# Guide Eigenstructure Assignment for Control System Design

In practical cases most of the time, measurement of the states is not possible so we have to use the output feedback controller for satisfying control performances. Also, in the end of this manuscript, an example is given for demonstrating the effectiveness of proposed method. Eigenstructure assignment one of the most powerful methods in multivariable control systems design.

According to the results of Konstantopoulos and Antsaklis and degrees of freedom in eigenstructure assignment method by using output feedback Duan, , in recent two decades using eigenstructure assignment has been developed in all field of multivariable control systems. Reconfigurable control design Konstantopoulos and Antsaklis, , Helicopter control systems Manness and Smiths, , Control reconfiguration in second order dynamic systems Wang et al.

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In recent years, some researches have been performed for such problems. Apkarian et al. He et al. However, this method can be used when all states are available. According to parametric eigenstructure assignment Duan, characteristics of enhanced LMI Shimomura et al.

Also, the method is developed for dynamical output feedback. By developing this method for dynamical output feedback, degrees of freedom in dynamical feedback can be used. A, B and C are known matrices with appropriate dimensions. By applying the static output feedback of the form:. We know that the eigenvalues of non-defective matrix are distinct and they have less sensitivity to parameter changes Duan, Assume that the closed loop matrix A cl is non-defective then, the Jordan form of A cl is diagonal. The Jordan form of A cl is:.

Assume A,B is controllable. According to the above preliminaries and the results of Duan , the following theorem for parametric eigenstructure assignment by using output feedback is described Duan, By applying the static output feedback of the following form to Eq. Consider the closed loop system Eq.

According to theorem 1 and Enhanced LMI 14 the following theorem is represented to solve the main problem. Theorem 2: Consider the system 1 that A,B 2 is controllable. There exist the output feedback of the form Eq.

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By substituting V instead of S in Eq. According to Eq. And the proof is completed. If the LMI 21 is feasible, then the main problem has a solution. Note that for solving main problem, satisfying all of the following conditions are important:. For controllable system 8 if suppose that B,C are full rank then static output feedback can assign max m,r eigenvalues and corresponding eigenvectors Konstantopoulos and Antsaklis, In general, we may desire to exercise some control over more than max m,r closed loop eigenvalues.

So we generalize this method by using dynamical feedback of the form:.

If combine the above controller with 1 , then closed loop system is obtained:. According to the above equations is clear that dynamical output feedback controller can be translated as static output feedback. By choosing appropriate n c , all of the eigenvalues can be assigned. Authors Authors and affiliations E. Shapiro J. Contributed Papers. This is a preview of subscription content, log in to check access. Harvey, C. AC, pp. MathSciNet Google Scholar.

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