ENGR 329 Lab Manualchem.engr.utc.edu/engr329/lab-manual/329-manual-01-Fall.doc · Web viewToday,...

Manual for

ENGR 329CONTROL SYSTEMS LABORATORY

Web site http://chem.engr.utc.edu/329

Dr. Jim Henry425-4398

[email protected]

College of Engineering and Computer ScienceUniversity of Tennessee at Chattanooga

Fall, 2001

Grateful acknowledgment is given to the following for support of this laboratory: Analog Devices CompanyCenter for Excellence in Computer Applications at UTCMicroMotion InstrumentsMicrosoft CorporationNational Instruments CorporationNational Science FoundationPlant Engineering Consultants

Contents

1: Schedule 22: Grading 43: Objectives & Guidelines 54: Project Information 75: Weekly Assignments 14Appendices 67

1: SCHEDULE

Part I -- System Identification

Week 1 Introduction, steady-state measurements, statistics, data acquisition software, cleaning up

Week 2 Steady-state operating curves, graphing & word processing software

Week 3 Week 3 ReportStep response measurements

Week 4 Week 4 ReportModeling software

Week 5 Modeling -- Approximate FOPDT

Week 6 Week 6 ReportFrequency response measurements

Week 7 Modeling frequency response

Week 8 Week 8 ReportPlant visit

Part II -- Control System Design

Week 9 Root locus plotting

Week 10 Week 10 ReportProportional control design

Week 11 Proportional control experiment

Week 12 Week 12 ReportPI control design

Week 13 PI control experiment

Week 14 Week 14 Report

If I treat you as you are, I will make you worse.If I treat you as though you are what you are capable of becoming, I help you become that.

--Goethe

1. Schedule

Schedule of Presentations & Reports

Pre-Lab Present-ation Report

1

2 B W

3 G V All members

4 All Teams Red, Blue, Greenthird member in each

5 B W

6 G V All Teams White, Violetfirst member in each

7 B W

8 All Teams Red, Blue, Greenfirst member in each

9

10 G V All Teams White, Violetthird member in each

11 B W

12 G V All Teams Red, Blue, Greensecond member in each

White, Violetfourth member in each

13 B W

14 All Teams White, Violetsecond member in each

Red, Blue, Greenfourth member in each

Pre-Lab presentation is to be given by 2 students; one from each of the teams indicated in the "Pre-Lab" column.

Presentations of results from all teams will be given on the days indicated in the "Presentation" column.

Reports are scheduled as indicated in the "Report" column. Within your team, you submit reports in order of the team assignment list given in lab. For example, "third member" in week 4 means that the person listed third in the team assignment list submits the Week 4 Report.

2: GRADING

The grading in ENGR 329 will reflect what is observed of your understanding of control systems operation. Evidence of this understanding can be observed in your

• ability to apply the principles to a physical system (performance in the laboratory and quality of results)

• ability to construct models to simulate the physical system (performance in modeling and quality of results)

• ability to interpret, describe and explain experimental and modeling work (reports and presentations)

The weights given will be35 points Physical laboratory

(attendance, performance)20 points Modeling laboratory

(attendance, performance)20 points Reports25 points Presentations

Physical and modeling lab will be graded on this scalepoints for Leadership

ContributionsParticipation CreativityCooperation Teamwork

0 points Absent

The semester grade will be determined by your point total90-up A80-89 B70-79 C65-69 D0-64 F

The following must be completed to receive a passing grade in the lab:2 reports, 3 presentations, 5 out of the 6 physical laboratories, 3 out of 4 of the modeling laboratories and a submitted report notebook.

All work done will receive credit if it is submitted before the last day of classes for the semester.

3: OBJECTIVES & GUIDELINES

ObjectivesThe main objectives of the laboratory experiences are to help you sharpen your skill in observing what happens to an engineering system and to accurately and completely describe what you observe.

Here is a diagram of the experiments and modeling to be done during the semester. Generally, in the first half of the semester, you will be conducting experiments and making observations on your system so that you can build a good linear FOPDT model of the system. A "good" model is one in which the results of the model calculations are in agreement with the experimentally observed results.

Observations on Laboratory System

Modeling to Approximate the System

Week 1-2Observations at steady state

SystemIdentification

Week 3Observations of dynamic response to a step input

Week 4-5Build a model to describe the system

Week 6Observations of dynamic response to a sine input

Week 7Improve a model to describe the system

ControlSystem

Week 11Verify & fine tune the design of the controller

Week 9-10Use a model to design a proportional controller

DesignWeek 13-14Verify & fine tune the design of the controller

Week 12Design a proportional-integral controller

In the second half of the semester, you will use this linear FOPDT model that you have built to design a control system to give a response of your system under certain operating scenarios. Then you will conduct experiments to see if your designs were valid and useful.

3: Objectives & Guidelines

Guidelines on Safety, Cleanliness, Conservation, Citizenship

We have had years of experience with no lost-time injuries in this lab. Let's all do our part to make this year another one. In the event that someone is injured in the lab and is bleeding, before you help them, put on latex gloves that are available in the lab. Have someone show you where they are.

These labs are not routinely cleaned by the custodial workers. We have to keep them clean ourselves. Always leave the lab cleaner when you leave than when you arrived. If the trash cans are full, set them in the hall to be emptied. If an empty trash can is outside the door, bring it into the lab.

Around the computer workstations, do not have food or drink. If you have food or drink elsewhere, please clean up your stuff. Recycle aluminum cans. Rinse them first if there's grunge in them.

Conserve resources and money by printing only what is necessary for effective learning. If you print something that you don't need, place the paper in the "one-side-good" recycle stack to be reused. (Put the good side up.) If you are printing a draft, please use paper from the "one-side-good" stack.

Printers are not instantaneous. This lab has one printer and many users. During heavy use times, plan twice and print once. This will reduce frustrations. In the event you don't get a printout instantly, re-read this paragraph.

If you have any suggestions to improve this lab, pass it on to an instructor or assistant.

4: PROJECT INFORMATION

XRC/248Crane Positioning System

PWM Motor CartM V T Xin

PC

The position control system consists of a cart on rails coupled to a DC electric motor. The torque of the motor is manipulated by the pulse width modulating power supply under the table. The input signal (0-100% of 10 volts DC) to the power supply comes from the computer. The torque moves the cart via a chain.

The position of the cart is detected by potentiometer ("pots" are variable resistors) that has 10 volts DC across the ends. The voltage on the wiper of the pot is sent to the computer.

Computer HardwareComputer DellRAM 128 megProcessor Intel Pentium IISpeed 450 MHzDisk 13.4 gigaData Aq. Board MIO-16-10E

4. Project Information

SRC/249

The motor speed control system consists of a 3-phase, 5 hp electric motor that is coupled to a DC generator. The speed of the motor is manipulated by the variable-voltage, variable-frequency power supply on the wall. The input signal (0-100 % of 10 volts DC) to the power supply comes from the computer. The generator produces about 85 volts which can be connected to one or two banks of light bulbs to provide a load for the generator.

The speed of the motor-generator set is detected by a photocell and chopper wheel. The pulses from the photocell are converted to a voltage (0-10 volts DC) that is sent to the computer. The computer converts this to RPMs for display & recording.

Operation

The Plexiglas lid for the motor-generator set must be closed for the motor to run.

To operate the pulse-to-voltage converter, its power must be turned on. The power switch for the converter is the little toggle switch on the aluminum box.



VRC/272

The voltage control system consists of a 3-phase, 5 hp electric motor that is coupled to a DC generator. The speed of the motor is manipulated by the variable-voltage, variable-frequency power supply on the wall. The input signal (0-100 % of 10 volts DC) to the power supply comes from the computer. The generator produces about 85 volts which can be connected to one or two banks of light bulbs to provide a load for the generator.

Operation

The Plexiglas lid for the motor-generator set must be closed for the motor to run.



TRC/303

The temperature control system consists of a reservoir of water which can be heated or cooled by two copper coils in the reservoir that have hot or cold water running through them. A video is available that shows the inner construction of the reservoir.

The hot water comes from an ordinary hot water heater. The flow rate of the hot water is varied by changing the speed of two pumps. The speed of the pumps' 3-phase motors are manipulated by variable-voltage, variable-frequency power supplies on the wall.

The input signal (0-100% of 10 volts DC) to the power supply comes from the computer. The cold water flow rate is varied with a manual valve; the cold water is simply tap water.

The temperature of the reservoir is determined by a "resistance temperature device" or RTD. An RTD is a 100 platinum resistor that changes resistance when it changes temperature. The RTD is connected to an Analog Devices signal conditioner that converts the resistance to a voltage (0-5 volts) for the computer to read.

OperationMake sure the water heater is plugged in. The circulating water pump must be operating for the experiment to function; plug it in and turn it on with the push-button switch. Adjust the cooling water supply to the desired flow rate by observing the rotameter on the wall.

Computer HardwareComputer TangentRAM 128 megProcessor Intel Pentium-ProSpeed 200 MHzDisk 3 GBData Aq. Board AT-MIO-16E-10


LRC/307

Reference: Smith & Corripio, pp. 135-142, 168-172

The level control system consists of a water tank. The water entering the tank is supplied through a manual valve and a rotameter flowmeter. The tank also has a variable-speed pump connected to the bottom that receives a signal from the computer telling it how fast to pump the water out. The computer sends a voltage (0-100% of 10 volts DC) to an Analog Devices signal conditioner that converts it to a 4-20 ma signal that the pump recognizes.

The level of water in the tank is detected by an air bubbler that is connected to a pressure transducer. The pressure required to push a bubble into the tank is the hydrostatic head due to the depth of the water. The pressure is converted by the transducer into a voltage (0-10 volts DC) that is read by the computer.



PRC/308

The pressure control system consists of a air blower powered by a 3-phase electric motor. The speed of the motor is manipulated by the variable-voltage, variable-frequency power supply on the wall. The input signal (0-100% of 10 volts DC) to the power supply comes from the computer.

The air goes into one or more of the outlet ducts. The pressure in the first manifold is detected by a pressure transducer. The pressure is converted by the transducer into a voltage (0-10 volts DC) that is read by the computer.



FRC/309

Reference: Smith & Corripio, pp. 268-270

The flow control system consists of a water pump powered by a 3-phase electric motor. The speed of the motor is manipulated by the variable-voltage, variable-frequency power supply on the wall. The input signal (0-100% of 10 volts DC) to the power supply comes from the computer.

The water goes into one or more of the outlet pipes. The flow in the first pipe is detected by a MicroMotion mass flow meter. The MicroMotion sends out a signal of 4-20 ma that is fed through a 500 resistor to convert it to a voltage (1-5 volt DC) that is read by the computer.


5.2 Weekly Assignments

5: WEEKLY ASSIGNMENTS

WEEK 1 INTRODUCTION

ObjectivesTo get an overview of the course. To receive assignments of a project and project team. To understand how to operate the data acquisition station. To understand what the input function and the output function are for the system. To observe steady-state performance of the system. To begin to make measurements on the system and analyze your data.

BackgroundEach system has some "input function" and some "output function." The input function is called M(t), it is usually a function of time. The output function is called C(t), it also is usually a function of time. The names of the functions come from the fact that later they are called the Manipulated variable and the Controlled variable. A diagram that shows the input-output relation is in Figure 2.

SystemM(t)

Input

C(t)

Output

Figure 2. Single-Input, Single-Output system diagram

This week and next week, you are to make measurements on your system, learn about the variability of the measurements and find the steady-state operating curve (SSOC) for your system.

Variations in Measured Quantities(ENGR 322)

Every time an experimental measurement is taken, there is some error associated with the measurement. Today you are to determine the error in measurements in your system. Do this by taking steady-state measurements of the output function, C(t), for a number of data points. Find the mean and standard deviation of the measurements you make. Report your results as mean±2x(standard deviation). This range will include the true value of the function at a confidence level of 95%. Be aware that the standard deviation may be different at different operating points. Software packages like Excel or Kaleidagraph can help a lot with the statistics.


The following graph shows how this statistical analysis could look. This graph is the measured output for a steady input.

15

16

17

18

100 105 110 115 120O

utpu

tTime

Output = 16.08 ± 0.27

Variation in Output Data

Figure 3. Output data varying with time

System Operating Curve

For each value of a constant value of the input function, there will be a value of the output function; this is called the steady-state value of the output for that value of the input. A graph of the output function (on the ordinate) versus the input function (on the abscissa) is called a steady-state operating curve.

An example of what steady-state operating curves look like is in Figure 4.

0

5

10

15

20

0 20 40 60 80 100

Outp

ut

Input

Figure 4. Example of a Steady-State Operating Curve


An example of what the steady-state operating curve (SSOC) looks for the Position system is in Figure 5. (Friction is the apparent cause of the hysteresis observed in the SSOC of the Position system.)

33

34

35

36

37

38

39

40

0 20 40 60 80 100

Outp

ut

Input

Figure 5. An example of Steady-State Operating Curve for the Position system

Lab Assignments

248 Position ControlTake data on the position of the cart. Make estimates of the error in

measuring position at several points.

249 Speed ControlTake data on the motor RPM. Make estimates of the error in

measuring RPM at several speeds.

272 Voltage ControlTake data on the generator's voltage. Make estimates of the error in

measuring voltage at several input values.

303 Temperature ControlTake data on the temperature of the water in the reservoir. Make

estimates of the error in measuring temperature at several temperatures.

307 Level ControlTake data on the height of the water in the tanks. Make estimates of

the error in measuring height at several heights.

308 Pressure ControlTake data on the pressure of the air in the tube. Make estimates of

the error in measuring air pressure at several pressures.


309 Flow ControlMeasure the flow of the water going through the flow meter. Make

estimates of the error in measuring water flow rate at several flow rates.

DATA ACQUISITION PROGRAM - LABVIEW

LabVIEW stands for "Laboratory Virtual Instrument Engineering Workstation." It is used in this lab for control of the experiments and data acquisition, analysis and presentation. If you want to know more, the LabVIEW manuals are in room 213, Grote Hall. LabVIEW works by utilizing data acquisition boards that are installed inside the computers.

LabVIEW Tutorial

Bring the "LabVIEW" window to the top by clicking on the "LabVIEW" box at the very bottom of the screen. Open the program for your system by choosing the name you wish to use. Today, you'll want the one that has the word "Manual" in the name.

You'll get a screen similar to the one in Figure 6, on the next page.

At the top of the window is the name of the controller program.

At the top of the Window is the "RUN" button. Click on it when you want to take data.

On the left is the "controller."The manual slide control: this directly controls the "input" of your systemNext to it is a slide indicator: it indicates the value of the "output" of your systemBelow it is a meter labeled "Controller Output (%)." It is the "output" of the controller. In terms of the diagrams in section 4 of this manual, it is the symbol "M" (for manipulated variable) for your system. (It is not the "output" of your system.)

On the right is a rectangle that is a strip chart recorder. The fact that this "controller" has a recorder connected to it is why the "RC" is in the nickname.

Above the recorder is a little box labeled "points" that tells you how many data readings have been made and recorded.On TRC/303 and LRC/307 there is another little box that says how often the data readings are made and recorded. You can change that value to whatever you wish.On FRC/309 and PRC/308 there is a slide switch to open and close the solenoid valve on the system.

And, finally, in the top left is the "Stop" button. When you want to stop making data readings and recording them, click on this button.


Figure 6. Manual LabVIEW Operating Panel

Procedure for getting steady-state system performance data

Click on the RUN button. Move the manual control slide on the left and observe the response of the system. The chart on the right of the screen is emulating a strip chart recorder.

When you want to stop taking data, click on the STOP button. It will then ask if you want to save the data on a disk file. If you say yes, you will be asked what name you want for the data file. Choose a name that is meaningful to you.

LabVIEW will ask you if you want to draw a time-response graph of the data. This will be a graph that has time on the abscissa and the input and output functions on the ordinate. With the LabVIEW time-response graph, you can use the cursor to find the numerical values of any point on the graph. Move the graph cursor to the point you want the value of and the x–value and y-value of the point will be in the cursor panel.

It will ask you if you want to draw an input-output graph of the data. This will be a graph that has the input function on the abscissa and the output function on the ordinate. Figures 4 and 5 have examples of this input-output graph. Again, you may use the cursor to find values on the graph.


You'll probably want to play around a bit before taking any serious data. If you've just been playing around, click "NO" to all of these requests.

When you are ready to get serious and want a steady-state performance curve, set the manual "input" value to the input value you wish to use. "Run" the LabVIEW program. Observe when the system reaches steady state and collect data until at least 30 data points are acquired. Click the STOP button and save the data, perform the statistical analysis, and make the graphs.

In Figure 7 is a diagram that depicts the flow of information is the LabVIEW environment. The computer operator and the equipment transmitters provide inputs into the program. The program provides outputs to the pieces of equipment, the computer monitor and to data files on computer disks.

LabVIEW Controller Program

Operator Inputs

Outputs to Equipment

Outputs to Data Files

Data PlotsSpread Sheet or Graphing programs

Equipment inputs

Figure 7. Information paths in laboratory experiments

After using LabVIEW, the controller program, the data can be read by other programs, such as spread sheet programs (Excel, for example) or graphing programs. Excel is available on all the computers in 329 lab.

About Disks

Disk File Suggestion: For all your data files that you save this week, start their names with "W1" (meaning week #1). Save the files on the "Student Files on the Web" directory in a folder for your team.

For safety of your data, always save your data on your own floppy disks as well as on "Student files on the Web".


WEEK 2 STEADY-STATE OPERATING DATA

ObjectivesTo continue to observe steady-state performance of the system. To understand any disturbance functions (loads) for the system. Introduction to word processing, equations & graphics for Microsoft Word.

Lab Assignments248 Position Control

Continue to get the parameters for your system.Section 001--Get operating curve for unpainted spring.Section 002--Get operating curve for painted spring.

249 Speed ControlSection 001--Get operating curve for one bank of lights.Section 002--Get operating curve for two banks of lights.

272 Voltage ControlSection 001--Get operating curve for one bank of lights.Section 002--Get operating curve for two banks of lights.

303 Temperature ControlSection 001--Get operating curve for CWS of 20 GPH.Section 002--Get operating curve for CWS of 30 GPH.

307 Level ControlSection 001--Section 002--

308 Pressure ControlSection 001--Get operating curve for one open duct.Section 002--Get operating curve for two open ducts.

309 Flow ControlSection 001--Get operating curve for two motor valves closed.Section 002--Get operating curve for one motor valves closed.


ResultsPlot a graph of the output function versus the input function. Probably five (5) to ten (10) data points will be useful for this. Indicate the error in the output function by putting 95% confidence error bars on the points you plot.

Put a copy of the SSOC and the results of your error analysis on the bulletin board near your laboratory system. These graphs also go in the lab report described below.

Disk File Suggestion: For all your data files that you save this week, start their names with "W2" (meaning week #2) and save them on "Student files on the Web" in your team's folder.

Word processing assignment

Prepare a lab report following the format on the next page. Each section in a report is to start on a new page.

Double space the text in paragraphs. Put page numbers at the bottom of the pages (except the cover page).

This assignment is due at the beginning of the next scheduled lab meeting.

If the report is lacking some substantial component, it will not receive a grade.

If you are smart, you will save this report on a disk and modify and build from it through the rest of the semester.


Week 3 ReportReport is due at the beginning of Week 3 lab meeting.

REPORT CONTENTS

Title PageIncludes "UTC," "Engineering 329," Title, Your Name, Your partners' names, Date

Introduction In the first paragraph, it tells briefly what was done and for what purpose. In the second paragraph, it tells how the report is organized.

Background and Theory Describes the engineering background of the lab, including equations and schematic diagrams

ProcedureDescribes what was done in the physical lab

ResultsDescribes what you observed, the data. Includes tables and graphs. Each table and graph must be explained.It builds on the "Procedure;" the "Procedure" section must describe how all the results in this section were obtained.It includes results of experiments: estimates of errors of the results, SSOC

Discussion Tells the significance of the experiment and the results. It builds on the "Results;" the "Results" section must include all the results that are discussed in this section. "No surprises"

Conclusions and RecommendationsDescribes what principles were demonstrated by the experimental results. It builds on the "Discussion;" the "Discussion" section must prepare the reader for all conclusions that are mentioned in this section. "No surprises"

AppendicesIncludes raw data, references & other things that interrupt the "flow" of the report. Anything that is in an appendix (except "references") must be mentioned someplace in the report.

AttachmentsInclude a sheet for each team member that describes the contribution to the work in the laboratory.

Disk File Suggestion: For all your report files that you save this week, start their names with "RW3" (meaning report for week #3), put them on the Web.


WEEK 3 STEP RESPONSE TESTING

Week 3 ReportSubmission of report.

ObjectivesTo observe experimentally the time response of the output function of the system to a step function input. To observe the system's steady-state gain, the system's response time and the system's dead time (if any). To make the observations for a number of different values for the size of the input step function and the initial steady-state values. To make the observations for a variety of system configurations, if appropriate.

Reference: Smith & Corripio, pp 308-315

The purpose of this lab is to get step response data as shown in Figure 8. Figure 8(a) shows a graph of the input step function, m(t). Notice the input function is not at zero when the step occurs. This is called the base line value of m(t). At a certain time, the "start" time, the input function makes a step increase, m.

(a) Step Input (b) Step Response (output)

Figure 8. Step response input and output functions

The system takes a certain time to respond to this input step. A typical response curve is shown in Figure 8(b). From this graph you can get these characteristics of the system: steady-state gain, response time and dead time. These are called the First-Order Plus Dead Time (FOPDT) parameters of the system.


Procedure for getting step response data

Open LabVIEW program labeled "(Step)". It is programmed to provide a step input to the system, You should get a panel somewhat like the one shown in Figure 9, below.

Choose a value of the "input" variable that you want to be the base line value. Choose a value that you want to be the step height of the "input" variable. Click on the RUN button. The chart on the right of the screen emulates a strip chart recorder.

After the system reaches steady state at the input base line condition, click on the "Step" switch to initiate the step input function. When you want to stop, click on the STOP button. As before, it will ask if you want to save the data on a disk file and if you want to draw a time-response graph of the data. If you've just been playing around, click "NO" to all of these requests.You'll probably want to play around a bit before taking any serious data. You'll want to try different values of the step parameters: base line value and step height.

Figure 9. Step function programmed controller panel

When you are ready to get some good data, set the parameter values and start the instrument (click on the RUN button). After the step response has ended, click the STOP button and save the data, if you wish, and make the time-response graph.


Lab AssignmentsFor all stations, get step response data for the same conditions as Week 1 & 2.

For the "fast" systems (Motor Speed, Air Pressure & Water Flow) you are to obtain the system parameters for steps in the "up" direction as well as steps in the "down" direction. Also, obtain system parameters for different size steps and at different parts of your operating range.

Results

The time-response graph should look similar to the one in Figure 8(b). S&C (p 220) describes three different ways to get the parameters (K, and to). These are called Fit 1, Fit 2 and Fit 3. You are to use any of these methods to find the parameters for your system.

As the experimental measurement of your output variable has error associated with it, so too does the experimental determination of gain, dead time and time constant (K, to, ). You are to get an estimate of these errors, also. A good way to do this is to make several measurements and then use Student's T statistics after your analysis by the different fits. Appendix A4 and the poster in Grote 213 (credit to Dr. Schonblom) and in the Freshman lab can help you with this.

Week 4 report described below will present and discuss the fits.



Week 4 Report

Drafts of Week 4 Report are due prior to next lab meeting.

WEEK 4 REPORT CONTENTS STEADY STATE AND STEP RESPONSE

Introduction

Theory & BackgroundDescription of system components & connections. Schematic diagram (like

S&C, Fig. 1-1.2). Input function(s) and output functionLaplace domain descriptions in terms of deviation variables, OLTF (like S&C, equation 3-3.2)Block diagram (like S&C, Figure 3-4.2)

ProcedureResultsCalibrations. Steady state performance curves. Experimental results for step

input (curves, gain, time constant & dead time for different conditions) (like S&C, Figure 6-17). Include the various Fits.

Each table and graph must be explained.Estimates of error in measurements and calculated results. Variance and

Student's T.Include a table which clearly presents your results.

DiscussionComparison with theory. Observations about the SSOC of the system

Conclusions

Recommendation

AppendicesPhysical properties (dimensions, etc.) of components & system

AttachmentsInclude a sheet for each team member that describes the contribution to the work in the laboratory since last reported.

Disk File Suggestion: For all your report files that you save this week, start their names with "RW4" (meaning report for week #4) and save them on "Student files on the Web" in your team's folder.


WEEK 4 MODELING

Week 4 ReportSubmission of final report and brief oral presentation (5 min limit).

WEEK 4 ORAL PRESENTATION CONTENTS

BackgroundBrief description of system, "input" and "output"Brief description of performance curves (SSOC)ResultsSample time-response graphs for experiment & parametersConclusions"The experimental results showed "

Some suggested slides for Week 4 PresentationBackgroundTheoryResultsConclusions

TheoryTransfer function

Parameters

ResultsTime response Experimental Parameters

BackgroundSystem

InputOutput

SSOCOperating Range

Conclusions

Show the SSOC, with axes and operating ranges

ObjectivesTo learn how to model dynamic systems with Excel. To observe the dynamic response of the mathematical model for a system. To observe the impact of parameter values on the dynamic response. To learn how to obtain modeling results in graphical and tabular form.

See the following pages for details on Excel software and this week's assignment.


FOR NEXT WEEK

Bring with you the results of the steady state and step response experiments. Specifically, you must have

Gain, Kto, dead time, (tau) first-order time constant

at your chosen operating point (both for step up and step down) and the values of the steady-state input and output parameters at the operating point.

Post these results also on the bulletin board in your area in the laboratory.

MODELING FOPDT - EXCEL

FOPDT

If the input function to a FOPDT is a step function, having a step of height=A occur at time t=td,

m (t ) A u( t td )

then the time response of the FOPDT system is

c(t) A u(t td to) K 1 e

t td to

where K is the gain, to is the dead-time and is the first-order time constant, or characteristic time, of the FOPDT system. The variables m(t) and c(t) in these equations are deviation variables.

We want to have Excel draw a graph of these equations so we can see the effect of the various parameters.


Excel Tutorial

Open Excel by double clicking on the Excel icon.

You'll see a spreadsheet open, named Worksheet 1. We're going to enter in three columns of numbers to have Excel plot for us. The columns are "Time," "Input" and "Output." Type each of these names in the top row of the spreadsheet in columns A, B and C, respectively. Move to the right in the spreadsheet by touching the "tab" key or using the key or by clicking where you want to move to.

In cell A2, enter a 0 (zero). In cell A3, enter this: =1.+A2

and then touch "return." This has put a formula in cell A3 that adds 1.0 to the value of cell A2.

We're going to put formulas in cells B2 and C2 for the functions m(t) and c(t). For now, we're going to use values for td, A, K, to and . Let's use 2, 3, 4, 5 and 6, respectively.

Click on B2 and put in this formula: =if(A2>2,3,0)

This says, "if the value of cell A2 is greater than 2, then this cell takes the value 3, else it takes the value 0."

Click on C2 and put in this formula: =if(A2>2+5,3*4*(1-exp(-(A2-2-5)/6)),0)

Now click on cell A3 and notice in the lower right corner of the cell there is a small square dot. Move your cursor to that square (your cursor will change shape into a ), click-and-drag your mouse down to near the bottom of your screen. Release the mouse and the formula will have been copied into all the cells you dragged over and the values of time will be in that column.

Now, do a similar thing in column B, dragging down to the same cell you dragged to in column A.

Now, do a similar thing in column C, dragging down to the same cell you dragged to in column A.

Now do a related thing. Click in the last time-value in column A and drag across to column C and release the mouse. All three cells should be selected. Now click on the square at the right bottom of that selection and drag it down (off the screen and it will scroll down) to a ways, maybe, say, to row 40.

Now you should have 3 columns with the values in them that we want. The values in row 40 should be 38, 3 and about 11.9.


Graphing

Scroll back to the top of the spreadsheet. Highlight the three columns we want to plot by clicking on the A at the top of column A and dragging over to the C above column C.

Now click on the graphing button that looks like this:

Then answer questions posed by the Chart-Wizard. ALWAYS CHOOSE SCATTER PLOT!

Using Parameters

Go back to the spreadsheet and enter a table of the parameters that we are interested in. Let's make a table like this

D E1 td= 22 A= 33 K= 44 to= 55 tau= 6

Now click on cell E1. Pull down the Insert menu to Name , and choose Define and release the mouse button. That cell now is defined as having the name shown ("td"). Repeat this for cells E2 through E5.

Now go back and click on cell B2. Click next to the "2" that represents the td, delete it and type "td" (without the quotes).

Do this also for the "3" in the formula. Replace with "A".

Repeat in the formula in C2. Replacing the 2, 3, 4, 5, and 6 with the parameter names.

Now, again highlight cells B2 and C2, then drag them down to row 40, as you did before.

Look at the chart. It should be the same.

Now go back to the parameter cells and change one of them and watch the graph change.

Save your Excel file on your floppy disk. You will need it again next week.


WEEK 5 MODELING -- APPROXIMATE LINEAR FOPDT MODEL

Objectives To learn how to model first-order-plus-dead-time (FOPDT) with Excel. To observe the dynamic response of the approximate linear FOPDT model for a system. To observe the impact of parameter values on the dynamic response. To adjust the linear FOPDT model parameters to get the model results to agree with the experimental results.

Modeling Assignments

Preliminary

There are two things to add to the Excel model you developed last week. Last week's model started at zero (0) for input and output. We want our model to be somewhere else in the operating range of the system. Last week's model did not take into account that the operating point at which we want to model our system is not at a value of zero (0) on the input signal. So, we need to add the baselines for our system to the Excel model.

Open last week's Excel model and add two items in the parameter list. Here is a suggestion as to what to add (the numerical values are just examples).

inbl= 30outbl= 3

Use the menu Insert/Name>/Define... to make both of these added parameter usable names in your formulas. Then in cell B2 add the "Input Baseline" as shown here:

=IF(A2>td,A,0)+inbl

And in cell C2 add the "Output Baseline" as shown here:

=IF(A2>td+to,A*K*(1-EXP(-(A2-td-to)/tau)),0)+outbl

Copy both of these formulas down to the bottom of your spreadsheet.

Save this file.

By the way, there is a file like the one you've developed available on the Internet at http://chem.engr.utc.edu/329.


Approximate FOPDT Method

Reference: Smith & Corripio, pp 308-315

Open the file developed on the previous page. Open the file with the experimental data in it. What you're going to do is to put these two files together.

Select the "model" file. Select columns B and C by clicking on the B at the top of the column and dragging across to the C at the top of the next column. Insert 2 columns by pulling down the Insert menu and choosing Columns.

Now select the "experimental data" file. Select the experimental data (time, input and output columns) by clicking at the top of the columns as above. Copy that data by pulling down the Edit menu and choosing Copy. Then go back to the "model" file, select columns A, B and C and paste this data. Paste the data by pulling down the Edit menu and choosing Paste.

Suggestion: Now save this file with a new name.

Adjust the length of the model columns (D and E) to the same as data columns (A, B and C). Do this by either dragging down or by using Edit/Clear. Now draw a graph of the data and the model by highlighting the first five columns and using ChartWizard as last week.

Now adjust the parameters so that the model curves fit the data curves as nearly as possible. Here is a suggested order of doing this:

• Set input baseline and output baseline so the baseline parts of the model curves agree with the experimental data.

• Set td and A so that the model input curve agrees entirely with the experimental input curve.

• Set K, to and so that the model output curve agrees with the experimental input curve as close as possible. Suggested values of these come from your previous analysis.

By the way, there is a file like the one you've developed available on the Internet at http://chem.engr.utc.edu/329.

Disk File Suggestion: For all your model & results files that you save this week, start their names with "W5" (meaning week #5) and save them on "Student files on the Web" in your team's folder.


Week 6 Report


WEEK 6 REPORT CONTENTS STEP RESPONSE MODELING

Introduction

Theory & BackgroundDescription of system components & connections.Schematic diagram (like S&C, Fig. 1-1.2). Input function(s) and output function.Laplace domain descriptions in terms of deviation variables, OLTF (like S&C, equation 3-3.2)Approximate FOPDT modelBlock diagram (like S&C, Figure 3-4.2)Steady state performance curves (SSOC's).

ModelingEquations

Procedure

ResultsExperimental results for step input (curves, gain, time constant & dead time

for different conditions) (like S&C, Figure 7-2.3).Modeling results for step input (curves)Direct comparison of experimental and modeling results.Each table and graph must be explained.Description of errors in results and estimates of magnitudes of error

DiscussionComparison with theory, modeling & experiments

Conclusions

Recommendation

AppendicesPhysical properties (dimensions, etc.) of components & system


Disk File Suggestion: Use file names beginning with "WR6"


WEEK 6 FREQUENCY RESPONSE

Week 6 Report Submission of final report and oral presentation (10 min limit).


BackgroundBrief description of system, "input" and "output"Brief description of performance curves (SSOC)Objective of controller designTheoryBrief review of system transfer function (FOPDT) (include parameter values)ModelingModel & parametersResultsSample time-response graphs for experiment & approximate modelConclusions"The experimental results showed The approximate model results showed ."

Some suggested slides for Week 6 PresentationBackgroundTheoryModelingResultsConclusions

TheoryTransfer function

Parameters

ResultsTime response Experimental Approximate model

BackgroundSystem

InputOutput

SSOCOperating Range

ModelingModel equations

Parameters

Conclusions

Show the SSOC, with axes and operating ranges


Objectives To observe experimentally the time response of the output function of the system to a sine function input at a variety of different frequencies. To observe the system's amplitude ratio and the system's phase lag. To make the observations for a number of different values for the amplitude and the average value of the input sine function. To make the observations for a variety of system configurations, as appropriate.

Reference: Smith & Corripio, pp 45-46, 53, Chapter 9

The purpose of this lab is to get frequency response data as shown in Figure 12.

Figure 12. Frequency response input and output functions

Figure 12(a) shows an input to the system that is a sine wave. The input function baseline is Mb. The peak-to-peak amplitude of the sine wave is m. The sine wave has a frequency, f, measured in Hertz (Hz). Hertz is the same as cycles per second.

A typical output function is shown in Figure 12(b). The output function will have the same frequency but may have a different peak-to-peak amplitude, c. The output function may also be delayed so that it lags in phase compared to the input function.

cm


PROCEDURE FOR GETTING SINE RESPONSE DATA

Open LabVIEW program labeled "(Sine)". This program emulates the operation of a programmable controller. You should get a panel somewhat like the one shown in Figure 13.

Figure 13. Sine wave input controller panel

Choose a value of the "input" variable, M(t), that you want to be the base line value. Choose a value that you want to be the sine amplitude height of the "input" variable. Choose the frequency of the sine wave that you want. Set these in the appropriate windows in the panel. Click on the RUN button. The chart on the right of the screen is emulating a strip chart recorder.

When you want to stop, click on the STOP button. As before, it will ask if you want to save the data on a disk file, if you want to draw a time-response graph of the data and if you want to draw an input-output graph of the data. If you've just been playing around, click "NO" to all of these requests.

You'll probably want to play around a bit before taking any serious data. You'll want to try different values of the sine parameters: frequency, sine wave amplitude and base line value.

When you are ready to get some good data, set the parameter values and start the instrument (click on the RUN button). After the sine response has ended,


click the STOP button and save the data, make the time-response graph AND the input-output graph. This latter graph is the famous and useful Lissajous figure.

When you complete a run you will get a graphs similar to those in Figure 14.

Figure 14. Low frequency sine response graphs and Lissajous figure

The very low frequency Lissajous figure is actually a portion of the same curve you got as the steady state operating curve in Weeks 1 & 2. Incidentally, the

slope of this line, , is the steady-state gain.

The amplitude ratio (AR) is the ratio of to or ( ). The phase shift

is the fraction of a cycle that the output signal graph lags behind the input

signal graph. It is "t" as a fraction of "T" multiplied by 360 or 360.

The amplitude ratio (AR) is the ratio of the vertical height to the horizontal height of the Lissajous figure. When the input and output are exactly in phase, the Lissajous figure is a single line as in Figure 14(b). This means the phase shift is 0.

At a higher frequency, the AR and phase shift are different. A Lissajous figure like Figure 15(b) results. The AR is determined by the ratio of the vertical height to the horizontal height of the oval.


Figure 15. Medium frequency sine response graphs and Lissajous figure

At an even higher frequency, the phase shift becomes even larger and the Lissajous figure leans over to the left. An example is shown in Figure 16.

Figure 16. Medium-high frequency sine response graphs and Lissajous figure

You will benefit by running experiments at about 10 different frequencies. A

good place to start is at a frequency of about = (notice that is

the frequency in cycles/unit time). Then you might run experiments at successively lower frequencies by approximately halving the frequency each time. Stop going to lower frequencies when the output is nearly in phase with the input.

Then you might run experiments at higher frequencies by approximately doubling your starting frequency and continue doubling the frequency on successive experiments. Stop going to higher frequencies when the output has no perceptible steady oscillation. "No perceptible" oscillation means that the amplitude in the output is smaller than twice the standard deviation of the output measurements found in weeks 1 & 2.


After you have AR and phase angle data for a number of frequencies, make two plots like those shown in Figure 17. These are Bode plots. Notice Figure 17(a) is a log-log plot and Figure 17(b) is a semi-log plot. Prepare the results for making the Bode plots by filling in a table of results such as below.

Frequency..(lowest)..........................................................

..(highest).

Amplitude Ratio......................................................................

..............

Phase Angle......................................................................

..............

Figure 17. Example Bode plot obtained from experimental data

There are three important values to get from a Bode plot.1. The order of the system. This is the negative of the slope of the

AR vs. Frequency plot at the high frequencies.2. The ultimate frequency. This is the frequency for which the

phase angle is -180°.3. The Kcu. This is the 1/(AR) at the ultimate frequency.

Hints:XRC/248, SRC/249, VRC/272, PRC/308 & FRC/309 are probably higher than 1st order.


LRC/307 & TRC/303 are probably about 1st order.


"The study of how [the amplitude ratio and the phase angle] vary as varies is an important part of automatic process control."

––Smith & Corripio, 1st ed, p. [95]

FOR NEXT WEEK

Bring with you the Bode Plot graphs. Specifically, you must haveApparent order, mThe ultimate frequency, fu

Kcu

Post these results also on the bulletin board in your area in the laboratory.


WEEK 7 MODELING FREQUENCY RESPONSE

Objectives

To compare the experimental Bode plots with the approximate FOPDT model's Bode plots for the system. To observe the impact of parameter values on the model's Bode plots. To learn how to generate sine responses with Excel. To observe the sine responses of the approximate mathematical model for a system.

Modeling Assignment

If the input function to a FOPDT is a sine function, having an amplitude = A and frequency = f, then

m(t) = A•sin(2ft)

then the steady oscillation part of the time response of the FOPDT system is

where K is the gain, to is the dead-time and is the first-order time constant, or characteristic time, of the FOPDT system. The variables m(t) and c(t) in these equations are deviation variables. This means that you have to add the input baseline and output baseline to the values to get them to agree with the experimental data.

The model equations for the Bode plots are

and

We want to have Excel draw a graph of these equations so we can see the effect of the various parameters and see what parameters best match our experimental results.

Comparing Bode Plot for Model and Experiment

From last week, you got some values of the Amplitude Ratio (AR) and Phase Angle (PA) for the sine inputs to your system at some different frequencies.


Enter these into a new Excel spreadsheet, like this: (These are ONLY example numbers)

A B C D E1 Freq AR PA2 0.011 50 -83 0.02 49 -254 0.05 40 -555 0.1 21 -886 0.2 10 -1507 0.5 4 -1908 1 2 -2559

You are leaving columns C and D blank for now; later you'll put the model equations in there. It will be useful if you enter column D by using a formula and setting it to column A. The formula for cell D1 is =A1.

For now, add to your spreadsheet, in columns G and H, the names and values of the approximate model parameters for your system: K, to and tau. (These are ONLY example numbers)

G H1 K= 7.22 to= 0.653 tau= 0.33

Now enter the values of frequency that you want to have for you model Bode plot. Suggestion: have these values' range a bit wider than your experimental frequency range. Put these in column A, starting BELOW where your last experimental value is, say cell A10. Copy these into column D, also.

In the cell in column C BELOW the last experimental point, let's say, C10, enter this formula for the amplitude ratio

=K/SQRT(1+(2*PI()*A10)^2*tau^2)

In the cell in column F BELOW the last experimental point, let's say, F10, enter this formula for the phase angle

=(ATAN(-2*PI()*A10*tau)-2*PI()*A10*to)*180/PI(0)

Now plot the amplitude ratio part of the Bode plot by highlighting columns A, B and C; choosing the ChartWizard, selecting Scatter and making the axes log-log.


Now plot the phase angle part of the Bode plot by highlighting columns D, E and F; choosing the ChartWizard, selecting Scatter and making the axes semi-log.

VARYING PARAMETERS

Now go back to the parameter cells and change one of them and watch the graph change. Find the values of the parameters that make the model agree well with the experimental results. Here is the suggested order:

Which Bode Graph

Parameter(s) to vary

Objective

AR vs f K get model's AR-vs-f curve to agree at low frequencies

AR vs f tau get model's AR-vs-f curve to agree near the "corner" of the curve

PA vs f to get model's PA-vs-f curve to agree at PA=-180

Save the file.Label the graphs.Print the graphs.

Save the file. These values are now the best estimates for your approximate model's parameters to fit your system.

Comparing Time Responses for Model and Experiment

We're going to build a time-response model in a file that has some experimental sine-response data in it. Begin by choosing the file that has data in it for the lowest frequency sine-input experiment that you did that has good data.

Open an Excel by double clicking on the Excel icon. Open the data file from Week 6 by using the File/Open... menu item.

Part of the spreadsheet will look like this:

A B C D E F G1 T im e ( s e c ) In p u t Va lu e ( % )P re s s u re ( cm - H 2 O )2 0 .0 1 1 5 0 0 .1 5 23 0 .1 8 5 0 0 .1 7 34 0 .1 9 6 5 0 0 .1 5 25 0 .2 1 3 5 0 0 .1 3 26 0 .2 1 3 5 0 0 .1 3 27 0 .2 1 3 5 0 0 .1 3 28 0 .2 1 3 5 0 0 .1 3 2


Save this now in a new file named "W7-something-or-other." Do this with the File/Save As... menu.

We're going to enter in two columns of numbers to have Excel plot for us. The columns are "Input" and "Output." Type each of these names in the top row of the spreadsheet in columns D and E, respectively. Move to the right in the spreadsheet by touching the "tab" key or using the key or by clicking where you want to move to.

We're going to put formulas in cells D2 and E2 for the functions m(t) and c(t). In columns F and G, we're going to put names and values for A, f, K, to,, input baseline and output baseline. Let's use 2, 3, 0.2, 0.3, 0.4, 40 and 11, respectively. So those columns will look like this:

F GA 2f 3K 0.2to 0.3tau 0.4in-baseline 40out-baseline 11

Use the Insert/Names >/Define to define the names for all these parameters.

Click on D2 and put in this formula: =A*sin(2*PI()*f*A2)+inbl

Click on E2 and put in this formula: =A*K/(SQRT(1+2*PI()*f*2*PI()*f*tau*tau))*SIN(2*PI()*f*A2

+ATAN(-2*PI()*f*tau)-2*PI()*f*to)+outbl

Now copy these formulas down the spreadsheet. Click in the last time-value in column D and drag across to column E and release the mouse. Both should be selected. Now click on the square at the right bottom of that selection and drag it down (off the screen and it will scroll down) to the end of the experimental data.

Now you should have 5 columns with the values in them that we want.

Adjust amplitude, frequency, "inbl" and "outbl" so the curves agree.

Label the curves and print the curves. Save your file.

Repeat these time response models for 2 additional frequencies. Use one at a frequency near the "corner" frequency and one at a frequency of one of the highest frequencies you did.



By the way, there are files like the one you've developed that are available on the Internet at http://chem.engr.utc.edu/329.


Week 8 Report


WEEK 8 REPORT CONTENTS FREQUENCY RESPONSE

Introduction

Theory & BackgroundDescription of system components & connectionsSchematic diagramInput function and output function, SSOC with operating rangesTime domain and Laplace domain descriptions, OLTFApproximate linear FOPDT modelBlock diagram Frequency response theory for FOPDT (like S&C, examples in 9-1)

ModelingEquations & methods used in modeling

Procedure

ResultsExperimental results. Each table and graph must be explained.Description of errors in results and estimates of magnitudes of errorExperimental and modeling results for sine input (typical curves, amplitude

ratios, phase lags)Experimental and modeling Bode plots (Kcu, u, order of the system) (like

S&C, Fig. 9-2.3)

DiscussionComparison among theory, experiment & modeling for sine input

Conclusions

Recommendation

AppendicesPhysical propertiesData curves & calculations




WEEK 8 REPORTS & PLANT VISIT

Week 8 Report Submission of final report and oral presentation (10 minute limit).


Brief system description, including input & output functionsReview of performance curves (SSOC)Description of frequency response experimentsSample time response graph: Transients, steady oscillation, amplitude ratio

(AR) & phase shiftExperimental Bode plots--order of the systemModeling approach (approximate FOPDT)Comparison of results of experiment & approximate modelModeling results' Bode plotComparisons of experimental results and approximate modeling resultsConclusion(s) about systemConclusion(s) about approximate model

Some suggested slides for Week 8 ReportBackgroundPrevious WorkModelingTheoryResultsConclusions

Previous WorkTransfer function

Parameters

ResultsTime response Experimental Approximate modelBode plotsKcu, u, fu, order of the system

BackgroundSystem

InputOutput

Block diagramSSOCOperating Range

ModelingModel equations

Parameters

Conclusions


WEEK 9 ROOT LOCUS PLOTTING

ROOT LOCUS PLOTTING PREPARATION FOR EXCEL

Reference: S&C, pages 368-386

For a FOPDT system, the system transfer function (TF) is

G(s) Ke t0s

s 1(9.1)

The system that you have experimental data for has these three parameters: Gain, K, dead time, to and first-order response time, . At this point, your system's values have been determined by you, and you are not to modify them in any future work.

Using Padé's approximation for the exponential term, the TF becomes

Doing the algebra to simplify it, we get

(9.2)

For a proportional feedback controller, the controller transfer function is

GC (s) K C (9.3)

So, the OLTF for a FOPDT with proportional control becomes

(9.4)

ROOT LOCUS PLOTTING OBJECTIVES

For your experimental system, find the Kcu , find the Kc that gives quarter decay response and the Kc that gives critical damped response.


Plot the Root Locus for the Proportional-Only Feedback control system with your system's parameters for the FOPDT model. Use your best estimate at those parameter values; those parameter values found with the frequency-response modeling would be a good choice.

Use the Root Locus plot and determine the value of Kc which gives you these different responses:

Response to step change in set-point Symbol Value

Critical Damping KCD

Quarter-decay (under damping) KQD

"Ultimate" (Marginal stability) Kcu

Include root locus plots for your system in the reports for weeks 10, 12 and 14


Week 10 Report


WEEK 10 REPORT CONTENTS PROPORTIONAL CONTROLLER DESIGN

Introduction

Theory & BackgroundDescription of system components & connectionsSchematic diagramInput function and output function, SSOC with operating rangesTheory & governing equations for system and proportional-only feedback

controllerTime domain and Laplace domain descriptions, OLTF, CLTF, characteristic

equationsQuarter decay tuning parametersBlock diagram for feedback control systemPrevious system results (gain, time constant, dead time, etc.)


ResultsFrequency response resultsExperimental results. Each table and graph must be explained.Description of errors in results and estimates of magnitudes of errorRoot locusModeling results for proportional-only control (Kcu, quarter decay and critical

damped tuning parameters)Design results for system performance with control

DiscussionComparison of Bode & root locus of system response with feedback control

ConclusionsValues of Kc for specified system response

Recommendation

AppendicesPhysical properties

AttachmentsInclude a sheet for each team member that describes the contribution to the

work in the laboratory since last reported.



WEEK 10 PROPORTIONAL CONTROL DESIGN

Week 10 Report Submission of final report and oral presentations (10 minute limits).


BackgroundBrief description of system, Diagram, "input" and "output"Brief description of performance curves (SSOC), Operating rangeBaseline Input and OutputObjective of controller designPrevious WorkBrief review of system transfer function (FOPDT or other) (include parameter values)Description of feedback control in your systemCLTF for your systemBode plots and resultsModelingRoot locusModel & parametersResultsKcu, Kc for quarter decay, Kc for critical damping, range of Kc for underdamped, range of Kc for overdampedConclusions"For a response,our system needs a proportional controller with a Kc of ."

Some suggested slides for Week 10 ReportBackgroundPrevious WorkTheoryModelingResultsConclusions

Previous WorkTransfer function

ParametersFeedback controlCLTF

ResultsKc results (table) including fu

from Root Locus

BackgroundSystem

InputOutput

SSOCBlock DiagramOperating RangeController objectives

ModelingRoot locusModel equations

ParametersFeedback control

Conclusions


Objectives To observe the operation and behavior in the time domain of your system's approximate linear FOPDT model with proportional control. To observe the response of a closed loop controlled system to a set point change. To observe the effect of the value of the proportional feedback gain, Kc. To observe the limits of stable operation of the closed loop system. To predict Kc for critically damped response, quarter decay response and Kcu.

Modeling

Reference: Smith & Corripio, pp 227-231, 304-330.

There is an Excel spreadsheet that models proportional-only control of an FOPDT system. They are available on the Internet at http://chem.engr.utc.edu/329.

Use this model and put in the FOPDT parameters for your system and parameters for a case where you have a change in set point (within the operating range of the output variable for your system) at a certain time. The "system" parameters are the ones that you have found for your system.

For the underdamped case, determine the value of Kc that gives sustained oscillation. This is Kcu.

For the underdamped case, put in a Kc that is the Kc for quarter-decay that you got from Root Locus or the formula in Smith & Corripio. Observe what the damping ratio is for this value of Kc.

Details

Find the largest reasonable Kc and the smallest reasonable Kc. Then run the model for about 10 values of Kc over this range from largest to smallest. For each of these values of Kc, run a model and plot a graph of the response. Make a note of the offset, decay ratio and frequency of oscillation (if any) for each value of Kc. Present these results as a graph, if that is useful. Print the graphs and fill in this table for your observed results.

Kc Decay Ratio Frequency Offset



For Next Week

Bring plots of the results of the approximate model for various values of Kc.

Complete a memo like the one on the next page for your customers.


Chattanooga Control Systems AssociatesMEMO

Date: ___________________

To: ______________ System Engineers

From: _____________Team of Control Engineers

Subject: Tuning parameters for your control system

Greetings,

We have done the experiments, analysis and modeling for your system with proportional feedback control. We have looked at the situation where you wish to operate consistently and safely as some set point and then wish to change the set point to another value and have the system respond quickly, consistently and safely. In our investigations, we have taken the initial set point to be ___________* and that the set point will be changed by a step of __________*.

Below are the recommendations for the variety of responses that we studied.

Using traditional Ziegler-Nichols tuning formulas, for quarter decay, based on "Fit 1" parameters we recommendKc = ________*

To obtain the initial set point, we recommend a "bias" or "Input baseline" of _________.*Please be aware that with any proportional controller, there will be offset. For this Kc, we estimate the offset to be __________*.

We also found values of the controller tuning parameter, Kc, by modeling the time response for various values of the controller gain. Here is a table of the various time responses, our recommended Kc and other aspects of the response. For all cases, the "Input baseline" should be the same as above.

Description of response

Recommended Kc

Our predicted decay ratio

Our predicted offset

Our predicted oscillation frequency

Our predicted time to settle out

units * (dimensionless) * * *Over damped

Critically dampedTenth decay

Quarter decay

Marginal stability ("Ultimate")

(other)

(other)

* -- note to control engineers: remember to put units


WEEK 11 PROPORTIONAL CONTROL EXPERIMENT

PROPORTIONAL CONTROL EXPERIMENTS

Objectives To observe the operation and behavior of the recommended design of a proportional control system. To observe the effect of the value of the proportional feedback gain, Kc. To observe the response to a closed loop controlled system to a set point change.

To determine the ultimate gain and ultimate period for the closed-loop system. To tune the controller with approximate modeling results for critically damped response, quarter decay and at the limit of stability.

Reference: Smith & Corripio, pp 227-231, 304-320

Preliminaries

Each team will make a presentation at the beginning of lab in which they make a statement as to what values of Kc they have designed to give their system quarter decay ratio to a step change in set point. Two minutes maximum presentation.

PROCEDURE FOR RUNNING THE PROPORTIONAL-ONLY CONTROLLER

Experiments

Open LabVIEW program labeled "(P-only)". This program emulates an automatic feedback controller. You should get a panel somewhat like the one shown in Figure 19. On this panel, you put the "set point" with the control slide on the left. The "set point" is the value you want for the output variable. Set the value you want for Kc, the proportional controller gain, with the knob or in the appropriate window. Also set the Base Line Motor Speed (%) that is the one for your system. Click on the RUN button.

Choose the value of Kc that your theory and approximate modeling predict will be good for you system. You can observe the system's response to a step change in set point by changing the set point.


Figure 19. Proportional-only automatic controller panel

The meter in the lower left is the controller output. It is the signal (the "manipulated" variable) sent by the controller to the system.

Using the values of Kc that you used in your approximate modeling results for various tests, observe the experimental system responses for the equivalent experiments.

Experimentally determine what value of Kc gives marginally stable operation (Kcu) and determine that frequency (u or fu, and Tu --see Smith & Corripio, p. 304-305). Use the Ziegler-Nichols formula on page 306 to design a quarter decay Kc. Run an experiment with this value of Kc and plot a graph of the response. Observe and report the offset, decay ratio and frequency of oscillation in the response.

Note about "Marginally Stable Behavior"

For linear systems (the subject of ENGR 328 and our Excel models), if Kc > Kcu , the output will be increasingly oscillatory without any bounds. For real systems (like in ENGR 329), the output can never grow without bounds because eventually the system will go outside of its operating range and reach a physical limit. Two examples of limits are (1) a power supply can only put out voltages within some limited range and (2) water level in a tank can not be negative.


So, the Kc for "marginally stable behavior" in the real world means that at smaller values of Kc, the oscillations are damped and for larger values of Kc, the oscillations are not damped. That is, the oscillations are sustained indefinitely.

Finally, run an experiment with a value of Kc that Root Locus predicted would be the critically damped and plot a graph of the response. Observe and report the offset, decay ratio and frequency of oscillation in the response, if any.

Organize your results on the table below.

Source of Kc value Kc Decay Ratio Frequency Offset

In plotting your time-response graphs, always include the set-point in your graphs.



Week 12 Report


WEEK 12 REPORT CONTENTS PROPORTIONAL CONTROLLER PERFORMANCE

Introduction -- An introduction to your system and this report

Theory & BackgroundDescription of system components & connectionsSchematic diagramInput function and output function, SSOC with operating rangesBlock diagram for feedback control. Theory & equations for Proportional feedback controllerSSOC (again) with controller operating linesTime domain and Laplace domain descriptions, OLTF, CLTF, characteristic

equations, Kcu, uBode diagrams. Quarter decay tuning parameters from theoryPrevious system results (gain, time constant, etc.)


ResultsRoot locus plotsPerformance of system with proportional-only control. (Week 11

experiments)Arrange results in tables in order of KcExperimental results. Each table and graph must be explained.Description of errors in results and estimates of magnitudes of error

DiscussionComparison of theory, modeling and behavior of experimental system

responses with proportional only control

ConclusionsValues of Kc, for specified system response

Recommendation

AppendicesData curves & calculations

AttachmentsInclude a sheet for each team member that describes the contribution to the

work in the laboratory since last reported.


WEEK 12 PI CONTROLLER DESIGN

Week 12 Report Oral presentations and submission of report.

Presentations given this week are to cover this material:System descriptionDiagrams (schematic and block)Input and Output functionsPrevious results

SSOCOperating rangeBaseline Input and Output

FOPDT parametersBode plotsfU by 3 methods

P-only designP-only root locus P-only results

best P-only controller response curves & parameters

ResultsConclusions

Objectives To design a PI controller for your system.

"Design a PI controller" mean choose a Kc and I to give the response that you want for your system. In this assignment,

1. Use one of the formulas in Smith & Corripio for I 2. Draw a root locus for this value of I

Draw another root locus for a larger IDraw another root locus for a smaller I


The quarter decay tuning parameters for a PI controller can be derived from Smith & Corripio, table on page 306 or 320. Use these values to see how your approximate model behaves with PI control.

You need to know a value of I. On page 320 of the Amazing Smith and Corripio, it suggests that I be equal to 3.33*to. That is, about three and a third times your dead time. If you know TU, the period of oscillation of your system at marginal stability, there is a formula on S&C, page 306. You can get TU from last week's results. So try one of these formulas for I.

ROOT LOCUS PLOTTING PREPARATION FOR EXCEL


Reference: S&C, pages 368-386

For a FOPDT system with Padé's approximation, the system transfer function (TF) is

(12.1)

For a proportional-integral feedback controller, the controller transfer function is

GC s KC 1 1

Is

KC

Is 1Is

(12.2)

So, the OLTF for a FOPDT with proportional integral control becomes

OLTF KC K t0

2 Is2 K I

t02

s K

It02

s3 It02

s

2 I s

(12.3)

Useful things to do with Excel: Plot the root locus for the problem. Label with the values of Kc on the Root Locus plot where the breakaway and crossover are. Draw the line from the origin to the point where Kc gives "quarter decay."

When you include Excel work: In the Theory section, include material that shows what you're doing and any work you do in preparation for Excel calculations. In the Appendix of your report, put tables of your input, results and the raw plots.






Repeat this for a value of i that is four times as large as before and again for a value of i that is one-fourth as large.

four times the i :





one-fourth the i :





Also, use the Ziegler-Nichols formulas to find the values of the PI tuning parameters for quarter-decay response to a step change in set-point.



WEEK 13 PI CONTROL EXPERIMENT

Objectives To observe the operation and behavior of your PI control system design. To observe the effect of the value of the proportional feedback gain, Kc and the integral time, I. To observe the limits of stable operation of the closed loop system. To observe the response to a closed loop controlled system to a set point change. To tune the controller with approximate modeling results for quarter decay response. To observe reset windup.


Preliminaries

Each team will make a presentation at the beginning of lab. In this presentation they make a statement as to what values of Kc and I they have designed to give their system quarter decay response to a step change in set point. Two minutes maximum presentation.

Also, bring with you the printed-out graphs from the experiments with a P-only controller.

PROCEDURE FOR RUNNING THE PI CONTROLLER

Experiments

Open LabVIEW program labeled "(PI)". This program emulates a proportional-integral feedback controller. You should get a panel somewhat like the one shown in Figure 21.

Again on this panel, you put the "set point" with the control slide on the left. The "set point" is the value you want for the output variable. Set the values you want for Kc, the proportional controller gain, and I, the integral or reset time, with the knobs or in the appropriate windows. Click on the RUN button.


Figure 21. PI automatic controller panel

Choose the values of Kc and I that your theory and approximate modeling predict will be good for you system. You can observe the system's response to a step change in set point by changing the set point. You can observe the system's response to a disturbance by changing disturbance input.

Using the values of Kc and I that your approximate modeling results gave for various system responses, observe the experimental system responses for the equivalent experiments. Check the decay ratio to see if agrees with your modeling.

Also, for each value of I that you chose, find the largest reasonable Kc and the smallest reasonable Kc. Then run the system for a number of values of Kc over this range from largest to smallest. For each of these values of Kc, run a test and plot a graph of the response. Make a note of the offset, decay ratio and frequency of oscillation (if any) for each value of Kc. Present these results as a graph, if that is useful.

Finally, observe reset windup by choosing a set point that is about 5% to 10% below the maximum operating point & then starting the LabVIEW program and plot a graph of the response. Observe and report the characteristics of the response.


Kc I Decay Ratio Frequency



Week 14 Report


WEEK 14 REPORT CONTENTS PI CONTROLLER PERFORMANCE

Introduction -- An introduction to your system and this reportBackgroundDescription of system components & connectionsSchematic diagramInput function and output function, SSOC with operating rangesBode plotsTime domain and Laplace domain descriptions, OLTF, CLTF, characteristic

equations, Kcu, u, fuPrevious system results (gain, time constant, P-only results, etc.)Theory & governing equations for PI feedback controllerBlock diagram for feedback control. Quarter decay tuning parameters from theorySSOC (again) with controller operating linesModelingEquations & methods used in modelingRoot locus plotsRouth and Direct Substitution resultsResultsExperimental results. Each table and graph must be explained.Description of errors in results and estimates of magnitudes of errorPerformance of experimental system with PI control (graphs of overshoot, etc.)Arrange results in tables in order of KcModeling results for PI controlReset windupDiscussionComparison of theory, modeling and behavior of experimental system

responses with P-only controlComparison of theory, modeling and behavior of experimental system

responses with PI controlConclusionsRecommendationAppendices Physical properties

Attachments Include a sheet for each team member that describes the contribution to the work in the laboratory since last reported.


WEEK 14

Week 14 Report Oral presentations and submission of report.

Presentations given this week are to cover this material:System descriptionDiagramInput and Output functionsPrevious results:

SSOCOperating rangesBaseline Input and OutputFOPDT parametersBode plots and resultsRouth and Direct substitution results

P-only designP-only root locus P-only resultsbest P-only controller response curves & parameters

PI controller results:PI design parameters, PI root locus, Routh and Direct substitution resultsPI experimental results, best PI controller response curves & parameters, reset windup,

graphs of overshootdecay ratiofrequencysettling timerise time

Results & conclusions

REJOICE

APPENDICES

1 References

2 Hints on Lab Reports

3 Oral presentation format, hints and grading

4 Student's t statistics

APPENDIX A1 REFERENCES

Smith, Carlos, and Armando Corripio, Principles and Practice of Automatic Process Control, 2nd Ed., John Wiley, 1997.

APPENDIX A2 HINTS ON LAB REPORT

Diagrams, Graphs, & Tables:The reason for putting diagrams, graphs and tables in a report is to organize information so that it

can be easily understood. However, the point that you want to convey in a diagram, graph or table may not be obvious to the reader or grader. You have to tell the reader or the grader what point you want him or her to see in the diagram, graph or table so that he or she doesn't miss that point. A good rule of thumb is to write 2 or 3 sentences for each diagram, graph or table to explain or describe the diagram, graph or table.

Figures:Try to size figures so they fit upright on the page. If you have to turn them sideways, remember

that the top of the figure goes to the left of the page.

See the examples below.

Top

BEST

axis

O.K . BAD

Appendices

Binder:The reports submitted are to have all your team's reports included in a binder. Put the newest in the front. Include the grading sheet with each report. Separate the reports with tabbed separators that are labeled to identify the reports.

"TJe":This is known as the "Thomas Jefferson Error." This is marked when

you write it's when you mean its. Both words are legitimate words, but they mean different things. It's is the contraction of the two words it is; its is a possessive pronoun that refers to an object.

Contents of "Theory & Background" (by Jay Ware)

Brief review of systemThis should include a schematic diagram of your system with control elements labeled using the standard symbols in the appendix of S&C. The schematic diagram does not have to be pictorial, but is to show the functional relationships among the various components of the system. The block diagram and governing equations or FOPDT equations should be included. Both time-domain and Laplace-domain equations are appropriate. The input and output functions should be clearly defined.

Discussion of Principles behind experimentThis should include the theoretical output for a given input. For examples, the step response for the step input and the steady oscillation response for a sine input. For the control experiments, discuss the theory of P-only or PI control and discuss typical responses. Include tuning parameter equations and discuss how changes in parameters affect the typical response.

Discussion of theory as applied to systemAll variables for your system should be defined. Where you know the values of parameters (from previous measurements or reports) these are to be described in this section. You should clearly point out what are the manipulated variable and the controlled variables.

Brief summary of theoretical responseThis should consist of an explanation of how the system should respond based on theory. In the "Discussion" section, this theoretical response should be compared with experimental and/or approximate modeling response.

Appendices

ENGINEERING LAB REPORT COMMENTS & GRADING

NAME:___________________________REPORT DATE:

TEAM:___________________________TITLE:

CONTENT (50%)Title Page -- Name and name of team members? Introduction -- Clear reason for report?

Background-enough to follow report? Introduces report?

Theory (diagrams, equations, calculations) Modeling (diagrams, equations) Results (tables, graphs, calculations) Discussion (follow from Results?) Conclusions (follow from Discussion?) Recommendations Appendices Evaluator's recommendations

Total points (out of 50)

FORM (50%)Appearance (margins, page #s) Type (size, quality, consistency) Graphs (clear, consistent) Tables (clear, consistent) Diagrams (clear, consistent) Language (word choice, significant figures)

Total points (out of 50) OVERALL COMMENTS

Appendices

EVALUATOR_____________________________ OVERALL GRADE

Appendices

APPENDIX A3

ORAL PRESENTATIONS

ORGANIZATIONEstablish Purpose: What is your objective in making presentation?Assume your audience does not know anything about your subject.Write Conclusion FirstOutline:

IntroductionName and name of team members State reason for presentationBackground-provide enough to follow talk

BodyPresent ideas Include strengths and weaknesses

SummarySummarize briefly State conclusion

Questions-be preparedSelect Information Based on Support of Conclusion-KISS

PREPARATIONVisual aids

Overhead, charts etc. - approximately 1/minute of talkone idea/slide

Include -title slide outline of talk conclusionDo not have complete sentences on your slidesMake your letters on the overheads at least 1/4 inch high.

If you can't get a printer to do that for you, do it by handor use the copy machine to enlarge your copy.

Everything on your slide must be important enough to be there.If it's not important, leave it off. If it's important, makeit clear and tell your audience about it.

Have a balance among the number of slides with words only, those with diagrams or graphs and those with tables

Practice--to have confidenceMemorize introduction and conclusions Make them strongPractice out loud to an empty roomPresent to other team members for critique and potential questions.

DELIVERYCommunicate Stay within time limitConfidence Be in controlMaintain eye contact-maintain eye contact-maintain eye contact

Appendices

ORAL PRESENTATION GRADE

NAME: DATE TEAM: SUBJECT:

CONTENT (50%)Introduction

Name and name of team members?_______State reason for presentation?______Background-enough to follow talk?_______

Body Summary

Summarize briefly?

State conclusion? Questions?

VISUAL AIDS (20%)Overhead?

Include -title slide? outline of talk? conclusion? # Text slides # Graphic slides # Tabular slides

DELIVERY (30%)Confidence Stay within time limit? start________end______time_____ Delivery In control/evidence of practice? Maintain eye contact?

OVERALL COMMENTS

EVALUATOR OVERALL GRADE

Appendices

APPENDIX A4

Students T Statistics

Maximum Errors99% Confidence

Error = (Xmax - Xmin)*t/nwhere Xmax - Xmin is the range of the measurements

n is the number of measurementst is the "Student's t"

Note: all the measurements have to be of the same quantity or variable

n t t/n2 40.0 20.03 7.0 2.34 4.5 1.15 3.7 0.756 3.4 0.56

Note: all the measurements have to be of the same quantity or variable

Appendices

Faculty Verification Statement

This course pack is original work. There is no disallowed use of copyright materials. I will defend and indemnify Barnes & Noble Bookstores, Inc., against any liability, claim or expense resulting from any third party claim of a copyright in any materials in this course pack.

Signature Date

Course

Appendices

GRADE RECORD STUDENT

PHONE

WEEK TOPIC GRADE DATE INITIALS

1 INTRO L P C C C T

2 SSOC L P C C C T

3 STEP RX L P C C C T

WK 3 REPORT /10

4 MODELING INTRO

5 MODELING FOPDT L P C C C T

6 FREQ RX L P C C C T

MID-TERM TOTAL POINTS /35

GRADE A | B | C | D | F31 28 24 23

7 MODELING F-R L P C C C T

8 PLANT VISIT

9 ROOT LOCUS L P C C C T

10 MODELING P-ONLY L P C C C T

11 P-ONLY L P C C C T

12 MODELING PI L P C C C T

13 PI L P C C C T

WRITTEN REPORT /10

ORAL REPORT #1 /10

ORAL REPORT #2 /10

PRE-LAB /5

SEMESTER TOTAL POINTS

GRADE A | B | C | D | F90 80 70 65

L P C C C T LEADERSHIP, PARTICIPATION, COOPERATION, CONTRIBUTIONS, CREATIVITY, TEAMWORK

(5 POINT MAX, EACH DAY)

SSOC

Operating Range Input Range

Output Range

Slope of SSOC in Operating Range

Variance of Output Measurements(typical)

SINE EXPERIMENTS

Estimated c (radians)

(frequency)

Amplitude Phasefreq. Ratio Shift

STEP EXPERIMENTS

Operating Point Input Baseline

Output Baseline

Estimates of FOPDT Parameters

Gain, K

Time constant,

Dead time, to

SINE MODELING

Best (rounded) FOPDT parameters

Gain, K

Time constant,

Dead time, to

STEP MODELING

Best (rounded) FOPDT parameters

Gain, K

Time constant,

Dead time, to

AR slope at high freq.

ƒu at phase = -180°

AR at ƒu

Kcu

ROOT LOCUS MODELING

P-only

PI MODELING

Kc (QD) (p.306)

Kcu

u

Kc (QD)

Kc (CD)

Root LocusIntegral time

PI

Kcu

u

Kc (QD)

Kc (CD)

P-only MODELING

Kc (QD) (p.306) Experimental

Kc damping

(QD) (p.306)

(QD) (RL)

ExperimentalKc damping

(QD) (p.306)

(QD) (RL)

P-only EXPERIMENT


Kc damping

(QD) (p.306)

(QD) (RL)

PI EXPERIMENT


Kc damping

(QD) (p.306)

(QD) (RL)

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