Page 121 - Textos de Matemática Vol. 43

ON OPTIMAL INFINITE-TIME REGULATION ON SO(3) ALESSANDRO SACCON, JOHN HAUSER AND A. PEDRO AGUIAR
Dedicated to Professor Fa´tima Silva Leite on the occasion of her 60th birthday.
Abstract. In this paper we investigate the problem of extending the infinite time horizon linear quadratic regulator (LQR) for systems evolving on the special orthogonal group SO(3). Applying Pontryagin’s Maximum Principle, we derive the necessary conditions for optimality and the associated Hamiltonian equations. For a particular class of cost functions that depend explicitly on the state and the input, we derive a closed-form solution of the optimal feedback and prove that the non di↵erentiable value function is the viscosity solution of an appropriate Hamilton-Jacobi-Bellman equation on SO(3).
1. Introduction
Motivated by the Linear Quadratic Regulator (LQR) problem, which addresses
the computation of the optimal feedback u?(t) that minimizes the cost functional (1.1) Z 1xT(⌧)Qx(⌧)+uT(⌧)Ru(⌧)d⌧,
0
(1.2) x˙(t)=Ax(t)+Bu(t), x(0)=x0 2Rn,
in this paper we focus on a similar problem but with the state of the system evolv- ing on the special orthogonal group SO(3). The weighting matrices Q and R a↵ect the closed loop behavior of the system, providing a penalty on the state and input, respectively. In this paper we propose a modification of the quadratic cost function in (1.1) such that it makes intrinsic sense on SO(3) and resembles on this manifold a quadratic cost. As far as the authors know, we could not find a solution to this problem in the literature with exception of our recent results reported in [6], but these ones deal with the kinematic equations expressed in the spatial frame. Closely related results can be found in the field of optimal control problems for systems evolving on Riemannian manifolds [3],[4],[5] and in the rich literature on the optimal
2000 Mathematics Subject Classification. 49Q, 22E.
Key words and phrases. Lie groups, Optimal Control, Rotational Matrices.
This work was supported by projects CONAV/FCT-PT (PTDC/EEA-CRO/113820/2009),
Co3-AUVs (EU FP7 no.231378), FCT-ISR/IST plurianual funding program, and the CMU- Portugal program. The work of the second author was supported by AFOSR FA9550-09-1-0470 and by an invited scientist grant from the Foundation for Science and Technology (FCT), Portugal. 109
given the linear system