离散Takagi-Sugeno模糊控制系统的稳定性研究
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摘要
模糊控制理论作为控制领域热点研究问题之一,近年来得到了长足的发展.由于模糊控制技术具有不需要精确的数学模型可以有效的利用专家知识且具有鲁棒性强等特点,从而成功地应用于模式识别,信号处理,智能机器,决策分析,医疗,财经等等.在模糊控制技术之中,最常用的控制模型是Takagi-Sugeno(T-S)模型.在该模型框架下,模糊系统的控制归结为常微分方程的稳定性问题。众所周知,系统的稳定性是人们首要考虑的问题,近年来人们利用各种方法不断的改进模糊系统的稳定性条件,取得了很多成果.在前人研究工作的基础上,本文通过设计新的Lyapunov函数和控制律给出了更加宽松的稳定性条件,并将其应用于鲁棒H_∞输出反馈控制问题,取得了良好的控制效果.
首先,定义了隶属度函数相关的矩阵函数,并应用此函数设计Lyapunov函数,得到了新的稳定性条件,新条件在保持系统的稳定性方面和现有的结果等价。但因为含有较少的变量和线性矩阵不等式,在一定程度上减小了计算复杂度.最后以仿真例子验证了所提出方法的有效性.
值得指出的是,为了获得保守性较小的条件,引进矩阵变量是一种非常重要和常用的方法.在研究离散模糊系统时,设计了新的Lyapunov函数,并对当前时刻和下一时刻的隶属度函数分别引进了矩阵变量,克服了以前文献只针对当前时刻的隶属度函数引进矩阵变量的缺点,获得了保守性更小的稳定性条件.将设计新控制律的方法推广至连续模糊系统,通过讨论隶属度函数的组合给出了新的稳定性条件,并且讨论了现存一些条件之间的关系.仿真例子表明由该方法所获得的稳定性条件是有效的,且具有更小的保守性.
虽然针对当前时刻和下一时刻的隶属度函数分别引进矩阵变量可以获得保守性较小的结果,但同时却增加了计算负担.为了在减小所得条件的保守性的同时能够一定程度的减小计算的复杂度,设计了另外一种方法-增加隶属度函数相关的矩阵函数的次数,随着次数的增加。线性矩阵不等式的个数也不断增加,但是每个不等式含有较少的变量,容易执行,从而一定程度上减小了计算负担。应用这种方法来处理状态反馈H_∞控制问题,不仅能够使得系统在较大的参数范围内稳定,而且能够获得良好的抗干扰能力.仿真结果表明了这种方法的有效性.
考虑到系统可能会出现内部不确定性,外部扰动和系统某个状态未知的情况,需要设计观测器.利用隶属度函数相关的矩阵函数设计了新的观测器和基于观测器的模糊H_∞控制律,来研究含有内部不确定性和外部扰动的离散非线性系统,得到了新的鲁棒H_∞稳定性条件.通过和现有结果的对比说明了本文所得到的稳定性条件具有较小的保守性.
As an active research field,the fuzzy control theory makes a rapid progress recently,which plays an important role in control fields.Fuzzy logic control is successfully applied to many control problems,such as pattern recognition,signal processing,machine intelligence,decision making,finance,medicine,and so on,because they do not need accurate mathematical models of the system and can cooperate with human experts' knowledge.The T-S fuzzy model is very important in the fuzzy control theory.Based on the model,the controlled system can be solved as differential equation.It is well known that the stability analysis is very important in system control and should be considered firstly.Due to the importance,it has drawn a large number of researchers' attention to study the problem.Based on the published results,this thesis gives more relaxed stability conditions by developing new Lyapunov function and controller which are constructed by a kind of matrix function.further more,it can deal with robust H_∞output feedback control problem effectively.
First,the basis-dependent matrix function is developed by some definitions.Using the matrix function,a new Lyapunov function is obtained and some stabilization conditions are proposed by applying the Lyapunov function.It is proved that the new condition are equivalent to the previous ones,however,since the new conditions contain less variables and LMIs than the previous ones,it needs less computational time than before.The final simulation shows the effect of the new condition.
It is worth pointing out that introducing slack variables is a useful and popular technique to reduce the conservatism of the obtained theorems.For discrete-time fuzzy system,the slack variables in the latest result are only introduced for the current time membership function,while in this thesis the slack variables are introduced for the current time membership function and next time membership function respectively and thus leading to less conservative results.For the continuous-time fuzzy system,new controller can be designed to reduce the conservatism. Further more,the relationship between the new result and the previous ones are discussed.Some examples are shown that the new theorems are less conservative than the published ones.
Introducing slack variables for the current time membership function and next time membership function respectively do can reduce the conservatism of the obtained theorem,however, the computational burden increases too.In order to reduce the conservatism of the of the obtained theorem,at the same time,decrease the computational burden,a new method-increasing the degree of the matrix function instead of introducing exterior variables is developed.It is shown that the conservatism of the obtained theorems is reduced as the degree of the matrix function increases,although the number of the variables and LMIs increases,each linear matrix inequality is simple and easy to be solved and hence,the computational burden is reduced.Applying the method to deal with the state feedback H_∞control problem,the controlled system can be stabilized with the parameter varying in a larger area,in addition,the controlled system has good H_∞performance.The simulations show this point.
At some time,there may be uncertainties in the system,some disturbance and some states of the system are unknown which needs observer to be observed.By applying the matrix function,new observer which contains previous ones as special cases,based on the observer the new H_∞controller are constructed to deal with this kind of system and new robustness H_∞stabilization conditions are obtained.The simulations show the proposed method is effective.
首先,定义了隶属度函数相关的矩阵函数,并应用此函数设计Lyapunov函数,得到了新的稳定性条件,新条件在保持系统的稳定性方面和现有的结果等价。但因为含有较少的变量和线性矩阵不等式,在一定程度上减小了计算复杂度.最后以仿真例子验证了所提出方法的有效性.
值得指出的是,为了获得保守性较小的条件,引进矩阵变量是一种非常重要和常用的方法.在研究离散模糊系统时,设计了新的Lyapunov函数,并对当前时刻和下一时刻的隶属度函数分别引进了矩阵变量,克服了以前文献只针对当前时刻的隶属度函数引进矩阵变量的缺点,获得了保守性更小的稳定性条件.将设计新控制律的方法推广至连续模糊系统,通过讨论隶属度函数的组合给出了新的稳定性条件,并且讨论了现存一些条件之间的关系.仿真例子表明由该方法所获得的稳定性条件是有效的,且具有更小的保守性.
虽然针对当前时刻和下一时刻的隶属度函数分别引进矩阵变量可以获得保守性较小的结果,但同时却增加了计算负担.为了在减小所得条件的保守性的同时能够一定程度的减小计算的复杂度,设计了另外一种方法-增加隶属度函数相关的矩阵函数的次数,随着次数的增加。线性矩阵不等式的个数也不断增加,但是每个不等式含有较少的变量,容易执行,从而一定程度上减小了计算负担。应用这种方法来处理状态反馈H_∞控制问题,不仅能够使得系统在较大的参数范围内稳定,而且能够获得良好的抗干扰能力.仿真结果表明了这种方法的有效性.
考虑到系统可能会出现内部不确定性,外部扰动和系统某个状态未知的情况,需要设计观测器.利用隶属度函数相关的矩阵函数设计了新的观测器和基于观测器的模糊H_∞控制律,来研究含有内部不确定性和外部扰动的离散非线性系统,得到了新的鲁棒H_∞稳定性条件.通过和现有结果的对比说明了本文所得到的稳定性条件具有较小的保守性.
As an active research field,the fuzzy control theory makes a rapid progress recently,which plays an important role in control fields.Fuzzy logic control is successfully applied to many control problems,such as pattern recognition,signal processing,machine intelligence,decision making,finance,medicine,and so on,because they do not need accurate mathematical models of the system and can cooperate with human experts' knowledge.The T-S fuzzy model is very important in the fuzzy control theory.Based on the model,the controlled system can be solved as differential equation.It is well known that the stability analysis is very important in system control and should be considered firstly.Due to the importance,it has drawn a large number of researchers' attention to study the problem.Based on the published results,this thesis gives more relaxed stability conditions by developing new Lyapunov function and controller which are constructed by a kind of matrix function.further more,it can deal with robust H_∞output feedback control problem effectively.
First,the basis-dependent matrix function is developed by some definitions.Using the matrix function,a new Lyapunov function is obtained and some stabilization conditions are proposed by applying the Lyapunov function.It is proved that the new condition are equivalent to the previous ones,however,since the new conditions contain less variables and LMIs than the previous ones,it needs less computational time than before.The final simulation shows the effect of the new condition.
It is worth pointing out that introducing slack variables is a useful and popular technique to reduce the conservatism of the obtained theorems.For discrete-time fuzzy system,the slack variables in the latest result are only introduced for the current time membership function,while in this thesis the slack variables are introduced for the current time membership function and next time membership function respectively and thus leading to less conservative results.For the continuous-time fuzzy system,new controller can be designed to reduce the conservatism. Further more,the relationship between the new result and the previous ones are discussed.Some examples are shown that the new theorems are less conservative than the published ones.
Introducing slack variables for the current time membership function and next time membership function respectively do can reduce the conservatism of the obtained theorem,however, the computational burden increases too.In order to reduce the conservatism of the of the obtained theorem,at the same time,decrease the computational burden,a new method-increasing the degree of the matrix function instead of introducing exterior variables is developed.It is shown that the conservatism of the obtained theorems is reduced as the degree of the matrix function increases,although the number of the variables and LMIs increases,each linear matrix inequality is simple and easy to be solved and hence,the computational burden is reduced.Applying the method to deal with the state feedback H_∞control problem,the controlled system can be stabilized with the parameter varying in a larger area,in addition,the controlled system has good H_∞performance.The simulations show this point.
At some time,there may be uncertainties in the system,some disturbance and some states of the system are unknown which needs observer to be observed.By applying the matrix function,new observer which contains previous ones as special cases,based on the observer the new H_∞controller are constructed to deal with this kind of system and new robustness H_∞stabilization conditions are obtained.The simulations show the proposed method is effective.
引文
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