### Abstract

We present a global solution for an optimal feedback control problem of the underactuated Heisenberg system or nonholonomic integrator. Set in the general framework of the Hamilton-Jacobi theory, this work demonstrates the potential applicability of our methodology to general underactuated optimal control problems. We incorporate the Heisenberg system into a typical optimal control formulation called the hard constraint problem, and transform into a two point boundary value problem for a Hamiltonian system. It is viewed as a canonical transformation in itself, to which we apply our recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory. It is first recognized that our previously developed procedure for solving fullyactuated optimal control problems is not directly applicable due to a singularity caused by underactuation. However, within the same framework of generating functions we are provided with a way to circumvent this singularity by algebraic manipulations linked with the underactuated coordinate. This results in a scalar transcendental equation whose solution ultimately leads to a nonlinear optimal feedback control law in an analytical form. We illustrate our solution by numerical examples.

Original language | English |
---|---|

Title of host publication | Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC |

Pages | 2687-2692 |

Number of pages | 6 |

Publication status | Published - 2006 Dec 1 |

Event | 45th IEEE Conference on Decision and Control 2006, CDC - San Diego, CA, United States Duration: 2006 Dec 13 → 2006 Dec 15 |

### Other

Other | 45th IEEE Conference on Decision and Control 2006, CDC |
---|---|

Country | United States |

City | San Diego, CA |

Period | 06/12/13 → 06/12/15 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Modelling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC*(pp. 2687-2692). [4177499]

}

*Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC.*, 4177499, pp. 2687-2692, 45th IEEE Conference on Decision and Control 2006, CDC, San Diego, CA, United States, 06/12/13.

**Globally optimal feedback control law of the underactuated Heisenberg system by generating functions.** / Park, Chandeok; Scheeres, Daniel J.; Guibout, Vincent; Bloch, Anthony.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Globally optimal feedback control law of the underactuated Heisenberg system by generating functions

AU - Park, Chandeok

AU - Scheeres, Daniel J.

AU - Guibout, Vincent

AU - Bloch, Anthony

PY - 2006/12/1

Y1 - 2006/12/1

N2 - We present a global solution for an optimal feedback control problem of the underactuated Heisenberg system or nonholonomic integrator. Set in the general framework of the Hamilton-Jacobi theory, this work demonstrates the potential applicability of our methodology to general underactuated optimal control problems. We incorporate the Heisenberg system into a typical optimal control formulation called the hard constraint problem, and transform into a two point boundary value problem for a Hamiltonian system. It is viewed as a canonical transformation in itself, to which we apply our recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory. It is first recognized that our previously developed procedure for solving fullyactuated optimal control problems is not directly applicable due to a singularity caused by underactuation. However, within the same framework of generating functions we are provided with a way to circumvent this singularity by algebraic manipulations linked with the underactuated coordinate. This results in a scalar transcendental equation whose solution ultimately leads to a nonlinear optimal feedback control law in an analytical form. We illustrate our solution by numerical examples.

AB - We present a global solution for an optimal feedback control problem of the underactuated Heisenberg system or nonholonomic integrator. Set in the general framework of the Hamilton-Jacobi theory, this work demonstrates the potential applicability of our methodology to general underactuated optimal control problems. We incorporate the Heisenberg system into a typical optimal control formulation called the hard constraint problem, and transform into a two point boundary value problem for a Hamiltonian system. It is viewed as a canonical transformation in itself, to which we apply our recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory. It is first recognized that our previously developed procedure for solving fullyactuated optimal control problems is not directly applicable due to a singularity caused by underactuation. However, within the same framework of generating functions we are provided with a way to circumvent this singularity by algebraic manipulations linked with the underactuated coordinate. This results in a scalar transcendental equation whose solution ultimately leads to a nonlinear optimal feedback control law in an analytical form. We illustrate our solution by numerical examples.

UR - http://www.scopus.com/inward/record.url?scp=39649094266&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=39649094266&partnerID=8YFLogxK

M3 - Conference contribution

SN - 1424401712

SN - 9781424401710

SP - 2687

EP - 2692

BT - Proceedings of the 45th IEEE Conference on Decision and Control 2006, CDC

ER -