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不确定时滞系统与非线性系统的自适应控制
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摘要
本文主要基于Lyapunov稳定性理论,以线性矩阵不等式(LMI)和最小二乘支撑向量机(LSSVM)为主要工具,研究了不确定时滞系统满足不同设计要求的各种鲁棒自适应控制问题和一类仿射非线性离散系统的自适应控制问题。全文主要有以下几个部分组成。
     第一部分:介绍时滞系统自适应控制的发展趋势和研究概况。在研究中所涉及到的一些主要数学方法和工具。然后提出本文研究的一些主要问题及所需的一些准备知识。
     第二部分:研究了两类不确定时变时滞系统的鲁棒自适应控制问题。首先,针对一类多变时滞的不确定系统,系统的时滞不确定性不要求满足匹配条件且上界未知,通过求解一个线性矩阵不等式和基于不确定性上界的估计值设计了一种鲁棒自适应状态反馈控制器,并证明了此控制器使得闭环系统和参数估计误差最终一致有界。最后,我们研究了一类含匹配变时滞状态扰动非线性系统的鲁棒自适应镇定问题。当系统的变时滞扰动的上界函数未知但满足所谓的线性增长条件,通过求解一个Hamilton-Jacobi不等式和Lyapunov-Krasovskii型泛函设计出了一种自适应鲁棒状态反馈控制器,并证明了此控制器使得闭环系统和参数估计误差一致最终有界且系统的状态一致渐近趋于零。
     第三部分:研究了一类同时包含匹配和不匹配不确定性时滞互联系统的分散自适应控制问题。首先针对一类含未知常时滞的不确定互联系统,假定互联项满足匹配和线性增长条件,但增益未知,同时假定匹配不确定性和扰动是有界的,但上界未知。通过采用自适应律估计这些未知的界并结合线性矩阵不等式方法设计了一种分散鲁棒自适应控制器,基于Lyapunov稳定性理论和Lyapunov-Krasovskii型泛函证明了此控制器使得闭环系统的状态最终一致趋于零。然后,我们将上述结论进一步推广到时变时滞情形,对变时滞满足不同的条件分别设计了相应的分散鲁棒自适应控制器,并且也得到了满意的结果。
     第四部分:研究了不确定时变时滞系统的自适应输出反馈控制问题。首先,研究了一类含变时滞扰动不确定系统的输出反馈自适应稳定控制问题。假定不确定性满足所谓的匹配条件且标称系统的传递函数矩阵是严格反馈正实的。基于Lyapunov稳定性理论和Lyapunov-Krasovskii型泛函给出了两种自适应输出反馈控制器设计方法,并采用不同的自适应律对系统的不确定性上界进行在线估计,证明了所设计的两种控制器均能使闭环系统最终一致有界且其中一种还能进一步保证系统状态一致渐近趋于零。其次,针对一类含不确定参数和多时变时滞互联项的大系统,在不确定参数矩阵上界未知的情况下,基于Lyapunov稳定性理论设计出了一种分散自适应输出反馈控制器,并证明了此控制器使得闭环系统全局指数一致收敛到一个有界球。
     第五部分:研究了变时滞系统的自适应滑动模控制问题。首先,我们研究了一类含不匹配不确定性和匹配外界扰动时变时滞系统的自适应滑动模控制方法。基于线性矩阵不等式给出了滑动平面存在的充分条件,并采用自适应控制策略估计扰动的未知上界从而使得闭环系统是全局渐近稳定的。在此基础上,还研究了执行器具有饱和特性的情形下,滑模自适应控制器的设计方法。该方法不仅取消了饱和界已知的假设,而且在考虑了各种不确定性和扰动的情况下,给出了滑动模态存在性和可达性的条件,从而保证了闭环系统的渐近稳定性。其次,针对一类含外界干扰和变时滞互联项的大系统,提出了一种基于Lyapunov稳定性理论的分散自适应滑动模跟踪控制策略,通过引入积分滑动模和能在线估计不确定扰动与时滞关联界的自适应算法,实现了对参考模型的跟踪控制。
     第六部分:针对一类仿射非线性离散系统,提出了一种基于最小二乘支撑向量机的自适应控制策略。同神经网络方法相比,具有一个隐层以上的回归支撑向量机除了可以逼近紧致集上任意非线性函数外,还具有自动确定模型复杂度的能力和较高的泛化能力。该方法最终归结为求解一个线性方程组。利用矩阵计算的一些技巧引入了迭代算法,从而避免矩阵的求逆运算。同时,当样本数据增加较多,通过引入遗忘机制来降低在线运算的计算复杂性,提高运算速度。
     本文对主要设计方案均进行了仿真研究,仿真结果表明,本文对不同类型的不确定时滞系统给出相应的自适应控制方案均能获得良好的控制效果。
Based on the Lyapunov stability theory,this dissertation is mainly on the design of the robust adaptive controllers for uncertain time delay systems subject to different assumptions by using linear matrix inequalities(LMIs),and the design of the adaptive control of a class of nonlinear discrete-time systems by the use of least square support vector machine algorithm (LSSVM).The main work of this dissertation consists of six parts as follows:
     In part one,the research development and general situation about the adaptive control for time delay systems are discussed first,and then the main mathematical tools and methods, which will be used in this dissertation,are stated.Furthermore,the main problems,which are studied in this dissertation,are introduced.Finally,some necessary preliminaries in this dissertation are given.
     In part two,the problem of robust adaptive controller design for two classes of uncertain time varying delay systems is studied.First,the problem of robust adaptive stabilization for a class of multiple time-delay uncertain systems is discussed.The time-delay uncertainties are assumed to satisfy the mismatching conditions and the values of its upper bounds are unknown.By using of the LMI and Lyapunov-Krosovskii functional we propose a memoryless adaptive state feedback controller,which can guarantee the closed-loop system is globally stable in the sense of uniform ultimate boundedness.Finally,the problem of adaptive stabilization for a class of nonlinear systems including matching time-varying delayed disturbance is discussed.The bound of the time-varying delayed state disturbances are unknown and are assumed to satisfy the linear growth conditions.By using of the Lyapunov stability theory and Lyapunov-Krosovskii functional we propose a robust adaptive state feedback controller,which can guarantee the closed-loop system is globally stable in the sense of uniform ultimate boundedness and the state trajectories are uniformly asymptotically to zero.
     In part three,the problem of decentralized robust adaptive control is considered for a class of uncertain large-scale time-delay systems in the presence of mismatched and matched uncertainties.The constant time delay is considered first.The interconnections are assumed to be bounded by a linear function of delayed states with unknown gains.The upper bounds of the matching uncertainties and perturbations are also assumed to be unknown.The adaptation laws are proposed to estimate such unknown bounds,and by making use of the LMI method,a class of decentralized robust adaptive controllers is constructed.Based on the Lyapunov stability theory and Lyapunov-Krasovskii functional,it is shown that the state trajectories of the large-scale systems are uniformly asymptotically to zero.Then the above results are extended to the situation with time varying delay.Different decentralized adaptive controllers are designed for time varying delay subject to different conditions.Satisfactory results are also achieved.
     In part four,the problem of adaptive output feedback stabilization control for uncertain time varying delay systems is studied.A class of uncertain dynamic system including time-varying delayed perturbations is discussed first.The uncertainties are assumed to satisfy the so-called matching conditions and the transfer function matrix of the nominal system is strictly feedback positive real.By using of the Lyapunov stability theory and Lyapunov-Krosovskii functional we propose two different output feedback adaptive controllers and two different adaptation laws to estimate the unknown upper bounds of the uncertainties of the system,which all can guarantee the closed-loop system is globally stable in the sense of uniform ultimate boundedness.In addition,one of them can guarantee the state trajectories of the system are uniformly asymptotically to zero.Then the problem of decentralized adaptive output feedback stabilization for a class of large-scale systems subject to uncertain parameters and multiple time-varying delays in the interconnections is studied. The interconnections are assumed to satisfy the matching conditions and the nominal system of each subsystems is strictly feedback positive real.By estimating the unknown bounds of the uncertain parameter matrices we propose an decentralized output feedback adaptive controller, which can guarantee the closed-loop system to converge,globally,uniformly and exponentially,to a bounded ball.
     In part five,the problem of adaptive sliding mode control for time varying delay systems is studied.A class of time varying delay systems with mismatched uncertainties and matched external perturbations is investigated first.Based on the Lyapunov theory,a sufficient condition derived in terms of LMI is given to guarantee the existence of the sliding surface. The adaptive control is used to overcome the unknown upper bound of perturbations.The globally asymptotic stability is also achieved for the proposed control methodology.Then the adaptive sliding mode control scheme with saturation actuator is considered.The approach not only removes the assumption that the bounds of the saturation are known,but also give the existence and reachable conditions of the sliding-mode.By theoretical analysis,it is shown that the asymptotical stability of the closed-loop system is guaranteed under the saturation input.Finally,the problem of robust tracking and model following is considered for a class of linear large-scale systems subject to time varying delay interconnections and external disturbances.Based on the Lyapunov stability theory,a decentralized adaptive sliding mode control scheme is proposed.An adaptation algorithm that can adapt the unknown upper bounds of the uncertainties and time varying delay interconnections is introduced as well as integral sliding mode,so that the tracking error decrease asymptotically to zero and reference model following is achieved.
     Finally,we introduce the use of recurrent least square support vector machine algorithm for the adaptive control of a class of nonlinear discrete-time systems.Comparing to neural networks,the regression support vector machines with one or more hidden layer not only can approximate any nonlinear function on the compact set but also has well generalization ability and the ability to adaptively determine the complexity of the model.The procedure is finally comes down to solve a set of linear equations in an iterative way.Advantage of the newly designed algorithm is that the computation of inverse matrix is avoided.The curse of dimensionality is also avoided by using the finite time window.
     In this dissertation,simulations are made for major design schemes.Simulation results also show the effectiveness of the proposed approaches.
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