Exploiting invariants in the numerical solution of multipoint boundary value problems for DAE. (English) Zbl 0947.65096
The paper presents a new approach to the numerical solution of boundary value problems (BVPs) for higher-index differential-algebraic equations (DAEs). Invariants known for the original DAE as well as invariants of the reduced index 1 formulation are exploited to stabilize initial value problem (IVP) integration, derivative generation, and the solution of nonlinear and linear systems by an enhanced multiple shooting method.
The paper concentrates on two important classes of BVPs for DAEs: parameter estimation in descriptor form models for multi-body systems, treatment of singular controls and state constraints in optimal control. Applications for two important problem classes are presented: parameter estimation in multi-body systems given in descriptor form, and singular and state-constrained optimal control problems. In particular, generalizations of the “internal numerical differentiation” technique to DAEs with invariants and a new multistage least squares decomposition technique for DAE boundary value problems are developed, which are implemented in the multiple shooting code PARFIT and in the collocation code COLFIT.
The paper concentrates on two important classes of BVPs for DAEs: parameter estimation in descriptor form models for multi-body systems, treatment of singular controls and state constraints in optimal control. Applications for two important problem classes are presented: parameter estimation in multi-body systems given in descriptor form, and singular and state-constrained optimal control problems. In particular, generalizations of the “internal numerical differentiation” technique to DAEs with invariants and a new multistage least squares decomposition technique for DAE boundary value problems are developed, which are implemented in the multiple shooting code PARFIT and in the collocation code COLFIT.
Reviewer: A.Dishliev (Sofia)
MSC:
| 65L80 | Numerical methods for differential-algebraic equations |
| 70F10 | \(n\)-body problems |
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
Keywords:
invariants; boundary value problems; higher-index differential-algebraic equations; parameter identification; descriptor form for multi-body systems; optimal control; singular control; state constraints; multiple shooting method; collocationSoftware:
COLNEWReferences:
| [1] | T. Alishenas, Zur numerischen Behandlung, Stabilisierung durch Projektion und Modellierung mechanischer Systeme mit Nebenbedingungen und Invarianten, Ph.D. thesis, Royal Institute of Technology, Stockholm, Sweden, 1992. |
| [2] | Th. Andrzejewski, H. G. Bock, E. Eich, and R. von Schwerin, Recent advances in the numerical integration of multibody systems, in Advanced Multibody System Dynamics, Simulation and Software Tools, W. Schiehlen, ed., Kluwer Academic Publishers, Dordrecht, Boston, London, 1993, pp. 127-151. · Zbl 0800.93581 |
| [3] | U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1988. · Zbl 0671.65063 |
| [4] | U. M. Ascher and L. R. Petzold, Projected Collocation for Higher‐Order Higher‐Index Differential‐Algebraic Equations, Technical report 91‐9, Department of Computer Science, University of British Columbia, Vancouver, British Columbia, Canada, 1991. · Zbl 0772.65049 |
| [5] | Ascher, U. M.Qin, H.Reich, S.Stabilization of DAEs and invariant manifoldsNumer. Math.671994pp. 131149 |
| [6] | Ascher, U. M.Spiteri, R. J.Collocation software for boundary value Differentialalgebraic equationsSIAM J. Sci. Comput.151994pp. 938952 |
| [7] | Baumgarte, J.Stabilization of constraints and integrals of motion in dynamical systemsComput. Math. Appl. Mech. Engrg.11976pp. 116 · Zbl 0262.70017 |
| [8] | H. G. Bock, Numerische Behandlung von zustandsbeschränkten und Chebyshev‐Steuerungsproblemen, Technical report, Carl‐Cranz‐Gesellschaft, Oberpfaffenhofen, Germany, 1981. |
| [9] | H. G. Bock, Recent advances in parameter identification techniques for O.D.E., in Numerical Treatment of Inverse Problems, P. Deuflhard and E. Hairer, eds., Progress in Scientific Computing 2, Birkhäuser, Boston, 1983. · Zbl 0516.65067 |
| [10] | H. G. Bock, Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen, Bonner Mathematische Schriften Nr. 183, University of Bonn, Bonn, Germany, 1987. · Zbl 0622.65064 |
| [11] | H. G. Bock, E. Eich, and J. P. Schlöder, Numerical solution of constrained least squares boundary value problems in Differential‐algebraic equations, in Numerical Treatment of Differential Equations, K. Strehmel, ed., Teubner, Leipzig, Germany, 1988. · Zbl 0682.65047 |
| [12] | H. G. Bock and R. von Schwerin, An Inverse Dynamics ADAMS‐Method for Constrained Multibody Systems, Preprint 93‐27, IWR, University of Heidelberg, Heidelberg, Germany, 1993. |
| [13] | K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial‐Value Problems in Differential‐Algebraic Equations, North-Holland, Amsterdam, 1989. · Zbl 0699.65057 |
| [14] | Bryson, A. E.Denham, W. F.Dreyfus, S. E.Optimal programming problems with inequality constraintsI, AIAA J.11963pp. 25442550 · Zbl 0142.35902 |
| [15] | P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School Monographs, No. 3, Yeshiva University, New York, 1964. |
| [16] | E. Eich, Projizierende Mehrschrittverfahren zur Numerischen Lösung von Bewegungsgleichungen Technischer Mehrkörpersysteme mit Zwangsbedingungen und Unstetigkeiten, Ph.D. thesis, University of Augsburg, Augsburg, Germany, 1991; Fortschritts‐Berichte VDI, Reihe 18, Nr. 109, VDI‐Verlag, Düsseldorf, Germany, 1992. · Zbl 0780.70001 |
| [17] | C. Führer, Differential‐algebraische Gleichungssysteme in mechanischen Mehrkörpersystemen, Ph.D. thesis, Munich University of Technology, München, Germany, 1988. · Zbl 0688.70001 |
| [18] | E. Griepentrog and R. März, Differential‐Algebraic Equations and Their Numerical Treatment, Teubner‐Text Math., 88, Teubner, Leipzig, 1986. · Zbl 0629.65080 |
| [19] | A. Griewank, On automatic differentiation, in Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, eds., Kluwer Academic Publishers, Dordrecht, Boston, London, 1989, pp. 83-108. · Zbl 0696.65015 |
| [20] | F. Grupp and W. Kortüm, Parameter identification of nonlinear descriptor systems, in Advanced Multibody System Dynamics, Simulation and Software Tools, W. Schiehlen, ed., Kluwer Academic Publishers, Dordrecht, Boston, London, 1993, pp. 457-462. · Zbl 0800.93580 |
| [21] | Hanke, M.On a least‐squares collocation method for linear Differential‐algebraic equationsNumer. Math.541988pp. 7990 · Zbl 0643.65042 |
| [22] | de Hoog, F. R.Mattheij, R. M. M.On dichotomy and well conditioning in BVPSIAM J. Numer. Anal.241987pp. 89105 · Zbl 0629.65084 |
| [23] | H. J. Kelley, R. E. Kopp, and H. G. Moyer, Singular extremals, in Topics in Optimization, G. Leitmann, ed., Academic Press, New York, 1967. |
| [24] | R. Lamour, Shooting Methods for Transferable DAE’s—Solution of Shooting Equation, Vol. XXIV, Banach Center Publications, Warsaw, Poland, 1990. · Zbl 0716.65070 |
| [25] | Lamour, R.A well‐posed shooting method for transferable DAE’sNumer. Math.591991pp. 815829 · Zbl 0723.65060 |
| [26] | Leimkuhler, B.Reich, S.Symplectic integration of constrained Hamiltonian systemsMath. Comp.631994pp. 589605 · Zbl 0813.65103 |
| [27] | Lentini, M.März, R.The conditioning of boundary value problems in transferable Differential‐algebraic equationsSIAM J. Numer. Anal.271990pp. 10011015 · Zbl 0702.65075 |
| [28] | Lentini, M.März, R.Conditioning and dichotomy in Differential algebraic equationsSIAM J. Numer. Anal.271990pp. 15191526 · Zbl 0732.65062 |
| [29] | März, R.On boundary value problems in Differential‐algebraic equationsAppl. Math. Comput.311989pp. 517537 · Zbl 0671.65065 |
| [30] | R. Mehlhorn, K. Lesch, and G. Sachs, A technique for improving stability and efficiency in singular control problems, in Proc. 1993 AIAA Guidance, Navigation, and Control Conf., Monterey, CA, 1993. |
| [31] | G. Sachs and K. Lesch, Periodic maximum range cruise with singular control, in Proc. 1990 AIAA Guidance, Navigation, and Control Conf., Portland, OR, 1990. |
| [32] | G. Sachs, K. Lesch, H. G. Bock, and M. Steinbach, Periodic optimal trajectories with singular control for aircraft with high aerodynamic efficiency, in Optimal Control: Calculus of Variations, Optimal Control Theory and Numerical Methods, R. Bulirsch, A. Miele, J. Stoer, and K. H. Well, eds., International Series in Numerical Mathematics, 111, Birkhäuser, Basel, Boston, Berlin, 1993, pp. 289-304. · Zbl 0790.93105 |
| [33] | J. P. Schlöder, Numerische Methoden zur Behandlung hochdimensionaler Aufgaben der Parameteridentifizierung, Bonner Mathematische Schriften Nr. 187, University of Bonn, Bonn, Germany, 1988. · Zbl 0639.65036 |
| [34] | V. Schulz, Ein effizientes Kollokationsverfahren zur numerischen Behandlung von Mehrpunktrandwertaufgaben in der Parameteridentifizierung und Optimalen Steuerung, Diploma thesis, University of Augsburg, Augsburg, Germany, 1990. |
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