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To the problem of solution symmetry for linear matrix differential equations
Abstract
The symmetry of Cauchy problem solution for linear matrix differential equations is under research in the present article. The coefficients of the equation in question are supposed to be analytical functions in some domain of the complex plane. We find a formula for high-order derivatives of an arbitrary solution of the equation. We prove the sufficient conditions for the symmetry of Cauchy problem solution for linear matrix differential equations on the basis of the devised formula. To check these conditions, we need to analyse the properties of the special matrix sequence. Since the sequence consists of the infinite number of elements, the check is difficult to implement. It is shown that if some requirements are met, then it is sufficient to check only first several elements of the sequence. The example of the linear matrix differential equation is given to illustrate how the proposed condition may be used in proving the solution symmetry. The obtained results may be used in solving various problems of the control theory.
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We derive upper and lower bounds for the trace of the solution of the time-varying linear matrix differential equation (t) = AH(t) P(t) + P(t) A(t) + Q(t). A practical numerical example is given to verify the bounds. The bounds obtained are useful since the considered equation
is encountered in a number of applications in systems and control theory.
In this paper, by using majorization inequalities, upper bounds on summations of eigenvalues (including the trace) of the solution for the Lyapunov matrix differential equation are obtained. In the limiting cases, the results reduce to bounds of the algebraic Lyapunov matrix equation. The effectiveness of the results are illustrated by numerical examples.
In this paper, the finite-time optimal control problem for time-invariant linear singularly perturbed systems is considered. The reduced-order pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed differential matrix Riccati equation of dimension n 1 + n 2 into the regular differential matrix Riccati equation pure-slow of dimension n 1 and the stiff differential matrix Riccati equation pure-fast of dimension n 2 . A formula is derived that produces the solution of the original singularly perturbed matrix Riccati differential equation in terms of solutions of the pure-slow and pure-fast reduced-order differential matrix Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear systems independently in pure-slow and pure-fast time scales. An example for a catalytic fluid reactor model has been include to demonstrate the utility of the method.
An exact method is presented for discretizing a constant-coefficient, non-square, matrix differential Riccati equation, whose solution is assumed to exist. The resulting discrete-time equation gives the values that have no error at discrete-time instants for any discrete-time interval. The method is based on a matrix fractional transformation, which is more general than existing ones, for linearizing the differential Riccati equation. A numerical example is presented to compare the proposed method with that based on gage invariance and bilinearization, which has better performances than the conventional forward-difference method.
In this technical note, we investigate a solution of the matrix differential Riccati equation that plays an important role in the linear quadratic optimal control problem. Unlike many methods in the literature, the approach that we propose employs the negative definite anti-stabilizing solution of the matrix algebraic Riccati equation and the solution of the matrix differential Lyapunov equation. An illustrative numerical example is provided to show the efficiency of our approach.
We consider the construction of solutions of terminal problems for multidimensional affine systems. We show that the terminal problem for a regular system in quasicanonical form can be reduced to a boundary value problem for a system of ordinary differential equations of lower order with right-hand side depending on a vector parameter. We prove a sufficient condition for the existence of a solution of the above-mentioned boundary value problem. A method for constructing a numerical solution is developed.
We propose efficient algorithms for solving large-scale matrix differential equations. In particular, we deal with the differential Riccati equations (DRE) and state the applicability to the differential Lyapunov equations (DLE). We focus on methods, based on standard versions of ordinary differential equations, in the matrix setting. The application of these methods yields algebraic Lyapunov equations (ALEs) with a certain structure to be solved in every step. The alternating direction implicit (ADI) algorithm and Krylov subspace based methods allow to exploit this special structure. However, a direct application of classic low-rank formulations requires the use of complex arithmetic. Using an -type decomposition of both, the right hand side and the solution of the equation, we avoid this problem. Thus, the proposed methods are a more practical alternative for large-scale problems arising in applications. Also, they make the application of higher order methods feasible. The numerical results show the better performance of the proposed methods compared to earlier formulations.
We suggest a method for solving the terminal control problem for multidimensional affine systems that are not linearizable by feedback. We prove a sufficient condition for the existence of a solution and present a numerical procedure for its construction.
The Drazin inverse solutions of the matrix equations , and are considered in this
paper. We use both the determinantal representations of the Drazin inverse
obtained earlier by the author and in the paper. We get analogs of the Cramer
rule for the Drazin inverse solutions of these matrix equations and using their
for determinantal representations of solutions of some differential matrix
equations, and , where the matrix is singular.
We use elementary methods and operator identities to solve linear matrix differential equations and we obtain explicit formulas for the exponential of a matrix. We also give explicit constructions of solutions of scalar homogeneous equations with certain initial values, called dynamic solutions, that play an important role in the solution of homogeneous and non-homogeneous matrix differential equations. We show that the same methods can be used to solve linear matrix difference equations. (c) 2006 Elsevier Inc. All rights reserved.
Periodic Lyapunov, Sylvester and Riccati differential equations have many important applications in the analysis and design of linear periodic control systems. For the numerical solution of these equations efficient numerically reliable algorithms based on the periodic Schur decomposition are proposed. The new multi-shot type algorithms compute periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to traditionally used one-shot periodic generator methods, the multi-shot type methods have the advantage to be able to address problems with large periods and/or unstable dynamics. Applications of the proposed techniques to compute several system norms are presented.
We revisit the canonical continuous-time and discrete-time matrix algebraic and matrix differential equations that play a central role in Lyapunov-based stability arguments. The goal is to generalize and extend these types of equations and subsequent analysis to dynamical systems on domains other than R or Z, called ¿time scales¿, e.g. nonuniform discrete domains or domains consisting of a mixture of discrete and continuous components. In particular, we compare and contrast a generalization of the algebraic Lyapunov equation and the dynamic Lyapunov equation in this time scales setting.
The paper presents closed form solutions for a class of matrix linear systems whose state variable is a matrix. The formulation evaluates the state response of the system in terms of the original system matrices. The proposed solutions naturally fit systems which are most conveniently described by matrix processes. Its formulation uses a compact notation referred as double matrix exponential functions, which is an extension of matrix exponential function, for aiding both intuition and mathematical manipulation. It is a straightforward extension of the solutions for ordinary vector linear systems studied in the past several decades and will play an important role in the design of matrix linear systems using original system matrices.
Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.
This SIAM edition is an unabridged, corrected republication of th Incluye bibliografía e índice
A lower bound for the determinant of the solution to the Lyapunov matrix differential equation is derived. It is shown that this bound is obtained as a solution to a simple scalar differential equation. In the limiting case where the solution to the Lyapunov differential equation becomes stationary, the result reduces to one of the existing bounds for the algebraic equation.
Теория обыкновенных дифференциальных уравнений. М.: Издательство иностранной литературы
- Э А Коддингтон
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Коддингтон Э.А., Левинсон Н. Теория обыкновенных дифференциальных
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Сер. «Естественные науки». 2016. № 3 25 19. Фетисов Д.А. Об одном методе решения терминальных задач для аффинных систем // Наука и образование On solving periodic differential matrix equations with applications to periodic system norms computation
- Мгту Вестник
- . Н Э Баумана
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. «Естественные науки». 2016. № 3 25
19. Фетисов Д.А. Об одном методе решения терминальных задач для аффинных
систем // Наука и образование. МГТУ им. Н.Э. Баумана. Электрон. журнал.
2013. № 11. DOI: 10.7463/1113.0622543 URL: http://technomag.bmstu.ru/
doc/622543.html
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