Coupled Sylvester Equations Research Articles

General matrix exponent solutions to the coupled derivative nonlinear Schrödinger equation on half-line

Generalized matrix exponential solutions to the coupled derivative nonlinear Schrödinger equation (DNLSE) are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices through Mathematica symbolic computations.

Solving state feedback control of fractional linear quadratic regulator systems using triangular functions

The present paper is useful for two reasons. First, as far as we know, the right operational matrix of the Riemann–Liouville fractional integral for triangular functions is introduced for the first time. So, hereafter, it can be applied in approximating any fractional optimal control problem in Caputo sense, over and over. Second, the rate of convergence of the triangular functions under a new-blown norm ‖.‖p,α in its corresponding space, denoted by Lp,α, is investigated. So far, the error analysis in Lp,α space has not been studied.In this article, both left and right operational matrices of the triangular functions for arbitrary fractional order integral α > 0 in Caputo sense are applied to approximate solutions of fractional linear optimal control systems which have a quadratic performance index. The necessary and sufficient optimality conditions are stated in a fractional two-point boundary value problem. This problem is converted to a set of coupled equations involving left and right Riemann–Liouville integral operators. Using these operational matrices of the triangular functions, an extension of the functions of control, state, co-state and other engaged functions is considered. This technique is a successful approach because of reducing such systems to a system of coupled Sylvester equations. Applying the definition of Kroneker product, a linear algebraic system containing the corresponding matrices of the original problem and the operational matrices of fractional order integral can be constructed. Accordingly, fractional linear quadratic optimal control can be solved indirectly. The advantage of this methodology, in addition to simplicity, is its low computational cost and its flexible precision. The simulation results confirm the reliability and validity of this method.

Generalized Matrix Exponential Solutions to the AKNS Hierarchy

Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponential solutions share the same linear structure.

BCR method for solving generalized coupled Sylvester equations over centrosymmetric or anti-centrosymmetric matrix

This paper introduces another version of biconjugate residual method (BCR) for solving the generalized coupled Sylvester matrix equations over centrosymmetric or anti-centrosymmetric matrix. We prove this version of BCR algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round-off errors. Furthermore, a method is provided for choosing the initial matrices to obtain the least norm solution of the problem. At last, some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed.

The static output feedback from the invariant point of view

This paper addresses the static output feedback (SOF) pole placement problem of linear minimal systems with |$m$| inputs, |$r$| outputs, |$n$| states and non-zero controllability and observability indices, as an eigenstructure assignment problem. A mosaic Hankel matrix having SOF-invariant properties is used to reformulate the coupled Sylvester equations. Necessary conditions for assignability and stabilizability as well as sufficient for linearity are presented together with an algorithm calculating the solution.

Solution of Equations of State and Output Feedback for a Line Transmission Via Invariant Subspaces

The objective of this paper is to find a solution by output feedback static for state matrices representing a transmission line whose parameters longitudinal do not take into account the effects of frequency. The transmission line is represented by a cascade of π circuits, and the results presented are based on the concept of (C, A, B)-invariance and on the coupled Sylvester equations. We present algorithms for computation of the output feedback matrix based on eigenvalues assignment techniques. The simulations are performed in MATLAB.

Static Output Feedback Stabilization using Invariant Subspaces and Sylvester Equations

This paper presents systematic computational algorithms for obtaining the output feedback gain matrix in linear systems stabilization problems. Based on the concept of (C,A,B)-invariant subspaces, introduced previously by the first author, that has related the existence of a gain matrix to the solution of coupled Sylvester equations, two algorithms are presented: 1) in the Syrmos-Lewis algorithm, a modification is proposed to provide a more adequate framework to numerical solution, and 2) by using orthogonal transformations, the Alexandridis-Paraskevopoulos algorithm is modified to overcome, in part, the Kimura condition. Numerical examples are provided to illustrate the application of the proposed algorithms.

Estabilização de sistemas descritores por realimentação de saídas via subespaços invariantes

Considera-se o problema da existência e do cálculo de uma matriz de realimentação de saídas que estabiliza fortemente um sistema descritor linear. Os resultados apresentados baseiam-se no conceito de (C,A,E,B)-invariância e de sua caracterização algébrica através de equações acopladas de Sylvester e de equações acopladas de tipo Lyapunov. Apresentam-se algoritmos para o cálculo da matriz de realimentação de saídas, baseados em posicionamento de autoestrutura ou na solução de desigualdades e igualdades matriciais lineares. Os resultados e algoritmos apresentados generalizam resultados anteriores propostos para sistemas lineares clássicos e para classes particulares de sistemas descritores lineares.

OUTPUT FEEDBACK DESIGN BY COUPLED LYAPUNOV-LIKE EQUATIONS

This note presents two coupled Lyapunov-like conditions under which a linear discrete-time system can be stabilized by static output feedback. The originality of these conditions is their relation to the well-known coupled Sylvester equations that describe both the (A, B) and (C, A)-invariance of a subspace. For systems verifying Kimura's condition, we show that output feedback stabilizing gain matrices can be computed through the successive resolution of two standard convex programming problems. Numerical results are provided to show the effectiveness of the proposed approach.

Classroom Note:Centrosymmetric Matrices

Attention is drawn to some earlier work on the odd--even properties of eigenvectors of centrosymmetric matrices.

Coupled and constrained sylvester equations in system design

Several design problems, including reduced observer and compensator design, output feedback, and finite transmission zero assignment, are examined using the vehicle of the coupled Sylvester equations. The coupling is generally provided through a third equation involving the solutions of the two linear Sylvester equations, thus serving as a constraint on the allowed solutions. The Sylvester approach allows the unification of algebraic and geometric approaches, and provides numerical design algorithms through the tool of the Hessenberg form.

Output feedback eigenstructure assignment using two Sylvester equations

The eigenstructure assignment problem with output feedback is studied for systems satisfying the condition p+m>n. The main tool used is the concept of (C, A, B)-invariance and two coupled Sylvester equations, the solution of which leads to the computation of an output stabilizing feedback. A computationally efficient algorithm for the solution of these two coupled equations, which leads to the computation of a desired output feedback, is presented. >

Computational design techniques for reduced-order compensators

The authors propose two computational techniques for the pole-placement problem using reduced-order compensators. Computationally efficient algorithms are presented for the observer-based compensator and the Brasch-Pearson compensator. The key tool for these algorithms is the concept of two coupled Sylvester equations, the solutions of which completely characterize the desired reduced-order compensators. >

Condition Estimates for Matrix Functions

A sensitivity theory based on Fréchet derivatives is presented that has both theoretical and computational advantages. Theoretical results such as a generalization of Van Loan’s work on the matrix exponential are easily obtained: matrix functions are least sensitive at normal matrices. Computationally, the central problem is to estimate the norm of the Fréchet derivative, since this is equal to the function’s condition number. Two norm-estimation procedures are given; the first is based on a finite-difference approximation of the Fréchet derivative and costs only two extra function evaluations. The second method was developed specifically for the exponential and logarithmic functions; it is based on a trapezoidal approximation scheme suggested by the chain rule for the identity $e^X = ( e^{X/2^n } )^{2^n } $. This results in an infinite sequence of coupled Sylvester equations that, when truncated, is uniquely suited to the “scaling and squaring” procedure for $e^X $ or the “inverse scaling and squaring” procedure for log X. Both the trapezoid approximation method and the more general finite-difference approach yield excellent condition estimates for a large class of problems taken from the literature. The problems in this set illustrate that condition estimates based on the Fréchet derivative have the virtue of reliability and general applicability.