The use of a quadratic performance index to design multivariable control systems

A quadratic performance index is considered as a generalized criterion for designing linear multivariable control systems. A method is developed for expressing the characteristic polynomial of the optimal system as an explicit function of the weighting matrix elements of the performance index. Since these elements are actually the design parameters it then becomes possible to obtain some general properties of the optimal systems as well as to select the weighting matrix elements, such that more conventional criteria on the system transient response are satisfied. The results are applied to the design of two multivariable systems; in one case, the plant is initially unstable and, in the second case, it is desired to have the system response follow that of a linear model.