Eigenstructure assignment in linear multivariable systems--A parametric solution

Abstract

Abstruct-This paper generalizes a recently reported method (11 of closed-loop eigenstructure assignment via state feedback in a linear multivariable system (with n states and r control inputs). By introducing a lemma on the differentiation of determinants, the class of assignable eigenvectors and generalized eigenvectors associated nith the assigned eigenvalues is explicitly described by a complete set of n r-dimensional free parameter vectors. This parametric characterization conveniently organizes the nonuniqueness of the solution of the eigenvalue-assignment problem and thereby protides an efficient means of further modifying the system dynamic response. A numerical example is worked out to demonstrate the feasibility of the method.

Robust Eigenvalue Assignment in Descriptor Systems via Output Feedback

“…Let Assumption A1 be satisfied, and T ∞ and V ∞ be the infinite left and right closed-loop eigenvector matrices given in (7) and (8), respectively. Then (1) All the output feedback controllers in the form of (2) for the descriptor linear system (1), which satisfy the first condition in Problem RPA can be parameterized by…”

“…It thus follows from Proposition 3.1 that the sensitivity measures of the closed-loop finite eigenvalues are given by (18). (6)), several ways have been given in [30] under the controllability condition of the open-loop system (1). General computational methods for such right coprime polynomial matrices can also be found in [21,31,32,33] and [34].…”

“…As is well known, solution to the problem of eigenvalue assignment in a multi-variable linear system is generally not unique. This fact was known in the early 70s but has only been fully revealed by recent parametric eigenstructure assignment approaches in [1][2][3][4][5][6][7][8][9][10][11][12]. The degrees of freedom provided by parametric eigenstructure assignment may be used to achieve some desired system specifications.…”

“…The degrees of freedom provided by parametric eigenstructure assignment may be used to achieve some desired system specifications. Such an idea has resulted in many applications of parametric eigenstructure assignment ( [1], and [13][14][15][16]). pole assignment in descriptor linear systems via state feedback control.…”

Parametric state-feedback control with model reduction

“…The problem is tackled in this paper using a parametrization of the class of linear state feedback controllers which assign a set of desired self-conjugate eigenvalues to the closed-loop system (Fahmy and O'Reilly 1982, 1983a, Roppenecker 1983, 1986. For ease of exposition, we make five simplifying though inessential assumptions, that the model to be reduced is least-order, that the algebraic multiplicity of the left half-plane transmission zeros is equal to their geometric multiplicity, that the observable closedloop eigenvalues are assigned distinct values not equal to the corresponding openloop eigenvalues or to the left half-plane transmission zeros.…”