Solution Of Lyapunov Matrix Equation Research Articles

Computation of the Solutions of Lyapunov Matrix Equations with Iterative Decreasing Dimension Method

The existence of a solution of continuous and discrete-time Lyapunov matrix equations was studied. Both Lyapunov matrix equations are transformed into a matrix-vector equation and the solution of the obtained new system was examined. The iterative decreasing dimension method (IDDM) was implemented for solving the generated matrix-vector equation. Computations have been done with Maple procedures that run the constituted algorithms.

Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems

In this paper, the problem of dwell time for the Hurwitz stability of switched linear systems is considered. Dwell time is determined based on the solution of Lyapunov matrix equation for the Hurwitz stability of switched linear differential systems. A numerical example illustrating the efficiency of theorem has been given.

Numerical method based on fiber bundle for solving Lyapunov matrix equation

In this paper, we firstly introduce the origin of Lyapunov matrix equation, and then the geometric methods for solving Lyapunov equation are given by using the Log-Euclidean metric and the fiber bundle-based Riemannian metric based on the manifold of positive definite Hermitian matrices. Then, the solution of Lyapunov matrix equation is presented by providing a natural gradient descent algorithm (NGDA), a Log-Euclidean descent algorithm (LGDA) and a Riemannian gradient algorithm based on fiber bundle (RGA). At last, the convergence speeds of the RGA, the NGDA and the LGDA are compared via two simulation examples. Simulation results show that the convergence speed of the RGA is faster than both of the LGDA and the NGDA.

Extending the Gauss–Huard method for the solution of Lyapunov matrix equations and matrix inversion

SummaryThe solution of linear systems is a recurrent operation in scientific and engineering applications, traditionally addressed via the LU factorization. The Gauss–Huard (GH) algorithm has been introduced as an efficient alternative in modern platforms equipped with accelerators, although this approach presented some functional constraints. In particular, it was not possible to reuse part of the computations in the solution of delayed linear systems or in the inversion of the matrix. Here, we adapt GH to overcome these two deficiencies of GH, yielding new algorithms that exhibit the same computational cost as their corresponding counterparts based on the LU factorization of the matrix. We evaluate the novel GH extensions on the solution of Lyapunov matrix equations via the LRCF‐ADI method, validating our approach via experiments with three benchmarks from model order reduction. Copyright © 2017 John Wiley & Sons, Ltd.

Improved neural solution for the Lyapunov matrix equation based on gradient search

By using the hierarchical identification principle, based on the conventional gradient search, two neural subsystems are developed and investigated for the online solution of the well-known Lyapunov matrix equation. Theoretical analysis shows that, by using any monotonically-increasing odd activation function, the gradient-based neural networks (GNN) can solve the Lyapunov equation exactly and efficiently. Computer simulation results confirm that the solution of the presented GNN models could globally converge to the solution of the Lyapunov matrix equation. Moreover, when using the power-sigmoid activation functions, the GNN models have superior convergence when compared to linear models.

A Comparative Study for New Estimates for Solution of Lyapunov Matrix Equation applied to Stability Analysis of Singularly Perturbed Systems

A Comparative Study for New Estimates for Solution of Lyapunov Matrix Equation applied to Stability Analysis of Singularly Perturbed Systems

Improved gradient-based neural networks for online solution of Lyapunov matrix equation

By adding different activation functions, a type of gradient-based neural networks is developed and presented for the online solution of Lyapunov matrix equation. Theoretical analysis shows that any monotonically-increasing odd activation function could be used for the construction of neural networks, and the improved neural models have the global convergence performance. For the convenience of hardware realization, the schematic circuit is given for the improved neural solvers. Computer simulation results further substantiate that the improved neural networks could solve the Lyapunov matrix equation with accuracy and effectiveness. Moreover, when using the power-sigmoid activation functions, the improved neural networks have superior convergence when compared to linear models.

Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

This paper studies the iterative solutions of Lyapunov matrix equations associated with Itô stochastic systems having Markovian jump parameters. For the discrete-time case, when the associated stochastic system is mean square stable, two iterative algorithms with one in direct form and the other one in implicit form are established. The convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system. For the continuous-time case, a transformation is first performed so that it is transformed into an equivalent discrete-time Lyapunov equation. Then the iterative solution can be obtained by applying the iterative algorithm developed for discrete-time Lyapunov equation. Similar to the discrete-time case, an implicit iteration is also proposed for the continuous case. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms are analyzed and compared. Numerical examples are worked out to validate the effectiveness of the proposed algorithms.

A first approximation of the quasipotential in problems of the stability of systems with random non-degenerate perturbations

The problem of a local description (near to a stationary point or orbit) of the quasipotential—the Lyapunov function, used when analysing the stability of a system with small non-degenerate random perturbations is considered. First approximations are constructed for quasipotentials in neighbourhoods of these invariant sets. The quadratic forms specifying these approximations are goverened by certain matrices. The construction of these matrices is reduced to the solution of Lyapunov matrix equations (which are algebraic in the case of stationary points, and differential with periodic coefficients in the case of orbits).

Preconditioned Krylov Subspace Methods for Lyapunov Matrix Equations

The authors study the iterative solution of Lyapunov matrix equations \[ AX + XA^T = - D^T D \] by reconditioned Krylov subspace methods. These solution techniques are of interest for problems leading to large and sparse matrices A as those arising from certain applications in large space structure control theory. We show how conjugate gradient (CG)-type methods for nonsymmetric linear systems can be applied to this type of equation utilizing the special structure when computing matrix-vector and inner products. In contrast to recently developed methods for such matrix equations based on Krylov subspaces associated with A, the authors implicitly work with the equivalent system of linear equations involving the Kronecker sum $M = A \otimes I_N + I_N \otimes A$. Motivation for this new approach comes from the observation that it allows the straightforward incorporation of preconditioners. Several preconditioners for such problems are presented and analyzed. In particular, since the solution matrix X is known to be symmetric, it is of interest to know which of the methods produces symmetric iterates in each step. It is proven that this is the case for alternating direction implicit (ADI)-type and (point) symmetric successive overrelaxation (SSOR) preconditioning in association with the quasiminimal residual (QMR) method. As numerical results show, it is essential to use reconditioning in association with Krylov subspace methods. Several preconditioners for such problems are presented and analyzed. Finally numerical examples are presented where the different preconditioners are compared.

Nonlinear operator equations and applications to modelling

The main purpose of this paper is to treat the problem of numerical solutions of Lyapunov matrix equations and nonlinear matrix equations such as the Riccati type which occur in many areas of applications such as in control theory, robotics, mathematical modelling and computer simulation, etc. The main approach is to obtain solutions of such type equations in a more direct manner as in contrast to completely interactive methods. Parallel processing of algorithms is made use of in the computation of solutions of such nonlinear matrix equations associated with nonlinear matrix differential equations arising in the study of nonlinear dynamical control systems.

Stability of Large Scale Systems Under Decentralized Control

The problem dealt with in this paper is the decentralized stabilization of linear continuous-time large scale systems (LSS) by static state feedbacks. The suggested approach consists in an additional decomposition of the local subsystems, usage of bounds on the solution of Lyapunov matrix equation, vector Lyapunov function and the comparison principle. Our main purpose is to overcome two important shortcomings of the existing approaches: the requirement for solution of Riccati (Lyapunov) matrix equations at each iteration, which may be time consuming and the a priori unawareness of this solution, involved in the stability condition. The main result is a sufficient stability condition, whose outstanding feature is flexibility. The validity of the approach and its capabilities are illustrated on the error regulation problem of a string of moving vehicles.

Formulae for the solution of Lyapunov matrix equations

A new algorithm for solving the Lyapunov matrix equation X − A∗XA = Q is proposed. The method proceeds by reducing to a special case for which an explicit formula is given. The technique is purely algebraic (i.e. involves no iteration), but does not involve the calculation of the characteristic polynomial of A or reduction to a canonical form. If Q is symmetric and the matrices are of type n × n, the number of multiplications and divisions required is about 4n4. Two simple devices are given whereby the technique can be extended to a wider class of linear matrix equations.

An algorithm for computing powers of a Hessenberg matrix and its applications

A simple algorithm for computing the first n powers of an n× n Hessenberg matrix with unit codiagonal or for evaluating a polynomial of degree ⩽ n in such a matrix is proposed in this paper. Several applications of the algorithm are mentioned, including the solution of Lyapunov matrix equations associated with stability problems.

Solution of Lyapunov matrix equation for continuous systems via Schwarz and Routh canonical forms

Iterative methods for solution of the Lyapunov matrix equation for continuous systems A'L+LA=−C based on prior reduction of A to Schwarz or Routh canonical form, are described. The methods are based on the effect which a similarity transformation on A has on the equation. A restriction on solubility is discussed.