### 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.

Original language | English |
---|---|

Pages (from-to) | 1069-1075 |

Number of pages | 7 |

Journal | Automatica |

Volume | 39 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2003 Jun 1 |

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Electrical and Electronic Engineering

## Fingerprint Dive into the research topics of 'Synthesis of H<sub>∞</sub> PID controllers: A parametric approach'. Together they form a unique fingerprint.

## Cite this

_{∞}PID controllers: A parametric approach.

*Automatica*,

*39*(6), 1069-1075. https://doi.org/10.1016/S0005-1098(03)00078-5