Unified Approach for Weighted Sensitivity Design of PID Controllers with Smith Predictors

Abstract

This paper combines a Smith Predictor with an approach for graphically determining all Proportional-Integral-Derivative (PID) controllers in either continuous-time (CT) or discrete-time (DT) domains that meet performance specifications expressed in a form of a weight on the sensitivity transfer function using Hardy-Space (H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> ) control. A Smith Predictor (SP) is often used when designing a controller for a system that exhibits a "relatively" large delay that may cause the system’s relative stability and/or performance to deteriorate. The PID controller gains, namely Proportional gain K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> , Integral gain K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> and Derivative gain K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> , will be determined graphically using only the frequency response of the systems’ components, i.e., plant with delay and SP structure. 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. The improvement can be observed even if there is a mismatch between the actual process and its corresponding SP model. By using the delta operator, the same procedure can be applied to either continuous or discrete time systems, hence a unified approach. The stability boundaries of the PID controller will be determent graphically where within the boundaries, nominal stability is guaranteed and weighted sensitivity requirements are met.