Comments on direct transcription solution of DAE constrained optimal control problems with two discretization approaches

There have been a number of results in the literature showing that a direct transcription numerical approach to optimal control problems could have a number of surprising and desirable behaviors that were not always predicted by the existing theory especially when differential algebraic equations are involved. Most of these results were developed with an implicit Runge-Kutta (IRK) discretization being used. It is important to know which of these observations hold for direct transcription software using different types of discretizations and which are discretization specific. This paper reexamines several of these questions but using an hp-pseudospectral code. It is seen that while philosophically the results are often similar to the IRK results, there are some differences that should be understood by users solving constrained optimal control problems. This paper also discusses a type of regularization for higher index problems that does not reduce the index.

[1]  M. R. Osborne,et al.  The numerical solution of differential equations , 1961 .

[2]  L. Biegler.  Nonlinear programming strategies for dynamic chemical process optimization , 2014, Theoretical Foundations of Chemical Engineering.

[3]  Anil V. Rao,et al.  GPOPS-II , 2014, ACM Trans. Math. Softw..

[4]  Stephen L. Campbell,et al.  Convergence of Nonconvergent IRK Discretizations of Optimal Control Problems with State Inequality Constraints , 2001, SIAM J. Sci. Comput..

[5]  Mohammad Mahdi Hosseini.  Adomian decomposition method for solution of nonlinear differential algebraic equations , 2006, Appl. Math. Comput..

[6]  Stephen L. Campbell,et al.  Direct transcription solution of high index optimal control problems and regular Euler-Lagrange equations , 2007 .

[7]  Mohammad Mahdi Hosseini.  Numerical Solution of Linear High-Index DAEs , 2004, ICCSA.

[8]  René Lamour,et al.  Differential-Algebraic Equations: A Projector Based Analysis , 2013 .

[9]  R. März,et al.  Index Criteria for Differential Algebraic Equations Arising from Linear-Quadratic Optimal Control Problems , 2006 .

[10]  Volker Mehrmann,et al.  Regularization of linear and nonlinear descriptor systems , 2011 .

[11]  Esmail Babolian,et al.  System of linear ordinary differential and differential-algebraic equations and pseudo-spectral method , 2010, Comput. Math. Appl..

[12]  John T. Betts,et al.  Practical Methods for Optimal Control and Estimation Using Nonlinear Programming , 2009 .

[13]  Anil V. Rao,et al.  Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method , 2010, TOMS.

[14]  F. Ghanbari,et al.  CONVERGENCE ANALYSIS OF THE PSEUDOSPECTRAL METHOD FOR LINEAR DAEs OF INDEX-2 , 2013 .

[15]  Stephen L. Campbell,et al.  Direct transcription solution of higher-index optimal control problems and the virtual index , 2007 .

[16]  M. M. Hosseini.  Pseudospectral method for numerical solution of DAEs with an error estimation , 2005, Appl. Math. Comput..

[17]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[18]  J.T. Betts,et al.  Direct transcription solution of inequality constrained optimal control problems , 2004, Proceedings of the 2004 American Control Conference.

[19]  W. Hager,et al.  An hp‐adaptive pseudospectral method for solving optimal control problems , 2011 .

[20]  L. Biegler,et al.  Control and Optimization with Differential-Algebraic Constraints , 2012 .

[21]  A. Engelsone,et al.  Order of Convergence in the Direct Transcription Solution of Optimal Control Problems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[22]  Volker Mehrmann,et al.  Differential-Algebraic Equations: Analysis and Numerical Solution , 2006 .

[23]  D. Jacobson,et al.  New necessary conditions of optimality for control problems with state-variable inequality constraints , 1971 .

[24]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[25]  John T. Betts,et al.  Optimal Low Thrust Trajectories to the Moon , 2003, SIAM J. Appl. Dyn. Syst..

[26]  Roswitha März.  Chapter 3: Notes on Linearization of DAEs and on Optimization with Differential-Algebraic Constraints , 2012 .

[27]  Lorenz T. Biegler,et al.  Advantages of Nonlinear-Programming-Based Methodologies for Inequality Path-Constrained Optimal Control Problems - A Numerical Study , 2008, SIAM J. Sci. Comput..

[28]  Volker Mehrmann,et al.  Chapter 2: Regularization of Linear and Nonlinear Descriptor Systems , 2012 .

[29]  J. Betts,et al.  Direct transcription solution of optimal control problems with higher order state constraints: theory vs practice , 2007 .

[30]  Anil V. Rao,et al.  Practical Methods for Optimal Control Using Nonlinear Programming , 1987 .

[31]  Lorenz T. Biegler,et al.  Convergence rates for direct transcription of optimal control problems using collocation at Radau points , 2008, Comput. Optim. Appl..

[32]  Lorenz T. Biegler,et al.  Simultaneous dynamic optimization strategies: Recent advances and challenges , 2006, Comput. Chem. Eng..