A matrix inequalities approach to H-infinity control in a behavioral framework

The behavioral framework for solving control problems was formally introduced by Willems in the mid-80s. Since then, this approach has gained widespread acceptance. Starting with the idea that dynamical systems may be viewed as sets of trajectories, the behavioral framework has enabled researchers to solve many control problems such as modeling, robustness, system analysis, observer and controller design. Several optimal control problems have been solved in a behavioral setting. In particular, solutions have been provided for the $H\infty$ control problem, through frequency domain methods and J-spectral factorizations.; In this dissertation, we develop a matrix inequalities approach to $H\infty$ control in the behavioral framework. We start by deriving a new and improved version of the behavioral bounded real lemma, which provides a link between an $H\infty$ condition and a linear matrix inequality (LMI). Previous results consisted of a matrix inequality in which the controller's equation did not explicitly appear. Our version provides a linear matrix inequality feasibility problem (LMIP), which explicitly involves the controller's equations, thereby yielding an algorithm for verifying that a specified controller solves a given $H\infty$ problem. We illustrate our result by applying the algorithm to a simplified car suspension design problem.; We obtain both an upper bound and a lower bound for the optimal $H\infty$ gain that a system can achieve. These bounds, which depend exclusively on the system's parameters, are the solutions of LMI eigenvalue problems (LMI-EVPs). We illustrate our results by solving several numerical examples.; We find a controller that achieves a given suboptimal gain for the system. That solution is in the form of a sufficient condition based upon feasibility of a Bilinear Matrix Inequality (BMIP). We then show how our method applies to the problem at hand when modeling uncertainties exist.