Automatic Controls Design Project
Report 1
Prepared For:
Georg Maue
ME 421, Section 1001
Department of Mechanical Engineering
University of Nevada, Las Vegas
Prepared By:
Piero Quino
Due Date:
01 March 2022
1
Quino, Piero
The model of the plant schematics is shown below
Input is and Output is
Give Data:
State variables of the system is
Deriving the equation of motion using the Free body diagram
The equation for the above J1 is
For J2
Differential Equation of motion is
For J3
Differential Equation of motion is
For J4
Differential Equation of motion is
For J5
Differential Equation of motion is
For J6
Differential Equation of motion is
Using the above differential equation for the free body diagram for the six inertial masses we resolve the equation in state variable form:
The Plant Model in the Transfer function form given below:
Compute Pole of the plant using the MATLAB and above developed transfer form of system.
Modelling of Plant
% Clear data and figures
clc
clea
close all
% Define Symbolic variable
syms s
% Define Parameters
J1=0.7;
J2=0.7;
J3=0.7;
J4=0.7;
J5=20;
J6=20;
K1=20;
K2=30;
D=50;
N1=12;
N2=16;
N3=20;
N4=30;
% Define Matrix A
A=[J1*s^2+D*s XXXXXXXXXX;...
0 J2*(N1^2/N2^2)*s^2+D*(N1^2/N2^2)*s XXXXXXXXXX;...
0 D*s J3*s^2+K1 0 0 0;...
XXXXXXXXXXJ4*(N3^2/N4^2)*s^2+D*(N3^2/N4^2)*s+K1*(N3^2/N4^2) 0 0;...
XXXXXXXXXXD*s-K1 J5*s^2+D*s+K2 0;...
XXXXXXXXXXD*s-K2 J6*s^2+D*s];
Determinant of A
disp('Characteristic Equation')
disp(expand(det(A)))
Characteristic Equation
(2401*s^12)/ XXXXXXXXXX*s^11)/ XXXXXXXXXX*s^10)/ XXXXXXXXXX*s^9)/ XXXXXXXXXX*s^8)/ XXXXXXXXXX*s^ XXXXXXXXXX*s^ XXXXXXXXXX*s^ XXXXXXXXXX*s^ XXXXXXXXXX*s^3
Compute Pole
s=tf('s');
Coeff=fliplr(eval(coeffs(det(A))));
sys=tf(1,Coeff);
Pole=pole(sys);
disp('Poles')
disp(Pole)
Poles
XXXXXXXXXX + 0.000023523959457i
XXXXXXXXXX - 0.000023523959457i
XXXXXXXXXX + 0.000000000000000i
XXXXXXXXXX + 5.345224838248488i
XXXXXXXXXX - 5.345224838248488i
XXXXXXXXXX + 0.000000000000000i
XXXXXXXXXX + 0.000000000000000i
XXXXXXXXXX + 0.000000000000000i
XXXXXXXXXX + 0.000000000000000i
Plot Root locus
figure
locus(sys)
Microsoft Word - DESIGN_22
Spring XXXXXXXXXXUNIVERSITY OF NEVADA, LAS VEGAS
DEPARTMENT OF MECHANICAL ENGINEERING
MEG XXXXXXXXXXAutomatic Controls
Design Project
Objective: The design project will give everyone in the class an opportunity to apply the knowledge
gained in class in a reasonably realistic setting. We will analyze plants, their dynamics and
other properties, and explore design strategies by which we can create a ‘good’ controller while
considering the existing constraints.
General Rules for all Reports
As Seniors, you will be graduating soon. Prepare the reports as you would for a supervisor at your place of
employment. Make the report as clear and transparent as possible.
Graphs and Figures
*
Figure 1 DC Motor with limiter
Every graph must have a descriptive Title. Label and Scale All axes. If a plot contains multiple lines,
you must add a legend explaining each curve. Add handwritten legends if needed. Do NOT paste Matlab
‘Scope’ images into the report, since they do not contain proper labeling.
Simulink Models:
Avoid overlapping and crossing lines a much as possible. Re-a
ange the icons so that a clear path from left
to right is visible.
Late Submissions:
You must submit all design project reports. You will receive a grade of “F” for the entire course if
any report is missing.
A penalty of 20% of the max. grade will be applied for each day after the submission deadline. Reports more
than 5 days late will be assigned a zero grade.
The schedule below lists due dates and assignments for the individual parts of the project. Due dates are
listed below
Week Due date Topic
5 Tue.
3/01
Report #1
Part 1: Model the plant assigned to you. Each plant has one input and one output variable.
Choose state variables, create free-body diagrams, and determine the plant’s differential
equation in state variable form, see examples on pages 10 and 11. Express the plant model
in transfer function form (by hand or better in Malab), and compute all plant poles.
If your model is nonlinear, e.g. the independent variable comprises sinusoidal or quadratic
terms, linearize the model equation about its operating point.
Part 2: Model the complete linear open-loop system including the plant. Specify input and
output variables, distu
ances, and transfer functions. The complete open-loop system
egins with a controller (model initially as gain K), followed by an amplifier (with limiter
in the nonlinear case), the actuator = DC motor (see also second lab handout File:
lab2v.pdf (in the FILES folder on Canvas) or the DC motor discussion in the textbook,
Chapter 2), and the system being controlled. No sensor is specified. Assume that the
controlled variable is directly available to the controller. Select an appropriately sized DC
servomotor (see instructions below) and amplifier to drive the plant.
Part 3: Create a Linear open-loop computer model as seen in Fig. 1 below, where the
plant is represented as the transfer function of part 1. Use Matlab. Do not yet define the
nonlinear elements (Limiter and Coulomb friction) shown in Fig. 1.
Submit: 1. The complete validated model of your plant, including all free-body diagrams
used to derive the state equations. Validation: Show that your plant is stable, i.e. that it has
NO poles in the right half of the s-plane, see below.
2. The plant model in transfer function format, see example below. If you compute the
transfer function and plant poles in Matlab (RECOMMENDED) please include your
Matlab commands script in the report.
3. Verify that the model is open-loop stable by computing all plant poles. Submissions
containing unstable plant poles are not accepted. List the Plant transfer function and all
plant poles. Any undamped oscillators in the plant will result in imaginary axis pole pairs.
However, if you discover unstable poles in the right half of the s-plane, please review your
plant model for e
ors. All assigned plants are open-loop stable and therefore cannot have
poles in the right half of the s-plane.
4. Validated Matlab model,
5. a plot of the open-loop step response, in Matlab. Please select the time scales so that
oth the transition and the steady state are visible.
Again: Submissions containing system models with rhp poles will not be accepted.
8 Tue
3/08
Report #2
Part 1: Using the validated plant model of report 1, create a
Nonlinear model Simulink model only. Place the limiter after the amplifier, see
Fig. 1(on Page 8)
If a limiter is not explicitly given in the manufacturer’s motor data sheet, choose it such
that it limits the actuator output at approx. 70% of its maximum cu
ent.
Part 2:
Using a unit step reference input, design a P-controller for approx. 20% overshoot (if your
plant is too poorly damped, document this fact and design for a larger overshoot. If your
plant has imaginary axis poles, the closed loop may be unstable with P-control for any gain
K. If the closed loop is unstable, demonstrate this fact by plotting the plant’s root locus).
Simulate and plot the feedback system step response with P-control for two scenarios:
(a) Linear Model : No Limiter
(b) Nonlinear Model with Limiter.
Show the complete block diagram of both linear and nonlinear feedback systems.
Verify that the loop has negative feedback. Also, compute and plot the closed loop
system response to an appropriately sized distu
ance step (r = XXXXXXXXXXPlace the
distu
ance between servo amplifier output and plant input, see Fig. 1 (on Page 8)
Notes on defining the Limiter: Physical significance: The limiter models the fact that no
eal actuator can deliver infinite power. Check your motor specifications sheet for the input
voltage range (typically +/- 10 Volts DC or similar). These values constitute the
VOLTAGE LIMITER in Fig. 1. Your servo-amplifier will also have a cu
ent limit (max.
cu
ent spec.) which you can enter in the model of Fig. 1 as a CURRENT LIMITER.
Model the limiter in Matlab Simulink.
Limiter Dynamics: Try the limiter at different load levels. You’ll observe that the control
loop will be linear as long as the voltage input to the amplifier is within the input voltage
ange (typically +/- 10 Volts DC or similar). Only when the voltage exceeds the limits will
you see clipping. Run your simulations at step sizes large enough that clipping is visible.
Graphing with Simulink: Use the SCOPE feature only while designing your control loop.
For submission, connect the variables you wish to plot to a SIMOUT block (located in
sinks). plot the results using the plot command. Please add a descriptive title to each plot,
label all axes, and add legends whenever you plot multiple variables in the same plot. Use
the Matlab legend or gtext command to label curves.
Here is a Matlab code example that plots two responses from a simulink model
ContinDiscrete.mdl.
sim('ContinDiscrete')
figure(1)
plot(ycd(:,1),ycd(:,2),':')
hold on
plot(ycd(:,1),ycd(:,3))
xlabel('Time (sec)');
ylabel('Output responses');
title(' Output Responses of Continuous vs. Discrete Control')
gtext('continuous controller')
gtext('discrete controller, T =.07')
grid on
Submit: Plant mathematical model and both Matlab models (a) and (b)
Plots: 1. Closed loop step responses for (a) and (b) in the same graph.
2. Closed loop distu
ance responses for (a) and (b) in the same graph.
For both plots, select the time scales so that both the transition and the steady state are
visible.
Re. Assignment #3:
Show detail in all