Abstract: A rational function for the transfer function model of a multivariable discrete control system is suggested and its controllers are obtained. There are two types of control systems depending on the nature of the disturbance. For tracking control systems, the disturbance is a set of set point changes. For regulating control systems, the disturbance is a vector ARIMA time series. The quadratic performance controllers for these systems are similar but opposite in nature. For tracking control systems, a two and a half degrees of freedom controller can be designed for an enhanced quadratic performance. This controller uses the future values of the disturbance which are the set points of the control system for further reduction of the error. The controller is particularly useful for nonminimum phase tracking control systems.

Abstract: Two linear quadratic tracking controllers and a minimal prototype controller are presented for the control of a discrete single input and single output (SISO) tracking control system. The minimal prototype controller is an unconstrained controller. Depending on the models of the set point and the plant transfer function, this controller might be desirable. But usually one would choose one of the two linear quadratic controllers which minimize the sum of squared errors between the output and the set point variables with a penalty on that of the input variable. The one degree of freedom (1-DOF) controller performs well, but for nonminimum phase systems the two and a half degrees of freedom (2.5-DOF) controller is the stronger one as it can suppress the inverse response of a non-minimum phase system. The 1-DOF controller gives the stochastic regulating controller counterpart known as the linear quadratic Gaussian controller. A digital control chip for implementation of the controllers is also disclosed.