. In particular, a separation structure is established between the design of (sub)optimal hold and sampler.