Department of Mechanical & Industrial Engineering Concordia University

MECH 371 Analysis and Design of Control Systems

Laboratory Manual

W. Xie, H. Hong, T. Wen, G. Huard 1/9/2017

Table of Contents

Lab 1: Control System Introduction, Familiarization with Lab Equipment and Instruments …………………….………………………………………………....7

Lab 2: Determination of DC Motor Dead-Band, Gain, Servo-Amplifier Gain,Torque/SpeedCharacteristic…….……………………………………...17

Lab 3: Time Response of Basic Closed-Loop System and Effect of Tachometer Feedback………………………………………………………………27

Lab 4: Frequency Response of Basic Closed-Loop DC Motor System……33

Lab 5: DC Motor Position Control with Cascade PID Compensation…….37

Lab 6: Time Response of Basic Closed-Loop Speed Control System and Effect of Torque Load………………………………………………………….…...42

Appendix A: Connect Oscilloscope to MS Excel………………………………………………….43 Appendix B: Alternate method to get one shot wave form…………………………………44 Appendix C: How to get Time Constant using Cursor measurement……………………45 Appendix D: Summary of MS150 Data- DC Motor System …………..………………….46

Labs are scheduled on an alternative week basis (every two weeks). Therefore, formal lab reports must be submitted every two weeks during your lab period.

Please submit your last lab report directly to your lab instructor at his/her office, one week after you have performed the lab.

No late lab reports will be accepted.

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General Laboratory Safety Rules

Follow Relevant Instructions • Before attempting to install, commission or operate equipment, all relevant

suppliers’/manufacturers’ instructions and local regulations should be understood and implemented.

• It is irresponsible and dangerous to misuse equipment or ignore instructions, regulations or warnings.

• Do not exceed specified maximum operating conditions (e.g. temperature, pressure, speed etc.).

Installation/Commissioning • Use lifting table where possible to install heavy equipment. Where manual lifting is

necessary beware of strained backs and crushed toes. Get help from an assistant if necessary. Wear safety shoes appropriate.

• Extreme care should be exercised to avoid damage to the equipment during handling and unpacking. When using slings to lift equipment, ensure that the slings are attached to structural framework and do not foul adjacent pipe work, glassware etc.

• Locate heavy equipment at low level. • Equipment involving inflammable or corrosive liquids should be sited in a containment area

or bund with a capacity 50% greater that the maximum equipment contents. • Ensure that all services are compatible with equipment and that independent isolators are

always provided and labeled. Use reliable connections in all instances, do not improvise. • Ensure that all equipment is reliably grounded and connected to an electrical supply at the

correct voltage. • Potential hazards should always be the first consideration when deciding on a suitable

location for equipment. Leave sufficient space between equipment and between walls and equipment.

• Ensure that equipment is commissioned and checked by a competent member of staff permitting students to operate it.

Operation • Ensure the students are fully aware of the potential hazards when operating equipment. • Students should be supervised by a competent member of staff at all times when in the

laboratory. No one should operate equipment alone. Do not leave equipment running unattended.

• Do not allow students to derive their own experimental procedures unless they are competent to do so.

Maintenance • Badly maintained equipment is a potential hazard. Ensure that a competent member of staff

is responsible for organizing maintenance and repairs on a planned basis.

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• Do not permit faulty equipment to be operated. Ensure that repairs are carried out competently and checked before students are permitted to operate the equipment.

Electricity • Electricity is the most common cause of accidents in the laboratory. Ensure that all members

of staff and students respect it. • Ensure that the electrical supply has been disconnected from the equipment before

attempting repairs or adjustments. • Water and electricity are not compatible and can cause serious injury if they come into

contact. Never operate portable electric appliances adjacent to equipment involving water unless some form of constraint or barrier is incorporated to prevent accidental contact.

• Always disconnect equipment from the electrical supply when not in use.

Avoiding Fires or Explosion • Ensure that the laboratory is provided with adequate fire extinguishers appropriate to the

potential hazards. • Smoking must be forbidden. Notices should be displayed to enforce this. • Beware since fine powders or dust can spontaneously ignite under certain conditions. Empty

vessels having contained inflammable liquid can contain vapor and explode if ignited. • Bulk quantities of inflammable liquids should be stored outside the laboratory in accordance

with local regulations. • Storage tanks on equipment should not be overfilled. All spillages should be immediately

cleaned up, carefully disposing of any contaminated cloths etc. Beware of slippery floors. • When liquids giving off inflammable vapors are handled in the laboratory, the area should

be properly ventilated. • Students should not be allowed to prepare mixtures for analysis or other purposes without

competent supervision.

Handling Poisons, Corrosive or Toxic Materials • Certain liquids essential to the operation of equipment, for example, mercury, are poisonous

or can give off poisonous vapors. Wear appropriate protective clothing when handling such substances.

• Do not allow food to be brought into or consumed in the laboratory. Never use chemical beakers as drinking vessels

• Smoking must be forbidden. Notices should be displayed to enforce this. • Poisons and very toxic materials must be kept in a locked cupboard or store and checked

regularly. Use of such substances should be supervised.

Avoid Cuts and Burns • Take care when handling sharp edged components. Do not exert undue force on glass or

fragile items.

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• Hot surfaces cannot, in most cases, be totally shielded and can produce severe burns even when not visibly hot. Use common sense and think which parts of the equipment are likely to be hot.

Eye/Ear Protection • Goggles must be worn whenever there is risk to the eyes. Risk may arise from powders,

liquid splashes, vapors or splinters. Beware of debris from fast moving air streams. • Never look directly at a strong source of light such as a laser or Xenon arc lamp. Ensure the

equipment using such a source is positioned so that passers-by cannot accidentally view the source or reflected ray.

• Facilities for eye irrigation should always be available. • Ear protectors must be worn when operating noisy equipment.

Clothing

• Suitable clothing should be worn in the laboratory. Loose garments can cause serious injury if caught in rotating machinery. Ties, rings on fingers etc. should be removed in these situations.

• Additional protective clothing should be available for all members of staff and students as appropriate.

Guards and Safety Devices • Guards and safety devices are installed on equipment to protect the operator. The

equipment must not be operated with such devices removed. • Safety valves, cut-outs or other safety devices will have been set to protect the equipment.

Interference with these devices may create a potential hazard. • It is not possible to guard the operator against all contingencies. Use commons sense at all

times when in the laboratory. • Before staring a rotating machine, make sure staff are aware how to stop it in an emergency. • Ensure that speed control devices are always set to zero before starting equipment.

First Aid

• If an accident does occur in the laboratory it is essential that first aid equipment is available and that the supervisor knows how to use it.

• A notice giving details of a proficient first-aider should be prominently displayed. • A short list of the antidotes for the chemicals used in the particular laboratory should be

prominently displayed.

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Standard lab safety must be followed in all laboratories

• First discuss your experiment regarding possible hazards or problems, with the demonstrator, or the MIE technical staff, or your professor.

• Do not work alone. Work with another person in a lab that has running machinery, machine tools, conveyors, hydraulics, lifting equipment, voltage hazards, or where chemicals are in use.

• Safety glasses must be worn in the vicinity of pneumatics, machine tools grinders, power saws, and drills.

• Users of lasers need special safety glasses for the particular wavelength of the laser. • No equipment or machine may be operated by anyone unless they have received adequate

instruction from a qualified instructor e.g. machine tools, hydraulics, chemicals, lasers, running machinery, robots. Undergraduate students may not use any machine or equipment unless a Department technical staff member Is present. Graduate students are the responsibility of their immediate academic supervisor.

• Workplace Hazardous Material training must be obtained before using chemicals or compressed gasses. Contact Dainius Juras tel: 848 3128 for training.

• All appropriate safety accessories (lab coats, safety glasses, gloves, etc.) must be used when handling chemicals. No open toe shoes are permitted in laboratories.

• No chemicals to be left unattended or unlabeled according to WHMIS. All chemicals must be stored properly.

• Long term unattended tests must be fail safe. When the university Is officially closed, you may not work in a lab unless your supervisor or a technical staff member is present.

Mr. Gilles Huard 8798

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• No eating in laboratories. • Major accidents and injuries must be reported at once to Security tel: 811, the Safety

Officer (tel: 3128), the Professor (Supervisor) or the Department Administrator (tel: 7975} should then be informed.

• During working hours all minor accidents should be reported to the Safety Officer (tel: 3128), the Professor (Supervisor) or the Department Administrator (tel: 7975).

• An "Incident Report" must be filled out by the person involved, for all accidents and injuries.

LABORATORY RULES Considering the large number of students attending the labs and in order for the lab to operate properly, the students are asked to abide by the following rules:

1. No smoking, eating, or drinking is permitted in the laboratory. 2. No students are allowed access to the instruments of the other course. 3. No equipment is allowed to be exchanged from one bench to another. 4. Upon entering and when leaving the laboratory students should check equipment against the

list posted at each station. 5. All damaged or missing equipment and cables must be reported immediately to the

demonstrator. Failure to do so will result in students being charged for damages or losses. 6. All data must be recorded in the laboratory paper and must be signed by the demonstrator. 7. Laboratory demonstrators are not permitted to admit any students other than those on their

class list. 8. Any student who is more than 30 minutes late will not be permitted into the laboratory.

Furthermore, repeated tardiness will not be tolerated. 9. After your laboratory session is completed all components, connecting jumpers, and cables

must be returned to their respective places. 10. No students are allowed access to parts in the cabinets. Your laboratory demonstrator will

provide you with all necessary parts.

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Lab 1: Control System Introduction, Familiarization with Lab Equipment & Instruments

Objectives To familiarization students with MS150 DC Motor Control Modules, instruments such as function generator and oscilloscope and to calibrate potentiometers, Op Amp and pre-amplifier. Introduction The purpose of the laboratory is to acquaint the student with a practical classical feedback, control system [specifically an electromechanical angular position control system using a DC motor], and to become familiar with the measurement of basic performance parameters of the system, both in the time-domain and in the sinusoidal frequency domain. A position control system (rather than any other variable) is used in this lab not only because of its wide application (e.g. position control in robotic manipulators, setting of hydraulic/pneumatic valves in process-control systems, positioning of directional antennas in communication systems etc.) but also because important operational characteristics such as overshoot may be directly (visually) observed when the controlled variable is the 'position'.

Figure 1.1 is a basic angular position control system. An Input Potentiometer 150H (input position transducer) translates the desired angular position θd into a proportional voltage Vi. A Servo-amplifier 150D which drives the motor and which, together with a DC motor in 150F, forms the 'servomotor'. The motor drives a mechanical load mainly consisting of a flywheel (representing a real load), through a Gear train (Gear box) in 150F which provides both amplification of the motor torque as well as speed reduction.

Figure 1-1 Basic DC Motor Angular Position Control System

Ki

150H

150A K1

150B top

𝑲𝑲𝒎𝒎

𝟏𝟏+𝑺𝑺𝝉𝝉𝑴𝑴

150D +150F

𝟏𝟏𝑺𝑺𝑺𝑺

150X (Gear)

Ko

150K

𝜽𝜽𝒅𝒅 𝜽𝜽𝒐𝒐 𝝎𝝎𝒎𝒎 𝑽𝑽𝒆𝒆

𝑽𝑽𝒊𝒊

−

𝑽𝑽𝒐𝒐

Kp

150C

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An Output Potentiometer 150K (output position transducer) translates the angular position θo, of the flywheel shaft, into a proportional voltage Vo. The device called the “Reference Comparator” 150A compares the voltage Vo with the reference input voltage Vi, which represent s the desired posit ion of the fly wheel, and generates the difference between them: Ve = Vi –Vo, then the

Figure 1-2 MS150 DC Motor Control System

PS150E DCM150F

SA150D

Oscilloscope Function Generator

GT150X

IP150H OP150K

PA150C OA150A

AU150B Top

PID150Y AU150B Bot

LU150L

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voltage Ve will represent the 'error' between the desired position and the actual position. The reference comparator is therefore also called an 'error detector'. The 'error' signal Ve , can be adjusted by K1 150B, then amplified by a pre-amplifier 150C and subsequently by a power-amplifier 150D, is used to drive the motor in such a sense as to reduce the 'error' itself. A system such as the one just described (shown in Figure 1.1) is called a closed loop, negative-feedback position control system.

The Modular DC Servotrainer MS150 used in the lab is designed to demonstrate the basic principles of a classical closed-loop negative feedback control system as shown in Figure 1-2: an electromechanical system using a DC motor which controls the angular-position of a shaft. The equipment consists of modular units for the motor, amplifiers etc., mounted on a baseplate. The various modules are positioned on a baseplate as shown above. Each station also includes a function generator and an Oscilloscope. Except for some main connections, interconnections between the various modules are made by the student, using banana-plug-ended patch cords which are provided in the laboratory. The power supply module 150E is permanently connected to the motor-tachometer module 150F and to the servo-amplifier module 150D. Terminals which provide a balanced +15/ 0 /-15 volt DC output are available on the power supply and servo-amp modules. A 3-wire harness is connected to distribute the ±15 volt supply to the operational amplifier, preamplifier and PID modules. The +15/ 0 /-15 supply voltages, available at terminals on the power supply module 150E, are also used to supply voltages to the Input & Output Potentiometers (150H & 150K) , which make up the "error channel".

Power Supply PS150E provides the ±15 volt DC power supplies through two sets of sockets. These sockets are used to operate small amplifiers and provide reference voltage. The Ammeter is used for monitoring motor overload. The AC outputs are not used in our experiments. The front panel is shown in Figure 1-3.

Potentiometers: showing in Figure 1-4. The module includes an Input Potentiometer IP150H (as an input position transducer), an Output Potentiometer OP150 K (output position transducer), and an Attenuator Unit AU150B containing two smaller potentiometers, which are used to adjust gains in the forward and feedback paths. The input and output pots are fitted with discs graduated (in degrees) on their shaft.

However, the output pot can be rotated continuously over 360º, whereas the input pot has a limited rotation of about ± 150°. Both these 'angular position transducers' are normally supplied with +15 and –15 volts, so that their outputs can vary linearly from zero to almost either of these limits as their shafts are rotated in either direction from a central (zero) position. Normal operation is symmetrical about this zero position. Note that in the output pot, a zero-voltage transition also

Figure 1-3 Power Supply: PS150E

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occurs at the + or –180° position, hence requiring operation which ensures output angular displacements within these limits. Assuming that the total voltage applied across the output pot is 30 volts, and the rotation is 360°, the position-to-voltage transducer sensitivity K0 will be 30 / 360 ≈ 0.083 volt / deg., or approx. 4.8 volts/radian. The input and output potentiometers should be calibrated to obtain their sensitivity constants and/or to confirm whether Ki ≈ Ko. The pots in the Attenuator unit are provided with knobs and scale graduations from 0 to 10. These pots can be used as voltage dividers and to obtain the very small voltages.

Operational Amplifier OA150A (Figure 1-5) is an op-amp normally connected as a unity-gain summing-inverter by means of the 3-position switch mounted on it. It is used as the angular-position-error detector. Since the unit is a summing amplifier, the feedback signal polarity must be reversed with respect to the reference signal, in order that the output will represent the error. The unit has three summing input terminals, and the output is available at two (or three) output sockets. The unit also has a zero-set control and a selector switch, which selects the feedback (normally resistive) within the unit. The selector switch is normally switched to the leftmost position indicating resistive feedback with unity gain. The op-amp must be zeroed before use. {ZERO PROCEDURE: With no input applied (input terminals#1, #2, #3 connect to ground), the Zero-Set control knob should be carefully adjusted until the output #6 is zero volt mean.} Experiment Procedure MS150 System is equipped with a DC motor, with a tachometer to measure angular velocity, turning potentiometer (designated as input pot and output pot ) to give and measure angular position, and power amplifier (also known as pre-amp and servo-amp) to drive the motor. The

Figure 1-5 Op-Amp OA150A

A: Input Pot IP150H B: Output Pot OP150K C: Attenuator AU150B Figure 1-4 Potentiometers

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command signal can be provided by the function generator or input pot, and the output of angular position or velocity can be measured by the oscilloscope. Figure 1-2 shows the MS150 system. In these experiments, we will begin with the close loop DC motor position control system setup, power source and DC voltage measurement, use of potentiometers (attenuator, input pot and output pot), Op-amp as a signal adder . Exp#1 Basic DC Motor Angular Position Control System Setup

1. Referring to Figure 1.1 block diagram, make connection as following Figure 1-6: Op Amp OP150A will be used as a “signal adder” to detect the error signal between “command” -IP150H and “real position”- OP150K. The error signal will be used to control how far and which direction motor to run. Please note that this error signal can’t directly drive motor. It has to be adjustable (by a Controller-AU150B used as a proportional controller). This control signal will be amplified by “Power amplifier” (Pre-Amp PA150C and Servo-Amp SA150D). This amplified power (voltage and current) will drive motor.

2. Check control stability: no function generator connection, adjust AU150B dial to 1 or less, turn IP150H to 45 degree clockwise, check if OP150K follow the IP150H clockwise. If not, re-check your connection, especial the IP150H and OP150K are cross-connected. If still doesn’t work, ask your lab demonstrator to check your connect. Make sure the system is a stable negative feedback position control system.

3. Increase the dial of AU150B, the system will become unstable, decrease the dial of AU150B, the system will be stabilized and will stop response when dial to 0.

4. Disconnect the wire at the #3 of IP150H and connect to Function Generator output as shown in dashed line. The function generator will be used as a command signal. Set function generator: Square wave, High: 3v, Low 0, Offset 1.5v. Frequency: 0.5Hz.

5. Oscilloscope setting to get a low frequency waveform and measurement. The signal can be displayed or not displayed by press button 1 and Button 2 above the scale knob. Press button 1: coupling: DC, invert: off, Probe setup: 1x. Press button 2: same as ch1, except for invert: on. To make signal display correctly on the screen by adjusting the Horizontal Scale (time scale:S) and Vertical Scale (voltage scale: V) knob. If function generator signal is 3V p-p, 1.5 offset, 0.5 Hz, to get maximum display of 2 cycle signal on screen, adjust position: baseline on bottom of screen, vertical scale: 1 v, horizontal scale:400ms, as shown in Figure 1-7.

6. Adjust Attenuator AU150B: dial to 1, capture the response image using excel as shown in Figure 1-10.

7. Repeat step 6 for dial adjusting to 2, and 0.5.

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Figure 1-6 Basic DC Motor Angular Position Control System connection

Figure 1-7 Oscilloscope setting and Screen capture

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Exp#2 Calibration of Input, Output Potentiometers 1) Apply +15 and – 15 volts to the Input and Output pots (150H and 150K) exactly as shown in Figure 1-8, noting the physical 'cross-connection' with respect to the pot terminal polarities*. Rotate the pot shafts until each output is zero volts. [Note that the Output pot shaft can be rotated only by turning the motor shaft which is between DCM150F and GT150X. DO NOT FORCE THE SHAFT WHICH IS CONNECTED TO THE OUTPUT POTENTIOMETER]. Check that the graduated disc attached to the pots indicates zero degree position, and the voltage output of each pots should be zero volt, if not, record the angle and use it as an offset. Don’t force to adjust disc to zero. *Note: This 'cross-connection' is necessary in the final setup (close loop control setup), since both pots are rotated in the same direction, their outputs will be with opposing polarities. Thus, if the outputs are summed (as is done in the lab by the operational amplifier module 150A), the op-amp output indicate the error in angular position between the two potentiometers. The op-amp thus serves as the error detector. If the two pots are physically identical, then setting both to the same angular position should result in zero output from the op-amp. The generation of the error signal is observed in the next step. 2) Rotate input pot shaft in steps and record the output voltage from IP150H #3(CH1 mean), fill out the following table. 3) Repeat step #2 for output pot, rotate motor shaft (not Disc) to change the disc position.

Figure 1-8 Input and output pot calibration setup

Figure 1-9 Input and Output pot

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Input Pot position (150H) 𝜃𝜃𝜃𝜃 ( Degree)

Voltage From #3 Vi CH1 (volt)

-120 -90 -60 -30 -10 0 10 30 60 90 120

Output Pot Position (150K) 𝜃𝜃𝜃𝜃 (Degree)

Voltage from #3 Vo CH2 (volt)

-170 -120 -90 -60 -30 -10 0 10 30 60 90 120 170

*turning the pot clockwise for positive polarity Exp#3 Observation of the Error Signal 1. Zero Op-Amp (Figure 1-10): Connect a Ground (0 volt) signal to one of the op-amp inputs (leave the other two inputs open). Then adjust the “zero set” knob so the output of the op-amp is zero. 2. Remove the ground signal from #1 of Op-amp. Connect the input and output potentiometer as in Figure 1-12. Rotate the output pot shaft approximately to V0= 1 V position. Use DMM to check Vo. 3. With the output potentiometer position left undisturbed, from start point position (0 degree), vary the input pot position by slowly turning the knob and observe the change in Op-amp output. Fill out the following table.

Figure 1-10 Zero Op-Amp

+15

0V -15V

CH2

Figure 1-11 Op-Amp as a Summing and Error Signal Block Diagram

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output pot Vo (DMM)

input Pot Position (Deg)

input Pot Vi CH1

Op-Amp Output Ve CH2

Erro (Cal) Ve= -(Vi+Vo)

Difference of Ve(cal) & Ve(real)

1v -170 1 -90 1 -45 1 0 1 45 1 90 1 170

Experiment Results Exp#1 Basic DC Motor Angular Position Control System Setup

1) Simulate the system using the block diagram as shown in Figure 1.1 by Matlab Simulink. Get 3 step (𝜃𝜃𝑑𝑑=40 deg.) responses of k1=0.5, k1=1, and k1=2 (Assume: 𝐾𝐾𝑖𝑖 = 𝐾𝐾𝑜𝑜 ≈0.088𝑉𝑉𝑜𝑜𝑉𝑉𝑉𝑉

𝑑𝑑𝑑𝑑𝑑𝑑≈ 4.8

𝑟𝑟𝑟𝑟𝑑𝑑, 𝐾𝐾𝑝𝑝 ≈ 10, 𝐾𝐾𝑚𝑚 = 𝐾𝐾𝑀𝑀 ∗ 𝐾𝐾𝑆𝑆𝑑𝑑𝑟𝑟𝑆𝑆 ≈ 20 ∗ 6 = 120 𝑣𝑣𝜃𝜃𝑣𝑣𝑣𝑣/𝑣𝑣𝜃𝜃𝑣𝑣𝑣𝑣, 𝜏𝜏𝑚𝑚 = 0.1𝑠𝑠, N=30).

2) Compare above simulated responses with experimental response and comments.

Exp#2 Calibration of Input, Output Potentiometers 1) Obtain the calibration curve of output voltage versus input angle in both directions from

zero, and hence calculate the sensitivities Ki and Ko of the two pots in volts/rad. If the two values are close to each other, the average value may be calculated.

2) What is the “cross connection” mean in experiment.

Figure 1-12 Op-Amp Calibration and Error Signal connection

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3) Can you explain which is input signal, which is output signal for IP150H. How can you give input, what is the unit, how and where can you get output, and what is the unit.

4) Can you explain which is input signal, which is output signal for OP150K. How can you give input, what is the unit of input, how and where can you get output signal, and what is the unit of output.

Exp#3 Observation of the Error Signal

1) Explain how to check if the op-amp is zero or not. Can you use DMM to do? Please explain in detail.

2) From experimental results table, explain how to make connection to get a signal subtraction. 3) Referring Figure 1-11 and 1-12, if we switch the connection of IP150H (#1 to +15, #2 to -

15). What is the effect on Ve error signal?

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Lab 2: Determination of DC Motor Dead-Band, Gain, Servo-Amplifier Gain, Torque/Speed Characteristic.

Objectives To familiar with the DC motor module, amplifiers and tachometer. To verify DC motor parameters, calibrate the gain of Pre-amplifier, Gain of servo-amplifier and tachometer. Introduction Preamplifier PA150C (Figure 1-6) is a low-power control amplifier which is used to provide the "deadband compensation" voltage, as well as a fixed forward-path gain Kp. The module has two summing input terminals and two output terminals. {An additional input terminal labelled "Tacho" may also be present.} A positive voltage applied to either input yields an amplified positive voltage at the upper output socket(3),the socket(4) staying near zero; a negative voltage applied to either input yields an amplified positive voltage at the lower output socket(4), the socket(3) staying zero. The two output terminals provide the positive voltage drive required as input for the servo-amplifier. Thus, if the output terminals are connected to the servoamplifier input terminals, the motor will reverse direction whenever the preamplifier input voltage changes polarity. With zero input, the voltages at both output sockets must be equal, and this condition must be achieved by adjusting the Zero Set control on the preamplifier. { PROCEDURE: Power on preamplifier. With the input terminals left open circuited, adjust the Zero Set knob until the differential voltage between the two output sockets is zero, i.e., until the voltages at the two output sockets are equalized}. The preamplifier must remain reasonably balanced for proper operation. Maximum output is about 12 volts, and the linear voltage gain is about 10 to 15 V/V. 'Deadband' occurs due to the presence of mechanical static-friction (Coulomb-friction) effects in the commutator brushes and in the bearings. The term 'deadband' which essentially is "the no-response of the motor until the servoamplifier[motor] input voltage Vm, exceeds a certain value Vd " [see Figure 2-2 (a)] occurs in both rotational directions. The 'deadband' prevents the modeling of the servomotor as a linear element. In the experimental equipment, the motor is 'linearized', by providing the servoamplifier input with a bias voltage Vb which is approximately equal to the deadband voltage. The required bias is obtained from a pre-amplifier which has the transfer

Figure 2-1 Preamplifier PA150C

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characteristic shown in Figure 2-2(b). The bias voltage Vb is somewhat less than Vd in order to prevent motor response due to spurious noise signals which may be present in the preamplifier output. At balance, identical output voltages of 1 to 1.5 volts should be obtained.

Servoamplifier SA150D is the power-amplifier which drives the motor. Its panel shows a simplified schematic of the amplifier. The left side of the panel contains two input terminals which accept only positive input signal voltages: A positive input voltage [exceeding the deadband voltage], when applied to one input terminal will rotate the motor in one direction, a similar positive voltage applied to the other terminal will produce reverse rotation. Negative inputs will have no effect. The panel also contains a set of ± 15v terminals which can be used by other units. The servoamplifier is already connected to the power supply unit by a cable, and does not require further power connections.(Figure 2-1) DC Motor DCM150F & Reduction Gear Tacho Unit GT150X consists of a DC motor mechanically coupled to a tachogenerator on high speed input end,( tachometer sensitivity is 0.025 Volts per radian-sec-1 and its output polarity can be reversed by appropriate patching), through a 30:1 reduction gear (a 90° worm gear assembly), to an output shaft on the other end. The output shaft is coupled to the Output Potentiometer through a coupling link. A top panel display can be switched to indicate speed in r/min or to

Figure 2-3 Servoamplifier SA150D

Vin =V1

Vin

=V2

Figure 2-4 DC Motor DCM150F coupled Tacho- gear GT150X, Loading Unit LU150L

Figure 2-2 (a) Motor Deadband Vd (b) Preamplifier Bias Vb

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monitor an external DC voltage. The motor is operated in the armature-controlled mode, through appropriate patch-cord connections made on the Servoamplifier. The motor is already connected to the Servoapmplifier by cables, and does not require further power connections. The motor is a permanent magnet type and has a single armature winding. Current flow through the armature is controlled by power amplifiers as in Figure 2-1 so that rotation in both directions is possible by using one, or both of the inputs. The input signals are provided by a specialized Pre-Amplifier Unit PA150C, which connected to inputs #1 and #2 on SA150D. As the motor accelerates the armature generates an increasing ‘back-emf’ Va tending to oppose the driving voltage Vin. The armature current is thus roughly proportional to (Vin – Va). If the speed drops (due to loading) Va reduces, the current increases and thus so does the motor torture. This tends to oppose the speed drop. This mode of control is called ‘armature-control’ and gives a speed proportional to Vin as in Figure 2-5 a. Due to brush friction, a certain minimum input signal is needed to start the motor rotating. Figure 2-5 b show how the speed varies with load torque.

Loading Unit LU150L An aluminum disc can be mounted on the extended motor shaft and when rotated between the poles of the magnet of the loading unit, the eddy currents generated have the effect of a brake. The strength of the magnetic brake can be controlled by the position of the magnet (Figure 2-4). Figure 2-6 show the approximate brake position characteristics of motor at 1000 rpm. For other speeds, the torque will be proportional to the speed.

Figure 2-5 a DC Motor Deadband Figure 2-5 b DC Motor Speed-Torque Character

Figure 2-6 Approximate Brake Characteristics at 1000rpm

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The armature-controlled DC Motor is used in the laboratory equipment. The motor is driven by a servoamplifier [the combination of the two being called a 'Servomotor’]. The transfer function can be written as follow: 𝐺𝐺𝑃𝑃(𝑆𝑆) = 𝜔𝜔𝑚𝑚

𝑉𝑉𝑚𝑚= 𝐾𝐾𝑚𝑚

1+𝑆𝑆𝜏𝜏𝑚𝑚 (2-1)

where 𝝎𝝎𝒎𝒎 is the output angular velocity, Vm is the motor input voltage(between #3 and #4 on SA150D), Km is the motor gain constant and 𝜏𝜏𝑚𝑚 is an equivalent electro-mechanical time constant. The two characteristic constants in (2-1) can be experimentally determined. The block diagram is shown in Figure 2-7. Experiment Procedure In these experiments, we will calibrate the Pre-amplifier gain, servo-amplifier gain, determine motor dead-band, investigate brake characteristics and servomotor time constant. Exp#1 Determination of Preamplifier Bias and Gain

1. Balance the Preamplifier and determine Preamplifier Bias: Power PA150C, connect a common signal (0V) to the inputs (input 1 and 2). Monitor (using the oscilloscope and its measurement feature, scope vertical position and scale should be same) both outputs (3 and 4), adjust the Balance Control (zero set knob) until both outputs have the same voltage, as shown in Figure 2-8, This voltage should be in the range of 1.0 to 1.5 volts and is the "bias" voltage which is intended for overcoming part of the system Deadband. The zero set knob should not be disturbed after balancing. Record the bias value.

2. Set up the circuit to obtain a small voltage signal: as shown in Figure 2-9 using the Attenuator modules. Note that the pots in the Attenuator are connected in cascade so that very small DC voltages required as input for the gain determination can be easily obtained. Set top pot 1.5 volt (after get 1.5V at #2, don’t touch the top knob any more), then bottom pot will yield an output of 0 to 1.5 volt over its entire knob-rotation range at socket 5. If -15V connect to # 3, then # 5 can obtain an output of 0~ -1.5V.

3. Preamplifier Gain: Disconnect the common signal from the pre-amp input #1, and connect the signal from the bottom pot #5 as show in Figure 2-10. Apply various voltages to the preamplifier input #1, check with scope CH1. Connect PA150C output #3 to scope CH2. Leave input #2 and output #4 unconnected. Vary input signal from 0~1.5 by turning bottom knob, (Oscilloscope: press “measure”, Source: CH1, Type: mean. Second side menu: Source: CH2, Type: mean, properly adjust voltage and time scale to get reading from Scope), record data at the following table. Then disconnect +15V on AU150B #3, and connect -15V to AU150B #3, thus a variable

Figure 2-7 Block Diagram: SA150D+DCM150F+ GT150X

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signal (varies from 0~ -1.5V) can be obtained at #5 on AU150B and at input #1 on PA150C, the amplified signal output to PA150C output #4,CH2 connect to #4 of PA150C, leave input 2 and output 3 unconnected, record data at the following table. (Fig. 2-10 Dash line)

Figure 2-9 Using AU150B get small variable Signal

Figure 2-8 Balance PA150C

Figure 2-10 Preamplifier Calibration Connection

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Input terminal #

Input Voltage CH1 Volt (mean)

Output Terminal #

Output Voltage CH2 Volt (mean)

1 -1.5 (real reading here) 4 1 -1.2 4 1 -1.0 4 1 -0.8 4 1 -0.6 4 1 -0.4 4 1 -0.2 ↑ ( ) 4 1 0

1 0.2 ↓ ( ) 3 1 0.4 3 1 0.6 3 1 0.8 3 1 1.0 3

1 1.2 3 1 1.5 3

Exp#2 Servomotor Gain, Deadband and Tachometer sensitivity Determination 1. Set up the circuit as shown in Figure 2-11. Apply 1.5 ~ 2.5 volts (using top attenuator AU150B)

#3 connect from +15v, #1 connect from 0V, #2 connect to input #1 on Servo-amp SA150D , Scope CH1 measure input voltage Vin at #1 on SA150D. DMM measure motor control voltage Vm between #3 and #4 on SA150D.

2. GT150X connection: #1 from 0V, #2 connects to #3, switch turn towards to #3 for display n (rpm) on LED, Scope CH2 measure tachometer voltage at #2.

3. Set Load unit LU150L at 0 position (unload status). 4. Gradually increase (adjust AU150B) input voltage allowing the motor to start to turn. Note that

the motor does not respond until the input voltage exceeds a certain threshold value Vd, which is the deadband voltage for one direction. Continue to increase the input voltage approximate to 1.1V (read mean from CH1 of scope Vin), read Vm from DMM, read n (rpm) on the LED display, read CH2 mean volts Vt and record all values on the following table.

5. Repeat step 3, increase input voltage approximate to 2.0V. 6. Disconnect #1 on SA150D, connect to #2 instead, the motor will run in opposite direction, repeat

step 4-5. Servo – Amp Tachometer Terminal #

Vin Volts (CH1) mean

Vm Volts (DMM)

n rpm (read LED)

𝜔𝜔=2𝜋𝜋𝜋𝜋/60 ( Rad/S)

Vtg Volts (CH2)mean

𝐾𝐾𝑣𝑣𝐾𝐾 = 𝑉𝑉𝑣𝑣/𝜔𝜔 Volts.S/rad

#1 (≈ 1.1) #1 (≈ 2.0) #2 (≈ 1.1) #2 (≈ 2.0)

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7. Put Load unit LU150L at position#10, repeat from above step #4, adjust Vin at 1.1V, 2v. fill the

following table Servo – Amp Tachometer Terminal #

Vin Volts (CH1) mean

Vm Volts (DMM)

n rpm (read LED)

𝜔𝜔=2𝜋𝜋𝜋𝜋/60 ( Rad/S)

Vtg Volts (CH2)mean

𝐾𝐾𝑣𝑣𝐾𝐾 = 𝑉𝑉𝑣𝑣/𝜔𝜔 Volts.S/rad

#1 (≈ 1.1) #1 (≈ 2.0) #2 (≈ 1.1) #2 (≈ 2.0)

Exp#3 Torque speed Characteristics investigate 1. Set up the circuit as shown in Figure 2-11. Follow the steps 1-2 of Exp#2.2. 2. Set Load Unit at position #0. Gradually increase (adjust AU150B) input voltage, read LED,

make speed n reach to maximum speed. Record all data in follow table of position #0. 3. Keep input voltage unchanged, set load unit LU150L at position #1, record all data, repeat until

position #10.

Figure 2-11 Servo-motor gain, DC Motor Deadband, gain, Time constancy, and load characteristics investigate connection

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Load Position

Servo – Amp Tachometer Vin Volts

(CH1) mean Vm Volts (DMM)

n rpm (read LED)

𝜔𝜔=2𝜋𝜋𝜋𝜋/60 ( Rad/S)

Vt Volts (CH2)mean

0 1 2 3 4 5 6 7 8 9 10

4. Set Load unit LU150L at position#0, decrease input voltage Vin until reach to above speed at

position#10(if above table at position#10 is 900rpm,adjust vin at position #0 until rpm reach to 1000), record all data in following table, repeat until position #10.

Load Position

Servo – Amp Tachometer Vin Volts

(CH1) mean Vm Volts (DMM)

n rpm (read LED)

𝜔𝜔=2𝜋𝜋𝜋𝜋/60 ( Rad/S)

Vt Volts (CH2)mean

0 1 2 3 4 5 6 7 8 9 10

5. Repeat step 4: set load unit LU150L at position#0, adjust Vin until rpm reach to the speed at

last position #10, record all data in following table, repeat until position #10. Load Position

Servo – Amp Tachometer Vin Volts

(CH1) mean Vm Volts (DMM)

n rpm (read LED)

𝜔𝜔=2𝜋𝜋𝜋𝜋/60 ( Rad/S)

Vt Volts (CH2)mean

0

1 2 3 4 5 6 7 8 9 10

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Exp#4 Servomotor Time Constant Determination:

1. Function Generator and Oscilloscope Setting: 1) The function generator is used to get a square-wave signal. Adjust Frequency(Period,

press again to switch to Period), Amplitude(HiLevel), Offset, (LoLevel) and Duty Cycle of these signals. Signal: Frequency: 0.3Hz; HiLevel:2.0v; Lolevel: 0v; Offset:1.0v.

2) Use the circuit shown in Figure 2-11. Function generator(dash line) replace AU150B #2(Green line)

3) Set Oscilloscope to operate in Roll Mode(400ms/div ~5 sec/div) which produce a scrolling trace. Adjust the position knob of CH1 and CH2 of DSO to bottom position(baseline at bottom) . Scope setting: Scale CH1: 500mv;CH 2:1.00V; Horizontal scale: 40ms.

2. Set load unit LU150L at position #0. Use the circuit shown in Figure 2-11. Squire wave signal replace AU150B #2, connect to #1 on Servo-Amp SA150D as shown in dashed line

3. Power on motor make sure motor run in one direction and full stop periodically. A trace of squire wave will roll on screen. Adjust oscilloscope Vertical Scale(volts/div) and time scale (sec/div) controls until the positive-going half-cycle of the square wave appears as a 'step' in the display [see Figure bellow]. Press Run/Stop ** button in Scope to display the rising wave form. Use the paired cursors to graphically determine the servomotor time constant 𝜏𝜏𝑀𝑀

by reading off the time corresponding to 63.2 % of the 'final' value*. Draw the display and mark the time constant and final value, fill out the following table.

4. Connect function generator to #2 of Servo-Amp and to make sure motor run in reverse direction and full stop periodically. Repeat the steps #2). Fill out table’s second row. Load unit set at position #0

5. Repeat Step #1, Set load unit LU150L at position #10, repeat step #2 and #3.fill following

table. Load unit set at position #10

V Function Gen. Vin (CH1) Tachometer (Vt CH2) Time Constant 𝜏𝜏𝑚𝑚 0~2 V -2 ~0V

Experiment Results Exp#1 Determination of Preamplifier Bias and Gain

1) Can you explain which terminal is input, which is output if we use AU150B as a voltage divider. Do we need a ground, explain why.

2) Obtain a plot of output voltage (terminal 3) versus input voltage from 0~ +1.0 V (input terminal #1). Next, on the same graph, plot other output (terminal #4) versus Vin from 0 ~ -1.0V (input terminal #1). A V-shaped characteristic will result if the preamplifier has been

V Function Gen. Vin (CH1) Tachometer (Vt CH2) Time Constant 𝜏𝜏𝑚𝑚 0~2 V -2~0V

25

well balanced. Find Kp (slope of plot). Find preamplifier bias voltage when Vin =0 from the plots.

Exp#2 Servomotor Deadband and Gain Determination

1) From the table, calculate 𝜔𝜔 and Kt, 𝜔𝜔 = 2𝜋𝜋𝜋𝜋60

(Rad/s), 𝐾𝐾𝑣𝑣 = 𝑉𝑉𝑉𝑉𝜔𝜔

(𝑉𝑉𝜃𝜃𝑣𝑣𝑣𝑣. 𝑆𝑆𝑅𝑅𝑟𝑟𝑑𝑑

) . 2) Plot 𝜔𝜔 versus Vm in both input terminal #1 and #2 in one graph. Find Vd (deadband) from

this plot. 3) Plot 𝜔𝜔 versus Vin, find Km for position#0. What is the difference compare with no

load(position#0), why. 4) Find Km for Position#10. What is the difference compare with no load(position#0), why. 5) Can you derive a block diagram model of armature-controlled DC motor, includes a load

torque in your block. (referring to Appendix B for all parameters). Clear indicate Km, 𝜏𝜏𝑚𝑚. 6) Run matlab Simulink with 5) block diagram.

Exp#3 Torque speed Characteristics

1) Plot 𝜋𝜋 (𝑟𝑟𝑟𝑟𝑚𝑚) versus position # to get the speed toque characteristics from all setting. (plot three curve in one graph).

Exp#4 Servomotor Time Constant Determination

1) From the recorded table and plot, get average time constant.

2) Find servomotor gain from the final value and input square-wave amplitude. Refer to (2-1) and Figure 2-7,

𝑉𝑉𝑉𝑉𝑉𝑉𝑖𝑖𝜋𝜋

= 𝐾𝐾𝑚𝑚∗𝐾𝐾𝑉𝑉1+𝑆𝑆𝜏𝜏𝑚𝑚

, when t∞, S0, 𝐾𝐾𝑚𝑚 = 𝑉𝑉𝑉𝑉𝑉𝑉𝑖𝑖𝜋𝜋∗𝐾𝐾𝑉𝑉

. Compare this Km with

obtained from Exp#2.2(Question# 4).

3) Can you explain why do you get different time constant and motor constant at different load?

4) Using above obtained parameters to simulate the DC motor with Matlab Simulink. Compare

with experimental plot, with #2.2 –6) Simulink result.

26

Lab 3: Time Response of Basic Closed-Loop System and Effect of Tachometer Feedback

Objectives To observe the time response of the closed-loop DC motor position control system, investigate the performance of second order system, and effect of tachometer feedback on the second-order system response. Introduction Basic Angular Position Control System: The block diagram of the basic system which is investigated is shown in Figure 3-1. The speed reducing gear coupled at the output shaft of the motor is represented as a block having the transfer function (1/SN), to indicate speed reduction as well as angular velocity - to - position conversion. Ki and Ko are the transfer functions of the Input and Output potentiometers, Kp is the pre-amp gain, and Km is the servo-motor gain respectively, which were obtained by calibration in the previous lab.

The op-amp 150A is used to sum multi-signals as “Reference Comparator” or “Error Detector”. The error voltage Ve is the difference between desired voltage Vi and real voltage Vo (or = 𝐾𝐾𝑖𝑖𝜃𝜃𝑖𝑖 −𝐾𝐾𝑜𝑜𝜃𝜃𝑜𝑜 ). The close loop transfer function of the system of Figure 3-1 (including Ki) may be obtained as:

150 X Gear Box

Figure 3-1 Basic Angular Position Control System

𝜽𝜽𝒅𝒅 𝜽𝜽𝒐𝒐

𝑲𝑲𝒎𝒎

𝟏𝟏 + 𝑺𝑺 𝝉𝝉𝑴𝑴

27

𝜃𝜃𝑜𝑜𝜃𝜃𝑖𝑖

=(𝐾𝐾𝑖𝑖𝐾𝐾𝑜𝑜

)𝐾𝐾𝑀𝑀𝐾𝐾𝑜𝑜𝑁𝑁𝜏𝜏𝑚𝑚

𝑆𝑆2+ 𝑆𝑆𝜏𝜏𝑚𝑚

+𝐾𝐾𝑀𝑀𝐾𝐾𝑜𝑜𝑁𝑁𝜏𝜏𝑚𝑚

=(𝐾𝐾𝑖𝑖𝐾𝐾𝑜𝑜

)𝜔𝜔𝑛𝑛2

𝑆𝑆2+2𝜁𝜁𝜔𝜔𝑛𝑛𝑆𝑆+𝜔𝜔𝑛𝑛2 (3-1)

Where the Natural Frequency is 𝜔𝜔𝜋𝜋 = �𝐾𝐾𝑀𝑀𝐾𝐾𝑜𝑜𝑁𝑁𝜏𝜏𝑚𝑚

(3-2)

𝐾𝐾𝑀𝑀 = 𝐾𝐾𝑝𝑝𝐾𝐾𝑚𝑚, and the Damping Ratio is 𝜁𝜁 = � 𝑁𝑁4𝐾𝐾𝑜𝑜𝐾𝐾𝑀𝑀𝜏𝜏𝑀𝑀

(3-3)

Transient time response specifications to a step input are defined as follows (refer to Figure 3-2):

Period Time: 𝑇𝑇 = 2𝜋𝜋

𝜔𝜔𝑑𝑑 where the damped natural frequency is 𝜔𝜔𝑑𝑑 = 𝜔𝜔𝜋𝜋�1 − 𝜁𝜁2 .

Rise time, tr: the time required for the response to rise from 0 ~ 100% of its final value for a underdamped second-order system.

Peak Time: 𝑣𝑣𝑝𝑝 = time taken to reach the first maximum, 𝑣𝑣𝑝𝑝 ≈𝜋𝜋𝜔𝜔𝑑𝑑

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

tr

tp

T

X1

X2

X3

P.O

𝜽𝜽(𝒕𝒕)

𝒕𝒕

𝜽𝜽𝟏𝟏

𝜽𝜽𝜽𝜽

𝜽𝜽𝜽𝜽

2%

ts

Figure 3-2 Time Response of Basic Angular Position control System

O.S

28

Percent Overshoot (P.O.): the maximum peak value of the response curve measured from unity.

𝑃𝑃.𝑂𝑂 = 100𝑒𝑒− 𝜋𝜋𝜋𝜋

�1−𝜋𝜋2 (3-4) Settling Time: the time required for the response curve to reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually 2% or 5%).

𝒕𝒕𝒔𝒔 ≈ 𝟒𝟒/𝜻𝜻𝝎𝝎𝒏𝒏 , (2% settling time)

The time-domain specifications are quite important since most control systems must exhibit an acceptable time response. Except for certain applications where oscillations cannot be tolerated, it is desirable that the transient response be sufficiently fast and be sufficiently damped. Thus, for a desirable transient response of a second-order system, the damping ratio must be between 0.4 and 0.8. Small values of 𝜁𝜁 (𝜁𝜁 < 0.4) yield excessive overshoot in the transient response and a system with a large value of 𝜁𝜁 (𝜁𝜁 > 0.8) responds sluggishly. An overshoot in the range of 2 to 6% is considered to be the optimum, a ‘range’ being necessary because setting the P.O. may involve a ‘trade-off’ with other specifications.

Note that an increase in KM, while providing an increase in the natural frequency (ie. speed of response or rise-time), will also result in a reduction in the damping ratio, thereby increasing the tendency towards instability (ie. larger overshoot and settling time). Thus, a 'trade-off’ exists between, say, the rise-time and the settling time. Furthermore, in the experimental setup, all of the above system parameters are constant and any adjustment capability can only be obtained through an effective variation in the forward-path gain. In the experimental setup, such a gain- variation is obtained by an attenuator which is ahead of the pre-amplifier (see Figure 3-3) to effectively reduce the gain of that amplifier (ie. 0< overall forward-path gain ≤ KM , in our case, KM=K1KpKm , where K1 is the potentiometer constant which was calibrated in previous lab and ranged from 0 to 1)

Basic angular position control system with velocity feedback:

The restrictive trade-off situation between 𝜁𝜁 and 𝝎𝝎𝒏𝒏in the basic system described above may be somewhat improved by using additional 'derivative feedback'. In obtaining the derivative of the output position signal, it is desirable to use a tachometer instead of physically differentiating the output signal. In our lab, the angular velocity of the motor 𝜔𝜔𝑚𝑚 (Tachometer feedback or Velocity feedback or Rate feedback) is introduced. In the laboratory system, a ‘tachogenerator’ (Tachometer) is physically coupled to one end of the motor. It produces a DC voltage output 𝑉𝑉𝑉𝑉 =𝐾𝐾𝑉𝑉𝜔𝜔𝑚𝑚 , which is used as an additional negative feedback signal as shown in Figure 3-3.

This system can be shown to have the same transfer function given by Eqn. (3-1) where 𝜔𝜔𝜋𝜋 remains unchanged but with 𝜁𝜁 now given by:

𝜁𝜁 = � 𝑁𝑁4𝐾𝐾𝑜𝑜𝐾𝐾𝑀𝑀𝜏𝜏𝑀𝑀

(1 + 𝐾𝐾𝑀𝑀𝐾𝐾2𝐾𝐾𝑉𝑉) (3-5)

29

The Damping Ratio is now multiplied by the factor (1 + KMK2Kt). Thus, 𝜁𝜁 can now be independently set for any given 𝝎𝝎𝒏𝒏 . In the laboratory setup, an attenuator (with pot constant k2) is used in cascade with the tachometer output, so that an effective adjustment range for Kt from zero to its full value is possible. For the basic system, optimum* step response should normally occur with the pot coefficients k1=0.4 and K2 = 0.02, respectively. It can be seen that velocity-feedback improves stability by introducing extra damping.

Experiment Procedure

Pre-lab: Please review Lab#1 and Lab #2 and connect a basic angular position control system with velocity feedback as shown in Figure 3-3. The lab equipment layout is shown in Figure 1-2. All the +15V and -15V and 0V voltage will be connected in the lab. The power supply PS150E, servo-amplifier SA150D and DC motor DMC150-F are internal connected. Please note that input pot IP150H and output pot OP150K must be cross connected (referring Figure 1-13). Make sure Ve is a voltage difference and not a voltage sum. The reference voltage of IP150H and OP150K is ±15V. Exp #1 Basic Closed-loop System Set Up Notes: Throughout the following experiments, it will be assumed that the op-amp and the pre-amp remain zeroed and balanced respectively and that the supplies to the input/output pots are cross-connected so that the op-amp is the difference between the input and output position signals. 1) Without power on, set up circuit as in Figure 3-3. Set the input and output pots to their mid-positions, indicating approximately zero output voltages. Also set the two pots in the Attenuator unit to K1 =0.5, K2 = 0. 2) Offset the reference input pot by about 30° and turn the power on, the output pot will rotate following the reference pot position if the system is functioning as a negative feedback system. If it does not, then the feedback signal polarities of position are incorrect and must be reversed as

Figure 3-3 Basic Angular Position Control System with Velocity Feedback

30

required until the system shows the proper position following response. (Letting K2=0, check position feedback first. Switch +15,-15 connect of IP150H to make sure system is controllable and stable. Then add K2=0.5, if system is unstable, switch the polarities of GB150X.) 3) Disconnect input pot IP150H terminal #3 from terminal #1 of OA150A, connect function generator to terminal #1 of OA150A. Set square wave, 4V pk-pk, frequency 0.3 Hz. Connect scope CH1 to Vi, (terminal #1 of OA150A), CH2 to Vo (terminal #2 of OA150A). Set the DSO time base to produce a scrolling trace (roll mode). Now observe the responses to step input with various settings of K1 and K2 which are the two control pots in the Attenuator module. Note that K1

effectively sets the forward-path gain (from zero to 1) while K2 sets the magnitude of the tachometer feedback signal. 4) With K2 remaining at zero (thereby removing the velocity-feedback loop): increase K1 in steps and observe the change in the transient response. Capture input and output in one plot for same K2=0 but with K1=0.1, 0.2, 0.4. Try to measure Tp, T and P.O for each case. (using run/stop and Cursor measurement). 5) Refer to Figure 3-2. record all data, capture the response image and sign data sheet by Lab

instructor before leave. Exp #2 Closed-loop System with Tachometer Feedback 1) Keep same procedure as Exp#1 steps 1) -3). 2) With K1 set at maximum (=1), observe the change in transient response as the tachometer (velocity) feedback is gradually introduced by increasing K2. Using RUN/STOP , capture plot for same K1=1, with K2=0.1, 0.2, 0.3 or 0.05(if the system shows too sluggish). Also try to measure Tp, T and P.O for each case. Exp #3 Closed-loop System Time Response 1) Keep same as Exp#1 steps 1)—3). 2) Select k1 and K2 which yields what you consider to be the 'best' step-response (approximately

10% of overshoot). From the displayed 'best' response curve, use the DPO cursors to graphically determine the Percentage Overshoot and use it to estimate the damping ratio. Capture image of this 'best' response for the report.

Experiment Results Exp#1 Basic Closed-loop System Set Up Write a summary of your observations in your report. Calculate 𝜔𝜔𝑑𝑑, 𝜔𝜔𝜋𝜋 and 𝜁𝜁 from your recorded T, Tp, and P.O for each case. Comment on your results. Exp#2 Closed-loop System with Tachometer Feedback Write a summary of your observations in your report. Calculate 𝜔𝜔𝑑𝑑, 𝜔𝜔𝜋𝜋 and 𝜁𝜁 from your recorded T, Tp, and P.O for each case. Pay attention to 𝜔𝜔𝜋𝜋 . Comment on your results.

31

Exp #3 Closed-loop System Time Response 1) Estimate two 𝜁𝜁 from your best response experiment by equation (3-4). Compare them with value calculated using equation (3-5)

𝜁𝜁 = � 𝑁𝑁4𝐾𝐾𝑜𝑜𝐾𝐾1𝐾𝐾𝑀𝑀𝜏𝜏𝑀𝑀

(1 + 𝐾𝐾1𝐾𝐾𝑀𝑀𝐾𝐾2𝐾𝐾𝑉𝑉)

where the selected values of K1 and K2 have been introduced to take into account the effective modified values of the gain KM= Kp*Km and the tachometer sensitivity Kt. Tabulate the results of your comparison. N = 30 is the output shaft gear ratio. You can find all the other parameters in your former experiment results. 2) Use Matlab Simulink to simulate the system as in Figure 3-3. K1 and K2 are the two values used in the lab (Exp#3, step #3, one is 10% of overshoot, other is 20% of overshoot). N = 30 is the output shaft gear ratio. You can find all other parameters in your former experiment results. Plot the simulated results and check the P.O. in graphs. Compare them with your experimental plot.

32

Lab 4: Frequency Response of Basic Closed-Loop DC Motor System

Objectives To study the frequency response of a basic closed-loop DC motor system by observing its natural response, and compare the experimental response with computer simulation response. Introduction The frequency response means the steady state response of a system to a sinusoidal input. The resulting output for a closed loop DC motor system is sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle. Consider the DC motor described by Equation (3-1),

𝜃𝜃𝑜𝑜(𝑠𝑠)𝜃𝜃𝑖𝑖(𝑠𝑠)

= 𝜔𝜔𝑛𝑛2

𝑆𝑆2+2𝜁𝜁𝜔𝜔𝑛𝑛𝑆𝑆+𝜔𝜔𝑛𝑛2 = 𝐺𝐺(𝑠𝑠) (4-1)

The input 𝜃𝜃𝑖𝑖(t) is sinusoidal and is given by:

0 10 20 30 40 50 60-4

-3

-2

-1

0

1

2

3

4

𝜃𝜃𝜃𝜃(𝑣𝑣)

𝜃𝜃𝜃𝜃(𝑣𝑣)

𝜃𝜃𝜃𝜃

Figure 4-1 Frequency response of closed loop system

𝜃𝜃(𝑣𝑣)

𝒕𝒕

𝜃𝜃𝜃𝜃

t T

33

𝜃𝜃𝜃𝜃(𝑣𝑣) = 𝜃𝜃𝑖𝑖 sin𝜔𝜔𝑣𝑣

If the system is stable, then the output 𝜃𝜃𝜃𝜃(𝑣𝑣) can be given by

𝜃𝜃𝜃𝜃(𝑣𝑣) = 𝜃𝜃 𝑑𝑑|𝐺𝐺(𝑗𝑗𝜔𝜔)| sin(𝜔𝜔𝑣𝑣 + 𝜙𝜙) Where 𝜙𝜙 = ∠G(jω) = tan−1 𝑖𝑖𝑚𝑚𝑟𝑟𝑑𝑑𝑖𝑖𝜋𝜋𝑟𝑟𝑟𝑟𝑖𝑖 𝑝𝑝𝑟𝑟𝑟𝑟𝑉𝑉 𝑜𝑜𝑜𝑜 𝐺𝐺(𝑗𝑗𝜔𝜔)

𝑟𝑟𝑑𝑑𝑟𝑟𝑉𝑉 𝑝𝑝𝑟𝑟𝑟𝑟𝑉𝑉 𝑜𝑜𝑜𝑜 𝐺𝐺(𝑗𝑗𝜔𝜔)

We can present frequency response characteristics in graphical forms, Bode Diagrams or Logarithmic Plots. A Bode Diagram consists of two graphs: one is a plot of the logarithm of the magnitude of a sinusoidal transfer function (20 log |𝐺𝐺(𝑗𝑗𝜔𝜔)| ), or called dB; the other is a plot of the phase angle (deg) or phase shift; both are plotted against the frequency in logarithmic scale.

An example of input and output sinusoidal waveform is shown in Figure 4-1. The output/input magnitude ratio: M (dB) =20 log |𝐺𝐺(𝑗𝑗𝜔𝜔)| = 20 𝑣𝑣𝜃𝜃𝐾𝐾 �𝜃𝜃𝑜𝑜

𝜃𝜃𝑖𝑖� (4-2)

100

101

102

-180

-135

-90

-45

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

-40

-30

-20

-10

0

10

20System: DCFrequency (rad/sec): 10.6Magnitude (dB): 10.3

System: DCFrequency (rad/sec): 15Magnitude (dB): 0.00225

Mag

nitu

de (d

B)

Figure 4-2 Bode diagram of closed loop DC motor System

Mp (in dB)

𝜔𝜔𝑝𝑝

34

Phase shift: 𝜙𝜙(degree) =− 360𝑣𝑣𝑇𝑇 (4-3)

Figure 4-2 shows the Bode Diagram of closed loop DC motor system. We can estimate the underdamped natural frequency 𝜔𝜔𝜋𝜋 and damping ratio 𝜁𝜁 by the asymptotic lines from Bode diagram. 𝑀𝑀𝑝𝑝(𝜃𝜃𝑑𝑑) = 20 log 1

2𝜁𝜁�1−𝜁𝜁2 , 𝜁𝜁 < 0.707 (4-4)

Experiment Procedure Exp#1 Set up circuit shown in Figure 3-3, make sure system is controllable and stable, use function generator to replace input pot 150H. Function generator setting: Squire wave, 4 volts peak-to peak, 0.5Hz, offset 0v. Get step response by setting K1, K2 which yielded your good step-response (25% of overshoot). Set the oscilloscope to read DC at 1 volt/div and adjust the sec/div setting until the waveform is scrolling, 100-500ms/div. Turn on the 'invert' for Ch2 of scope. Exp#2 After getting 35% of overshoot, keep all setting unchanged, only adjust function generator controls to obtain a sine wave output. Exp#3 Keep the peak-to-peak input voltage magnitude unchanged, manually change frequency from 0.1 to 10 Hz, find the resonance peak, more readings will have to be taken near the peak so that it is well defined in a plot. Conversely, less readings may be taken in regions where the response is 'flat'. At each frequency fin, 'freeze' the signal and use cursors to find the input period T, pk-pk 𝜃𝜃𝜃𝜃, pk-pk 𝜃𝜃𝜃𝜃, and the phase-shift time t between input and output waveforms. Tabulate the results as shown in Table 4-1. Experiment Results Exp#1 Calculate the output/input Magnitude Ratio M (dB) and the Phase shift Ф (degrees) at each frequency and put them into the table above. For an input 𝜃𝜃𝜃𝜃(𝑣𝑣) and an output 𝜃𝜃𝜃𝜃(𝑣𝑣) which lags the input, the Ф (degrees) and M (dB) may be calculated by (4-3) and (4-4). Plot the Magnitude Ratio M (dB) and the Phase-lag Ф (degrees) against the radian-frequency ω=2πf, using two-cycle, semi-logarithmic graph paper. Example M and Ф plots are shown in Figure 4-2. Typical data points are also shown to emphasize the need to take more readings at frequencies where rapid changes occur. Note: A distinct peak will not be obtained if the system is set for near critical damping.

Exp#2 Estimation of undamped natural frequency ωn and damping ratio ζ from the resulting frequency response plot. The magnitude (dB) – frequency data points plotted on semi-log graph paper can be used to obtain system parameters such as ζ and ωn, as shown in Figure 4-2. Use asymptotic lines to estimate the ωn and the peak (if any) to find the ζ by (4-4). Compare the result of ζ with the corresponding value calculated earlier in Lab#3.

Exp#3 Use Matlab M scripts to plot the Bode diagram given the system parameters as in the previous lab. The K1 and K2 are chosen in this lab. Find the 𝜔𝜔𝜋𝜋 and 𝜁𝜁 from the Bode diagram.

35

Table 4-1

Fin (Hz)

T (sec)

t (sec)

𝜃𝜃𝜃𝜃 (pk-pk) (volt)

𝜃𝜃𝜃𝜃 (pk-pk) (volt)

𝜔𝜔 (rad/S)

M ( dB)

Phase (deg)

0.1 0.5 1 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.5 4.0 5.0 6.0 7.0 8.0 9.0 10.0

36

Lab 5: DC Motor Position Control with Cascade PID Compensation

Objectives To investigate PID controller and cascaded PID with tachometer feedback, compare the experimental response with computer simulation response. Introduction 'Compensation' is the modification of system (plant) performance characteristics so that they conform to certain desired specifications. This is accomplished by effectively changing the transfer function (more specifically, the OLTF) of the system, by introducing a ‘compensator’ block at some suitable point in the closed loop. The compensator is usually located near the input comparator, since the signal levels are low there and hence the compensator can be a low-power device. In cascade Proportional-Integral-Derivative (PID) compensation, the time-integral and time-derivative of the comparator output are obtained and added to that output itself and the composite signal is used as the actuating signal (refer to Figure 5-1).

In the laboratory setup, a Proportional-Integral-Derivative amplifier unit (called PID unit PID150Y) is used in the forward path, following the reference comparator, for the investigation of cascade compensation. The Proportional-Integral-Derivative unit PID150Y is a three-mode control amplifier. It provides three operational paths (P+I) or (P+D) or (P+I+D). The block diagram is shown in Figure 5-2. Switching possibilities can be readily seen on the simplified schematic shown on the faceplate of the unit (see Figure 5-2).

This amplifier has the following transfer function:

Gc(S) = K [1 + (1/sTi) + sTd] (5-1) where the proportional gain K and the integral and derivative time constants Ti and Td can be varied over specified ranges by means of three calibrated knobs on the unit. [The gain K can be varied from 0.11 to 11 in two decade ranges. The Integral Time Constant Ti can be set from 0.11 to 11 seconds (in two decade ranges) and the Derivative Time Constant Td can be set from 2 milliseconds to 220 milliseconds in two ranges. Also, the Integral and Derivative functions can be independently switched on or off as required.]

37

Figure 5-1 DC Motor Position Control with Cascade PID + Velocity Feedback Compensation

Ki

150H

150A

PID

150Y PID

𝑲𝑲𝒎𝒎

𝟏𝟏+𝑺𝑺𝝉𝝉𝑴𝑴

150D +150F

𝟏𝟏𝑺𝑺𝑺𝑺

150X

Kt

150X

K2

150B

Ko

150K

𝜽𝜽𝒅𝒅 𝜽𝜽𝒐𝒐

𝝎𝝎𝒎𝒎

𝑽𝑽𝒆𝒆 𝑽𝑽𝒊𝒊

− −

𝑽𝑽𝒐𝒐

𝑽𝑽𝒕𝒕

Kp

150C

Vc

150C

Figure 5-2 PID150Y Module

38

Now consider the system with the PID unit in the forward path, but with the tachometer feedback removed (with the PID module parameters K, Ti and Td set, K2=0, this will correspond to cascade PID compensation).

Using 𝐺𝐺 = 𝐺𝐺𝑐𝑐𝐺𝐺𝑝𝑝 =𝐾𝐾𝐾𝐾𝑀𝑀[1+ 1

𝑠𝑠𝑇𝑇𝑖𝑖+𝑠𝑠𝑇𝑇𝑑𝑑]

𝑠𝑠𝑁𝑁(1+𝑠𝑠𝜏𝜏𝑀𝑀) and 𝐻𝐻 = 𝐾𝐾𝑂𝑂 , the CLTF is given by 𝑇𝑇 = 𝐾𝐾𝑖𝑖

𝐺𝐺1+𝐺𝐺𝐺𝐺

.

The equivalent unit-feedback transfer function 𝐺𝐺𝑢𝑢𝑜𝑜𝑠𝑠𝑐𝑐 = 𝑇𝑇1−𝑇𝑇

may be found, assuming 𝐾𝐾𝑖𝑖𝐾𝐾𝑜𝑜

= 1 by:

𝐺𝐺𝑢𝑢𝑜𝑜𝑠𝑠𝑐𝑐(𝑠𝑠) =𝐾𝐾𝐾𝐾𝑜𝑜𝐾𝐾𝑀𝑀𝑇𝑇𝑑𝑑[𝑠𝑠2+ 1

𝑇𝑇𝑑𝑑𝑠𝑠+ 1

𝑇𝑇𝑖𝑖𝑇𝑇𝑑𝑑]

𝑠𝑠2 𝑁𝑁(1+𝑠𝑠𝜏𝜏𝑀𝑀) (5-2)

Equation (5-2) clearly shows that (a) the system ‘Type’ has been changed to Type 2, and (b) a pair of zeros has been introduced. ie: the system will now have zero steady state error for both step and ramp inputs. However, its transient response will depend on the location of the roots of the system characteristic equation [ie: closed loop poles].

Experiment Procedure Pre-lab: Please review Lab#1, Lab#2 and Lab#3, connect a basic angular position control system with PID controller and cascade velocity feedback as shown in Figure 5-1. The lab equipment layout is shown in Figure 1-6. All the +15V and -15V and 0V voltage have to be connected. The power supply PS150E, servo-amplifier SA150D and DC motor DMC150-F are internal connected. Please note that input pot IP150H and output pot OP150K must be cross connected (referring Figure 1-11), make sure Ve is a voltage difference not a voltage sum. The reference voltage of IP150H and OP150K is ±15V. Expt. #1 Dc Motor Position Control With PID Compensation Notes: Throughout the following experiments, it will be assumed that the op-amp and the pre-amp remain zeroed and balanced respectively and that the supplies to the input/output pots are cross-connected so that the op-amp is the difference between the input and output position signals. Check system is controllable and stable, then replace input pot with function generator. Set up the circuit shown in Figure 5.1. Set K2 to zero to eliminate the velocity feedback. Adjust the function generator controls to obtain a square wave output of about 4 volts peak-to-peak, symmetrical about the zero volt baseline, at approximately 0.4 Hz. Notes: For the following each steps, record your observations (Stop/Run scope, using curser measurement to get Peak time Tp, overshoot P. O., steady state error ∆𝜃𝜃𝑒𝑒. capture display, or drawing at a blank paper, for your report), and comment on them. 1. Proportional Compensation: switch out the Integral (Ti=∞) and Derivative (Td=0) paths, and switch in only the Proportional path. Observe the change in 'step' (square-wave) response of the

39

angular-position output as the proportional compensator gain K is varied from 0.1 to 1. Momentarily switch the input waveform to a triangular-wave and observe the change in the "follower" (ramp) response as K is changed. (Let: K=0.1, 0.2) 2. Proportional-Integral Compensation*: Switch in the Integral path. Set the proportional gain K=0.1, and observe the effect on the output responses for square wave and triangular wave input when Ti is set to various values, Let ( 𝑇𝑇𝑖𝑖 = 0.5, 1, 10 ). 3. Proportional-Derivative Compensation*: With K=0.1, switch out the Integral path and switch in the Derivative path instead. Observe the effect on the output responses for square wave and triangular wave input when Td is set to various values. Decrease the proportional gain if necessary, to reduce noise. Let ( 𝑇𝑇𝑑𝑑 = 2, 20 , 200𝑚𝑚𝑠𝑠 ). 4. Proportional-Integral-Derivative Compensation*: Next, switch in the Integral path again. The compensation is now a "PID". Observe the effect on the 'step' (square-wave) response and triangular-wave response when the gain K, Ti and Td are set to various values. Let a) (K=0.1, 𝑇𝑇𝑖𝑖 = 0.5, 𝑇𝑇𝑑𝑑 = 20 𝑚𝑚𝑠𝑠 ).

b) (K=0.1, 𝑇𝑇𝑖𝑖 = 10, 𝑇𝑇𝑑𝑑 = 20 𝑚𝑚𝑠𝑠 ). c) (K=0.1, 𝑇𝑇𝑖𝑖 = 0.5 , 𝑇𝑇𝑑𝑑 = 200 𝑚𝑚𝑠𝑠 ). d) (K=0.1, , 𝑇𝑇𝑖𝑖 = 10, 𝑇𝑇𝑑𝑑 = 200 𝑚𝑚𝑠𝑠 ).

Expt. #2 DC Motor Position Control With PID Compensation and Tachometer feedback Cascade Compensation with Tachometer feedback: Velocity feedback compensation can be introduced in addition to any of the cascade compensation schemes given in Expt#1,steps 1, 2, 3 and 4 above, by means of pot coefficient K2. Note that the tachometer feedback is now applied directly to the preamplifier (PA150C, input#2) in an internal loop which is also called a "minor" feedback loop. Observe the effect of increasing K2 in each of the above cases. Record your observations and comment on them.

a) P+ Tach: (K=0.1, K2=0) ;(K=0.1, K2=0.1); (K=0.1, K2=0.2) b) PI+Tach: (K=0.1, 𝑇𝑇𝑖𝑖 = 10 𝐾𝐾2 = 0) ; (K=0.1, 𝑇𝑇𝑖𝑖 = 10 𝐾𝐾2 = 0.1) ; (K=0.1, 𝑇𝑇𝑖𝑖 = 10 𝐾𝐾2 =

0.2)

Setting K, Ti, Td a. Step Response Tp, P.O. , S.S. e b. Ramp Response 1. P a. Squire f=200mhz v=4v p-p offset=0 b. Tran.

K=0.1

P K=0.2 PI

40

Experiment Results Expt. #1 1). Derive the transfer Function 𝜃𝜃𝜃𝜃 (real position of DC motor)/ 𝜃𝜃𝜃𝜃 (desired position of DC motor) from Figure 5.1 given K2=0. 2). In Expt.#1, draw a Root-Locus plot for each step 1,2, 3 and 4 using system parameters and the value in the experiments. Comment on them. 3). In Expt.#1, Simulate block diagram Figure 5.1 for each step 1,2, 3 and 4 using system parameters and the value in the experiments, compare it with the experimental results. 4). In Expt.#1, get step response of the transfer function from question #1 for each step 1,2, 3 and 4 using system parameters and the value in the experiments. Compare the results with Question #3 and experimental results. Expt. #2 1). Derive the transfer Function 𝜃𝜃𝜃𝜃 (real position of DC motor)/ 𝜃𝜃𝜃𝜃 (desired position of DC motor) from Figure 5.1 with velocity feedback K2. 2). In Expt.#2, Simulate block diagram Figure 5.1 with K2 using system parameters and the value in the experiments of each case, compare them with the experimental results. 3). In Expt.#2, get step response of the transfer function from question #1 for each case. Compare the results with Expt#2 Question #2 and experimental results.

41

Lab 6: Time Response of Basic Closed-Loop Speed Control System and Effect of Torque Load

42

Appendix A: Connect Oscilloscope to MS Excel When turn on oscilloscope, the desktop of computer will display :

This means the computer connected to Scope. Now link scope to Excel. Click ADD_INS, then first

Icon , click “Identify”:

Click to capture scope Image:

43

Appendix B: Alternate method to get one shot wave form

1) Scope setting: Scale: Ch1: 1.0V; Ch2: 2V, Horiaontal time scale: 40 ms,

2) Press Scope “Trigger Menu”: Type: Edge, Source: CH1, Slope: Rising (Falling depending on waveform), Mode: Auto, Couple: DC. Trigger level set to +1.0v for rising wave or -1.0v for falling wave.

3) Run motor by turn power. Press Scope: Single, A rising (or falling waveform) will display and freeze on screen. Now you can use cursor to measure it.

44

Appendix C: How to get Time Constant using Cursor measurement

From Scope, Press Cursor: two vertical cursor with cross bars will display for paired measurement: using “Mulipurpose a or b” to move cursor, press “Menu 1 or 2”, the cursor will move along signal of CH1 or CH2. The ∆𝑥𝑥𝑥𝑥𝑥𝑥 𝑚𝑚𝑠𝑠 is the time measurement of these two cursors. The ∆𝑥𝑥𝑥𝑥𝑥𝑥 𝑚𝑚𝑉𝑉 is the voltage measurement of these two cross bars. Put “a” to initial point of signal CH2, “b” to final or stead state of tachometer, The ∆8.96𝑉𝑉 is Vt as following graph:

1) Refer to graph above, find the cursor of 63.2% of Vt( e.i. if Vt=8.96v, the 63.2% of Vt is 8.96*0.632=5.66 v, move second cursor “b” to ∆5.68𝑉𝑉), The ∆29.6𝑚𝑚𝑠𝑠 is the time constant. Draw a graph of this waveform and measurement.

2) Press Menu 1, the cursor will move along signal of CH1, refer to graph bellow, using “Muliti purpose a” to move cursor to low signal of CH1, “b” to high signal of CH1, The ∆3.96𝑉𝑉 is Vm as shown in following graph:

45

Appendix D: Summary of MS150 Data- DC System:

46