LPV系统鲁棒变增益控制研究及其应用
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摘要
变增益控制一直是工程界最为常用的控制方法之一,应用领域十分广泛,特别是在飞行控制系统设计领域。传统的变增益控制设计方法缺乏保证系统在整个工作区域内的稳定性理论证明,同时设计步骤繁琐,已逐渐不能适应目前系统要求。LPV系统是一类重要的时变系统,由于能够描述一类实际动态系统本身存在的非线性和时变特性,所以应用LPV系统进行变增益设计可以避免传统变增益出现的问题。
论文研究了LPV系统的鲁棒变增益控制,分析了目前鲁棒变增益控制中存在的保守性,并展开研究,得到了具有更低保守性的方法,并将部分研究结果应用到一类无尾式飞行器的控制系统的设计中。主要研究内容和创新点如下：
(1)研究了鲁棒变增益控制中的模型问题,即系统的LPV建模。介绍了常用非线性系统向LPV系统转化的方法,针对目前方法存在的保守性,提出了一种改进的非线性系统LPV的表示方法,并且通过比较不同方法与原非线性系统可达集大小的方式验证了改进方法较之目前方法具有更小的保守性；同时研究了一般LPV系统的多胞形表示,根据矩阵的高阶奇异值分解,给出了一般LPV系统向多胞形转化的实现步骤。
(2)研究了多胞形系统基于二次Lyapunov函数的鲁棒变增益控制方法。针对目前基于变参数几何位置的调度策略存在的保守性,提出了一种基于模型广义距离的调度策略,通过计算当前时刻模型与多胞形顶点模型间的Gap度量,然后进行凸分解,进而进行变增益控制,仿真结果验证了较之目前的调度策略具有更小的保守性。
(3)针对二次Lyapunov函数方法存在的保守性,从两个方面研究了基于参数依赖Lyapunov函数的鲁棒变增益控制方法。首先,针对多胞形系统,提出了一种多松弛变量的参数依赖Lyapunov函数方法,将Lyapunov函数矩阵与系统矩阵解耦,降低了保守性；其次,针对仿射参数依赖LPV系统,提出了一种仿射参数依赖Lyapunov函数方法,考虑了变参数的变化速率,降低了保守性。
(4)研究了基于线性分式变换的鲁棒变增益控制方法。阐述了基于线性分式变换的鲁棒变增益的基本思想,研究了一类LPV系统的降维线性分式变换方法,并且给出了降维变换的思路和步骤；针对目前线性分式变换鲁棒变增益控制方法存在的保守性,基于S-过程提出了一种保守性更小的鲁棒变增益控制分析和综合的方法,该方法通过对全块标量矩阵的选择,采用双线性矩阵不等式,降低了由于对标量矩阵过多约束限制带来的保守性,同时给出了控制器实现的迭代线性矩阵不等式算法。
(5)研究了LPV系统的鲁棒变增益多目标控制。将扩展线性矩阵不等式方法和参数依赖Lyapunov函数方法应用到LPV系统的多通道多目标控制中,降低了单一Lyapunov函数方法带来的保守性,同时研究了一类双线性矩阵不等式的求解问题,提出了一种简单有效的迭代LMI算法,进一步降低了鲁棒变增益多目标控制的保守性。
(6)将LPV系统鲁棒变增益控制方法应用于一类无尾式飞行器的飞控系统的设计中。建立了飞行器的LPV模型,采用Lyapunov函数方法和线性分式变换方法分别设计了飞行器的纵向和横侧向的鲁棒变增益控制器,并根据系统存在的不确定性,检验了控制器的鲁棒性,仿真结果验证了方法的有效性和可行性。
总之,论文始终围绕着LPV系统的鲁棒变增益控制展开研究,得到了一些新的结果,并且将部分研究结果应用于飞行器的控制系统设计中,为鲁棒变增益控制在高性能飞行器中的应用做了有意义的尝试。
Gain-Scheduling (GS) is one of the most famous methods in the engineering and is applied in many fields, especially for flight control systems. Due to lack of stability theory prove within operation range and burdensome process, the traditional GS is not compatible for modern system gradually, Linear Parameter-Varying (LPV) system is an important time-varying system, GS based on LPV system can overcome the shortcomings of the traditional GS, because it can describe nonlinear and time-varying characteristics of a class of dynamic systems.
The robust GS theory based on LPV system is investigated, and the conservatism of current methods is analyzed and studied, some new methods which have smaller conservatism are obtained. And partial results are applied to the flight control system of a class of tailless aircraft. The main research contents and contributions are listed as follows:
(1) Model problems of robust GS, LPV system modeling, are studied. The transformation methods of nonlinear system to LPV system are introduced, and aim at the conservatism of these methods, an improved method is proposed. The performance of models got by different methods is evaluated using reachable set and the results testify the proposed method has smaller conservatism. At the same time, method of polytope representation of the general LPV system based on high-dimensional singular value decomposition is studied, and the implementation steps are given.
(2) Robust GS based on quadratic Lyapunov functions is studied and a new GS strategy is proposed because of the conservatism of the current GS strategy which based on parameter geometry distance. Aim at polytope, the new strategy based on Gap-Metric does convex decomposition through generalized distance between the system and its vertex systems. The simulation result testifies the new one has smaller conservatism.
(3) In order to reduce the conservatism of the quadratic Lyapunov function method, the parameter dependent Lyapunov function method is studied from two aspects. Firstly, aim at the polytope, a multi-slack variables parameter dependent Lyapunov function method is proposed, this new method can decouple the Lyapunov function and system matrix and the conservatism is reduced. Secondly, aim at the affine parameter dependent LPV system, an affine parameter dependent Lyapunov function method is proposed. The variations of the variable parameter are considered and the conservatism is reduced.
(4) The robust GS based on linear fractional transformation (LFT) is studied and the basic idea of this method is introduced. The LFT representation of polynomial LPV and the dimensionality reduction methods and steps are given. A less conservative analysis and synthesis method based on S-procedure is proposed. The bilinear matrix inequality is used in this method because less constraint on full block scaling matrix and the iterative LMI algorithm to the BMI is given.
(5) The multi-objective robust GS based on LPV system is studied, the extended LMI and parameter dependent Lyapunov function, which can reduce the conservatism of the multi-channel single Lyapunov function method, are applied into LPV system multi-objective control using multi-channel principle. A class of BMI problem is studied, a new simple and effective iterative LMI algorithm is proposed which can further reduce the conservatism.
(6) The results obtained in the dissertation are applied to flight control system of a class of tailless aircraft. The LPV model of the aircraft is developed, the robust GS controllers of longitudinal and lateral are designed using Lyapunov function method and LFT method respectively, and in accordance with system uncertainty the robustness of the controller is verified. The simulation results prove the feasibility and efficiency of the methods.
In a word, the dissertation is focus on the robust GS control based on LPV system, and some new results are obtaied, and then the results are applied to flight control system. The meaningful attempt of robust GS to the high-performance aircraft is done.
论文研究了LPV系统的鲁棒变增益控制,分析了目前鲁棒变增益控制中存在的保守性,并展开研究,得到了具有更低保守性的方法,并将部分研究结果应用到一类无尾式飞行器的控制系统的设计中。主要研究内容和创新点如下：
(1)研究了鲁棒变增益控制中的模型问题,即系统的LPV建模。介绍了常用非线性系统向LPV系统转化的方法,针对目前方法存在的保守性,提出了一种改进的非线性系统LPV的表示方法,并且通过比较不同方法与原非线性系统可达集大小的方式验证了改进方法较之目前方法具有更小的保守性；同时研究了一般LPV系统的多胞形表示,根据矩阵的高阶奇异值分解,给出了一般LPV系统向多胞形转化的实现步骤。
(2)研究了多胞形系统基于二次Lyapunov函数的鲁棒变增益控制方法。针对目前基于变参数几何位置的调度策略存在的保守性,提出了一种基于模型广义距离的调度策略,通过计算当前时刻模型与多胞形顶点模型间的Gap度量,然后进行凸分解,进而进行变增益控制,仿真结果验证了较之目前的调度策略具有更小的保守性。
(3)针对二次Lyapunov函数方法存在的保守性,从两个方面研究了基于参数依赖Lyapunov函数的鲁棒变增益控制方法。首先,针对多胞形系统,提出了一种多松弛变量的参数依赖Lyapunov函数方法,将Lyapunov函数矩阵与系统矩阵解耦,降低了保守性；其次,针对仿射参数依赖LPV系统,提出了一种仿射参数依赖Lyapunov函数方法,考虑了变参数的变化速率,降低了保守性。
(4)研究了基于线性分式变换的鲁棒变增益控制方法。阐述了基于线性分式变换的鲁棒变增益的基本思想,研究了一类LPV系统的降维线性分式变换方法,并且给出了降维变换的思路和步骤；针对目前线性分式变换鲁棒变增益控制方法存在的保守性,基于S-过程提出了一种保守性更小的鲁棒变增益控制分析和综合的方法,该方法通过对全块标量矩阵的选择,采用双线性矩阵不等式,降低了由于对标量矩阵过多约束限制带来的保守性,同时给出了控制器实现的迭代线性矩阵不等式算法。
(5)研究了LPV系统的鲁棒变增益多目标控制。将扩展线性矩阵不等式方法和参数依赖Lyapunov函数方法应用到LPV系统的多通道多目标控制中,降低了单一Lyapunov函数方法带来的保守性,同时研究了一类双线性矩阵不等式的求解问题,提出了一种简单有效的迭代LMI算法,进一步降低了鲁棒变增益多目标控制的保守性。
(6)将LPV系统鲁棒变增益控制方法应用于一类无尾式飞行器的飞控系统的设计中。建立了飞行器的LPV模型,采用Lyapunov函数方法和线性分式变换方法分别设计了飞行器的纵向和横侧向的鲁棒变增益控制器,并根据系统存在的不确定性,检验了控制器的鲁棒性,仿真结果验证了方法的有效性和可行性。
总之,论文始终围绕着LPV系统的鲁棒变增益控制展开研究,得到了一些新的结果,并且将部分研究结果应用于飞行器的控制系统设计中,为鲁棒变增益控制在高性能飞行器中的应用做了有意义的尝试。
Gain-Scheduling (GS) is one of the most famous methods in the engineering and is applied in many fields, especially for flight control systems. Due to lack of stability theory prove within operation range and burdensome process, the traditional GS is not compatible for modern system gradually, Linear Parameter-Varying (LPV) system is an important time-varying system, GS based on LPV system can overcome the shortcomings of the traditional GS, because it can describe nonlinear and time-varying characteristics of a class of dynamic systems.
The robust GS theory based on LPV system is investigated, and the conservatism of current methods is analyzed and studied, some new methods which have smaller conservatism are obtained. And partial results are applied to the flight control system of a class of tailless aircraft. The main research contents and contributions are listed as follows:
(1) Model problems of robust GS, LPV system modeling, are studied. The transformation methods of nonlinear system to LPV system are introduced, and aim at the conservatism of these methods, an improved method is proposed. The performance of models got by different methods is evaluated using reachable set and the results testify the proposed method has smaller conservatism. At the same time, method of polytope representation of the general LPV system based on high-dimensional singular value decomposition is studied, and the implementation steps are given.
(2) Robust GS based on quadratic Lyapunov functions is studied and a new GS strategy is proposed because of the conservatism of the current GS strategy which based on parameter geometry distance. Aim at polytope, the new strategy based on Gap-Metric does convex decomposition through generalized distance between the system and its vertex systems. The simulation result testifies the new one has smaller conservatism.
(3) In order to reduce the conservatism of the quadratic Lyapunov function method, the parameter dependent Lyapunov function method is studied from two aspects. Firstly, aim at the polytope, a multi-slack variables parameter dependent Lyapunov function method is proposed, this new method can decouple the Lyapunov function and system matrix and the conservatism is reduced. Secondly, aim at the affine parameter dependent LPV system, an affine parameter dependent Lyapunov function method is proposed. The variations of the variable parameter are considered and the conservatism is reduced.
(4) The robust GS based on linear fractional transformation (LFT) is studied and the basic idea of this method is introduced. The LFT representation of polynomial LPV and the dimensionality reduction methods and steps are given. A less conservative analysis and synthesis method based on S-procedure is proposed. The bilinear matrix inequality is used in this method because less constraint on full block scaling matrix and the iterative LMI algorithm to the BMI is given.
(5) The multi-objective robust GS based on LPV system is studied, the extended LMI and parameter dependent Lyapunov function, which can reduce the conservatism of the multi-channel single Lyapunov function method, are applied into LPV system multi-objective control using multi-channel principle. A class of BMI problem is studied, a new simple and effective iterative LMI algorithm is proposed which can further reduce the conservatism.
(6) The results obtained in the dissertation are applied to flight control system of a class of tailless aircraft. The LPV model of the aircraft is developed, the robust GS controllers of longitudinal and lateral are designed using Lyapunov function method and LFT method respectively, and in accordance with system uncertainty the robustness of the controller is verified. The simulation results prove the feasibility and efficiency of the methods.
In a word, the dissertation is focus on the robust GS control based on LPV system, and some new results are obtaied, and then the results are applied to flight control system. The meaningful attempt of robust GS to the high-performance aircraft is done.
引文
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