具有不等式路径约束的微分代数方程系统的动态优化
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摘要
针对具有不等式路径约束的微分代数方程(Differential-algebraic equations, DAE)系统的动态优化问题,通常将DAE中的等式路径约束进行微分处理,或者将其转化为点约束或不等式约束进行求解.前者需要考虑初值条件的相容性或增加约束,在变量间耦合度较高的情况下这种转化求解方法是不可行的;后者将等式约束转化为其他类型的约束会增加约束条件,增加了求解难度.为了克服该缺点,本文提出了结合后向差分法对DAE直接处理来求解上述动态优化问题的方法.首先利用控制向量参数化方法将无限维的最优控制问题转化为有限维的最优控制问题,再利用分点离散法用有限个内点约束去代替原不等式路径约束,最后用序列二次规划(Sequential quadratic programming, SQP)法使得在有限步数的迭代下,得到满足用户指定的路径约束违反容忍度下的KKT (Karush Kuhn Tucker)最优点.理论上证明了该算法在有限步内收敛.最后将所提出的方法应用在具有不等式路径约束的微分代数方程系统中进行仿真,结果验证了该方法的有效性.
For dynamic optimization of differential algebraic systems with inequality path constraints, the equality constraint in differential-algebraic equations(DAE) is often differentiated or transformed into point constraints or inequality constraints to solve. As for the former, the compatibility of initial conditions needs to be checked or more constraints are added, making the original optimization problem infeasible in some cases. For the latter, the way that equality constraint is converted to other types of constraints increases the difficulty of solving. In order to overcome the above problems, a new method is proposed to solve the above problem directly based on the backward differentiation formula. Firstly, the control vector parameterization is used to convert the optimal control problem of infinite dimensions into a finite dimensional one. Then, a set of interior-time points by using the pointwise discretization method are used to replace the original inequality path constraints. Finally, the sequential quadratic programming(SQP) is used to locate the Karush Kuhn Tucker(KKT) points within finite-stepped iterations. Proof is provided for the finite convergence of the algorithm. The dynamic optimization algorithm is applied to the differential algebraic equation systems with inequality path constraint,and simulation studies are carried out to verify the effectiveness of the proposed method for the differential algebraic systems with inequality path constraints.
For dynamic optimization of differential algebraic systems with inequality path constraints, the equality constraint in differential-algebraic equations(DAE) is often differentiated or transformed into point constraints or inequality constraints to solve. As for the former, the compatibility of initial conditions needs to be checked or more constraints are added, making the original optimization problem infeasible in some cases. For the latter, the way that equality constraint is converted to other types of constraints increases the difficulty of solving. In order to overcome the above problems, a new method is proposed to solve the above problem directly based on the backward differentiation formula. Firstly, the control vector parameterization is used to convert the optimal control problem of infinite dimensions into a finite dimensional one. Then, a set of interior-time points by using the pointwise discretization method are used to replace the original inequality path constraints. Finally, the sequential quadratic programming(SQP) is used to locate the Karush Kuhn Tucker(KKT) points within finite-stepped iterations. Proof is provided for the finite convergence of the algorithm. The dynamic optimization algorithm is applied to the differential algebraic equation systems with inequality path constraint,and simulation studies are carried out to verify the effectiveness of the proposed method for the differential algebraic systems with inequality path constraints.
引文
1 Chen Mei-Rong, Guo Yi-Nan, Gong Dun-Wei, Yang Zhen.A novel dynamic multi-objective robust evolutionary optimization method. Acta Automatica Sinica, 2017, 43(11):2014-2032(陈美蓉,郭一楠,巩敦卫,杨振.一类新型动态多目标鲁棒进化优化方法.自动化学报, 2017, 43(11):2014-2032)
2 Chen Long, Liu Quan-Li, Wang Lin-Qing, Zhao Jun, Wang Wei. Data-driven prediction on performance indicators in process industry:a survey. Acta Automatica Sinica, 2017,43(6):944-954(陈龙,刘全利,王霖青,赵珺,王伟.基于数据的流程工业生产过程指标预测方法综述.自动化学报, 2017, 43(6):944-954)
3 Ding Jin-Liang, Yang Cui-E, Chen Li-Peng, Chai Tian-You.Dynamic multi-objective optimization algorithm based on reference point prediction. Acta Automatica Sinica, 2017,43(2):313-320(丁进良,杨翠娥,陈立鹏,柴天佑.基于参考点预测的动态多目标优化算法.自动化学报, 2017, 43(2):313-320)
4 Bienstock D. Optimal control of cascading power grid failures. In:Proceedings of the 2010 IEEE Conference on Decision and Control and European Control Conference, New York, USA:IEEE, 2010. 2166-2173
5 Chomat M, Schreier L, Bendl J. Optimal control of input rectifier in voltage-Source inverter supplied from unbalanced power grid. In:Proceedings of the 2006 IEEE International Symposium on Industrial Electronics, New York,USA:IEEE, 2006. 1042-1045
6 Liu P, Li G, Liu X. Fast engineering optimization:a novel highly effective control parameterization approach for industrial dynamic processes. ISA Transactions, 2015, 58:248-254
7 Jie X, Huang Y, Lou H H. A probability distribution estimation based method for dynamic optimization. AIChE Journal, 2010, 53(7):1805-1816
8 Hirmajer T, Balsa-Canto E, Banga J R. DOTcvpSB, a software toolbox for dynamic optimization in systems biology.BMC Bioinformatics, 2009, 10(1):199
9 Bredies K, Lorenz D A, Maass P. An optimal control problem in medical image processing. In:Proceedings of the IFIP-TC7 Conference, Turin, Italy:DBLP, 2005:249-259
10 Barve H A, Banavar R N. Energy-optimal control of a particle in a dielectrophoretic system. In:Proceedings of the2010 IEEE Conference on Decision&Control. New York,USA:IEEE, 2010:3353-3358
11 Barve H A, Banavar R N. Energy-optimal control of a particle in a dielectrophoretic system. In:Proceedings of the2010 IEEE Conference on Decision&Control. New York,USA:IEEE, 2010:3353-3358
12 Sun Yong, Zhang Mao-Rui, Liang Xiao-Ling. Improved gauss pseudospectral method for solving nonlinear optimal control problem with complex constraints. Acta Automatica Sinica, 2013, 39(5):672-678(孙勇,张卯瑞,梁晓玲.求解含复杂约束非线性最优控制问题的改进Gauss伪谱法.自动化学报, 2013, 39(5):672-678)
13 Biegler L T. An overview of simultaneous strategies for dynamic optimization. Chemical Engineering&Processing Process Intensification, 2007, 46(11):1043-1053
14 Biegler L T. Nonlinear programming:concepts, algorithms,and applications to chemical processes. Society for Industrial and Applied Mathematics, 2010.
15 Peng Hai-Jun, Gao Qiang, Wu Zhi-Gang, Zhong Wan-Xie. A mixed variable variational method for optimal control problems with applications in aerospace control. Acta Automatica Sinica, 2011, 37(10):1248-1255(彭海军,高强,吴志刚,钟万勰.求解最优控制问题的混合变量变分方法及其航天控制应用.自动化学报, 2011, 37(10):1248-1255)
16 Chachuat B, Mitsos A, Barton P I. Optimal design and steady-state operation of micro power generation employing fuel cells. Chemical Engineering Science, 2005, 60(16):4535-4556
17 Betts J T, Huffman W P. Application of sparse nonlinear programming to trajectory optimization. Journal of Guidance Control Dynamics, 1992, 15(1):198-206
18 Fu J, Faust J M M, Chachuat B, Mitsosc1 A. Local optimization of dynamic programs with guaranteed satisfaction of path constraints. Automatica, 2015, 62(C):184-192
19 Hu Yun-Qing, Liu Xing-Gao, Xue An-Ke. A penalty method for solving inequality path constrained optimal control problems. Acta Automatica Sinica, 2013, 29(12):1996-2001(胡云卿,刘兴高,薛安克.带不等式路径约束最优控制问题的惩罚函数法.自动化学报, 2013, 39(12):1996-2001)
20 Qian Ji-Xin, Song Chun-Yue, Wang Ke-Xin, Cheng Yang.Nonlinear Predictive Control, Beijing:Science Press, 2015.(钱积新,宋春跃,王可心,陈扬.非线性预测控制,北京:科学出版社, 2015.)
21 Pontryagin L S, Boltyanskii V G, Gamkrelidze R V,Mishchenko E F. The mathematical theory of optimal processes. Interscience, 1962.
22 Fabien B C. A technique for the direct optimization of dynamic systems described by differential-algebraic equations.Optimal Control Applications&Methods, 2007, 29(6):445-466
23 Brenan K E, Campbell S L, Petzold L R. Numerical solution of initial-value problems in differential-algebraic equations.North-Holland, 1989.
24 Styczen K, Drag P. A modified multipoint shooting feasibleSQP method for optimal control of DAE systems. In:Proceedings of Computer Science and Information Systems,New York, USA:IEEE, 2011. 477-484
25 Feehery W F, Barton P I. Dynamic simulation and optimization with inequality path constraints. Computers&Chemical Engineering, 1996, 20(3):169-176
26 Jacobson D, Lele M. A transformation technique for optimal control problems with a state variable inequality constraint. IEEE Transactions on Automatic Control, 1969,14(5):457-464
27 Vassiliadis V S, Sargent R W H, Pantelides C C. Solution of a class of multistage dynamic optimization problems. 2.problems with path constraints. Industrial&Engineering Chemistry Research, 1994, 10(9):2122-2133
28 Floudas C A, Stein O. The adaptive convexification algorithm:a feasible point method for semi-infinite programming. SIAM Journal on Optimization, 2008, 18(4):1187-1208
29 Chachuat B. Nonlinear and dynamic optimization:from theory to practice. Automatic Control Laboratory EPFL,2007, 107(Spring):192-193
30 Rehbock V, Teo K L, Jennings L S, Lee C S. An exact penalty function approach to all-time-step constrained discrete-time optimal control problems. Applied Mathematics&Computation, 1992, 49(2-3):215-230
31 Loxton R C, Teo K L, Rehbock V, Yiu K F C. Optimal control problems with a continuous inequality constraint on the state and the control. Automatica, 2009, 45(10):2250-2257
32 Liu X, Hu Y, Feng J, Liu K. A novel penalty approach for nonlinear dynamic optimization problems with inequality path constraints. IEEE Transactions on Automatic Control,2014, 59(10):2863-2867
33 Liu P, Li X, Liu X, Hu Y. An improved smoothing techniquebased control vector parameterization method for optimal control problems with inequality path constraints. Optimal Control Applications&Methods, 2016, 38(4):586-600
34 Chen T W C, Vassiliadis V S. Inequality path constraints in optimal control:a finite iterationε-convergent scheme based on pointwise discretization. Journal of Process Control, 2005, 15(3):353-362
35 Biegler L T. An overview of simultaneous strategies for dynamic optimization. Chemical Engineering&Processing Process Intensification, 2007, 46(11):1043-1053
36 Vassiliadis V. Computational solution of dynamic optimization problems with general differential-algebraic constraints by. Journal of Guidance Control&Dynamics, 1993, 15(2):457-460
37 Teo K L, Goh C J, Wong K H. A unified computational approach to optimal control problems. Longman Scientific and Technical, 1991:2763-2774
38 Martin R, Teo K L. Optimal control of drug administration in cancer chemotherapy. World Scientific, 1994.
39 Gunn D J, Thomas W J. Mass transport and chemical reaction in multifunctional catalyst systems. Chemical Engineering Science, 1965, 20(2):89-100
40 Hirmajer T, Fikar M, Balsa-Canto E, Banga J R. DOTcvp:dynamic optimization toolbox with control vector parameterization approach. 2008.
41 Irizarry R. A generalized framework for solving dynamic optimization problems using the artificial chemical process paradigm:applications to particulate processes and discrete dynamic systems. Chemical Engineering Science, 2005,60(21):5663-5681
42 Huang Y J, Reklaitis G V, Venkatasubramanian V. Model decomposition based method for solving general dynamic optimization problems. Computers&Chemical Engineering, 2002, 26(6):863-873
2 Chen Long, Liu Quan-Li, Wang Lin-Qing, Zhao Jun, Wang Wei. Data-driven prediction on performance indicators in process industry:a survey. Acta Automatica Sinica, 2017,43(6):944-954(陈龙,刘全利,王霖青,赵珺,王伟.基于数据的流程工业生产过程指标预测方法综述.自动化学报, 2017, 43(6):944-954)
3 Ding Jin-Liang, Yang Cui-E, Chen Li-Peng, Chai Tian-You.Dynamic multi-objective optimization algorithm based on reference point prediction. Acta Automatica Sinica, 2017,43(2):313-320(丁进良,杨翠娥,陈立鹏,柴天佑.基于参考点预测的动态多目标优化算法.自动化学报, 2017, 43(2):313-320)
4 Bienstock D. Optimal control of cascading power grid failures. In:Proceedings of the 2010 IEEE Conference on Decision and Control and European Control Conference, New York, USA:IEEE, 2010. 2166-2173
5 Chomat M, Schreier L, Bendl J. Optimal control of input rectifier in voltage-Source inverter supplied from unbalanced power grid. In:Proceedings of the 2006 IEEE International Symposium on Industrial Electronics, New York,USA:IEEE, 2006. 1042-1045
6 Liu P, Li G, Liu X. Fast engineering optimization:a novel highly effective control parameterization approach for industrial dynamic processes. ISA Transactions, 2015, 58:248-254
7 Jie X, Huang Y, Lou H H. A probability distribution estimation based method for dynamic optimization. AIChE Journal, 2010, 53(7):1805-1816
8 Hirmajer T, Balsa-Canto E, Banga J R. DOTcvpSB, a software toolbox for dynamic optimization in systems biology.BMC Bioinformatics, 2009, 10(1):199
9 Bredies K, Lorenz D A, Maass P. An optimal control problem in medical image processing. In:Proceedings of the IFIP-TC7 Conference, Turin, Italy:DBLP, 2005:249-259
10 Barve H A, Banavar R N. Energy-optimal control of a particle in a dielectrophoretic system. In:Proceedings of the2010 IEEE Conference on Decision&Control. New York,USA:IEEE, 2010:3353-3358
11 Barve H A, Banavar R N. Energy-optimal control of a particle in a dielectrophoretic system. In:Proceedings of the2010 IEEE Conference on Decision&Control. New York,USA:IEEE, 2010:3353-3358
12 Sun Yong, Zhang Mao-Rui, Liang Xiao-Ling. Improved gauss pseudospectral method for solving nonlinear optimal control problem with complex constraints. Acta Automatica Sinica, 2013, 39(5):672-678(孙勇,张卯瑞,梁晓玲.求解含复杂约束非线性最优控制问题的改进Gauss伪谱法.自动化学报, 2013, 39(5):672-678)
13 Biegler L T. An overview of simultaneous strategies for dynamic optimization. Chemical Engineering&Processing Process Intensification, 2007, 46(11):1043-1053
14 Biegler L T. Nonlinear programming:concepts, algorithms,and applications to chemical processes. Society for Industrial and Applied Mathematics, 2010.
15 Peng Hai-Jun, Gao Qiang, Wu Zhi-Gang, Zhong Wan-Xie. A mixed variable variational method for optimal control problems with applications in aerospace control. Acta Automatica Sinica, 2011, 37(10):1248-1255(彭海军,高强,吴志刚,钟万勰.求解最优控制问题的混合变量变分方法及其航天控制应用.自动化学报, 2011, 37(10):1248-1255)
16 Chachuat B, Mitsos A, Barton P I. Optimal design and steady-state operation of micro power generation employing fuel cells. Chemical Engineering Science, 2005, 60(16):4535-4556
17 Betts J T, Huffman W P. Application of sparse nonlinear programming to trajectory optimization. Journal of Guidance Control Dynamics, 1992, 15(1):198-206
18 Fu J, Faust J M M, Chachuat B, Mitsosc1 A. Local optimization of dynamic programs with guaranteed satisfaction of path constraints. Automatica, 2015, 62(C):184-192
19 Hu Yun-Qing, Liu Xing-Gao, Xue An-Ke. A penalty method for solving inequality path constrained optimal control problems. Acta Automatica Sinica, 2013, 29(12):1996-2001(胡云卿,刘兴高,薛安克.带不等式路径约束最优控制问题的惩罚函数法.自动化学报, 2013, 39(12):1996-2001)
20 Qian Ji-Xin, Song Chun-Yue, Wang Ke-Xin, Cheng Yang.Nonlinear Predictive Control, Beijing:Science Press, 2015.(钱积新,宋春跃,王可心,陈扬.非线性预测控制,北京:科学出版社, 2015.)
21 Pontryagin L S, Boltyanskii V G, Gamkrelidze R V,Mishchenko E F. The mathematical theory of optimal processes. Interscience, 1962.
22 Fabien B C. A technique for the direct optimization of dynamic systems described by differential-algebraic equations.Optimal Control Applications&Methods, 2007, 29(6):445-466
23 Brenan K E, Campbell S L, Petzold L R. Numerical solution of initial-value problems in differential-algebraic equations.North-Holland, 1989.
24 Styczen K, Drag P. A modified multipoint shooting feasibleSQP method for optimal control of DAE systems. In:Proceedings of Computer Science and Information Systems,New York, USA:IEEE, 2011. 477-484
25 Feehery W F, Barton P I. Dynamic simulation and optimization with inequality path constraints. Computers&Chemical Engineering, 1996, 20(3):169-176
26 Jacobson D, Lele M. A transformation technique for optimal control problems with a state variable inequality constraint. IEEE Transactions on Automatic Control, 1969,14(5):457-464
27 Vassiliadis V S, Sargent R W H, Pantelides C C. Solution of a class of multistage dynamic optimization problems. 2.problems with path constraints. Industrial&Engineering Chemistry Research, 1994, 10(9):2122-2133
28 Floudas C A, Stein O. The adaptive convexification algorithm:a feasible point method for semi-infinite programming. SIAM Journal on Optimization, 2008, 18(4):1187-1208
29 Chachuat B. Nonlinear and dynamic optimization:from theory to practice. Automatic Control Laboratory EPFL,2007, 107(Spring):192-193
30 Rehbock V, Teo K L, Jennings L S, Lee C S. An exact penalty function approach to all-time-step constrained discrete-time optimal control problems. Applied Mathematics&Computation, 1992, 49(2-3):215-230
31 Loxton R C, Teo K L, Rehbock V, Yiu K F C. Optimal control problems with a continuous inequality constraint on the state and the control. Automatica, 2009, 45(10):2250-2257
32 Liu X, Hu Y, Feng J, Liu K. A novel penalty approach for nonlinear dynamic optimization problems with inequality path constraints. IEEE Transactions on Automatic Control,2014, 59(10):2863-2867
33 Liu P, Li X, Liu X, Hu Y. An improved smoothing techniquebased control vector parameterization method for optimal control problems with inequality path constraints. Optimal Control Applications&Methods, 2016, 38(4):586-600
34 Chen T W C, Vassiliadis V S. Inequality path constraints in optimal control:a finite iterationε-convergent scheme based on pointwise discretization. Journal of Process Control, 2005, 15(3):353-362
35 Biegler L T. An overview of simultaneous strategies for dynamic optimization. Chemical Engineering&Processing Process Intensification, 2007, 46(11):1043-1053
36 Vassiliadis V. Computational solution of dynamic optimization problems with general differential-algebraic constraints by. Journal of Guidance Control&Dynamics, 1993, 15(2):457-460
37 Teo K L, Goh C J, Wong K H. A unified computational approach to optimal control problems. Longman Scientific and Technical, 1991:2763-2774
38 Martin R, Teo K L. Optimal control of drug administration in cancer chemotherapy. World Scientific, 1994.
39 Gunn D J, Thomas W J. Mass transport and chemical reaction in multifunctional catalyst systems. Chemical Engineering Science, 1965, 20(2):89-100
40 Hirmajer T, Fikar M, Balsa-Canto E, Banga J R. DOTcvp:dynamic optimization toolbox with control vector parameterization approach. 2008.
41 Irizarry R. A generalized framework for solving dynamic optimization problems using the artificial chemical process paradigm:applications to particulate processes and discrete dynamic systems. Chemical Engineering Science, 2005,60(21):5663-5681
42 Huang Y J, Reklaitis G V, Venkatasubramanian V. Model decomposition based method for solving general dynamic optimization problems. Computers&Chemical Engineering, 2002, 26(6):863-873