代数Riccati矩阵方程解的估计和解的迭代算法及其应用

发布时间：2018-04-04 11:08

本文选题：矩阵界　切入点：迭代算法　出处：《湘潭大学》2017年博士论文

【摘要】：在自动控制、工程计算、固体力学、参数的识别、生物工程等许多领域,都涉及控制系统的设计、控制与优化。在控制系统的设计过程中,稳定性、能控性和能观性等三大重要性质需要重点考虑,以确保最后得到的产品能满足各项规定的性能指标。探讨这些特性常常可转化为求解相关的矩阵方程。尤其是控制系统中的最优控制与稳定性等一些重要特性的研究常常可归结为Riccati矩阵方程的求解及其上、下界的估计。我们将探讨控制系统中的代数Riccati矩阵方程解的界的估计及其数值算法,根据得到的解的界讨论了其在冗余最优控制中的一些具体应用。第一章中,介绍了代数Riccati矩阵方程的应用背景和研究现状,及冗余最优控制的研究现状。给出本文所涉及的记号和定义。第二章中,利用不等式的放缩技巧和控制不等式等性质,研究了连续代数Ricc ati矩阵方程解的上界估计,改进了近期已有的结果。进一步将该上界应用于冗余最优控制系统中,当控制输入增加的时候,给出了几个控制器增益减少的条件。用实例验证了所得结果的有效性。第三章中,利用M-矩阵的逆矩阵性质,特征值不等式和控制不等式等性质,研究了连续耦合代数Riccati矩阵方程解的上界估计。当连续耦合代数Riccati矩阵方程退化为连续代数Riccati矩阵方程时,该结果也改进了近期已有的一些结论。第四章中,利用离散耦合代数Riccati矩阵方程的等价形式,运用不等式技巧,对称矩阵的一些性质等,把耦合项作为一个整体,得到了离散耦合代数Riccati矩阵方程解的上、下界估计,理论上改进了近期已有的一些结果。进一步利用得到的上、下界估计,Frobenius范数的性质与不动点定理,给出了离散耦合代数Riccati矩阵方程解的存在唯一性条件。利用矩阵序列收敛的定义和Cauchy序列的特点,设计了离散耦合代数Riccati矩阵方程解的不动点迭代算法。数值例子验证了所得结果的有效性.第五章中,在比第四章所获结果更强的限定条件下,根据离散耦合代数Riccati矩阵方程的等价形式,运用非负矩阵的性质,不等式的技巧,M-矩阵的逆矩阵性质来解矩阵不等式,得到了离散耦合代数Riccati矩阵方程解的更好的上、下界估计。再利用得到的上、下界估计,Cauchy-Schwarts不等式及不动点定理,给出了离散耦合代数Riccati矩阵方程解的存在唯一性条件。进一步,设计了离散耦合代数Riccati矩阵方程解的不动点迭代算法。数值例子验证了所得结果的优越性和有效性。
[Abstract]:In many fields, such as automatic control, engineering calculation, solid mechanics, parameter identification, bioengineering and so on, it involves the design, control and optimization of control system.In the design of the control system, three important properties, namely, stability, controllability and observability, need to be considered to ensure that the final product can meet the performance index of various regulations.The discussion of these properties can often be transformed into solving related matrix equations.In particular, the study of some important properties such as optimal control and stability in control systems can be attributed to the solution of Riccati matrix equations and the estimation of upper and lower bounds.We will discuss the estimates of the bounds of solutions of algebraic Riccati matrix equations in control systems and their numerical algorithms. Based on the bounds of the obtained solutions, some concrete applications in redundant optimal control are discussed.In the first chapter, the application background and research status of algebraic Riccati matrix equation and the research status of redundant optimal control are introduced.The notations and definitions involved in this paper are given.In chapter 2, the upper bound estimate of the solution of the continuous algebraic Ricc ati matrix equation is studied by using the scaling technique of the inequality and the property of the control inequality, and the recent results are improved.Furthermore, the upper bound is applied to the redundant optimal control system. When the control input is increased, several conditions for the gain reduction of the controller are given.An example is given to verify the validity of the obtained results.In chapter 3, the upper bound estimates of the solutions of the continuous coupled algebraic Riccati matrix equation are studied by using the inverse matrix property, eigenvalue inequality and control inequality of the M- matrix.When the continuous coupled algebraic Riccati matrix equation degenerates to the continuous algebraic Riccati matrix equation, this result also improves some recent conclusions.In chapter 4, by using the equivalent form of discrete coupled algebraic Riccati matrix equation, using inequality technique, some properties of symmetric matrix, and taking the coupling term as a whole, the upper and lower bound estimates of the solution of discrete coupled algebraic Riccati matrix equation are obtained.Some recent results have been improved theoretically.Furthermore, by using the properties and fixed point theorems of the upper and lower bound estimators, the existence and uniqueness conditions of solutions for discrete coupled algebraic Riccati matrix equations are given.Based on the definition of matrix sequence convergence and the characteristics of Cauchy sequence, a fixed point iterative algorithm for the solution of discrete coupled algebraic Riccati matrix equation is designed.A numerical example is given to verify the validity of the obtained results.In chapter 5, under the condition that the results obtained in chapter 4 are stronger than those obtained in chapter 4, according to the equivalent form of discrete coupled algebraic Riccati matrix equation, using the property of nonnegative matrix and the technique of inequality, the inverse matrix property of M- matrix is used to solve the matrix inequality.A better upper and lower bound estimate for the solution of discrete coupled algebraic Riccati matrix equation is obtained.Using the Cauchy-Schwarts inequality and fixed point theorem, the existence and uniqueness conditions of solutions for discrete coupled algebraic Riccati matrix equations are given.Furthermore, a fixed point iterative algorithm for solving discrete coupled algebraic Riccati matrix equations is designed.Numerical examples demonstrate the superiority and validity of the obtained results.
【学位授予单位】：湘潭大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.6

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