In this phase, the filter uses the system's physical model to project the state forward in time. Error Covariance Prediction: is the state transition matrix, is the control input matrix, is the estimation error uncertainty, and is the process noise covariance. Phase 2: Update (Measurement Update) Once a physical sensor measurement ( ) arrives, the filter corrects its prediction. Calculate Kalman Gain: Update State Estimate: Update Error Covariance: is the measurement matrix, is the sensor noise covariance, and is the Kalman Gain. If sensor noise ( ) is very high, Kkcap K sub k
Demystifying the Kalman Filter: A Beginner's Guide with MATLAB Examples
to force the filter to trust your live measurements more. If your tracking is too erratic, increase or decrease
It transitions from basic averages and moving averages to the actual Kalman Filter equation. In this phase, the filter uses the system's
% Matrix Kalman Filter: Tracking Position and Velocity clear all; close all; clc; dt = 0.1; % Time step (seconds) N = 100; % Number of samples % System Matrices A = [1 dt; 0 1]; % State transition matrix (Pos = Pos + Vel*dt) H = [1 0]; % Measurement matrix (we only measure position) Q = [0.01 0; 0 0.01]; % Process noise covariance R = 2.25; % Measurement noise covariance (std dev of 1.5 meters) % Initial States x_est = [0; 0]; % Initial [Position; Velocity] estimate P = eye(2); % Initial error covariance matrix % Simulated True Trajectory true_pos = zeros(N, 1); measurements = zeros(N, 1); est_pos = zeros(N, 1); est_vel = zeros(N, 1); current_pos = 0; current_vel = 5; % True velocity is 5 m/s for k = 1:N % Update true positions and create noisy measurements current_pos = current_pos + current_vel * dt; true_pos(k) = current_pos; measurements(k) = current_pos + sqrt(R)*randn(); % --- 1. PREDICT --- x_pred = A * x_est; P_pred = A * P * A' + Q; % --- 2. KALMAN GAIN --- K = (P_pred * H') / (H * P_pred * H' + R); % --- 3. UPDATE --- x_est = x_pred + K * (measurements(k) - H * x_pred); P = (eye(2) - K * H) * P_pred; % Save data est_pos(k) = x_est(1); est_vel(k) = x_est(2); end % Plotting results figure; subplot(2,1,1); plot(true_pos, 'g', 'LineWidth', 2); hold on; plot(measurements, 'r.', 'MarkerSize', 8); plot(est_pos, 'b--', 'LineWidth', 2); title('Position Tracking'); ylabel('Position (m)'); legend('True', 'Noisy Sensor', 'Kalman Estimate'); grid on; subplot(2,1,2); plot(repmat(current_vel, N, 1), 'g', 'LineWidth', 2); hold on; plot(est_vel, 'b--', 'LineWidth', 2); title('Velocity Estimation'); ylabel('Velocity (m/s)'); xlabel('Time Steps'); legend('True', 'Kalman Estimate'); grid on; Use code with caution. Why Phil Kim’s Approach Works for Beginners
This article provides an in-depth breakdown of the foundational concepts taught in Phil Kim's book, the core mathematical equations of the filter, and a complete MATLAB example to help you implement your very first Kalman filter. Why Use a Kalman Filter? The Core Intuition Imagine you are driving a car through a long, dark tunnel.
To understand the mathematics behind the code, you must grasp the four fundamental concepts that dictate how a Kalman filter processes information: State Vector ( Calculate Kalman Gain: Update State Estimate: Update Error
This article breaks down the core concepts of the Kalman filter, explains why Phil Kim's approach is so effective, and provides a foundational MATLAB example to get you started. What is a Kalman Filter and Why Do We Need It?
To expand your tracking project further, consider exploring sensor fusion algorithms on the MathWorks Aerospace and Defense Hub or dive into advanced filtering toolbox tutorials directly on the MATLAB Signal Processing Documentation Page.
The book starts with simple, one-dimensional problems (like estimating a constant voltage or a simple weight) where the math involves basic algebra rather than complex matrix multiplication. % Matrix Kalman Filter: Tracking Position and Velocity
You can look at your speedometer and estimate your position based on how fast you've been driving (dead reckoning). However, wheel slip and wind resistance will cause small errors to accumulate over time, leading your estimate to drift away from reality (system uncertainty).
function [voltMeas, voltEst, P] = SimpleKalman(z) % SIMPLEKALMAN: A basic 1D Kalman Filter implementation. % Input 'z' is the raw, noisy measurement. persistent initialized x P A H Q R % Initialize variables on the very first run if isempty(initialized) x = 10; % Initial guess of the state (Volts/Temp) P = 1; % Initial estimation error covariance A = 1; % System matrix (state does not inherently change) H = 1; % Measurement matrix Q = 0.001; % Process noise covariance (system is stable) R = 0.5; % Measurement noise covariance (sensor fluctuates) initialized = true; end % 1. Predict Phase x_pred = A * x; P_pred = A * P * A' + Q; % 2. Update Phase (Kalman Gain) K = (P_pred * H') / (H * P_pred * H' + R); % 3. Update State Estimate with Measurement 'z' x = x_pred + K * (z - H * x_pred); % 4. Update Error Covariance P = (1 - K * H) * P_pred; % Return outputs voltMeas = z; voltEst = x; end Use code with caution. Step 2: Test the Filter with a Simulation Script
The Kalman filter has several key components: