The existence uniqueness and the fixed iterative algorithm of the solution for the discrete coupled algebraic Riccati equation

In this article, applying the properties of M-matrix and non-negative matrix, utilising eigenvalue inequalities of matrix's sum and product, we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic Riccati equation (DCARE). Secondly, we discuss the solution existence uniqueness condition of the DCARE using the developed upper and lower matrix bounds and a fixed point theorem. Thirdly, a new fixed iterative algorithm of the solution for the DCARE is shown. Finally, the corresponding numerical examples are given to illustrate the effectiveness of the developed results.     

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.     

Matrix bounds of the discrete ARE solution

This paper provides new lower and upper matrix bounds of the solution to the discrete algebraic Riccati equation. The lower bound always works if the solution exists. The upper bounds are presented in terms of the solution of the discrete Lyapunov equation and its upper matrix bound. The upper bounds are always calculated if the solution of the Lyapunov equation exists. A numerical example shows that the new bounds are tighter than previous results in many cases.     

The new solution bounds of the discrete algebraic Riccati equation and their applications in redundant control inputs systems

Redundant control inputs play an important part in engineering and are often used in $H_2$ control problems, dynamic control allocation, quadratic performance optimal control, and many uncertain systems. In this paper, by the equivalent form of the discrete algebraic Riccati equation (DARE), we propose new upper and lower bounds of the solution for the equivalent DARE. Compared with some existing work on this topic, the new bounds are more tighter. Next, when increasing the columns of the input matrix, we give the applications of these new upper and lower solution bounds to obtain a sufficient condition for strictly decreasing feedback controller gain. Finally, corresponding numerical examples illustrate the effectiveness of our 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.     

The generalized continuous algebraic Riccati equation and impulse-free continuous-time LQ optimal control

The purpose of this paper is to investigate the role that the so-called constrained generalized Riccati equation plays within the context of continuous-time singular linear–quadratic (LQ) optimal control. This equation has been defined following the analogy with the discrete-time setting. However, while in the discrete-time case the connections between this equation and the linear–quadratic optimal control problem has been thoroughly investigated, to date very little is known on these connections in the continuous-time setting. This note addresses this point. We show, in particular, that when the continuous-time constrained generalized Riccati equation admits a solution, the corresponding linear–quadratic problem admits an impulse-free optimal control. We also address the corresponding infinite-horizon LQ problem for which we establish a similar result under the additional constraint that there exists a control input for which the cost index is finite.     

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.     

Novel insights on the stabilising solution to the continuous-time algebraic Riccati equation

In the present paper we present a closed-form solution, as a function of the closed-loop poles, for the continuous-time algebraic Riccati equations (CAREs) related to single-input single-output systems with non-repeated poles. The proposed solution trades the standard numerical algorithm approach for one based on a spectral factorisation argument, offering potential insight into any control technique based on a CARE and its solution. As an example, we present the equivalence of two fairly recent control over network results. Furthermore we apply the proposed result to the formula for the optimal regulator gain matrix k (or equivalently the Luenberger's observer gain l) and present an example. Finally, we conclude by discussing the possible extension of the proposed closed-form solution to the repeated eigenvalues case and to the case when the CARE is related to multiple-input multiple-output systems.     

离散时间代数Riccati方程解矩阵的特征值分析

针对离散时间代数Riccati方程DTARE的唯一对称正定解X的特征值,通过矩阵的恒等变形,给出了一种新的分析方法.最后获得解X的极值特征值的上界和下界,以及解X的特征值的和———迹的一个下界.     

Stable Hermitian solutions of discrete algebraic Riccati equations

Stable and Lipschitz stable hermitian solutions of the discrete algebraic Riccati equations are characterized, in the complex as well as in the real case.This paper was written while the first author visited The College of William and Mary.Partially supported by NSF Grant DMS-8802836 and by the Binational United States-Israel Science Foundation.     

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.     

Intervals of solutions of the discrete-time algebraic Riccati equation

If two solutions YZ of the DARE are given then the set of solutions X with YXZ can be parametrized by invariant subspaces of the closed loop matrix corresponding to Y. The paper extends the geometric theory of Willems from the continuous-time to the discrete-time ARE making the weakest possible assumptions.     

一类离散时间代数Riccati矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在线性二次优化问题中遇到的含未知矩阵之逆的离散时间代数Riccati矩阵方程(DTARME)转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求DTARME的对称解的双迭代算法。双迭代算法仅要求DTARME有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定。数值算例表明双迭代算法是有效的。     

On the existence of maximal solution for generalized algebraic Riccati equations arising in stochastic control

Existence of maximal solution is proved for a generalized version of the well-known standard algebraic Riccati equations which arise in certain stochastic optimal control problems.     

On the numerical solution of a structured nonsymmetric algebraic Riccati equation

A class of nonsymmetric algebraic Riccati equation, where one of the two linear coefficients is block diagonal, is studied. These equations arise in the modeling of an adaptive MMAP[K]/PH[K]/1$MMAP\left[K\right]/PH\left[K\right]/1$ queue. Some theoretical results are proved, and two new algorithms are introduced that exploit the diagonal structure of the linear coefficient.     

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.     

Algorithm for solving algebraic Riccati equation which has singular Hamiltonian matrix

The algorithm of construction of the solution of ARE, with Hamiltonian matrix having zero eigenvalues, is developed. The algorithm generalizes the Schur method on ARE with singular Hamiltonian matrix and could be used for J-factorization of matrix polynomial, which has zero roots.     

Existence and uniqueness of unmixed solutions of the discrete-time algebraic Riccati equation

A solution X of a discrete-time algebraic Riccati equation is called unmixed if the corresponding closed-loop matrix Φ(X) has the property that the common roots of det(sI−Φ(X)) and det(I−sΦ(X)^*) (if any) are on the unit circle. A necessary and sufficient condition is given for existence and uniqueness of an unmixed solution such that the eigenvalues of Φ(X) lie in a prescribed subset of .     

Strong solutions of the discrete-time algebraic Riccati equation

Conditions are given under which a solution of the DARE is positive semidefinite if and only if all the eigenvalues of its associated closed-loop matrix are in the closed unit disc.