How to Use Kalman Filter in MATLAB: A Step-by-Step Guide for Beginners
Kalman filter is a powerful technique for estimating the state of a dynamic system from noisy measurements. It can be used for applications such as tracking, navigation, sensor fusion, and data analysis. In this article, you will learn the basics of Kalman filter and how to implement it in MATLAB using some simple examples.
What is Kalman Filter
Kalman filter is a recursive algorithm that updates the estimate of the state vector and its covariance matrix based on the previous estimate and the new measurement. The state vector represents the variables of interest in the system, such as position, velocity, orientation, etc. The covariance matrix represents the uncertainty or error in the state estimate. The measurement is the observation of the system from a sensor or a model.
The Kalman filter consists of two steps: prediction and update. In the prediction step, the algorithm projects the state estimate and its covariance matrix to the next time step using a state transition model and a process noise model. In the update step, the algorithm incorporates the new measurement and its covariance matrix using a measurement model and a measurement noise model. The result is an improved state estimate and its covariance matrix that minimize the mean squared error.
How to Implement Kalman Filter in MATLAB
To implement Kalman filter in MATLAB, you need to define four matrices: A, B, C, and D. These matrices represent the state transition model, the control input model, the measurement model, and the direct feedthrough model respectively. You also need to define two vectors: x and u. These vectors represent the initial state estimate and the control input respectively. Finally, you need to define two matrices: P and Q. These matrices represent the initial state covariance matrix and the process noise covariance matrix respectively.
Once you have defined these variables, you can use the built-in function kalman to create a Kalman filter object. This object has two methods: predict and correct. These methods correspond to the prediction and update steps of the Kalman filter algorithm. You can use these methods in a loop to process a sequence of measurements and obtain a sequence of state estimates.
Kalman Filter Examples in MATLAB
To illustrate how to use Kalman filter in MATLAB, let's consider two examples: a one-dimensional constant velocity model and a two-dimensional constant acceleration model.
Example 1: One-Dimensional Constant Velocity Model
In this example, we assume that the system has one state variable: position (x). The system has a constant velocity (v) that is unknown. The system has no control input. The system has one measurement variable: position (z). The measurement is corrupted by Gaussian noise with zero mean and standard deviation (sigma).
The following code shows how to implement Kalman filter for this example:
% Define parameters
v = 1; % true velocity
sigma = 0.1; % measurement noise standard deviation
dt = 0.1; % time step
N = 100; % number of measurements
% Define matrices
A = [1 dt]; % state transition matrix
B = ; % control input matrix
C = ; % measurement matrix
D = ; % direct feedthrough matrix
% Define vectors
x = ; % initial state estimate
u = ; % control input
% Define matrices
P = ; % initial state covariance matrix
Q = ; % process noise covariance matrix
R = [sigma^2]; % measurement noise covariance matrix