A systematic framework is constructed to consider robustness issues in sampled-data systems. It is based on the sampled-data frequency domain which consists of operator valued transfer functions. Operator valued frequency responses are also developed, which connect the usual discrete time and continuous time frequency domains in a common setting.
Robust stabilization to structured linear time invariant (LTI) perturbations is examined. A necessary and sufficient condition for robust stability is derived. The obtained condition is infinite dimensional and therefore computational tools are developed to evaluate it: the technique encloses the system stability radius between converging upper and lower bounds that can be chosen to be arbitrarily tight. Each of the bounds corresponds to a finite dimensional problem. An example is used to demonstrate that, even in the case of unstructured uncertainty, the traditional small gain theorem can be extremely conservative when used to assess LTI robustness in a sampled-data context.
Robust performance of sampled-data systems is also a main emphasis of this work. Robust performance is considered to structured linear periodic perturbations, where the periodicity is that of the nominal sampled-data system. A precise characterization of robust performance is developed in the sampled-data frequency domain, and this condition has finite dimensional features. To address computation, a related sufficient condition for robust performance is constructed and a convex optimization is proposed for its evaluation. It is demonstrated that this sufficient condition is an exact robust performance condition to a class of quasi periodic perturbations. A complementary condition is also considered in terms of minimizing the weighted Hilbert-Schmidt norm of the sampled-data transfer function.