ON THE MINIMAL NONNEGATIVE SOLUTION OF ONSYMMETRIC ALGEBRAIC RICCATI EQUATION

We study perturbation bound and structured condition number about the minimalnonnegative solution of nonsymmetric algebraic Riccati equation,obtaining a sharp per-turbation bound and an accurate condition number.By using the matrix sign functionmethod we present a new method for finding the minimal nonnegative solution of this al-gebraic Riccati equation.Based on this new method,we show how to compute the desiredM-matrix solution of the quadratic matrix equation X~2-EX-F=0 by connecting itwith the nonsymmetric algebraic Riccati equation,where E is a diagonal matrix and F isan M-matrix.     

离散时间代数Riccati方程的解矩阵的下界与上界

本文讨论离散时间代数Ｒｉｃｃａｔｉ方程ＡＴＸＡ－Ｘ－（ＡＴＸＢ＋Ｌ）（Ｒ＋ＢＴＸＢ）＾－１（ＬＴ＋ＢＴＸＡ）＋Ｑ＝０的唯一对称正定解的上界和下界。     

A Parameterized Class of Complex Nonsymmetric Algebraic Riccati Equations

In this paper, by introducing a definition of parameterized comparison matrix of a given complex square matrix, the solvability of a parameterized class of complex nonsymmetric algebraic Riccati equations (NAREs) is discussed. The existence and uniqueness of the extremal solutions of the NAREs is proved. Some classical numerical methods can be applied to compute the extremal solutions of the NAREs, mainly including the Schur method, the basic fixed-point iterative methods, Newton's method and the doubling algorithms. Furthermore, the linear convergence of the basic fixed-point iterative methods and the quadratic convergence of Newton's method and the doubling algorithms are also shown. Moreover, some concrete parameter selection strategies in complex number field for the doubling algorithms are also given. Numerical experiments demonstrate that our numerical methods are effective.     

The King-Werner method for solving nonsymmetric algebraic Riccati equation

For the nonsymmetric algebraic Riccati equation arising from transport theory, we concern about solving its minimal positive solution. In [1], Lu transferred the equation into a vector form and pointed out that the minimal positive solution of the matrix equation could be obtained via computing that of the vector equation. In this paper, we use the King-Werner method to solve the minimal positive solution of the vector equation and give the convergence and error analysis of the method. Numerical tests show that the King-Werner method is feasible to determine the minimal positive solution of the vector equation.     

Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution

We consider the initial value problem for a nonsymmetric matrix Riccati differential equation, where the four coefficient matrices form an M-matrix. We show that for a wide range of initial values the Riccati differential equation has a global solution X(t) on [0,∞) and X(t) converges to the stable equilibrium solution as t goes to infinity.     

线性等式约束系统广义Riccati代数方程的求解*

本文基于定常离散LQ控制问题的动力学方程、价值泛函及系统的约束方程,根据极大值原理,给出了线性等式约束系统下的广义Riccati方程,进而对上述方程进行了深入的探讨,并给出了相应的数值例题。     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Solving the nonnegative solution for a (shifted) nonsymmetric algebraic Riccati equation in the critical case

We are interested in computing the nonnegative solution of a nonsymmetric algebraic Riccati equation arising in transport theory. The coefficient matrices of this equation have two parameters c and α. There have been some iterative methods presented by Lu in [13] and Bai et al. in [2] to solve the minimal positive solution for or . While the equation has a unique nonnegative solution when c=1 and α=0, all the methods presented by Lu and Bai cannot be used to find the nonnegative solution. To cope with this problem, a shifted technique is used in this paper to transform the original Riccati equation into a new one so that all the methods can be effectively employed to solve the nonnegative solution. Numerical experiments are given to illustrate the results.     

变量代换法和几类里卡提方程的通解

研究了几类里卡提方程,利用变量代换法得到了方程的通解的几组充分性条件.     

ON SIGN FUNCTION AND SQUARE ROOT METHODS FOR SOLUTION OF REAL ALGEBRAIC RICCATI EQUATIONS

In this paper, we first justify the theory of the square root algorithm presented in [1], [2] by simple application of the matrix sign function, then we present an iterative square root algorithm for the continuous time algebraic Riccati equation.     

广义四元数的一次代数方程

本文运用广义四元数代数的矩阵表示讨论了两类广义四元数的一次代数方程的解问题，并得到了这两类代数方程有唯一解、无穷多解，无解的判别条件。     

Condition numbers of algebraic Riccati equations in the Frobenius norm

Consider the continuous-time algebraic Riccati equation (CARE) and the discrete-time algebraic Riccati equation (DARE) which arise in linear control and system theory. It is known that appropriate assumptions on the coefficient matrices guarantee the existence and uniqueness of Hermitian positive semidefinite stabilizing solutions. In this note, we apply the theory of condition developed by Rice to define condition numbers of the CARE and DARE in the Frobenius norm, and derive explicit expressions of the condition numbers in a uniform manner. Both the complex case and real case are considered, and connections to certain existing condition numbers of the CARE and DARE are discussed.     

一类幂级数求和函数的代数方程方法

建立幂级数和函数相关的代数方程,给出形如sum from n=o to ∞ anxn(其中an为以n为变元的多项式)的幂级数求和函数的一种方法.     

一类Riccati方程的推广

把 Riccati方程 y′=Py2 + Qy+ R推广成 Riccati型方程 :f′( y) dydx=Pf 2 ( y) + Qf ( y) + R.并给出其可积的条件及其对应的通积分 .     

Monotonicity and convexity properties of matrix Riccati equations

Using a Fréchet-derivative-based approach some monotonicity,convexity/concavity and comparison results concerning strictlyunmixed solutions of continuous- and discrete-time algebraicRiccati equations are obtained; it turns out that these solutionsare isolated and smooth functions of the input data. Similarly,it is proved that the solutions of initial value problems forboth Riccati differential and difference equations are smoothand monotonic functions of the input data and of the initial value. They are also convex or concave functions with respectto certain matrix coefficients.     

一类广义Riccati方程的三个可积判据

考虑一类广义Riccati方程,通过函数变换,在所给条件下,将这类方程等价地化为变量分离方程,从而得到了该方程可积的三个充分性判据,并给出方程通解的参数表达形式,扩大了Riccati方程的可解性范围.     

Minimal ranks of some quaternion matrix expressions with applications

Suppose that p(X, Y) = A − BX − X^(∗)B^(∗) − CYC^(∗) and q(X, Y) = A − BX + X^(∗)B^(∗) − CYC^(∗) are quaternion matrix expressions, where A is persymmetric or perskew-symmetric. We in this paper derive the minimal rank formula of p(X, Y) with respect to pair of matrices X and Y = Y^(∗), and the minimal rank formula of q(X, Y) with respect to pair of matrices X and Y = −Y^(∗). As applications, we establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.     

关于超越系数的Riccati方程亚纯解的增长性

该文讨论了Riccati方程亚纳解的增长性问题,证明了我们曾给出的解的高阶增长级的上界稍加修改后即为一种最佳上界     

关于一类广义Riccati方程三个可积判据的注记

通过变量变换的方法,将文[1]中一类广义Riccati方程的三个充分性判据统一起来,并加以推广.充分利用参数λ,κ的可变性揭示该结果与现有Riccati方程可积性间的关系,扩大了Riccati方程的可积性范围.     

A new two‐phase structure‐preserving doubling algorithm for critically singular M‐matrix algebraic Riccati equations

Among numerous iterative methods for solving the minimal nonnegative solution of an M‐matrix algebraic Riccati equation, the structure‐preserving doubling algorithm (SDA) stands out owing to its overall efficiency as well as accuracy. SDA is globally convergent and its convergence is quadratic, except for the critical case for which it converges linearly with the linear rate 1/2. In this paper, we first undertake a delineatory convergence analysis that reveals that the approximations by SDA can be decomposed into two components: the stable component that converges quadratically and the rank‐one component that converges linearly with the linear rate 1/2. Our analysis also shows that as soon as the stable component is fully converged, the rank‐one component can be accurately recovered. We then propose an efficient hybrid method, called the two‐phase SDA, for which the SDA iteration is stopped as soon as it is determined that the stable component is fully converged. Therefore, this two‐phase SDA saves those SDA iterative steps that previously have to have for the rank‐one component to be computed accurately, and thus essentially, it can be regarded as a quadratically convergent method. Numerical results confirm our analysis and demonstrate the efficiency of the new two‐phase SDA. Copyright © 2015 John Wiley & Sons, Ltd.