Optimization on Quadratic Matrix Lie Groups

Conference on Decision and Control, 2006

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E 3 , the spheres S 3 and the hyperboloids H 3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

IEEE Transactions on Automatic Control, 2013

Many nonlinear systems of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. Examples range from aircraft and underwater vehicles to quantum mechanical systems. In this paper, we develop an algorithm for solving continuous time optimal control problems for systems evolving on (noncompact) Lie groups. This algorithm generalizes the projection operator approach for trajectory optimization originally developed for systems on vector spaces. Notions for generalizing system theoretic tools such as Riccati equations and linear and quadratic system approximations are developed. In this development, the covariant derivative of a map between two manifolds plays a key role in providing a chain rule for the required Lie group computations. An example optimal control problem on SO(3) is provided to highlight implementation details and to demonstrate the effectiveness of the method.

2010

Let S(A) denote the orbit of a complex or real matrix A under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix A 0 by the sum of matrices in S(A 1 ), . . . , S(A N ) in the sense of finding the Euclidean least-squares distance

Proceedings of the 44th IEEE Conference on Decision and Control, 2005

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E 3 , the spheres S 3 and the hyperboloids H 3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

Applied Numerical Mathematics, 2001

In recent years several numerical methods have been developed to integrate matrix differential systems of ODEs whose solutions remain on a certain Lie group throughout the evolution. In this paper some results, derived for the orthogonal group in by , will be extended to the class of quadratic groups including the symplectic and Lorentz group. We will show how this approach also applies to ODEs on the Stiefel manifold and the orthogonal factorization of the Lorentz group will be derived. Furthermore, we will consider the numerical solution of important problems such as the Penrose regression problem, the calculation of Lyapunov exponents of Hamiltonian systems, the solution of Hamiltonian isospectral problems. Numerical tests will show the performance of our numerical methods.

2000

In this paper we consider numerical methods for solving nonlinear equations on matrix Lie groups. Recently Owren and Welfert (Technical Report Numerics, No 3/1996, Norwegian University of Science and Technology, Trondheim, Norway, 1996) have proposed a method where the original nonlinear equation F (Y)= 0 is transformed into a nonlinear equation on the Lie algebra of the group, thus Newton-type methods may be applied which require the evaluation of exponentials of matrices.

8th IFAC Symposium on Nonlinear Control Systems, 2010

In this paper, we investigate a generalization of the infinite time horizon linear quadratic regulator (LQR) for systems evolving on the special orthogonal group SO(3). Using Pontryagin's Maximum Principle, we derive the necessary conditions for optimality and the associated Hamiltonian equations. For a special class of weighting matrices, we show that the optimal feedback can be computed explicitly and we prove that the non differentiable value function is the viscosity solution of an appropriate Hamiltn-Jacobi-Bellman equation on SO(3). For arbitrary positive definite weighting matrices, numerical simulations allow us to explore the relationship between the optimal trajectories and weighting matrices, and in particular to highlight nontrivial non differentiability properties of the value function.

2011

This paper discusses key implementation details required for computing the solution of a continuous-time optimal control problem on a Lie group using the projection operator approach. In particular, we provide the explicit formulas to compute the time-varying linear quadratic problem which defines the search direction step of the algorithm. We also show that the projection operator approach on Lie groups generates a sequence of adjoint state trajectories that converges, as a local minimum is approached, to the adjoint state trajectory of the first order necessary conditions of the Pontryagin's Maximum Principle, placing it between direct and indirect optimization methods. As illustrative example, an optimization problem on SO(3) is introduced and numerical results of the projection operator approach are presented, highlighting second order converge rate of the method.

Electronic Journal of Linear Algebra, 2015

For given Z, B ∈ C^{n\times k}, the problem of finding A ∈ C^{n\times n}, in some prescribed class W, that minimizes ||AZ − B|| (Frobenius norm) has been considered by different authors for distinct classes W. Here, this minimization problem is studied for two other classes, which include the symmetric Hamiltonian, symmetric skew-Hamiltonian, real orthogonal symplectic and unitary conjugate symplectic matrices. The problem of minimizing ||A − A˜||, where A˜ is given and A is a solution of the previous problem, is also considered (as has been done by others, for different classes W). The key idea of this contribution is the reduction of each one of the above minimization problems to two independent subproblems in orthogonal subspaces of C^{n\times n}. This is possible due to the special structures under consideration. Matlab codes are developed, and numerical results of some tests are presented.

Most of the configuration space of mechanical and non-mechanical problems in physical world involve matrix Lie groups. Especially these Lie groups provide a mathematically rich formulation for studying a variety of motion control problems involving rotating and translating bodies. While control systems have been specialized to different matrix Lie groups, study of optimal control of these systems by minimizing the input cost function has been a tempting research area for physicists and mathematicians. The smallest exceptional Lie group G_2 and its Lie algebra g_2 act locally as the symmetries when a ball performs rolling motion on a larger ball restricted by the radius of the smaller ball being one third of the bigger one. Here we take a left-invariant control system specified on G_2 and study its optimal control as well as stability of the resulting dynamics.