联立法中全局和局部正交配置算法
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摘要
化工过程的动态优化控制命题大多可以写成微分代数混合方程组(differential algebraic equations)的形式,联立法是求解该类命题的一种重要的数值方法。目前联立法中常用的离散方法是局部法,其中有限元正交配置(orthogonal collocation on finite elements)具有精度高、计算量小、稳定性好等优点。然而伪谱法(pseudo-spectral method)作为一种全局法,在离散中也有独特的优点,特别是具有指数级的收敛速度和较高精度,而且产生的NLP规模较小。本文分别以有限元正交配置法和伪谱法代表局部法和全局法比较其原理,并讨论离散配置点以及其在两种方法上的不同应用,针对离散后两种方法产生的NLP,分别提出判据以保证足够的自由度,最后用连续DAEs与不连续优化控制两个例子进一步比较这两种方法得出,如果命题平滑采用PS方法具有更好的收敛速度。
Most of the dynamic optimal control problems in chemical engineering can be written in the form of differential algebraic equations(DAEs).Simultaneous approach is an important method for solving these problems.The discretization strategies often used in this approach are local methods,for instance, orthogonal collocation on finite elements(OCFE),which has many advantages.Pseudo-spectral(PS),as a global method,has its own unique properties.It may offer a rapid convergence rate for the approximation of analytic functions and has high precision and low computational effort with a simple structure.As the representations of local method and global method respectively,OCFE and PS were compared.The collocation points and their different distributions were presented,and the degree of freedom(DOF) of non-linear programming(NLP) after discretization was discussed,as well as the criteria were offered to ensure the DOF of the NLPs.At last,a continuous case and a discontinuous case were studied,and it was concluded that if the problem was smooth enough then the convergence of PS was better than that of OCFE.
Most of the dynamic optimal control problems in chemical engineering can be written in the form of differential algebraic equations(DAEs).Simultaneous approach is an important method for solving these problems.The discretization strategies often used in this approach are local methods,for instance, orthogonal collocation on finite elements(OCFE),which has many advantages.Pseudo-spectral(PS),as a global method,has its own unique properties.It may offer a rapid convergence rate for the approximation of analytic functions and has high precision and low computational effort with a simple structure.As the representations of local method and global method respectively,OCFE and PS were compared.The collocation points and their different distributions were presented,and the degree of freedom(DOF) of non-linear programming(NLP) after discretization was discussed,as well as the criteria were offered to ensure the DOF of the NLPs.At last,a continuous case and a discontinuous case were studied,and it was concluded that if the problem was smooth enough then the convergence of PS was better than that of OCFE.
引文
[1]Kameswaran Shivakumar,Biegler Lorenz T.Simultaneous dynamic optimization strategies:recent advances and challenges.Computers and Chemical Engineering,2006, 30:1560-1575
[2]Huntington Geoffrey Todd.Advancement and analysis of a Gauss pseudo-spectral transcription for optimal control problems[D].United States:Massachusetts Institute of Technology,2007
[3]Biegler Lorenz T,Cervantes Arturo M,Andreas Wachter. Advances in simultaneous strategies for dynamic process optimization.Chemical Engineering Science,2002,57: 575-593
[4]Kameswaran Shivakumar,Biegler Lorenz T.Convergence rates for direct transcription of optimal control problems using collocation at Radau points.Comput.Optim.,2008, 41:81-126
[5]Tong Kewei(童科伟),Zhou Jianping(周建平),He Linshu(何麟书).Legendre Gauss pseudospectral method for solving optimal control problem.Acta Aeronautica ET Astronautica Sinica(航空学报),2008,29:1531-1538
[6]Gong Qi,Kang Wei,Ross I Michael.A pseudospectral method for the optimal control of constrained feedback linearizable systems.IEEETransoction on Automatic Control,2006,51:1115-1129
[7]Canuto C,Hussaini M Y.Spectral Methods in Fluid Dynamics.Berlin:Springer-Verlag,1988:53-67
[8]Liu Lubo(刘鲁波),Chen Xiaofei(陈晓非),Wang Yanbin(王彦宾).Chebyshev pseudospectral method for seismic wave field modeling.Northwestern Seismological Journal(西北地震学报),2007,29:18-25
[9]Elnagar Gamal N,Kazemi M A.A cell-averaging Chebyshev spectral method for the controlled duffing oscillator.Applied Numerical Mathematics,1995,18: 461-471
[10]Shen Jie,Tang Tao.Spectral and High-order Methods with Applications.Beijing:Science Press,2006:7-17
[11]Ascher Urt M,Petzold Linda R.Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations.United States:Baker& Taylor Books,1997: 270-276
[12]Logsdon J S,Biegler L T.Accurate solution of differentialalgebraic optimization problems.Ind.Engng.Chem., 1989,28:1628-1639
[13]Ciarlet P G,Iserles A,Kohn R V,Wright M H.A Practical Guide to Pseudospectral Methods.New York: Cambridge University Press,1996:23-28
[14]Haire E,Wanner G.Solving Ordinary Differential Equations(Ⅱ):Stiff and Differential-algebraic Problems. 2nd ed.Berlin/Heibelbeg:Springer-Verlag,Science Press,2006:71-83
[15]Kenneth Holmstrom,Anders O Goran,Marcus M Edvall.User's guide for TOMLAB 7[OL], Inc.TOMLAB,2008 http://tomopt.com/tomlab/ download/productsheets.php
[16]Ross I M,Fahroo F.Pseudospectral knotting methods for solving optimal control problems.Journal of Guidance, Control,and Dynamics,2004,27:397-405
[17]Ross I M,Fahroo F.A direct method for solving nonsmooth optimal control problems//Luis Basanez,Juan A de la Puente//Proceedings of the 15th world congress of the international federation on automatic control.Spain: Barcelona,2003:477-482
[2]Huntington Geoffrey Todd.Advancement and analysis of a Gauss pseudo-spectral transcription for optimal control problems[D].United States:Massachusetts Institute of Technology,2007
[3]Biegler Lorenz T,Cervantes Arturo M,Andreas Wachter. Advances in simultaneous strategies for dynamic process optimization.Chemical Engineering Science,2002,57: 575-593
[4]Kameswaran Shivakumar,Biegler Lorenz T.Convergence rates for direct transcription of optimal control problems using collocation at Radau points.Comput.Optim.,2008, 41:81-126
[5]Tong Kewei(童科伟),Zhou Jianping(周建平),He Linshu(何麟书).Legendre Gauss pseudospectral method for solving optimal control problem.Acta Aeronautica ET Astronautica Sinica(航空学报),2008,29:1531-1538
[6]Gong Qi,Kang Wei,Ross I Michael.A pseudospectral method for the optimal control of constrained feedback linearizable systems.IEEETransoction on Automatic Control,2006,51:1115-1129
[7]Canuto C,Hussaini M Y.Spectral Methods in Fluid Dynamics.Berlin:Springer-Verlag,1988:53-67
[8]Liu Lubo(刘鲁波),Chen Xiaofei(陈晓非),Wang Yanbin(王彦宾).Chebyshev pseudospectral method for seismic wave field modeling.Northwestern Seismological Journal(西北地震学报),2007,29:18-25
[9]Elnagar Gamal N,Kazemi M A.A cell-averaging Chebyshev spectral method for the controlled duffing oscillator.Applied Numerical Mathematics,1995,18: 461-471
[10]Shen Jie,Tang Tao.Spectral and High-order Methods with Applications.Beijing:Science Press,2006:7-17
[11]Ascher Urt M,Petzold Linda R.Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations.United States:Baker& Taylor Books,1997: 270-276
[12]Logsdon J S,Biegler L T.Accurate solution of differentialalgebraic optimization problems.Ind.Engng.Chem., 1989,28:1628-1639
[13]Ciarlet P G,Iserles A,Kohn R V,Wright M H.A Practical Guide to Pseudospectral Methods.New York: Cambridge University Press,1996:23-28
[14]Haire E,Wanner G.Solving Ordinary Differential Equations(Ⅱ):Stiff and Differential-algebraic Problems. 2nd ed.Berlin/Heibelbeg:Springer-Verlag,Science Press,2006:71-83
[15]Kenneth Holmstrom,Anders O Goran,Marcus M Edvall.User's guide for TOMLAB 7[OL], Inc.TOMLAB,2008 http://tomopt.com/tomlab/ download/productsheets.php
[16]Ross I M,Fahroo F.Pseudospectral knotting methods for solving optimal control problems.Journal of Guidance, Control,and Dynamics,2004,27:397-405
[17]Ross I M,Fahroo F.A direct method for solving nonsmooth optimal control problems//Luis Basanez,Juan A de la Puente//Proceedings of the 15th world congress of the international federation on automatic control.Spain: Barcelona,2003:477-482
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