April  2011, 29(2): 647-670. doi: 10.3934/dcds.2011.29.647

Numerical procedure for optimal control of higher index DAEs

1. 

Institute of Automatic Control and Robotics, Warsaw University of Technology, 02-525 Warsaw, Poland

Received  September 2009 Revised  March 2010 Published  October 2010

The paper deals with optimal control problems described by higher index DAEs. We introduce a numerical procedure for solving these problems. The procedure, based on the appropriately defined adjoint equations, refers to an implicit Runge--Kutta method for differential--algebraic equations. Assuming that higher index DAEs can be solved numerically the gradients of functionals defining the control problem are evaluated with the help of well--defined adjoint equations. The paper presents numerical examples related to index three DAEs showing the validity of the proposed approach.
Citation: Radoslaw Pytlak. Numerical procedure for optimal control of higher index DAEs. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 647-670. doi: 10.3934/dcds.2011.29.647
References:
[1]

K. Balla, Linear subspaces for linear DAE's of index 1, Comput. Math. Appl., 32 (1996), 81-86.

[2]

K. Balla, Boundary conditions and their transfer for differential algebraic equations of index 1, Comput. Math. Appl., 31 (1996), 1-5.

[3]

K. Balla and R. März, Transfer of boundary conditions for DAE's of index 1, SIAM J. Numer. Anal., 33 (1996), 2318-2332.

[4]

K. Balla and R. März, "An Unified Approach to Linear Differential Algebraic Equations and Their Adjoint Equations," Institute of Mathematics Technical Report, Humboldt University, Berlin, 2000.

[5]

K. Balla, K. and R. März, Linear differential algebraic equations and their adjoint equations, Results in Mathematica, 37 (2000), 13-35.

[6]

K. E. Brenan, S. L. Campbell and L. R. Petzold, "Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations," North-Holland, New York, 1989.

[7]

Y. Cao, S. Li, L. Petzold and R. Serben, Adjoint sensitivity analysis for differential-algebraic equations: Part I, The adjoint DAE system and its numerical solution, SIAM J. Sci. Comp., 24 (2000), 1076-1089.

[8]

Y. Cao, S. Li and L. Petzold, Adjoint sensitivity analysis for differential-algebraic equations: Algorithms and software, J. Comput. Appl. Math., 149 (2002), 171-191.

[9]

W. F. Frenery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differential-algebraic systems, Appl. Numer. Math., 25 (1997), 41-54.

[10]

C. W. Gear, Differential-algebraic equation index transformations, SIAM J. Sci. Stat. Comput., 9 (1988), 39-47.

[11]

C. W. Gear, Differential algebraic equations, indices and integral algebraic equations, SIAM J. Numer. Anal., 27 (1990), 1527-1534.

[12]

D. R. A. Giles, "A Comparison of Three Problem-Oriented Simulation Programs for Dynamic Mechanical Systems," Thesis, Univ. Waterloo, Ontario, 1978.

[13]

E. Hairer, Ch. Lubich and M. Roche, "The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods," Lecture Notes in Mathematics, 1409, Springer-Verlag, Berlin, Heidelberg, 1989.

[14]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II," Springer-Verlag, Berlin Heidelberg New York, 1996.

[15]

A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers," preprint, UCRL-JRNL-200037, 2004.

[16]

L. Kershenbaum, D. Q. Mayne, R. Pytlak and R. B. Vinter, Receding horizon control, in "Advances in Model-Based Predictive Control, Oxford, September 20-21, Conference Proceedings," 1993.

[17]

S. Li and L. R. Petzold, Software and algorithms for sensitivity analysis of large-scale differential-algebraic systems, J. Comput. Appl. Math., 125 (2000), 131-145.

[18]

S. Li, L. R. Petzold and W. Zhu, Sensitivity analysis of differential-algebraic equations: A comparison of methods on a special problem, Appl. Numer. Math., 32 (2000), 161-174.

[19]

T. Maly, and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems, Appl. Numer. Math., 20 (1997), 57-79.

[20]

D. W. Manning, "A Computer Technique for Simulating Dynamic Multibody Systems Based on Dynamic Formalizm," Thesis, Univ. Waterloo, Ontario, 1981.

[21]

R. März, Differential algebraic systems anew, Applied Numetical Mathematics, 42 (2002), 315-335.

[22]

R. März, Characterizing differential algebraic equations without the use of derivative arrays, Int. J. Comp. & Mathem. with Appl., 50 (2005), 1141-1156.

[23]

M. D. R. de Pinho and R. Vinter, Necessary conditions for optimal control problems involving nonlinear differential algebraic equations, J. Math. Anal. Appl., 212 (1997), 493-516.

[24]

R. Pytlak, Optimal control of differential-algebraic equations, Proceed. of the 33rd IEEE CDC, Orlando, Florida, (1994), 951-956.

[25]

R. Pytlak, "Numerical Methods for Optimal Control Problem With State Constraints," Lecture Notes in Mathematics, 1707, Springer-Verlag, 1999.

[26]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 1: First order approximations, J. Optimization Theory and Applications, 134 (2007), 61-75.

[27]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 2: Necessary optimality conditions, J. Optimization Theory and Applications, 134 (2007), 77-90.

[28]

R. Pytlak and R. B. Vinter, A feasible directions algorithm for optimal control problems with state and control constraints: Convergence analysis, SIAM J. Control and Optimization, 36 (1998), 1999-2019.

[29]

R. Pytlak and R. B. Vinter, A feasible directions type algorithm for optimal control problems with state and control constraints: Implementation, J. Optimization Theory and Applications, 101 (1999), 623-649.

[30]

W. Schiehlen (ed.), "Mulitbody Systems Handbook," Springer-Verlag, Berlin, 1990.

show all references

References:
[1]

K. Balla, Linear subspaces for linear DAE's of index 1, Comput. Math. Appl., 32 (1996), 81-86.

[2]

K. Balla, Boundary conditions and their transfer for differential algebraic equations of index 1, Comput. Math. Appl., 31 (1996), 1-5.

[3]

K. Balla and R. März, Transfer of boundary conditions for DAE's of index 1, SIAM J. Numer. Anal., 33 (1996), 2318-2332.

[4]

K. Balla and R. März, "An Unified Approach to Linear Differential Algebraic Equations and Their Adjoint Equations," Institute of Mathematics Technical Report, Humboldt University, Berlin, 2000.

[5]

K. Balla, K. and R. März, Linear differential algebraic equations and their adjoint equations, Results in Mathematica, 37 (2000), 13-35.

[6]

K. E. Brenan, S. L. Campbell and L. R. Petzold, "Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations," North-Holland, New York, 1989.

[7]

Y. Cao, S. Li, L. Petzold and R. Serben, Adjoint sensitivity analysis for differential-algebraic equations: Part I, The adjoint DAE system and its numerical solution, SIAM J. Sci. Comp., 24 (2000), 1076-1089.

[8]

Y. Cao, S. Li and L. Petzold, Adjoint sensitivity analysis for differential-algebraic equations: Algorithms and software, J. Comput. Appl. Math., 149 (2002), 171-191.

[9]

W. F. Frenery, J. E. Tolsma and P. I. Barton, Efficient sensitivity analysis of large-scale differential-algebraic systems, Appl. Numer. Math., 25 (1997), 41-54.

[10]

C. W. Gear, Differential-algebraic equation index transformations, SIAM J. Sci. Stat. Comput., 9 (1988), 39-47.

[11]

C. W. Gear, Differential algebraic equations, indices and integral algebraic equations, SIAM J. Numer. Anal., 27 (1990), 1527-1534.

[12]

D. R. A. Giles, "A Comparison of Three Problem-Oriented Simulation Programs for Dynamic Mechanical Systems," Thesis, Univ. Waterloo, Ontario, 1978.

[13]

E. Hairer, Ch. Lubich and M. Roche, "The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods," Lecture Notes in Mathematics, 1409, Springer-Verlag, Berlin, Heidelberg, 1989.

[14]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II," Springer-Verlag, Berlin Heidelberg New York, 1996.

[15]

A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers," preprint, UCRL-JRNL-200037, 2004.

[16]

L. Kershenbaum, D. Q. Mayne, R. Pytlak and R. B. Vinter, Receding horizon control, in "Advances in Model-Based Predictive Control, Oxford, September 20-21, Conference Proceedings," 1993.

[17]

S. Li and L. R. Petzold, Software and algorithms for sensitivity analysis of large-scale differential-algebraic systems, J. Comput. Appl. Math., 125 (2000), 131-145.

[18]

S. Li, L. R. Petzold and W. Zhu, Sensitivity analysis of differential-algebraic equations: A comparison of methods on a special problem, Appl. Numer. Math., 32 (2000), 161-174.

[19]

T. Maly, and L. R. Petzold, Numerical methods and software for sensitivity analysis of differential-algebraic systems, Appl. Numer. Math., 20 (1997), 57-79.

[20]

D. W. Manning, "A Computer Technique for Simulating Dynamic Multibody Systems Based on Dynamic Formalizm," Thesis, Univ. Waterloo, Ontario, 1981.

[21]

R. März, Differential algebraic systems anew, Applied Numetical Mathematics, 42 (2002), 315-335.

[22]

R. März, Characterizing differential algebraic equations without the use of derivative arrays, Int. J. Comp. & Mathem. with Appl., 50 (2005), 1141-1156.

[23]

M. D. R. de Pinho and R. Vinter, Necessary conditions for optimal control problems involving nonlinear differential algebraic equations, J. Math. Anal. Appl., 212 (1997), 493-516.

[24]

R. Pytlak, Optimal control of differential-algebraic equations, Proceed. of the 33rd IEEE CDC, Orlando, Florida, (1994), 951-956.

[25]

R. Pytlak, "Numerical Methods for Optimal Control Problem With State Constraints," Lecture Notes in Mathematics, 1707, Springer-Verlag, 1999.

[26]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 1: First order approximations, J. Optimization Theory and Applications, 134 (2007), 61-75.

[27]

R. Pytlak, Optimal control of differential-algebraic equations of higher index, Part 2: Necessary optimality conditions, J. Optimization Theory and Applications, 134 (2007), 77-90.

[28]

R. Pytlak and R. B. Vinter, A feasible directions algorithm for optimal control problems with state and control constraints: Convergence analysis, SIAM J. Control and Optimization, 36 (1998), 1999-2019.

[29]

R. Pytlak and R. B. Vinter, A feasible directions type algorithm for optimal control problems with state and control constraints: Implementation, J. Optimization Theory and Applications, 101 (1999), 623-649.

[30]

W. Schiehlen (ed.), "Mulitbody Systems Handbook," Springer-Verlag, Berlin, 1990.

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