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 & 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), 2015. Google Scholar

[2]

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

[3]

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

[4]

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

[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. doi: 10.1007/s11081-007-9000-8. Google Scholar

[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, (2000), 245. Google Scholar

[7]

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

[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. doi: 10.1016/S0377-0427(00)00305-8. Google Scholar

[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. doi: 10.1007/s11075-016-0119-6. Google Scholar

[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. doi: 10.1016/j.cam.2006.02.024. Google Scholar

[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, (2012), 17. doi: 10.1137/9781611972252.ch2. Google Scholar

[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. doi: 10.1002/oca.957. Google Scholar

[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. doi: 10.1016/S0167-6911(98)00043-7. Google Scholar

[14]

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

[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. doi: 10.1016/j.apnum.2006.03.012. Google Scholar

[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, (2005). Google Scholar

[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, (1995). Google Scholar

[18]

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

[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. doi: 10.1142/S0219876213500199. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[23]

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

[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. doi: 10.13001/1081-3810.1466. Google Scholar

[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, (2002). Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[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), 1. Google Scholar

[31]

A. Steinbrecher, M001 - The simple pendulum,, Preprint 2015/26, (2015). Google Scholar

[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. doi: 10.1007/s10107-004-0559-y. Google Scholar

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), 2015. Google Scholar

[2]

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

[3]

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

[4]

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

[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. doi: 10.1007/s11081-007-9000-8. Google Scholar

[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, (2000), 245. Google Scholar

[7]

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

[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. doi: 10.1016/S0377-0427(00)00305-8. Google Scholar

[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. doi: 10.1007/s11075-016-0119-6. Google Scholar

[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. doi: 10.1016/j.cam.2006.02.024. Google Scholar

[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, (2012), 17. doi: 10.1137/9781611972252.ch2. Google Scholar

[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. doi: 10.1002/oca.957. Google Scholar

[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. doi: 10.1016/S0167-6911(98)00043-7. Google Scholar

[14]

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

[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. doi: 10.1016/j.apnum.2006.03.012. Google Scholar

[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, (2005). Google Scholar

[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, (1995). Google Scholar

[18]

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

[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. doi: 10.1142/S0219876213500199. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[23]

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

[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. doi: 10.13001/1081-3810.1466. Google Scholar

[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, (2002). Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[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), 1. Google Scholar

[31]

A. Steinbrecher, M001 - The simple pendulum,, Preprint 2015/26, (2015). Google Scholar

[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. doi: 10.1007/s10107-004-0559-y. Google Scholar

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