# Sanjay Lall

- Professor of Electrical Engineering
- Professor of Aeronautics and Astronautics
- Information Systems Laboratory
- Stanford University

### Robust Control Synthesis in the Time Domain

This thesis investigates the synthesis of controllers for time varying systems in order to satisfy an induced 2-norm closed loop performance bound. This performance criterion is a generalisation of the well known H-infinity norm criterion used in the frequency domain for analysis and synthesis of linear time invariant control systems. A number of different time varying system frameworks are considered, for which there are no frequency domain counterparts. One such class is that of aperiodic sampled-data systems, that is continuous time systems connected to a discrete controller via sampling and hold devices. Multiple generalised sampling and hold devices, which may be aperiodic and asynchronous, are permitted within the framework considered in this thesis. Using game theory, necessary and sufficient conditions are given for the existence of controllers satisfying a prespecified performance bound for such multi-rate sampled-data systems, and expressions for such a controller are given if one exists. In the state feedback case this result is generalised to include time varying systems with possible discontinuities in the state vector, of which sampled-data systems are a special case. Another inherently time domain problem investigated in this work is the moving horizon H-infinity controller. The moving horizon problem was originally formulated as a method of stabilising time varying systems without requiring information about the system matrices over all future time, by minimising a quadratic cost function up to some finite time ahead at each instant. In this thesis the method is generalised so that a differential game with quadratic cost function is solved up to a finite time ahead. Explicit constructions are given in both the state and output feedback case corresponding to this min-max problem, based upon state estimation using information from a finite time in the past. It is shown that with the moving horizon controller it is possible to synthesise controllers which are not only stabilising but also satisfy some prespecified closed loop induced norm bound over all future time. The moving horizon problem was originally formulated as a purely continuous time problem. In this thesis it is generalised also to the case when the controller is updated only on discrete, possibly overlapping, intervals. It is shown that, if the terminal weights satisfy certain conditions, then it is possible to synthesise controllers which are gamma-feasible by solving separate dynamic games on each finite interval.The pdf for this paper is not yet posted; please email lall@stanford.edu if you would like a copy.