Analysis and robust control of hybrid continuous/discrete time systems

Hybrid Continuous/Discrete Time Systems are systems in which one part evolves in continuous time and another in discrete time. Important examples of such systems are digital (sampled-data) control systems, in which a continuous time plant is controlled by a digital controller using sample and hold devices. Traditionally, such systems have been discretized and treated as discrete-time systems thereby ignoring their continuous time (or inter-sample) behavior. The traditional approach is not applicable in modern theories of robust control such as ${\cal H}\sp\infty$ or $\ell\sp1,$ where for continuous time plants it is more natural that plant and signal uncertainties are given in continuous time. Therefore, to pose robust control problems with sampled-data controllers, one needs to consider the continuous time behavior of these systems. The continuous time behavior of sampled-data systems is in general periodically time varying. A new framework is developed for continuous-time periodic systems which has many of the useful qualities of the time invariant theory such as transforms and constant state space realizations. The framework is based on a lifting technique which establishes a strong correspondence between periodically time-varying systems in continuous time, and certain types of infinite dimensional time-invariant systems in discrete time. The utility of this framework is shown by posing and providing exact and constructive solutions to both the ${\cal H}\sp\infty$ and the $\ell\sp1$ sampled-data control problems, where the performance specifications are given in terms of continuous time uncertainties. A natural generalization is given for the ${\cal H}\sp2$ problem in the case of time-varying systems. This is used to pose the ${\cal H}\sp2$ problem for sampled-data systems, and a complete solution is also obtained