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.

Keywords: 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://archive.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://archive.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://archive.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, Volume 14 (2008) no. 2, pp. 294-317. doi : 10.1051/cocv:2007054. http://archive.numdam.org/articles/10.1051/cocv:2007054/

[1] On abnormal extremals for Lagrange variational problems. J. Math. Systems Estim. Control 1 (1998) 87-118. | MR | Zbl

and ,[2] 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

and ,[3] Collocation software for boundary-value ODE's. ACM Trans. Math. Software 7 (1981) 209-222. | Zbl

, and ,[4] 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

, and ,[5] 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

, and ,[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] Singular trajectories and their role in control theory, Mathématiques & applications 40. Springer-Verlag-Berlin-Heidelberg-New York (2003). | MR | Zbl

and ,[9] Applied Optimal Control. Ginn and Company (1969).

and ,[10] 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

, and ,[11] 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

, , and ,[12] Geometric Control of Mechanical Systems, Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics 49. Springer-Verlag (2004). | MR | Zbl

and ,[13] Asymptotic stabilization of minimum phase nonlinear systems. IEEE Trans. Automat. Control 36 (1991) 1122-1137. | MR | Zbl

and ,[14] Inversion in indirect optimal control2003).

and ,[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] Direct trajectory optimization by a Chebyshev pseudo-spectral method. J. Guid. Control Dyn 25 (2002) 160-166.

and ,[17] Differentially flat systems with inequality constraints: An approach to real-time feasible trajectory generation. J. Guid. Control Dyn 24 (2001) 219-227.

, and ,[18] Flatness and defect of nonlinear systems: introductory theory and examples. Int. J. Control 61 (1995) 1327-1361. | MR | Zbl

, , and ,[19] A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (1999) 922-937. | MR | Zbl

, , and ,[20] User's Guide for NPSOL 5.0: A Fortran Package for Nonlinear Programming. Systems Optimization Laboratory, Stanford University, Stanford, CA 94305 (1998).

, , and ,[21] Direct trajectory optimization using nonlinear programming and collocation. AIAA J. Guid. Control 10 (1987) 338-342. | Zbl

and ,[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] A new computational approach to real-time trajectory generation for constrained mechanical systems, in IEEE Conference on Decision and Control (2000).

, and ,[27] Real-time constrained trajectory generation applied to a flight control experiment, in Proc. of the IFAC World Congress (2002).

, and ,[28] Abnormal minimizers. SIAM J. Control Optim 32 (1994) 1605-1620. | MR | Zbl

,[29] 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.

, , , , , and ,[30] Collocation and inversion for a reentry optimal control problem, in Proc. of the 5th Intern. Conference on Launcher Technology (2003).

, and ,[31] Nonlinear Dynamical Control Systems. Springer-Verlag (1990). | MR | Zbl

and ,[32] Flatness and higher order differential model representations in dynamic optimization. Comput. Chem. Eng 26 (2002) 385-400.

and ,[33] Inversion based constrained trajectory optimization2001).

, and ,[34] Pseudospectral methods for optimal motion planning of differentially flat systems2002).

and ,[35] Exploiting higher-order derivatives in computational optimal control, in Proc. of the 2002 IEEE Mediterranean Conference (2002).

, and ,[36] Trajectory optimization based on differential inclusion. J. Guid. Control Dyn 17 (1994) 480-487. | MR | Zbl

,[37] Method for automatic costate calculation. J. Guid. Control Dyn 19 (1996) 1252-1261. | Zbl

and ,[38] Time-optimal control of axi-symmetric rigid spacecraft using two controls. J. Guid. Control Dyn 22 (1999) 682-694.

and ,[39] Differentially Flat Systems. Control Engineering Series, Marcel Dekker (2004). | Zbl

and ,[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] Direct and indirect methods for trajectory optimization. Ann. Oper. Res 37 (1992) 357-373. | MR | Zbl

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