本文選題：矩陣界　切入點(diǎn)：迭代算法　出處：《湘潭大學(xué)》2017年博士論文

【摘要】：在自動(dòng)控制、工程計(jì)算、固體力學(xué)、參數(shù)的識(shí)別、生物工程等許多領(lǐng)域,都涉及控制系統(tǒng)的設(shè)計(jì)、控制與優(yōu)化。在控制系統(tǒng)的設(shè)計(jì)過程中,穩(wěn)定性、能控性和能觀性等三大重要性質(zhì)需要重點(diǎn)考慮,以確保最后得到的產(chǎn)品能滿足各項(xiàng)規(guī)定的性能指標(biāo)。探討這些特性常�？赊D(zhuǎn)化為求解相關(guān)的矩陣方程。尤其是控制系統(tǒng)中的最優(yōu)控制與穩(wěn)定性等一些重要特性的研究常�？蓺w結(jié)為Riccati矩陣方程的求解及其上、下界的估計(jì)。我們將探討控制系統(tǒng)中的代數(shù)Riccati矩陣方程解的界的估計(jì)及其數(shù)值算法,根據(jù)得到的解的界討論了其在冗余最優(yōu)控制中的一些具體應(yīng)用。第一章中,介紹了代數(shù)Riccati矩陣方程的應(yīng)用背景和研究現(xiàn)狀,及冗余最優(yōu)控制的研究現(xiàn)狀。給出本文所涉及的記號(hào)和定義。第二章中,利用不等式的放縮技巧和控制不等式等性質(zhì),研究了連續(xù)代數(shù)Ricc ati矩陣方程解的上界估計(jì),改進(jìn)了近期已有的結(jié)果。進(jìn)一步將該上界應(yīng)用于冗余最優(yōu)控制系統(tǒng)中,當(dāng)控制輸入增加的時(shí)候,給出了幾個(gè)控制器增益減少的條件。用實(shí)例驗(yàn)證了所得結(jié)果的有效性。第三章中,利用M-矩陣的逆矩陣性質(zhì),特征值不等式和控制不等式等性質(zhì),研究了連續(xù)耦合代數(shù)Riccati矩陣方程解的上界估計(jì)。當(dāng)連續(xù)耦合代數(shù)Riccati矩陣方程退化為連續(xù)代數(shù)Riccati矩陣方程時(shí),該結(jié)果也改進(jìn)了近期已有的一些結(jié)論。第四章中,利用離散耦合代數(shù)Riccati矩陣方程的等價(jià)形式,運(yùn)用不等式技巧,對(duì)稱矩陣的一些性質(zhì)等,把耦合項(xiàng)作為一個(gè)整體,得到了離散耦合代數(shù)Riccati矩陣方程解的上、下界估計(jì),理論上改進(jìn)了近期已有的一些結(jié)果。進(jìn)一步利用得到的上、下界估計(jì),Frobenius范數(shù)的性質(zhì)與不動(dòng)點(diǎn)定理,給出了離散耦合代數(shù)Riccati矩陣方程解的存在唯一性條件。利用矩陣序列收斂的定義和Cauchy序列的特點(diǎn),設(shè)計(jì)了離散耦合代數(shù)Riccati矩陣方程解的不動(dòng)點(diǎn)迭代算法。數(shù)值例子驗(yàn)證了所得結(jié)果的有效性.第五章中,在比第四章所獲結(jié)果更強(qiáng)的限定條件下,根據(jù)離散耦合代數(shù)Riccati矩陣方程的等價(jià)形式,運(yùn)用非負(fù)矩陣的性質(zhì),不等式的技巧,M-矩陣的逆矩陣性質(zhì)來解矩陣不等式,得到了離散耦合代數(shù)Riccati矩陣方程解的更好的上、下界估計(jì)。再利用得到的上、下界估計(jì),Cauchy-Schwarts不等式及不動(dòng)點(diǎn)定理,給出了離散耦合代數(shù)Riccati矩陣方程解的存在唯一性條件。進(jìn)一步,設(shè)計(jì)了離散耦合代數(shù)Riccati矩陣方程解的不動(dòng)點(diǎn)迭代算法。數(shù)值例子驗(yàn)證了所得結(jié)果的優(yōu)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.
【學(xué)位授予單位】：湘潭大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.6

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