Unified Approach for Robust Stability Design of a PID Controller with a Smith Predictor

This paper combines a Smith Predictor (SP) with a unified approach for graphically determining all Proportional-Integral-Derivative (PID) controller gains in either continuous-time (CT) or discrete-time (DT) domains. The PID controller will meet robust stability condition expressed in a form of weight on the complementary sensitivity transfer function. A SP is often utilized for a system that exhibits a “relatively” large delay that may cause a system's relative stability and/or performance to deteriorate. The PID controller gains, namely proportional gain ($K_{p}$), integral gain ($K_{i}$) and derivative gain ($K_{d}$), will be determined graphically using only the frequency response of a system's components, i.e., the plant with deadtime and SP structure. It will be shown that the inclusion of a SP along with a PID controller can significantly improve stability margins and/or performance when compared to relying solely on a PID controller even when there is a mismatch between the process and the SP dynamics models. The improvement can be observed for all the perturbed processes in the uncertainty pool. By using the unified approach delta operator, the same procedure can be applied to either continuous or discrete time systems, hence a unified approach. In this paper, the PID controller coefficients have been also found for non-unity feedback systems. The robust stability boundaries of the PID controller will be determined graphically where within the common boundaries, nominal stability is guaranteed and the robust stability requirements are met.