## Abstract

In the first part of this work a general discrete-time dynamical not necessarily linear system with additive not necessarily Gaussian noises is considered and the problem of estimating the unknown states using the past history of the known states is addressed. The proposed approach is based on the concept of conditionally-orthogonal projection, where the conditional expectation operator given the known states' history, rather than the unconditional expectation, is used to define orthogonality. This approach, called conditionally-linear approach, is a direct extension of the standard linear filtering approach, but is more adequate than the latter for on-line implementation since it allows for the filter gains to be function of the incoming observations. In the case of an underlying discretization of a continuous-time system with Gaussian additive measurement noises, the proposed discrete-time algorithm is formally shown to be asymptotically equivalent to the continuous-time optimal non-linear filter, when the discretization step is very small. For conditionally-Gaussian systems, this filter coincides with the standard conditionally-Gaussian filter. Moreover, its application to conditionally-linear non-Gaussian systems is straightforward. The latter result is the basis for the second part of the work where the mode estimation problem for a special class of jump-linear systems is addressed. It is shown that the state-space equations of the mode can be developed as a conditionally-linear non-Gaussian model given the continuous states. Assuming the continuous states to be known, a suboptimal mode-estimator is developed by applying the general conditionally-linear filter to the mode equations. One highlight of the estimator is that it only requires the first two moments of the mode-measurement noise. On the other hand, the optimal non-linear filter (Wonham filter) requires a complete knowledge of the probability distribution of that noise. Both filters are compared on a simple numerical example via Monte-Carlo simulations, which confirm the asymptotic optimal behavior of the proposed filter in the Gaussian case. The case of partial information in the continuous states, which can not be handled with the Wonham filter, is also considered. Using the state augmentation technique, the model equations are recast in a form that fits the full information case, and an approximated Conditionally-Linear filter is applied for estimating the mode. As an numerical aerospace example, the problem of gyro failure detection from accurate spacecraft attitude measurements is considered and the filter performance are illustrated via extensive Monte-Carlo simulations.

Original language | English |
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Title of host publication | Collection of Technical Papers - AIAA Guidance, Navigation, and Control Conference 2006 |

Publisher | American Institute of Aeronautics and Astronautics Inc. |

Pages | 1794-1825 |

Number of pages | 32 |

ISBN (Print) | 1563478196, 9781563478192 |

DOIs | |

State | Published - 1 Jan 2006 |

Externally published | Yes |

Event | AIAA Guidance, Navigation, and Control Conference 2006 - Keystone, CO, United States Duration: 21 Aug 2006 → 24 Aug 2006 |

### Publication series

Name | Collection of Technical Papers - AIAA Guidance, Navigation, and Control Conference 2006 |
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Volume | 3 |

### Conference

Conference | AIAA Guidance, Navigation, and Control Conference 2006 |
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Country/Territory | United States |

City | Keystone, CO |

Period | 21/08/06 → 24/08/06 |

## ASJC Scopus subject areas

- General Engineering