Kalman filtering

Dan Simon

June 01, 2001

Dan SimonJune 01, 2001

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.

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.

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. Many physical processes, such as a vehicle driving along a road, a satellite orbiting the earth, a motor shaft driven by winding currents, or a sinusoidal 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:

Equation 1

Output equation:

Equation 2

In the above equations A, B, and C are matrices; k is the time index; x is called the state of the system; u is a known input to the system; y is the measured output; and w and z are the noise. The variable w is called the process noise, and z is called the measurement noise. Each of these quantities are (in general) vectors and therefore contain more than one element. The vector x contains all of the information about the present state of the system, but we cannot measure x directly. Instead we measure y, which is a function of x that is corrupted by the noise z. We can use y to help us obtain an estimate of x, but we cannot necessarily take the information from y at face value because it is corrupted by noise. The measurement is like a politician. We can use the information that it presents to a certain extent, but we cannot afford to grant it our total trust.

For example, suppose we want to model a vehicle going in a straight line. We can say that the state consists of the vehicle position p and velocity v. The input u is the commanded acceleration and the output y is the measured position. Let's say that we are able to change the acceleration and measure the position every T seconds. In this case, elementary laws of physics say that the velocity v will be governed by the following equation:

Equation 3

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 value for 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:

Equation 4

where Equation 4a is the velocity noise. A similar equation can be derived for the position p:

Equation 5

where Equation 5a is the position noise. Now we can define a state vector x that consists of position and velocity:

Equation 6

Finally, knowing that the measured output is equal to the position, we can write our linear system equations as follows:

Equation 7

zk is the measurement noise due to such things as instrumentation errors. If we want to control the vehicle with some sort of feedback system, we need an accurate estimate of the position p and the velocity v. In other words, we 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 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. We have to assume that the average value of w is zero and the average value of z is zero. We have to further assume that no correlation exists between w and z. That is, at any time k, wk, and zk are independent random variables. Then the noise covariance matrices Sw and Sz are defined as:

Process noise covariance

Equation 8

Measurement noise covariance

Equation 9

where wT and zT indicate the transpose of the w and z random noise vectors, and E(•) means the expected value.

Now we are finally in a position to look at the Kalman filter equations. There are many alternative but equivalent ways to express the equations. One of the formulations is given as follows:

Equation 10

That's the Kalman filter. It consists of three equations, each involving matrix manipulation. In the above equations, a -1 superscript indicates matrix inversion and a T superscript indicates matrix transposition. The K matrix is called the Kalman gain, and the P matrix is called the estimation error covariance.

The state estimate Equation 10a equation is fairly intuitive. The first term used to derive the state estimate at time k + 1 is just A times the state estimate at time k, plus B times the known input at time k. This would be the state estimate if we didn't have a measurement. In other words, the state estimate would propagate in time just like the state vector in the system model. The second term in the Equation 10a equation is called the correction term and it represents the amount by which to correct the propagated state estimate due to our measurement.

Inspection of the K equation shows that if the measurement noise is large, Sz will be large, so K will be small and we won't give much credibility to the measurement y when computing the next Equation 10a. On the other hand, if the measurement noise is small, Sz will be small, so K will be large and we will give a lot of credibility to the measurement when computing the next .

Vehicle navigation
Now consider the vehicle navigation problem that we looked at earlier. A vehicle is traveling along a road. The position is measured with an error of 10 feet (one standard deviation). The commanded acceleration is a constant 1 foot/sec2. The acceleration noise is 0.2 feet/sec2 (one standard deviation). The position is measured 10 times per second (T = 0.1). How can we best estimate the position of the moving vehicle? In view of the large measurement noise, we can surely do better than just taking the measurements at face value.

Since T = 0.1, the linear model that represents our system can be derived from the system model presented earlier in this article as follows:

Equation 11

Because the standard deviation of the measurement noise is 10 feet, the Sz matrix is simply equal to 100.

Now we need to derive the Sw matrix. Since the position is proportional to 0.005 times the acceleration, and the acceleration noise is 0.2 feet/sec2, the variance of the position noise is (0.005)2¥(0.2)2 = 10-6. Similarly, since the velocity is proportional to 0.1 times the acceleration, the variance of the velocity noise is (0.1)2¥(0.2)2 = 4¥10-4. Finally, the covariance of the position noise and velocity noise is equal to the standard deviation of the position noise times the standard deviation of the velocity noise, which can be calculated as (0.005¥0.2)¥(0.1¥0.2) = 2¥10-5. We can combine all of these calculations to obtain the following matrix for :

Equation 12

Now we just initialize Equation 16 as our best initial estimate of position and velocity, and we initialize P0 as the uncertainty in our initial estimate. Then we execute the Kalman filter equations once per time step and we are off and running.

I simulated the Kalman filter for this problem using Matlab. The results are shown in the accompanying figures. Figure 1 shows the true position of the vehicle, the measured position, and the estimated position. The two smooth curves are the true position and the estimated position, and they are almost too close to distinguish from one another. The noisy-looking curve is the measured position.

Figure 2 shows the error between the true position and the measured position, and the error between the true position and the Kalman filter's estimated position. The measurement error has a standard deviation of about 10 feet, with occasional spikes up to 30 feet (3 sigma). The estimated position error stays within about two feet.

Figure 3 shows a bonus that we get from the Kalman filter. Since the vehicle velocity is part of the state x, we get a velocity estimate along with the position estimate. Figure 4 shows the error between the true velocity and the Kalman filter's estimated velocity.

The Matlab program that I used to generate these results is shown in Listing 1. Don't worry if you don't know Matlab-it's an easy-to-read language, almost like pseudocode, but with built-in matrix operations. If you use Matlab to run the program you will get different results every time because of the random noise that is simulated, but the results will be similar to the figures shown here.

Listing 1: Kalman filter simulation

function kalman(duration, dt)

% function kalman(duration, dt)
% Kalman filter simulation for a vehicle travelling along a road.
% duration = length of simulation (seconds)
% dt = step size (seconds)

measnoise = 10; % position measurement noise (feet)
accelnoise = 0.2; % acceleration noise (feet/sec^2)

a = [1 dt; 0 1]; % transition matrix
b = [dt^2/2; dt]; % input matrix
c = [1 0]; % measurement matrix
x = [0; 0]; % initial state vector
xhat = x; % initial state estimate

Sz = measnoise^2; % measurement error covariance
Sw = accelnoise^2 * [dt^4/4 dt^3/2; dt^3/2 dt^2]; % process noise cov
P = Sw; % initial estimation covariance

% Initialize arrays for later plotting.
pos = []; % true position array
poshat = []; % estimated position array
posmeas = []; % measured position array
vel = []; % true velocity array
velhat = []; % estimated velocity array

for t = 0 : dt: duration,
% Use a constant commanded acceleration of 1 foot/sec^2.
    u = 1;
    % Simulate the linear system.
    ProcessNoise = accelnoise * [(dt^2/2)*randn; dt*randn];
    x = a * x + b * u + ProcessNoise;
    % Simulate the noisy measurement
    MeasNoise = measnoise * randn;
    y = c * x + MeasNoise;
    % Extrapolate the most recent state estimate to the present time.
    xhat = a * xhat + b * u;
    % Form the Innovation vector.
    Inn = y - c * xhat;
    % Compute the covariance of the Innovation.
    s = c * P * c' + Sz;
    % Form the Kalman Gain matrix.
    K = a * P * c' * inv(s);
    % Update the state estimate.
    xhat = xhat + K * Inn;
    % Compute the covariance of the estimation error.
    P = a * P * a' - a * P * c' * inv(s) * c * P * a' + Sw;
    % Save some parameters for plotting later.
    pos = [pos; x(1)];
    posmeas = [posmeas; y];
    poshat = [poshat; xhat(1)];
    vel = [vel; x(2)];
    velhat = [velhat; xhat(2)];

% Plot the results
close all;
t = 0 : dt : duration;

plot(t,pos, t,posmeas, t,poshat);
xlabel('Time (sec)');
ylabel('Position (feet)');
title('Figure 1 - Vehicle Position (True, Measured, and Estimated)')

plot(t,pos-posmeas, t,pos-poshat);
xlabel('Time (sec)');
ylabel('Position Error (feet)');
title('Figure 2 - Position Measurement Error and Position Estimation Error');

plot(t,vel, t,velhat);
xlabel('Time (sec)');
ylabel('Velocity (feet/sec)');
title('Figure 3 - Velocity (True and Estimated)');

xlabel('Time (sec)');
ylabel('Velocity Error (feet/sec)');
title('Figure 4 - Velocity Estimation Error');

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