Constrained Control: Polytopic Techniques

There is a resurgence of interest in the control of dynamic systems with hard constraints on states and controls and many significant advances have been made. A major reason for the success of model predictive control, which, with over 2000 applications, is the most widely used modern control technique, is precisely its ability to handle effectively hard constraints. But there are also other important developments. The solution of the constrained linear quadratic regulator problem has been characterized permitting, at least in principle, explicit determination of the value function and the optimal state feedback controller. Maximal output admissible sets have been effectively harnessed to provide easily implementable regulation and control of constrained linear systems. The solution of the robust, constrained time-optimal control problem has also been characterized. A common feature of all these advances is their reliance on polytopic techniques. Knowledge of robust controllability sets is required in model predictive control of constrained dynamic systems; these sets are polytopes when the system being controlled is linear and the constraints polytopic. In other problems, such as robust time optimal control and (unconstrained) $1 optimal control, the value function itself is polytopic. Partitioning of the state space into polytopes is required for the characterization of the solution of the constrained linear quadratic regulator problem for which the value function is piecewise quadratic and the optimal control piecewise affine. It is possible that polytopic computation may become as useful a tool for the