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## Kalman filter

Books Magazines 1. Nonfiction 1. Collectibles 8. Advertising 4. Radio, Phonograph, TV, Phone 4. Additional approaches include belief filters which use Bayes or evidential updates to the state equations. A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the "simple" Kalman filter, the Kalman—Bucy filter , Schmidt's "extended" filter, the information filter , and a variety of "square-root" filters that were developed by Bierman, Thornton, and many others. Perhaps the most commonly used type of very simple Kalman filter is the phase-locked loop , which is now ubiquitous in radios, especially frequency modulation FM radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic communications equipment.

Kalman filters are based on linear dynamical systems discretized in the time domain. They are modeled on a Markov chain built on linear operators perturbed by errors that may include Gaussian noise. The state of the system is represented as a vector of real numbers.

At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on the system if they are known. Then, another linear operator mixed with more noise generates the observed outputs from the true "hidden" state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the key difference that the hidden state variables take values in a continuous space as opposed to a discrete state space as in the hidden Markov model.

There is a strong analogy between the equations of the Kalman Filter and those of the hidden Markov model.

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A review of this and other models is given in Roweis and Ghahramani , [15] and Hamilton , Chapter In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the framework of the Kalman filter.

This means specifying the following matrices:. At time k an observation or measurement z k of the true state x k is made according to. Many real dynamical systems do not exactly fit this model. In fact, unmodeled dynamics can seriously degrade the filter performance, even when it was supposed to work with unknown stochastic signals as inputs. The reason for this is that the effect of unmodeled dynamics depends on the input, and, therefore, can bring the estimation algorithm to instability it diverges. On the other hand, independent white noise signals will not make the algorithm diverge.

The problem of distinguishing between measurement noise and unmodeled dynamics is a difficult one and is treated in control theory under the framework of robust control. The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state.

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The Kalman filter can be written as a single equation, however it is most often conceptualized as two distinct phases: "Predict" and "Update". The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also known as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from the current timestep.

In the update phase, the current a priori prediction is combined with current observation information to refine the state estimate. This improved estimate is termed the a posteriori state estimate. Typically, the two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation. However, this is not necessary; if an observation is unavailable for some reason, the update may be skipped and multiple prediction steps performed.

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Likewise, if multiple independent observations are available at the same time, multiple update steps may be performed typically with different observation matrices H k. The formula for the updated a posteriori estimate covariance above is valid for the optimal K k gain that minimizes the residual error, in which form it is most widely used in applications. Proof of the formulae is found in the derivations section, where the formula valid for any K k is also shown. That is, all estimates have a mean error of zero. Practical implementation of the Kalman Filter is often difficult due to the difficulty of getting a good estimate of the noise covariance matrices Q k and R k.

Extensive research has been done in this field to estimate these covariances from data. One practical approach to do this is the autocovariance least-squares ALS technique that uses the time-lagged autocovariances of routine operating data to estimate the covariances. It follows from theory that the Kalman filter is the optimal linear filter in cases where a the model perfectly matches the real system, b the entering noise is white uncorrelated and c the covariances of the noise are exactly known.

Several methods for the noise covariance estimation have been proposed during past decades, including ALS, mentioned in the section above. After the covariances are estimated, it is useful to evaluate the performance of the filter; i. If the Kalman filter works optimally, the innovation sequence the output prediction error is a white noise, therefore the whiteness property of the innovations measures filter performance.

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Several different methods can be used for this purpose. Consider a truck on frictionless, straight rails. Initially, the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We show here how we derive the model from which we create our Kalman filter. From Newton's laws of motion we conclude that.

Another way to express this, avoiding explicit degenerate distributions is given by. At each time step, a noisy measurement of the true position of the truck is made. If the initial position and velocity are not known perfectly, the covariance matrix should be initialized with suitable variances on its diagonal:. The filter will then prefer the information from the first measurements over the information already in the model.

Then the Kalman filter may be written:. A similar equation holds if we include a non-zero control input.

From above, the four equations needed for updating the Kalman gain are as follows:. Since the gain matrices depend only on the model, and not the measurements, they may be computed offline. Since the measurement error v k is uncorrelated with the other terms, this becomes.

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