非线性系统最优控制的改进逐次逼近法研究
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摘要
近年来,非线性系统的最优控制问题成为系统与控制领域研究的热点问题。对于非线性系统,其最优控制问题将不可避免的导致求解Hamilton-Jacobi- Bellman (HJB)方程或非线性两点边值问题。鉴于非线性HJB方程和非线性两点边值问题通常情况下无法求得解析解,众多的近似求解方法被引入以谋求近似的求解非线性HJB方程或非线性两点边值问题。逐次逼近法通过构造非线性两点边值问题的线性两点边值问题序列,使难于求解的非线性两点边值问题得到了近似的迭代求解。所设计的近似最优控制律具有线性状态反馈加开环近似补偿器的形式。逐次逼近法与其它非线性系统最优控制的近似方法相比,具有较好的收敛性,且所需的计算量较小,有着很好的实际应用前景。本文提出了一种非线性系统最优控制的改进逐次逼近法,主要对逐次逼近法作了两方面的改进。首先,根据仿射非线性系统的特殊结构特征,改进了线性两点边值问题序列的构造,以使得所设计的次优控制律对于仿射非线性具有闭环线性反馈的近似最优补偿器形式,具有较好的鲁棒性。其次,构造了闭环系统序列,使得所构造的线性两点边值问题序列的状态序列在每次迭代中都更接近于非线性系统关于二次型性能指标的最优轨迹,从而提高了迭代算法的收敛性。
本文首先综述了非线性系统最优控制的研究现状,着重介绍了逐次逼近法的数学来源及其在非线性系统最优控制领域的应用成果。在此基础上,经过研究分析,提出了一种具有更好鲁棒性和收敛性的改进逐次逼近算法,以解决非线性系统的最优控制问题。本文的研究内容概括如下:
1.基于改进逐次逼近法研究一类仿射非线性系统的最优控制问题。通过使用改进方法,将非线性两点边值问题转化为线性两点边值问题序列,从而使难于求解的非线性两点边值问题可以迭代的得到近似求解。通过截取有限的迭代结果,得到原非线性系统最优控制问题的一种近似解,并得到具有完全闭环线性反馈形式的近似最优控制律。同时,证明了所构造的迭代序列的收敛性,并通过实例仿真验证了方法的有效性。
2.基于改进逐次逼近法研究一类Lipschitz连续的非线性系统的最优控制问题。首先将系统动态方程中的非线性项中的仿射非线性部分分离出来,即将满足Lipschitz条件的非线性函数分离成仿射和非仿射两部分。然后结合使用原方法和改进方法来解决这类非线性系统的最优控制问题。通过构造线性非齐次两点边值问题序列,近似的求解了非线性两点边值问题。截取有限的迭代求解结果,得到的近似最优控制律包含线性反馈项和开环的非线性近似补偿项。并证明了所构造的迭代序列的收敛性。仿真实例验证了方法的有效性。
3.研究非线性相似组合系统的最优控制问题。首先使用一种相似组合系统的简化方法,将非线性相似组合系统的最优控制问题转化为一类仿射非线性大系统的最优控制问题。然后,应用改进逐次逼近法设计其闭环线性反馈的近似最优控制律,并证明了所提出的迭代算法的收敛性。仿真实例验证了方法的有效性和时效性。
4.研究非线性互联大系统的最优控制问题。基于改进逐次逼近法,将大规模非线性互联两点边值问题转化为一组线性两点边值问题序列。通过迭代求解,得到包含线性反馈项和开环的非线性近似补偿项的近似最优控制律,并证明了迭代算法的收敛性。仿真实例验证了方法的有效性和时效性。
5.研究一类仿射非线性系统的最优跟踪问题。根据系统关于参考信号的误差性能指标,最优跟踪问题归结于非线性两点边值问题的求解问题。引入改进逐次逼近法,最优跟踪问题得到解决,设计了包含线性反馈项和关于参考信号外系统状态的线性前馈项的近似最优跟踪控制律。并通过设计关于外系统的降维状态观测器,解决了所设计的次优控制律的物理可实现问题。仿真实例验证了方法的有效性。
6.研究一类仿射非线性系统的最优滑模的设计问题。通过引入改进逐次逼近法,对非线性系统设计了虚拟的闭环线性反馈的近似最优控制律,然后根据线性滑模的设计方法,设计了非线性系统的最优切换面,并得到了基于最优滑模的变结构控制律。仿真实例验证了方法的有效性。
7.总结本文的主要工作,展望今后的研究研究方向。
In recent years, the optimal control problem of nonlinear systems has been one of the most challenging problems in system and control. For nonlinear systems, its optimal control problem always gives rise to nonlinear Hamilton-Jacobi-Bellman equation or nonlinear two-point boundary value problem, both of which are hard to solve in general. Therefore many approximation approaches was introduced to solve this problem. The successive approximation approach is a familiar one. By transforming the nonlinear two-point boundary value problem into a sequence of linear two-point boundary value problem, the nonlinear two-point boundary value problem is solved iteratively by using this approach. And an approximate optimal control law consisting of a linear state feedback term and an open-loop approximate compensator term is designed. Comparing with the other approximate approaches, the successive approximation approach takes lower computational loads, and still with a good convergence. Therefore it has a good prospect of application. This dissertation presents an improved successive approximation approach by improving on the successive approximation approach in two aspects. Firstly, for affine nonlinear systems, the transformation from nonlinear two-point boundary value problem to the linear sequence of it is rebuilt to design the approximate optimal control law purely in close-loop linear state feedback form. As a result, the close-loop system under this control law becomes more robust. Secondly, the state sequence of the linear two-point boundary value problem sequence approaches the optimal state trajectory more rapidly during the iteration procedure by introducing a close-loop system sequence. Thus the astringency of the iteration procedure is increased.
In this dissertation, the history of the development in the optimal control of nonlinear systems is firstly reviewed. And the latest research tendency and the main methods are summarized. By working on the mathematic origin of the successive approximation approach and its application to the nonlinear optimal control problem, a more robust and convergent approach named improved successive approximation approach is proposed. The major results of this dissertation are summarized as follows.
1. Based on the improved successive approximation approach, the optimal control for a class of affine nonlinear systems is studied. By using the improved approach, the nonlinear two-point boundary value problem, which is the necessary condition of the optimal control problem, is transformed into a linear two-point boundary value problem sequence that is easy to solve. An approximate solution is obtained by truncating the sequence to a finite iteration, and then an approximate optimal control law with purely close-loop linear state feedback form is designed. Meanwhile, the convergence of the iteration is ensured by proving that the constructed sequence is uniformly convergent to the nonlinear two-point boundary value problem. Also a simulation example is employed to test the validity of the iteration algorithm proposed and its superiority upon the original approach.
2. Based on the improved successive approximation approach, the optimal control for a class of Lipschitz-continues nonlinear systems is studied. By separating the affine nonlinear part with the other non-affine nonlinear part, the improved approach together with the original approach is employed to solve this problem. An approximate optimal control law with both a linear state feedback term and an open-loop nonlinear compensator is designed. And the convergence is also proved. Its superiority upon using the original approach alone is shown by a simulation example.
3. The optimal control of the nonlinear similar composite system is studied. By using some decoupling methodology, the nonlinear similar composite system is transformed into an affine nonlinear system. Then optimal control of the affine nonlinear system, which is equivalent to the nonlinear similar composite system, is familiar with the problem studied in chapter 1. A close-loop state feedback optimal control is designed, and the convergence also is proved. A simulation example is employed to test the validity and efficiency of the algorithm proposed.
4. The optimal control of the nonlinear interconnected large-scale system is studied. By using the improved successive approximation approach, the problem is solved iteratively. An approximate optimal control law with both a linear state feedback term and a nonlinear compensator is designed, and the convergence is also proved. A simulation example is employed to test the validity and efficiency of the algorithm proposed.
5. The optimal tracking control of a class of affine nonlinear systems is studied. With respect to the error-based quadratic performance index, the optimal control problem is still come down to a nonlinear two-point boundary value problem. By introducing the improved approach, the problem is solved iteratively, and an approximate control law with both a linear state feedback term and a reference model state feedforward term is designed.
6. The optimal sliding mode control of a class of affine nonlinear systems is studied. By introducing the improved approach, a virtual approximate optimal control of the nonlinear system is designed. Then, according to the theory of linear switching manifold design methodology, a linear optimal switching manifold is designed. And a sliding mode control law is obtained based on the designed linear optimal switching manifold.
7. The conclusions are made. And the direction for the future study is indicated.
本文首先综述了非线性系统最优控制的研究现状,着重介绍了逐次逼近法的数学来源及其在非线性系统最优控制领域的应用成果。在此基础上,经过研究分析,提出了一种具有更好鲁棒性和收敛性的改进逐次逼近算法,以解决非线性系统的最优控制问题。本文的研究内容概括如下:
1.基于改进逐次逼近法研究一类仿射非线性系统的最优控制问题。通过使用改进方法,将非线性两点边值问题转化为线性两点边值问题序列,从而使难于求解的非线性两点边值问题可以迭代的得到近似求解。通过截取有限的迭代结果,得到原非线性系统最优控制问题的一种近似解,并得到具有完全闭环线性反馈形式的近似最优控制律。同时,证明了所构造的迭代序列的收敛性,并通过实例仿真验证了方法的有效性。
2.基于改进逐次逼近法研究一类Lipschitz连续的非线性系统的最优控制问题。首先将系统动态方程中的非线性项中的仿射非线性部分分离出来,即将满足Lipschitz条件的非线性函数分离成仿射和非仿射两部分。然后结合使用原方法和改进方法来解决这类非线性系统的最优控制问题。通过构造线性非齐次两点边值问题序列,近似的求解了非线性两点边值问题。截取有限的迭代求解结果,得到的近似最优控制律包含线性反馈项和开环的非线性近似补偿项。并证明了所构造的迭代序列的收敛性。仿真实例验证了方法的有效性。
3.研究非线性相似组合系统的最优控制问题。首先使用一种相似组合系统的简化方法,将非线性相似组合系统的最优控制问题转化为一类仿射非线性大系统的最优控制问题。然后,应用改进逐次逼近法设计其闭环线性反馈的近似最优控制律,并证明了所提出的迭代算法的收敛性。仿真实例验证了方法的有效性和时效性。
4.研究非线性互联大系统的最优控制问题。基于改进逐次逼近法,将大规模非线性互联两点边值问题转化为一组线性两点边值问题序列。通过迭代求解,得到包含线性反馈项和开环的非线性近似补偿项的近似最优控制律,并证明了迭代算法的收敛性。仿真实例验证了方法的有效性和时效性。
5.研究一类仿射非线性系统的最优跟踪问题。根据系统关于参考信号的误差性能指标,最优跟踪问题归结于非线性两点边值问题的求解问题。引入改进逐次逼近法,最优跟踪问题得到解决,设计了包含线性反馈项和关于参考信号外系统状态的线性前馈项的近似最优跟踪控制律。并通过设计关于外系统的降维状态观测器,解决了所设计的次优控制律的物理可实现问题。仿真实例验证了方法的有效性。
6.研究一类仿射非线性系统的最优滑模的设计问题。通过引入改进逐次逼近法,对非线性系统设计了虚拟的闭环线性反馈的近似最优控制律,然后根据线性滑模的设计方法,设计了非线性系统的最优切换面,并得到了基于最优滑模的变结构控制律。仿真实例验证了方法的有效性。
7.总结本文的主要工作,展望今后的研究研究方向。
In recent years, the optimal control problem of nonlinear systems has been one of the most challenging problems in system and control. For nonlinear systems, its optimal control problem always gives rise to nonlinear Hamilton-Jacobi-Bellman equation or nonlinear two-point boundary value problem, both of which are hard to solve in general. Therefore many approximation approaches was introduced to solve this problem. The successive approximation approach is a familiar one. By transforming the nonlinear two-point boundary value problem into a sequence of linear two-point boundary value problem, the nonlinear two-point boundary value problem is solved iteratively by using this approach. And an approximate optimal control law consisting of a linear state feedback term and an open-loop approximate compensator term is designed. Comparing with the other approximate approaches, the successive approximation approach takes lower computational loads, and still with a good convergence. Therefore it has a good prospect of application. This dissertation presents an improved successive approximation approach by improving on the successive approximation approach in two aspects. Firstly, for affine nonlinear systems, the transformation from nonlinear two-point boundary value problem to the linear sequence of it is rebuilt to design the approximate optimal control law purely in close-loop linear state feedback form. As a result, the close-loop system under this control law becomes more robust. Secondly, the state sequence of the linear two-point boundary value problem sequence approaches the optimal state trajectory more rapidly during the iteration procedure by introducing a close-loop system sequence. Thus the astringency of the iteration procedure is increased.
In this dissertation, the history of the development in the optimal control of nonlinear systems is firstly reviewed. And the latest research tendency and the main methods are summarized. By working on the mathematic origin of the successive approximation approach and its application to the nonlinear optimal control problem, a more robust and convergent approach named improved successive approximation approach is proposed. The major results of this dissertation are summarized as follows.
1. Based on the improved successive approximation approach, the optimal control for a class of affine nonlinear systems is studied. By using the improved approach, the nonlinear two-point boundary value problem, which is the necessary condition of the optimal control problem, is transformed into a linear two-point boundary value problem sequence that is easy to solve. An approximate solution is obtained by truncating the sequence to a finite iteration, and then an approximate optimal control law with purely close-loop linear state feedback form is designed. Meanwhile, the convergence of the iteration is ensured by proving that the constructed sequence is uniformly convergent to the nonlinear two-point boundary value problem. Also a simulation example is employed to test the validity of the iteration algorithm proposed and its superiority upon the original approach.
2. Based on the improved successive approximation approach, the optimal control for a class of Lipschitz-continues nonlinear systems is studied. By separating the affine nonlinear part with the other non-affine nonlinear part, the improved approach together with the original approach is employed to solve this problem. An approximate optimal control law with both a linear state feedback term and an open-loop nonlinear compensator is designed. And the convergence is also proved. Its superiority upon using the original approach alone is shown by a simulation example.
3. The optimal control of the nonlinear similar composite system is studied. By using some decoupling methodology, the nonlinear similar composite system is transformed into an affine nonlinear system. Then optimal control of the affine nonlinear system, which is equivalent to the nonlinear similar composite system, is familiar with the problem studied in chapter 1. A close-loop state feedback optimal control is designed, and the convergence also is proved. A simulation example is employed to test the validity and efficiency of the algorithm proposed.
4. The optimal control of the nonlinear interconnected large-scale system is studied. By using the improved successive approximation approach, the problem is solved iteratively. An approximate optimal control law with both a linear state feedback term and a nonlinear compensator is designed, and the convergence is also proved. A simulation example is employed to test the validity and efficiency of the algorithm proposed.
5. The optimal tracking control of a class of affine nonlinear systems is studied. With respect to the error-based quadratic performance index, the optimal control problem is still come down to a nonlinear two-point boundary value problem. By introducing the improved approach, the problem is solved iteratively, and an approximate control law with both a linear state feedback term and a reference model state feedforward term is designed.
6. The optimal sliding mode control of a class of affine nonlinear systems is studied. By introducing the improved approach, a virtual approximate optimal control of the nonlinear system is designed. Then, according to the theory of linear switching manifold design methodology, a linear optimal switching manifold is designed. And a sliding mode control law is obtained based on the designed linear optimal switching manifold.
7. The conclusions are made. And the direction for the future study is indicated.
引文
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