## Posts

### Estimating the state of a dc motor

This example will focus on estimating the angular position \( \theta \), angular velocity \( \dot{\theta} \) and armature current \( i \) of a DC motor with a linear Kalman filter. When modeling DC motors, it is important to mention that certainly nonlinear models are superior to linear ones. For didactic purposes, the following widely used linear model is sufficient for the time being.

### Cubature Kalman Filter

The Cubature Kalman Filter (CKF) is the newest representative of the sigma-point methods. The selection of sigma points in the CKF is slightly different from the Unscented Kalman Filter (UKF) and is based on the Cubature rule. As for the UKF, the CKF follows the idea that it is easier to approximate a probability function than to linearize a nonlinear function.

### Multiple Model State Estimation

Central for the performance of a model-based state estimator is how well the model corresponds to the process to be observed. In practice, the question arises whether the process to be estimated always behaves exactly according to one model or whether the process changes between different models. The solution is to take this into account by using multiple models. In the literature this aspect is called multiple model. The challenge is both how exactly, i.e. only one model or the mixture of several models, and according to which rules the process changes into the different models.

### Normalized Innovation Squared (NIS)

The Normalized Innovation Squared (NIS) metric allows to check whether the Kalman filter is consistent with the measurement residual \( \nu (k) \) and the associated innovation covariance matrix \( \mathbf{S}(k) \).

### Root Mean Square Error (RMSE)

To evaluate the performance of state estimators, the estimation error \( \tilde{\mathbf{x}}(k) = \mathbf{x}(k) - \mathbf{\hat{x}}(k) \) is evaluated. The root mean square error (RMSE), which is a widely used quality measure, is suitable for this purpose. The basis is the estimation error \(\tilde{\mathbf{x}}(k)\) for each time step \( k \in {1…K} \).

### In simulation

For a simulation with a length of \( K\) time increments the RMSE is averaged by \( N\) Monte Carlo runs in order to achieve a high statistical significance \[ \text{RMSE}(\tilde{\mathbf{x}}(k)) = \sqrt{\frac{1}{N} \sum^N_{i=1} (\tilde{x}^i_1(k)^2 + … + \tilde{x}^i_n(k)^2)} \] where \( n = \text{dim}(\tilde{\mathbf{x}}(k)) \) is.

### Normalized Estimation Error Squared (NEES)

A desired property of a state estimator is that it is able to indicate the quality of the estimate correctly. This ability is called consistency of a state estimator and has a direct impact on the estimation error \( \tilde{\mathbf{x}}(k) \), i.e. an inconsistent state estimator does not provide the most optimal result. A state estimator should be able to indicate the quality of the estimate correctly, because an increase in sample size leads to a growth in information content and the state estimate \( \hat{\mathbf{x}}(k) \) shall be as close as possible to the true state \( \mathbf{x}(k) \). This results from the requirement that a state estimator shall be unbiased. Mathematically speaking, this is expressed by the expected value of the estimation error \( \tilde{\mathbf{x}}(k) \) being zero \[ E [ \mathbf{x}(k) - \hat{\mathbf{x}}(k) ] = E [ \tilde{\mathbf{x}}(k) ] \overset{!}{=} 0 \]

### Discretization of linear state-space model

In practice, the discretization of the state-space equations is of particular importance.

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