Quadratic Optimization and Linear Filtering

Abstract

Up to now, we have shed light on control methods which require the modes placement of the matrix of the closed-loop system in a stability domain, whether an open half-space in continuous-time or an open disk in discrete-time. However, no recommendation is given regarding the positioning of these modes: should they be close to the border? or far away? In § 7.2, we show how the gains of the control can be obtained by means of the solution of a quadratic optimization problem. In § 7.3, we show how the gain of the observer can be obtained by means of the solution of a linear filtering problem, after introducing a probabilistic framework to account for perturbations and measurement errors. We detail the results in the discrete-time case, and we state them more briefly in the continuous-time case in § 7.4.