积微分系统最优控制问题的参数化方法
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摘要
最优控制在各工程技术领域、经济管理和资源分配等实际应用部门有非常广泛的应用。然而，大多数实际问题是在无限维空间内讨论的，往往太复杂很难求出它的解析解。因此，最优控制问题的数值计算研究就显得非常重要。
本文利用控制参数化方法研究了动态系统在固定时间区间[O，T]上的时滞积微分系统最优控制问题的解法。这一方法是先将时间区间[O，T]分成许多子区间，控制变量在这些相应的子区间内用逐段常数函数来逼近，于是，最优控制问题便可以由一系列最优参数选择问题来逼近。关于最优参数选择问题的求解应用到Pontryagin极大值原理，将微分约束转嫁到目标函数的梯度计算中，从而使问题变为以目标泛函为目标，只含等式约束平和不等式约束的标准的非线性规划问题，这样，便可用成熟的最优化方法求解。因此，最优控制问题的求解关键是将其转化为最优参数选择问题，而最优参数选择问题的求解关键是计算目标函数的梯度。按照这一思路，本文先推导出了时滞积微分系统最优参数选择问题目标函数的梯度公式，再利用控制参数化方法把最优控制问题转化为一系列最优参数选择问题。这样，就构造出了时滞积微分系统最优控制问题的一种新的数值解法，最后证明了此算法的收敛性。
文章还讨论了一种特殊时滞积微分系统最优参数选择问题，并且推导出了目标函数的梯度计算公式。
Optimal control theory is used widely in the application field of engineer technology, economic management and resource allocation etc. However, most practical problems are discussed in infinite - dimensional spaces, so which too complex to obtain analytical solution. Therefore, to find the numerical solution of optimal control problem is very important.
This paper devised computational algorithms using concept of the control parametrization for optimal control problems governed by delay integro-differential system on the fixed timeinterval [0,T] .The method partitions the interval [0,T] into several subintervals and controlvariables are approximated by piecewise constant functions with the instants of switching preassigned by the corresponding partition. Then, an optimal control problem is approximated by a series of corresponding optimal parameter selection problems. In order to find the solution of optimal parameter selection problem, we Shift differential constraint to the object function gradient calculation which must use Pontryagin Maximum Principle. In this way, the problem be transformed a standard nonlinear programming problem which only involved equality and inequality constraints. Hence, we can solve it by method of optimization.
Therefore, the key to solve the optimal control problem is transforming it into the optimal parameter selection problem, and the key to solve the optimal parameter selection problem is deriving the gradient formula for objective function. Based on this idea, we have been derived the gradient formula for the optimal parameter selection problem governed by integro-differential system first. Then, the optimal control problem was transformed a series of optimal parameter selection problems by method of control parametrization. In this way, we have constructed a new numerical method for optimal control problems governed by delay integro-differential system. Finally, we have proved that the solution of optimal parameter selection in a specified interval converges to the original optimal control problem.
And, the gradient formula for the objective function of optimal parameter selection problem governed by a special delay integro-differential system have been derived in the thesis.
本文利用控制参数化方法研究了动态系统在固定时间区间[O，T]上的时滞积微分系统最优控制问题的解法。这一方法是先将时间区间[O，T]分成许多子区间，控制变量在这些相应的子区间内用逐段常数函数来逼近，于是，最优控制问题便可以由一系列最优参数选择问题来逼近。关于最优参数选择问题的求解应用到Pontryagin极大值原理，将微分约束转嫁到目标函数的梯度计算中，从而使问题变为以目标泛函为目标，只含等式约束平和不等式约束的标准的非线性规划问题，这样，便可用成熟的最优化方法求解。因此，最优控制问题的求解关键是将其转化为最优参数选择问题，而最优参数选择问题的求解关键是计算目标函数的梯度。按照这一思路，本文先推导出了时滞积微分系统最优参数选择问题目标函数的梯度公式，再利用控制参数化方法把最优控制问题转化为一系列最优参数选择问题。这样，就构造出了时滞积微分系统最优控制问题的一种新的数值解法，最后证明了此算法的收敛性。
文章还讨论了一种特殊时滞积微分系统最优参数选择问题，并且推导出了目标函数的梯度计算公式。
Optimal control theory is used widely in the application field of engineer technology, economic management and resource allocation etc. However, most practical problems are discussed in infinite - dimensional spaces, so which too complex to obtain analytical solution. Therefore, to find the numerical solution of optimal control problem is very important.
This paper devised computational algorithms using concept of the control parametrization for optimal control problems governed by delay integro-differential system on the fixed timeinterval [0,T] .The method partitions the interval [0,T] into several subintervals and controlvariables are approximated by piecewise constant functions with the instants of switching preassigned by the corresponding partition. Then, an optimal control problem is approximated by a series of corresponding optimal parameter selection problems. In order to find the solution of optimal parameter selection problem, we Shift differential constraint to the object function gradient calculation which must use Pontryagin Maximum Principle. In this way, the problem be transformed a standard nonlinear programming problem which only involved equality and inequality constraints. Hence, we can solve it by method of optimization.
Therefore, the key to solve the optimal control problem is transforming it into the optimal parameter selection problem, and the key to solve the optimal parameter selection problem is deriving the gradient formula for objective function. Based on this idea, we have been derived the gradient formula for the optimal parameter selection problem governed by integro-differential system first. Then, the optimal control problem was transformed a series of optimal parameter selection problems by method of control parametrization. In this way, we have constructed a new numerical method for optimal control problems governed by delay integro-differential system. Finally, we have proved that the solution of optimal parameter selection in a specified interval converges to the original optimal control problem.
And, the gradient formula for the objective function of optimal parameter selection problem governed by a special delay integro-differential system have been derived in the thesis.
引文
[1] 毛云英．动态系统与最优控制[M]，北京：高等教育出版社，1994．
[2] 王日爽．泛函分析与最优化理论[M]，北京：北京航天航空大学出版社，2003．
[3] 王朝珠，秦化淑．最优控制理论[M]，北京：科学出版社，2003．
[4] 巴扎拉，．希蒂著；王化存，张春柏译．非线性规划[M]，贵州：贵州人民出版社，1985．
[5] 王长清．近代解析应用数学基础[M]，西安：西安电子科技大学出版社，2001．
[6] 王高雄，周之铭，朱思铭，王寿松．常微分方程[M]，北京：高等教育出版，1982．
[7] 邢继祥，张春蕊，徐洪泽．最优控制应用基础[M]，北京：科学出版社，2003．
[8] 陈宝林．最优化理论与算法[M]，北京：清华大学出版社，2005．
[9] 李国勇，张翠平，郭红戈，曲兵妮．最优控制理论及参数优化[M]，北京：国防工业出版社，2006．
[10] 张学铭，李训经，陈祖浩．最优控制系统的微分方程理论[M]，北京：高等教育出版社，1989．
[11] 席少霖，赵凤治．最优化计算方法[M]，上海：上海科学技术出版社，1983．
[12] 袁亚湘．非线性规划数值方法[M]，上海：上海科学技术出版社，1992．
[13] 袁亚湘，孙文瑜．最优化理论与方法[M]，北京：科学出版社，2006．
[14] 程鹏，王艳东．现代控制理论基础[M]，北京：北京航天航空出版社，2004．
[15] 谢政，李建平，汤泽滢．非线性最优化[M]，长沙：国防科学技术出版社，2003．
[16] 薛昌兴．实变函数和泛函分析(上)[M]，北京：高等教育出版社，1993．
[17] A. Siburian, V. Rehbock. Improving the robustness of optimal controls with respect to inequality [J], Nonlinear Analysis, 2001, Vol 47, 5659—5670.
[18] C. Z. Wu, K. L. Teo, Yi Zhao and W. Y. Yan. Solving an identification problem as an impulsive optimal parameter selection problem [J], Computers and Mathematics with Applications, 2005, Vol50, 217—229.
[19] C. Z. Wu, K. L. Teo, Yi Zhao and W. Y. Yan. An optimal control problem involving impulsive integrodifferential systems[J], Optimization Methods and Software, June 2007, Vol. 22, No. 3, 531-549.
[20] D. C. Jiang, K. L.Teo, W. Y. Yan. A new computational method for the functional inequality constrained minimax optimazation problem [J], Computers Math Applic, Dec 1995, Volume 3, 2310-2315.
[21] H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings. Control parametrization enhancing technique for optimal discrete-valued control problems[J], Automatica, 1999, Volume 35 1401—1407.
[22] H. W. J. Lee, K. L.Teo, V. Rehbock and L. S. Jennings. Control parametrization enhancing technique for time optimal control problems [J], Dynamical syst. App. 1977, Vol 6, 243—261.
[23] H. W. J. Lee, K. L. Teo, X. Q. Cai. An optimal control approach to nonlinear mixrd programming problems[J], Computers Math Applic, August 1998, Vol. 36, 3. pp87—105.
[24] H. Brunner and J. D. Lambert. Stability of numerical methods for volterra integro-differential equations [J], Computing, 1974, Vol 12, 75-89.
[25] J. T. Edwards, N. J. Ford and J. A. Roberts. The numerical simulation of the qualitative behaviour of volterra integro-differential equations [M]. In J. Levesley, I. J. Anderson, and J. C. Mason, editors, Proceedings of Algorithms for Approx-imation Ⅳ, pages 86-93, University of Huddersfield, 2002.
[26] K.L.Teo, C.J.Goh and K.H.wong. A Unified Computational Approach to Optimal Control Problems [M], New York, Longman (1991).
[27] K.H.Wong,L.S.Jennings and F.Benyah. Control parametrization method for fee planning time optimal control problems with time-delayed arguments [J], Nonlinear Analysis, 2001, Vol 47, 5679—5689.
[28] Klas Adolfsson, Mikael Enelund,Stig Larsson. Adaptive discretization of an integro-differential equation with a weakly singular convolution kerne [J], Comput.Methods Appl. Mesh.Engrg. 2003, Vol 192, 5285—5304.
[29] K.L.Teo, L.S.Jennings, H.W.J.Lee, V.Rehbock. The control parametrization enhancing transform for constrained optimal control problems [J], J.Austral.Math.Soc.Ser.B,1999, Vol 40,314—335.
[30] K.L.Teo, Control parametrization enhancing transform to optimal control problems [J], Nonlinear Analysis, 2005,Vol 63, e2223—e2236.
[31] Le Ping Bao. A Class of Nonlinear Impulsive Integral Differential Equations and Optimal control .[Degree thesis of Master]. GuiYang: GuiZhou University, 2006.
[32] M.Schlegel, K.Stockmann, T.Binder, W.marquardt. Dynamic optimization using adaptive control vector parametrization[J], Computers and Chemical Engineering, 2005,Vol 29,1731—1751.
[33] V.Rehbock K.L.Teo,and L.S.Jennings, ,.A computational procedure for suboptimal robust controls[J], Dynamics and Control, 1992, Vol 2,331-348.
[34] Xiang Xiaoling and Kuang Huawu. Delay systems and optimal controls[J], Acta Mathematics Applicates Sinica, 2000, Vol 16, 27-35.
[35] Y.Liu, K.L.Teo, L.S. Jennings and S. Wang. On a class of optimal control problems with state jumps[J], Journal of Optimization Theury and Applications, July1998, Vol. 98, No. 1, pp. 65-82.
[2] 王日爽．泛函分析与最优化理论[M]，北京：北京航天航空大学出版社，2003．
[3] 王朝珠，秦化淑．最优控制理论[M]，北京：科学出版社，2003．
[4] 巴扎拉，．希蒂著；王化存，张春柏译．非线性规划[M]，贵州：贵州人民出版社，1985．
[5] 王长清．近代解析应用数学基础[M]，西安：西安电子科技大学出版社，2001．
[6] 王高雄，周之铭，朱思铭，王寿松．常微分方程[M]，北京：高等教育出版，1982．
[7] 邢继祥，张春蕊，徐洪泽．最优控制应用基础[M]，北京：科学出版社，2003．
[8] 陈宝林．最优化理论与算法[M]，北京：清华大学出版社，2005．
[9] 李国勇，张翠平，郭红戈，曲兵妮．最优控制理论及参数优化[M]，北京：国防工业出版社，2006．
[10] 张学铭，李训经，陈祖浩．最优控制系统的微分方程理论[M]，北京：高等教育出版社，1989．
[11] 席少霖，赵凤治．最优化计算方法[M]，上海：上海科学技术出版社，1983．
[12] 袁亚湘．非线性规划数值方法[M]，上海：上海科学技术出版社，1992．
[13] 袁亚湘，孙文瑜．最优化理论与方法[M]，北京：科学出版社，2006．
[14] 程鹏，王艳东．现代控制理论基础[M]，北京：北京航天航空出版社，2004．
[15] 谢政，李建平，汤泽滢．非线性最优化[M]，长沙：国防科学技术出版社，2003．
[16] 薛昌兴．实变函数和泛函分析(上)[M]，北京：高等教育出版社，1993．
[17] A. Siburian, V. Rehbock. Improving the robustness of optimal controls with respect to inequality [J], Nonlinear Analysis, 2001, Vol 47, 5659—5670.
[18] C. Z. Wu, K. L. Teo, Yi Zhao and W. Y. Yan. Solving an identification problem as an impulsive optimal parameter selection problem [J], Computers and Mathematics with Applications, 2005, Vol50, 217—229.
[19] C. Z. Wu, K. L. Teo, Yi Zhao and W. Y. Yan. An optimal control problem involving impulsive integrodifferential systems[J], Optimization Methods and Software, June 2007, Vol. 22, No. 3, 531-549.
[20] D. C. Jiang, K. L.Teo, W. Y. Yan. A new computational method for the functional inequality constrained minimax optimazation problem [J], Computers Math Applic, Dec 1995, Volume 3, 2310-2315.
[21] H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings. Control parametrization enhancing technique for optimal discrete-valued control problems[J], Automatica, 1999, Volume 35 1401—1407.
[22] H. W. J. Lee, K. L.Teo, V. Rehbock and L. S. Jennings. Control parametrization enhancing technique for time optimal control problems [J], Dynamical syst. App. 1977, Vol 6, 243—261.
[23] H. W. J. Lee, K. L. Teo, X. Q. Cai. An optimal control approach to nonlinear mixrd programming problems[J], Computers Math Applic, August 1998, Vol. 36, 3. pp87—105.
[24] H. Brunner and J. D. Lambert. Stability of numerical methods for volterra integro-differential equations [J], Computing, 1974, Vol 12, 75-89.
[25] J. T. Edwards, N. J. Ford and J. A. Roberts. The numerical simulation of the qualitative behaviour of volterra integro-differential equations [M]. In J. Levesley, I. J. Anderson, and J. C. Mason, editors, Proceedings of Algorithms for Approx-imation Ⅳ, pages 86-93, University of Huddersfield, 2002.
[26] K.L.Teo, C.J.Goh and K.H.wong. A Unified Computational Approach to Optimal Control Problems [M], New York, Longman (1991).
[27] K.H.Wong,L.S.Jennings and F.Benyah. Control parametrization method for fee planning time optimal control problems with time-delayed arguments [J], Nonlinear Analysis, 2001, Vol 47, 5679—5689.
[28] Klas Adolfsson, Mikael Enelund,Stig Larsson. Adaptive discretization of an integro-differential equation with a weakly singular convolution kerne [J], Comput.Methods Appl. Mesh.Engrg. 2003, Vol 192, 5285—5304.
[29] K.L.Teo, L.S.Jennings, H.W.J.Lee, V.Rehbock. The control parametrization enhancing transform for constrained optimal control problems [J], J.Austral.Math.Soc.Ser.B,1999, Vol 40,314—335.
[30] K.L.Teo, Control parametrization enhancing transform to optimal control problems [J], Nonlinear Analysis, 2005,Vol 63, e2223—e2236.
[31] Le Ping Bao. A Class of Nonlinear Impulsive Integral Differential Equations and Optimal control .[Degree thesis of Master]. GuiYang: GuiZhou University, 2006.
[32] M.Schlegel, K.Stockmann, T.Binder, W.marquardt. Dynamic optimization using adaptive control vector parametrization[J], Computers and Chemical Engineering, 2005,Vol 29,1731—1751.
[33] V.Rehbock K.L.Teo,and L.S.Jennings, ,.A computational procedure for suboptimal robust controls[J], Dynamics and Control, 1992, Vol 2,331-348.
[34] Xiang Xiaoling and Kuang Huawu. Delay systems and optimal controls[J], Acta Mathematics Applicates Sinica, 2000, Vol 16, 27-35.
[35] Y.Liu, K.L.Teo, L.S. Jennings and S. Wang. On a class of optimal control problems with state jumps[J], Journal of Optimization Theury and Applications, July1998, Vol. 98, No. 1, pp. 65-82.
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