Pid Control Stability Analysis

Proportional-integral-derivative (PID) controllers have been widely utilized in feedback con-trol system because of their simplicity in controller structure, robustness to modeling errors anddisturbances, and easy implementation. When the feedback control is used, the stabilities ofclosed loop system must be considered at first. PID controllers can control only certain class ofplants. Therefore it is necessary to analysis the existence of stabilizing PID controllers. On theother hand, It turns out that much insight into the PID control can be obtained by analyzing thestability region, which is the set of controller parameters that give stable closed-loop systems. Theproblem of great importance, both theoretically and practically.In this paper, for SISO system, the properties of stabilizing PID controllers in parameter spaceis studied. A fast method for determining the stabilizing PID controllers in parameter is proposed.It is shown that the number of unstable poles of closed loop is relative with integral term Ki.Furthermore, several cases with PID controllers are given by using the method of root locus andexamples are given for illustration. For MIMO system, the PID controller synthesis are inves-tigated. We convert PID controller synthesis into a linear matrix inequality (LMI) H_∞controlproblem. Under certain technical hypothesis, the problem is converted into convex inequalities.Necessary and sufficient conditions for the existence of the PID controllers which make the closedsystem stable are available.