SHOC

Hybrid Optimal Control

Optimal control of Hybrid Ordinary Differential Systems

 [Presentation] [Papers] [Code] [Contacts]

This project consists in providing generic algorithms with both symbolic and numerical modules to solve nonlinear optimal control problems.

We consider a general nonlinear dynamical system which state is described by the solution of the following ODE:

We want to present a generic algorithm for controlling the system (1) from an initial state X0 at time t=0 to a final state Xf at unspecified time tf using the admissible control functions u that take values in a convex and compact polyhedral set Umof Rm, in such a way that:

is minimized. The polytope Um is defined as the convex hull of its vertices:

Affine

We first consider general linear systems. The system (1) is replaced by the following one: X'(t)=AX(t) + Bu(t). We then provide a generic algorithm with both symbolic and numerical modules. Especially we propose new algorithm under-approximating  the controllable domain in view of its analytical resolution in the context of singular subarcs. New efficient methods computing a block Kalman canonical decomposition and the optimal solutions are also presented.

Hybridization

Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given an optimal sequence of states, we  are then able to traverse the automaton till the target, locally insuring the optimality.

[Papers]

For further informations about this work, have a look at:

[Source Code]

You can also download our source code:

• Dimension 2, a first step towards generic algorithms (src, preview), Kevin Hamon.

• A full algorithm to solve any linear optimal control problem in 2D whose dynamic is described by: X'(t)=Bu(t). Admissible controls take values in any 2 dimensional polytope.

•  In any dimension (src)
Available in this library:
• Geometric tools for polytope manipulation.
• Hybrid computation: modelisation of nonlinear systems by hybrid systems.
• Controllability.
• Optimal solutions computation (in progress).
(Full description and user's guide soon available)

To run it, you need:

• Maple at least 8.0

[Contacts]

 Aude RONDEPIERRE Laboratoire de Modélisation et Calcul B.P. 53 -- 51, av. des Mathématiques, 38041 Grenoble, France. Aude.Rondepierre@imag.fr http://ljk.imag.fr/membres/Aude.Rondepierre Jean-Guillaume DUMAS Laboratoire de Modélisation et Calcul B.P. 53 -- 51, av. des Mathématiques, 38041 Grenoble, France Jean-Guillaume.Dumas@imag.fr http://ljk.imag.fr/membres/Jean-Guillaume.Dumas