Design of robust PID controllers

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Design of robust PID controllers

Ackermann, J. / Kaesbauer, D.

A SISO (Single-Input, Single-Output) process is modeled by multi-model approach, that means by several models which describe the process in the different operating points. The problem is to design a robust PID (Proportional plus Integrating plus Derivating) controller that simultaneously stabilizes the process in all working points. Beyond stabilization the aim is to shift all poles to a desired region in the complex plane, e.g. to a shifted half-plane that guarantees a specified settling time of the responses. The parameter space approach gives a new perspective of a recent result. The importance of singular frequencies is explained and yields a new approach for calculating stability and stability polygons. It is shown that the simple geometric structure applies to half planes (Hurwitz and sigma₀-stability) and circles with arbitrary real center and radius (Bandwidth circle, Schur stability, approximate performance measure for discrete time systems). The same line of proof shows, that other stability regions do not have the simple structure of their boundaries in controller parameter space. Robust design by simultaneous Gamma-stabilization of several operating conditions of a plant essentially reduces to finding the intersection of convex polygons. A car steering control example shows that the neighborhood of a previous solution can be analyzed for possibilities of improvement. The example also lead to a different new solution set in an unexpected region of the PID parameter space.