Bounds for the eigenvalues of the continuous algebraic Riccati equation

By using singular value decomposition and majorisation inequalities, we propose new upper and lower bounds for summations of eigenvalues (including the trace) of the solution of the continuous algebraic Riccati equation. These bounds improve and extend some of the previous results. Finally, we give corresponding numerical examples to illustrate the effectiveness of our results.     

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

In 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.     

Stabilisation of time-varying linear systems via Lyapunov differential equations

This article studies stabilisation problem for time-varying linear systems via state feedback. Two types of controllers are designed by utilising solutions to Lyapunov differential equations. The first type of feedback controllers involves the unique positive-definite solution to a parametric Lyapunov differential equation, which can be solved when either the state transition matrix of the open-loop system is exactly known, or the future information of the system matrices are accessible in advance. Different from the first class of controllers which may be difficult to implement in practice, the second type of controllers can be easily implemented by solving a state-dependent Lyapunov differential equation with a given positive-definite initial condition. In both cases, explicit conditions are obtained to guarantee the exponentially asymptotic stability of the associated closed-loop systems. Numerical examples show the effectiveness of the proposed approaches.     

Lyapunov 矩阵微分方程解的新估计

采用控制不等式方法,并结合正规矩阵的相关性质,我们给出系统矩阵A是正规矩阵的Lyapunov矩阵微分方程解的特征值的和(包括迹)的界.在极限情况下,这些结果可以变为Lyapunov矩阵代数方程的界.数值算例表明该结果的有效性.     

New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

The discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.     

Upper solution bounds of the continuous coupled algebraic Riccati matrix equation

In this article, by using some matrix identities, we construct the equivalent form of the continuous coupled algebraic Riccati equation (CCARE). Further, with the aid of the eigenvalue inequalities of matrix's product, by solving the linear inequalities utilising the properties of M-matrix and its inverse matrix, new upper matrix bounds for the solutions of the CCARE are established, which improve and extend some of the recent results. Finally, a corresponding numerical example is proposed to illustrate the effectiveness of the derived results.     

Solution of matrix Riccati differential equation for nonlinear singular system using genetic programming

In this paper, we propose a novel approach to find the solution of the matrix Riccati differential equation (MRDE) for nonlinear singular systems using genetic programming (GP). The goal is to provide optimal control with reduced calculation effort by comparing the solutions of the MRDE obtained from the well known traditional Runge Kutta (RK) method to those obtained from the GP method. We show that the GP approach to the problem is qualitatively better in terms of accuracy. Numerical examples are provided to illustrate the proposed method.      

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1?d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.     

Recursive approximate solution to time-varying matrix differential Riccati equation: linear and nonlinear systems

An approximate solution is proposed for linear-time-varying (LTV) systems based on Taylor series expansion in a recursive manner. The intention is to present a fast numerical solution with reduced sampling time in computation. The proposed procedure is implemented on finite-horizon linear and nonlinear optimal control problem. Backward integration (BI) is a well known method to give a solution to finite-horizon optimal control problem. The BI performs a two-round solution: first one elicits an optimal gain and the second one completes the answer. It is very important to finish the backward solution promptly lest in practical work, system should not wait for any action. The proposed recursive solution was applied for mathematical examples as well as a manipulator as a representative of complex nonlinear systems, since path planning is a critical subject solved by optimal control in robotics.     

Bounds for the solution of the Lyapunov matrix equation — A unified approach

In recent years, several bounds have been reported for the solution of the continuous and the discrete Lyapunov equations. Using the unified Lyapunov equation, we give in this paper bounds for the solution of this equation. In the limiting cases, the bounds reduce to existing bounds for both the continuous and discrete Lyapunov equations.     

Upper and lower matrix bounds of the solution for the discreteLyapunov equation

New matrix bounds of the solution for the discrete algebraic matrix Lyapunov equation are established. The upper and lower eigenvalue bounds such as each eigenvalue including the extreme ones, the trace, and the determinant are also determined by these matrix bounds for the same solution. The present schemes are tighter as compared with the majority of existing results     

Finite-time stability analysis of a class of nonlinear time-varying systems: a numerical algorithm

The paper investigates the finite-time stability (FTS) analysis of a very general class of nonlinear time-varying systems. The FTS of the considered system, whose vector field consists of a nonlinear part which can be sublinear or superlinear, and a linear part which can be time-varying, has not been fully studied before. By estimating the bound of the norm of the considered system's states with the generalised Gronwall–Bellman inequality, a sufficient criterion is established to guarantee the FTS of the considered system. To facilitate checking the criterion in practice, a novel numerical algorithm is proposed by numerically solving certain differential equations. Therefore, the FTS of the considered class of nonlinear time-varying systems can be easily analysed by the numerical algorithm. Further considering the numerical errors in the practical numerical computation, we strictly prove the credibility and programmability of the numerical algorithm in theory. Finally, three numerical examples are provided to illustrate the effectiveness the proposed results.     

New matrix bounds,an existence uniqueness and a fixed-point iterative algorithm for the solution of the unified coupled algebraic Riccati equation

In this paper, combining the equivalent form of the unified coupled algebraic Riccati equation (UCARE) with the eigenvalue inequalities of a matrix's sum and product, using the properties of an M-matrix and its inverse matrix, we offer new lower and upper matrix bounds for the solution of the UCARE. Furthermore, applying the derived lower and upper matrix bounds and a fixed-point theorem, an existence uniqueness condition of the solution of the UCARE is proposed. Then, we propose a new fixed-point iterative algorithm for the solution of the UCARE. Finally, we present a corresponding numerical example to demonstrate the effectiveness of our results.     

New lower matrix bounds for the solution of the continuous algebraic lyapunov equation

New lower matrix bounds are derived for the solution of the continuous algebraic Lyapunov equation (CALE). Following each bound derivation, an iterative algorithm is proposed to derive tighter matrix bounds. In comparison to existing results, the presented results are more concise and are always valid when the CALE has a non‐negative definite solution. We finally give numerical examples to show the effectiveness of the derived bounds and make comparisons with existing results. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms

This paper deals with the free nonlinear dynamical system [xdot](t)=A x(t)+h(t, x(t)), t≥t ₀, x(t ₀)=x ₀, where A is an n×n-matrix and h a nonlinear vector function with h(t, u)=o(∥u∥). As the first novel point, a lower bound for the asymptotic behaviour on the solution x(t) is derived. Two methods are applied to determine the optimal two-sided bounds, where one of the methods is the differential calculus of norms. In this context, the second novel point enters; it consists of a new strategy to significantly reduce the computation time for the determination of the optimal constants in the two-sided bounds. The obtained results are especially of interest in engineering and cannot be obtained by the methods used so far.     

The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order

Using the method of upper and lower solutions in reverse order, we present an existence theorem for a linear fractional differential equation with nonlinear boundary conditions.     

求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix

Some new estimates for the eigenvalue decay rate of the Lyapunov equation AX+XA^T=B with a low rank right-hand side B are derived. The new bounds show that the right-hand side B can greatly influence the eigenvalue decay rate of the solution. This suggests a new choice of the ADI-parameters for the iterative solution. The advantage of these new parameters is illustrated on second order damped systems with a low rank damping matrix.     

Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations

In this paper, we propose upper bounds for the sum of the maximal eigenvalues of the solutions of the continuous coupled algebraic Riccati equation (CCARE) and the discrete coupled algebraic Riccati equation (DCARE), which are then used to infer upper bounds for the maximal eigenvalues of the solutions of each Riccati equation. By utilizing the upper bounds for the maximal eigenvalues of each equation, we then derive upper matrix bounds for the solutions of the CCARE and DCARE. Following the development of each bound, an iterative algorithm is proposed which can be used to derive tighter upper matrix bounds. Finally, we give numerical examples to demonstrate the effectiveness of the proposed results, making comparisons with existing results.