This example shows how to generate C code for a MATLAB Kalman filter function, kalmanfilter
, which estimates the position of a moving object based on past noisy measurements. It also shows how to generate a MEX function for this MATLAB code to increase the execution speed of the algorithm in MATLAB.
There are no prerequisites for this example.
kalmanfilter
FunctionThe kalmanfilter
function predicts the position of a moving object based on its past values. It uses a Kalman filter estimator, a recursive adaptive filter that estimates the state of a dynamic system from a series of noisy measurements. Kalman filtering has a broad range of application in areas such as signal and image processing, control design, and computational finance.
The Kalman estimator computes the position vector by computing and updating the Kalman state vector. The state vector is defined as a 6-by-1 column vector that includes position (x and y), velocity (Vx Vy), and acceleration (Ax and Ay) measurements in a 2-dimensional Cartesian space. Based on the classical laws of motion:
The iterative formula capturing these laws are reflected in the Kalman state transition matrix "A". Note that by writing about 10 lines of MATLAB code, you can implement the Kalman estimator based on the theoretical mathematical formula found in many adaptive filtering textbooks.
type kalmanfilter.m
% Copyright 2010 The MathWorks, Inc. function y = kalmanfilter(z) %#codegen dt=1; % Initialize state transition matrix A=[ 1 0 dt 0 0 0;... % [x ] 0 1 0 dt 0 0;... % [y ] 0 0 1 0 dt 0;... % [Vx] 0 0 0 1 0 dt;... % [Vy] 0 0 0 0 1 0 ;... % [Ax] 0 0 0 0 0 1 ]; % [Ay] H = [ 1 0 0 0 0 0; 0 1 0 0 0 0 ]; % Initialize measurement matrix Q = eye(6); R = 1000 * eye(2); persistent x_est p_est % Initial state conditions if isempty(x_est) x_est = zeros(6, 1); % x_est=[x,y,Vx,Vy,Ax,Ay]' p_est = zeros(6, 6); end % Predicted state and covariance x_prd = A * x_est; p_prd = A * p_est * A' + Q; % Estimation S = H * p_prd' * H' + R; B = H * p_prd'; klm_gain = (S \ B)'; % Estimated state and covariance x_est = x_prd + klm_gain * (z - H * x_prd); p_est = p_prd - klm_gain * H * p_prd; % Compute the estimated measurements y = H * x_est; end % of the function
The position of the object to track are recorded as x and y coordinates in a Cartesian space in a MAT file called position_data.mat
. The following code loads the MAT file and plots the trace of the positions. The test data includes two sudden shifts or discontinuities in position which are used to check that the Kalman filter can quickly re-adjust and track the object.
load position_data.mat
hold; grid;
Current plot held
for idx = 1: numPts z = position(:,idx); plot(z(1), z(2), 'bx'); axis([-1 1 -1 1]); end title('Test vector for the Kalman filtering with 2 sudden discontinuities '); xlabel('x-axis');ylabel('y-axis'); hold;
Current plot released
ObjTrack
FunctionThe ObjTrack.m
function calls the Kalman filter algorithm and plots the trajectory of the object in blue and the Kalman filter estimated position in green. Initially, you see that it takes a short time for the estimated position to converge with the actual position of the object. Then, three sudden shifts in position occur. Each time the Kalman filter readjusts and tracks the object after a few iterations.
type ObjTrack
% Copyright 2010 The MathWorks, Inc. function ObjTrack(position) %#codegen % First, setup the figure numPts = 300; % Process and plot 300 samples figure;hold;grid; % Prepare plot window % Main loop for idx = 1: numPts z = position(:,idx); % Get the input data y = kalmanfilter(z); % Call Kalman filter to estimate the position plot_trajectory(z,y); % Plot the results end hold; end % of the function
ObjTrack(position)
Current plot held
Current plot released
The codegen
command with the -config:lib
option generates C code packaged as a standalone C library.
Because C uses static typing, codegen
must determine the properties of all variables in the MATLAB files at compile time. Here, the -args
command-line option supplies an example input so that codegen
can infer new types based on the input types.
The -report
option generates a compilation report that contains a summary of the compilation results and links to generated files. After compiling the MATLAB code, codegen
provides a hyperlink to this report.
z = position(:,1); codegen -config:lib -report -c kalmanfilter.m -args {z}
Code generation successful: To view the report, open('codegen/lib/kalmanfilter/html/report.mldatx').
The generated C code is in the codegen/lib/kalmanfilter/
folder. The files are:
dir codegen/lib/kalmanfilter/
. kalmanfilter_data.c .. kalmanfilter_data.h .gitignore kalmanfilter_initialize.c buildInfo.mat kalmanfilter_initialize.h codeInfo.mat kalmanfilter_ref.rsp codedescriptor.dmr kalmanfilter_rtw.mk compileInfo.mat kalmanfilter_terminate.c defines.txt kalmanfilter_terminate.h examples kalmanfilter_types.h html rtw_proj.tmw interface rtwtypes.h kalmanfilter.c kalmanfilter.h
kalmanfilter.c
Functiontype codegen/lib/kalmanfilter/kalmanfilter.c
/* * File: kalmanfilter.c * * MATLAB Coder version : 5.1 * C/C++ source code generated on : 24-Aug-2020 19:16:51 */ /* Include Files */ #include "kalmanfilter.h" #include "kalmanfilter_data.h" #include "kalmanfilter_initialize.h" #include <math.h> #include <string.h> /* Variable Definitions */ static double x_est[6]; static double p_est[36]; /* Function Definitions */ /* * Arguments : const double z[2] * double y[2] * Return Type : void */ void kalmanfilter(const double z[2], double y[2]) { static const short R[4] = { 1000, 0, 0, 1000 }; static const signed char a[36] = { 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1 }; static const signed char iv[36] = { 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 }; static const signed char c_a[12] = { 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 }; static const signed char iv1[12] = { 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 }; double b_a[36]; double p_prd[36]; double B[12]; double Y[12]; double x_prd[6]; double S[4]; double b_z[2]; double a21; double a22; double a22_tmp; double d; double d1; int i; int k; int r1; int r2; signed char Q[36]; if (!isInitialized_kalmanfilter) { kalmanfilter_initialize(); } /* Copyright 2010 The MathWorks, Inc. */ /* Initialize state transition matrix */ /* % [x ] */ /* % [y ] */ /* % [Vx] */ /* % [Vy] */ /* % [Ax] */ /* [Ay] */ /* Initialize measurement matrix */ for (i = 0; i < 36; i++) { Q[i] = 0; } /* Initial state conditions */ /* Predicted state and covariance */ for (k = 0; k < 6; k++) { Q[k + 6 * k] = 1; x_prd[k] = 0.0; for (i = 0; i < 6; i++) { r1 = k + 6 * i; x_prd[k] += (double)a[r1] * x_est[i]; d = 0.0; for (r2 = 0; r2 < 6; r2++) { d += (double)a[k + 6 * r2] * p_est[r2 + 6 * i]; } b_a[r1] = d; } } for (i = 0; i < 6; i++) { for (r2 = 0; r2 < 6; r2++) { d = 0.0; for (r1 = 0; r1 < 6; r1++) { d += b_a[i + 6 * r1] * (double)iv[r1 + 6 * r2]; } r1 = i + 6 * r2; p_prd[r1] = d + (double)Q[r1]; } } /* Estimation */ for (i = 0; i < 2; i++) { for (r2 = 0; r2 < 6; r2++) { d = 0.0; for (r1 = 0; r1 < 6; r1++) { d += (double)c_a[i + (r1 << 1)] * p_prd[r2 + 6 * r1]; } B[i + (r2 << 1)] = d; } for (r2 = 0; r2 < 2; r2++) { d = 0.0; for (r1 = 0; r1 < 6; r1++) { d += B[i + (r1 << 1)] * (double)iv1[r1 + 6 * r2]; } r1 = i + (r2 << 1); S[r1] = d + (double)R[r1]; } } if (fabs(S[1]) > fabs(S[0])) { r1 = 1; r2 = 0; } else { r1 = 0; r2 = 1; } a21 = S[r2] / S[r1]; a22_tmp = S[r1 + 2]; a22 = S[r2 + 2] - a21 * a22_tmp; for (k = 0; k < 6; k++) { i = k << 1; d = B[r1 + i]; d1 = (B[r2 + i] - d * a21) / a22; Y[i + 1] = d1; Y[i] = (d - d1 * a22_tmp) / S[r1]; } for (i = 0; i < 2; i++) { for (r2 = 0; r2 < 6; r2++) { B[r2 + 6 * i] = Y[i + (r2 << 1)]; } } /* Estimated state and covariance */ for (i = 0; i < 2; i++) { d = 0.0; for (r2 = 0; r2 < 6; r2++) { d += (double)c_a[i + (r2 << 1)] * x_prd[r2]; } b_z[i] = z[i] - d; } for (i = 0; i < 6; i++) { d = B[i + 6]; x_est[i] = x_prd[i] + (B[i] * b_z[0] + d * b_z[1]); for (r2 = 0; r2 < 6; r2++) { r1 = r2 << 1; b_a[i + 6 * r2] = B[i] * (double)c_a[r1] + d * (double)c_a[r1 + 1]; } for (r2 = 0; r2 < 6; r2++) { d = 0.0; for (r1 = 0; r1 < 6; r1++) { d += b_a[i + 6 * r1] * p_prd[r1 + 6 * r2]; } r1 = i + 6 * r2; p_est[r1] = p_prd[r1] - d; } } /* Compute the estimated measurements */ for (i = 0; i < 2; i++) { d = 0.0; for (r2 = 0; r2 < 6; r2++) { d += (double)c_a[i + (r2 << 1)] * x_est[r2]; } y[i] = d; } /* of the function */ } /* * Arguments : void * Return Type : void */ void kalmanfilter_init(void) { int i; for (i = 0; i < 6; i++) { x_est[i] = 0.0; } /* x_est=[x,y,Vx,Vy,Ax,Ay]' */ memset(&p_est[0], 0, 36U * sizeof(double)); } /* * File trailer for kalmanfilter.c * * [EOF] */
You can accelerate the execution speed of the kalmanfilter
function that is processing a large data set by using the codegen
command to generate a MEX function from the MATLAB code.
kalman_loop
Function to Process Large Data SetsFirst, run the Kalman algorithm with a large number of data samples in MATLAB. The kalman_loop
function runs the kalmanfilter
function in a loop. The number of loop iterations is equal to the second dimension of the input to the function.
type kalman_loop
% Copyright 2010 The MathWorks, Inc. function y=kalman_loop(z) % Call Kalman estimator in the loop for large data set testing %#codegen [DIM, LEN]=size(z); y=zeros(DIM,LEN); % Initialize output for n=1:LEN % Output in the loop y(:,n)=kalmanfilter(z(:,n)); end;
Now time the MATLAB algorithm. Use the randn
command to generate random numbers and create the input matrix position
composed of 100,000 samples of (2x1) position vectors. Remove all MEX files from the current folder. Use the MATLAB stopwatch timer (tic
and toc
commands) to measure how long it takes to process these samples when running the kalman_loop
function.
clear mex delete(['*.' mexext]) position = randn(2,100000); tic, kalman_loop(position); a=toc;
Next, generate a MEX function using the command codegen
followed by the name of the MATLAB function kalman_loop
. The codegen
command generates a MEX function called kalman_loop_mex
. You can then compare the execution speed of this MEX function with that of the original MATLAB algorithm.
codegen -args {position} kalman_loop.m which kalman_loop_mex
/tmp/Bdoc20b_1465442_111858/tpcea1ddd1/coder-ex53054096/kalman_loop_mex.mexa64
Now, time the MEX function kalman_loop_mex
. Use the same signal position
as before as the input, to ensure a fair comparison of the execution speed.
tic, kalman_loop_mex(position); b=toc;
Notice the speed execution difference using a generated MEX function.
display(sprintf('The speedup is %.1f times using the generated MEX over the baseline MATLAB function.',a/b));
The speedup is 17.5 times using the generated MEX over the baseline MATLAB function.