**SHOC
**

**Hybrid Optimal Control
**

[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 X_{0} at time t=0 to a
final state
X_{f} at unspecified time t_{f} using the admissible
control functions u
that take values in a convex and compact polyhedral set U_{m}of
R^{m}, in such a way that:

is minimized. The polytope U_{m} is defined as the convex
hull of its vertices:

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.

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

- Hybrid Optimal Control

Jean-Guillaume Dumas, Aude Rondepierre

International Journal of Control, soumis. - Piecewise
Affine Systems Controllability and Hybrid Optimal Control (ps,
pdf)

Aude Rondepierre

ICINCO'05: International Conference on Informatics in Control, Automation and Robotics, September 2005.

- Algorithms
for Hybrid Optimal Control. Symbolic/Numeric Control of Affine
Dynamical Systems.

Aude Rondepierre and Jean-Guillaume Dumas

ISSAC '05: International Symposium on Symbolic and Algebraic Computation, July 2005.

- Algorithms
for Hybrid Optimal Control.

Aude Rondepierre and Jean-Guillaume Dumas

Rapport de Recherche IMAG-ccsd-00004191, arXiv math.OC/0502172. Février 2005.

- Contrôle optimal et algorithme de calcul des trajectoires. Kevin Hamon.

Technical report. IMAG, September 2004

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)

- Geometric tools for polytope manipulation.

- Hybrid computation: modelisation of nonlinear systems by hybrid systems.
- Controllability.

- Optimal solutions computation (in progress).

To run it, you need:

- Maple at least 8.0
- Qhull

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 |