A geometric integrator for optimal control problems

SEBASTIÁN J. FERRARO

Madrid

Workshop; Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series; 2015

Instituto de Ciencias Matemáticas, Madrid

(Observación: el resumen contiene muchas fórmulas en LaTeX, y Sigeva no admite el caracter de barra invertida. Se ha copiado aquí haciendo varios cambios para mejorar la legibilidad.)In this talk we will explore an approach to the geometric integration of optimal trajectories of mechanical systems [1]. Given a mechanical system with a (regular) Lagrangian function L:TQ ->R, its controlled Euler--Lagrange equations are, in local coordinates,[frac{d}{dt}frac{partial L}{partial dot{q}^A}-frac{partial L}{partial q^A}=u_A,]where $u=(u_A) in U subset R^n$ is in an open subset of R^n, the set of control parameters. Given a cost function C: TQxU -> R, we seek to minimize $int_{t_0}^{t_f} C(q^A,dot{q}^A,u_A) $. We can see this problem as a Lagrangian system where the second-order Lagrangian $tilde{L}: T^{(2)}Q -> R$ is given by[tilde{L}(q^A,dot{q}^A,ddot{q}^A)=C(q^A,dot{q}^A,frac{d}{dt}frac{partial L}{partialdot{q}^A}-frac{partial L}{partial q^{A}})](see for example [2]).In previous approaches (see for instance [3, 4, 5]), the theory of discrete variational mechanics for second-order systems was derived using a discrete Lagrangian $L_d: QxQxQ -> R$, where 3 points are used to approximate the positions, velocities and accelerations. Instead of this, we will use a different kind of discrete Lagrangian $L_d: TQxTQ -> R$ and explain some of the advantages.The existence and regularity of the exact discrete Lagrangian will be discussed, as well as the notions of discrete Legendre transforms.Some concrete examples and simulations will be shown, corresponding to the optimal control of fully actuated systems.References[1] Leonardo Colombo, Sebastián Ferraro, and David Martín de Diego. Geometric integrators for higher-order variational systems and their application to optimal control. Preprint arXiv:1410.5766.[2] Anthony M. Bloch. Nonholonomic mechanics and control, volume 24 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York, 2003. With the collaboration of J. Baillieul, P. Crouch and J. Marsden, With scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov, Systems and Control.[3] Roberto Benito, Manuel de León, and David Martín de Diego. Higher-order discrete Lagrangian mechanics. Int. J. Geom. Methods Mod. Phys., 3(3):421-436, 2006.[4] Leonardo Colombo, David Martín de Diego, and Marcela Zuccalli. On variational integrators for optimal control of mechanical control systems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106(1):161-171, 2012.[5] Leonardo Colombo, David Martín de Diego, and Marcela Zuccalli. Higher-order discrete variational problems with constraints. J. Math. Phys., 54(9):093507, 17, 2013.[6] J. E. Marsden and M. West. Discrete mechanics and variational integrators. Acta Numer., 10:357-514, 2001.