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An adaptive generalized extended Kalman filter for real-time identification of structural systems, state and input based on sparse measurement

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Abstract

Extended Kalman filtering with unknown input (EKF-UI) is often used to estimate the structural system state, parameters and unknown input in structural health monitoring. However, the real-time performance of EKF-UI is bound to whether the measurement equation has a direct feedthrough of unknown input, which great limits its application scope. Based on the zero-order-hold assumption and random walk assumption of unknown input, a novel adaptive discrete state equation is derived in this paper. The new equation establishes a connection between the current state and the current input and allows the adjustment of the sensitivity matrix of the unknown input. Then, based on the adaptive discrete state equation and minimum variance unbiased estimation principle, an adaptive generalized extended Kalman filter with unknown input is derived. The proposed algorithm eliminates the limitation that the real-time performance is restricted by whether the measurement equation has a direct feedthrough of the input and realizes the optimization of the state and input estimates in the sense of minimum variance. To demonstrate the feasibility of the proposed method, numerical example of a shear frame structure with Bouc–Wen hysteresis nonlinearity and experimental test of a five-story shear frame are conducted. The comparison with existing methods shows the advantages of the proposed method.

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Data in this study are available from the corresponding author upon reasonable request.

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Funding

This work was supported by the 111 Project of Hubei Province (Grant Number 2021EJD026).

Author information

Authors and Affiliations

  1. Hubei Key Laboratory of Disaster Prevention and Mitigation, Yichang, 443002, China

    Jinshan Huang

  2. College of Civil Engineering and Architecture, China Three Gorges University, Yichang, 443002, China

    Jinshan Huang

  3. Department of Civil Engineering, Xiamen University, Xiamen, 361005, China

    Ying Lei

  4. Department of Disaster Mitigation for Structures, Tongji University, Shanghai, 200092, China

    Xianzhi Li

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  1. Jinshan Huang
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Correspondence to Xianzhi Li.

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Appendices

Appendix 1: The detailed derivation process of Eqs. (56) and (57)

Considering Eq. (35), The partial derivative of \({P}_{{\varvec{f}}}\) with respect to \({{\varvec{B}}}_{k+1}^{\text{opt}}\) is:

$$ \begin{aligned} \frac{{\partial P_{{\varvec{f}}} }}{{\partial {\varvec{B}}_{k + 1}^{\text{opt}} }} & = \frac{{\partial tr\left( {{\varvec{S}}_{k + 1} \tilde{\varvec{R}}_{k + 1} {\varvec{S}}_{k + 1}^{T} } \right)}}{{\partial \tilde{\varvec{R}}_{k + 1} }}:\frac{{\partial \tilde{\varvec{R}}_{k + 1} }}{{\partial \varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} }_{k + 1|k}^{{{\varvec{ZZ}}}} }}:\frac{{\partial \varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} }_{k + 1|k}^{{{\varvec{ZZ}}}} }}{{\partial {\varvec{B}}_{k + 1}^{\text{opt}} }} - 2\frac{{\partial tr\left[ {\left( {{\varvec{S}}_{k + 1} {\varvec{T}}_{k + 1} - {\mathbf{I}}} \right){\varvec{\varGamma}}_{k + 1}^{T} } \right]}}{{\partial {\varvec{T}}_{k + 1} }}:\frac{{\partial {\varvec{T}}_{k + 1} }}{{\partial {\varvec{B}}_{k + 1}^{\text{opt}} }} \\ & = 2{\varvec{H}}_{k + 1|k}^{T} {\varvec{S}}_{k + 1}^{T} \left\{ {{\varvec{S}}_{k + 1} {\varvec{H}}_{k + 1|k} \left[ {{\varvec{B}}_{k + 1}^{\text{opt}} \left( {\hat{\varvec{P}}_{k|k}^{{{\varvec{ff}}}} + {\varvec{Q}}_{k}^{u} } \right) - \left( {{\varvec{A}}_{k} \hat{\varvec{P}}_{k|k}^{{{\varvec{Zf}}}} + {\varvec{B}}_{k}^{zoh} \hat{\varvec{P}}_{k|k}^{{{\varvec{ff}}}} } \right)} \right] -{\varvec{\varGamma}}_{k + 1} } \right\} \\ \end{aligned} $$
(80)

Considering Eq. (45), the partial derivative of \({P}_{{\varvec{Z}}}\) with respect to \({{\varvec{B}}}_{k+1}^{\text{opt}}\) is:

$$ \begin{aligned} \frac{{\partial P_{{\varvec{Z}}} }}{{\partial {\varvec{B}}_{k + 1}^{\text{opt}} }} & = \frac{{\partial tr\left( {{\varvec{L}}_{k + 1} \tilde{\varvec{R}}_{k + 1} {\varvec{L}}_{k + 1}^{T} } \right)}}{{\partial \tilde{\varvec{R}}_{k} }}:\frac{{\partial \tilde{\varvec{R}}_{k + 1} }}{{\partial \varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} }_{k + 1|k}^{{{\varvec{ZZ}}}} }}:\frac{{\partial \varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} }_{k + 1|k}^{{{\varvec{ZZ}}}} }}{{\partial {\varvec{B}}_{k + 1}^{\text{opt}} }} \\ & \quad + \frac{{\partial tr\left( { - 2\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} }_{k + 1|k}^{{{\varvec{ZZ}}}} {\varvec{H}}_{k + 1|k}^{T} {\varvec{L}}_{k + 1}^{T} + \varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} }_{k + 1|k}^{{{\varvec{ZZ}}}} } \right)}}{{\varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} }_{k|k - 1}^{{{\varvec{ZZ}}}} }}:\frac{{\partial \varvec{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{P} }_{k + 1|k}^{{{\varvec{ZZ}}}} }}{{\partial {\varvec{H}}_{k + 1|k}^{T} }} \\ & \quad - 2\frac{{\partial tr\left[ {\left( {{\varvec{L}}_{k + 1} {\varvec{T}}_{k + 1} - {\varvec{B}}_{k + 1}^{\text{opt}} } \right){\varvec{\varLambda}}_{k + 1}^{T} } \right]}}{{\partial {\varvec{T}}_{k + 1} }}:\frac{{\partial {\varvec{T}}_{k + 1} }}{{\partial {\varvec{B}}_{k + 1}^{\text{opt}} }} \\ & = 2\left( {{\varvec{L}}_{k + 1} {\varvec{H}}_{k + 1|k} - {\mathbf{I}}} \right)^{T} \left\{ {\left( {{\varvec{L}}_{k + 1} {\varvec{H}}_{k + 1|k} - {\mathbf{I}}} \right)\left[ {{\varvec{B}}_{k + 1}^{\text{opt}} \left( {\hat{\varvec{P}}_{k|k}^{{{\varvec{ff}}}} + {\varvec{Q}}_{k}^{u} } \right) - \left( {{\varvec{A}}_{k} \hat{\varvec{P}}_{k|k}^{{{\varvec{Zf}}}} + {\varvec{B}}_{k}^{zoh} \hat{\varvec{P}}_{k|k}^{{{\varvec{ff}}}} } \right)} \right] -{\varvec{\varLambda}}_{k + 1} } \right\} \\ \end{aligned} $$
(81)

Appendix 2: Theorem on the inverse of block matrix

Theorem: Let the square matrix \({\varvec{N}}=\left[\begin{array}{cc}{\varvec{A}}& {\varvec{B}}\\ {\varvec{C}}& {\varvec{D}}\end{array}\right]\) be invertible, and its sub-block square matrix \({\varvec{A}}\) is invertible, then \({{\varvec{E}}=\left({\varvec{D}}-{\varvec{C}}{{\varvec{A}}}^{-1}{\varvec{B}}\right)}^{-1}\) exists and the inverse matrix of \({\varvec{N}}\) is [45]:

$${{\varvec{N}}}^{-1}={\left[\begin{array}{cc}{\varvec{A}}& {\varvec{B}}\\ {\varvec{C}}& {\varvec{D}}\end{array}\right]}^{-1}=\left[\begin{array}{cc}{{\varvec{A}}}^{-1}+{{\varvec{A}}}^{-1}{\varvec{B}}{\varvec{E}}{\varvec{C}}{{\varvec{A}}}^{-1}& -{{\varvec{A}}}^{-1}{\varvec{B}}{\varvec{E}}\\ -{\varvec{E}}{\varvec{C}}{{\varvec{A}}}^{-1}& {\varvec{E}}\end{array}\right]$$
(82)

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Huang, J., Lei, Y. & Li, X. An adaptive generalized extended Kalman filter for real-time identification of structural systems, state and input based on sparse measurement. Nonlinear Dyn 112, 5453–5476 (2024). https://doi.org/10.1007/s11071-023-09251-7

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