Adaptive Kalman filtering in Golang

go get github.com/konimarti/kalman

• Adaptive Kalman filtering with Rapid Ongoing Stochastic covariance Estimation (ROSE)

• A helpful introduction to how Kalman filters work, can be found here.

• Kalman filters are based on a state-space representation of linear, time-invariant systems:

  The next state is defined as

  $$x(t+1)=A_d\ast x(t)+B_d\ast u(t)$$

  where A_d is the discretized prediction matrix and B_d the control matrix. x(t) is the current state and u(t) the external input. The response (measurement) of the system is y(t):

  $$y(t)=C\ast x(t)+D\ast u(t)$$

Using the standard Kalman filter

	// create filter
	filter := kalman.NewFilter(
		lti.Discrete{
			Ad, // prediction matrix (n x n)
			Bd, // control matrix (n x k)
			C,  // measurement matrix (l x n)
			D,  // measurement matrix (l x k)
		},
		kalman.Noise{
			Q, // process model covariance matrix (n x n)
			R, // measurement errors (l x l)
		},
	)

	// create context
	ctx := kalman.Context{
		X, // initial state (n x 1)
		P, // initial process covariance (n x n)
	}

	// get measurement (l x 1) and control (k x 1) vectors
	..

	// apply filter
	filteredMeasurement := filter.Apply(&ctx, measurement, control)
}

Results with standard Kalman filter

See example here.

Results with Rapid Ongoing Stochasic covariance Estimation (ROSE) filter

See example here.

Math behind the Kalman filter

• Calculation of the Kalman gain and the correction of the state vector ~x(k) and covariance matrix ~P(k): $${}^y(k)=C{\ast }^x(k)+D\ast u(k)dy(k)=y(k){-}^y(k)K(k){=}^P(k)\ast C^T\ast (C{\ast }^P(k)\ast C^T+R(k){)}^{(}-1)x(k){=}^x(k)+K(k)\ast dy(k)P(k)=(I-K(k)\ast C){\ast }^P(k)$$
• Then the next step is predicted for the state ^x(k+1) and the covariance ^P(k+1): $${}^x(k+1)=Ad\ast x(k)+Bd\ast u(k{)}^P(k+1)=Ad\ast P(k)\ast A{d}^T+Gd\ast Q(k)\ast G{d}^T$$

Credits

This software package has been developed for and is in production at Kalkfabrik Netstal.