8  Some additional topics

Reference: Simon (2006).

The extra topics discussed are

  1. Verifying that a Kalman filter is operating reliably

  2. Multiple model estimation

  3. Reduced-order filtering

  4. Robust Kalman filtering

  5. Delayed measurements data

These topics are related to system identification with low-fidelity data. Hence, they are studied.

8.1 Verifying Kalman filter performance

The Kalman filter performance can be verified using the knowledge of statistics of the innovations.

Innovations is defined as \(\mathbf{r}_k = \left(\mathbf{y}_k-\mathbf{H}_k\hat{\mathbf{x}}_k^-\right)\). It is obtained using one-step a priori update.

Innovations can be thought of as the part of the measurement that contains new information and that is used to update the state estimate.

A nonzero innovations allows the measurement to affect the state estimate. The innovations \(\mathbf{r}_k\) can be written in terms of estimation error as

\[ \begin{align} \mathbf{r}_k &= \mathbf{y}_k - \mathbf{H}_k\hat{\mathbf{x}}_k^- \\ &= \left(\mathbf{H}_k\mathbf{x}_k + \mathbf{v}_k\right) - \mathbf{H}_k\hat{\mathbf{x}}_k^- \\ &= \mathbf{H}_k\left(\mathbf{x}_k-\hat{\mathbf{x}}_k^-\right)+\mathbf{v}_k \\ &= \mathbf{H}_k\epsilon_k+\mathbf{v}_k \end{align} \tag{8.1}\]

It is shown in the reference that the innovations is a zero-mean white-noise stochastic process with covariance defined as

\[ E\left[\mathbf{r}_k\mathbf{r}_i^T\right] = \mathbf{H}_k\mathbf{P}_k^-\mathbf{H}_k^T+\mathbf{R}_k \tag{8.2}\]

Kalman filter is verified during operation by computing the innovations’s mean and covariance. If it is colored, nonzero-mean or has the wrong covariance, then there is something wrong with the filter. The most likely reason for such a discrepancy is a modeling error.

8.2 Multiple model estimation

  • Suppose the exact system dynamics model is unknown or the system model changes depending on some unknown factors. Then, multiple Kalman filters (one for each possible model) is used and the state estimates are combined to obtain a refined state estimate.

  • To perform that, Bayes rule is used. Overall idea is as follows,

    • Suppose, there are \(N\) possible systems chosen for the given dynamics. Let, the set \(p\) is defined as the set of \(\left(\mathbf{F},\mathbf{G},\mathbf{H},\mathbf{Q},\mathbf{R}\right)\).

    • A probability value is assigned for each state set \(p\) and the probability of given measurement following a particular dynamics defined by set \(p_j\) is given by Bayes rule as

      \[ Pr(p_j|\mathbf{y}_k) = \frac{pdf(\mathbf{y}_k|p_j)Pr(p_j)}{\sum_{i=1}^Npdf(\mathbf{y}_k|p_i)Pr(p_i)} \]

  • Then the refined state estimate is obtained as the weighted sum of estimate from each dynamics model with weight being the probability value as

    \[ \begin{align} \hat{\mathbf{x}}_k^- &= \sum_{j=1}^NPr(p_j|\mathbf{y}_k)\hat{\mathbf{x}}_{kj}^-\\ \mathbf{P}_k^- &= \sum_{j=1}^NPr(p_j|\mathbf{y}_k)P_{kj}^- \end{align} \]

8.3 Reduced-order Kalman filtering

  • If a user wants to estimate only a subset of the state vector, then a reduced-order filter can be designed. This is used in a real-time application where computational effort is a main consideration (eg. weather forecasting).

  • Various approaches for reduced order modeling was proposed. Like, if the dynamics model of an underlying system can be reduced to a lower-order model that approximates the full-order model, then this lower-order model can be used with standard Kalman filter. Another approach is to work with subset of states, instead of full states.

  • In the reference, two approaches were described

    • Anderson’s approach to reduced-order filtering
    • Reduced-order Schmidt-Kalman filter

  • Details can be referred when needed

8.4 Robust Kalman filtering

  • The main assumption of Kalman filtering is that the system model and noise statistics are known. If any of these assumptions are violated then the filter estimates can degrade.

  • The concept is that the quantities such as the system matrix \(\mathbf{F}_k\), the measurement matrix \(\mathbf{H}_k\) and the noise covariances \(\mathbf{Q}_k,\mathbf{R}_k\) are considered with having uncertainties.

  • In the reference, the uncertainties in the noise covariances were considered and the system derivation was provided. Some additional references were specified. This is a place where PINNs can be made dominant.

8.5 Delayed measurements and synchronization errors

  • In decentralized filtering systems, observations are often collected at various physical locations, and then transmitted in bulk to a central processing computer. Here, the measurements may not arrive at the processing computer synchronously. The computer may receive measurements out of sequence.

  • Various approaches were taken in the literature to handle this problem and one of the approaches is described in the reference.

  • Let, the time sequence be \(0,1,\dots,k\). The overall idea is to perform the Kalman filter state estimation using the existing sequential measurements data to estimate the state \(\hat{\mathbf{x}}_k\). The estimate of state is given as \[ \hat{\mathbf{x}}_k = E\left[\mathbf{x}_k | \mathbf{y}_1,\dots,\mathbf{y}_k\right] \]

  • If a new measurement at time \(k_0 < k\) comes in, then the new measurement is used to obtain an updated state and covariance estimate at time \(k_0\), and then use those quantities to update the estimate and covariance at time \(k\).

  • For this, an alternate form of Kalman filter, a statistical derivation of the Kalman filter, was described and with that, the delayed measurement Kalman filter was developed.

  • The algorithm for delayed measurement filter was given in the reference.