Constrained discrete optimal control on Lie groups

Course overview

Constraints are inherent in any control design problem, and incorporating constraints upfront during the design stage is beneficial but a challenging task. The course will expose the participants to constrained discrete optimal control techniques for mechanical and aerospace systems using models derived from the discrete mechanics approach. The underlying configuration manifold of many mechanical systems is a Lie group. An extension of Boltyanski’s Discrete Maximum Principle (DMP) (restricted to Euclidean spaces) to Lie groups will be presented as the central tool to solve these problems. The DMP on Lie groups yields necessary conditions for an optimal control history that appears as a two-point boundary value problem. Techniques to solve such two-point boundary value problems will also be discussed.

Topics

1. Tools from convexity (5 hours)

   • Introduction to basic mathematical tools in convexity - convex sets, convex hulls,

   • covering sets, separating hyperplanes, separation of convex cones.

2. Discrete mechanics (5 hours)

   • The motivation and need to adopt the discrete mechanics approach in obtaining discrete models of mechanical systems.

   • The variational approach to deriving a discrete integrator for a mechanical system.

   • The Lagrangian and Hamiltonian viewpoints and relevant mathematical objects.

3. Discrete Maximum Principle on Lie Groups (7 hours)

   • Boltyanski's Maximum Principle for discrete dynamical systems with constraints on Euclidean spaces.

   • The solution procedure; the extension to Lie groups and the Maximum Principle on Lie Groups.

4. Numerical techniques (4 hours)

   • Two numerical schemes to solve the two-point BVPs will be discussed:

     • the multiple shooting technique and

     • the Stochastic Approximation algorithm.