This paper presents the role of vector relative degree in the formulation of stationarity conditions of optimal control problems for affine control systems. After translating the dynamics into a normal form, we study the hamiltonian structure. Stationarity conditions are rewritten with a limited number of variables. The approach is demonstrated on two and three inputs systems, then, we prove a formal result in the general case. A mechanical system example serves as illustration.
Mots-clés : optimal control, inversion, adjoint states, normal form
@article{COCV_2008__14_2_294_0,
author = {Petit, Nicolas and Chaplais, Fran\c{c}ois},
title = {Inversion in indirect optimal control of multivariable systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {294--317},
publisher = {EDP-Sciences},
volume = {14},
number = {2},
year = {2008},
doi = {10.1051/cocv:2007054},
mrnumber = {2394512},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2007054/}
}
TY - JOUR AU - Petit, Nicolas AU - Chaplais, François TI - Inversion in indirect optimal control of multivariable systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 294 EP - 317 VL - 14 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007054/ DO - 10.1051/cocv:2007054 LA - en ID - COCV_2008__14_2_294_0 ER -
%0 Journal Article %A Petit, Nicolas %A Chaplais, François %T Inversion in indirect optimal control of multivariable systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 294-317 %V 14 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007054/ %R 10.1051/cocv:2007054 %G en %F COCV_2008__14_2_294_0
Petit, Nicolas; Chaplais, François. Inversion in indirect optimal control of multivariable systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 2, pp. 294-317. doi : 10.1051/cocv:2007054. http://www.numdam.org/articles/10.1051/cocv:2007054/
[1] and , On abnormal extremals for Lagrange variational problems. J. Math. Systems Estim. Control 1 (1998) 87-118. | MR | Zbl
[2] and , A new efficient method for optimization of a class of nonlinear systems without Lagrange multipliers. J. Optim. Theor. Appl 97 (1998) 11-28. | MR | Zbl
[3] , and , Collocation software for boundary-value ODE's. ACM Trans. Math. Software 7 (1981) 209-222. | Zbl
[4] , and , Numerical solution of boundary value problems for ordinary differential equations. Prentice Hall Series in Computational Mathematics Prentice Hall, Inc., Englewood Cliffs, NJ (1988). | MR | Zbl
[5] , and , Numerical solution of boundary value problems for ordinary differential equations, Classics in Applied Mathematics 13. Society for Industrial and Applied Mathematics (SIAM) (1995). | MR | Zbl
[6] , Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn 21 (1998) 193-207. | Zbl
[7] , Practical methods for optimal control using nonlinear programming, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001). | MR | Zbl
[8] and , Singular trajectories and their role in control theory, Mathématiques & applications 40. Springer-Verlag-Berlin-Heidelberg-New York (2003). | MR | Zbl
[9] and , Applied Optimal Control. Ginn and Company (1969).
[10] , and , 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. | MR | Zbl
[11] , , and , 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). | MR | Zbl
[12] and , Geometric Control of Mechanical Systems, Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics 49. Springer-Verlag (2004). | MR | Zbl
[13] and , Asymptotic stabilization of minimum phase nonlinear systems. IEEE Trans. Automat. Control 36 (1991) 1122-1137. | MR | Zbl
[14] and , Inversion in indirect optimal control2003).
[15] , A Chebyshev finite difference method for solving a class of optimal control problems. Int. J. Comput. Math 80 (2003) 883-895. | MR | Zbl
[16] and , Direct trajectory optimization by a Chebyshev pseudo-spectral method. J. Guid. Control Dyn 25 (2002) 160-166.
[17] , and , Differentially flat systems with inequality constraints: An approach to real-time feasible trajectory generation. J. Guid. Control Dyn 24 (2001) 219-227.
[18] , , and , Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control 61 (1995) 1327-1361. | MR | Zbl
[19] , , and , A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (1999) 922-937. | MR | Zbl
[20] , , and , User's Guide for NPSOL 5.0: A Fortran Package for Nonlinear Programming. Systems Optimization Laboratory, Stanford University, Stanford, CA 94305 (1998).
[21] and , Direct trajectory optimization using nonlinear programming and collocation. AIAA J. Guid. Control 10 (1987) 338-342. | Zbl
[22] , Nonlinear Control Systems. Springer, New York, 2nd edn. (1989).
[23] , Nonlinear Control Systems II. Springer, London-Berlin-Heidelberg (1999). | MR
[24] , Optimization by vector spaces methods. Wiley-Interscience (1997). | MR | Zbl
[25] , Real-time optimal trajectory generation for constrained systems. Ph.D. thesis, California Institute of Technology (2003).
[26] , and , A new computational approach to real-time trajectory generation for constrained mechanical systems, in IEEE Conference on Decision and Control (2000).
[27] , and , Real-time constrained trajectory generation applied to a flight control experiment, in Proc. of the IFAC World Congress (2002).
[28] , Abnormal minimizers. SIAM J. Control Optim 32 (1994) 1605-1620. | MR | Zbl
[29] , , , , , and , 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.
[30] , and , Collocation and inversion for a reentry optimal control problem, in Proc. of the 5th Intern. Conference on Launcher Technology (2003).
[31] and , Nonlinear Dynamical Control Systems. Springer-Verlag (1990). | MR | Zbl
[32] and , Flatness and higher order differential model representations in dynamic optimization. Comput. Chem. Eng 26 (2002) 385-400.
[33] , and , Inversion based constrained trajectory optimization2001).
[34] and , Pseudospectral methods for optimal motion planning of differentially flat systems2002).
[35] , and , Exploiting higher-order derivatives in computational optimal control, in Proc. of the 2002 IEEE Mediterranean Conference (2002).
[36] , Trajectory optimization based on differential inclusion. J. Guid. Control Dyn 17 (1994) 480-487. | MR | Zbl
[37] and , Method for automatic costate calculation. J. Guid. Control Dyn 19 (1996) 1252-1261. | Zbl
[38] and , Time-optimal control of axi-symmetric rigid spacecraft using two controls. J. Guid. Control Dyn 22 (1999) 682-694.
[39] and , Differentially Flat Systems. Control Engineering Series, Marcel Dekker (2004). | Zbl
[40] , Optimal motion design using inverse dynamics. Technical report, Konrad-Zuse-Zentrum für Informationstechnik Berlin (1997).
[41] . Trajectory generation for nonlinear control systems. Ph.D. thesis, California Institute of Technology (1996).
[42] and , Direct and indirect methods for trajectory optimization. Ann. Oper. Res 37 (1992) 357-373. | MR | Zbl
Cité par Sources :
