EKF – Extended (nonlinear) Kalman filter

Block SymbolLicensing group: MODEL

Function Description
The block implements a nonlinear state estimator known as Extended Kalman filter. The goal is to provide estimates of unmeasurable state quantities of a nonlinear dynamic system described by a state space model $\mathrm{d}x∕\mathrm{d}t=f\left(x,u\right)+w\left(t\right),y=h\left(x,u\right)+v\left(t\right)$ for a continuous-time case and $x\left(k+1\right)=f\left(x\left(k\right),u\left(k\right)\right)+w\left(k\right),y\left(k\right)=h\left(x\left(k\right),u\left(k\right)\right)+v\left(k\right)$ for the case of a discrete-time system. The variables $w,v$ are the process and observation noises which are both assumed to be zero mean multivariate Gaussian processes with covariance $Q$ and $R$ specified in the block parameters. The Extended Kalman filter is the nonlinear version of the Kalman filter which linearizes the state and output equations about the current working point. It is a predictor-corrector type algorithm which switches between open-loop prediction using the state equation and correction of the estimates by directly measured output quantities. The measurements can be supplied to the filter non-equidistantly in an arbitrary execution period of the block.

The prediction step is run in each execution period and solves the state equation by numerical integration, starting from an initial value $x0$ and initial covariance $P0$. Various numerical methods, chosen by the user specified parameter $solver$, are available to perform the integration of the vector state differential equation. A special choice of $solver=1$ signalizes the discrete-time system case for which the numerical integration reduces to simple evaluation of the recursive formula given by the first-order difference equation in $x\left(k+1\right)=f\left(x\left(k\right),u\left(k\right)\right)$. Apart from the state vector, also its covariance matrix $P$ is propagated in time, capturing the uncertainty of the estimates in the form of their (co)variances. Please refer to the documentation of the NSSM block for more details about the available numerical integration algorithms.

The filtering correction step takes place whenever the input of the block is set to $nz>0$. This signalizes that new vector of measurements is available at the $z$ input and it is used to correct the state and its covariance estimates from the prediction step. Multiple right sides of the output equation can be implemented in the cooperating REXLANG block. This may be useful e.g. for systems equipped with various sensors providing their data asynchronously to each other (and with respect to the block execution times) with different sampling periods. For the setting $nz=0$, the user algorithm signalizes no output data available in the current execution period, forcing the filter to extrapolate the state estimates by performing the prediction step only.

The Extended Kalman filter is generally not an optimal filter in the sense of minimization of the mean-squared error of the obtained state estimates. However, it provides modest performance for sufficiently smooth nonlinear systems and is considered to be a de facto standard solution for nonlinear estimation. A special case is obtained by setting linear state and output equations in the cooperating REXLANG block. This case leads to standard linear Kalman filter which is stochastically optimal for the formulated state estimation problem.

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