Algorithm 1. Scheme of the proposed damage identification and state tracking method

- Initialization at $t_0$: $${\widehat{\mathit{x}}}_0={{\mathit{L}}_0}^{\text{T}}\mathbb{E}\left[{\mathit{x}}_0\right]$$ $${\mathit{P}}_0={{\mathit{L}}_0}^{\text{T}}\mathbb{E}\left[\left({\mathit{x}}_0-\mathbb{E}\left[{\mathit{x}}_0\right]\right){\left({\mathit{x}}_0-\mathbb{E}\left[{\mathit{x}}_0\right]\right)}^{\text{T}}\right]{\mathit{L}}_0$$ $${}^j\mathit{x}_0={\widehat{\mathit{x}}}_0$$ $${}^j\omega _0=p({\mathit{y}}_0|{}^j\mathit{x}_0)\hspace{1em}\hspace{1em}\hspace{1em}j=1,\dots ,N_s$$ $${\widehat{\mathbf{\phi }}}_{l,0}=\mathbb{E}\left[{\mathbf{\phi }}_{l,0}\right]$$ $${\mathit{P}}_0^{ss}=\mathbb{E}\left[\left({\mathbf{\phi }}_{l,0}-{\widehat{\mathbf{\phi }}}_{l,0}\right){\left({\mathbf{\phi }}_{l,0}-{\widehat{\mathbf{\phi }}}_{l,0}\right)}^{\text{T}}\right]$$ -Recursive computation at $t_k=t_1,\dots ,T_{end}$; $j=1,\dots ,N_s$

Prediction stage: Draw samples ${}^j\mathit{x}_k^{-}~ p({\mathit{x}}_k|{}^j\mathit{x}_{k-1})$ Push particles toward the region of high probability through an EKF $${}^j\mathit{P}_k^{-}={\mathit{F}}_k{}^j\mathit{P}_{k-1}{\mathit{F}}_k^T+\mathit{W}$$ $${}^j\mathit{G}_k={}^j\mathit{P}_k^{-}{\mathit{H}}^T{\left(\mathit{H}{}^j\mathit{P}_k^{-}{\mathit{H}}^T+\mathit{V}\right)}^{-1}$$ $${}^j\mathit{x}_k={}^j\mathit{x}_k^{-}+{}^j\mathit{G}_k\left({\mathit{y}}_k-{\mathit{H}}^T{}^j\mathit{x}_k^{-}\right)$$ $${}^j\mathit{P}_k={}^j\mathit{P}_k^{-}-{}^j\mathit{G}_k\mathit{H}{}^j\mathit{P}_k^{-}$$

Update stage: Evolve weights $${}^j\omega _k={}^j\omega _{k-1} p({\mathit{y}}_k|{}^j\mathit{x}_k)$$ Resampling $${}^{j}u~\mathfrak{U}\left[0,1\right]$$ $$\text{find} m \text{s}.\text{t}. {\displaystyle \sum }_{i=1}^{m-1}{}^i\omega _k<{}^{j}u<{\displaystyle \sum }_{i=1}^m{}^i\omega _k$$ $${}^j\mathit{x}_k={}^m\mathit{x}_k$$ $${}^j\omega _k^{*}=\frac{1}{N_s}$$ Compute expected value $${\widehat{\mathit{x}}}_k={\displaystyle \sum }_{j=1}^{N_s}{}^j\omega _k^{*}{}^j\mathit{x}_k$$ Predict sub-space and associated covariance $${\mathbf{\phi }}_{l,k}^{-}={\widehat{\mathbf{\phi }}}_{l,k-1}$$ $${\mathit{P}}_k^{ss-}={\mathit{P}}_{k-1}^{ss}+{\mathit{W}}^{ss}$$ Calculate Kalman filter gain for updating sub-space $${\mathit{G}}_k^{ss}={\mathit{P}}_k^{ss-}{\mathit{H}}_k^{ss\text{T}}{\left({\mathit{H}}_k^{ss}{\mathit{P}}_k^{i-}{\mathit{H}}_k^{ss\text{T}}+\mathit{V}\right)}^{-1}$$ Update sub-space and associated covariance $${\widehat{\mathbf{\phi }}}_{l,k}={\mathbf{\phi }}_{l,k}^{-}+{\mathit{G}}_k^{ss}\left({\mathit{y}}_k-{\mathit{H}}_k^{ss}{\mathbf{\phi }}_{l,k}^{-}\right)$$ $${\mathit{P}}_k^{ss}={\mathit{P}}_k^{ss-}-{\mathit{G}}_k^{ss}{\mathit{H}}_k^{ss}{\mathit{P}}_k^{ss-}$$