Abstract
An algorithm is described for computing time-varying controls and time-invariant parameters to optimize the performance of a dynamic system whose behaviour is described by differential-algebraic equations (DAEs) of arbitrary index.
The problem formulation deals with a multistage system, in which each stage has its own describing equations. It also accommodates end-point, interior point, and path constraints which can be equalities or inequalities.
The algorithm uses a parameterization of the controls in terms of a given set of basis functions. Path constraints are dealt with via integral constraints. The problem is thus converted into a nonlinear programming problem, for which the objective and constraint functions are evaluated by integration of the system equation of the sensitivity equations.
The special problems arising when the DAE system has index greater than one are explicitly dealth with, using automatic differentiation to generate the necessary additional equations, thus reducing the system index. A constraint stabilization technique for the resulting low index system is proposed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bachmann, R., L. Brüll, T. Mrziglod, and U. Pallaske, “On methods for reducing the index of differential-algebraic equations”, Comput. chem. Engng., 14, pp. 1271–1273 (1990).
Brenan, K.E., S.L. Campbell, and L.R. Petzold, “Numerical solution of initial-value problems in differential-algebraic equations”, North Holland, New York (1989).
Campbell, S.L., “A computational method for general higher index nonlinear singular systems of differential equations” , IMA CS Transactions Sci. Computing, pp. 178–180 (1988).
Chung Y. and A.W. Westerberg, “A proposed numerical algorithm for solving nonlinear index problems”, Ind. Eng. Chem. Res., 29, pp. 1234–1239 (1990).
Cuthrell, J.E. and L.T. Biegler, “On the optimization of differential-algebraic process systems”, AIChE Journal, 33, pp. 1257–1270 (1987).
Gear, C.W., “Differential-algebraic equation index transformations”, SIAM J. Sci. Stat. Comput., 9, pp. 39–47 (1988).
Gear, C.W. and L.R. Petzold, “ODE methods for the solution of differential/algebraic systems”, SIAM J. Numer. Anal., 21, pp. 716–728 (1984).
Goh, C.J. and K.L. Teo, “Control parameterization: a unified approach to optimal control problems with general constraints”, Automatica, 24, pp. 3–18 (1988).
Gritsis, D.M., C.C. Pantelides, and R.W.H. Sargent, “Optimal control of systems described by index-two differential-algebraic equations”, submitted to SIAM J. Sci. Stat. Comput. (1992).
Horwitz, L.B., and P.E. Sarachik, “A computational technique for calculating the optimal control signal for a specific class of problems” , Record of the 2nd Asilomar Conference on Circuits and Systems, pp. 537–540 (1968).
Mattsson, S.E. and G. Söderlind, “Index reduction in differential-algebraic equations using dummy derivatives”, SIAM J. Sci. Stat. Comput., bf 14 pp. 677–692 (1993).
Morison, K.R., and R.W.H. Sargent, “Optimization of multistage processes described by differential-algebraic equations”, Lecture Notes in Mathematics, 1230, pp. 86–102, Springer-Verlag, Berlin (1986).
Pantelides, C.C, “The consistent initialization of differential-algebraic equations”, SIAM J. Sci. Stat. Comput., 7, pp. 720–733 (1988).
Pollard, G.P. and R.W.H. Sargent, “Off-line computation of optimal controls for a plate distillation column”, Automatica, 6, pp. 59–76 (1970).
Renfro, J.C., A.M. Morshedi, and O.A. Absjornsen, “Simultaneous optimization and solution of systems described by differential-algebraic equations”, Comput. chem. Engng., 11, pp. 503–517 (1987).
Rosen, O. and R. Luus, “Evaluation of gradients for piecewise constant optimal control” Comput. chem. Engng., 15, pp. 273–281 (1991).
Sargent, R.W.H. and G.R. Sullivan, “The development of an efficient optimal control package”, Proc. 8th IFIP Conference on Optimization Techniques, Würzburg, 1977. Springer-Verlag, Berlin (1978).
Tsang, T.H., D.M. Himmelblau, and T.F. Edgar, “Optimal control via collocation and nonlinear programming”, Int. J. Control., 21, pp. 763–768 (1975).
Vasantharajan, S. and L.T. Biegler, “Simultaneous strategies for optimization of differential-algebraic systems with enforcement of error criteria”, Comput. chem. Engng., 14, pp. 1083–1100 (1990).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Pantelides, C.C., Sargent, R.W.H., Vassiliadis, V.S. (1994). Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations. In: Bulirsch, R., Kraft, D. (eds) Computational Optimal Control. ISNM International Series of Numerical Mathematics, vol 115. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-8497-6_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8497-6_15
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-5015-4
Online ISBN: 978-3-0348-8497-6
eBook Packages: Springer Book Archive