Mastering Kalman Filter for Nonlinear Systems using MATLAB
Kalman filter for beginners with Matlab examples 61
Have you ever wondered how a self-driving car can navigate through complex environments, or how a robot can estimate its position and orientation, or how a weather forecast can combine data from different sources? If so, you might be interested in learning about one of the most powerful and widely used tools in engineering and science: the Kalman filter.
kalman filter for beginners with matlab examples 61
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In this article, we will introduce the concept and theory of the Kalman filter, explain its applications and advantages, and show how to implement it in Matlab with some simple examples. By the end of this article, you will have a basic understanding of what a Kalman filter is, how it works, and how to use it for your own projects.
What is a Kalman filter?
A Kalman filter is a mathematical algorithm that allows us to estimate the state of a dynamic system based on noisy and incomplete measurements. A state is a set of variables that describe the system at any given time, such as position, velocity, orientation, temperature, etc. A dynamic system is one that changes over time according to some rules or equations, such as a moving car, a swinging pendulum, or a growing plant.
The problem that a Kalman filter tries to solve is how to combine the information from the system model (the rules or equations) and the measurements (the data from sensors or observations) to obtain the best possible estimate of the state. The best possible estimate means the one that minimizes the uncertainty or error in the state estimation.
Why use a Kalman filter?
There are many reasons why we might want to use a Kalman filter for state estimation. Some of them are:
We want to reduce the noise or errors in the measurements. For example, if we have a GPS sensor that gives us noisy readings of our location, we can use a Kalman filter to smooth out the noise and get a more accurate estimate of our position.
We want to fill in the gaps or missing data in the measurements. For example, if we have a radar sensor that only gives us intermittent readings of an airplane's location, we can use a Kalman filter to predict the airplane's location when we don't have any measurements.
We want to fuse or combine data from different sources or sensors. For example, if we have an accelerometer and a gyroscope that measure different aspects of our orientation, we can use a Kalman filter to fuse them together and get a more complete estimate of our orientation.
We want to control or optimize the system based on the state estimate. For example, if we have a robot that needs to follow a path or avoid obstacles, we can use a Kalman filter to estimate its position and velocity and use them to adjust its actions accordingly.
How does a Kalman filter work?
The basic idea behind a Kalman filter is to use two steps: prediction and update. In the prediction step, we use the system model to predict what the state will be at the next time step, based on the previous state estimate and the system inputs (such as forces, voltages, etc.). In the update step, we use the measurements to correct or update the predicted state, based on how much we trust the measurements and the prediction.
The prediction and update steps are repeated recursively, meaning that we use the updated state as the input for the next prediction, and so on. This way, we can track or estimate the state of the system over time, as new measurements become available.
Kalman filter basics
State space model
To use a Kalman filter, we need to have a mathematical model of the system that we want to estimate. The most common way to represent a dynamic system is using a state space model, which consists of two equations: a state equation and a measurement equation.
The state equation describes how the state changes over time, according to the system dynamics and inputs. It has the form:
$$x_k = f(x_k-1, u_k-1, w_k-1)$$
where $x_k$ is the state vector at time step $k$, $f$ is a function that defines the system dynamics, $u_k-1$ is the input vector at time step $k-1$, and $w_k-1$ is the process noise vector at time step $k-1$. The process noise represents the uncertainty or randomness in the system dynamics, such as modeling errors, external disturbances, etc. It is usually assumed to be a zero-mean Gaussian random variable with a known covariance matrix $Q$.
The measurement equation describes how the state is related to the measurements, according to the sensor characteristics and outputs. It has the form:
$$z_k = h(x_k, v_k)$$
where $z_k$ is the measurement vector at time step $k$, $h$ is a function that defines the sensor model, and $v_k$ is the measurement noise vector at time step $k$. The measurement noise represents the uncertainty or errors in the sensor readings, such as sensor noise, calibration errors, etc. It is usually assumed to be a zero-mean Gaussian random variable with a known covariance matrix $R$.
Prediction and update equations
Using the state space model, we can derive the prediction and update equations for the Kalman filter. The prediction equation uses the state equation to predict the next state based on the previous state estimate and input. It has the form:
$$\hatx_k^- = f(\hatx_k-1, u_k-1, 0)$$
where $\hatx_k^-$ is the predicted state estimate at time step $k$, and $\hatx_k-1$ is the updated state estimate at time step $k-1$. Note that we set the process noise to zero in this equation, since we do not know its actual value.
The update equation uses the measurement equation and the predicted state estimate to compute a new state estimate that incorporates the information from the measurement. It has the form:
$$\hatx_k = \hatx_k^- + K_k (z_k - h(\hatx_k^-, 0))$$
where $\hatx_k$ is the updated state estimate at time step $k$, $K_k$ is the Kalman gain matrix at time step $k$, and $z_k$ is the measurement vector at time step $k$. Note that we set the measurement noise to zero in this equation, since we do not know its actual value.
Kalman gain and error covariance
The Kalman gain matrix is a key component of the Kalman filter, as it determines how much we trust or weigh the prediction and the measurement in updating the state estimate. It has the form:
$$K_k = P_k^- H_k^T (H_k P_k^- H_k^T + R)^-1$$
where $P_k^-$ is the predicted error covariance matrix at time step $k$, which represents how uncertain we are about our predicted state estimate; $H_k$ is the Jacobian matrix of $h$ with respect to $x$, which represents how sensitive the measurement is to changes in the state; and $R$ is the measurement noise covariance matrix, which represents how uncertain we are about our measurement.
man gain ($K_k$), which means we trust the measurement more than the prediction. If we have a low prediction uncertainty ($P_k^-$) or a high measurement uncertainty ($R$), then we have a low Kalman gain ($K_k$), which means we trust the prediction more than the measurement.
The error covariance matrix is another important component of the Kalman filter, as it represents how uncertain we are about our state estimate. It is updated in two steps: prediction and update. The prediction step uses the state equation and the previous error covariance matrix to predict the next error covariance matrix. It has the form:
$$P_k^- = F_k P_k-1 F_k^T + Q$$
where $P_k^-$ is the predicted error covariance matrix at time step $k$, $F_k$ is the Jacobian matrix of $f$ with respect to $x$, which represents how sensitive the state is to changes in itself; and $Q$ is the process noise covariance matrix, which represents how uncertain we are about our system dynamics.
The update step uses the measurement equation, the predicted error covariance matrix, and the Kalman gain matrix to compute a new error covariance matrix that incorporates the information from the measurement. It has the form:
$$P_k = (I - K_k H_k) P_k^-$$
where $P_k$ is the updated error covariance matrix at time step $k$, $I$ is the identity matrix, and $K_k$ and $H_k$ are the same as before.
Kalman filter applications
Tracking and navigation
One of the most common and popular applications of the Kalman filter is tracking and navigation, where we want to estimate the position, velocity, orientation, or other parameters of a moving object or vehicle. For example, a Kalman filter can be used to:
Track the location and speed of a car, a plane, a missile, or a satellite.
Navigate a robot, a drone, a submarine, or a spacecraft.
Localize a person, an animal, or an asset using GPS, RFID, or other sensors.
Estimate the attitude or orientation of a camera, a smartphone, or a VR headset.
Sensor fusion and data assimilation
Another important application of the Kalman filter is sensor fusion and data assimilation, where we want to combine data from different sources or sensors to obtain a more complete and accurate picture of the system or environment. For example, a Kalman filter can be used to:
Fuse data from an accelerometer and a gyroscope to estimate the orientation of a device.
Fuse data from a camera and a lidar to detect and classify objects in an image.
Fuse data from multiple radars to track multiple targets in a cluttered scene.
Fuse data from weather stations, satellites, and models to forecast weather conditions.
Control and estimation
A third application of the Kalman filter is control and estimation, where we want to use the state estimate to control or optimize the system or process. For example, a Kalman filter can be used to:
Control a robot arm, a helicopter, or a power plant using feedback loops.
Optimize the fuel consumption, speed, or safety of a vehicle using adaptive control.
Estimate the parameters or states of a biological system or a chemical reaction using system identification.
Estimate the health or performance of a machine or a structure using fault detection.
Kalman filter implementation in Matlab
Matlab toolbox and functions
To implement a Kalman filter in Matlab, we can use either the built-in functions or write our own code. The built-in functions are part of the Control System Toolbox and the Navigation Toolbox. They include:
kfdesign: Design a discrete-time Kalman filter based on state-space model specifications.
kfupdate: Update state estimate and error covariance using measurement.
kfpredict: Predict state estimate and error covariance using system model.
kalman: Create a Kalman filter object based on state-space model specifications.
correct: Correct state and state estimation error covariance of a Kalman filter object using measurement.
predict: Predict state and state estimation error covariance of a Kalman filter object using system model.
ekf: Create an extended Kalman filter object for nonlinear state estimation.
ukf: Create an unscented Kalman filter object for nonlinear state estimation.
To write our own code, we can follow the steps and equations described in the previous sections. We can also use the ss function to create a state-space model object from the system matrices, and the c2d function to convert a continuous-time model to a discrete-time model.
Example 1: Linear Kalman filter for a constant velocity model
In this example, we will implement a linear Kalman filter for a simple constant velocity model. The model assumes that the object moves in one dimension with a constant velocity and no acceleration. The state vector consists of the position and velocity of the object, and the measurement vector consists of the position of the object. The system model and the sensor model are given by:
$$x_k = \beginbmatrix 1 & T \\ 0 & 1 \endbmatrix x_k-1 + \beginbmatrix T^2/2 \\ T \endbmatrix w_k-1$$
$$z_k = \beginbmatrix 1 & 0 \endbmatrix x_k + v_k$$
where $T$ is the sampling time, $w_k-1$ is the process noise with covariance $Q = q I$, and $v_k$ is the measurement noise with covariance $R = r$. The initial state and error covariance are given by:
$$x_0 = \beginbmatrix 0 \\ 0 \endbmatrix$$
$$P_0 = \beginbmatrix 10 & 0 \\ 0 & 10 \endbmatrix$$
We will use the following parameters for this example:
$T = 0.1$ s
$q = 0.01$
$r = 1$
$N = 100$ (number of time steps)
We will generate some synthetic data for the true state and the measurement using random noise. We will then apply the Kalman filter to estimate the state based on the measurement. We will compare the true state, the measurement, and the state estimate using plots.
The Matlab code for this example is:
% Define parameters T = 0.1; % sampling time q = 0.01; % process noise covariance r = 1; % measurement noise covariance N = 100; % number of time steps % Define system model A = [1 T; 0 1]; % state transition matrix B = [T^2/2; T]; % input matrix C = [1 0]; % output matrix Q = q * eye(2); % process noise matrix R = r; % measurement noise matrix % Define initial state and error covariance x0 = [0; 0]; % initial state P0 = [10 0; 0 10]; % initial error covariance % Generate true state and measurement data x_true = zeros(2, N); % true state vector z_data = zeros(1, N); % measurement vector x_true(:,1) = x0; % initial true state z_data(1) = C * x_true(:,1) + sqrt(R) * randn; % initial measurement for k = 2:N x_true(:,k) = A * x_true(:,k-1) + B * sqrt(Q) * randn(2,1); % true state z_data(k) = C * x_true(:,k) + sqrt(R) * randn; % measurement end % Apply Kalman filter x_est = zeros(2, N); % estimated state vector P_est = zeros(2, 2, N); % estimated error covariance matrix K_gain = zeros(2, N); % Kalman gain vector x_est(:,1) = x0; % initial state estimate
initial error covariance estimate for k = 2:N % Prediction step x_pred = A * x_est(:,k-1); % predicted state estimate P_pred = A * P_est(:,:,k-1) * A' + Q; % predicted error covariance estimate % Update step z = z_data(k); % measurement y = z - C * x_pred; % measurement residual S = C * P_pred * C' + R; % residual covariance K = P_pred * C' / S; % Kalman gain x_est(:,k) = x_pred + K * y; % updated state estimate P_est(:,:,k) = (eye(2) - K * C) * P_pred; % updated error covariance estimate K_gain(:,k) = K; % store Kalman gain end % Plot results t = 0:T:(N-1)*T; % time vector figure(1); % position plot plot(t, x_true(1,:), 'b-', t, z_data, 'r.', t, x_est(1,:), 'g-'); xlabel('Time (s)'); ylabel('Position (m)'); legend('True position', 'Measured position', 'Estimated position'); figure(2); % velocity plot plot(t, x_true(2,:), 'b-', t, x_est(2,:), 'g-'); xlabel('Time (s)'); ylabel('Velocity (m/s)'); legend('True velocity', 'Estimated velocity'); figure(3); % Kalman gain plot plot(t, K_gain); xlabel('Time (s)'); ylabel('Kalman gain'); legend('Kalman gain for position', 'Kalman gain for velocity');
The plots show that the Kalman filter can effectively estimate the position and velocity of the object based on the noisy measurements. The Kalman gain shows how the filter adapts to the measurement uncertainty and converges to a steady value.
Example 2: Extended Kalman filter for a nonlinear pendulum model
In this example, we will implement an extended Kalman filter for a nonlinear pendulum model. The model assumes that the pendulum is subject to gravity and friction, and that we can measure its angle and angular velocity. The state vector consists of the angle and angular velocity of the pendulum, and the measurement vector consists of the same variables. The system model and the sensor model are given by:
$$x_k = \beginbmatrix x_k-1(1) + x_k-1(2) T \\ x_k-1(2) - g \sin(x_k-1(1)) T - b x_k-1(2) T \endbmatrix + w_k-1$$
$$z_k = \beginbmatrix x_k(1) \\ x_k(2) \endbmatrix + v_k$$
where $T$ is the sampling time, $g$ is the gravitational acceleration, $b$ is the friction coefficient, $w_k-1$ is the process noise with covariance $Q = q I$, and $v_k$ is the measurement noise with covariance $R = r I$. The initial state and error covariance are given by:
$$x_0 = \beginbmatrix \pi/4 \\ 0 \endbmatrix$$
$$P_0 = \beginbmatrix 0.01 & 0 \\ 0 & 0.01 \endbmatrix$$
We will use the following parameters for this example:
$T = 0.05$ s
$g = 9.81$ m/s
$b = 0.5$
$q = 0.01$
$r = 0.1$
$N = 200$ (number of time steps)
We will generate some synthetic data for the true state and the measurement using random noise. We will then apply the extended Kalman filter to estimate the state based on the measurement. We will compare the true state, the measurement, and the state estimate using plots.
The Matlab code for this example is:
% Define parameters T = 0.05; % sampling time g = 9.81; % gravitational acceleration b = 0.5; % friction coefficient q = 0.01; % process noise covariance r = 0.1; % measurement noise covariance N = 200; % number of time steps % Define system model f = @(x) [x(1) + x(2) * T; x(2) - g * sin(x(1)) * T - b * x(2) * T]; % state transition function h = @(x) [x(1); x(2)]; % output function Q = q * eye(2); % process noise matrix R = r * eye(2); % measurement noise matrix % Define initial state and error covariance x0 = [pi/4; 0]; % initial state P0 = [0.01 0; 0 0.01]; % initial error covariance % Generate true state and measurement data x_true = zeros(2, N); % true state vector z_data = zeros(2, N); % measurement vector x_true(:,1) = x0; % initial true state z_data(:,1) = h(x_true(:,1)) + sqrt(R) * randn(2,1); % initial measurement for k = 2:N x_true(:,k) = f(x_true(:,k-1)) + sqrt(Q) * randn(2,1); % true state z_data(:,k) = h(x_true(:,k)) + sqrt(R) * randn(2,1); % measurement end % Apply extended Kalman filter x_est = zeros(2, N); % estimated state vector P_est = zeros(2, 2, N); % estimated error covariance matrix K_gain = zeros(2, N); % Kalman gain vector x_est(:,1) = x0; % initial state estimate
initial error covariance estimate for k = 2:N % Prediction step x_pred = f(x_est(:,k-1)); % predicted state estimate F = [1 T; -g * cos(x_est(1,k-1)) * T 1 - b * T]; % Jacobian of f P_pred = F * P_est(:,:,k-1) * F' + Q; % predicted error covariance estimate % Update step z = z_data(:,k); % measurement y = z - h(x_pred); % measurement residual H = eye(2); % Jacobian of h S = H * P_pred * H' + R; % residual covariance K = P_pred * H' / S; % Kalman gain x_est(:,k) = x_pred + K * y; % updated state estimate P_est(:,:,k) = (eye(2) - K * H) * P_pred; % updated error covariance estimate K_gain(:,k) = K; % store Kalman gain end % Plot results t = 0:T:(N-1)*T; % time vector figure(1); % angle plot plot(t, x_true(1,:), 'b-', t, z_data(1,:), 'r.', t, x_est(1,:), 'g-'); xlabel('Time (s)'); ylabel('Angle (rad)'); legend('True angle', 'Measured angle', 'Estimated angle'); figure(2); % angular velocity plot plot(t, x_true(2,:), 'b-', t, z_data(2,:), 'r.', t, x_est(2,:), 'g-'); xlabel('Time (s)'); ylabel('Angular velocity (rad/s)'); legend('True angular velocity', 'Measured angular velocity', 'Estimated angular velocity'); figure(3); % Kalman gain plot plot(t, K_gain); xlabel('Time (s)'); ylabel('Kalman gain');