Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T00:56:27.367Z Has data issue: false hasContentIssue false
  • English
  • Français

Inversion in indirect optimal controlof multivariable systems

Published online by Cambridge University Press:  20 March 2008

François Chaplais  and
Nicolas Petit
François Chaplais
Affiliation:
Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 35 rue Saint-Honoré, 77305 Fontainebleau Cedex, France; [email protected]
Nicolas Petit
Affiliation:
Centre Automatique et Systèmes, École Nationale Supérieure des Mines de Paris, 60 bd Saint-Michel, 75272 Paris Cedex 06, France; [email protected]
Get access

Abstract

This paper presents the role of vector relative degree in theformulation of stationarity conditions of optimal control problemsfor affine control systems. After translating the dynamics into anormal form, we study the Hamiltonian structure. Stationarityconditions are rewritten with a limited number of variables. Theapproach is demonstrated on two and three inputs systems, then, weprove a formal result in the general case. A mechanical systemexample serves as illustration.

Keywords

Optimal controlinversionadjoint statesnormal form
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 2 , April 2008 , pp. 294 - 317
Copyright
© EDP Sciences, SMAI, 2008

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

Agrachev, A.A. and Sarychev, A.V., On abnormal extremals for Lagrange variational problems. J. Math. Systems Estim. Control 1 (1998) 87–118.
Agrawal, S.K. and Faiz, N., A new efficient method for optimization of a class of nonlinear systems without Lagrange multipliers. J. Optim. Theor. Appl 97 (1998) 11–28. CrossRef
Ascher, U.M., Christiansen, J. and Russel, R.D., Collocation software for boundary-value ODE's. ACM Trans. Math. Software 7 (1981) 209–222. CrossRef
U.M. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical solution of boundary value problems for ordinary differential equations. Prentice Hall Series in Computational Mathematics Prentice Hall, Inc., Englewood Cliffs, NJ (1988).
U.M. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical solution of boundary value problems for ordinary differential equations, Classics in Applied Mathematics 13 . Society for Industrial and Applied Mathematics (SIAM) (1995).
Betts, J.T., Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn 21 (1998) 193–207. CrossRef
J.T. Betts, Practical methods for optimal control using nonlinear programming, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001).
B. Bonnard and M. Chyba, Singular trajectories and their role in control theory, Mathématiques & applications 40 . Springer-Verlag-Berlin-Heidelberg-New York (2003).
A.E. Bryson and Y.C. Ho, Applied Optimal Control. Ginn and Company (1969).
Bulirsch, R., Montrone, F. and Pesch, H.J., Abort landing in the presence of windshear as a minimax optimal control problem, part 2: Multiple shooting and homotopy. J. Optim. Theor. Appl 70 (1991) 223–254. CrossRef
R. Bulirsch, E. Nerz, H.J. Pesch and O. von Stryk, Combining direct and indirect methods in optimal control: Range maximization of a hang glider, in Optimal Control, R. Bulirsch, A. Miele, J. Stoer and K.H. Well Eds., International Series of Numerical Mathematics, Birkhäuser 111 (1993).
F. Bullo and A.D. Lewis, Geometric Control of Mechanical Systems, Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics 49 . Springer-Verlag (2004).
Byrnes, C.I. and Isidori, A., Asymptotic stabilization of minimum phase nonlinear systems. IEEE Trans. Automat. Control 36 (1991) 1122–1137. CrossRef
F. Chaplais and N. Petit, Inversion in indirect optimal control, in Proc. of the 7th European Control Conf (2003).
El-Kady, M., Chebyshev, A finite difference method for solving a class of optimal control problems. Int. J. Comput. Math 80 (2003) 883–895. CrossRef
Fahroo, F. and Ross, I.M., Direct trajectory optimization by a Chebyshev pseudo-spectral method. J. Guid. Control Dyn 25 (2002) 160–166. CrossRef
Faiz, N., Agrawal, S.K. and Murray, R.M., Differentially flat systems with inequality constraints: An approach to real-time feasible trajectory generation. J. Guid. Control Dyn 24 (2001) 219–227. CrossRef
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control 61 (1995) 1327–1361. CrossRef
Fliess, M., Lévine, J., Martin, P. and Rouchon, P., Lie-Bäcklund, A approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (1999) 922–937. CrossRef
P.E. Gill, W. Murray, M.A. Saunders and M.A. Wright, User's Guide for NPSOL 5.0: A Fortran Package for Nonlinear Programming. Systems Optimization Laboratory, Stanford University, Stanford, CA 94305 (1998).
Hargraves, C. and Paris, S., Direct trajectory optimization using nonlinear programming and collocation. AIAA J. Guid. Control 10 (1987) 338–342. CrossRef
A. Isidori, Nonlinear Control Systems. Springer, New York, 2nd edn. (1989).
A. Isidori, Nonlinear Control Systems II. Springer, London-Berlin-Heidelberg (1999).
D.G. Luenberger, Optimization by vector spaces methods. Wiley-Interscience (1997).
M.B. Milam, Real-time optimal trajectory generation for constrained systems. Ph.D. thesis, California Institute of Technology (2003).
M.B. Milam, K. Mushambi and R.M. Murray, A new computational approach to real-time trajectory generation for constrained mechanical systems, in IEEE Conference on Decision and Control (2000).
M.B. Milam, R. Franz and R.M. Murray, Real-time constrained trajectory generation applied to a flight control experiment, in Proc. of the IFAC World Congress (2002).
Montgomery, R., Abnormal minimizers. SIAM J. Control Optim 32 (1994) 1605–1620. CrossRef
R.M. Murray, J. Hauser, A. Jadbabaie, M.B. Milam, N. Petit, W.B. Dunbar and R. Franz, Online control customization via optimization-based control, in Software-Enabled Control, Information technology for dynamical systems, T. Samad and G. Balas Eds., Wiley-Interscience (2003) 149–174.
T. Neckel, C. Talbot and N. Petit, Collocation and inversion for a reentry optimal control problem, in Proc. of the 5th Intern. Conference on Launcher Technology (2003).
H. Nijmeijer and A.J. van der Schaft, Nonlinear Dynamical Control Systems. Springer-Verlag (1990).
Oldenburg, J. and Marquardt, W., Flatness and higher order differential model representations in dynamic optimization. Comput. Chem. Eng 26 (2002) 385–400. CrossRef
N. Petit, M.B. Milam and R.M. Murray, Inversion based constrained trajectory optimization, in 5th IFAC Symposium on Nonlinear Control Systems (2001).
I.M. Ross and F. Fahroo, Pseudospectral methods for optimal motion planning of differentially flat systems, in Proc. of the 41th IEEE Conf. on Decision and Control (2002).
I.M. Ross, J. Rea and F. Fahroo, Exploiting higher-order derivatives in computational optimal control, in Proc. of the 2002 IEEE Mediterranean Conference (2002).
Seywald, H., Trajectory optimization based on differential inclusion. J. Guid. Control Dyn 17 (1994) 480–487. CrossRef
Seywald, H. and Kumar, R.R., Method for automatic costate calculation. J. Guid. Control Dyn 19 (1996) 1252–1261. CrossRef
Shen, H. and Tsiotras, P., Time-optimal control of axi-symmetric rigid spacecraft using two controls. J. Guid. Control Dyn 22 (1999) 682–694. CrossRef
H. Sira-Ramirez and S.K. Agrawal, Differentially Flat Systems. Control Engineering Series, Marcel Dekker (2004).
M.C. Steinbach, Optimal motion design using inverse dynamics. Technical report, Konrad-Zuse-Zentrum für Informationstechnik Berlin (1997).
M.J. van Nieuwstadt. Trajectory generation for nonlinear control systems. Ph.D. thesis, California Institute of Technology (1996).
von Stryk, O. and Bulirsch, R., Direct and indirect methods for trajectory optimization. Ann. Oper. Res 37 (1992) 357–373. CrossRef