时滞及非线性系统最优输出跟踪控制研究
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摘要
本文分别研究了非线性系统、时滞非线性时滞系统、非线性互联大系统和一类仿射非线性系统的最优输出跟踪控制问题，讨论了控制律的物理可实现问题。本文的主要研究内容可概括如下。
1．首先综述了与研究背景相关的时滞系统、非线性系统的特点及研究方法，然后对最优控制问题与跟踪问题的相互关系和发展状况进行了介绍，在此基础上对时滞及非线性系统最优输出跟踪控制的主要研究方法和最新动态进行了系统分析。
2．研究参考输入动态特性由外系统给出的非线性系统的最优输出跟踪问题。针对利用极大值原理研究时滞非线性系统最优输出跟踪控制问题时导出的两点边值问题，运用灵敏度参数方法进行转化，将求解最优输出跟踪控制的既含有超前项又含有时滞项的原两点边值问题问题化为一族等价的不含超前项和时滞项的两点边值问题。这族两点边值问题可通过迭代方法依次求得各阶两点边值问题的解，从而获得原两点边值问题的等价解。引入了降维状态观测器，给出了物理可实现的最优输出跟踪控制律，并利用仿真研究验证了方法的有效性。
3．研究了非线性系统的最优输出跟踪控制问题。对具有一般形式的非线性系统在零点处进行级数展开，使之成为一阶线性项和高阶非线性项分离的形式。利用灵敏度法，对最优跟踪控制问题所导致的非线性两点边值问题进行级数展开，将原问题转化为一族包含已知非线性级数展开项的线性两点边值问题。详细研究了复合函数的高阶导数，给出了非线性项的各阶级数展开项的算法，代入并逐阶求解就得到由线性反馈和补偿项组成的最优跟踪控制律。并利用仿真研究验证了方法的有效性。
4．研究了含有非线性互联耦合项的动态大系统最优跟踪控制问题，提出一种次优控制律的设计方法。在系统中引入灵敏度参数ε，关于灵敏度参数ε在ε=0处展开Maclaurin级数。将求解高阶耦合的非线性两点边值问题简化为求解一族解耦的线性两点边值问题序列。该族线性微分方程问题的解，构成含有非线性互联耦合项的动态大系统最优跟踪控制问题非线性共态向量序列，最优控制律由线性反馈项和非线性共态向量的和组成。在最优控制律中截取共态向量和的前有限项，导出了一种次优控制律的设计算法。最后对其进行了仿真研究。
5．研究了一类仿射非线性系统的最优控制问题，改进了唐等提出的逐次逼近法。根据最优控制理论，这类仿射非线性系统最优控制问题的必要条件为无法解析求解的非线性两点边值问题。通过引入非线性微分方程近似求解的逐次逼近法，将非线性两点边值问题转化为一组线性两点边值问题序列。在构造线性两点边值问题序列时，采用了齐次的构造技巧。迭代求解所构造的齐次线性两点边值问题序列，设计出了具有线性闭环反馈形式的最优控制律。截取有限的迭代结果，设计出一种近似的最优控制律，给出了近似最优控制律的设计算法。并通过实例仿真验证了方法的有效性。
6．文章最后总结了论文的主要工作，并对下一步的研究方向进行了展望。
The dissertation studies the OOT problem for nonlinear systems, time-delay andnonlinear systems, nonlinear interconnected large-scale systems and a class of affinenonlinear systems, and then dicusses the optimal control law is physically realizable.The main contents are given as follows:
1. Firstly, the characteristics and research methods of time-delayed systems andnonlinear systems which are relative to research objects have been summarized. Nextthe relationship and development of optimal control problem and tracking problemhave been introduced. Then based on the prepareed work, the relative studies on theOOT problem for time-delay and nonlinear systems up to now and the main methodsare given in detail.
2. The OOT problem of the nonlinear systems whose reference input is generallyproduced by an exosystem is studied. With the Maximum Value Principle, thetwo-point boundary value (TPBV) problems for the OOT of the time-delay andnonlinear systems have been obtained. By introducing a sensitivity parameter, theoriginal TPBV problem for solving the OOT control with delay and advance terms istransformed into a series of TPBV problems without delay or advance terms. Then bysolving the two-point boundary value problem sequence recursively, TPBV isobtained. A reference input observer is introduced such that the optimal control law isphysically realizable. Simulation examples are employed to test the validity of thepresented sensitivity algorithm.
3. The OOT for nonlinear systems is considered. The nonlinear systems ingeneral form are expanded at zero and become to a form in which the one-order linearterms separated from the high-order nonlinear terms. Using a sensitivity parameter approach, and expanding the nonlinear two-point boundary value (TPBV) problemsled by the OOT control problem with respect to the sensitivity parameter, the originalnonlinear TPBV problem is transformed into a series of linear TPBV problemscontaining known low-order nonlinear terms. Algorithm of nonlinear expanding termsis obtained and substituting them into the recursion formula of adjoint vectorequations, the OOT control law consisting of linear terms and a nonlinearcompensation term can be approximately obtained. A simulation example is employedto test the validity of the presented algorithm.
4. Considering the optimal output tracking (OOT) problem for nonlinearinterconnected large-scale dynamic systems, an approach for designing suboptimalcontrol law is proposed. By introducing a sensitivity parameterε,εis expanded atzero. By using the approach, the high order, coupling, nonlinear two-point boundaryvalue (TPBV) problem is transformed into a sequence of linear decoupling TPBVproblems. By solving this sequence of linear differential equations, costate vectorsequence of nonlinear interconnected, coupling large-scale dynamic systems isobtained. Then optimal control law consists of linear feedback term and nonlinear costatevector sum. By intercepting the frontal finite terms of costate vector sum, we obtaineda suboptimal control law and its algorithm. A simulation example illustrates theefficiency of the optimal output tracking control law designed.
5. This paper considers an optimal control for a class of affine nonlinear systems. Animproved successive approximation approach is proposed. By introducing the SuccessiveApproximation Approach of the nonlinear differential equation theory, a linear two-pointboundary value problem sequence is constructed to approximate the nonlinear two-point boundaryvalue problem. A technique is employed to make sure the constructed linear two-point boundaryvalue problems of the approximating sequence homogeneous, Iteratively solving thisapproximating sequence, optimal control law of linear closed-loop feedback form is designed. Analgorithm is presented to design certain approximate optimal control law by truncating thefinite iteration. An illustrative example shows the validity of the algorithm
6. Finally, the main work in this dissertation is summarized and a proposition isindicated on the research work in the future.
1．首先综述了与研究背景相关的时滞系统、非线性系统的特点及研究方法，然后对最优控制问题与跟踪问题的相互关系和发展状况进行了介绍，在此基础上对时滞及非线性系统最优输出跟踪控制的主要研究方法和最新动态进行了系统分析。
2．研究参考输入动态特性由外系统给出的非线性系统的最优输出跟踪问题。针对利用极大值原理研究时滞非线性系统最优输出跟踪控制问题时导出的两点边值问题，运用灵敏度参数方法进行转化，将求解最优输出跟踪控制的既含有超前项又含有时滞项的原两点边值问题问题化为一族等价的不含超前项和时滞项的两点边值问题。这族两点边值问题可通过迭代方法依次求得各阶两点边值问题的解，从而获得原两点边值问题的等价解。引入了降维状态观测器，给出了物理可实现的最优输出跟踪控制律，并利用仿真研究验证了方法的有效性。
3．研究了非线性系统的最优输出跟踪控制问题。对具有一般形式的非线性系统在零点处进行级数展开，使之成为一阶线性项和高阶非线性项分离的形式。利用灵敏度法，对最优跟踪控制问题所导致的非线性两点边值问题进行级数展开，将原问题转化为一族包含已知非线性级数展开项的线性两点边值问题。详细研究了复合函数的高阶导数，给出了非线性项的各阶级数展开项的算法，代入并逐阶求解就得到由线性反馈和补偿项组成的最优跟踪控制律。并利用仿真研究验证了方法的有效性。
4．研究了含有非线性互联耦合项的动态大系统最优跟踪控制问题，提出一种次优控制律的设计方法。在系统中引入灵敏度参数ε，关于灵敏度参数ε在ε=0处展开Maclaurin级数。将求解高阶耦合的非线性两点边值问题简化为求解一族解耦的线性两点边值问题序列。该族线性微分方程问题的解，构成含有非线性互联耦合项的动态大系统最优跟踪控制问题非线性共态向量序列，最优控制律由线性反馈项和非线性共态向量的和组成。在最优控制律中截取共态向量和的前有限项，导出了一种次优控制律的设计算法。最后对其进行了仿真研究。
5．研究了一类仿射非线性系统的最优控制问题，改进了唐等提出的逐次逼近法。根据最优控制理论，这类仿射非线性系统最优控制问题的必要条件为无法解析求解的非线性两点边值问题。通过引入非线性微分方程近似求解的逐次逼近法，将非线性两点边值问题转化为一组线性两点边值问题序列。在构造线性两点边值问题序列时，采用了齐次的构造技巧。迭代求解所构造的齐次线性两点边值问题序列，设计出了具有线性闭环反馈形式的最优控制律。截取有限的迭代结果，设计出一种近似的最优控制律，给出了近似最优控制律的设计算法。并通过实例仿真验证了方法的有效性。
6．文章最后总结了论文的主要工作，并对下一步的研究方向进行了展望。
The dissertation studies the OOT problem for nonlinear systems, time-delay andnonlinear systems, nonlinear interconnected large-scale systems and a class of affinenonlinear systems, and then dicusses the optimal control law is physically realizable.The main contents are given as follows:
1. Firstly, the characteristics and research methods of time-delayed systems andnonlinear systems which are relative to research objects have been summarized. Nextthe relationship and development of optimal control problem and tracking problemhave been introduced. Then based on the prepareed work, the relative studies on theOOT problem for time-delay and nonlinear systems up to now and the main methodsare given in detail.
2. The OOT problem of the nonlinear systems whose reference input is generallyproduced by an exosystem is studied. With the Maximum Value Principle, thetwo-point boundary value (TPBV) problems for the OOT of the time-delay andnonlinear systems have been obtained. By introducing a sensitivity parameter, theoriginal TPBV problem for solving the OOT control with delay and advance terms istransformed into a series of TPBV problems without delay or advance terms. Then bysolving the two-point boundary value problem sequence recursively, TPBV isobtained. A reference input observer is introduced such that the optimal control law isphysically realizable. Simulation examples are employed to test the validity of thepresented sensitivity algorithm.
3. The OOT for nonlinear systems is considered. The nonlinear systems ingeneral form are expanded at zero and become to a form in which the one-order linearterms separated from the high-order nonlinear terms. Using a sensitivity parameter approach, and expanding the nonlinear two-point boundary value (TPBV) problemsled by the OOT control problem with respect to the sensitivity parameter, the originalnonlinear TPBV problem is transformed into a series of linear TPBV problemscontaining known low-order nonlinear terms. Algorithm of nonlinear expanding termsis obtained and substituting them into the recursion formula of adjoint vectorequations, the OOT control law consisting of linear terms and a nonlinearcompensation term can be approximately obtained. A simulation example is employedto test the validity of the presented algorithm.
4. Considering the optimal output tracking (OOT) problem for nonlinearinterconnected large-scale dynamic systems, an approach for designing suboptimalcontrol law is proposed. By introducing a sensitivity parameterε,εis expanded atzero. By using the approach, the high order, coupling, nonlinear two-point boundaryvalue (TPBV) problem is transformed into a sequence of linear decoupling TPBVproblems. By solving this sequence of linear differential equations, costate vectorsequence of nonlinear interconnected, coupling large-scale dynamic systems isobtained. Then optimal control law consists of linear feedback term and nonlinear costatevector sum. By intercepting the frontal finite terms of costate vector sum, we obtaineda suboptimal control law and its algorithm. A simulation example illustrates theefficiency of the optimal output tracking control law designed.
5. This paper considers an optimal control for a class of affine nonlinear systems. Animproved successive approximation approach is proposed. By introducing the SuccessiveApproximation Approach of the nonlinear differential equation theory, a linear two-pointboundary value problem sequence is constructed to approximate the nonlinear two-point boundaryvalue problem. A technique is employed to make sure the constructed linear two-point boundaryvalue problems of the approximating sequence homogeneous, Iteratively solving thisapproximating sequence, optimal control law of linear closed-loop feedback form is designed. Analgorithm is presented to design certain approximate optimal control law by truncating thefinite iteration. An illustrative example shows the validity of the algorithm
6. Finally, the main work in this dissertation is summarized and a proposition isindicated on the research work in the future.
引文
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