强γ_k-γ_（cl）H_∞镇定及若干相关问题研究
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摘要
基于H_∞范数最小化的控制理论是鲁棒控制领域最有效的工具之一,而强镇定控制器设计对控制理论与控制工程都具有重大的意义。由于研究之初忽略了控制器自身的稳定性,现有的主流控制器设计方法,包括LQG/H_∞等,都无法保证所设计控制器的稳定性,而另一方面,迄今为止的大多数强镇定问题研究成果存在诸多不足,如局限于特殊对象或特殊问题,依赖于非线性优化问题的求解等,缺乏普适性框架和有效的求解手段。
本文从H_∞性能指标优化角度入手,定义了强γk-γcl H_∞镇定、同时γk-γcl H_∞控制问题,将现有强镇定相关的若干重要问题纳入统一的框架考虑。随后,论文以状态空间描述和H_∞控制状态空间解为技术架构,结合经典代数Riccati方程方法与线性矩阵不等式方法,充分利用矩阵不等式约束中的自由度,系统深入地研究了强γk-γclH_∞镇定及相关的一系列关键问题。
在第三章,我们将描述系统方法引入线性时不变对象H_∞镇定问题,提出并证明了H_∞稳定的新充要条件——扩展有界实引理。随后,我们将该技术应用于凸多面体型/仿射型线性参数不确定系统的H_∞分析与综合问题,利用构造参数依赖的Lyapunov函数,分别得到相应的H_∞稳定条件,进而通过约束松弛变量,得到基于线性矩阵不等式约束的H_∞状态反馈控制器设计方法。
第四章中,我们通过互质分解和Youla参数化描述,证明了r≥2对象同时γk-γclH_∞控制问题等价于r-1关联对象的同时强γk-γcl H_∞镇定问题。利用有界实引理,我们提出了基于一个标准代数Riccati方程和若干线性矩阵不等式约束的n阶强γk-γclH_∞镇定条件,证明了该条件包含之前文献中的修正代数Riccati方程方法,并给出了基于直接线性矩阵不等式优化的控制器设计方法。在此基础上,通过放宽了另一个代数Riccati方程约束,我们得到一个改进n阶强γk-γcl H_∞镇定条件,利用路径跟踪(path-following)思想给出基于迭代线性矩阵不等式优化的控制器设计方法。
在第五章,我们将第三章中证明的扩展有界实引理引入强镇定问题研究,利用松弛变量提供的自由度,证明了n阶强γk-γcl H_∞镇定的新充分条件,该条件大大改善了第四章中的直接线性矩阵不等式形式镇定结果,随后,利用所有次优H_∞控制器的参数化描述,我们证明了任意l×n(l≥2)阶强γk-γcl H_∞控制器设计可以转化为一个n阶强γk-γcl H_∞控制器设计问题,并在此基础上提出嵌套式高阶控制器构造算法,为解决强镇定问题中可能的高阶控制器困难提供了一种有效的手段。对包含双线性矩阵不等式约束的优化问题,我们结合路径跟踪和互迭代思想给出了有效的n阶、l×n(l≥2)阶控制器设计算法。
在第四、五章研究成果的基础上,我们在第六章中从参数化控制器描述的结构入手,通过引入自由矩阵,提出并证明了基于广义H_∞中心控制器形式的n阶强γk-γclH_∞镇定条件,证明若干最新的直接线性矩阵不等式镇定条件都可视为该条件的特殊情况。随后我们系统分析了有界实引理条件在处理强镇定相关问题上的保守性,利用第三章中扩展有界实引理条件进一步迭代改进设计结果,并展示了新镇定方法如何与第五章中的嵌套式高阶控制器构造方法结合以克服强镇定控制器的阶数问题。此外,我们还给出了两对象同时γk-γcl H_∞控制器设计的有效算法。
H_∞control is one of the most efficient tools in robust control,while the strong stabilization problem has great significance in both control theory and practice.As neglected initially,it is well known that most of the current state-of-the-art design techniques,including LQG/H_∞,can not guarantee the stability of controller.On the other hand,due to different kinds of limitations,e.g.,considering only SISO plants or only special kinds of problems,depending on the solutions of nonlinear optimization problems,most results on strong stabilization to date can not be generalized and efficiently solved.
Considering the strong stabilization problem with H_∞performance index,we first define strongγ_k-γ_(cl) H_∞stabilization and simultaneousγ_k-γ_(cl) H_∞control problems,which include strong H_∞stabilization,stable H_∞stabilization,strong stabilization,simultaneous H_∞stabilization and simultaneous stabilization.Then by taking advantages of both ARE and LMI methods and exploring the freedom lying in the matrix inequality constraints,we obtain several results on strongγ_k-γ_(cl) H_∞stabilization and some related problems,which can be summarized as follows.
In Chapter 3,by introducing the descriptor system method into the H_∞stabilization problem for LTI systems,we prove a new necessary and sufficient condition for H_∞stabilization, i.e.,the extend bounded real lemma.Then this technique is applied to the H_∞analysis and synthesis problems for linear polytopic/affine uncertain systems.By constructing parameter-dependent Lyapunov functions,we obtain the corresponding H_∞stabilization conditions,which lead to direct H_∞state feedback controller design methods by refraining the slack variables.
In Chapter 4,by utilizing coprime factorization and Youla parameterization,we first prove that the simultaneousγ_k-γ_(cl) H_∞stabilization problem for r(r≥2) plants is equivalent to stronglyγ_k-γ_(cl) H_∞stabilize r-1 associated plants.Then based on the bounded real lemma,a sufficient condition for strongγ_k-γ_(cl) H_∞stabilization is proposed,which leads to the design procedures that involve an algebraic Riccati equation and some LMI constraints. We also prove that this condition include the modified ARE method in[1].Based on this result,an improved sufficient condition on strong H_∞stabilization as well as its dual form is derived,which extends the other ARE constraint into an inequality constraint.Two path-following algorithms are proposed to solve the associated optimization problems with BMI constraints,which lead to full order controllers.
In Chapter 5,by utilizing the extended bounded real lemma,a new sufficient condition is proposed for n-th order strongγ_k-γ_(cl) H_∞stabilization.This new condition fully relax the H_∞constraint on the controller and greatly improves the direct LMI design results.Then by exploring the parameterization of all suboptimal H_∞controllers,it's also proved that the design of an l×n-th order(l≥2) strongγ_k-γ_(cl) H_∞controller can be transformed into that of an n-th order controller for an associated plant,which tackles the possible order problem.Despite the BMI constraints in resulting optimization problems,the path-following and alternative iteration methods are adopted to formulate procedures for both n-th and l×n-th order(l≥2) controller design.
In Chapter 6,based on the results of Chapter 4 and 5,a new LMI method is proposed for the strongγ_k-γ_(cl) H_∞stabilization problem by modifying the controller form,which leads to a so-called generalized H_∞central controller form.It is shown that several recently developed methods can be seen as its special cases.Meanwhile,we analyze the conservatism of the bounded real lemma when additional constraints have to be taken into account,and adopt the extended bounded real lemma to further improve the LMI design results iteratively.For the possible order problem,the nested construction method can also be applied.For the case when r=2,we present an efficient algorithm for designing a simultaneousγ_k-γ_(cl) H_∞controller.
本文从H_∞性能指标优化角度入手,定义了强γk-γcl H_∞镇定、同时γk-γcl H_∞控制问题,将现有强镇定相关的若干重要问题纳入统一的框架考虑。随后,论文以状态空间描述和H_∞控制状态空间解为技术架构,结合经典代数Riccati方程方法与线性矩阵不等式方法,充分利用矩阵不等式约束中的自由度,系统深入地研究了强γk-γclH_∞镇定及相关的一系列关键问题。
在第三章,我们将描述系统方法引入线性时不变对象H_∞镇定问题,提出并证明了H_∞稳定的新充要条件——扩展有界实引理。随后,我们将该技术应用于凸多面体型/仿射型线性参数不确定系统的H_∞分析与综合问题,利用构造参数依赖的Lyapunov函数,分别得到相应的H_∞稳定条件,进而通过约束松弛变量,得到基于线性矩阵不等式约束的H_∞状态反馈控制器设计方法。
第四章中,我们通过互质分解和Youla参数化描述,证明了r≥2对象同时γk-γclH_∞控制问题等价于r-1关联对象的同时强γk-γcl H_∞镇定问题。利用有界实引理,我们提出了基于一个标准代数Riccati方程和若干线性矩阵不等式约束的n阶强γk-γclH_∞镇定条件,证明了该条件包含之前文献中的修正代数Riccati方程方法,并给出了基于直接线性矩阵不等式优化的控制器设计方法。在此基础上,通过放宽了另一个代数Riccati方程约束,我们得到一个改进n阶强γk-γcl H_∞镇定条件,利用路径跟踪(path-following)思想给出基于迭代线性矩阵不等式优化的控制器设计方法。
在第五章,我们将第三章中证明的扩展有界实引理引入强镇定问题研究,利用松弛变量提供的自由度,证明了n阶强γk-γcl H_∞镇定的新充分条件,该条件大大改善了第四章中的直接线性矩阵不等式形式镇定结果,随后,利用所有次优H_∞控制器的参数化描述,我们证明了任意l×n(l≥2)阶强γk-γcl H_∞控制器设计可以转化为一个n阶强γk-γcl H_∞控制器设计问题,并在此基础上提出嵌套式高阶控制器构造算法,为解决强镇定问题中可能的高阶控制器困难提供了一种有效的手段。对包含双线性矩阵不等式约束的优化问题,我们结合路径跟踪和互迭代思想给出了有效的n阶、l×n(l≥2)阶控制器设计算法。
在第四、五章研究成果的基础上,我们在第六章中从参数化控制器描述的结构入手,通过引入自由矩阵,提出并证明了基于广义H_∞中心控制器形式的n阶强γk-γclH_∞镇定条件,证明若干最新的直接线性矩阵不等式镇定条件都可视为该条件的特殊情况。随后我们系统分析了有界实引理条件在处理强镇定相关问题上的保守性,利用第三章中扩展有界实引理条件进一步迭代改进设计结果,并展示了新镇定方法如何与第五章中的嵌套式高阶控制器构造方法结合以克服强镇定控制器的阶数问题。此外,我们还给出了两对象同时γk-γcl H_∞控制器设计的有效算法。
H_∞control is one of the most efficient tools in robust control,while the strong stabilization problem has great significance in both control theory and practice.As neglected initially,it is well known that most of the current state-of-the-art design techniques,including LQG/H_∞,can not guarantee the stability of controller.On the other hand,due to different kinds of limitations,e.g.,considering only SISO plants or only special kinds of problems,depending on the solutions of nonlinear optimization problems,most results on strong stabilization to date can not be generalized and efficiently solved.
Considering the strong stabilization problem with H_∞performance index,we first define strongγ_k-γ_(cl) H_∞stabilization and simultaneousγ_k-γ_(cl) H_∞control problems,which include strong H_∞stabilization,stable H_∞stabilization,strong stabilization,simultaneous H_∞stabilization and simultaneous stabilization.Then by taking advantages of both ARE and LMI methods and exploring the freedom lying in the matrix inequality constraints,we obtain several results on strongγ_k-γ_(cl) H_∞stabilization and some related problems,which can be summarized as follows.
In Chapter 3,by introducing the descriptor system method into the H_∞stabilization problem for LTI systems,we prove a new necessary and sufficient condition for H_∞stabilization, i.e.,the extend bounded real lemma.Then this technique is applied to the H_∞analysis and synthesis problems for linear polytopic/affine uncertain systems.By constructing parameter-dependent Lyapunov functions,we obtain the corresponding H_∞stabilization conditions,which lead to direct H_∞state feedback controller design methods by refraining the slack variables.
In Chapter 4,by utilizing coprime factorization and Youla parameterization,we first prove that the simultaneousγ_k-γ_(cl) H_∞stabilization problem for r(r≥2) plants is equivalent to stronglyγ_k-γ_(cl) H_∞stabilize r-1 associated plants.Then based on the bounded real lemma,a sufficient condition for strongγ_k-γ_(cl) H_∞stabilization is proposed,which leads to the design procedures that involve an algebraic Riccati equation and some LMI constraints. We also prove that this condition include the modified ARE method in[1].Based on this result,an improved sufficient condition on strong H_∞stabilization as well as its dual form is derived,which extends the other ARE constraint into an inequality constraint.Two path-following algorithms are proposed to solve the associated optimization problems with BMI constraints,which lead to full order controllers.
In Chapter 5,by utilizing the extended bounded real lemma,a new sufficient condition is proposed for n-th order strongγ_k-γ_(cl) H_∞stabilization.This new condition fully relax the H_∞constraint on the controller and greatly improves the direct LMI design results.Then by exploring the parameterization of all suboptimal H_∞controllers,it's also proved that the design of an l×n-th order(l≥2) strongγ_k-γ_(cl) H_∞controller can be transformed into that of an n-th order controller for an associated plant,which tackles the possible order problem.Despite the BMI constraints in resulting optimization problems,the path-following and alternative iteration methods are adopted to formulate procedures for both n-th and l×n-th order(l≥2) controller design.
In Chapter 6,based on the results of Chapter 4 and 5,a new LMI method is proposed for the strongγ_k-γ_(cl) H_∞stabilization problem by modifying the controller form,which leads to a so-called generalized H_∞central controller form.It is shown that several recently developed methods can be seen as its special cases.Meanwhile,we analyze the conservatism of the bounded real lemma when additional constraints have to be taken into account,and adopt the extended bounded real lemma to further improve the LMI design results iteratively.For the possible order problem,the nested construction method can also be applied.For the case when r=2,we present an efficient algorithm for designing a simultaneousγ_k-γ_(cl) H_∞controller.
引文
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