UKF State Estimation

In order to illustrate the UKF for state-estimation, we provide a new application example corresponding to noisy time-series estimation.

In this example, the UKF is used to estimate an underlying clean time-series corrupted by additive Gaussian white noise. The time-series used is the Mackey-Glass-30 chaotic series. The clean times-series is first modeled as a nonlinear autoregression

$$\hspace{-.2in} \begin{split}x_k &= f\bigl(x_{k-1},...x_{k-M}, {\bf w}\bigr) + v_k \\ \end{split}$$ (19)

where the model $f$ (parameterized by ${\bf w}$) was approximated by training a feedforward neural network on the clean sequence. The residual error after convergence was taken to be the process noise variance.

Next, white Gaussian noise was added to the clean Mackey-Glass series to generate a noisy time-series $y_k = x_k + n_k$. The corresponding state-space representation is given by:

$${{\bf x}_k} = \qquad F( {\bf x}_{k-1}, {\bf w}) \qquad \qquad \qquad+ B \cdot v_{k-1}$$

$$\left[ \begin{array}{c} {x_k}\\ {x_{k-1}}\\ \vdots \\ {x_{k-M+1}} \end{array} \right] = \left[ \begin{array}{c} f({x_{k-1}},\ldots,{x_{k-M}}, {\bf w}) ...... \begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \\ \end{array} \right ] \cdot {v_{k-1}}$$

$${y_k} = \left [ \begin{matrix}{1}&{0}&\cdots &{0}\end{matrix} \right ] \cdot {\bf x}_k + n_k$$ (20)

In the estimation problem, the noisy-time series $y_k$ is the only observed input to either the EKF or UKF algorithms (both utilize the known neural network model). Note that for this state-space formulation both the EKF and UKF are order $L^2$ complexity. Figure 2 shows a sub-segment of the estimates generated by both the EKF and the UKF (the original noisy time-series has a 3dB SNR). The superior performance of the UKF is clearly visible.

Figure: Estimation of Mackey-Glass time-series with the EKF and UKF using a known model. Bottom graph shows comparison of estimation errors for complete sequence.