Kalman Filter For Beginners With Matlab Examples [patched] Download Top -

Introduction

The Kalman filter is a mathematical algorithm used to estimate the state of a system from noisy measurements. It is widely used in various fields such as navigation, control systems, signal processing, and econometrics. The Kalman filter is a powerful tool for estimating the state of a system, and it has many applications in real-world problems.

What is a Kalman Filter?

A Kalman filter is a algorithm that uses a combination of prediction and measurement updates to estimate the state of a system. It is a recursive algorithm, meaning that it uses the previous estimates to compute the current estimate. The Kalman filter consists of two main steps:

1. Prediction step: In this step, the algorithm predicts the state of the system at the next time step using the current state estimate and the system dynamics.
2. Measurement update step: In this step, the algorithm updates the state estimate using the measurement data and the predicted state.

Key Components of a Kalman Filter

The Kalman filter has several key components:

1. State transition model: This model describes how the state of the system changes over time.
2. Measurement model: This model describes how the measurements are related to the state of the system.
3. Process noise: This represents the uncertainty in the state transition model.
4. Measurement noise: This represents the uncertainty in the measurement data.
5. Initial state estimate: This is the initial estimate of the state of the system.

How Does a Kalman Filter Work?

The Kalman filter works as follows:

1. Initialize the state estimate and the covariance of the state estimate.
2. Predict the state of the system at the next time step using the state transition model.
3. Predict the covariance of the state estimate using the process noise and the state transition model.
4. Measure the system at the next time step.
5. Update the state estimate using the measurement data and the measurement model.
6. Update the covariance of the state estimate using the measurement noise and the measurement model.
7. Repeat steps 2-6.

MATLAB Example

Here is a simple MATLAB example of a Kalman filter:

% Define the state transition model
A = [1 1; 0 1];
% Define the measurement model
H = [1 0];
% Define the process noise
Q = [0.01 0; 0 0.01];
% Define the measurement noise
R = [1];
% Define the initial state estimate
x0 = [0; 0];
% Define the initial covariance of the state estimate
P0 = [1 0; 0 1];
% Generate some measurement data
t = 0:0.1:10;
x_true = sin(t);
y = x_true + randn(size(t));
% Run the Kalman filter
x_est = zeros(size(t));
P_est = zeros(size(t));
x_est(1) = x0(1);
P_est(1) = P0(1,1);
for i = 2:length(t)
    % Predict the state
    x_pred = A*x_est(i-1);
    P_pred = A*P_est(i-1)*A' + Q;
% Update the state estimate
    y_measurement = y(i);
    innovation = y_measurement - H*x_pred;
    S = H*P_pred*H' + R;
    K = P_pred*H'/S;
    x_est(i) = x_pred + K*innovation;
    P_est(i) = P_pred - K*H*P_pred;
end
% Plot the results
plot(t, x_true, 'b', t, x_est, 'r')
xlabel('Time')
ylabel('State')
legend('True state', 'Estimated state')

This example estimates the state of a simple system using a Kalman filter. Introduction The Kalman filter is a mathematical algorithm

Download MATLAB Code

You can download the MATLAB code used in this example from the following link:

[Insert link to download MATLAB code]

Conclusion

1. What is a Kalman Filter?

Imagine you are trying to track the position of a moving car. Your GPS gives you a noisy reading (maybe off by a few meters). Your knowledge of physics tells you the car should be moving smoothly. Which one do you trust?

The Kalman Filter (KF) is the mathematical tool that answers this question. It is an optimal recursive algorithm that combines:

• Noisy measurements (e.g., sensor data)
• Predicted states (e.g., based on velocity)

...to produce a smoother, more accurate estimate of a system's true state.

Beginner’s Takeaway: It’s a "smart averaging" technique that learns from past predictions and new measurements to filter out noise.

3. Kalman Filter Equations

Predict: x̂_k-1 = A x̂_k-1 + B u_k-1 P_k-1 = A P_k-1 A^T + Q

Update: K_k = P_k H^T (H P_k-1 H^T + R)^-1 x̂_k = x̂_k + K_k (z_k - H x̂_k-1) P_k = (I - K_k H) P_k Prediction step : In this step, the algorithm

Comments:

• K_k is the Kalman gain: weight given to measurement vs. prediction.
• P is the state covariance: uncertainty of estimate.
• Initialization: x̂_0 and P_0 must be set.

5. MATLAB Example – Tracking a Moving Object

Let’s implement a 1D Kalman Filter to track a car moving at constant velocity.

The Output

When you run the code, you will see:

1. Green Line: The perfect path.
2. Red Dots: The noisy, messy sensor data.
3. Blue Line: The Kalman Filter estimate. Notice how it smooths out the noise but eventually locks onto the true path!

The Intuition: "Who do I trust?"

Imagine you are in a tunnel trying to guess how fast your car is going.

1. Prediction (The Model): You look at your speedometer. It says you are going 50 mph. But you know the speedometer isn't perfect; it might be off by 2-3 mph.
2. Measurement (The Sensor): You look at a radar sign on the side of the road. It says you are going 52 mph. But the radar also has errors; it might be off by 1-2 mph.

The Question: Are you going 50, 52, or something in between?

The Kalman Filter Answer: You should take a weighted average. If you trust the radar more than your speedometer, you put more weight on the radar's number. If you trust the speedometer more, you weight that.

The "magic" of the Kalman Filter is that it calculates exactly how much weight to give each source based on math, giving you the statistically optimal estimate of the true speed.

Part 4: MATLAB Example 2 – Tracking a Falling Object (With Acceleration)

Now let's try a more realistic example: a ball falling under gravity. The state will be [Position; Velocity] and the acceleration (gravity) is known.

In this example, we use the Extended Kalman Filter (EKF) logic but simplified—because gravity is a known input.

%% Kalman Filter Example 2: Falling Object with Gravity
clear; clc; close all;
%% Simulation parameters
dt = 0.01;               % 10 ms time step
t_end = 2;               % 2 seconds of fall
t = 0:dt:t_end;
N = length(t);
g = -9.81;               % Gravity (m/s^2) Key Components of a Kalman Filter The Kalman
%% True dynamics (with no noise)
true_pos = 0.5 * g * t.^2;      % s = 0.5gt^2
true_vel = g * t;                % v = g*t
%% Noisy measurement (measuring position only)
meas_noise_std = 0.5;            % 0.5 meter noise
measurements = true_pos + meas_noise_std * randn(1, N);
%% Kalman Filter
x_est = [0; 0];                  % [pos; vel]
P_est = eye(2) * 1;
% State transition with known input (gravity)
% x(k+1) = Fx(k) + Bu(k)
F = [1, dt;
0, 1];
B = [0.5*dt^2; dt];              % Control input matrix for acceleration
u = g;                           % Control input (gravity)
H = [1, 0];                      % Measure only position
Q = [0.001, 0; 0, 0.001];        % Process noise (small)
R = meas_noise_std^2;            % Measurement noise
stored_x = zeros(2, N);
for k = 1:N
% Prediction with known input
x_pred = F * x_est + B * u;
P_pred = F * P_est * F' + Q;
% Measurement update
z = measurements(k);
y = z - H * x_pred;
S = H * P_pred * H' + R;
K = P_pred * H' / S;
x_est = x_pred + K * y;
P_est = (eye(2) - K * H) * P_pred;
stored_x(:, k) = x_est;

end
%% Plotting
figure;
plot(t, true_pos, 'g-', 'LineWidth', 2); hold on;
plot(t, measurements, 'r.', 'MarkerSize', 4);
plot(t, stored_x(1,:), 'b-', 'LineWidth', 2);
xlabel('Time (s)'); ylabel('Position (m)');
title('Tracking a Falling Object with Kalman Filter');
legend('True Position', 'Noisy Measurements', 'Kalman Estimate');
grid on;
rmse_raw = sqrt(mean((measurements - true_pos).^2));
rmse_kalman = sqrt(mean((stored_x(1,:) - true_pos).^2));
fprintf('Raw sensor RMSE: %.3f m\n', rmse_raw);
fprintf('Kalman filter RMSE: %.3f m\n', rmse_kalman);

Observations: The Kalman filter knows gravity is pulling the object, so even if the sensor says the ball is moving up (due to noise), the filter corrects it because it trusts the physics model.