Design of digital two- and three-term controllers for discrete-time multivariable systems

This paper presents simple methods for the design of digital two-term (PI, PD) and threo-term (PID) controllers for pole placement in discrete-time multivariable systems, where the controllers act directly on tho available system outputs. Conditions are first found for controllability of the augmented system incorporating integral and derivative state vectors, Then it is shown that for an m-inpuK-output system of order n, the PID controller can in general place up to 3m+l∞1poles of the (n + 2l)the order closed-loop system arbitrarily. The PI and PD controllers can each place up to 2m+l∞1 poles of the (n + l)order closed-loop system. Due to integral action, the PID and PI controllers also ensure that in the steady state, the system outputs follow step references in the presence of inaccessible constant disturbances and in the face of system parameter variations. The advantages of including derivative action in the controllers aro discussed briefly. The PID, PI and PD controllers can be ' tuned ' to improve closed-loop transient responses. The design methods are programmed in a conversational mode on a PDP 11/10 digital computer with a graphic display for interactive computer-aided design studies. A PI controller for a two-shaft turbojet is designed using the computer program.