多项式参数依赖系统鲁棒控制有关问题研究
|
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。
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 analysis
Consider the time-varying single parameter polynomial dependent system
The 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 system
A 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 γ
4. 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.
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。
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 analysis
Consider the time-varying single parameter polynomial dependent system
The 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 system
A 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 γ
4. 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.
引文
[1]陈善本,范颖晖.具L2有界不确定性系统线性二次鲁棒最优控制：LQL方法[M].北京：科学出版社：2004.
[2]郑大钟.线性系统理论.北京：清华大学出版社,2003.
[3]褚健,俞立,苏宏业.鲁棒控制理论及应用[M].杭州：浙江大学出版社,2000.4.
[4]梅生伟,申铁龙,刘志康.现代鲁棒控制理论与应用[M].北京：清华大学出版社,2003.4.
[5]王枞.控制系统的理论与应用[M].北京：北京邮电大学出版社,2005.3.
[6]史忠科.鲁棒控制理论[M].北京：国防工业出版社,2003.
[7]俞立.鲁棒控制--线性矩阵不等式处理方法[M].北京：清华大学出版社,2002.12.
[8]黄琳.稳定性与鲁棒性的理论基础[M].北京：科学出版社,2003.
[9]M.Dettori,C.W. Scherer.Robust stability analysis systems using full block S-Procedure for parameter dependent[C].Proceedings of the 37th IEEE Conference on Decision & Control,Tampa, Florida USA, December 1998:2798-2799.
[10]Eric Feron Pierre Apkarian Pascal Gahinet.S-procedure for the analysis of control systems with parametric uncertainties via parameter-dependent Lyapunov functions[C]. Proceeding of the 2004 American Control Conference, Ssatle Wathlngton,1996:968-972.
[11]Stephen Boyd, Eric Feron. History of linear matrix inequalities in control theory[C]. Proceedings of the American Control Conference,1994:31-34.
[12]S.P.Boyd,L.EI Ghaoui,E.Feron,V.Balakrishnan. Linear matrix inequalities in system and control theory[M].Philadelphia:SLAM,1994.
[13]P.Gahinet, A.Nemirovskii, A.J.Laub, et al. The LMI Control Toolbox[C]. In Proceedings of the 33rd Conference on Decision and Control,Lake Buena, Vista,1994:2038-2041.
[14]Tetsuya Iwasaki.Control System Design via LMIs[J]. J. SICE,1995,34(3):164-169.
[15]P.Gahinet,P.Apkarian.A linear matrix inequality approach to H∞ control[J]. International Journal of Robust and Nonlinear Control,1994,4(4):421-448.
[16]Balakrishnan,V.Linear matrix inequalities in robust control:a brief survey[C].In Proc.Math. Thy of Netw. and Sys., Indiana,2002.
[17]G.Becker, A.Packard, D. Philbrick, G.Balas. Control of parametrically-dependent linear systems:a single quadratic Lyapunov approach[C].Proceedings of the American Control C onference,San Francisco,1993:2795-2799.
[18]G.Becker,A.Packard. Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback[J].Systems&Control Leters,1994,23:205-215.
[19]A.Packard, P.Gahinet, G.Becker. Self-Scheduled H Control of linear parameter-varying systems:a design example[J]. Automatica,1995,31:1251-1261.
[20]E.Feron,P.Apkarian,P.Gahinet.Analysis and synthesis of robust control systems via parameter-dependent Lyapunov function [J].IEEE Trans. on Automat. Control,1996,41(7):1041-1046.
[21]J.C.Geromel, M.C.de Oliveira, L.Hsu. LMI characterization of structural and robust stability[J].Linear Algebra and Its Applications,1998,285(1-3):69-80.
[22]M.C.de Oliveira,J.Bernussou,J.C.Geromel.A new discrete-time stability condition[J]. Systems & Control Letters,1999,37(4):261-265.
[23]Andrew G.Sparks.Analysis of affine parameter-varying systems using parameter-dependent Lyapunov functions[C].Proceedings of the 36th IEEE Conference on Decision and Control,San Diego,IEEE,1997:990-991.
[24]Guang-Hong,Kai-Yew Lum.Comparisons among robust stability criteria for linear systems with affine parameter uncertainties[J].Automatica,2007,43:491-498.
[25]V.J.S. Leite,P.L.D.Peres.An improved LMI condition for robust D-stability of uncertain polytopic systems[J].IEEE Trans.on Automatic Control,2003,48(3):500-504.
[26]P.-A.Bliman. A convex approach to robust stability for linear systems with uncertain scalar parameters[J]. SI AM J. Control and Optimization,2004,42:2016-2042.
[27]X.Zhang, P.Tsiotras,P.-A.Bliman. Multi-parameter dependent Lyapunov functions for the stability analysis of LTI parameter-dependent systems[C].In IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 2004.
[28]Parrilo P A. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[D].California:California Institute ofTechnology,2000.
[29]S.Prajna,A.Papachristodoulou,P.A.Parrilo.Introducing SOSTOOLS:A general purpose sum of squares programming solver[C].Proc.of the 41 st IEEE Conference on Decision and Control, Las Vegas,Nevada,USA,2002:741-746.
[30]Prajna S,Papachristodoulou A.Seiler P,et al.Sostools:Sum of squares optimization toolbox for Matlab[R].California:California Institude ofTechnology,2004.
[31]S.Prajna,A.Papachristodoulou,Fen Wu.Nonlinear control synthesisby sum of squares optimization:a Lyapunov-based approach[C].Proc.of the 5th Asia Control Conference,2004:157-165.
[32]Topcu U,Packard A,Seiler P.Local stability analysis using simulations and sum-of-squares programming[J].Automatica,2008,44:2669-2675.
[33]Christian Ebenbauer,FrankAllgower.Analysis and design of polynomial control systems using dissipation inequalities and sum of squares[C].Computers and Chemical Engineering,2006,30:1590-1602.
[34]F.Wu, S.Prajna.SOS-based solution approach to polynomial LPV system analysis and synthesis problems[J]. International Journal of Control,2005,78(8):600-611.
[35]Zachary William Jarvis-Wloszek.Lyapunov based analysis and controller synthesis for polynomial systems using Sum-of-Squares optimization[D].University of California, Berkeley. Fall 2003. Submitted.
[36]Weehong Tan.Nonlinear control analysis and synthesis using Sum-of-Squares programming[D].University of California,Berkeley,Spring 2006,Submitted.
[37]Erin Aylward,Pablo A.Parrilo,Jean-Jacques E.Slotine.Algorithmic search for contraction metrics via SOS programming[C].Proceedings of the 2006 American Control Conference,Minneapolis,Minnesota,USA,June,2006:3001-3006.
[38]G.Chesi,A.Garulli,A.Tesi,A.Vicino.Robust stability of polytopic systems via polynomially parameter-dependent Lyapunov functions[C].In IEEE Conference on Decision and Control,Maui,Hawaii,USA,December 2003.
[39]G.Chesi,A.Garulli,A.Tesi,A.Vicino.Polynomially parameter-dependent Lyapunov functions for robust H1 performance analysis[C].In IF AC World Congress, Prague,Czech Republic,2005.
[40]Vinicius F.Montagner,Pedro L.D. Peres. H∞ parameter-dependent state feedback control of linear time-varying systems in polytopic domains[C].In Proceedings of the 44th IEEE Conference on Decision and Control and the European Control Conference,Seville, 2005:5006-5011.
[41]G.Chesi,A.Garulli,A.Tesi.A.Vicinoa. Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions[J].Automatica,2007,43(2):309-316.
[42]Chesi,G.,Garulli,A.,Tesi,A.,Vicino, A. Robust stability for polytopic systems via polynomially parameter-dependent Lyapunov functions[C].42nd IEEE CDC,Maui,Hawaii,2003b:4670-4675.
[43]Geromel,J.C.,Colaneri,P. Robust stability of time varying polytopic systems[J].Systems and Control Letters,2005,55(1):81-85.
[44]Leite,V.J.S.,Peres,P.L.D.An improved LMI condition for robust D-stability of uncertain polytopic systems[J].IEEE Transactions on Automatic Control,2003,48(3):500-504.
[45]Peaucelle,Arzelier,Bachelier,Bernussou.A new robust -stability condition for real convex polytopic uncertainty[J].Systems and Control Letters,2000,40:21-30.
[46]Fen Wu,Stephen Prajna.A new solution approach to polynomial LPV system analysis and synthesis[C].Proceeding of the 2004 American Control Conference,Boston, Massachusetts June,2004:1362-1367.
[47]P.Tsiotras.P.-A.Bliman.An exact stability analysis test for single-parameter polynomially-dependent linear systems[J].IEEE Trans. Automat. Control,2006,51(7): 1161-1164.
[48]Y Wang,L.Xie,C.E.de Souza. Robust control of a Class of uncertain nonli near-systems[J]. Systems & Control Leters,1992,19(2):139-149.
[49]Khargonekar,P.,Petersen,I. R,Zhou,K. Robust stabilization of uncertain linear systems: quadratic stabilizatility and H∞ control theory [J]. IEEE Transactions on Automatic Control,1990,35:356-361.
[50]Tetsuya Iwasaki.Robust performance analysis for systems with Structure Uncertainty[J].Inernational Journal of Robust and Nonlinear Control,1996,6(1):85-99.
[51]Chesi,G.,Garulli,A.,Tesi,A.,Vicino.A. Robust analysis of LFR systems through homogeneous polynomial Lyapunov functions[J].IEEE Transactions on Automatic Control,2004b,49(7):1211-1216.
[52]Ke-You Zhao, Michael J.Grimble, and Jakob Stoustrup. Average H2 Performance and Maximal Parameter Perturbation Radius for Uncertain Systems[J].Society for Industrial and Applied Mathematics,1999,37(6),1742-1750.
[53]Xiping Zhang. Parameter-dependent Lyapunov functions and stability analysis of linear parameter-dependent dynamical systems[D].Ph D dissertation,School of Aerospace Engineering Georgia Institute of Technology,October 2003.
[54]Gahinet P,Nemimvski A,ALaub,et al. The LMI control toolbox[C].Proc of 33rd Conf on Decision and Control,Lake Buena Vista,1994:2038-2041.
[55]韩小宝,王仲生,芦玉华.基于凸多面体构造的多项式LPV系统性能分析[J].计算机仿真,2009,26(8)：92-95.
[56]王凡,沈小林,路旭.一类多项式族的鲁棒稳定性分析[J].中北大学学报(自然科学版),2009,30(5)：439-442.
[57]H.R.Karimi,B.Moshiri,P.Jabehdar Maralani,B.Lohmann.Adaptive Hf,-control design for a class of LPV systems[J]. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville,Spain,2005:7918-7923.
[58]Didier Henriona,Denis Arzelier,Dimitri Peaucelle.Positive polynomial matrices and improved LMI robustness conditions[J].Automatica,2003,39:1479-1485.
[59]Henrion,D.,Lasserre,J.-B.Convergent relaxations of polynomial matrix inequalities and static output feedback[J].Transactions on Automatic Control,2006,51 (2):192-202.
[60]Trofino,A. Parameter dependent Lyapunov functions for a class of uncertain linear systems:A LMI approach[C].Proceedings of 38th IEEE CDC,Phoenix, Arizona,1999:2341-2346.
[61]Dasgupta,S.,Chockalingam,G.,Anderson,B.D.O.,Fu,M. Lyapunov functions for uncertain systems with applications to the stability of time varying systems[J].IEEE Transactions on Automatic Control,1994,41:93-106.
[62]吴健荣,杨成梧.广义系统渐近稳定性的一种新的Lyapunov判别方法[J].应用数学,2003,16(2)：92-96.
[63]C.E.de Souza, A.Trofino,J.de Oliveira. Parametric Lyapunov function approach to H2 analysis and control of linear parameter-dependent systems[J]. IEE Proc.-Control Theory Appl,2003,150(9).
[64]W. Xie. H2 gain scheduled state feedback for LPV system with new LMI formulation[J]. IEE Proc.-Control Theory Appl.,2005,152(6).
[65]Job C.Ostveen, Curtain. Riccati equations for strongly stabilizable bounded linear systems[J].Automatica,1998,34(8):953-967.
[66]Masubuchi,I.,Kamitane,Y.,Ohara,A.Suda,N. H∞ control for descriptor systems:a matrix inequalities approach[J].Automatica,1997,33:669-673.
[67]Miaomiao Ma, Hong Chen. LMI Based H∞ Control for Constrained Linear Systems with Norm-Bounded Uncertainties[C].Proceedings of the 6th World Congress on Control and Automation, Dalian,China,2006,6.
[68]RicardoC.L.F.Oliveira,,MauricioC.,de Oliveira,Pedro L.D.Peres.Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions[J].Systems&ControlLetters,2008,57:680-689.
[69]Ebihara,Y,Hagiwara,T.On the degree of polynomial parameter-dependent Lyapunov functions for robust stability of single parameter-dependent LTI systems:A counter-example to Barmish's conjecture[J].Automatica,2006,42(9):1599-1603.
[70]Ricardo C.L.F.Oliveira,Pedro L.D. Peres.LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions[J].Systems & Control Letters,2006,55(1):52-61.
[71]Graziano Chesi.Time-invariant uncertain systems:A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions[J].Automatica,2010,46(2):471-474.
[72]Hajer Bouzaouache,Naceur Benhadj Braiek.On the stability analysis of nonlinear systems using polynomial Lyapunov functions[J].Mathematics and Computers in Simulation,2008,76(5):316-329.
[73]Wilfried Gilbert,Didier Henrion,Jacques Bernussou,David Boyer.Polynomial LPV synthesis applied to turbo fan engines [J]. Control Engineering Practice,2010,18(9):1077-1083.
[74]Ping Li,James Lam,Graziano Chesi. On the synthesis of linear Hoo filters for polynomial systems[J].Systems & Control Letters,2012,61(1):31-36.
[75]Hamid Reza Karimi,Peter Maass.A convex optimisation approach to robust observer-based Hoo control design of linear parameter-varying delayed systems[J].Int.J.Modelling Identification and Control,2008,4(3):226-241.
[76]Baoyong Zhang,James Lamb,Shengyuan Xu.Deconvolution filtering for stochastic systems via homogeneous polynomial Lyapunov functions[J].Signal Processing,2009,89(4):605-614.
[77]P.Tsiotras,P.-A.Bliman. An Exact Stability Analysis Test for Single-Paramete Polynomially-Dependent Linear Systems.
[78]Leonardo Amaral Mozelli,Reinaldo Martinez Palhares. Stability analysis of linear time-varying systems:Improving conditions by adding more information about parameter variation[J].Systems & Control Letters,2011,60(5):338-343.
[79]Parrilo P A. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[D].California:California Institute of Technology,2000.
[80]M. S. Mahomoud. Linear parameter-varying discrete time-delay systems:stability and L2-Gain Controllers[J].Int.J.Contro,2000,73:481-494.
[81]D.J.Stilwell,W.J.Rugh. Stability and L2 Gain Properties of LPV Systems[J].Automatica, 2002,38:1601-1606.
[82]D.F.Coutinho,M.Fu,A.Trofino,P.Danes. L2-Gain analysis and control of uncertain nonlinear systems With bounded disturbance inputs[J].Int.J.Robust Nonlinear Control,2008,18:88-110.
[83]Gahinet,P.,Apkarian,P.,Chilali,M. Affine parameter-dependent Lyapunov functions and real parametric uncertainty[J].IEEE Transactions on Automatic Control,1996,41 (3):436-442.
[84]D.Peaucelle,Y.Ebihara,D.Arzelier,H.Tomomichi.General polynomial parameter-dependent lyapunov functions for polytopic uncertain systems[C].In International Symposium on Mathematical Theory of Networks and Systems, Kyoto, July 2006.
[85]D.Peaucelle,D.Arzelier,O.Bachelier,J.Bernussou. A new robust D-stability condition for real convex polytopic uncertainty[J].Systems & Control Letters,2000,40(1):21-30.
[86]Gahinet P.,et al. LMI control toolbox[M].The mathworks Inc,1994.
[87]邵东南,马鸿,张毅.正定矩阵的性质及应用[J].沈阳大学学报,1999(2)：60-62.
[88]Tetsuya Iwasaki. Control system design via LMIs[J].J.SICE,1995,34(3):164-169.
[89]王涛,肖建,李冀昆.变频调速非线性L2增益控制[J].电力系统及其自动化学报,2009,21(1)：16-19.
[90]许强,倪一峰.感应电动机LPV直接转矩控制系统研究[J].微电机,2008,08：21-24.
[91]K. Trangbaek. Linear Parameter Varying Control of Induction Motom[D]. Aalborg University,2001.
[92]姚兴佳,周洁,邓英,衣传宝,郭庆鼎.风力发电机组的LPV H∞控制器设计[J].沈阳工业大学学报,2008,30(5)：514-519.
[93]姚兴佳,周洁,邓英,衣传宝,郭庆鼎.双馈式变速变距风电机组LPV控制器设计[J].太阳能学报,2009,30(12)：1608-1613.
[94]沈艳霞,朱芸,纪志成.LPV动态补偿的风能转换系统变桨距控制[J].控制理论与应用,2009,26(11)：1282-1288.
[95]张鹏,赵克友.线性时不变参数多项式系统鲁棒指数稳定性的LMI方法[J].青岛大学学报(工程技术版),2009,24(1)：25-29.
[2]郑大钟.线性系统理论.北京：清华大学出版社,2003.
[3]褚健,俞立,苏宏业.鲁棒控制理论及应用[M].杭州：浙江大学出版社,2000.4.
[4]梅生伟,申铁龙,刘志康.现代鲁棒控制理论与应用[M].北京：清华大学出版社,2003.4.
[5]王枞.控制系统的理论与应用[M].北京：北京邮电大学出版社,2005.3.
[6]史忠科.鲁棒控制理论[M].北京：国防工业出版社,2003.
[7]俞立.鲁棒控制--线性矩阵不等式处理方法[M].北京：清华大学出版社,2002.12.
[8]黄琳.稳定性与鲁棒性的理论基础[M].北京：科学出版社,2003.
[9]M.Dettori,C.W. Scherer.Robust stability analysis systems using full block S-Procedure for parameter dependent[C].Proceedings of the 37th IEEE Conference on Decision & Control,Tampa, Florida USA, December 1998:2798-2799.
[10]Eric Feron Pierre Apkarian Pascal Gahinet.S-procedure for the analysis of control systems with parametric uncertainties via parameter-dependent Lyapunov functions[C]. Proceeding of the 2004 American Control Conference, Ssatle Wathlngton,1996:968-972.
[11]Stephen Boyd, Eric Feron. History of linear matrix inequalities in control theory[C]. Proceedings of the American Control Conference,1994:31-34.
[12]S.P.Boyd,L.EI Ghaoui,E.Feron,V.Balakrishnan. Linear matrix inequalities in system and control theory[M].Philadelphia:SLAM,1994.
[13]P.Gahinet, A.Nemirovskii, A.J.Laub, et al. The LMI Control Toolbox[C]. In Proceedings of the 33rd Conference on Decision and Control,Lake Buena, Vista,1994:2038-2041.
[14]Tetsuya Iwasaki.Control System Design via LMIs[J]. J. SICE,1995,34(3):164-169.
[15]P.Gahinet,P.Apkarian.A linear matrix inequality approach to H∞ control[J]. International Journal of Robust and Nonlinear Control,1994,4(4):421-448.
[16]Balakrishnan,V.Linear matrix inequalities in robust control:a brief survey[C].In Proc.Math. Thy of Netw. and Sys., Indiana,2002.
[17]G.Becker, A.Packard, D. Philbrick, G.Balas. Control of parametrically-dependent linear systems:a single quadratic Lyapunov approach[C].Proceedings of the American Control C onference,San Francisco,1993:2795-2799.
[18]G.Becker,A.Packard. Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback[J].Systems&Control Leters,1994,23:205-215.
[19]A.Packard, P.Gahinet, G.Becker. Self-Scheduled H Control of linear parameter-varying systems:a design example[J]. Automatica,1995,31:1251-1261.
[20]E.Feron,P.Apkarian,P.Gahinet.Analysis and synthesis of robust control systems via parameter-dependent Lyapunov function [J].IEEE Trans. on Automat. Control,1996,41(7):1041-1046.
[21]J.C.Geromel, M.C.de Oliveira, L.Hsu. LMI characterization of structural and robust stability[J].Linear Algebra and Its Applications,1998,285(1-3):69-80.
[22]M.C.de Oliveira,J.Bernussou,J.C.Geromel.A new discrete-time stability condition[J]. Systems & Control Letters,1999,37(4):261-265.
[23]Andrew G.Sparks.Analysis of affine parameter-varying systems using parameter-dependent Lyapunov functions[C].Proceedings of the 36th IEEE Conference on Decision and Control,San Diego,IEEE,1997:990-991.
[24]Guang-Hong,Kai-Yew Lum.Comparisons among robust stability criteria for linear systems with affine parameter uncertainties[J].Automatica,2007,43:491-498.
[25]V.J.S. Leite,P.L.D.Peres.An improved LMI condition for robust D-stability of uncertain polytopic systems[J].IEEE Trans.on Automatic Control,2003,48(3):500-504.
[26]P.-A.Bliman. A convex approach to robust stability for linear systems with uncertain scalar parameters[J]. SI AM J. Control and Optimization,2004,42:2016-2042.
[27]X.Zhang, P.Tsiotras,P.-A.Bliman. Multi-parameter dependent Lyapunov functions for the stability analysis of LTI parameter-dependent systems[C].In IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 2004.
[28]Parrilo P A. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[D].California:California Institute ofTechnology,2000.
[29]S.Prajna,A.Papachristodoulou,P.A.Parrilo.Introducing SOSTOOLS:A general purpose sum of squares programming solver[C].Proc.of the 41 st IEEE Conference on Decision and Control, Las Vegas,Nevada,USA,2002:741-746.
[30]Prajna S,Papachristodoulou A.Seiler P,et al.Sostools:Sum of squares optimization toolbox for Matlab[R].California:California Institude ofTechnology,2004.
[31]S.Prajna,A.Papachristodoulou,Fen Wu.Nonlinear control synthesisby sum of squares optimization:a Lyapunov-based approach[C].Proc.of the 5th Asia Control Conference,2004:157-165.
[32]Topcu U,Packard A,Seiler P.Local stability analysis using simulations and sum-of-squares programming[J].Automatica,2008,44:2669-2675.
[33]Christian Ebenbauer,FrankAllgower.Analysis and design of polynomial control systems using dissipation inequalities and sum of squares[C].Computers and Chemical Engineering,2006,30:1590-1602.
[34]F.Wu, S.Prajna.SOS-based solution approach to polynomial LPV system analysis and synthesis problems[J]. International Journal of Control,2005,78(8):600-611.
[35]Zachary William Jarvis-Wloszek.Lyapunov based analysis and controller synthesis for polynomial systems using Sum-of-Squares optimization[D].University of California, Berkeley. Fall 2003. Submitted.
[36]Weehong Tan.Nonlinear control analysis and synthesis using Sum-of-Squares programming[D].University of California,Berkeley,Spring 2006,Submitted.
[37]Erin Aylward,Pablo A.Parrilo,Jean-Jacques E.Slotine.Algorithmic search for contraction metrics via SOS programming[C].Proceedings of the 2006 American Control Conference,Minneapolis,Minnesota,USA,June,2006:3001-3006.
[38]G.Chesi,A.Garulli,A.Tesi,A.Vicino.Robust stability of polytopic systems via polynomially parameter-dependent Lyapunov functions[C].In IEEE Conference on Decision and Control,Maui,Hawaii,USA,December 2003.
[39]G.Chesi,A.Garulli,A.Tesi,A.Vicino.Polynomially parameter-dependent Lyapunov functions for robust H1 performance analysis[C].In IF AC World Congress, Prague,Czech Republic,2005.
[40]Vinicius F.Montagner,Pedro L.D. Peres. H∞ parameter-dependent state feedback control of linear time-varying systems in polytopic domains[C].In Proceedings of the 44th IEEE Conference on Decision and Control and the European Control Conference,Seville, 2005:5006-5011.
[41]G.Chesi,A.Garulli,A.Tesi.A.Vicinoa. Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions[J].Automatica,2007,43(2):309-316.
[42]Chesi,G.,Garulli,A.,Tesi,A.,Vicino, A. Robust stability for polytopic systems via polynomially parameter-dependent Lyapunov functions[C].42nd IEEE CDC,Maui,Hawaii,2003b:4670-4675.
[43]Geromel,J.C.,Colaneri,P. Robust stability of time varying polytopic systems[J].Systems and Control Letters,2005,55(1):81-85.
[44]Leite,V.J.S.,Peres,P.L.D.An improved LMI condition for robust D-stability of uncertain polytopic systems[J].IEEE Transactions on Automatic Control,2003,48(3):500-504.
[45]Peaucelle,Arzelier,Bachelier,Bernussou.A new robust -stability condition for real convex polytopic uncertainty[J].Systems and Control Letters,2000,40:21-30.
[46]Fen Wu,Stephen Prajna.A new solution approach to polynomial LPV system analysis and synthesis[C].Proceeding of the 2004 American Control Conference,Boston, Massachusetts June,2004:1362-1367.
[47]P.Tsiotras.P.-A.Bliman.An exact stability analysis test for single-parameter polynomially-dependent linear systems[J].IEEE Trans. Automat. Control,2006,51(7): 1161-1164.
[48]Y Wang,L.Xie,C.E.de Souza. Robust control of a Class of uncertain nonli near-systems[J]. Systems & Control Leters,1992,19(2):139-149.
[49]Khargonekar,P.,Petersen,I. R,Zhou,K. Robust stabilization of uncertain linear systems: quadratic stabilizatility and H∞ control theory [J]. IEEE Transactions on Automatic Control,1990,35:356-361.
[50]Tetsuya Iwasaki.Robust performance analysis for systems with Structure Uncertainty[J].Inernational Journal of Robust and Nonlinear Control,1996,6(1):85-99.
[51]Chesi,G.,Garulli,A.,Tesi,A.,Vicino.A. Robust analysis of LFR systems through homogeneous polynomial Lyapunov functions[J].IEEE Transactions on Automatic Control,2004b,49(7):1211-1216.
[52]Ke-You Zhao, Michael J.Grimble, and Jakob Stoustrup. Average H2 Performance and Maximal Parameter Perturbation Radius for Uncertain Systems[J].Society for Industrial and Applied Mathematics,1999,37(6),1742-1750.
[53]Xiping Zhang. Parameter-dependent Lyapunov functions and stability analysis of linear parameter-dependent dynamical systems[D].Ph D dissertation,School of Aerospace Engineering Georgia Institute of Technology,October 2003.
[54]Gahinet P,Nemimvski A,ALaub,et al. The LMI control toolbox[C].Proc of 33rd Conf on Decision and Control,Lake Buena Vista,1994:2038-2041.
[55]韩小宝,王仲生,芦玉华.基于凸多面体构造的多项式LPV系统性能分析[J].计算机仿真,2009,26(8)：92-95.
[56]王凡,沈小林,路旭.一类多项式族的鲁棒稳定性分析[J].中北大学学报(自然科学版),2009,30(5)：439-442.
[57]H.R.Karimi,B.Moshiri,P.Jabehdar Maralani,B.Lohmann.Adaptive Hf,-control design for a class of LPV systems[J]. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville,Spain,2005:7918-7923.
[58]Didier Henriona,Denis Arzelier,Dimitri Peaucelle.Positive polynomial matrices and improved LMI robustness conditions[J].Automatica,2003,39:1479-1485.
[59]Henrion,D.,Lasserre,J.-B.Convergent relaxations of polynomial matrix inequalities and static output feedback[J].Transactions on Automatic Control,2006,51 (2):192-202.
[60]Trofino,A. Parameter dependent Lyapunov functions for a class of uncertain linear systems:A LMI approach[C].Proceedings of 38th IEEE CDC,Phoenix, Arizona,1999:2341-2346.
[61]Dasgupta,S.,Chockalingam,G.,Anderson,B.D.O.,Fu,M. Lyapunov functions for uncertain systems with applications to the stability of time varying systems[J].IEEE Transactions on Automatic Control,1994,41:93-106.
[62]吴健荣,杨成梧.广义系统渐近稳定性的一种新的Lyapunov判别方法[J].应用数学,2003,16(2)：92-96.
[63]C.E.de Souza, A.Trofino,J.de Oliveira. Parametric Lyapunov function approach to H2 analysis and control of linear parameter-dependent systems[J]. IEE Proc.-Control Theory Appl,2003,150(9).
[64]W. Xie. H2 gain scheduled state feedback for LPV system with new LMI formulation[J]. IEE Proc.-Control Theory Appl.,2005,152(6).
[65]Job C.Ostveen, Curtain. Riccati equations for strongly stabilizable bounded linear systems[J].Automatica,1998,34(8):953-967.
[66]Masubuchi,I.,Kamitane,Y.,Ohara,A.Suda,N. H∞ control for descriptor systems:a matrix inequalities approach[J].Automatica,1997,33:669-673.
[67]Miaomiao Ma, Hong Chen. LMI Based H∞ Control for Constrained Linear Systems with Norm-Bounded Uncertainties[C].Proceedings of the 6th World Congress on Control and Automation, Dalian,China,2006,6.
[68]RicardoC.L.F.Oliveira,,MauricioC.,de Oliveira,Pedro L.D.Peres.Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions[J].Systems&ControlLetters,2008,57:680-689.
[69]Ebihara,Y,Hagiwara,T.On the degree of polynomial parameter-dependent Lyapunov functions for robust stability of single parameter-dependent LTI systems:A counter-example to Barmish's conjecture[J].Automatica,2006,42(9):1599-1603.
[70]Ricardo C.L.F.Oliveira,Pedro L.D. Peres.LMI conditions for robust stability analysis based on polynomially parameter-dependent Lyapunov functions[J].Systems & Control Letters,2006,55(1):52-61.
[71]Graziano Chesi.Time-invariant uncertain systems:A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions[J].Automatica,2010,46(2):471-474.
[72]Hajer Bouzaouache,Naceur Benhadj Braiek.On the stability analysis of nonlinear systems using polynomial Lyapunov functions[J].Mathematics and Computers in Simulation,2008,76(5):316-329.
[73]Wilfried Gilbert,Didier Henrion,Jacques Bernussou,David Boyer.Polynomial LPV synthesis applied to turbo fan engines [J]. Control Engineering Practice,2010,18(9):1077-1083.
[74]Ping Li,James Lam,Graziano Chesi. On the synthesis of linear Hoo filters for polynomial systems[J].Systems & Control Letters,2012,61(1):31-36.
[75]Hamid Reza Karimi,Peter Maass.A convex optimisation approach to robust observer-based Hoo control design of linear parameter-varying delayed systems[J].Int.J.Modelling Identification and Control,2008,4(3):226-241.
[76]Baoyong Zhang,James Lamb,Shengyuan Xu.Deconvolution filtering for stochastic systems via homogeneous polynomial Lyapunov functions[J].Signal Processing,2009,89(4):605-614.
[77]P.Tsiotras,P.-A.Bliman. An Exact Stability Analysis Test for Single-Paramete Polynomially-Dependent Linear Systems.
[78]Leonardo Amaral Mozelli,Reinaldo Martinez Palhares. Stability analysis of linear time-varying systems:Improving conditions by adding more information about parameter variation[J].Systems & Control Letters,2011,60(5):338-343.
[79]Parrilo P A. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[D].California:California Institute of Technology,2000.
[80]M. S. Mahomoud. Linear parameter-varying discrete time-delay systems:stability and L2-Gain Controllers[J].Int.J.Contro,2000,73:481-494.
[81]D.J.Stilwell,W.J.Rugh. Stability and L2 Gain Properties of LPV Systems[J].Automatica, 2002,38:1601-1606.
[82]D.F.Coutinho,M.Fu,A.Trofino,P.Danes. L2-Gain analysis and control of uncertain nonlinear systems With bounded disturbance inputs[J].Int.J.Robust Nonlinear Control,2008,18:88-110.
[83]Gahinet,P.,Apkarian,P.,Chilali,M. Affine parameter-dependent Lyapunov functions and real parametric uncertainty[J].IEEE Transactions on Automatic Control,1996,41 (3):436-442.
[84]D.Peaucelle,Y.Ebihara,D.Arzelier,H.Tomomichi.General polynomial parameter-dependent lyapunov functions for polytopic uncertain systems[C].In International Symposium on Mathematical Theory of Networks and Systems, Kyoto, July 2006.
[85]D.Peaucelle,D.Arzelier,O.Bachelier,J.Bernussou. A new robust D-stability condition for real convex polytopic uncertainty[J].Systems & Control Letters,2000,40(1):21-30.
[86]Gahinet P.,et al. LMI control toolbox[M].The mathworks Inc,1994.
[87]邵东南,马鸿,张毅.正定矩阵的性质及应用[J].沈阳大学学报,1999(2)：60-62.
[88]Tetsuya Iwasaki. Control system design via LMIs[J].J.SICE,1995,34(3):164-169.
[89]王涛,肖建,李冀昆.变频调速非线性L2增益控制[J].电力系统及其自动化学报,2009,21(1)：16-19.
[90]许强,倪一峰.感应电动机LPV直接转矩控制系统研究[J].微电机,2008,08：21-24.
[91]K. Trangbaek. Linear Parameter Varying Control of Induction Motom[D]. Aalborg University,2001.
[92]姚兴佳,周洁,邓英,衣传宝,郭庆鼎.风力发电机组的LPV H∞控制器设计[J].沈阳工业大学学报,2008,30(5)：514-519.
[93]姚兴佳,周洁,邓英,衣传宝,郭庆鼎.双馈式变速变距风电机组LPV控制器设计[J].太阳能学报,2009,30(12)：1608-1613.
[94]沈艳霞,朱芸,纪志成.LPV动态补偿的风能转换系统变桨距控制[J].控制理论与应用,2009,26(11)：1282-1288.
[95]张鹏,赵克友.线性时不变参数多项式系统鲁棒指数稳定性的LMI方法[J].青岛大学学报(工程技术版),2009,24(1)：25-29.