【作者】 原明亭；

【导师】 赵克友；

【作者基本信息】 山东大学 ， 控制理论与控制工程， 2013， 博士

【摘要】 目前,经过几十年的发展,鲁棒控制理论已取得了十分丰富的研究成果。在对不确定系统的研究过程中,大部分国内外学者是在二次稳定的基础上寻求问题的解决方法。但由于二次稳定性要求对于所有允许的不确定参数,存在一个统一的Lyapunov函数,因此所得的结果不可避免地引入了较大的保守性。针对此问题,在二十世纪九十年代中期出现了采用参数依赖的Lyapunov函数对不确定系统的鲁棒稳定性进行分析,并取得了一系列创造性的研究成果,但不管是仿射参数相关Lyapunov函数还是多项式相关Lyapunov函数,多应用于凸多面体型系统。但在工程实际中,系统对参数多是复杂的非线性依赖关系,有时还是多参数非线性依赖,而复杂的非线性函数又可用多项式函数任意逼近,因而对参数多项式型不确定系统进行系统稳定性分析、干扰抑制、控制器设计等有关鲁棒控制问题进行研究,是非常有意义的工作。论文针对参数多项式型不确定系统,进行了以下4部分研究：1.研究了单参数多项式依赖不确定系统的鲁棒稳定性问题,并得到相应的稳定性判据。(1)研究了时不变多项式单参数依赖线性连续时间不确定系统的鲁棒稳定性问题针对时不变多项式单参数系统x(t)=A(δ)x(t),其中A(δ)=A0+δA1+δ2A2+…+δLaALa呈参数阶次La的多项式形式,Aα={δ∈R:|δ|≤α}为给定α>0限定的不确定参数集取V(x,δ)=xTP(δ)x为系统备选Lyapunov函数,P(δ):=P0+δP1+δ2P2+…δLpPLp为多项式参数依赖形式。通过引入任意备选正定矩阵组S(x,δ):=∑i=1L(δix)TSi(δix)和任意备选反对称矩阵组T(x,δ):=∑i=1L(δix)TTix,当A0∈H时,若存在{P∈Sn,i=0,1,…,L},{Si∈Sn+,i=1,2,…,L}及{Ti∈Tn,i=1,2,…,L},得出了系统基于LMI的鲁棒渐近稳定和指数率γ鲁棒渐进稳定定理。(2)研究了时变多项式单参数依赖线性连续时间不确定系统的鲁棒稳定性问题。对时变多项式单参数系统x(t)=A(δ(t))x(t), A(δ(t)):=A0+δ(t)A1+δ(t)2A2+…+δ(t)LαALα为时变参数δ(t)的多项式参数依赖形式,Λα,β={δ(·):|δ|≤α,|δ|≤β|为时变参数δ(t)的给定了变化上限α和变化率上限β的不确定参数集。采用与时不变系统同样的方法,得到了系统基于LMI的鲁棒一致渐近稳定性定理和鲁棒指数率γ一致渐近稳定性定理。给出了当参数满足|δ(t)|≤α*与|δ|≤β时,系统是全局渐近稳定时参数变化的上届α*。(3)研究了时不变多项式单参数依赖线性离散时间不确定系统鲁棒稳定性问题。对时不变单参数离散系统xk+1=A(δ)xk,A(δ)和Λα。具有与时不变连续系统一样的约定,采用与时不变系统同样的方法,得到了系统基于LMI的鲁棒Schur稳定定理和鲁棒γ收敛率Schur稳定定理。以上对多项式单参数依赖不确定系统稳定性问题,将它们转化为一组线性矩阵不等式的可行性问题,给出了充分性判据。一般当Lp较大时效果比较明显,但一般与Lα相等即可；Lp>0时比Lp=0时保守性小,说明应用多项式参数依赖Lyapunov法比二次稳定性降低了保守性。数值算例验证了所得结论的有效性。2.研究了多项式单参数依赖系统的L2增益对时变单参数多项式依赖系统：矩阵皆以多项式形式依赖时变参数δ(t),简化考虑在Lb,Lc,Ld≤La的情况下,给定L2增益指标γ>0,在系统稳定前提下给出了判断系统是否满足该指标定理,并推广到时不变多项式参数依赖系统系统L2增益性能γ定理和二次稳定情况下系统L2增益性能y定理。当L2增益指标γ未知时,给出了求解系统L2增益指标定理,给出了参数依赖型的状态反馈控制器改善系统干扰抑制性能。3.设计了时变不确定参数的鲁棒增益可调控制器对时变多项式参数依赖系统设计了K=-εB2T(P0+∑δ(t)Pi)的控制律,在保证闭环系统是渐近稳定前提下,使得从扰动输入w(t)到被调输出z(t)的闭环传递函数T(s)的H∞范数小于γ。4.研究了多项式多参数依赖系统的鲁棒稳定性及系统稳定的参数边界以时不变两参数多项式型不确定系统为例研究多项式多参数系统鲁棒性能,定义系统矩阵参数δ=[δ1δ2]T∈R2,Λα={δ=[δ1δ2]T∈R2:|δi|≤α,i=1,2}为界α>0所定义的参数集。在已知A00∈H,构建了两参数多项式依赖Lyapunov函数将系统稳定性问题转化为线性矩阵不等式的可行性问题,给出了充分性判据和推论,并确定了系统的稳定边界,其所得结论适用于单参数多项式型和两参数仿射型不确定系统,适用范围较广。总之,论文研究了多项式型参数依赖不确定系统的鲁棒控制问题,根据Lyapunov稳定性原理,构建了多项式型参数依赖Lyapunov函数,采用S-procedure技巧、松弛变量方法及LMI技术得到有关结论,与二次稳定性理论、常用的参数依赖Lypunov函数方法相比,降低了保守性。更多还原

【Abstract】 At present, after decades of development, the robust control theory has made very rich research achievements. In the course of research for uncertain system, most of the domestic and overseas scholars solve the problem based on the quadratic stability. But the quadratic stability requirements a unity of Lyapunov function for all allow uncertain parameters, which the results will inevitably be introduced large conservatism. This issue has caused more and more attention from scholars.To solve this problem, Haddad, Gahinet and Feron etc in the mid1990s began to try to use parameters dependent Lyapunov function to solve uncertain system robust stability analysis. On this foundation, many scholars have made a series of creative research achievements, developed a new research direction for robust control, such as slack variable method, polynomial method with positive coefficient, D-Scale method, sum of squares method and so on.Now, the application of the parameters dependent Lyapunov function has been quite common, such as affined parameters dependent Lyapunov function and polynomial Lyapunov function. These methods applied to convex polytopic systems.However, some problems of not convex can be solved by linear matrix inequality (LMI) form, and some nonlinear problems can also be solved through the polynomial approximation. So the application of polynomial parameters-dependent Lyapunov functions in the uncertain systems not only has important theoretical significance, but also can further improve the possibility that the robust control theory is applied to industry. But research achievements of this field are very limited and the research results are not formed work detailed summary. Therefore, Our study will further promote the development of parameters dependent Lyapunov stability theory in this field, and it will inject new vitality for the development of robust control theory.In this paper, the robust control problem has been studied for parameters polynomial uncertain systems. The main contents:1. The robust stability analysis research of the single parameter polynomial type uncertain systems,and get the corresponding stability theorem.(1)study the time-invariant single parameter polynomial dependent linear continuous time uncertain systems robust stabi lity problem.Consider the time-invariant parameter polynomial dependent linear system that can be described by state-space equations of the form x(t)=A(δ)x(t),which A(δ):=A0+δA1+δ2A2+…+δLaALa is a polynomial parameter of order La and depended on the parameter δ.Aα={δ∈R:|δ|≤α} is an uncertain parameter set which is limited by α>O.Taken V(x,δ)=xTP(δ)x for the system candidate Lyapunov function and P(δ):=P0+δP1+δ2P2+…+δLpPLp is polynomial parameter depended on the parameter δ. Used by S-Procedure method to process parameter constraints.As any alternative positive definite matrix group S(x,δ):=∑i=1L(δix)TSi(δix) and any alternative negative definite matrix group T(x,δ):=∑i=1L(δix)TTix,we present a robust stability sufficient conditions of the system if there exist {Pi∈Sn,i=0,1,...,L},{Si∈Sn+,i=1,2...,L} and {Ti∈Tn,i=1,2,...,L} which the robust stability of the system is transformed into feasibility test of a pair of linear matrix inequalities (LMIs).(2)Study the robust stability of the time-varying single parameter polynomial dependent linear continuous uncertain system.Consider the time-varying parameter polynomial-dependent linear system that can be described by state-space equations of the form x(t)=A(δ(t))x(t)which A(δ(t)):=A0+δ(t)A1+δ(t)2A2+…+δ(t)LaALa is a polynomial parameter of order Lα and depended on the parameter δ(t).A,β={δ(·):|δ|≤α,|δ|≤β}.The theorem of a robust uniform-asymptotically stability and robust uniform-asymptotically stable with exponential rate γ are given.When the system is robust uniform-asymptotically stability under the condition of|δ|(t)|≤α*|δ|≤β,we present a method to seek the parameter vatiational α(3)Study the robust stability of the time-invariant single parameter polynomial linear discrete uncertain system.Consider the discrete uncertain system that can be described by state-space equations of the form xk+1=A(δ)xk which A(δ)=A0+δA1+δ2A2+…+δLaALa and Aa={δ∈R:|δ|≤α}. We use the same analytical method as the time-invariant single parameter polynomial dependent continuous time systems, the roubst stable problem is transformed into a set of linear matrix inequalities feasibility problem, the sufficient criteria is given.2. Study the single parameter polynomial dependent system L2gain analysisConsider the time-varying single parameter polynomial dependent systemThe matrices are polynomial parameter and depended on the parameter δ(t). In the case of Lb,Lc,Ld≤La, given γ>0, we propose a theorem for the system with given L2gain γ which is a LMI condition involving parameter-dependent Lyapunov. The theorem may be extended to be applied to the time-invariant system and discrete uncertain system.3. Design robust gain-acheduling controller of time-varying parameter polynomial dependent uncertain systemA time-varying parameter polynomial dependent system state-space model is considered We design a controller’s control law for the given scalar γ>0to guarantee the closed loop system asymptotically stability and let the system has the L2gain γ4. Study the robust stability of multi-parameters polynomial dependent uncertain system Consider the linear time-invariant two parameters polynomial uncertain system where x(·):[0,+∞)→Rn is the state, the systems matrix A(δ) satisfied while Aij∈R nxn. and α>0in system, as Aα={δ=[δ1δ2]T∈R2:|δi|≤α,i=1,2) Known A00∈H,taken V(x,δ)=xTP(δ)x for the system candidate Lyapunov function and which has two parameters. We propose a robust stability sufficient conditions of the system which the robust stability of the system is transformed into feasibility test of a pair of linear matrix inequalities (LMIs).The theorem may be applied to single parameter polynomial dependent system and discrete uncertain system.更多还原