基于Delta算子的统一代数Lyapunov方程解的上下界

基于Delta算子描述,统一研究了连续代数Lyapunov方程(CALE)和离散代数Lyapunov方程(DALE)的定界估计问题.采用矩阵不等式方法,给出了统一的代数Lyapunov方程(UALE)解矩阵的上下界估计,在极限情形下可分别得到CALE和DALE的估计结果.计算实例表明了本文方法的有效性.     

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.     

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.     

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     

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.     

广义系统具有正定解的Lyapunov方程

本文研究线性广义系统有正定解的Lyapunov方程，给出广义系统稳定等价于Lyapunov方程有正定解，进一步研究了广义系统R－能观，稳定和Lyapunov方程存在正定解三者之间的关系。基于该Lyapunov方程，给出广义系统允许（正则，稳定，无脉冲）的等价条件。     

Common Lyapunov function approach to matrix root clustering

This paper proposes an alternative route to the matrix root clustering problem, which enables to use a result on the common Lyapunov function for solving the problem efficiently. A necessary and sufficient condition is obtained in terms of the existence of a common positive definite solution to a set of Lyapunov inequalities for eigenvalues of a matrix to lie in a prescribed subregion of the complex plane. Applications to root clustering in sector regions are shown for illustration.     

Unified type non-stationary Lyapunov matrix equation — Simultaneous eigenvalue bounds

Simultaneous eigenvalue bounds for the solution of the unified non-stationary Lyapunov matrix equation are presented. When the solution becomes stationary, the results reduce to bounds of the unified type algebraic Lyapunov equation. In the limiting cases, the results reduce to bounds for the solution of the differential and difference Lyapunov equations. The bounds given in this paper are a generalization of some existing bounds obtained separately for the continuous and discrete type stationary and non-stationary Lyapunov equations.     

Generalized Lyapunov equation and factorization of matrix polynomials

This paper presents an algorithm for the construction of a solution of the generalized Lyapunov equation. It is proved that the polynomial matrix factorization relative to the imaginary axis may be reduced to the successive solution of Lyapunov equations, i.e. the factorization is reduced to the solution of a sequence of generalized Lyapunov equations, not to the solution of generalized Riccati equation.     

On solving the Lyapunov and Stein equations for a companion matrix

When the matrix A is in companion form, the essential step in solving the Lyapunov equation PA + A^TP = −Q involves a linear n × n system for the first column of the solution matrix P. The complex dependence on the data matrices A and Q renders this system unsuitable for actual computation. In this paper we derive an equivalent system which exhibits simpler dependence on A and Q as well as improved complexity and robustness characteristics. A similar results is obtained also for the Stein equation P − A^TPA = Q.     

Bounds for the eigenvalues of the solution of the unified algebraic Riccati matrix equation

In recent years, several eigenvalues bounds have been investigated separately for the solutions of the continuous and the discrete Riccati and Lyapunov matrix equations. In this paper, lower bounds for the eigenvalues of the solution of the unified Riccati equation (relatively to continuous and discrete cases), are presented. In the limiting cases, the results reduce to some new bounds for both the continuous and discrete Riccati equation.     

Lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations

In this paper, we propose lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations. We give comparisons between the parallel estimates. Finally, we give examples showing that our bounds can be better than the previous results for some cases.     

A computational solution for the matrix riccati equation using laplace transforms

The solution for the finite-time matrix Riccati equation is presented in this paper. The solution to the Riccati equation is obtained in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and solved using Laplace transforms and the computation is done by the digital computer.

A numerical example for the proposed method is given.     

Iterative procedure for solution of sylvester generalized matrix equation

An iteration procedure for solving Sylvester generalized matrix equation is proposed in this paper. The sufficient conditions of stability of the iteration procedure for solving this equation are obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 183–186, May–June, 2000.     

Modular array structure for non-restoring square root circuit

Square root is an operation performed by the hardware in recent generations of processors. The hardware implementation of the square root operation is achieved by different means. One of the popular methods is the non-restoring algorithm. In this paper, the classical non-restoring array structure is improved in order to simplify the circuit. This reduction is done by eliminating a number of circuit elements without any loss in the precision of the square root or the remainder. For a 64-bit non-restoring circuit the area of the suggested circuit is about 44% smaller than that of a conventional non-restoring array circuit. Furthermore, in order to create an environment for modular design of the non-restoring square root circuit, a number of modules are suggested. Using these modules it is possible to construct any square root circuit with an arbitrary number of input bits. The suggested methodology results in an expandable design with reduced-area. Analytical and simulation results show that the delay of the proposed circuit, for a 64-bit radicand, is 80% less than that of a conventional non-restoring array circuit.     

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.     

An iterative algorithm for a least squares solution of a matrix equation

In this article, we propose an iterative algorithm to compute the minimum norm least-squares solution of AXB+CYD=E, based on a matrix form of the algorithm LSQR for solving the least squares problem. We then apply this algorithm to compute the minimum norm least-squares centrosymmetric solution of min _X ‖AXB?E‖ _F . Numerical results are provided to verify the efficiency of the proposed method.     

A new type of recurrent neural networks for real-time solution of Lyapunov equation with time-varying coefficient matrices

A new kind of recurrent neural network is presented for solving the Lyapunov equation with time-varying coefficient matrices. Different from other neural-computation approaches, the neural network is developed by following Zhang et al.'s design method, which is capable of solving the time-varying Lyapunov equation. The resultant Zhang neural network (ZNN) with implicit dynamics could globally exponentially converge to the exact time-varying solution of such a Lyapunov equation. Computer-simulation results substantiate that the proposed recurrent neural network could achieve much superior performance on solving the Lyapunov equation with time-varying coefficient matrices, as compared to conventional gradient-based neural networks (GNN).     

Calculating the square root with arbitrary order of convergence

In this paper an iterative algorithm has been presented for calculating the square root of a real number with arbitrary order of convergence using formulae derived by applying binomial theorem. The primary objective is to reduce the number of division operations required.