Brief Synthesis of H∞ PID controllers: A parametric approach

Authors:
Ming-Tzu Ho
Affiliations:
Engineering Science Department, National Cheng Kung University, 1, University Road, Tainan 701, Taiwan
Venue:
Automatica (Journal of IFAC)
Year:
2003

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Abstract

This paper considers the problem of synthesizing proportional-integral-derivative (PID) controllers for which the closed-loop system is internally stable and the H"~-norm of a related transfer function is less than a prescribed level for a given single-input single-output plant. It is shown that the problem to be solved can be translated into simultaneous stabilization of the closed-loop characteristic polynomial and a family of complex polynomials. It calls for a generalization of the Hermite-Biehler theorem applicable to complex polynomials. It is shown that the earlier PID stabilization results are a special case of the results developed here. Then a linear programming characterization of all admissible H"~ PID controllers for a given plant is obtained. This characterization besides being computationally efficient reveals important structural properties of H"~ PID controllers. For example, it is shown that for a fixed proportional gain, the set of admissible integral and derivative gains lie in a union of convex sets.