Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T03:39:25.314Z Has data issue: false hasContentIssue false
  • English
  • Français

A set oriented approach to global optimal control

Published online by Cambridge University Press:  15 March 2004

Oliver Junge  and
Hinke M. Osinga
Oliver Junge
Affiliation:
Institute for Mathematics, University of Paderborn, 33095 Paderborn, Germany; junge@upb.de.
Hinke M. Osinga
Affiliation:
Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UK; H.M.Osinga@bristol.ac.uk.
Get access

Abstract

We describe an algorithm for computing the value function for “all source, single destination” discrete-time nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graph-theoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path problem on a finite graph. The method is illustrated by two numerical examples, namely a single pendulum on a cart and a parametrically driven inverted double pendulum.

Keywords

Global optimal controlvalue functionset oriented methodshortest path.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 2 , April 2004 , pp. 259 - 270
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Brooks and T. Lozano-Pérez, A subdivision algorithm in configuration space for findpath with rotation. IEEE Systems, Man and Cybernetics 15 (1985) 224-233.
M. Broucke, A geometric approach to bisimulation and verification of hybrid systems, in HSCC 1999, LNCS, F.W. Vaandragerand and J.H. van Schuppen Eds., Springer 1569 (1999) 61-75.
M. Broucke, M.D. Di Benedetto, S. Di Gennaro and A. Sangiovanni-Vincentelli, Theory of optimal control using bisimulations, in HSCC 2000, LNCS, N. Lynch and B. Krogh Eds., Springer 1790 (2000) 89-102.
M. Broucke, M.D. Di Benedetto, S. Di Gennaro and A. Sangiovanni-Vincentelli, Optimal control using bisimulations: Implementation, in HSCC 2001, LNCS, M.D. Di Benedetto and A. Sangiovanni-Vincentelli Eds., Springer 2034 (2001) 175-188.
T.H. Cormen, C.E. Leierson and R.L. Rivest, Introduction to Algorithms. Cambridge, Mass. MIT Press, New York McGraw-Hill (1990).
Dellnitz, M. and Hohmann, A., A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75 (1997) 293-317. CrossRef
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO – Set oriented numerical methods for dynamical systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler Ed., Springer (2001) 145-174.
Dijkstra, E.W., Note, A on Two Problems in Connection with Graphs. Numer. Math. 5 (1959) 269-271. CrossRef
M. Falcone, Numerical solution of Dynamic Programming equations, in Viscosity solutions and deterministic optimal control problems, M. Bardi and I. Capuzzo Dolcetta Eds., Birkhäuser (1997).
Galias, Z., Interval methods for rigorous investigations of periodic orbits. Int. J. Bifur. Chaos 11 (2001) 2427-2450. CrossRef
Grüne, L., Adaptive Grid Scheme, An for the discrete Hamilton-Jacobi-Bellman Equation. Numer. Math. 75 (1997) 319-337.
P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, User's Guide for NPSOL (Version 4.0): a Fortran package for nonlinear programming, Report SOL 86-2, Systems Optimization Laboratory, Stanford University (1986).
J. Hauser and H.M. Osinga, On the geometry of optimal control: the inverted pendulum example, in Proc. Amer. Control Conf., Arlington VA (2001) 1721-1726. CrossRef
Jadbabaie, A., Yu, J. and Hauser, J., Unconstrained receding horizon control of nonlinear systems. IEEE Trans. Automat. Control 46 (2001) 776-783. CrossRef
O. Junge, Rigorous discretization of subdivision techniques, in Proc. Int. Conf. Differential Equations Equadiff 99, B. Fiedler, K. Gröger and J. Sprekels Eds., World Scientific 2 (2000) 916-918.
Polymenakos, L.C., Bertsekas, D.P. and Tsitsiklis, J.N., Implementation of efficient algorithms for globally optimal trajectories. IEEE Trans. Automat. Control 43 (1998) 278-283. CrossRef
Schiele, K., On the stabilization of a parametrically driven inverted double pendulum. Z. Angew. Math. Mech. 77 (1997) 143-146. CrossRef
Sethian, J.A. and Vladimirsky, A., Ordered upwind methods for static Hamilton-Jacobi equations. Proc. Nat. Acad. Sci. USA 98 (2001) 11069-11074. CrossRef
E.D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Texts in Applied Mathematics 6, Springer (1998).
Szolnoki, D., Viability kernels and control sets. ESAIM: COCV 5 (2000) 175-185. CrossRef
Tsitsiklis, J.N., Efficient algorithms for globally optimal trajectories. IEEE Trans. Automat. Control 40 (1995) 1528-1538. CrossRef
O. von Stryk, User's Guide for DIRCOL (Version 2.1): a direct collocation method for the numerical solution of optimal control problems. TU Darmstadt (2000).