Design of optimal multivariable linear control systems

The paper presents a method of designing optimal linear multivariable control systems with quadratic performance indices. The procedure is based on Pontryagin's maximum principle, which is first used to give the costate and system equations in state-space form. Using the Laplace transform method, it is then shown that the optimum trajectory equation contains unstable terms which may be eliminated by a simple choice of the terminal values of the costate variables. With this choice of terminal values, the optimal control strategy is obtained, which ensures a stable system whilst minimising a quadratic measure of performance over the infinite interval. The design procedure is illustrated by example, and the optimum controller is compared with a finite-interval optimum controller. Thus, it is concluded that the optimum controllers produced by the present design procedure will give a good approximation to most finite-interval controllers.