Kyong B. Lim

Robust eigensystem assignment for flexible structures

An improved method is developed for eigenvalue and eigenvector placement of a closed-loop control system using either state or output feedback. The method basically consists of three steps. First, the singular value or QR decomposition is used to generate an orthonormal basis that spans admissible eigenvector space corresponding to each assigned eigenvalue. Second, given a unitary matrix, the eigenvector set that best approximates the given matrix in the least-square sense and still satisfies eigenvalue constraints is determined. Third, a unitary matrix is sought to minimize the error between the unitary matrix and the assignable eigenvector matrix. For use as the desired eigenvector set, two matrices, namely, the open-loop eigenvector matrix and its closest unitary matrix, are proposed. The latter matrix generally encourages both minimum conditioning and control gains. In addition, the algorithm is formulated in real arithmetic for efficient implementation. To illustrate the basic concepts, numerical examples are included.

Comparison of controller designs for an experimental flexible structure

Control system design and hardware testing are addressed for an experimental structure displaying the characteristics of a typical large flexible spacecraft. The practical aspects associated with designing and implementing various control design methodologies for a real system are described, and the results are given. The design methodologies under investigation include linear-quadratic-Gaussian (LQG) control, static and dynamic dissipative control, and H/sub infinity / optimal control. The merit of each design is based on its capacity for vibration suppression, its stability robustness characteristics with respect to unmodeled dynamics, and its ease of design and implementation. Among the three controllers considered, it is shown, through computer simulation and laboratory experiments, that the dynamic dissipative controller gives the best results.<<ETX>>

Eigenvalue and eigenvector derivatives of a nondefective matrix

A novel approach is introduced to address the problem of the existence of differentiable eigenvectors for a nondefective matrix that may have repeated eigenvalues. The existence of eigenvector derivatives for a unique set of continuous eigenvectors corresponding to a repeated eigenvalue is rigorously established for nondefective and analytic matrices. A numerically implementable method is then developed to compute the differentiable eigenvectors associated with repeated eigenvalues. The solutions of eigenvalue and eigenvector derivatives for repeated eigenvalues are then derived. An example is given to illustrate the validity of formulations developed in this paper.

Robust eigenvalue assignment with maximum tolerance to system uncertainties

For a linear time-invariant system with a feedback controller, the closed-loop eigenvalues perturb due to system uncertainties. Given an allowable tolerance for the closed-loop eigenvalue perturbation, an algorithm is developed to obtain a state feedback controller that maximizes the uncertainty tolerance of the open-loop system matrix. The design procedure is based on an existing eigenvalue assignment technique using Sylvester’s equation. A robustness condition is derived to guarantee satisfaction of a specified closed-loop perturbation tolerance. Finally, an iterative algorithm is presented for easy numerical implementation to compute the robust controller, and a numerical example is given for illustration.

Eigenvector derivatives of repeated eigenvalues using singular value decomposition

An explicit formula is obtained for the first-order eigenvector derivative that corresponds to the eigenvector of a repeated eigenvalue, in the case of the nonself-adjoint eigenvalue problem. This method applies to the class of nondefective problems whose first eigenvalue derivatives of the repeated eigenvalues are distinct. A singular-value decomposition approach is used to compute four requisite bases for eigenspaces, as well as to keep track of the dimensions of state variables and the conditioning of the state equations.

Integrated Controls-Structures Design: A Practical Design Tool For Modern Spacecraft

An integrated controls-structures design approach is developed for a class of flexible spacecraft. The integrated design problem is posed in the form of simultaneous optimization of both the structural and the control design variables. The approach is demonstrated by application to integrated design of a geostationary platform and to a ground-based flexible structure experiment. The numerical results obtained indicate that the integrated design approach can yield spacecraft designs that have substantially superior performance over the conventional design approach wherein the structural design and control design are performed sequentially.

Eigensystem assignment with output feedback

A new approach for the eigenvalue assignment of linear, first-order, time-invariant systems using output feedback is developed. The approach can assign the maximum allowable number of closed-loop eigenvalues through output feedback provided that the system is fully controllable and observable, and both the input influence and output influence matrices are full rank. First, a collection of bases for the space of attainable closed-loop eigenvectors is generated using the Singular Value Decomposition or QR Decomposition techniques. Then, an algorithm based on subspace intersections is developed and used to compute the corresponding coefficients of the bases, and the required output feedback gain matrix. Moreover, the additional freedom provided by the multi-inputs and multi-outputs beyond the eigenvalue assignment is characterized for possible exploitation. A numerical example is given to demonstrate the viability of the proposed approach.

H-infinity norm sensitivity formula with control system design applications

An analytic formula for the sensitivity of singular value peak variation with respect to parameter variation is derived. As a corollary, the derivative of the //«, norm of a stable transfer function with respect to a parameter is presented. It depends on the first derivative of the transfer function with respect to the parameter. If the transfer function has a linear system realization whose matrices depend on the parameter, then an analytic formula for this derivative is derived, and an efficient algorithm for calculating the //<» norm sensitivity is described. Examples are given that provide numerical verification of the //oo norm sensitivity formula and that demonstrate its utility in designing control systems satisfying //oo norm constraints.

Robust control design framework for substructure models

A framework for designing control systems directly from substructure models and uncertainties is proposed. The technique is based on combining a set of substructure robust control problems by an interface stiffness matrix which appears as a constant gain feedback. Variations of uncertainties in the interface stiffness are treated as a parametric uncertainty. It is shown that multivariable robust control can be applied to generate centralized or decentralized controllers that guarantee performance with respect to uncertainties in the interface stiffness, reduced component modes and external disturbances. The technique is particularly suited for large, complex, and weakly coupled flexible structures.

Experimental robust control studies on an unstable magnetic suspension system

This study is an experimental investigation of the robustness of various controllers designed for the Large Angle Magnetic Suspension Test Fixture (LAMSTF). Both analytical and identified nominal model are used for designing controllers along with two different types of uncertainty models. Robustness refers to maintaining tracking performance under analytical model errors and dynamically induced eddy currents, while external disturbances are not considered. Results show that incorporating robustness into analytical models gives significantly better results. However, incorporating incorrect uncertainty models may lead to poorer performance than not designing for robustness at all. Designing controllers based on accurate identified models gave the best performance. In fact, incorporating a significant level of robustness into an accurate nominal model resulted in reduced performance. This paper discusses an assortment of experimental results in a consistent manner using robust control theory.