### Abstract

The solution is given to a time-varying optimal state feedback problem with stochastic disturbances. The system is composed of a plant and disturbance model represented by polynomials in the delay operator, z(-1), leading to a solution involving spectral factorisation of operator equations and Diophantine operator equations. The cost function is over infinite time and the assumption is made that the system is time-varying for T steps into the future from the current sample and time-invariant thereafter. For a time-invariant system over infinite time, the optimal controller is a constant state-feedback matrix gain. Thus, with the assumption of time-invariance from T to, the feedback gain may be calculated using constant system polynomials. The solution of the spectral factors and Diophantine equations may then be computed recursively, for a scalar plant, working from T steps ahead to the current time. The controller calculated for the current time is then applied to the system. If the input non-linearity of a plant is represented in time-varying form, the time-varying ideas may be used to produce a nonlinear controller for the system. The example in this paper is for a smooth saturation non-linearity represented by a tanh function. Simulation results are given and it is clear that performance gains over a time-invariant controller are possible.

Original language | English |
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Pages | 3495-3500 |

Number of pages | 6 |

DOIs | |

Publication status | Published - Dec 2003 |

Event | 42nd IEEE Conference on Decision and Control 2003 - Maui, United States Duration: 9 Dec 2003 → 12 Dec 2003 |

### Conference

Conference | 42nd IEEE Conference on Decision and Control 2003 |
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Country | United States |

City | Maui |

Period | 9/12/03 → 12/12/03 |

### Keywords

- time varying
- optimal control
- non-linear system
- control systems
- time varying systems
- stochastic processes
- state feedback
- polynomials
- nonlinear equations
- nonlinear control systems
- delay
- cost function

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## Cite this

Grimble, M. J., & Martin, P. (2003).

*Time varying optimal control of a non-linear system*. 3495-3500. Paper presented at 42nd IEEE Conference on Decision and Control 2003, Maui, United States. https://doi.org/10.1109/CDC.2003.1271689