Simulation of a Motor and Cart System
L01
Lauren Hayes
Completed February 15, 2024
Submitted February 15, 2024
1
Summary:
The experiment aimed to replicate the lab's cart system manually using state equations and
Euler’s method in MATLAB. It involved simulating both nonlinear friction and linear
approximations, adjusting step sizes, and varying feedback gain and input amplitudes. Results
were plotted to justify sampling time, analyze system responses to step inputs, and compare
linear and nonlinear simulations. The report included code documentation for transparency. The
experimental setup was based on specified parameters, including Ra, Kt, Kb, and J. Overall, the
study provided insights into system dynamics and the impact of feedback gain and input
amplitude on cart position in the simulated environment.
Block Diagram:
Block diagram of the cart system – from the Lab Manual
2
Procedure:
From the block diagram above, the state equations describing the system were generated. The
equations, along with the known parameters, were transferred into a MATLAB script. When
writing the script, one decision to be made was the proper step interval to use (that is, the change
in time, dt). A test and compare method was used. The value of 0.005 was settled upon due to its
delivery of clean results. This value makes sense as Euler’s method will achieve a cleaner
approximation the more values are calculated within a specific time interval.
dt = 0.05
dt = 0.025
dt = 0.005 (chosen)
3
Results:
It was found that once in steady state, linear and nonlinear did not differ in any of the cases.
However, in the time that it took to reach steady state, the linear approximation exhibited a much
faster response, especially in cases II and III. The nonlinear did have a lower percent overshoot,
as can be seen in cases I and II.
A feedback gain of five produced a faster response but with an overshoot, whereas a feedback
gain of two produced no overshoot but a slower response time. The highest feedback gain with
the lowest input magnitude produced the most identical response between the nonlinear and
linear approximations. Only in cases I and II did the nonlinear position ever exceed that of the
linear.
CASE I
4
CASE II
5
CASE III
6
CASE IV
7
Appendix:
%-------------------------------------------------------------------------% This script performs simulations of a linear and a nonlinear system
% subject to different feedback gains and step input magnitudes. It tests
% different sampling times to find an appropriate time step for the
% simulations. The system models a cart with parameters such as moment of
% inertia, voltage constant, resistance, torque constant, and feedback gain.
%-------------------------------------------------------------------------% Parameters:
% - J: Moment of inertia (4 Kg-m^2)
% - kb: Voltage constant (volt/rad/sec)
% - R: Resistance (Ohms)
% - kt: Torque constant (N-m/amp)
% - k0: Feedback gain
%-------------------------------------------------------------------------% Simulation Time:
% - t_start: Start time
% - t_end: End time
% - step_size: Fixed step size for simulation
% - feedback_gains: Feedback gain values to test
% - input_magnitudes: Step input magnitudes to test
%-------------------------------------------------------------------------% Author: Lauren Hayes
% Date: 2/15/2024
%-------------------------------------------------------------------------% Start
close all; % Close all existing figures
clc; % Clear the command window
% Define system parameters
J = 8 * 10^-4; % Moment of inertia (4 Kg-m^2)
kb = 0.1; % Voltage constant (volt/rad/sec)
R = 2; % Resistance (Ohms)
kt = 0.1; % Torque constant (N-m/amp)
k0 = 4; % Feedback gain
% Define simulation time parameters
t_start = 0; % Start time
t_end = 10; % End time
dt = 0.005; % Fixed step size for simulation
% Define feedback gains and input magnitudes
feedback_gains = [5, 2]; % Feedback gain values to test
input_magnitudes = [1, 2]; % Step input magnitudes to test
8
% Loop over different feedback gains
for gain_idx = 1:length(feedback_gains)
k0 = feedback_gains(gain_idx); % Current feedback gain value
% Loop over different step input magnitudes
for input_idx = 1:length(input_magnitudes)
% Reset initial conditions for each simulation
theta = zeros(1, (t_end - t_start) / dt + 1); % Initialize angle array for linear simulation
omega = zeros(1, (t_end - t_start) / dt + 1); % Initialize angular velocity array for linear
simulation
THETA = zeros(1, (t_end - t_start) / dt + 1); % Initialize angle array for nonlinear
simulation
OMEGA = zeros(1, (t_end - t_start) / dt + 1); % Initialize angular velocity array for
nonlinear simulation
% Simulate linear system
for t = 1:length(theta)-1
Vr = 2 * input_magnitudes(input_idx); % Step input for linear simulation
x = 2.53 * (theta(t) / 3); % Calculation for x in linear simulation
Vs = 0.4 * x; % Calculation for Vs in linear simulation
Vi = k0 * (Vr - Vs); % Calculation for Vi in linear simulation
Pa = 2 * Vi; % Calculation for Pa in linear simulation
Va = Pa - kb * (omega(t)); % Calculation for Va in linear simulation
i = Va / R; % Calculation for current in linear simulation
f = omega(t) * 0.01; % Calculation for friction in linear simulation
T = (i * kt) - f; % Calculation for torque in linear simulation
omegaPrime = T / J; % Calculation for angular acceleration in linear simulation
theta(t+1) = theta(t) + dt * omega(t); % Update angle using Euler integration in linear
simulation
omega(t+1) = omega(t) + dt * omegaPrime; % Update angular velocity using Euler
integration in linear simulation
end
% Simulate nonlinear system
for t = 1:length(THETA)-1
VR = 2 * input_magnitudes(input_idx); % Step input for nonlinear simulation
X = 2.53 * (THETA(t) / 3); % Calculation for X in nonlinear simulation
VS = 0.4 * X; % Calculation for VS in nonlinear simulation
VI = k0 * (VR - VS); % Calculation for VI in nonlinear simulation
% Apply condition for PA calculation
if -5 <= VI && VI <= 5
PA = 2 * VI; % Calculation for PA within range in nonlinear simulation
elseif VI < -5
PA = -10; % Calculation for PA when VI is less than -5 in nonlinear simulation
9
else
PA = 10; % Calculation for PA when VI is greater than 5 in nonlinear simulation
end
VA = PA - kb * (OMEGA(t)); % Calculation for VA in nonlinear simulation
I = VA / R; % Calculation for current in nonlinear simulation
% Apply condition for F calculation
if OMEGA(t) < 0
F = OMEGA(t) * 0.008 - 0.02; % Calculation for friction when angular velocity is
negative in nonlinear simulation
elseif OMEGA(t) > 0
F = OMEGA(t) * 0.008 + 0.02; % Calculation for friction when angular velocity is
positive in nonlinear simulation
else
F = 0; % No friction when angular velocity is zero in nonlinear simulation
end
T = (I * kt) - F; % Calculation for torque in nonlinear simulation
OMEGAPrime = T / J; % Calculation for angular acceleration in nonlinear simulation
THETA(t+1) = THETA(t) + dt * OMEGA(t); % Update angle using Euler integration in
nonlinear simulation
OMEGA(t+1) = OMEGA(t) + dt * OMEGAPrime; % Update angular velocity using
Euler integration in nonlinear simulation
end
% Plot results for linear and nonlinear simulations
figure;
plot((0:length(theta)-1) * dt, theta, 'b', 'LineWidth', 2); hold on; % Plot linear simulation
results
plot((0:length(THETA)-1) * dt, THETA, 'r--', 'LineWidth', 2); % Plot nonlinear simulation
results
title(sprintf('Feedback Gain: %d, Input Magnitude: %d', k0,
2*input_magnitudes(input_idx))); % Set plot title
xlabel('Time'); % Set x-axis label
ylabel('Position (cm)'); % Set y-axis label
legend('Linear', 'Nonlinear'); % Set legend
grid on; % Add grid to the plot
end
end
10
11