Lyapunov Matrix Differential Equation Research Articles

Numerical Covariance Evaluation for Linear Structures Subject to Non-Stationary Random Inputs

Random vibration analysis is a mathematical tool that offers great advantages in predicting the mechanical response of structural systems subjected to external dynamic loads whose nature is intrinsically stochastic, as in cases of sea waves, wind pressure, and vibrations due to road asperity. Using random vibration analysis is possible, when the input is properly modeled as a stochastic process, to derive pieces of information about the structural response with a high quality (if compared with other tools), especially in terms of reliability prevision. Moreover, the random vibration approach is quite complex in cases of non-linearity cases, as well as for non-stationary inputs, as in cases of seismic events. For non-stationary inputs, the assessment of second-order spectral moments requires resolving the Lyapunov matrix differential equation. In this research, a numerical procedure is proposed, providing an expression of response in the state-space that, to our best knowledge, has not yet been presented in the literature, by using a formal justification in accordance with earthquake input modeled as a modulated white noise with evolutive parameters. The computational efforts are reduced by considering the symmetry feature of the covariance matrix. The adopted approach is applied to analyze a multi-story building, aiming to determine the reliability related to the maximum inter-story displacement surpassing a specified acceptable threshold. The building is presumed to experience seismic input characterized by a non-stationary process in both amplitude and frequency, utilizing a general Kanai–Tajimi earthquake input stationary model. The adopted case study is modeled in the form of a multi-degree-of-freedom plane shear frame system.

Lower and upper bounds of the solution for the Lyapunov matrix differential equation and an application in input-output finite-time stability of linear systems

In control theory and practical engineering fields, such as the controllability, observability, input–output finite-time stability of the linear systems, it is a significant problem to study the properties of the solution for the Lyapunov matrix differential equation where there are no restrictions on the system matrix. In this paper, by constructing an equivalent form of the Lyapunov matrix differential equation and utilizing some important matrix eigenvalue inequalities, lower and upper bounds for the matrix solution of the Lyapunov matrix differential equation that remove the strict restrictions for the system matrix are proposed. As an application in linear systems, we show that our bounds can be used to discuss the input–output finite-time stability for linear systems. Finally, we give some corresponding numerical examples to illustrate the effectiveness and superiority of our results.

Upper and lower bounds for the solution of the Lyapunov matrix differential equation

In control systems, the study of controllability, observability and stability can often be transformed into the estimation of the upper and lower bounds for the solution of the Lyapunov matrix differential equation. In this paper, by using the identity transformation of the Lyapunov matrix differential equation and some important eigenvalues inequalities, we obtain new upper and lower matrix bounds of the Lyapunov matrix differential equation. The random numerical examples indicate that the derived bounds are better than the existing one in most cases, which show the effectiveness and superiority of our results.

On the Ψ − Conditional Asymptotic Stability of a Nonlinear Lyapunov Matrix Differential Equation with Integral Term as Right Side

Abstract The aim of this paper is to give sufficient conditions for Ψ − conditional asymptotic stability of the Ψ − bounded solutions of a nonlinear Lyapunov matrix differential equation with integral term as right side.

A numerical approach based on Bernstein collocation method: Application to differential Lyapunov and Sylvester matrix equations

In this paper, we apply the Bernstein collocation method to construct the solution set of the Sylvester matrix differential equation (Sy-MDE) which involves the Lyapunov matrix differential equation. The method depends on the collocation method and Bernstein polynomials. The main advantage of the proposed method is that by using this method Sy-MDE reduces to a linear system of algebraic equations which can be solved by using an appropriate iterative method. We analyze the error and give a theorem that bounds the error. We also give the residual correction procedure to estimate the error. By using the procedure, we obtain a new approximate solution, namely a corrected Bernstein collocation solution. To illustrate how the proposed method is applied, several examples are given. Numerical experiments show the effectiveness and accuracy of the method for solving such types of Sy-MDE.

The Ψ–Asymptotic Equivalence of the Lyapunov Matrix Differential Equations with Integral Term as Right Side and Modified Argument

Abstract Using the notion of strict h-contraction, existence results for Ψ-asymptotic equivalence of two pairs of (Lyapunov) matrix differential equations with integral term as right side and modified argument are given.

On the Ψ − Conditional Exponential Asymptotic Stability of a Nonlinear Lyapunov Matrix Differential Equation with Integral Term as Right Side

Abstract The aim of this paper is to give sufficient conditions for Ψ − conditional exponential asymptotic stability of the Ψ − bounded solutions of a nonlinear Lyapunov matrix differential equation with integral term as right side.

The Ψ−asymptotic equivalence of the Lyapunov matrix differential equations with modified argument

Using the notion of strict h−contraction, existence results for Ψ−asymptotic equivalence of two pairs of (Lyapunov) matrix differential equations with modified argument are given.

On the Ψ−uniform asymptotic stability of nonlinear Lyapunov matrix differential equations

This paper deals with obtaining (necessary and) sufficient conditions for Ψ− uniform asymptotic stability of solutions of nonlinear Lyapunov matrix differential equations.

On the $Psi$-instability of a nonlinear Lyapunov matrix differential equation with integral term as right side

The aim of this paper is to give sufficient conditions for $Psi$-instability of trivial solution of a nonlinear Lyapunov matrix differential equation with integral term as right side.

An iterative algorithm for robust simulation of the Sylvester matrix differential equations

This paper proposes a new effective pseudo-spectral approximation to solve the Sylvester and Lyapunov matrix differential equations. The properties of the Chebyshev basis operational matrix of derivative are applied to convert the main equation to the matrix equations. Afterwards, an iterative algorithm is examined for solving the obtained equations. Also, the error analysis of the propounded method is presented, which reveals the spectral rate of convergence. To illustrate the effectiveness of the proposed framework, several numerical examples are given.

Robust stabilization of non-linear non-autonomous control systems with periodic linear approximation

Abstract The paper deals with the problem of stabilizing the equilibrium states of a family of non-linear non-autonomous systems. It is assumed that the nominal system is a linear controlled system with periodic coefficients. For the nominal controlled system, a new method for constructing a Lyapunov function in the quadratic form with a variable matrix is proposed. This matrix is defined as an approximate solution of the Lyapunov matrix differential equation in the form of a piecewise exponential function based on partial sums of a W. Magnus series. A stabilizing control in the form of a linear feedback with a piecewise constant periodic matrix is constructed. This control simultaneously stabilizes the considered family of systems. The estimates of the domain of attraction of an asymptotically stable equilibrium state of a closed-loop system that are common for all systems are obtained. A numerical example is given.

On the Ψ-asymptotic equivalence of the Ψ-bounded solutions of two Lyapunov matrix differential equations

Using Schauder Tychonoff fixed point theorem and the technique of Kronecker product of matrices, we prove existence results for Ψ-asymptotic equivalence of the Ψ-bounded solutions of two Lyapunov matrix differential equations.

The eigenvalue product bounds of the Lyapunov matrix differential equation and the stability of a class of time-varying nonlinear system

The Lyapunov matrix differential equation plays an important role in many scientific and engineering fields. In this paper, we first give a class relation between the eigenvalue of functional matrix derivative and the derivative of functional matrix eigenvalue. Using this relation, we convert the Lyapunov matrix differential equation into an eigenvalue differential equation. Further, by the Schur theorem, combining Hölder’s integral inequality with arithmetic–geometric average inequality, we provide several lower and upper bounds on the eigenvalue product for the solution of the Lyapunov matrix differential equation. As an application in control and optimization, we show that our bounds can be used to discuss the stability of a class of time-varying nonlinear systems. Finally, we illustrate the superiority and effectiveness of the derived bounds by a numerical example.

On ψ Bounded Solutions for a Nonlinear Lyapunov Matrix Differential Equation on 𝕉

Abstract Using Banach and Schauder - Tychono fixed point theorems, existence results for a nonlinear Lyapunov matrix differential equation on 𝕉 are given. The obtained results generalize and extend the results from [5] and [18].

Upper and lower eigenvalue summation bounds of the Lyapunov matrix differential equation and the application in a class time-varying nonlinear system

ABSTRACTIn this paper, we first show a class relation between the eigenvalue of functional matrix derivative and the derivative of function matrix eigenvalue. Applying the relation, we transform the time-varying linear matrix differential equation into eigenvalue differential equation. Furthermore, by using singular value decomposition and majorisation inequalities, we derive upper and lower bounds on eigenvalue summation of the solution for the Lyapunov matrix differential equation, which improve the recent results. As an application in control and optimisation, we show that our bounds could be used to discuss the stability of a class time-varying nonlinear system. Finally, we give a corresponding numerical example to show the superiority and effectiveness of the derived bounds.

Solution of the Lyapunov matrix differential equations by the frequency method

Methods for solving the Lyapunov matrix differential and algebraic equations in the time and frequency domains are considered. The solutions of these equations are finite and infinite Gramians of various forms. A feature of the proposed new approach to the calculation of Gramians is the expansion of the Gramians in a sum of matrix bilinear or quadratic forms that are formed using Faddeev's matrices, where each form is a solution of the linear differential or algebraic equation corresponding to an eigenvalue of the matrix or to a combination of such eigenvalues. An example illustrating the calculation of finite and infinite Gramians is discussed.

Lower Eigenvalue Bounds on Summation for the Solution of the Lyapunov Matrix Differential Equation

AbstractIn this paper, we offer several lower bounds on eigenvalue summation for the solution of the Lyapunov matrix differential equation applying index matrix eigenvalue inequalities and Hölder inequality. Further, we give a numerical example to illustrate the effectiveness of the derived bounds.

On the Ψ-Conditional Exponential Asymptotic Stability of Nonlinear Lyapunov Matrix Differential Equations

Abstract It is proved (necessary and) sufficient conditions for Ψ– conditional exponential asymptotic stability of the trivial solution of nonlinear Lyapunov matrix differential equations

Lower bounds on eigenvalue summation for the solution of the Lyapunov matrix differential equation

Lower bounds on eigenvalue summation for the solution of the Lyapunov matrix differential equation