Linear Perturbation Theory for Structured Matrix Pencils Arising in Control Theory (original) (raw)
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Eigenvalue characterization of some structured matrix pencils under linear perturbation
The electronic journal of linear algebra, 2024
We study the effect of linear perturbations on three families of matrix pencils. The matrix pairs of the first two families are Hermitian/skew-Hermitian with special 3 × 3 block cases appeared in continuous-time control, and the matrix pairs of the third family are special 3 × 3 non-Hermitian block matrices appeared in discrete-time control. For the first family of matrix pencils and more general cases of the second family of matrix pencils, based on the properties of the involved matrices, we obtain some upper or lower bounds on the set of eigenvalues of linearly perturbed matrix pencils which are on the imaginary axis. Studying a special 3 × 3 block matrix pencil, which is associated with continuous-time control, leads us to some linear perturbation that do not preserve (properly) the structure of the matrices. This, in turn, leads to a numerical technique for finding the nearest Hermitian/skew-Hermitian matrix pencil which can satisfy conditions such that, for some nonzero real perturbation parameter, some or all of its eigenvalues lie on the imaginary axis. We also study the linearly perturbed matrix pencils, associated with discrete-time control, using an one-to-one equivalence between the matrix pencil of continuous-time problem and the matrix pencil of discrete-time problem.
Perturbation Analysis of Linear Control Problems
IFAC Proceedings Volumes, 1987
In this paper we present a non-local perturbation analysis of the basic problems arising in the analysis and design of linear control systems: 1° Determination of eigenvalues and eigenvectors of a matrix; 2° Transformation of linear systems into canonical form; 3° Computation of a matrix exponential; 4° Pole assignment synthesis of linear systems; 5° solution of Riccati, Sylvester and Lyapunov continuous and discrete algebraic matrix equations. Most of the obtained perturbation bounds are unimprovable.
SIAM Journal on Matrix Analysis and Applications, 2003
Matrix pencils under the strict equivalence and matrix pairs under the state feedback equivalence are considered. It is known that a matrix pencil (or a matrix pair) smoothly dependent on parameters can be reduced locally to a special typically more simple form, called the versal deformation, by a smooth change of parameters and a strict equivalence (or feedback equivalence) transformation. We suggest an explicit recurrent procedure for finding the change of parameters and equivalence transformation in the reduction of a given family of matrix pencils (or matrix pairs) to the versal deformation. As an application, this procedure is applied to the analysis of the uncontrollability set in the space of parameters for a one-input linear dynamical system. Explicit formulae for a tangent plane to the uncontrollability set at its regular point and the perturbation of the uncontrollable mode are derived. A physical example is given and studied in detail.
On the Optimal Control Computation of Linear Systems
Journal of the Indonesian Mathematical Society, 2012
In this paper, we consider a numerical method for designing optimal control on Linear Quadratic Regulator (LQR) problem. In the optimal control design process through Pontryagin Maximum Principle (PMP), we obtain a system of differential equations in state and costate variables. This system lacks of initial condition on the adjoint variables, and this situation creates classic difficulty for solving optimal control problems. This paper proposes a constructive method to approximate the initial condition of the adjoint system.
Perturbation of linear control systems
Linear Algebra and its Applications, 1989
We study the variation of the controllability indices and the Jordan structure of a pair of matrices (A, I?), under small perturbations.
IEEE Transactions on Automatic Control, 1980
~lrstraet-h a new formulation of the disaete-time unearquadratic lem; the mathematical aspects, together with the notation, OPW eoobol problem, the spectrom of a bounded d-adjoht Hilbert are relegated to the Appendix. space m W which can be hm as Of a Tqlitz The Hilbert space operator, which plays a crucial role operator and a compact -tion, has been shown to elucidate the unddybg stroctnre of the problem. A new and efficient algorithm for in the discrete-time linear-quadratic optimal control probthe overall weighting matrix W of the form eompntingthisspeetnrmispreseoted lem, can be defined as follows. Consider a factorization of x: rip( -m, 2)-+ZRrn(co, t ) . 2 Paper by z. Manitius, chairman of the The operator we shall focus our attention on can be Systems Committee. Contract F44620-%(-03, t)=&'%, (54 7 1 G0067. Systems, University of Southern California, L o s Angeles, CA 9ooo7. He E k Jonckheere was with the Department of Electrical Engineering-%( -00, t ) : I$( -00, t)+l&(co, t ) . (5b) is now with the Philips Research Laboratory, Brussels, Belgium. I . , . M. Silverman is with the Department of Electrical Engineering-It is easily seen that %(co, t ) does not depend on the Systems, University of California, Los Angeles, CA 5". particular factorization of W, and that it is bounded and Manuscript