This paper deals with the robust control of a basic current-controlled magnetic bearing by an output dynamical compensator. Based on a parametric approach for eigenstructure assignment and a linearized model of the magnetic bearing system, a general explicit parametric expression for all the first-order dynamical compensators assigning desired nondefective closed-loop eigenstructure is obtained. This parametric expression is expressed in terms of the closed-loop eigenvalues and a free parameter. Through optimizing this free parameter and the closed-loop eigenvalues, a dynamical compensator is obtained which gives insensitive closed-loop eigenvalues, effectively attenuates the effect of the disturbance, and uses small control effort. To ensure desired closed-loop dynamical performance, the closed-loop eigenvalues are optimized within some desired regions on the left-half complex plane. Measures are also taken to ensure the robust stability of the closed-loop system in the case of large system parameter perturbations. The proposed approach is applied to a flywheel which is supported by four current-controlled active magnetic bearings. Nonlinear simulation and experimental results show the effect of the proposed approach.

A complete parametric approach is proposed for eigenstructure assignment in descriptor linear systems via output feedback, which assigns arbitrary rank (E) finite relative eigenvalues to the closea-loop system and guarantees the closed-loop regularity. General parametric expressions for the normalized left and right closed-loop eigenvector matrices are obtained on the basis of a recent parametric solution for the generalized Sylvester matrix equation. These expressions contain two groups of parameter vectors which represent the degree of freedom existing in the solution of the closed-loop eigenvectors. A pair of general complete expressions for the output feedback gain is also presented, containing the part of parameter vectors existing in the closed-loop eigenvectors and another extra two parameter matrices. The sub-conventional case is especially treated, in which there exists a one-to-one mapping relation between the closed--loop eigenvector matrices and the output feedback gain. This approach does not impose any restriction on the closed-loop eigenvalues, but generalizes and improves some of the existing results. An illustrative example Shows the effect 0f the proposed appgoach.

A new complete Parametric approach for eigenstructure assignment in. controllable descriptor linear systems via static state feedback control is proposed. General omplete parametric expressions in direct closed forms for the closed-looP eigen-vectors associated with the finite closed-loop eigenvalues, and two simple complete parametric solutions for the feedback gain matrix K are established. The approach guarantees closed-loop regularitY, arbitrary as signment of rank (E) finite closed-loop eigenvalues with arbitrary given algebraic and geometric multiPlicities, and realizes elimination of all possible initial time impulsive responses. Moreover, it does not impose any condition on the closed-loop finite eigenvalues, suits both the continuous-time and discrete-time descriptor systems, and overeomes the draw-backs of some previous works. Based on this approach, general Parametric exprcssions for the closed-loop left eigeavector matrix associated with the finite closed-loop eigenvalues are also established, and it is shown that the closed-loop system can be transgormed equivalently into a canonical form that is clearly compoNed of a annamical part and a non-dynamical part. with the help of this canonical form, a general analytical parametric expression of the closed-loop system is presented in terms of the closed-loop eigenvector matrices and the closed-loop Jordan matrix. The Method of Companion'is introduced, from the point of view of cormputation, to cope with inverses of all the matrices involved in this paper. Two examples illustrate the advantages of the proposed approaches.

Two new simple, complete. analytical. and restrietion-free soiUtions with complete and explicit freetion of the matrix equation AV+BW=VF are proposed. Here IA BJ is know and is control-lable, and F is in the Jordan form with arbitrary given eigenvalues-Based on the proposed solutions of this matrix equation. a complete parametric approach for eigenstructure assignment in liner. systems via state feedback is proposed, and two new algorithms are presented. The Proposed solutions of the lilatrix equation and the eigenstructure assign-merit result are generalizations of some previous results and are simpler and more effectice.

A simple, direct, complete and explicit parametric solution of the generalized Sylyester matrix equation AB+BW=EVF is proposed for matrix F in the Jordan form with arbitrary eigenvalues using the Smith canonical form of the matrix[A-sE B]. The obtained solution Is in a clear, neat form and may have important applications in descriptor linear system theory.