The EKF and its Flaws

Consider the basic state-space estimation framework as in Equations 1 and 2. Given the noisy observation $\mathbf{y}_k$, a recursive estimation for ${\bf x}_k$ can be expressed in the form (see [6]),

$$\hat{{\bf x}}_k = ( {\bf x}_k) + {\mathcal{K}}_k \cdot\left[ {{\bf y}_k} - (\mbox{prediction of } {{\bf y}_k}) \right]$$ (8)

This recursion provides the optimal minimum mean-squared error (MMSE) estimate for ${\bf x}_k$ assuming the prior estimate $\hat{\bf x}_{k-1}$ and current observation ${\bf y}_k$ are Gaussian Random Variables (GRV). We need not assume linearity of the model. The optimal terms in this recursion are given by

$$\hat{{\bf x}}_{k}^- = E[F(\hat{{\bf x}}_{k-1},{\bf v}_{k-1})]$$ (9)

$${\mathcal{K}}_k = {\bf P}_{ {\bf x}_k {\bf y}_k} {\bf P}_{\tilde{ {\bf y}}_k\tilde{ {\bf y}}_k}^{-1}$$ (10)

$$\hat{{\bf y}}_k^- = E[H(\hat{{\bf x}}_{k}^-,{\bf n}_k)],$$ (11)

where the optimal prediction of ${\bf x}_k$ is written as $\hat{{\bf x}}_{k}^-$, and corresponds to the expectation of a nonlinear function of the random variables $\hat{{\bf x}}_{k-1}$ and ${\bf v}_{k-1}$ (similar interpretation for the optimal prediction $\hat{{\bf y}}_k^-$). The optimal gain term ${\mathcal{K}}_k$ is expressed as a function of posterior covariance matrices (with $\tilde{{\bf y}}_k = {\bf y}_k - \hat{{\bf y}}_k^-$). Note these terms also require taking expectations of a nonlinear function of the prior state estimates.

The Kalman filter calculates these quantities exactly in the linear case, and can be viewed as an efficient method for analytically propagating a GRV through linear system dynamics. For nonlinear models, however, the EKF approximates the optimal terms as:

$$\hat{{\bf x}}_{k}^- \approx F(\hat{{\bf x}}_{k-1},\bar{{\bf v}})$$ (12)

$${\mathcal{K}}_k \approx \hat{{\bf P}}_{ {\bf x}_k {\bf y}_k}\hat{{\bf P}}_{\tilde{ {\bf y}}_k\tilde{ {\bf y}}_k}^{-1}$$ (13)

$$\hat{{\bf y}}_k^- \approx H(\hat{{\bf x}}_{k}^-,\bar{\mathbf{n}}),$$ (14)

where predictions are approximated as simply the function of the prior mean value for estimates (no expectation taken)¹ The covariance are determined by linearizing the dynamic equations $({\bf x}_{k+1} \approx {\bf A} {\bf x}_{k} + {\bf B} {{\bf v}_k, {\bf y}_k} \approx {\bf C}{\bf x_k} + {\bf D}{\bf n}_k)$, and then determining the posterior covariance matrices analytically for the linear system. In other words, in the EKF the state distribution is approximated by a GRV which is then propagated analytically through the ``first-order'' linearization of the nonlinear system. The readers are referred to [6] for the explicit equations. As such, the EKF can be viewed as providing ``first-order'' approximations to the optimal terms². These approximations, however, can introduce large errors in the true posterior mean and covariance of the transformed (Gaussian) random variable, which may lead to sub-optimal performance and sometimes divergence of the filter. It is these ``flaws'' which will be amended in the next section using the UKF.