### Abstract

We present a non-model based approach for asymptotic, locally exponentially stable attainment of the optimal open-loop control sequence for unknown, discrete-time linear systems with a scalar input, where not even the dimension of the system is known. This control sequence minimizes the finite-time horizon cost function, which is quadratic in the measured output and in the input. We make no assumptions on the stability of the unknown system, but we do assume that the system is reachable. The proposed algorithm employs the multi-variable discrete-time extremum seeking approach to minimize the cost function, extending results established for the scalar discrete-time extremum seeking method. Simulation results show that the Hessian's condition number, used as a measure of the optimization problem's level of difficulty, increases with both the system's level of instability and the length of the finite horizon for a scalar system. Thus, we suggest solving well-conditioned, shorter time horizon optimal control problems to obtain good initial control estimates for longer time horizon problems. We also show that the algorithm accommodates input constraints by employing the projection operator.

Original language | English (US) |
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Article number | 6426052 |

Pages (from-to) | 5717-5722 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

DOIs | |

State | Published - Dec 1 2012 |

Event | 51st IEEE Conference on Decision and Control, CDC 2012 - Maui, HI, United States Duration: Dec 10 2012 → Dec 13 2012 |

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

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## Cite this

*Proceedings of the IEEE Conference on Decision and Control*, 5717-5722. [6426052]. https://doi.org/10.1109/CDC.2012.6426052