New upper matrix bounds of the solution for perturbed continuous coupled algebraic Riccati matrix equation

In this paper, if the coefficient matrices in the continuous coupled algebraic Riccati equation (CCARE) undergo perturbations, with the aid of the equivalent form for the perturbation of the CCARE and the classical eigenvalue inequalities, we observe new upper matrix bounds for the perturbation of the CCARE through solving the linear inequalities. Finally, we present corresponding numerical examples to show the effectiveness of the derived results.     

The improved upper solution bounds of the continuous coupled algebraic Riccati matrix equation

In this paper, combining some special eigenvalue inequalities of matrix’s product and sum with the equivalent form of the continuous coupled algebraic Riccati equation (CCARE), we construct linear inequalities. Then, in terms of the properties of M-matrix and its inverse matrix, through solving the derived linear inequalities, we offer new upper matrix bounds for the solution of the CCARE, which improve some of the recent results. Finally, we present a corresponding numerical example to show the effectiveness of the given results.     

Lower solution bounds of the continuous coupled algebraic Riccati matrix equation

In this paper, applying eigenvalue sum inequality of symmetric matrix and the properties of M-matrix and its inverse matrix, we introduce new lower matrix bounds for the solution of the continuous coupled algebraic Riccati equation. Finally, we give corresponding numerical examples to demonstrate the effectiveness of the derived results.     

Two new upper bounds of the solution for the continuous algebraic Riccati equation and their application

In this paper, for the solution of the continuous algebraic Riccati equation(CARE), we derived two new upper matrix bounds. Compared with the existing results, the newly obtained bounds are less conservative and more practical, which means that the condition for the existence of the upper bounds derived here is much weaker. The advantage of the results is shown by theoretical analysis and numerical examples. Moreover, in redundant optimal control, when we increase the columns of the input matrix, some sufficient conditions are presented to strictly decrease the largest singular value of the feedback matrix by utilizing these upper bounds.We also give some examples to illustrate the effectiveness of these sufficient conditions.     

On upper bounds for the unified algebraic Riccati equation

In recent years, several eigenvalues, norms and determinants bounds have been investigated separately for the solutions of continuous and discrete Riccati equations. In this paper, an upper bound for solution of the unified Riccati equation is presented. In the limiting cases, the result reduces to a new upper bound for the solution of continuous and discrete Riccati equation.     

Solution bounds of the continuous Riccati matrix equation

A new approach is proposed for estimating the solution of the continuous algebraic Riccati equation (CARE). Upper and lower solution bounds of the CARE are presented. Comparisons show that the present bounds are more general and/or tighter than existing results.     

Some bounds of the solution of the algebraic Riccati equation

Bounds on extremal eigenvalues and lower and upper bounds of the trace for the solution of the algebraic Riccati equation are presented. Through two examples, it is shown that, in some special cases, the presented bounds can be better than the results recently published     

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.     

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.     

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.     

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.     

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

Novel bounds are proposed for the extreme and lower half eigenvalues of the solution matrix for the algebraic Riccati equation. The formulae giving these bounds can easily be applied to determine the region where the eigenvalues lie, and the bounds have the added advantage of being sharper in some cases than the previously proposed ones, as some realistic examples will show. The proposed bounds find many applications which are pointed out in the text.     

Upper matrix bounds for the discrete algebraic Riccati matrixequation

We propose upper matrix bounds for the discrete algebraic Riccati matrix equation. We show that our results are less restrictive than the previous bounds. Finally, we give numerical examples in order to verify the effectiveness of our results     

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.     

A note on eigenvalue bounds in algebraic Riccati equation

In this note, an upper bound on the maximum eigenvalue of the solution matrixKof the algebraic Riccati equation is established. The approach outlined also results in several lower bounds, which are more general than those derived in [1], for some of the largest eigenvalues ofK.     

The limiting form of the continuous algebraic Riccati equation

In this paper solution-preserving transformations of algebraic Riccati equations are examined. As an illustrative example we deal with the limiting cost of the linear-quadratic cheap control problem.     

A nonrecursive algebraic solution for the discrete Riccati equation

Equations for the optimal linear control and filter gains for linear discrete systems with quadratic performance criteria are widely documented. A nonrecursive algebraic solution for the Riccati equation is presented. These relations allow the determination of the steady-state solution of the Riccati equation directly without iteration. The relations also allow the direct determination of the transient solution for any particular time without proceeding recursively from the initial conditions. The method involves finding the eigenvalues and eigenvectors of the canonical state-costate equations.     

A note on the solution of the algebraic Riccati equation

In this note, we consider the problem of solving the algebraic Riccati equation (ARE) arising in the optimal control theory, Several cases are studied. The solution here is, unlike the usual ones, related directly to the controllability and the observability matrices, the weighting matrices and the resulting closed-loop eigenvalues. As a result, it only requires few matrix operations to obtain the solution.     

Upper bounds for the solution of the discrete Riccati equation

Upper bounds for individual eigenvalues and for summations of eigenvalues including the trace of the solution of the discrete algebraic Riccati equation are presented. Some are new, and some supplement bounds in the literature     

A numerical solution of the stochastic discrete algebraic Riccati equation

This article proposes two algorithms for solving a stochastic discrete algebraic Riccati equation which arises in a stochastic optimal control problem for a discrete-time system. Our algorithms are generalized versions of Hewer’s algorithm. Algorithm I has quadratic convergence, but needs to solve a sequence of extended Lyapunov equations. On the other hand, Algorithm II only needs solutions of standard Lyapunov equations which can be solved easily, but it has a linear convergence. By a numerical example, we shall show that Algorithm I is superior to Algorithm II in cases of large dimensions. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008