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.