基于T-S模型的非线性系统模糊控制器设计及应用
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摘要
随着科学技术的迅猛发展,传统控制理论已很难解决复杂非线性系统的建模、分析和设计问题。而模糊控制技术由于具有控制器设计简单、适用于许多非线性系统以及鲁棒性强等特点,自上世纪80年代以来,在控制理论和工程实践方面获得了很大的进展。但是由于模糊控制系统本质上的非线性,使得其稳定性分析与性能设计尚缺乏严格的理论基础,因此难以形成系统化的综合方法。1985年,日本学者Takagi和Sugeno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,使模糊控制系统的稳定性分析上升到一个新的理论高度。特别是近年来与并行分布补偿(PDC)算法以及线性矩阵不等式(LMI)技术的结合,使其成为非线性系统稳定性分析和控制器设计的一种有效处理方法。
本文的研究以非线性系统为主,同时考虑不确定性、时滞、随机跳变等因素的影响。基于Lyapunov稳定性理论,采用并行分布补偿算法,研究了一类非线性系统的模糊控制问题,内容涉及滤波器设计、保性能控制、区域极点配置、非脆弱控制、约束控制、鲁棒性能分析以及鲁棒镇定等方面。所得结果均可统一到一组线性矩阵不等式的处理框架中,由于内点算法的提出,使得求解十分方便。
本文的主要工作包括以下几个方面:
(1)在外部干扰和测量噪声统计特性未知的情况下,为连续非线性系统提供一种新的H_∞滤波器设计方法。首先利用T-S模糊模型对该系统进行建模,然后通过引入一个附加矩阵,解除Lyapunov矩阵和系统矩阵的耦合关系,获得具有较小保守性的结果。最后通过仿真验证方法的可行性。
(2)从两方面改进了一类不确定非线性系统的模糊保性能控制器设计方法。第一部分利用分段光滑Lyapunov函数的思想,导出保证系统鲁棒稳定的保性能控制律的存在条件。所得结果避免了寻找公共Lyapunov矩阵解的困难,具有较小的保守性。第二部分基于扩展稳定性条件和等价投影定理,并结合系统响应速度的考虑,给出系统以衰减率α鲁棒渐近稳定的充分条件及具有衰减率α的保性能控制律设计方法。
(3)针对一类具有范数有界时变参数不确定性的非线性系统,采用T-S模糊控制方法并结合闭环系统动态性能的考虑,分别给出了连续情形和离散情形下,系统具有圆盘极点约束的鲁棒保性能控制问题。结果表明,所设计的控制器不仅使系统具有良好的稳态性能,同时也获得了满意的过渡过程特性。
(4)讨论了非线性时滞系统的鲁棒性能分析和综合问题。首先,在放宽稳定性条件的基础上,研究一类同时具有状态时滞和控制时滞的不确定离散非线性系统的鲁棒H_∞保性能控制问题。其次,考虑在控制器存在可加性摄动时,离散非线性时滞系统的非脆弱LQ /H_∞控制器设计方法。所设计的控制器不仅能够保证闭环模糊系统的时滞依赖稳定性,而且还能使系统达到一定的H_∞干扰抑制水平。最后,通过构造一个适当的模糊Lyapunov函数,并结合自由权矩阵的方法,给出一种新的时滞依赖稳定性判据。该判据充分考虑了时滞上下界的信息,并且取消了对时滞导数小于1的限制,比以往结果具有更低的保守性。
(5)从约束控制、H_∞控制、鲁棒控制等几方面入手,采用T-S模糊模型,对非线性Markov跳变系统的稳定性分析和控制器设计作了初步的探讨。基于随机Lyapunov稳定性理论,并采用状态反馈形式,给出跳变系统相应的模糊控制器设计方法。与一般的状态反馈控制器相比,这里设计的控制器增益都是依赖于Markov跳变模态的。
最后对全文进行概括总结,并指出了有待进一步研究和完善的问题。
With the rapid development of science technology, the problem of modeling, analyzing and designing for complex nonlinear systems is very hard to be solved by traditional control theory. Since the eighties of the twentieth century, fuzzy control technology has obtained great evolution in control theory and engineering practice becaue that it is easier to design controller, suitable for applying in many nonlinear control systems and has stronger robust characteristic. However, due to the essential nonlinearity of fuzzy control system, the stability analysis and performance design for fuzzy control system are still short of rigorous theory foundation. As a result, the systematic synthesis approach is difficult to be formed. In 1985, the Takagi-Sugeno (T-S) fuzzy model proposed by Takagi and Sugeno brings far-reaching impact for fuzzy control theory and its applications, and it also makes stability analysis of fuzzy control system to a new theoretical height. In particular, under the framework of parallel distributed compensation (PDC) scheme and linear matrix inequality (LMI) technique, an effective design methodology has been developed to study the stability analysis and controller design for nonlinear systems.
The work of this dissertation mainly investigates nonlinear systems, including those systems subject to uncertainty, time delay and stochastic jumps. Based on the Lyapunov stability theory, some fuzzy control problems for a class of nonlinear systems are studied by applying the parallel distributed compensation scheme. The research contents relate to filter design, guaranteed cost control, regional pole assignment, non-fragile control, constraints control, robust performance analysis and robust stabilization and so on. All obtained results can be reduced to a feasible problem of linear matrix inequalities. Due to the interior-point algorithm, the solutions of LMIs are solved very easily.
The main research works of this dissertation are as follows:
(1) Under the assumptions that the statistical characteristic of the external disturbance and the measurement noise is unknown, a new design method of H_∞filter for continuous-time nonlinear system is provided. At first, the T-S fuzzy model is utilized to approximate the controlled plant. Then, by introducing an additional slack variable, a less conservative result is obtained, in which the Lyapunov matrix and system matrices are decoupled. Finally, a simulation example is verified the feasibility of the proposed method.
(2) The approach of fuzzy guaranteed cost controller design is improved for a class of uncertain nonlinear systems from two aspects. In the first part, by using the idea of piecewise Lyapunov function, the existing condition of guaranteed cost control law which ensures the robust stability of systems is derived. The obtained results avoid the difficulty for seeking a common Lyapunov matrix, so it has less conservativeness. In the second part, based on the relaxed stability condition and reciprocal projection lemma, as well as combining with the response rate of system, the design approach of guaranteed cost control law with decay rateαis given, which guarantees that the closed-loop uncertain fuzzy system is robust asymptotic stability.
(3) For a class of nonlinear systems in the presence of norm-bounded and time-varying parameter uncertainties, the problem of robust guaranteed cost control with disk pole constraints is investigated. By applying T-S fuzzy control approach and considering the dynamic performance of the closed-loop systems, the controller design method is given both for continuous-time and discrete-time case. The results show that the designed controller can make the system achieve better stable behavior, and obtain the satisfied transient performance at the same time.
(4) The problem of robust performance analysis and synthesis for nonlinear time delay systems is discussed. Firstly, on the basis of the relaxed stability condition, the problem of robust H_∞guaranteed cost control is considered for a class of uncertain discrete-time nonlinear systems with bounded time delays in both state and control input. Secondly, in the presence of the additive controller gain perturbations, a design procedure of non-fragile LQ /H_∞controller is provided for discrete-time nonlinear systems. The designed controller can not only guarantee the delay-dependent stability of the closed-loop fuzzy system, but also make the system achieve a H_∞disturbance attenuation level. Finally, by constructing an appropriate type of fuzzy weighting-dependent Lyapunov functional and adopting the free-weighting matrix method, a new delay-dependent stability criterion is derived for nonlinear systems with interval time-varying delay. This criterion fully considers the time delay information with its lower bound and upper bound. Furthermore, the limit that the derivative of time-varying delay must be smaller than one is not required. Therefore, the obtained results are less conservative than the existing ones.
(5) From the aspect of constraints control, H_∞control and robust control, the stability analysis and controller design are primarily discussed for nonlinear systems with Markovian jumping parameters by applying T-S fuzzy models. Based on the stochastic Lyapunov stability theory, the corresponding fuzzy controller design methods are given in the form of state feedback. Different from the generic state feedback controller, the controller gains designed here are all depend on the Markovian jump modes.
Finally, the conclusion of this paper is given. Upon this conclusion, some further research work and existing issues will be pointed out.
本文的研究以非线性系统为主,同时考虑不确定性、时滞、随机跳变等因素的影响。基于Lyapunov稳定性理论,采用并行分布补偿算法,研究了一类非线性系统的模糊控制问题,内容涉及滤波器设计、保性能控制、区域极点配置、非脆弱控制、约束控制、鲁棒性能分析以及鲁棒镇定等方面。所得结果均可统一到一组线性矩阵不等式的处理框架中,由于内点算法的提出,使得求解十分方便。
本文的主要工作包括以下几个方面:
(1)在外部干扰和测量噪声统计特性未知的情况下,为连续非线性系统提供一种新的H_∞滤波器设计方法。首先利用T-S模糊模型对该系统进行建模,然后通过引入一个附加矩阵,解除Lyapunov矩阵和系统矩阵的耦合关系,获得具有较小保守性的结果。最后通过仿真验证方法的可行性。
(2)从两方面改进了一类不确定非线性系统的模糊保性能控制器设计方法。第一部分利用分段光滑Lyapunov函数的思想,导出保证系统鲁棒稳定的保性能控制律的存在条件。所得结果避免了寻找公共Lyapunov矩阵解的困难,具有较小的保守性。第二部分基于扩展稳定性条件和等价投影定理,并结合系统响应速度的考虑,给出系统以衰减率α鲁棒渐近稳定的充分条件及具有衰减率α的保性能控制律设计方法。
(3)针对一类具有范数有界时变参数不确定性的非线性系统,采用T-S模糊控制方法并结合闭环系统动态性能的考虑,分别给出了连续情形和离散情形下,系统具有圆盘极点约束的鲁棒保性能控制问题。结果表明,所设计的控制器不仅使系统具有良好的稳态性能,同时也获得了满意的过渡过程特性。
(4)讨论了非线性时滞系统的鲁棒性能分析和综合问题。首先,在放宽稳定性条件的基础上,研究一类同时具有状态时滞和控制时滞的不确定离散非线性系统的鲁棒H_∞保性能控制问题。其次,考虑在控制器存在可加性摄动时,离散非线性时滞系统的非脆弱LQ /H_∞控制器设计方法。所设计的控制器不仅能够保证闭环模糊系统的时滞依赖稳定性,而且还能使系统达到一定的H_∞干扰抑制水平。最后,通过构造一个适当的模糊Lyapunov函数,并结合自由权矩阵的方法,给出一种新的时滞依赖稳定性判据。该判据充分考虑了时滞上下界的信息,并且取消了对时滞导数小于1的限制,比以往结果具有更低的保守性。
(5)从约束控制、H_∞控制、鲁棒控制等几方面入手,采用T-S模糊模型,对非线性Markov跳变系统的稳定性分析和控制器设计作了初步的探讨。基于随机Lyapunov稳定性理论,并采用状态反馈形式,给出跳变系统相应的模糊控制器设计方法。与一般的状态反馈控制器相比,这里设计的控制器增益都是依赖于Markov跳变模态的。
最后对全文进行概括总结,并指出了有待进一步研究和完善的问题。
With the rapid development of science technology, the problem of modeling, analyzing and designing for complex nonlinear systems is very hard to be solved by traditional control theory. Since the eighties of the twentieth century, fuzzy control technology has obtained great evolution in control theory and engineering practice becaue that it is easier to design controller, suitable for applying in many nonlinear control systems and has stronger robust characteristic. However, due to the essential nonlinearity of fuzzy control system, the stability analysis and performance design for fuzzy control system are still short of rigorous theory foundation. As a result, the systematic synthesis approach is difficult to be formed. In 1985, the Takagi-Sugeno (T-S) fuzzy model proposed by Takagi and Sugeno brings far-reaching impact for fuzzy control theory and its applications, and it also makes stability analysis of fuzzy control system to a new theoretical height. In particular, under the framework of parallel distributed compensation (PDC) scheme and linear matrix inequality (LMI) technique, an effective design methodology has been developed to study the stability analysis and controller design for nonlinear systems.
The work of this dissertation mainly investigates nonlinear systems, including those systems subject to uncertainty, time delay and stochastic jumps. Based on the Lyapunov stability theory, some fuzzy control problems for a class of nonlinear systems are studied by applying the parallel distributed compensation scheme. The research contents relate to filter design, guaranteed cost control, regional pole assignment, non-fragile control, constraints control, robust performance analysis and robust stabilization and so on. All obtained results can be reduced to a feasible problem of linear matrix inequalities. Due to the interior-point algorithm, the solutions of LMIs are solved very easily.
The main research works of this dissertation are as follows:
(1) Under the assumptions that the statistical characteristic of the external disturbance and the measurement noise is unknown, a new design method of H_∞filter for continuous-time nonlinear system is provided. At first, the T-S fuzzy model is utilized to approximate the controlled plant. Then, by introducing an additional slack variable, a less conservative result is obtained, in which the Lyapunov matrix and system matrices are decoupled. Finally, a simulation example is verified the feasibility of the proposed method.
(2) The approach of fuzzy guaranteed cost controller design is improved for a class of uncertain nonlinear systems from two aspects. In the first part, by using the idea of piecewise Lyapunov function, the existing condition of guaranteed cost control law which ensures the robust stability of systems is derived. The obtained results avoid the difficulty for seeking a common Lyapunov matrix, so it has less conservativeness. In the second part, based on the relaxed stability condition and reciprocal projection lemma, as well as combining with the response rate of system, the design approach of guaranteed cost control law with decay rateαis given, which guarantees that the closed-loop uncertain fuzzy system is robust asymptotic stability.
(3) For a class of nonlinear systems in the presence of norm-bounded and time-varying parameter uncertainties, the problem of robust guaranteed cost control with disk pole constraints is investigated. By applying T-S fuzzy control approach and considering the dynamic performance of the closed-loop systems, the controller design method is given both for continuous-time and discrete-time case. The results show that the designed controller can make the system achieve better stable behavior, and obtain the satisfied transient performance at the same time.
(4) The problem of robust performance analysis and synthesis for nonlinear time delay systems is discussed. Firstly, on the basis of the relaxed stability condition, the problem of robust H_∞guaranteed cost control is considered for a class of uncertain discrete-time nonlinear systems with bounded time delays in both state and control input. Secondly, in the presence of the additive controller gain perturbations, a design procedure of non-fragile LQ /H_∞controller is provided for discrete-time nonlinear systems. The designed controller can not only guarantee the delay-dependent stability of the closed-loop fuzzy system, but also make the system achieve a H_∞disturbance attenuation level. Finally, by constructing an appropriate type of fuzzy weighting-dependent Lyapunov functional and adopting the free-weighting matrix method, a new delay-dependent stability criterion is derived for nonlinear systems with interval time-varying delay. This criterion fully considers the time delay information with its lower bound and upper bound. Furthermore, the limit that the derivative of time-varying delay must be smaller than one is not required. Therefore, the obtained results are less conservative than the existing ones.
(5) From the aspect of constraints control, H_∞control and robust control, the stability analysis and controller design are primarily discussed for nonlinear systems with Markovian jumping parameters by applying T-S fuzzy models. Based on the stochastic Lyapunov stability theory, the corresponding fuzzy controller design methods are given in the form of state feedback. Different from the generic state feedback controller, the controller gains designed here are all depend on the Markovian jump modes.
Finally, the conclusion of this paper is given. Upon this conclusion, some further research work and existing issues will be pointed out.
引文
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