TY - JOUR
T1 - Synthesis of H∞ PID controllers
T2 - A parametric approach
AU - Ho, Ming Tzu
N1 - Funding Information:
The author would like to thank the anonymous reviewers for their valuable comments and suggestions. This work was supported by the National Science Council of Taiwan under Grant NSC 89-2218-E-006-034.
PY - 2003/6
Y1 - 2003/6
N2 - 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.
AB - 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.
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U2 - 10.1016/S0005-1098(03)00078-5
DO - 10.1016/S0005-1098(03)00078-5
M3 - Article
AN - SCOPUS:0038034165
SN - 0005-1098
VL - 39
SP - 1069
EP - 1075
JO - Automatica
JF - Automatica
IS - 6
ER -