2016, 6(4): 447-472. doi: 10.3934/naco.2016020

Solving higher index DAE optimal control problems

1. 

Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695-8205

2. 

Mathematisches Institut, Universität Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany

Received  October 2016 Revised  November 2016 Published  December 2016

A number of methods have been proposed for solving optimal control problems where the process being optimized is described by a differential algebraic equation (DAE). However, many of these methods require special circumstances to hold or the user to have special software. In this paper we go over many of these options and discuss what is usually necessary for them to be successful. We use a nonlinear index three control problem to illustrate many of our observations..
Citation: Stephen Campbell, Peter Kunkel. Solving higher index DAE optimal control problems. Numerical Algebra, Control and Optimization, 2016, 6 (4) : 447-472. doi: 10.3934/naco.2016020
References:
[1]

R. Altmann and J. Heiland, Simulation of multibody systems with servo constraints through optimal control, Oberwolfach Preprint OWP 2015-18, 2015.

[2]

J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comp. Meth. Appl. Mech. Eng., 1 (1972), 1-16.

[3]

J. T. Betts, Practical Methods for Optimal Control and Estimation using Nonlinear Programming, 2nd ed., Philadelphia, SIAM, 2010. doi: 10.1137/1.9780898718577.

[4]

J. T. Betts and S. O. Erb, Optimal low thrust trajectories to the moon, SIAM J. Appl. Dyn. Syst., 2 (2003), 144-170. doi: 10.1137/S1111111102409080.

[5]

J. T. Betts, S. L. Campbell and A. Engelsone, Direct transcription solution of optimal control problems with higher order state constraints: theory vs practice, Optim. Eng., 8 (2007), 1-19. doi: 10.1007/s11081-007-9000-8.

[6]

H. G. Bock, M. M. Diehl, D. B. Leineweber and J. P. Schlöder, A direct multiple shooting method for real-time optimization of nonlinear DAE processes, Nonlinear model predictive control (Ascona, 1998), Progr. Systems Control Theory, 26, Birkhöuser, Basel, (2000), 245-267.

[7]

K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1996.

[8]

C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real time control, J. Comp. Appl. Math., 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8.

[9]

S. L. Campbell and J. T. Betts, Comments on direct transcription solution of constrained optimal control problems with two discretization approaches, Numerical Algorithms, 73 (2016), 807-838. doi: 10.1007/s11075-016-0119-6.

[10]

S. L. Campbell and R. März, Direct transcription solution of high index optimal control problems and regular Euler-Lagrange equations, J. Comp. Appl. Math., 202 (2007), 186-202. doi: 10.1016/j.cam.2006.02.024.

[11]

S. L. Campbell, P. Kunkel and V. Mehrmann, Regularization of linear and nonlinear descriptor systems, in Control and Optimization with Differential-Algebraic Constraints(eds. L. T. Biegler, S. L. Campbell and V. Mehrmann) SIAM, Philadelphia, (2012), 17-34. doi: 10.1137/9781611972252.ch2.

[12]

C. L. Darby, W. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502. doi: 10.1002/oca.957.

[13]

A. L. Dontchev and W. W. Hager, A new approach to Lipschitz continuity in state constrained optimal control, Syst. Control Lett., 35 (1998), 137-143. doi: 10.1016/S0167-6911(98)00043-7.

[14]

A. L. Dontchev and W. W. Hager, Lipschitzian stability for state constrained nonlinear optimal control, SIAM J. Control Optim., 36 (1998), 698-718. doi: 10.1137/S0363012996299314.

[15]

A. Engelsone, S. L. Campbell and J. T. Betts, Direct transcription solution of higher-index optimal control problems and the virtual index, Appl. Numer. Math., 57 (2007), 281-296. doi: 10.1016/j.apnum.2006.03.012.

[16]

A. Engelsone, S. L. Campbell, and J. T. Betts, Order of convergence in the direct transcription solution of optimal control problems, Proc. IEEE Conf. Decision Control - European Control Conference, Seville, Spain, 2005.

[17]

W. F. Feehery, J. R. Banga and P. I. Barton, A novel approach to dynamic optimization of ODE and DAE systems as high-index problems, AICHE annual meeting, Miami, 1995.

[18]

W. F. Feehery and P. I. Barton, Dynamic simulation and optimization with inequality path constraints, Comp. Chem. Eng., 20 (1996), S707-S712.

[19]

F. Ghanbari and F. Goreishi, Convergence analysis of the pseudospectral method for linear DAEs of index-2, Int. J. Comp. Methods, 10 (2013), 1350019-1-1350019-20. doi: 10.1142/S0219876213500199.

[20]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Numer. Math., 87 (2000), 247-280. doi: 10.1007/s002110000178.

[21]

D. H. Jacobsen, M. M. Lele and J. L. Speyer, New necessary conditions of optimality for control problems with state variable inequality constraints, J. Math. Anal. Appl., 35 (1971), 255-284.

[22]

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, European Mathematical Society, Zürich, 2006. doi: 10.4171/017.

[23]

P. Kunkel and V. Mehrmann, Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index, Math. Control Signal, 20 (2008), 227-269. doi: 10.1007/s00498-008-0032-1.

[24]

P. Kunkel and V. Mehrmann, Formal adjoints of linear DAE operators and their role in optimal control, Electron. J. Linear Algebra, 22 (2011), 672-693. doi: 10.13001/1081-3810.1466.

[25]

P. Kunkel, V. Mehrmann and I. Seufer, GENDA: A software package for the numerical solution of general nonlinear differential-algebraic equations, Institut für Mathematik, TU Berlin Technical Report 730, Berlin, Germany, 2002.

[26]

P. Kunkel, V. Mehrmann and R. Stöver, Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index, Numer. Math., 98 (2004), 277-304. doi: 10.1007/s00211-004-0534-9.

[27]

P. Kunkel and R. Stöver, Symmetric collocation methods for linear differential-algebraic boundary value problems, Numer. Math., 91 (2002), 475-501. doi: 10.1007/s002110100315.

[28]

R. Lamour, R. März and E. Weinmüller, Boundary-value problems for differential algebraic equations: a survey, Surveys in Differential Algebraic Equations III, Springer, 2015.

[29]

R. Pytlak, Runge-Kutta based procedure for the optimal control of differential-algebraic equations, J. Optim. Theory Appl., 97 (1998), 675-705. doi: 10.1023/A:1022698311155.

[30]

A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Trans. Math. Software, 37 (2010), 22:1-22:39.

[31]

A. Steinbrecher, M001 - The simple pendulum, Preprint 2015/26, Technische Universität Berlin, Institut of Mathematik, 2015.

[32]

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.

show all references

References:
[1]

R. Altmann and J. Heiland, Simulation of multibody systems with servo constraints through optimal control, Oberwolfach Preprint OWP 2015-18, 2015.

[2]

J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comp. Meth. Appl. Mech. Eng., 1 (1972), 1-16.

[3]

J. T. Betts, Practical Methods for Optimal Control and Estimation using Nonlinear Programming, 2nd ed., Philadelphia, SIAM, 2010. doi: 10.1137/1.9780898718577.

[4]

J. T. Betts and S. O. Erb, Optimal low thrust trajectories to the moon, SIAM J. Appl. Dyn. Syst., 2 (2003), 144-170. doi: 10.1137/S1111111102409080.

[5]

J. T. Betts, S. L. Campbell and A. Engelsone, Direct transcription solution of optimal control problems with higher order state constraints: theory vs practice, Optim. Eng., 8 (2007), 1-19. doi: 10.1007/s11081-007-9000-8.

[6]

H. G. Bock, M. M. Diehl, D. B. Leineweber and J. P. Schlöder, A direct multiple shooting method for real-time optimization of nonlinear DAE processes, Nonlinear model predictive control (Ascona, 1998), Progr. Systems Control Theory, 26, Birkhöuser, Basel, (2000), 245-267.

[7]

K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, SIAM, Philadelphia, PA, 1996.

[8]

C. Büskens and H. Maurer, SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real time control, J. Comp. Appl. Math., 120 (2000), 85-108. doi: 10.1016/S0377-0427(00)00305-8.

[9]

S. L. Campbell and J. T. Betts, Comments on direct transcription solution of constrained optimal control problems with two discretization approaches, Numerical Algorithms, 73 (2016), 807-838. doi: 10.1007/s11075-016-0119-6.

[10]

S. L. Campbell and R. März, Direct transcription solution of high index optimal control problems and regular Euler-Lagrange equations, J. Comp. Appl. Math., 202 (2007), 186-202. doi: 10.1016/j.cam.2006.02.024.

[11]

S. L. Campbell, P. Kunkel and V. Mehrmann, Regularization of linear and nonlinear descriptor systems, in Control and Optimization with Differential-Algebraic Constraints(eds. L. T. Biegler, S. L. Campbell and V. Mehrmann) SIAM, Philadelphia, (2012), 17-34. doi: 10.1137/9781611972252.ch2.

[12]

C. L. Darby, W. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Applications and Methods, 32 (2011), 476-502. doi: 10.1002/oca.957.

[13]

A. L. Dontchev and W. W. Hager, A new approach to Lipschitz continuity in state constrained optimal control, Syst. Control Lett., 35 (1998), 137-143. doi: 10.1016/S0167-6911(98)00043-7.

[14]

A. L. Dontchev and W. W. Hager, Lipschitzian stability for state constrained nonlinear optimal control, SIAM J. Control Optim., 36 (1998), 698-718. doi: 10.1137/S0363012996299314.

[15]

A. Engelsone, S. L. Campbell and J. T. Betts, Direct transcription solution of higher-index optimal control problems and the virtual index, Appl. Numer. Math., 57 (2007), 281-296. doi: 10.1016/j.apnum.2006.03.012.

[16]

A. Engelsone, S. L. Campbell, and J. T. Betts, Order of convergence in the direct transcription solution of optimal control problems, Proc. IEEE Conf. Decision Control - European Control Conference, Seville, Spain, 2005.

[17]

W. F. Feehery, J. R. Banga and P. I. Barton, A novel approach to dynamic optimization of ODE and DAE systems as high-index problems, AICHE annual meeting, Miami, 1995.

[18]

W. F. Feehery and P. I. Barton, Dynamic simulation and optimization with inequality path constraints, Comp. Chem. Eng., 20 (1996), S707-S712.

[19]

F. Ghanbari and F. Goreishi, Convergence analysis of the pseudospectral method for linear DAEs of index-2, Int. J. Comp. Methods, 10 (2013), 1350019-1-1350019-20. doi: 10.1142/S0219876213500199.

[20]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Numer. Math., 87 (2000), 247-280. doi: 10.1007/s002110000178.

[21]

D. H. Jacobsen, M. M. Lele and J. L. Speyer, New necessary conditions of optimality for control problems with state variable inequality constraints, J. Math. Anal. Appl., 35 (1971), 255-284.

[22]

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution, European Mathematical Society, Zürich, 2006. doi: 10.4171/017.

[23]

P. Kunkel and V. Mehrmann, Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index, Math. Control Signal, 20 (2008), 227-269. doi: 10.1007/s00498-008-0032-1.

[24]

P. Kunkel and V. Mehrmann, Formal adjoints of linear DAE operators and their role in optimal control, Electron. J. Linear Algebra, 22 (2011), 672-693. doi: 10.13001/1081-3810.1466.

[25]

P. Kunkel, V. Mehrmann and I. Seufer, GENDA: A software package for the numerical solution of general nonlinear differential-algebraic equations, Institut für Mathematik, TU Berlin Technical Report 730, Berlin, Germany, 2002.

[26]

P. Kunkel, V. Mehrmann and R. Stöver, Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index, Numer. Math., 98 (2004), 277-304. doi: 10.1007/s00211-004-0534-9.

[27]

P. Kunkel and R. Stöver, Symmetric collocation methods for linear differential-algebraic boundary value problems, Numer. Math., 91 (2002), 475-501. doi: 10.1007/s002110100315.

[28]

R. Lamour, R. März and E. Weinmüller, Boundary-value problems for differential algebraic equations: a survey, Surveys in Differential Algebraic Equations III, Springer, 2015.

[29]

R. Pytlak, Runge-Kutta based procedure for the optimal control of differential-algebraic equations, J. Optim. Theory Appl., 97 (1998), 675-705. doi: 10.1023/A:1022698311155.

[30]

A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Trans. Math. Software, 37 (2010), 22:1-22:39.

[31]

A. Steinbrecher, M001 - The simple pendulum, Preprint 2015/26, Technische Universität Berlin, Institut of Mathematik, 2015.

[32]

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57. doi: 10.1007/s10107-004-0559-y.

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