Ingeniero

DAN

SIMON

f e a t u r e
72
JUNE 2001

Kalman Filtering
Originally developed for use in spacecraft navigation, the Kalman filter turns out to be useful for many applications. It is mainly used to estimate system states that can only be observed indirectly or inaccurately by the system itself.

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iltering is desirable in many situations in engineering and embedded systems. Forexample, 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 devicessuch 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 varianceof 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 avehicle 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.

Linear systems
In order to use a Kalman filter to remove noise from a signal, the process that we are measuring must be able to be described by a linear system. Manyphysical processes, such as a vehicle driving along a road, a satellite orbiting the earth, a motor shaft driven by winding currents, or a sinusoidal
Embedded Systems Programming

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.radio-frequency carrier signal, can be approximated as linear systems. A linear system is simply a process that can be described by the following two equations: State equation:

vk +1 = vk + Tuk
That is, the velocity one time-step from now (T seconds from now) will be equal to the present velocity plus the commanded acceleration multiplied by T. But the previous equation does not give a precise valuefor vk+1. Instead, the velocity will be perturbed by noise due to gusts of wind, potholes, and other unfortunate realities. The velocity noise is a random variable that changes with time. So a more realistic equation for v would be:
~ vk +1 = vk + Tuk + vk

need a way to estimate the state x. This is where the Kalman filter comes in.

The Kalman filter theory and algorithm
Suppose we have alinear 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 measureit. 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...