# Pole Assignment

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Then, by choosing appropriate free parameters in the parametric solutions of a class of linear matrix equations, the resulting feedback gain matrix can simultaneously minimize some quadratic performance index and assign the poles of the closed-loop systems to the desired locations.

The advantages of the proposed approach are summarized as follows.

Because of the common existence of parameters drift and delay effect in engineering systems (see, e.g., [9–13]), the feedback controllers designed by theoretical method are often subject to parameter perturbations after a period of running time.

But if the perturbation satisfies certain conditions, the linear quadratic optimal controller can still guarantee the stability of the closed-loop system.

It is well known in the literature (see, e.g., [1]) that the linear quadratic optimal controller possesses very good robustness properties; for example, it has at least a 6 d B gain margin and a phase margin for single input linear system [2].

On the other hand, as a nonoptimal design approach, the pole assignment approach can arbitrarily assign the poles of the closed-loop system to any place so that a very satisfactory transient performance of the closed-loop system can be achieved.

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Then, the closed-loop system (5) consisting of the linear system (1) and the linear state feedback (3) is asymptotically stable; namely, all the eigenvalues of the closed-loop system matrix of the quadratic optimal controller can ensure that the closed-loop system has very good robustness properties.

Moreover, for single-input linear systems, it has been proven that the closed-loop system has a phase margin of 60 degrees and a magnitude margin of by the feedback gain, the resulting closed-loop system remains asymptotically stable.

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