Lower bounds on the solution of Lyapunov matrix and algebraic Riccati equations

In this note various lower bounds for all the eigenvalues of the solution matrixKof the Lyapunov matrix equation are established. A special case of this result is a generalization of that presented in [1]-[3], where lower bounds for the maximum and minimum eigenvalues ofKare given. Moreover, the approach used here enables one to establish various lower bounds for some of the (largest) eigenvalues of the solution matrix of the algebraic Riccati equation.     

Residual bounds for discrete-time Lyapunov equations

This paper presents a novel backward error criterion together with a sensitivity measure for assessing the solution accuracy of discrete-time Lyapunov equations. The conventional method involves standard perturbation and sensitivity results for certain associated linear systems. Such an approach, however, ignores the structure of the linear system and is often inconclusive. The method considered in this paper takes full account of the underlying structure of the problem and provides a meaningful measure of accuracy     

On norm bounds for algebraic Riccati and Lyapunov equations

New lower bounds on the spectral norms of the positive definite solutions to the continuos and discrete algebraic matrix Riccati and Lyapunov equations are derived. These bounds are much easier to compute than previously available bounds and appear to be considerably tighter in many cases.     

Improved bounds for the eigenvalues of solutions of Lyapunov equations

Bounds on the solution and on the eigenvalues of the solution of the discrete and of the continuous matrix Lyapunov equations are presented. These bounds are nontrivial in the case of system matrices having multiple eigenvalues. Especially in the latter case, these bounds are better than bounds found in the literature. Further, these bounds are sharp, i.e., there exist systems such that the lower and upper bounds coincide yielding the solution or its eigenvalues, respectively.     

Eigenvalue bounds in the Lyapunov and Riccati matrix equations

Two related theorems of Strang [1] are extended to provide upper and lower bounds on the eigenvalues of the Lyapunov and Riccati matrices, given byQ=AB^{H}+BAandR=AB^{H}+BA+2AHA, whereAandHare Hermitian, positive definite, complex matrices. We discuss inversion to obtain eigenvalue bounds on the matrixAfor the usual case in whichQ , R, andHare known.     

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.     

Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case

We present two new bounds for the eigenvalues of the solutions to a class of continuous- and discrete-time Lyapunov equations. These bounds hold for Lyapunov equations with symmetric coefficient matrices and right-hand side matrices of low rank. They reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix.     

On the estimation of solution bounds of the generalized Lyapunov equations and the robust root clustering for the linear perturbed systems

This paper measures the solution bounds for the generalized Lyapunov equations (GLE). By making use of linear algebraic techniques, we estimate the upper and lower matrix bounds for the solutions of the above equations. All the proposed bounds are new, and it is also shown that the majority of existing bounds are the special cases of these results. Furthermore, according to these bounds, the problem of robust root clustering in sub-regions of the complex plane for linear time-invariant systems subjected to parameter perturbations is solved. The tolerance perturbation bounds for robust clustering in the given sub-regions are estimated. Compared to previous results, the feature of these tolerance bounds is that they are independent of the solution of the GLE.     

Upper bounds for the eigenvalues of the solution of the discretealgebraic Lyapunov equation

Upper bounds for summations including the trace, and for products including the determinant, of the eigenvalues of the solution of the discrete algebraic Lyapunov equation (DALE) are presented. All bounds are derived from the matrix series solution of the DALE. The majority of the bounds are tighter than those in the literature     

Frequency domain Lyapunov equations and performance bounds for differential linear repetitive processes

Repetitive processes are characterised by a series of passes through a set of dynamics defined over a finite fixed duration with explicit interaction between successive outputs. In this paper, a new Lyapunov equation based stability condition is developed for one subclass and used to construct bounds on expected system performance.     

New estimates for solutions of Lyapunov equations

Results for estimating the solution of differential and algebraic Lyapunov matrix equations are obtained, and some of the well-known results are generalized     

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 eigenvalues of the solution of the discrete Riccati and Lyapunov equations and the continuous Lyapunov equation

We present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete Riccati and Lyapunov matrix equations and the continuous Lyapunov matrix equation. Nearly all of our bounds for the discrete Riccati equation are new. The bounds for the discrete and continuous Lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation.     

Comments on "On some bounds in the algebraic Riccati and Lyapunov equations"

Results presented by Moil in the paper, which give some bounds for the trace and the determinant of the positive definite solution to the algebraic Riccati matrix equation, are shown to be erroneous, and suggestions to remedy these errors are made.     

A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU-GPU platforms

We describe a hybrid Lyapunov solver based on the matrix sign function, where the intensive parts of the computation are accelerated using a graphics processor (GPU) while executing the remaining operations on a general-purpose multi-core processor (CPU). The initial stage of the iteration operates in single-precision arithmetic, returning a low-rank factor of an approximate solution. As the main computation in this stage consists of explicit matrix inversions, we propose a hybrid implementation of Gauß-Jordan elimination using look-ahead to overlap computations on GPU and CPU. To improve the approximate solution, we introduce an iterative refinement procedure that allows to cheaply recover full double-precision accuracy. In contrast to earlier approaches to iterative refinement for Lyapunov equations, this approach retains the low-rank factorization structure of the approximate solution. The combination of the two stages results in a mixed-precision algorithm, that exploits the capabilities of both general-purpose CPUs and many-core GPUs and overlaps critical computations. Numerical experiments using real-world data and a platform equipped with two Intel Xeon QuadCore processors and an Nvidia Tesla C1060 show a significant efficiency gain of the hybrid method compared to a classical CPU implementation.     

Simultaneous eigenvalue lower bounds for the Lyapunov matrixequation

Simultaneous eigenvalue lower bounds for the solution of the Lyapunov equation are presented. The n-summation and the n -product bounds generalize existing bounds such as those for the trace and the determinant. These bounds are stronger than the majority of the relevant bounds shown in the literature     

A matrix inequality associated with bounds on solutions of algebraic Riccati and Lyapunov equations

A new proof is presented for the inequality,tr (XY) leq parallel X parallel_{2} cdot tr Y. This argument is valid under the condition thatYbe real symmetric nonnegative definite;Xmay be any square 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.