On the design of optimal constrained dynamic compensators for linear constant systems

Abstract

The design of linear time-invariant dynamic compensators of fixed dimensionality <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> , which are to be used for the regulation of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order linear time-invariant plant, is dealt with. A modified quadratic cost criterion is employed in which a quadratic penalty on the system state as well as all compensator gains is used; the effects of the initial state are averaged out. The optimal compensator gains are specified by a set of simultaneous nonlinear matrix algebraic equations. The numerical solution of these equations would specify the gain matrices of the dynamic compensator. The proposed method may prove useful in the design of low-order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> compensators for high-order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> plants that have few <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> outputs, so that the dimension of the compensator is less than that obtained through the use of the associated Kalman-Bucy filter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> or the Luenberger observer <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n - r</tex> .