基于T-S模型的非线性系统的模糊控制
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摘要
与传统控制相比,模糊控制具有两大不可比拟的优点:其一,它在许多应用中可以有效且便捷地实现人的控制策略和经验;其二,它不需要知道被控对象精确的数学模型就可实现较好的控制。日本学者Takagi T和Sugeno M在1985年提出的Takagi-Sugeno (T-S)模糊模型,给模糊控制理论研究及应用带来了深远的影响,使模糊系统稳定性分析上升到新的理论高度。
本文分别基于T-S线性模型和T-S双线性模型,根据Lyapunov稳定性理论、鲁棒控制理论和H∞控制理论,结合线性矩阵不等式(LMI)技术,深入研究了模糊系统稳定性分析和稳定化控制问题。
主要工作有以下几个方面:
1.对带有时变时滞的T-S连续时滞模糊系统,考虑以前在推导过程中常被忽略的有用项而可能带来保守性这一问题,构造适当的模糊Lyapunov函数,得到了开环模糊系统时滞相关渐近稳定的充分条件。然后根据并行分布补偿算法,设计了状态反馈控制器,得到了闭环系统时滞相关渐近稳定的充分条件。
2.对于带有不确定的T-S连续时滞模糊系统,基于模糊Lyapunov函数方法,引入带有时变时滞的模糊自由权值矩阵,给出了模糊系统时滞相关鲁棒H∞稳定的新的充分条件。自由权值矩阵的引入,去掉了时滞量导数小于1的约束条件。
3.考虑公共Lyapunov函数方法的保守性,基于模糊Lyapunov函数方法和并行分布补偿算法,分别研究了时滞离散模糊系统的H∞稳定性和带有不确定的时滞离散模糊的鲁棒H∞稳定性。引入松弛变量,以线性矩阵不等式的形式分别给出了系统稳定的充分条件和控制器的设计方法。
4.研究了一类用T-S双线性模型表示的非线性系统的稳定性分析和控制问题。首先,给出了带有不确定多输入模糊双线性系统鲁棒H∞稳定化的充分条件;然后,研究了一类输入和状态都带有时变时滞的模糊双线性系统的时滞相关稳定控制并给出了控制器的设计方法;最后,考虑了基于静态输出反馈控制的模糊系统的稳定控制问题,以LMI形式给出了系统稳定的充分条件。
5.考虑离散模糊双线性系统的稳定控制问题。首先,基于公共Lyapunov函数,给出了多输入离散模糊系统渐近稳定的充分条件;然后,研究了输入和状态都带有时滞的离散模糊系统,采用LMI处理方法,提出了模糊控制器的存在条件及设计方法;最后,基于分段Lyapunov函数及切换模型,同时考虑同一个子空间内不同模糊子系统之间的相互作用,得到了系统渐近稳定的充分条件。
6.在控制器存在加性摄动的情况下,分别研究了带有时滞的连续、离散模糊双线性系统的非脆弱保性能控制。根据并行分布补偿算法,分别设计了模糊非脆弱保性能控制器,使得闭环模糊系统是渐近稳定的,而且对于一个给定的二次型性能指标,保证闭环系统的性能指标不超过给定的性能指标的上界。
7.研究了一类用T-S双线性模型表示的非线性关联大系统的分散鲁棒控制问题。根据Lyapunov稳定性理论和分散控制方法,得到了关联大系统分散鲁棒镇定的充分条件及模糊控制器的设计方法。
最后,对全文进行了概括性总结,并指出了有待进一步研究和完善的问题。
Compared with classical control systems, fuzzy control systems have the following two unmatched advantages. First, it can be easy to realize effectively human control strategies and experience in many applications. Second, it can achieve better control performance in the absence of the accurate mathematic model for the controlled system. In 1985, Takagi T and Sugeno M proposed the Takagi-Sugeno (T-S) fuzzy model, which brings far-researching impact on fuzzy control theory and its application, and makes the stability analysis of fuzzy systems to a new theoretical height. Combining with the Lyapunov stability theory, robust control theory and H∞control theory, using the Linear Matrix Inequality (LMI), this thesis discussed the stability and stabilization problems of fuzzy systems based on T-S linear model and T-S bilinear model in detail, respectively.
The main research works in this thesis can be described as follows:
1. By constructing new appropriate fuzzy Lyapunov function and considering the useful terms (which are ignored in previous methods) when estimating the upper bound of the derivative of Lyapunov function, the delay-dependent stability criteria of open-loop fuzzy systems with time-varying delay are derived. Then, based on parallel distributed compensation (PDC) scheme, the delay-dependent stabilization conditions of the closed-loop systems are derived and the corresponding state feedback controller can be obtained, respectively.
2. Based on fuzzy Lyapunov function and fuzzy free-weighting matrices with time-varying delay, some new sufficient conditions for robust H∞stabilization of uncertain continuous-time fuzzy system with time-varying delay are given. The fuzzy free-weighting matrices are introduced, the purpose of which is to relax the constraint of the derivatives of time-delay.
3. In order to reduce the conservation of the common Lyapunov function,based on fuzzy Lyapunov function and PDC scheme, the H∞stability analysis of fuzzy delay discrete-time systems and robust H∞stability analysis of uncertain fuzzy delay discrete-time systems are studied, respectively. Introducing the relax matrix, the fuzzy controllers design is presented in the form of LMIs, respectively.
4. Stability analysis and synthesis of continuous-time nonlinear systems based on the T-S bilinear model are investigated. By using Lyapunov function and PDC scheme, some sufficient conditions for multiple inputs fuzzy bilinear systems to be robust H∞stable are derived. Then, the delay-dependent stability of fuzzy systems with time-varying delay both in state and input is discussed and an LMI based controller design method is obtained. Finally, based on the static output-feedback controller, some sufficient conditions for the fuzzy bilinear systems to be stable are obtained in the form of LMIs.
5. The stability of discrete-time fuzzy bilinear systems based on the T-S bilinear model is analyzed. By using Lyapunov function, some sufficient conditions for multiple inputs discrete-time fuzzy systems to be stable are derived. Then, the problem of stability analysis and synthesis of fuzzy systems with time-delay both in state and input are discussed. Sufficient conditions are derived for stabilization and are formulated in the form of LMIs. Finally, based on the piecewise Lyapunov function and switching fuzzy model, considered the interactions among the fuzzy subsystems in each subregion, the relaxed stabilization conditions are obtained for the fuzzy bilinear systems.
6. In the presence of the additive controller gain perturbations, the non-fragile guaranteed cost control problems for the continuous-time and discrete-time fuzzy bilinear systems with time-delay in both state and input are considered, respectively. Based on the PDC approach, the non-fragile guaranteed cost state feedback controllers design is obtained, such that the closed-loop systems are asymptotically stable and the closed-loop performance is no more than a certain upper bound of a given quadratic cost function.
7. The problem of decentralized robust control for a nonlinear interconnected system composed by T-S bilinear models with interconnections is considered. Based on the Lyapunov stability analysis theory and decentralized control theory, some sufficient robust stabilization conditions are derived for the whole closed-loop fuzzy interconnected systems.
Finally,some concluding remarks are given, and the future research works are pointed out.
本文分别基于T-S线性模型和T-S双线性模型,根据Lyapunov稳定性理论、鲁棒控制理论和H∞控制理论,结合线性矩阵不等式(LMI)技术,深入研究了模糊系统稳定性分析和稳定化控制问题。
主要工作有以下几个方面:
1.对带有时变时滞的T-S连续时滞模糊系统,考虑以前在推导过程中常被忽略的有用项而可能带来保守性这一问题,构造适当的模糊Lyapunov函数,得到了开环模糊系统时滞相关渐近稳定的充分条件。然后根据并行分布补偿算法,设计了状态反馈控制器,得到了闭环系统时滞相关渐近稳定的充分条件。
2.对于带有不确定的T-S连续时滞模糊系统,基于模糊Lyapunov函数方法,引入带有时变时滞的模糊自由权值矩阵,给出了模糊系统时滞相关鲁棒H∞稳定的新的充分条件。自由权值矩阵的引入,去掉了时滞量导数小于1的约束条件。
3.考虑公共Lyapunov函数方法的保守性,基于模糊Lyapunov函数方法和并行分布补偿算法,分别研究了时滞离散模糊系统的H∞稳定性和带有不确定的时滞离散模糊的鲁棒H∞稳定性。引入松弛变量,以线性矩阵不等式的形式分别给出了系统稳定的充分条件和控制器的设计方法。
4.研究了一类用T-S双线性模型表示的非线性系统的稳定性分析和控制问题。首先,给出了带有不确定多输入模糊双线性系统鲁棒H∞稳定化的充分条件;然后,研究了一类输入和状态都带有时变时滞的模糊双线性系统的时滞相关稳定控制并给出了控制器的设计方法;最后,考虑了基于静态输出反馈控制的模糊系统的稳定控制问题,以LMI形式给出了系统稳定的充分条件。
5.考虑离散模糊双线性系统的稳定控制问题。首先,基于公共Lyapunov函数,给出了多输入离散模糊系统渐近稳定的充分条件;然后,研究了输入和状态都带有时滞的离散模糊系统,采用LMI处理方法,提出了模糊控制器的存在条件及设计方法;最后,基于分段Lyapunov函数及切换模型,同时考虑同一个子空间内不同模糊子系统之间的相互作用,得到了系统渐近稳定的充分条件。
6.在控制器存在加性摄动的情况下,分别研究了带有时滞的连续、离散模糊双线性系统的非脆弱保性能控制。根据并行分布补偿算法,分别设计了模糊非脆弱保性能控制器,使得闭环模糊系统是渐近稳定的,而且对于一个给定的二次型性能指标,保证闭环系统的性能指标不超过给定的性能指标的上界。
7.研究了一类用T-S双线性模型表示的非线性关联大系统的分散鲁棒控制问题。根据Lyapunov稳定性理论和分散控制方法,得到了关联大系统分散鲁棒镇定的充分条件及模糊控制器的设计方法。
最后,对全文进行了概括性总结,并指出了有待进一步研究和完善的问题。
Compared with classical control systems, fuzzy control systems have the following two unmatched advantages. First, it can be easy to realize effectively human control strategies and experience in many applications. Second, it can achieve better control performance in the absence of the accurate mathematic model for the controlled system. In 1985, Takagi T and Sugeno M proposed the Takagi-Sugeno (T-S) fuzzy model, which brings far-researching impact on fuzzy control theory and its application, and makes the stability analysis of fuzzy systems to a new theoretical height. Combining with the Lyapunov stability theory, robust control theory and H∞control theory, using the Linear Matrix Inequality (LMI), this thesis discussed the stability and stabilization problems of fuzzy systems based on T-S linear model and T-S bilinear model in detail, respectively.
The main research works in this thesis can be described as follows:
1. By constructing new appropriate fuzzy Lyapunov function and considering the useful terms (which are ignored in previous methods) when estimating the upper bound of the derivative of Lyapunov function, the delay-dependent stability criteria of open-loop fuzzy systems with time-varying delay are derived. Then, based on parallel distributed compensation (PDC) scheme, the delay-dependent stabilization conditions of the closed-loop systems are derived and the corresponding state feedback controller can be obtained, respectively.
2. Based on fuzzy Lyapunov function and fuzzy free-weighting matrices with time-varying delay, some new sufficient conditions for robust H∞stabilization of uncertain continuous-time fuzzy system with time-varying delay are given. The fuzzy free-weighting matrices are introduced, the purpose of which is to relax the constraint of the derivatives of time-delay.
3. In order to reduce the conservation of the common Lyapunov function,based on fuzzy Lyapunov function and PDC scheme, the H∞stability analysis of fuzzy delay discrete-time systems and robust H∞stability analysis of uncertain fuzzy delay discrete-time systems are studied, respectively. Introducing the relax matrix, the fuzzy controllers design is presented in the form of LMIs, respectively.
4. Stability analysis and synthesis of continuous-time nonlinear systems based on the T-S bilinear model are investigated. By using Lyapunov function and PDC scheme, some sufficient conditions for multiple inputs fuzzy bilinear systems to be robust H∞stable are derived. Then, the delay-dependent stability of fuzzy systems with time-varying delay both in state and input is discussed and an LMI based controller design method is obtained. Finally, based on the static output-feedback controller, some sufficient conditions for the fuzzy bilinear systems to be stable are obtained in the form of LMIs.
5. The stability of discrete-time fuzzy bilinear systems based on the T-S bilinear model is analyzed. By using Lyapunov function, some sufficient conditions for multiple inputs discrete-time fuzzy systems to be stable are derived. Then, the problem of stability analysis and synthesis of fuzzy systems with time-delay both in state and input are discussed. Sufficient conditions are derived for stabilization and are formulated in the form of LMIs. Finally, based on the piecewise Lyapunov function and switching fuzzy model, considered the interactions among the fuzzy subsystems in each subregion, the relaxed stabilization conditions are obtained for the fuzzy bilinear systems.
6. In the presence of the additive controller gain perturbations, the non-fragile guaranteed cost control problems for the continuous-time and discrete-time fuzzy bilinear systems with time-delay in both state and input are considered, respectively. Based on the PDC approach, the non-fragile guaranteed cost state feedback controllers design is obtained, such that the closed-loop systems are asymptotically stable and the closed-loop performance is no more than a certain upper bound of a given quadratic cost function.
7. The problem of decentralized robust control for a nonlinear interconnected system composed by T-S bilinear models with interconnections is considered. Based on the Lyapunov stability analysis theory and decentralized control theory, some sufficient robust stabilization conditions are derived for the whole closed-loop fuzzy interconnected systems.
Finally,some concluding remarks are given, and the future research works are pointed out.
引文
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