周期黎卡提微分方程正定解的存在性

讨论了标准的周期黎卡提微分方程,给出了其存在埃尔米特周期正定（HPPD）解的一个完整的充分必要条件。准确地说,在经过一个适当的状态空间基底变换后该条件通过能稳性和能检测性概念表述。结果表明,当HP－PD解存在时,它或者是唯一的,或者有无限多个。这一结果可以看作是Richardson和Kwong的结果对周期时变情况的扩展。     

代数Riccati方程解的存在性条件

首先综述代数 Riccati 方程解的存在性条件, 然后对于该方程存在唯一正定最优解的充分必要条件给出严格证明. 最后利用这一条件, 纠正了鲁棒分散控制器设计中的一些错误结果.     

New Upper Matrix Bounds with Power Form for the Solution of the Continuous Coupled Algebraic Riccati Matrix Equation

In this paper, using the structure and coefficient matrix of the continuous coupled algebraic Riccati matrix equation (CCARE), we firstly construct positive definite matrices with power form. Then, applying the variant of the CCARE and inequalities of positive definite matrices, utilizing the characteristics of special matrices and eigenvalue inequalities, we propose new upper matrix bounds with power form for the solution of the CCARE, which improve and extend some of the recent results. Finally, we give corresponding numerical examples to illustrate the effectiveness of the derived results.     

具不定二次项的一般代数RICCATI方程解的存在性问题

讨论具有不定二次项的一般代数RICCATI方程(GARE)的实对称镇定解的存在性问题,利用GARE相应的微分方程解的性质,建立了GARE实对称镇定解存在性的充分条件.     

正倒向随机微分方程与一类线性二次随机最优控制问题

讨论一类正倒向随机微分方程解的存在唯一性及其对应的一类线性二次随机最优控制问题,利用单调性方法证明了一类特殊的正倒向随机微分方程解的存在唯一性定理,利用该结果研究一类耦合了一个倒向随机微分方程的线性随机控制系统广义最优指标随机控制问题,得到由正倒向随机微分方程的解所表示的唯一最优控制的显式表达式,并得到精确的线性反馈及其对应的Riccati方程.     

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.     

Explicit Solution for a Class of Discrete‐Time Algebraic Riccati Equations

In the present paper we obtain an explicit closed‐form solution for the discrete‐time algebraic Riccati equation (DTARE) with vanishing state weight, whenever the unstable eigenvalues are distinct. We discuss links to current algorithmic solutions and observe that the AREs in such a class solve on one hand the infimal signal‐to‐noise ratio (SNR) problem, whilst on the other hand, in their dual form they solve a Kalman filter problem. We then extend the main result to an example case of the optimal linear quadratic regulator gain. We relax some of the assumptions behind the main result and conclude with possible future directions for the present work.     

Riccati代数方程中矩阵P与R,Q关系的研究

本文对Riccati代数方程中矩阵P与R,Q的关系进行了研究.给出了矩阵R、Q变化时方程解P的变化范围.这对最优控制、鲁棒控制、容错控制及H∞控制等有着重要的意义.     

广义代数Riccati方程正解的研究

研究一类广义代数Riccati方程,该方程在随机控制领域有重要的应用,因而被许多学者所研究.通过引进能稳性的概念,给出了该方程有唯一正解的一个充分必要条件,所得结果不仅改进了以往的结论,也推广了确定性代数Riccati方程的相应性质.     

Solution of matrix Riccati differential equation for nonlinear singular system using genetic programming

In this paper, we propose a novel approach to find the solution of the matrix Riccati differential equation (MRDE) for nonlinear singular systems using genetic programming (GP). The goal is to provide optimal control with reduced calculation effort by comparing the solutions of the MRDE obtained from the well known traditional Runge Kutta (RK) method to those obtained from the GP method. We show that the GP approach to the problem is qualitatively better in terms of accuracy. Numerical examples are provided to illustrate the proposed method.      

New Upper and Lower Eigenvalue Bounds for the Solution of the Continuous Algebraic Riccati Equation

In this paper, applying majorization inequalities, new upper and lower bounds for summation of eigenvalues (including the trace) of the solution of the continuous algebraic Riccati equation are presented. Corresponding numerical examples illustrate that our new bounds extend some of the recent results.     

An Iterative Method for Optimal Feedback Control and Generalized HJB Equation

In this paper, a new iterative method is proposed to solve the generalized Hamilton-Jacobi-Bellman (GHJB) equation through successively approximate it. Firstly, the GHJB equation is converted to an algebraic equation with the vector norm, which is essentially a set of simultaneous nonlinear equations in the case of dynamic systems. Then, the proposed algorithm solves GHJB equation numerically for points near the origin by considering the linearization of the non-linear equations under a good initial control guess. Finally, the procedure is proved to converge to the optimal stabilizing solution with respect to the iteration variable. In addition, it is shown that the result is a closed-loop control based on this iterative approach. Illustrative examples show that the update control laws will converge to optimal control for nonlinear systems.      

具有时滞的线性奇异系统镇定的Riccati方程方法

本文考虑了具时滞的线性奇异系统的镇定问题，利用Ｒｉｃｃａｔｉ方程方法给出了系统无记忆反馈控制器的设计，并得到了系统可经状态反馈镇定的条件。     

广义预测控制中Diophantine方程的显式解

利用被控对象的离散差分方程与其状态空间能观标准型之间的关系,推导出广义预测控制中Diophantine方程的显式解,从而避免了其递推求解,使广义预测控制的应用更加方便。     

Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems

A new approach to study the indefinite stochastic linear quadratic (LQ) optimal control problems, which we called the “equivalent cost functional method”, is introduced by Yu (2013) in the setup of Hamiltonian system. On the other hand, another important issue along this research direction, is the possible state feedback representation of optimal control and the solvability of associated indefinite stochastic Riccati equations. As the response, this paper continues to develop the equivalent cost functional method by extending it to the Riccati equation setup. Our analysis is featured by its introduction of some equivalent cost functionals which enable us to have the bridge between the indefinite and positive-definite stochastic LQ problems. With such bridge, some solvability relation between the indefinite and positive-definite Riccati equations is further characterized. It is remarkable the solvability of the former is rather complicated than the latter, hence our relation provides some alternative but useful viewpoint. Consequently, the corresponding indefinite linear quadratic problem is discussed for which the unique optimal control is derived in terms of state feedback via the solution of the Riccati equation. In addition, some example is studied using our theoretical results.     

Systolic算法和结构求解代数Riccati方程

本文基于矩阵的符号函数法，提出了一种Ｕ－Ｄ分解算法和脉动（Ｓｙｓｔｏｌｉｃ）结构有效地求解代数Ｒｉｃｃａｔｉ方程以及用固定大小的方形阵列解决大型问题的方法。     

Singular optimal control for stochastic linear quadratic singular system using ant colony programming

In this article, singular optimal control for stochastic linear singular system with quadratic performance is obtained using ant colony programming (ACP). To obtain the optimal control, the solution of matrix Riccati differential equation is computed by solving differential algebraic equation using a novel and nontraditional ACP approach. The obtained solution in this method is equivalent or very close to the exact solution of the problem. Accuracy of the solution computed by the ACP approach to the problem is qualitatively better. The solution of this novel method is compared with the traditional Runge Kutta method. An illustrative numerical example is presented for the proposed method.     

Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.This work has been partially supported by Spanish DGICYT Grant No. PB91-O535.     

一类不确定组合大系统的稳定性：Riccati方程方法

本文研究了一类不确定线性组合大系统的二次稳定性问题，该类系统含有时变的未知但有界的不确定性参数，给出了完全用子系统描述的不确定组合大系统的稳定性充分条件，做为主要结果的特例，给出了不含不确定参数的组合大系统的用子系统描述的稳定性条件。     

一种求解非线性随机微分方程的算法及其实现

提出了一种求解非线性随机微分方程的算法 ,该算法具有简单、通用且易于实现的特点 .文中给出算法的详细推导过程及其实现 .文末还给出了采用该算法求解两个典型模型的实验结果 ,表明了该算法的正确性和可行性 .本算法为研究噪声背景下的非线性系统提供了一个有力的工具