### Abstract

This paper focuses on the joint input and state estimation for linear stochastic discrete-time systems. To obtain unique estimators that are optimal in the sense of being both least-squares and minimum-variance unbiased (MVU) and their recursive solution, an objective function of the sum of squared errors of outputs is first minimized with respect to an extended state vector including states and unknown inputs. A recursive solution of the extended state vector is then derived with the aid of matrix manipulations. Finally, appropriate matrix decomposition and transformation are used to extract the recursive solutions only involving the current states and unknown inputs from the recursive solution obtained in the previous step. This approach avoids excessive computation due to the dimensions of the matrices that increase with time. As a result, the resulting recursive solution is referred to as a general Kalman filter with unknown inputs (GKF-UI), which covers the most general case of unknown inputs so far in the literature without resorting to transforming the outputs and/or unknown inputs. The unknown inputs to be estimated by the proposed GKF-UI could be arbitrary signals without any prior information and the properties of the proposed GKF-UI are also examined. A numerical example with partial feedthrough from three unknown inputs is used to demonstrate the effectiveness and accuracy of the proposed algorithm.

Original language | English |
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Title of host publication | Proceedings of the 30th Chinese Control and Decision Conference, CCDC 2018 |

Place of Publication | China |

Publisher | IEEE, Institute of Electrical and Electronics Engineers |

Pages | 5423-5428 |

Number of pages | 6 |

ISBN (Electronic) | 9781538612439 |

DOIs | |

Publication status | Published - 6 Jul 2018 |

Event | 30th Chinese Control and Decision Conference, CCDC 2018 - Shenyang, China Duration: 9 Jun 2018 → 11 Jun 2018 |

### Conference

Conference | 30th Chinese Control and Decision Conference, CCDC 2018 |
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Country | China |

City | Shenyang |

Period | 9/06/18 → 11/06/18 |

### Fingerprint

### Cite this

*Proceedings of the 30th Chinese Control and Decision Conference, CCDC 2018*(pp. 5423-5428). China: IEEE, Institute of Electrical and Electronics Engineers. https://doi.org/10.1109/CCDC.2018.8408075

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*Proceedings of the 30th Chinese Control and Decision Conference, CCDC 2018.*IEEE, Institute of Electrical and Electronics Engineers, China, pp. 5423-5428, 30th Chinese Control and Decision Conference, CCDC 2018, Shenyang, China, 9/06/18. https://doi.org/10.1109/CCDC.2018.8408075

**A Kalman filtering for linear discrete-time system with unknown inputs.** / Li, Weiran; Pan, Jie; Li, Yanjun; Pan, Shuwen; Liu, Yan.

Research output: Chapter in Book/Conference paper › Conference paper

TY - GEN

T1 - A Kalman filtering for linear discrete-time system with unknown inputs

AU - Li, Weiran

AU - Pan, Jie

AU - Li, Yanjun

AU - Pan, Shuwen

AU - Liu, Yan

PY - 2018/7/6

Y1 - 2018/7/6

N2 - This paper focuses on the joint input and state estimation for linear stochastic discrete-time systems. To obtain unique estimators that are optimal in the sense of being both least-squares and minimum-variance unbiased (MVU) and their recursive solution, an objective function of the sum of squared errors of outputs is first minimized with respect to an extended state vector including states and unknown inputs. A recursive solution of the extended state vector is then derived with the aid of matrix manipulations. Finally, appropriate matrix decomposition and transformation are used to extract the recursive solutions only involving the current states and unknown inputs from the recursive solution obtained in the previous step. This approach avoids excessive computation due to the dimensions of the matrices that increase with time. As a result, the resulting recursive solution is referred to as a general Kalman filter with unknown inputs (GKF-UI), which covers the most general case of unknown inputs so far in the literature without resorting to transforming the outputs and/or unknown inputs. The unknown inputs to be estimated by the proposed GKF-UI could be arbitrary signals without any prior information and the properties of the proposed GKF-UI are also examined. A numerical example with partial feedthrough from three unknown inputs is used to demonstrate the effectiveness and accuracy of the proposed algorithm.

AB - This paper focuses on the joint input and state estimation for linear stochastic discrete-time systems. To obtain unique estimators that are optimal in the sense of being both least-squares and minimum-variance unbiased (MVU) and their recursive solution, an objective function of the sum of squared errors of outputs is first minimized with respect to an extended state vector including states and unknown inputs. A recursive solution of the extended state vector is then derived with the aid of matrix manipulations. Finally, appropriate matrix decomposition and transformation are used to extract the recursive solutions only involving the current states and unknown inputs from the recursive solution obtained in the previous step. This approach avoids excessive computation due to the dimensions of the matrices that increase with time. As a result, the resulting recursive solution is referred to as a general Kalman filter with unknown inputs (GKF-UI), which covers the most general case of unknown inputs so far in the literature without resorting to transforming the outputs and/or unknown inputs. The unknown inputs to be estimated by the proposed GKF-UI could be arbitrary signals without any prior information and the properties of the proposed GKF-UI are also examined. A numerical example with partial feedthrough from three unknown inputs is used to demonstrate the effectiveness and accuracy of the proposed algorithm.

KW - Kalman filtering

KW - least-squares estimation

KW - minimum-variance unbiased filter

KW - unknown input estimation

UR - http://www.scopus.com/inward/record.url?scp=85050873313&partnerID=8YFLogxK

U2 - 10.1109/CCDC.2018.8408075

DO - 10.1109/CCDC.2018.8408075

M3 - Conference paper

SP - 5423

EP - 5428

BT - Proceedings of the 30th Chinese Control and Decision Conference, CCDC 2018

PB - IEEE, Institute of Electrical and Electronics Engineers

CY - China

ER -