The following presentation is largely based on Tim Sullivan’s UQ book which I highly recommend. It gives clear and concise mathematical presentations of various topics in UQ. Suppose we have a state-observation model that is linear and additive. The state/variable under scrutiny here is linked to a degradation phenomenon, moreover the degradation dynamics here are supposed to be linear. For non-linear dynamics, non-parametric methods exist, especially using particle filters. For the moment let the system be defined as:

\[\begin{cases} x_k &= F_k x_{k-1} + G_k u_k + \xi_k \\ y_k &= H_k x_k + \zeta_k, \end{cases}, \quad \forall k \in \mathbb{N},\]

where $F_k \in \mathbb{R}^{N \times N}$ and $G_k \in \mathbb{R}^{N \times M}$ are the state transition and control matrices, $u_k \in \mathbb{R}^M$ are control inputs, and $\xi_k \sim \mathcal{N}(0, Q_k)$ represents Gaussian process noise. The observation map $H_k : \mathbb{R}^N \to \mathbb{R}^J$ has noise $\zeta_k \sim \mathcal{N}(0, R_k)$. This map is linked to the process of measurement of the state variables, and it is supposed that it only varies linearly with respect to the true state variables and is corrupted by noise, which is usually the case for sensors. Define the conditioned state as:

\[x_{k|j} := (x_k | y_1, \ldots, y_j), \quad \forall j \leq k.\]

The Kálmán filter is one of the first historical methods for perfoming online hybrid modeling, iteratively estimates the state $x_k$ and its covariance $C_k$ using two steps: prediction and correction. Initialization is given by:

\[x_{0|0} \sim \mathcal{N}(\widehat{x}_{0|0}, C_{0|0}) = \mathcal{N}(m_0, Q_0).\]

Prediction Step

Given the prior estimate , the predicted distribution is $\mathcal{N}(x_{k-1| k-1}, C_{k-1 |k-1})$:

\[\widehat{x}_{k|k-1} = F_k \widehat{x}_{k-1|k-1} + G_k u_k,\; C_{k|k-1} = F_k C_{k-1|k-1} F_k^T + Q_k\]

Correction Step

Using the observation $y_k$, the posterior is updated via the Kálmán gain:

\[K_k = C_{k|k-1} H_k^T \left(H_k C_{k\lvert k-1} H_k^T + R_k\right)^{-1}.\]

The posterior mean and covariance are then:

\[\widehat{x}_{k|k} = \widehat{x}_{k|k-1} + K_k (y_k - H_k \widehat{x}_{k|k-1}),\; C_{k|k} = (I_N - K_k H_k) C_{k|k-1}.\]

These formulas are derived from Gaussian conditioning and variational principles.

Remaining Useful Life prediction

For prognostics, the RUL distribution is is Gaussian. Starting at time $t_P$ of the asset degradation state, the future state distribution is propagated:

\[x_{P+\ell|P} \sim \mathcal{N}(\widehat{x}_{P+\ell|P}, C_{P+\ell|P}),\]

The RUL probability is then:

\[\mathbb{P}(\text{RUL}(a, t_P, c) \leq t_\ell | y_1, \ldots, y_P) = \mathbb{P}(x_{P+\ell|P} \geq c) = \int_{[c, \infty)} \gamma_{P+\ell|P}(z) \, dz,\]

where $\gamma_{ P+\ell P}$ is the posterior Gaussian density and the resulting integral is computed via numerical quadrature, accuracy depending largely on the dimensionality of the state vector.