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Explicit solution and eigenvalue bounds in the Lyapunov matrix equation
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In this article, an explicit solution to the algebraic Lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices, which enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the covariance matrix.Abstract:
An explicit solution to the algebraic Lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices. This enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the controllability matrix. Based on this explicit formula, upper and lower bounds for each eigenvalue of the solution are derived, which always give nontrivial estimates.read more
Citations
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Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case
TL;DR: In this paper, the eigenvalues of the solutions to a class of continuous and discrete-time Lyapunov equations with symmetric coefficient matrices and right-hand side matrices of low rank were studied.
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Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results
TL;DR: In this paper, the relations of a number of bounds for the solutions of the algebraic Riccati and Lyapunov equations that have been reported during the last two decades are investigated.
Journal ArticleDOI
Design of decentralized observation schemes for large-scale interconnected systems: some new results
TL;DR: The obtained algorithm is shown to considerably improve upon the existing results for the decentralized observer design problem, and the demonstration of how the observer gains can be tailored to the existing interconnection pattern within the overall system.
Journal ArticleDOI
Qualitative analysis and decentralized controller synthesis for a class of large-scale systems with symmetrically interconnected subsystems
TL;DR: An analysis of some important qualitative properties of such symmetrically interconnected systems focussing on the spectrum characterization, controllability and observability, and the solutions of the algebraic Riccati equation and the matrix Lyapunov equation is conducted.
Journal ArticleDOI
The solution to matrix equation AX+XTC=B
TL;DR: In this note, the problem of solution to the matrix equation AX+XTC=B is considered by the Moore–Penrose generalized inverse matrix and a general solution to this equation is obtained.
References
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Journal ArticleDOI
Upper and lower bounds on the solution of the algebraic Riccati equation
K. Yasuda,Kazumasa Hirai +1 more
TL;DR: In this article, the extremal eigenvalues of the positive definite solution of the Riccati equation were derived for discrete algebraic matrix Riccaci equations. But these estimations appear to appear to be considerably tighter than previously available results in many cases.
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Formulae for the solution of Lyapunov matrix equations
TL;DR: In this paper, a new algorithm for solving the Lyapunov matrix equation X − A∗XA = Q is proposed, which is purely algebraic and does not involve the calculation of the characteristic polynomial of A or reduction to a canonical form.
Journal ArticleDOI
On the Lyapunov matrix equation
TL;DR: In this paper, a fundamental inequality which is satisfied by the extremal eigenvalues of the matrices Q and P, provided A is a stability matrix, is established, which is extremely useful in determination of suboptimal controllers for the minimum time problem.
Journal ArticleDOI
Comparison of numerical methods for solving Liapunov matrix equations
I. S. Pace,Stephen Barnett +1 more
TL;DR: In this article, a comparison of most published methods for solving the linear matrix equations which arise when a quadratic form Liapunov function is applied to a constant linear system (continuous or discrete, real or complex).
Journal ArticleDOI
A note on eigenvalue bounds in algebraic Riccati equation
TL;DR: In this paper, an upper bound on the maximum eigenvalue of the solution matrix K of the algebraic Riccati equation is established, and several lower bounds are derived for some of the largest eigenvalues of K.