Stable polyhedra in parameter space.

*(English)*Zbl 1022.93016This nicely written paper focuses on robust design of low order controllers for linear dynamical systems described by polynomial equations. The paper is motivated by a recent striking result by Ho, Datta and Bhattacharyya stating that the stability region of a PID controller in the controller parameter space is a convex polygon as soon as the proportional gain is fixed. The original proof used the Hermite-Biehler theorem. In this paper an alternate proof is derived by studying geometrical properties of the stability boundary in the controller parameter space. The result is also generalized to a slightly larger class of problems including robustness analysis. An interesting application to simultaneous stabilization is also described, where the whole set of robust PID parameters with given proportional gain is obtained by computing intersections of convex polygons.

Reviewer: Didier Henrion (Toulouse)

##### MSC:

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93D09 | Robust stability |

93D21 | Adaptive or robust stabilization |

##### Keywords:

robust stability; PID controller; computer-aided control system design; stability region; simultaneous stabilization
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\textit{J. Ackermann} and \textit{D. Kaesbauer}, Automatica 39, No. 5, 937--943 (2003; Zbl 1022.93016)

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##### References:

[1] | Ackermann, J.; Bartlett, A.; Kaesbauer, D.; Sienel, W.; Steinhauser, R., Robust control: systems with uncertain physical parameters, (1993), Springer London |

[2] | Datta, A.; Ho, A.; Bhattacharya, S., Structure and synthesis of PID controllers, (2000), Springer London |

[3] | Ho, M., Datta, A., & Bhattacharya, S. (1998). Design of P, PI and PID controller for interval plants. In Proceedings of the American control conference, Philadelphia (pp. 2496-2501). |

[4] | Munro, N., & Solyemez, M.T. (2000). Fast calculation of stabilizing PID controllers for uncertain parameter systems. In Proceedings of 3rd IFAC symposium on robust control design, Prague, Czech Republic, 2000. |

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