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.

It is proved in this note that all Hurwitz Polynomials of order not less than from two simply connected Borel cones in the polynomial parameter space. Based on this result, adge theorems for Huiwitz sta-bility of general polyhedrons of polynomials and boundary theorems for Hurwitz stability of compact sets of and polynomials are obtaned. Both cases of families of polynomials with dependent and independent coefficients are considered. Different from the previous ones, our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomical farnilies and do not require the convexity or the connectivity of the set of polynomials. Moreover, and bouodary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine.

The problem of eigenstructure assignment in multivariable linear systems by decentralizde output feedback is considered. By using a complete parametric solution of a generalized Sylvester matrix equa-tion, parametric reprssentations of both the left and right closed-loop elgenvectors and generalized eigenvectors as well as the feedback gaino with respect to the closed-loop eigenvalues and two series of partially free parameter vector are established. The obtained result does not require any conditions on the closed-loop eigenvalues, and generalize some previous results in thi area. An example shows the effect of the proposed approach.

Based on a proposed Cornplete paratmetric approach for eigenstructure assignment in muktivatiable linear systems via dynanaical. COmpensators, insightful parametrizations of the closed loop eigenvalues sensitivities to the perturbed elements in the open loop System matrices are obtained, and an effective algorithm for eigenvalue assignment with minimum. sensitivity in multivariable linear Systems via dynamical Compeasators is then proposed. The algorithm does not Contain'going back' procedures, and allows the closad loop eigenvalues to be Convenientlv optimized within desired regions. A nuniierical example demonstrates its effect, simplicitY and numerical property.

A simple complete analytical restriction-free paraauetrlc Solution With complete and explicit freedom of matrix equation AV+BW=EVF is presented. Based on the presented solutlon of this eqnation, an approach for elgenstructure assignment for coatinuous des criptor system EX=AX+BU via descriptor-variable feedback u=kr is proposed; parametric forms of the gain matrix and the elgenvectors associated with the assigned firiite closed loop elgenvalues are given. The approach is shown to be better than that of Fahmy and 0'Reilly (1989, Int. J. Contr., 49. 129-143) in the sense that it not only possesses the functions of the approach of Fahmy and 0'Reilly (1989, Int. J. Contr.. 49, 129-143), but also has the following three advantages: (1) it eliminates the requurements on the assigned finite closed loop poles; (2) it provides more design parameters: and (3) it needs less computational work.

This paper considers the robust control of an active radial magnetic bearing system, having a homopolar, external rotor topology, which is used to support an annular fiber composite flywheel rim. A first-order dynamical compensator, which uses only position feedback information, is used for control, its design being based on a linearized one-dimensional second-order model which is treated as an interval system in order to cope with parameter uncertainties. Through robust stability analysis, a parameterization of all first-order robustly stabilizing dynamical compensators for the interval system is initially obtained. Then, by appropriate selection of the free parameters in the robust controller, the H2 norm of the disturbance-output transfer function is made arbitrarily small over the system parameter intervals, and the norm of the input-output transfer function is made arbitrarily close to a lower bound. Simulation and experimental results demonstrate both stability and performance robustness of the developed controller.