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

Answered 7 days AfterMar 07, 2022

The input and output response against the internal plant for a given plant system model is shown below with a given set of input and output values. The limiter is not explicitly given in the manufacturer’s motor datasheet, therefore chosen such that it limits the actuator output at approx. 70% of...

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