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George Mason University

Signals and Systems I

Spring 2018

Laboratory Project #5

Assigned: March 26, 2018

Due Date: Laboratory Section on Week of April 9, 2018

Description:

Applying Control Theory to our understanding of Signals and Systems Control Theory is

another major application of this course. Control Theory allows the designer to institute

measures that will counteract any destabilizing forces in a system ensuring that the system

performs as designed. Utilizing the steps below on an LTI system is a cornerstone of Control

Theory.

Lab Report Your report for this lab will consist of all the analytical (i.e, pencil/paper)

work, MATLAB plots and code, and relevant explanations. Each student must do his or her

own work on this lab. However, you may ask other students or any of the teaching staff for

advice.

1

Prelab

In this lab you will be introduced to feedback and control systems. Read the MATLAB help

files on pzplot and ltiview. A quick introduction to feedback and control systems can

also be found in your text under section 4.7 Application to Feedback and Controls.

2

2.1

Understanding Block Diagrams

Feedback & Controllers

(a) A typical system can be modeled within certain constraints as linear, constant-coefficient,

differential equation (LCCDE). A transfer function that relates the input to the output can then be created as shown in the block diagram below. Note that in controls

it is typical to call the system a plant and to label its transfer function as P (s).

Figure 1: Block Diagram.

What is the relationship between X(s), Y (s) and P (s)?

Instructor Verification (separate page)

(b) Figure 2 shows the addition of a unit feedback loop. Calculate the transfer function

T (s) for this new system. Start by determining the value of e(t) in terms of x(t) and

y(t). Next, find the output y(t) in terms of e(t). Use these equation to eliminate e(t)

to find Tf (s) =

Y (s)

X(s) .

Figure 2: Block Diagram with Feedback.

How does the addition of the feedback impact the poles and zeros of the plants

transfer function, P (s)?

Instructor Verification (separate page)

(c) A controller C(s) has been added to the system as shown in Figure 3 below. Using

the same process as in section 2(b) above calculate the new system transfer function

Tcf (s).

Figure 3: Block Diagram with Controller and Feedback.

What is the impact of adding the controller C(s) to the system?

Instructor Verification (separate page)

2.2

Transfer Functions

To examine how the feedback loop and controller impacts a system we will examine the

mechanical system as shown in Example 1.12 of the textbook.

Figure 4: Mechanical System.

Summing the forces provides the system equation:

M y¨(t) + B y?(t) + Ky(t) = x(t)

(a) Determine the transfer function P (s) =

Y (s)

X(s)

(1)

using the system equation in Eq. (1).

Instructor Verification (separate page)

(b) Adding a PID controller to our system provides an opportunity to tune the system

performance to meet a set of required specifications. A PID controller would consist

of three parts: a Proportional control (Kp ), an Integral control (Ki ), and a Derivative

control (Kd ). For u(t) in Figure 3 this would be

Z

d

u(t) = Kp e(t) + Ki e(t)dt + Kd e(t)

(2)

dt

This yields the transfer function

C(s) =

3

3.1

Ki

U (s)

= Kp +

+ Kd s

E(s)

s

(3)

PID Controllers

Understanding Key Performance Parameters

To better understand the performance specifications consider a cruise control on a car. As

you are waiting on a red light you set your cruise control to the maximum speed limit for

the road (or a little over?). When the light turns green you start the cruise control. Figure 5

below captures your cars performance in the first few seconds.

Figure 5: Design Parameters for Unit Step Response.

You are interested in your car accelerating quickly off the start so you are particularly

interested in how fast your car gets up to speed or, as shown on the graph, the rise time tr .

response time tr = time from 10% of stead-state value to 90%.

You are not too anxious to attract the attention of the police so you dont want to

overshoot your set speed by too much. The next criteria would constrain overshoot.

% Overshoot os% = 100 * (peak value – steady state value)/(steady state value)

As your cruise control tries to close in on the steady state speed you do not want the

oscillation to go on for too long so you the next criteria is the settling time ts .

settling time ts = when the oscillations are all under +/- 2%

Finally, the car settles in on its steady state value which, you would hope, is what you

have set. Therefore you would like to check for any steady state error

steady state error errss = steady state value (ys s) – input (x(t))

3.2

Impacts of PID Controllers

The PID controller shown in Section 2.2(b) consisted of three parts: a Proportional control

(Kp ), an Integral control (Ki ) and a Derivative control (Kd ). Each of these constants,

Kp , Ki , and Kd , will have an impact on the systems unit step response in a positive way:

A proportional control, Kp , reduces the rise time and decreases the steady-state error.

An integral control, Ki , eliminates the steady-state error

A derivative control, Kd , increases the stability of the system, reduces overshoot, and

improves the transient response.

Unfortunately, each type of controller impacts all key parameters as shown in the table

below. Determining the values of Kp , Ki , and Kd is an iterative process.

Control

Kp

Ki

Kd

Rise Time

Decrease

Decrease

Small Change

Overshoot

Increase

Increase

Decrease

Settling Time

Small Change

Increase

Decrease

S-S Error

Decrease

Eliminate

No Change

Your job as a system control designer is to design a controller using the least number of

parts. Note that you are not required to use all three controls in a design. Typically you

will find the system performs well with just a Proportional controller (P), a combination

of a Proportional and Integral (PI), or even a Proportional and Derivative (PD) controller.

The choice depends on the specifications that need to be met and the cost of implementing

the controller. We will ignore the implementation costs.

Returning to the system depicted in Section 2.2 above recall that the system equation

was given as

M y¨(t) + B y?(t) + Ky(t) = x(t)

(4)

Using the transfer function found in Section 2.2(a), make the following assignments:

M = 1 kg

B = 10 N s/m

K = 20 N/m

3.3

Tuning a PID Controller

The standard way to set the parameters of the PID controller is to view the systems

performance to a unit step input. Since x(t) = 1, you would hope that the output, y(t)

would also be 1.

We will use three different controllers to see how they perform on the system. MATLAB

provides two extremely useful functions that reduces the work required for this process.

The first is the pole-zero plot, pzplot.

Suppose you have a transfer function defined by

H(s) =

s2

3s + 1

+ 5s + 4

(5)

To plot its poles and zeros, enter

>> h=tf([3 1],[1 5 4]);

>> pzplot(h);

% Refer to Lab 4 if you have questions

The plot depicts poles as x and zeroes as o.

The second function is ltiview. This function provides a wealth of information so it is

a little more involved. To call up the LTI Viewer type

>> ltiview(h);

Click on the help button at the top and select LTI Viewer Help. You will need to learn

how to select a Step plot style (e.g., shows the unit step response of the system), and to

show the key characteristics of the system (Peak Response, Settling Time, Rise Time, and

Steady State).

(a) The first step is to set a baseline where there is no feedback or controller as depicted

in Figure 1. Using the P (s) you found in Section 2.2(a) and the values from Section

3.2, plot the poles and zeros of the system. Next, plot the unit step response and

calculate the key characteristics of the system: Peak Response, Settling Time, Rise

Time, and Steady State. Note these values for future comparison.

Design Criteria:

Your design criteria for this lab is to reduce the overshoot to less than 10% (e.g., no higher

than 1.1), establish a settling point within 2% of the input (e.g., between 0.98 and 1.02)

and as fast a resoponse time as you can obtain.

(b) Now consider using a Proportional controller. Using the transfer function Tcf you

found in Section 2.2(b), substitute for the values for P (s) given in Section 3.2 and set

Ki and Kd equal to zero. Since the input x(t) is 1, we would want the output to y(t)

to be 1. Using the value for yss you found in Section 3.3(a), start by setting Kp as

follows and change until the setpoint (i.e., the input) equals to output:

Kp =

1

yss

(6)

Note: if the resolution too low you can change both the time vector by clicking on

Edit ? Viewer Preferences ? Parameters. From there select Define Vector

under Time Vector and enter the time vector [start : step : end] values.

Make a pole-zero plot and compare the result with the baseline in Section 3.3(a)

above. What are the key parameters (Peak Response, Settling Time, Rise Time, and

Steady State) and which ones have improved and which have fallen father away from

ideal?

(c) Use a PD controller for comparison. Keep Kp at the same value as above and try

several values for Kd (as a minimum try Kd = 1, 5, 10 and 20). Make a pole-zero plot

and an ltiview plot and compare the key parameters with the Proportional controller

in section 3.3(b). What value of Kd provided the best response of the system in your

opinion and explain why.

(d) Next try a PI controller. This time set Kp at 1/3rd the value as above and try several

values for Ki (as a minimum try Ki = 50, 70 and 90). Make a pole-zero plot and an

ltiview plot and compare the key parameters with the PI controller in section 3.3(c)

What value of Ki provided the best overall response of the system in your opinion

and explain why.

Lab #5

ECE 220: Spring 2018

Instructor Verification Sheet

For each verification, be prepared to explain your answer and respond to other related questions that the lab TAs or professors might ask. Turn this page in along with your lab report.

Name:

Date of Lab:

Part 2.1(a): What is the relationship between X(s), Y (s) and P (s)?

Part 2.1(b): How does the addition of the feedback impact the poles and zeros of the plants

transfer function, P (s)?

Part 2.1(c): What is the impact of adding the controller C(s) to the system?

Part 2.2(a): For the given mechanical system, what is the transfer function P (s) =

Y (s)

X(s)

Lab Report Format

Signals and Systems 1

ECE 220 Spring 2018

Each lab report that you submit for this class is a formal report that is well-prepared, carefully

written, and follows the format given below. The report must include descriptions and analysis

of all assigned portions of the lab. Lab reports are to be typed and submitted through the class

Blackboard website. Once you have completed your report, please submit it via Blackboard prior

to the start of the next lab. Thus, Lab #1s report is due prior to the start of Lab #2.

General Guidelines

Your report should be typewritten, neat and well-organized.

The report must follow the format given below. All analytical work and calculations made

should be clearly explained.

MATLAB code is to be placed in report appendices.

It is expected that the report will be grammatically correct with no spelling errors. Points

will be deducted for poor grammar and/or spelling errors. You are encouraged to use a spell

checker.

Explanations that describe your work must be included in the report. Plots that are submitted

within the report must include properly labeled axes and a title. Each plot should be given a

figure number and a caption. Points will be deducted if you include unnecessary graphs and

plots.

Within the text of your report, when referring to a particular plot, refer to it by number.

Detailed Lab Report Format

Your report must contain the following sections. Subsections may be added for purposes of

clarification.

1. Title Page: This page contains an identification of the Laboratory by number and title. It

also must include the name and G-Number of the author and the date of submission.

2. Introduction: This section contains a description of the purpose and objectives of the

laboratory project. Do not simply copy text from the assignment. The Introduction should

summarize the topics covered in the laboratory and a brief summary of the key results

obtained.

3. Main Body: This section is the most detailed part in your report and it will generally contain

subsections. Note that it should not contain any MATLAB code. What it does contain is a

description of all results obtained during the lab as well as any figures that were generated

with MATLAB or other sources. If theoretical calculations are required, these should be detailed

and, where appropriate, compared with the experimental portion of the lab.

4. Conclusions: In this Section, summarize the conclusions that you made once the results

were obtained. The conclusions should be tied back to the objectives of the Lab Project.

This should be a concise section that focuses on the important results obtained and lists the

conclusions that resulted from these results.

5. References/Appendices: Include a description of any external references that you used

(e.g. web pages, technical papers, etc.) during the lab. The appendices should also contain

listings of your MATLAB code. You should document your code with comment lines, where

appropriate. This holds for both the primary code as well as any scripts and subroutines that

you have written. Remember, if a grader has trouble understanding your MATLAB code,

you will not receive full credit for that portion of your report. Adding carefully chosen

comments within the code will make it much more likely that your code is fully understood.

1

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