Discrete-time pole placement with stable controller.

*(English)*Zbl 0766.93022Summary: The classical polynomial approach to pole placement involves the computation of a particular solution of a given Diophantine equation. The solution corresponds to a proper controller whose order is fixed by the number of nonminimum phase zeros of the plant transfer function, by its order, and by the desired closed-loop characteristic polynomial. This controller might be unstable for some desired closed-loop characteristic polynomials, even though the plant can be stabilized by a stable controller. Now, unstable controllers are known to deteriorate closed- loop system performances, compared with stable ones. Hence, we propose a procedure to build, for a specific class of plants, a stable pole placement controller when the classical pole placement controller is unstable. The properties of both stable and unstable controllers are compared via simulations. It appears that, for the considered plants, the stable controller improves at least one of the following properties, compared with its unstable counterpart: step response transient, gain margin, measurement noise rejection. The results obtained with our procedure are also compared with the results obtained via Middleton’s approach, which consists in modifying the desired closed-loop poles so as to obtain a stable and minimum phase controller with the classical pole placement method. Finally, our pole placement design with stable controller is applied to solve a particular simultaneous partial pole placement problem.

##### MSC:

93B55 | Pole and zero placement problems |

93C55 | Discrete-time control/observation systems |

93C05 | Linear systems in control theory |

93C62 | Digital control/observation systems |

##### Keywords:

closed-loop system
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\textit{M. Kinnaert} and \textit{V. Blondel}, Automatica 28, No. 5, 935--943 (1992; Zbl 0766.93022)

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