Proceedings ArticleDOI
Algebraic solvability tests for linear matrix inequalities
Carsten W. Scherer
- pp 349-354
TLDR
In this paper, the Riccati inequality is reduced to an algebraic Lyapunov inequality with a matrix whose eigenvalues are located at the imaginary axis, and necessary conditions for the solvability of such inequalities are derived.Abstract:
Discusses algebraic tests for the solvability of the indefinite linear matrix inequality (LMI) (A*P+PA+Q/B*P+S* PB+S/R)/spl ges/0 which arises in the general LQ problem and in H/sub /spl infin//-control. The author presents a new geometric algorithm which allows one to directly reduce the LMI to a certain algebraic Riccati inequality (ARI). Under a mild regularity assumption the author describes how to further reduce the Riccati inequality to an indefinite Lyapunov inequality with a matrix whose eigenvalues are located at the imaginary axis. Finally, the author derives new general necessary conditions for the solvability of such Lyapunov inequalities and discusses cases under which these conditions are also sufficient. >read more
Citations
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Journal ArticleDOI
Spectral factorization with imaginary-axis zeros☆
TL;DR: In this paper, the existence and calculation of Hermitian solutions of a linear matrix inequality corresponding to the spectral factorization of a proper rational spectral density was studied, and it was shown explicitly that each such imaginary-axis eigenvalue defines a fixed part of all solutions.
Journal ArticleDOI
Design of unknown input observer for nonlinear systems with time-varying delays
TL;DR: The effectiveness of the proposed unknown input observer is shown with numerical results, which confirm that it estimates the states quite precisely for nonlinear time-delay systems.
Journal ArticleDOI
A complete algebraic solvability test for the nonstrict Lyapunov inequality
TL;DR: In this paper, the authors provide a complete algebraic test for verifying the existence of a Hermitian solution X of the nonstrict Lyapunov inequality A ∗ X + XA + Q ⩾ 0.
Proceedings ArticleDOI
Algebraic solvability tests for the nonstrict Lyapunov inequality
TL;DR: In this article, an algebraic test for verifying the existence of a Hermitian solution X of the nonstrict Lyapunov inequality A/sup */X+XA+Q/spl ges/0.
References
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Journal ArticleDOI
Least squares stationary optimal control and the algebraic Riccati equation
TL;DR: In this paper, the optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated, and the integrand of the performance criterion is allowed to be fully quadratically in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem.
Journal ArticleDOI
Method of Centers for Minimizing Generalized Eigenvalues
Stephen Boyd,Laurent El Ghaoui +1 more
TL;DR: In this article, the authors consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables.
Journal ArticleDOI
An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices
TL;DR: In this paper, the authors considered the problem of finding an optimal solution to a convex diferentiable problem with a positive definite constraint, where positive definiteness can be verified numerically via Cholesky decomposition.
Journal ArticleDOI
The Matrix Equation $AX + XB = C$
TL;DR: In this article, a comprehensive theory of the matrix linear equation $AX + XB = C$ is presented, where the equation is viewed as a vector equation in the vector space of all $m \times n$ matrices.
Journal ArticleDOI
Spectral factorization via Hermitian pencils
David J. Clements,Keith Glover +1 more
TL;DR: The spectral factorization problem is solved for state-space systems via results on the canonical forms and inertia of Hermitian matrix pencils and algebraic results give a deflation method for spectralfactorization.