【作者】 鲍志勇；

【导师】 彭勇；

【作者基本信息】 燕山大学 ， 仪器科学与技术， 2021， 博士

【摘要】 多项式模糊模型对非线性系统进行建模时,部分非线性项以多项式的形式在模型中得以保留,从而可以描述更精确和更宽泛的复杂非线性系统。多项式模糊控制系统中的隶属函数描述了系统重要的非线性信息,在稳定性分析中缺乏对隶属函数的分析不可避免的会使得稳定分析结果保守。本文以多项式模糊系统为研究对象,基于非并行分布补偿概念设计状态反馈、静态输出等多项式模糊控制器,研究了融合隶属函数的稳定性分析、降低多项式模糊切换系统驻留时间边界下限、时滞离散多项式模糊正系统稳定性及正性分析等问题。本文主要研究工作如下:首先研究多项式模糊控制系统下的融合隶属函数技术。在非并行分布补偿概念下设计状态反馈和静态输出反馈多项式模糊控制器。在稳定性分析层面,分别构建了Chebyshev隶属函数和Lagrange隶属函数以降低稳定条件的保守性。对于Chebyshev隶属函数,从逼近误差范数的角度证明了Chebyshev隶属函数相对现有逼近隶属函数具有更小的逼近误差。在划分的子域下设计了分段Chebyshev隶属函数并与前件变量的边界信息相结合构成新的融合隶属函数技术,以获得放松的稳定性分析结果。对于Lagrange隶属函数,运用Lagrange插值理论将插值基函数与采样点的隶属度进行加权和操作,并以多项式项的形式融合到稳定条件中。从而解决了分段线性、Taylor级数隶属函数的插值点必须为系统变量两端端点的限制和Chebyshev隶属函数基函数的选择问题。其次研究多项式模糊切换系统稳定性分析问题。通过多项式模糊模型描述更广泛的切换非线性系统的动力学特征,并基于非并行分布补偿概念设计切换多项式模糊控制器。通过融合隶属函数技术放松了稳定条件的保守性,并减小了切换系统驻留时间的边界下限。另外,提出多重交换算法以解决多变量Chebyshev隶属函数的算法实现问题。最后研究时滞离散多项式模糊正系统的稳定性和正性条件保守问题。将时滞离散正非线性系统的动力学特征通过多项式模糊模型进行描述,并基于非并行分布补偿设计概念设计多项式模糊控制器。在稳定性和正性分析方面,设计线性余正Lyapunov函数并将隶属函数关系约束信息和隶属函数固有性质信息通过松弛矩阵融合到稳定条件中,以降低稳定条件的保守性。进而,综合利用逼近隶属函数、逼近误差、逼近隶属函数上下边界、前件变量边界等信息,以进一步有效降低稳定分析结果的保守性。更多还原

【Abstract】 The polynomial fuzzy model can make the nonlinear term be preserved in the model in the form of polynomial,which can describe the more accurate and wider complex nonlinear systems.The membership functions are the essential nonlinear information in the polynomial fuzzy systems.Therefore,the lack of analysis of the membership functions in the stability analysis will inevitably make the resulting conservative.This thesis takes polynomial fuzzy-model-based systems as the research object,designs state feedback,static output polynomial fuzzy controllers under the concept of non-parallel distribution compensation.The membership-function-dependent stability analysis,the reduction of the lower bounds of the dwell time of polynomial fuzzy-model-based switched systems,and the problem of stability and positivity analysis of discrete polynomial fuzzy-model-based systems with time delay are investigated.The main research includes as following:Firstly,this thesis derives the membership-functiond-dependent stability analysis of polynomial fuzzy-model-based systems with membership functions.The state feedback and static output feedback polynomial fuzzy controllers are designed under the concept of non-parallel distribution compensation.For the stability analysis,the Chebyshev membership functions and Lagrange membership functions are derived to reduce the conservativeness of the stability conditions.For the Chebyshev membership functions,from the perspective of the approximation error’s norm,it is proved that the Chebyshev membership functions have a minor approximation error compared with the existing approximated membership functions.Design a piecewise Chebyshev membership function under the divided subdomains and combine it with the boundary information of the premise variables to form a new membership-function-dependent technology to obtain a relaxed stability analysis result.On the other hand,for the Lagrange membership functions,the Lagrange interpolation theory is used to carry out the weighted sum operation of the interpolation basis function and the membership grade of the sample points and take it into the stability conditions in the form of a polynomial term.This solves the problem of the restriction that the interpolation points of the piecewise linear/Taylor series membership functions must be two endpoints of the system states and the choice of the basis functions of the Chebyshev membership functions.Secondly,the stability analysis of the polynomial fuzzy-model-based switched systems is studied.A polynomial fuzzy model is used to model a wider switched nonlinear system.Based on the concept of non-parallel distribution compensation,a switching polynomial fuzzy controller is derived.The conservativeness of the stability conditions is relaxed by the membership-function-dependent technology,and the lower bounds of the dwell time of the switched systems are reduced.In addition,a multiple exchange algorithm is proposed to solve the algorithm implementation problem of the multivariable Chebyshev membership functions.Finally,the stability and positivity of the discrete polynomial fuzzy-model-based positive system with time delay are studied.The dynamic of the discrete positive nonlinear system with time delay is described by a polynomial fuzzy model,and a polynomial fuzzy controller is designed based on the concept of non-parallel distribution compensation design.In terms of stability and positivity analysis,a linear co-positive Lyapunov function is designed.The membership function relation constraint information and the inherent property information of the membership functions are merged into the stability conditions through the slack matrix to reduce the conservativeness.Furthermore,information such as the approximation of the membership functions,the approximation error,the upper and lower boundaries of the membership functions,and the boundaries of the premise variables are comprehensively used to further effectively reduce the conservativeness of the stability analysis results.更多还原