Kalman Filtering
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10-01-2011, 04:06 PM


Filtering is desirable in many situations in engineering and embedded systems. For example, radio communication signals are corrupted with noise. A good filtering algorithm can remove the noise from electromagnetic signals while retaining the useful information. Another example is power supply voltages. Uninterruptible power supplies are devices that filter line voltages in order to smooth out undesirable fluctuations that might otherwise shorten the lifespan of electrical devices such as computers and printers. The Kalman filter is a tool that can estimate the variables of a wide range of processes. In mathematical terms we would say that a Kalman filter estimates the states of a linear system. The Kalman filter not only works well in practice, but it is theoretically attractive because it can be shown that of all possible filters, it is the one that minimizes the variance of the estimation error. Kalman filters are often implemented in embedded control systems because in order to control a process, you first need an accurate estimate of the process variables. This article will tell you the basic concepts that you need to know to design and implement a Kalman filter. I will introduce the Kalman filter algorithm and we’ll look at the use of this filter to solve a vehicle navigation problem. In order to control the position of an automated vehicle, we first must have a reliable estimate of the vehicle’s present position. Kalman filtering provides a tool for obtaining that reliable estimate.

The Kalman filter theory and algorithm

Suppose we have a linear system model as described previously. We want to use the available measurements y to estimate the state of the system x. We know how the system behaves according to the state equation, and we have measurements of the position, so how can we determine the best estimate of the state x? We want an estimator that gives an accurate estimate of the true state even though we cannot directly measure it. What criteria should our estimator satisfy? Two obvious requirements come to mind.

First, we want the average value of our state estimate to be equal to the average value of the true state. That is, we don’t want our estimate to be biased one way or another. Mathematically, we would say that the expected value of the estimate should be equal to the expected value of the state.

Second, we want a state estimate that varies from the true state as little as possible. That is, not only do we want the average of the state estimate to be equal to the average of the true state, but we also want an estimator that results in the smallest possible variation of the state estimate.

Mathematically, we would say that we want to find the estimator with the smallest possible error variance. It so happens that the Kalman filter is the estimator that satisfies these two criteria. But the Kalman filter solution does not apply unless we can satisfy certain assumptions about the noise that affects our system. Remember from our system model that w is the process noise and z is the measurement noise.

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14-12-2012, 12:18 PM

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