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CE110 Servo Trainer

© TecQuipment Ltd 2010Do not reproduce or transmit this document in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system without the express permission of TecQuipment Limited.

TecQuipment has taken care to make the contents of this manual accurate and up to date. However, if you find any errors, please let us know so we can rectify the problem.

TecQuipment supply a Packing Contents List (PCL) with the equipment. Carefully check the contents of the package(s) against the list. If any items are missing or damaged, contact TecQuipment or the local agent.

PW/PE/ajp/db/0710

TECQUIPMENT CE110 SERVO TRAINER

CONTENTS

SECTION PAGE

1.0 INTRODUCTION 1-1

1.1 General 1-1 1.2 CE110 Servo Trainer 1-3 1.3 Electrical Installation, Operating Conditions and

Maintenance 1-6

2.0 CONTROL THEORY 2-1

2.1 Fundamentals of Control Theory 2-1 2.1.1 Introduction 2-1 2.1.2 Control Principles 2-2

2.2 Advanced Principles of Control 2-13 2.2.1 Introduction 2-13 2.2.2 Servo System Modelling: Speed Control System 2-15 2.2.3 Servo System Modelling: Position Control System 2-17 2.2.4 Actuator and Sensor Characteristics 2-18 2.2.5 Measurement of System Characteristics 2-22 2.2.6 Controller Design: Angular Velocity Control 2-24 2.2.7 Controller Design: Angular Position Control 2-28 2.2.8 Controller Design: Disturbance Rejection 2-29

2.3 Advanced Principles of Control: Non-Linear System Elements

2-32

2.3.1 Amplifier Saturation 2-32 2.3.2 Amplifier Dead-Zone 2-34 2.3.3 Anti-Dead-Zone (Inverse Dead-Zone) 2-35 2.3.4 Hysteresis (Backlash) 2-36 2.3.5 Composite Non-Linearities 2-39

3.0 DIGITAL CONTROL TECHNIQUES 3-1

3.1 Fundamental Digital Control Principles 3-1 3.1.1 Representation of a Digital Controller 3-2

3.2 Software Implementation of a Three Term Controller 3-4 3.2.1 Proportional Control 3-5 3.2.2 Proportional and Integral Control 3-7 3.2.3 Proportional, Integral and Derivative Control 3-10

3.3 Implementation of Computer Control 3-12

i


ii

CONTENTS

SECTION PAGE

4.0 EXPERIMENTATION 4-1

4.1 Introduction 4-1 4.2 Experiment 1: Basic Tests and Transducer Calibration 4-3 4.3 Experiment 2: Response Calculating and

Measurements

4-11 4.4 Experiment 3: Proportional Control of Servo Trainer

Speed

4-15 4.5 Experiment 4: Proportional plus Integral Control of

Servo Trainer Speed

4-22 4.6 Experiment 5: Disturbance Cancellation and Feed-

Forward Control

4-29 4.7 Experiment 6: Angular Position Control: Proportional

Control

4-31 4.8 Experiment 7: Angular Position Control: Velocity

Feedback

4-36 4.9 Experiment 8: Angular Position Control and the

Influence of Non-Linearity

4-40 4.10 Experiment 9: Non-Linear System Characteristics 4-44

5.0 RESULTS AND COMMENTS 5-1 5.1 Experiment 1: Results and Comments 5-1 5.2 Experiment 2: Results and Comments 5-7 5.3 Experiment 3: Results and Comments 5-11 5.4 Experiment 4: Results and Comments 5-14 5.5 Experiment 5: Results and Comments 5-20 5.6 Experiment 6: Results and Comments 5-24 5.7 Experiment 7: Results and Comments 5-26 5.8 Experiment 8: Results and Comments 5-31 5.9 Experiment 9: Results and Comments 5-35

APPENDIX 1 BLANK EXPERIMENT CIRCUIT DIAGRAM A1-1


SECTION 1.0 INTRODUCTION 1.1 General The CE110 Servo Trainer is one of a unique range of products designed specifically for the theoretical study and practical investigation of basic and advanced control engineering principles. This includes the analysis of static and dynamic systems using analogue and/or digital techniques. A typical system configuration is shown in Figure 1.1 where a CE110 is shown adjacent to a CE120 Controller.

Figure 1.1 CE110 Servo Trainer adjacent to CE120 Controller

The CE110 Servo Trainer relates specifically to velocity control and angular position control problems as they would typically occur in industry. It may also, however, be used as a practical introduction to the design, operation and application of control systems in general.

The CE110 is an intrinsically safe, adaptable and self-contained facility for students of control engineering to investigate and compare a wide range of

Page 1-1


functional control system configurations. In particular with the CE110, they can examine the control of the velocity of a rotating shaft with differing loads and inertia's. An additional facility is available to engage, via an electrically operated clutch, an additional load shaft equipped with a gearbox and angular position sensor. This extends the experimental possibilities to position control. The CE110 includes a set of typical, user-adjustable non-linear elements which are associated with servo-control. These elements may be set up to demonstrate a wide range of practical non-linear phenomena.

IMPORTANT The CE110 is supplied for operation at the local mains supply voltage, either 110/120 V or 220/240V, unless otherwise indicated at the time of ordering. The set voltage is shown on the Test Certificate Supplied with the CE110 or on the

Serial Number Plate to be found at the rear of the unit.

Section 2 of this manual gives a step by step development of the fundamental and advanced control theory required to support the educational use of the CE110. This enables the performance of a particular Servo-System configuration to be either predicted in the case of an existing system or, at the design stage, the settings needed to achieve the desired (optimum) performance specifications. The CE110 is designed to operate with external analogue, digital or other standard industrial control elements. TecQuipment also make the optional CE120 Controller and the CE122 Digital Interface to work with the CE110. TecQuipment supply the CE120 and CE122 with the CE2000 software (see their relevant user guides). This allows the units to do open and closed-loop control investigations on any other item of laboratory equipment with compatible analogue inputs and outputs. The CE2000 software includes pre-written files that work with the CE110 and the experiments in this user guide. As an alternative, the CE110 may be controlled by any other compatible analogue or digital controller. However, it will be necessary to make the relevant amendments to the operating procedures and connection diagrams given in the manuals.

Page 1-2


1.2 CE110 Servo Trainer

Figure 1.2 CE110 Servo Trainer

The CE110 Servo Trainer is shown in Figure 1.2. It comprises a motor driven rotating shaft upon which is mounted, (from left to right): i. An inertial load flywheel ii. A tachometer to measure the shaft speed iii. A generator which provides an electrically variable load upon the

motor. iv. An electrically driven motor which provides the motive power which

rotates the shaft. v. An electrically operated clutch to enable the motor driven shaft to be

connected to a secondary shaft called here the position output shaft, which connects to:-

vi. A 30:1 ratio reduction gearbox vii. An output shaft position sensor and calibrated visual indicator.

Adjacent to the visual indicator of output shaft position is a manually operated position dial which can be used for setting desired (set-point) angular positions

Page 1-3


The CE110 includes power amplifiers for the drive motor and load generator and power supplies/signal conditioning circuits for the associated speed and velocity sensors. The motor speed is determined by the voltage applied to the drive amplifier input socket on the front panel. Likewise, the generator load is determined by the external load input. Both inputs are arranged to operate in the range ±10V (0 to 10V in the case of the generator). The shaft velocity sensor and the output shaft position sensor are sealed to give outputs calibrated in the range ±10V. A door at the rear of the left hand side allows access to change the size of the inertial load by adding or removing the inertia discs supplied. For safety, a micro-switch mounted in the door disables the drive amplifier when the access door is open or not fully latched. In addition to the main rotating components, a further facility for investigating servo-mechanism control is provided in the form of a set of typical servo-system non-linear elements. These are situated at the top of the unit and, as shown in Figure 1.3, from left to right comprise:- i. An anti-dead-zone block, to eliminate any dead-zone deliberately

introduced or inherent in the CE110 motor. ii. A dead-zone block, to introduce additional dead-zone so it may be

simulated and studied. iii. A saturation block, to allow servo-drive amplifier saturation to be

simulated and studied. iv. A hysteresis block, to allow gearbox and servo-drive train backlash to

be simulated and studied. The operation of the non-linear units is discussed in detail in Section 2.3 of this manual. The front panel of the CE110, shown in Figure 1.3, provides a schematic functional outline of the unit as well as providing quick and easy access, via 2mm sockets, to both the individual transducers and the motor and generator control circuits.

Page 1-4


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Figure 1.3 CE110 Front Panel

Page 1-5


Connection between external power supplies, control modules/measuring instruments and the motor/transducer circuits of the CE110 are made via 2mm sockets mounted on the front panel. The connecting leads supplied with the CE110 enable the user to make circuit/unit interconnections and so assemble a wide range of functional control systems as required. To readily facilitate the connection of the CE110 to most standard laboratory equipment and instrumentation, adapters are supplied to convert the 2mm format of the CE range to either a 4mm/or to a BNC format. 1.3 Electrical Installation, Operating Conditions and Maintenance

Manufacturer TecQuipment Ltd, Bonsall Street, Long Eaton, Nottingham NG10 2AN,

ENGLAND Importer The manufacturer

Electrical Connection

WARNING

The electrical supply must be connected to the apparatus through aswitch, circuit-breaker or plug and socket. The apparatus must beconnected to earth.

Connect the apparatus to an electrical supply using the cord or cables provided with the apparatus. Refer to the following colour code to identify the individual conductors: GREEN AND YELLOW: EARTH (E or ) BROWN LIVE BLUE NEUTRAL

Maintenance and inspection

A qualified person must carry out electrical maintenance.Ensure that the following procedures are followed. 1. Assume the apparatus is energised until it is known to be isolated from the electrical supply. 2. Use insulated tools where there are possible electrical hazards. 3. Check the insulation of the cord or external cables. Replace if damaged. 4. Confirm that the apparatus earth circuit is complete. 5. Find out the reason that a fuse blew, or a circuit breaker tripped, before replacing or resetting. 6. Confirm that a replacement component is compatible with the item being replaced.

Page 1-6


Periodically inspect the apparatus to ensure that there is no visible damage. Pay particular attention to connectors, switches, indicators, fuse holders and cables. If a fuse needs to be changed, use the following procedure: 1. Switch off the apparatus and disconnect it from the mains. 2. Remove the fuse and replace it with the exact type specified. 3. Reconnect the unit to the mains supply. 4. Switch on and ensure that the unit works correctly. 5. If the fuse fails again contact the importer or TecQuipment for advice.

Handling instructions

Net weight: 17 kg. Ensure that correct handling procedures are used when moving this apparatus.

Operating Conditions

Storage temperature range –25°C to +55°C (when packaged for transport)

Safe operating temperature range +5°C to +40°C Safe operating relative humidity range

30 % to 95 % (non-condensing)

Operating environment Laboratory environment Supply voltage (nominal) 230 V 115 V Current (maximum) 500 mA 1 A Frequency 50/60 Hz Fuse type T1.6 A 20 mm

ceramic (see IEC 60127-III)

T3.15 A 20 mm ceramic

(see IEC 60127-III)Supply type TN–S (refer to IEC 60364)

Noise Levels

The maximum sound pressure levels measured for this apparatus are fewer than 70 dB(A).

Spare Parts Refer to the ‘Packing Contents List’ for any spare parts that are supplied with the apparatus. Contact TecQuipment or the importer if any other spare parts are needed.

Page 1-7


Page 1-8


SECTION 2.0 CONTROL THEORY 2.1 Fundamentals of Control Theory 2.1.1 Introduction The object of this section is to provide an introduction to control engineering principals by firstly considering the operating characteristics of the individual elements used in typical control engineering systems. It then further considers the performance of these elements when combined to form a complete control engineering system. The text includes the development of control theory relating to servo mechanism control in velocity and positional control systems. This is considered essential in ensuring that the student both understands and is able to explain the results obtained from the practical investigations contained in Section 4 of this manual. This also allows the initial controller setting for the individual systems to be set or established as directed. It then helps in the analysis of how the systems actually respond to various steady state and transient operating criteria. The primary object of the CE110 Servo Trainer, of which this manual forms an important part, is to provide a practical environment in which to study and understand the control of a servo-system. These systems occur widely throughout all branches of industry to such an extent that a grounding in servo mechanism control forms a basic component of a control engineer's training. A simple but widespread industrial application of servo control is the regulation at a constant speed of an industrial manufacturing drive system. For example, in the production of strip plastic, a continuous strip of material is fed through a series of work stations. The speed at which the strip is fed through must be precisely controlled at each stage. Similar examples exist where accurate position control is required. A popular example is the position control of the gun turret on a battle tank, which must be capable of both rapid aiming, target tracking and rejection of external disturbances. The following theory and examples are based upon the need to maintain a selected speed or position of a rotating shaft under varying conditions.

Page 2-1


2.1.2 Control Principles Consider a simple system where a motor is used to rotate a load, via a rigid shaft, at a constant speed, as shown in Figure 2.1.

Flywheel (theoretical load)

Coupling shaft Coupling shaft

Shaft speed

Drive MotorGenerator

LoadDriveMotor

Amplifier

Figure 2.1 Simple Motor & Load System

The load will conventionally consist of two elements, i. A flywheel or inertial load, which will assist in removing rapid

fluctuations in shaft speed and, ii. An electrical generator from which electrical power is removed by a

load. Under equilibrium conditions with a constant shaft speed, we have

Electrical Mechanical power absorbed by thepower supplied generator and frictionalto motor losses

=

Page 2-2


When this condition is achieved the system is said to be in equilibrium since the shaft speed will be maintained for as long as both the motor input energy and the generator and frictional losses remain unchanged. If the motor input and/or the load were to be changed, whether deliberately or otherwise, the shaft speed would self-adjust to achieve a new equilibrium. That is, the speed would increase if the input power exceeded the losses or reduce in speed if the losses exceeded the input power. When operated in this way the system is an example of an open-loop control system, because no information concerning shaft speed is fed back to the motor drive circuit to compensate for changes in shaft speed. The same configuration exists in many industrial applications or as part of a much larger and sophisticated plant. As such the load and losses may be varied by external effects and considerations which are not directly controlled by the motor/load arrangement. In such a system an operator may be tasked to observe any changes in the shaft speed and make manual adjustments to the motor drive when the shaft speed is changed. In this example the operator provides; a) The measurement of speed by observing the actual speed against a

calibrated scale. b) The computation of what remedial action is required by using their

knowledge to increase or decrease the motor input a certain amount. c) The manual effort to accomplish the load adjustment, required to

achieve the desired changes in the system performance, or by adjusting the supply to the motor.

Again, reliance is made on the operators experience and concentration to achieve the necessary adjustment with minimum delay and disturbance to the system. This manual action will be time consuming and expensive, since an operator is required whenever the system is operating. Throughout a plant, even of small size, many such operators would be required giving rise to poor efficiency and high running costs. This may cause the process to be an uneconomic proposition, if it can be made to work at all!

Page 2-3


There are additional practical considerations associated with this type of manual control of a system in that an operator cannot maintain concentration for long periods of time and also that they may not be able to respond quickly enough to maintain the required system parameters. A more acceptable system is to use a transducer to produce an electrical signal which is proportional to the shaft speed. Electronic circuits would then generate an Error Signal which is equal to the difference between the Measured Signal and the Reference Signal. The Reference Signal is chosen to achieve the shaft speed required. It is also termed the Set Point (or Set Speed in the case of a servo speed control system).

The Error Signal is then used , with suitable power amplification, to drive the motor and so automatically adjust the actual performance of the system. The use of a signal measured at the output of a system to control the input condition is termed Feedback. In this way the information contained in the electrical signal concerning the shaft speed, whether it be constant or varying, is used to control the motor input to maintain the speed as constant as possible under varying load conditions. This is then termed a Closed-Loop Control System because the output state is used to control the input condition.

Figure 2.2 shows a typical arrangement for a closed-loop control system which includes a feedback loop.

Page 2-4


Drive Motor

Inertia load(flywheel)

Reference signal(Set speed)

Generator

Feedbackcontroller

Motordrive

amplifier

error

Shaft speed signal

Differencingamplifier

Actuation Signal

Figure 2.2 Closed-Loop Control System including Feedback Loop

The schematic diagram shown in Figure 2.3 represents the closed-loop control system described previously.

Figure 2.3 Schematic Representation of Closed-Loop Control System

Page 2-5


Next, consider the situation when the system is initially in equilibrium and then the load is caused to increase by the removal of more energy from the generator. With no immediate change in the motor input, the shaft speed will fall and the Error signal increase. This will in turn increase the supply to the motor and the shaft speed will increase automatically. As the speed is being returned to the original Set Speed value, the Error signal reduces causing the energy supplied to the motor to also reduce. Eventually the supply to the motor would become so small that it cannot drive the load and so stalls. In practice the actual motor torque would reduce until a new equilibrium was produced where the motor torque equalled the load torque and the Error achieves a new constant value. The difference between the Actual speed and the Set speed is termed the Steady State Error of the system. If the Gain of the amplifier was increased, the Steady State Error would be reduced but not totally removed, for exactly the same reasons as given previously. If the Gain were to be increased too much the possibility of Instability may be introduced. This will become evident by the shaft speed oscillating and the input of the motor changing rapidly.

Figure 2.4 Proportional Control Amplifier Gain Characteristics

Page 2-6


The system described previously is said to have Proportional Feedback since the Gain of the amplifier is constant. This means that the ratio of the output to the input is constant once selected. Figure 2.4 shows the characteristic of a typical Proportional Control Amplifier with the Gain set at different levels, increasing from K1 to K5.

In order to maintain a non-zero input to the motor drive, there must always be a non-zero error signal at the input to the proportional amplifier. Hence, on its own Proportional Control cannot maintain the shaft speed at the desired level with zero error, other than by manual adjustment of the Reference. Moreover, proportional gain alone would not be able to compensate fully for any changes made to the operating conditions. Operating with zero Error may, however, be achieved by using a controller which is capable of Proportional and Integral Control - (PI). Figure 2.5 shows a typical schematic diagram of a PI Controller.

Figure 2.5 Schematic of PI Controller. The Proportional Amplifier in this circuit has the same response as that shown previously in Figure 2.4 (K1 to K5). An Integrating Amplifier is designed such that its output is proportional to the integral of the input. Figure 2.6 shows the typical response of an Integrating Amplifier supplied with a varying input signal.

Page 2-7


Figure 2.6 Typical Response of an Integrating Amplifier supplied with Varying Input Signal

From Figure 2.6 it can be seen that, a) When the input is zero the output remains constant. b) When the input is positive the output ramps upwards at a rate

controlled by the actual magnitude of the input and also the gain of the integrator.

c) When the input is negative the output ramps downwards at a rate controlled by the actual magnitude of the input and also the gain of the integrator.

d) If the input itself is damping or changing in any way then the output will follow an integral characteristic, again following the criteria given in (a) and (b) above.

e) When a change in input polarity occurs the output responds in the manner described above, starting at the instantaneous output value at which the change occurred.

f) The magnitudes achieved at the output are dependent on the magnitude of the input signal and also the time allowed for the

Page 2-8


damping to occur. In a practical integrator, the output signal is also limited by the voltage of the power supply to the integrator itself.

Effectively, when a constant DC signal is supplied to the input of an Integrating Amplifier its output will 'ramp' at a constant rate. Whether it ramps up or down is determined by whether the input polarity is either positive or negative. By arranging the polarity of the Error signal in a control system correctly, the output from the integrator can be configured to always drive the system in the correct direction so as to minimise (zero) the Error. In practice, an integrator would be used, as shown in Figure 2.5, with proportional amplification to give an overall system response of the required characteristic. The overall response of the PI Controller to a step change in Set Speed (or the shaft speed conditions due to the load increasing) is the combined effects of its two circuits, as shown in Figure 2.7.

Figure 2.7 Overall Response of the PI Controller to a step change in Set Speed

Consider the system described previously by Figure 2.2, where the load rate is increased by the load generator, but now with a PI Controller in the Feedback Loop. As before, the Proportional Amplifier on its own will leave an Error at the instance of the change in speed. However, with the Integrator output signal

Page 2-9


increasing, ramping upwards in response to this error, the supply to the drive motor and the motor torque will correspondingly increase. The shaft speed will rise until the Set speed is achieved and the Error is zero. At this condition the motor and loads are equal and the system is in equilibrium. This new operating condition will be maintained until another disturbance causes the speed to change once again, whether upwards or downwards, and the controller automatically adjusts it's output to compensate. In practice the PI Controller constantly monitors the system performance and makes the necessary adjustments to keep it within specified operating limits. The amount of Integral Action will affect the response capability of the system to compensate for a change. Figure 2.8 shows the typical response of a system with constant Proportional and varying levels of Integral Action.

Figure 2.8 Typical System Response with Constant Proportional and Varying Integral Action.

With an intermediate level of Integral Action the system moves quickly, with minimum overshoot, to the Set Level value. In the example shown, the value of Integral Action chosen is said to achieve Critical Damping.

Page 2-10


With a low level of Integral Control there is a very slow response giving rise to a distinct time delay between when a change occurs and when the control circuit re-establishes the Set Level again. This type of system is said to be Over Damped. With a high level of Integral Control the response of the system may be so fast that it overshoots the required value and then oscillates about that point under Integral Action until it finally settles down to the steady state condition, if at all. Note that, in the example given, the time for the system to settle down is greater than when the Integral Control value was small. This type of system is said to be Under Damped. For large levels of Integral Control, the system oscillations of the under damped system might grow and become unstable. In general, a) Any increase in the amount of integral action would cause the system

to accelerate more quickly in the direction required to reduce the Error and have a tendency to increase instability.

b) Decreasing the integral action would cause the system to respond more slowly to disturbances and so take longer to achieve equilibrium.

Where fast response is required with minimum overshoot a Three-Term Controller is used. This consists of the previous PI Controller with a Differential Amplifier included to give a PID ( or Three-Term ) Controller. The performance of a Differential Amplifier is that the output is the differential of the input. Figure 2.9 shows the characteristic of a Differentiator supplied with a square wave input. Each time the input level is reversed the output responds by generating a large peak which then decays to zero until the next change occurs. In a practical Differentiator the maximum peak value would be achieved at the power supply rail voltage levels to the Differentiator itself. In a PID Controller the polarity of the output would be configured to actually oppose any change and thereby dampen the response of the system. The gain of the Differentiator would control the amount of damping provided, both in amplitude and duration.

Page 2-11


Figure 2.9 Differentiator supplied with Squarewave Input The damping required for the situation described in Figure 2.8 could also, therefore, be achieved by including a Differentiator in the control loop to suppress the high acceleration caused by the Integrator without affecting it's ability to remove the Error. It is the balance between the Integral and Differential Action which now controls the overall system response to a step change in Set Level. The speed and manner with which a system can overcome disturbances is termed the Transient Response. By careful selection of the parameters of the proportional, integral and differential amplifiers it is possible to produce a system Transient Response to suit the specific application. This section so far has only dealt with control engineering principles in a very basic way so that the CE110 Servo Trainer can be used by students and engineers new to control engineering without them having to be familiar with the mathematics. It is possible to verify these principles by setting up suitable test circuits with the CE110 and the CE120 and then confirming the various system responses. Section 2.2 builds upon these fundamental principles and introduces the advanced topics of mathematical modelling, system tuning and predicting

Page 2-12


system performance. This includes the more complex control of shaft position in an output shaft of a reduction gearbox by varying the motor drive in the input shaft of the gearbox. 2.2 Advanced Principles of Control 2.2.1 Introduction In this Section we build on the introductory material of Section 2.1 and describe more advanced methods for the analysis and control of the Servo Trainer. The ability to analyse a system, real or otherwise, is especially important in establishing the relevant design parameters for new plant or in predicting the performance of existing equipment which is to operate under new conditions. Being able to predict the performance of any complex engineering system in advance of its construction and operation will both reduce costs and also minimise project development time. The ability to represent a control situation using mathematical equations also allows computers to be used as an invaluable development tool for the engineer. The computer, once programmed to respond in exactly the same way as the chosen system, can thoroughly 'test' or simulate that system under all possible operating conditions, both quickly and cheaply. For some equipment it may only be possible to simulate certain operating conditions since in real life the actual condition cannot be safely or economically reproduced, e.g. the landing on the moon could only be achieved after the equipment had been designed and built, and yet the engineers had the confidence to commit vast resources to the development and construction project as well as gaining experience in advance through the use of simulators. Most importantly, they were able to commit the safety of humans to man the vehicles.

Page 2-13


Figure 2.10 Servo Control System: Clutch Disengaged

Page 2-14


Page 2-15

2.2.2 Servo System Modelling: Speed Control System NOTE: This theory describes how you could find the characteristics of the servo trainer from the individual properties of its elements including the electrical properties of the motor. This is only for reference, as this is impossible without taking the equipment apart, which TecQuipment do not recommend, as it would cancel your guarantee. Initially, consider the servo control system with the clutch disengaged. In this configuration the system is a speed control process which can be represented as shown in Figure 2.10 The system model is determined by relating the torque supplied by the motor (τm) to that required to drive the load generator, the flywheel and frictional losses. This can be expressed as,

τm = Load Torque + Frictional Torque + Inertial Torque

The load torque can be considered as a torque which is proportional to the load control voltage (vl) while the frictional torque can be considered as a torque which is proportional to the shaft speed ω . The inertial torque is

determined by the flywheel inertia and the shaft acceleration ddtω . Thus

τ ω ν ωm b k I

d

dt= + +l l

2.1 Where b = Friction coefficient of rotating components kl= Gain constant of load/generator Ι = Inertia of flywheel The motor electrical circuit is governed by the equation

( )ν t Ri Ldi

dt bemf= + + ν

2.2


Page 2-16

Where ν(t) is the motor input voltage R is the motor armature resistance L is the armature inductance i is the armature current and νbemf is the motor back emf The back emf and the motor torque can be written in terms of the motor constant km, thus

ν ω

τ

bemf m

m m

k

k i

=

=

2.3

Combining Equations 2.1, 2.2 and 2.3 by taking Laplace transforms and eliminating variables yields the transfer function relating the output speed ω(s) to the input voltage v(s) and the load voltage vl(s)

( )

( )( )( )

( )( )ω ν ν( ) ( )s

k s

sI b sL R k

k R sL

sI b sL R ksm

m m

=+ + +

−+

+ + +2 2l

l

2.4

The transfer function simplifies if the inductance L of the armature circuit is assumed to be small compared with the inertia of the flywheel. This gives the first order transfer function

( ) ( )ω ν νs

k s

Ts

k s

Tsm=

+−

+' ' (

1 1l l )

2.5a Where time constant T is given by

TIR

bR km

=+ 2

and

kk

bR kmm

m

' =+ 2

kk R

bR km

'll=+ 2

2.5b


Frequently, we will consider the situation when the servo-control system only has an inertial load. In this case vl(s) = 0 and Equation 2.5 simplifies to

( ) ( )ω νsk

Tssm=

+'

( )1

2.6

2.2.3 Servo-System Modelling: Position Control System With the electric clutch engaged, the gearbox and output position shaft are connected to the main shaft as shown in Figure 2.11

Figure 2.11 Servo Control System: Clutch Engaged The output shaft position (θ), is related to the main shaft velocity (ω) by:

θ ω( )

( )s

s

s=

30

2.7

Where the constant '30' is associated with the 30:1 reduction in speed through the gearbox. Note that the addition of the gearbox load will also change the gain and time constant characteristics of Equations 2.5 and 2.6. Equations 2.5 and 2.6 are used together to provide the system model of the servo-control system dynamics.

Page 2-17


2.2.4 Actuator and Sensor Characteristics When the servo-control system is used as a feedback control system the motor speed, ω, is controlled (or actuated) by adjusting the applied voltage to the motor drive amplifier, v. Likewise, the shaft speed and angular position are sensed by transducers which produce output voltages. yω and yθ which are proportional to the shaft velocity,ω, and position, θ, respectively.

Figure 2.12 Schematic Representation of Servo Control Feedback System The overall system may be represented schematically as shown in Figure 2.12. The motor voltage, v, and the shaft speed, ω, are related by a steady state actuator characteristic which is assumed to be linear (more will be said of this assumption in section 2.3). The velocity sensor and angular position sensor also have linear characteristics, as shown in Figures 12.13a, b, and c.

Page 2-18


Figure 2.13a Speed vs Motor Input Voltage

Figure 2.13b Sensor Output vs Shaft Speed

Page 2-19


Figure 2.13c Sensor Output vs Shaft Position If ki, kω, kθ are the motor, velocity sensor and angle sensor gain constants respectively, then

ω ν

ω

θ

ω ω

θ θ

=

=

=

k

y k

y k

i

2.8 Note that ki is, as stated previously a steady state gain constant which, from Equation 2.5, is equal to the gain k'm obtained from the modelling exercise. Combining Equations 2.6 and 2.8 gives the standard first order system transfer function.

( ) ( ) ( )y sG

Tsv sω =

+1

1

2.9

Page 2-20


Page 2-21

Where G k ki1 = ω , is the steady state gain of the transfer function from the

In addition, the sensed output shaft position lated to the sensed velocity yω by

input drive voltage, v, to the sensed shaft position, yω.

yθ is re

( ) ( )y sG

sy sθ ω= 2

2.10

where

Gk

k2 30= θ

ω

hen the overall transfer function for the servo-control system can be drawn

as in Figure 2.14 and written thus:

2.10b

T

( ) )( ( )y sG G

s Tsv sθ =

+1 2

1

2.11

Figure 2.14 Overall Transfer Function for Servo Control System

se in loading due to the earbox, the value of G1, T will be changed when the clutch is engaged and e gearbox and output position shaft are connected.

Again it should be noted that, because of the increagth


2.2.5 Measurement Of System Characteristics Motor and Sensor Characteristics The motor steady state characteristic, and the speed sensor characteristics are obtained by running the motor at various velocities and recording the corresponding voltages. These are then plotted to obtain the characteristics, as shown in Figure 2.13. The angular position sensor is likewise obtained by rotating the output shaft (using the motor) to various positions, recording the corresponding voltages and plotting to obtain the characteristics. Note that all the servo-control system characteristics are approximately linear. The output and gains will, however, change slightly over a period of time. This phenomenon is known as drift and is not uncommon in industrial sensors and actuators. The motor characteristics will change significantly according to operating conditions. Specifically, the gain G1, and the time constant T will change when the clutch connecting the gearbox and output position shaft is activated. Also, the servo-control system allows for the inertial load to be varied by altering the flywheel thickness (mass) by adding or removing discs. This will alter the inertia I and hence (via Equation 2.5b) the system time constant, T.

Page 2-22


System Dynamic Characteristics: Step Response Method. For a first order system, like the servo-control transfer function for shaft speed, the gain G1 and time constant T can be obtained from a step response test as follows: With reference to Figure 2.15, the gain is determined by applying a step change, with amplitude U, to the input of a system. The final, or steady state, value of the output will be the product U x G1, from which the gain can be relatively determined. The time constant T is defined as the time required for the step response of the system to reach 0.632 of its final value.

Figure 2.15 Step Response

This method is generally easy to use, and gives reasonably accurate results, provided the system characteristic is known to be first order. System Dynamic Characteristics: Direct Calculation An alternative to step response testing is to measure the system characteristics individually and then use Equations 2.5b to calculate the gain and time

Page 2-23


constant of the process. This method requires a knowledge of the system model (from Sections 2.2.2 and 2.2.3) and the ability to make basic measurements of system parameters. In the case of the servo-control system and with reference to Equations 2.5b, it is possible to determine the parameters by either experimentation, direct measurements, or use of manufacture's data sheets (in the case of the motor characteristics). In practice however, the time required and inaccuracy of certain measurements (especially the friction coefficient, b) mean that direct calculation of the system dynamic characteristics would only be undertaken if a detailed simulation of the process was required. We will use step response testing methods in this manual. 2.2.6 Controller Design: Angular Velocity Control Figure 2.16 represents a velocity control system in block diagram form.

Figure 2.16 Velocity Control System The aim of the feedback controller is twofold. First it is to bring the output speed, yω, into correspondence with the reference speed yr. This necessitates finding ways of making the error, e, under steady operating conditions. The second aim of the controller is to alter the dynamic behaviour of the servo-system to improve the speed of response to changes in the reference speed. This requires us to find ways of altering the system dynamic response via feedback control. We will consider the steady operating performance separately below.

Page 2-24


Page 2-25

Steady State Errors

A main reason for applying feedback control to a system is to bring the system output into correspondence with some desired reference value. The theory developed in Section 2.1.2 'Control Principles' has already explained that there is often some difference between the reference and the actual output. In this Section we see how these errors are quantified when the steady state has been reached. The steady error ,ess, is a measure of how well a controller performs in this respect. The steady state is defined as,

( )[ ]e esst

=→οοlim t

e s

2.12

Where the error, e(t), is the difference between the reference Set Speed value and the actual output, as shown in Figure 2.16. Equation 2.12 can be re-written in the frequency domain as,

( )[ ]e ssss

=→ο

lim .

2.13

For a constant set speed or reference input yr, the steady state error (from Figure 2.16) is,

( ) ( )ey

K s G sssr

s=

+

→οlim

1 1.

2.14 Where K(s) is the controller transfer function and G(s) is the servo system transfer function. If proportional control only is used then,

( )K s Kp=

and,

( )ey

K G sssr

ps=

+

→ο

lim1 1.

2.15


Thus proportional control for the servo control system will involve a steady error which is inversely proportional to the gain, Kp. If proportional plus integral (PI) control is used,

( )K s KK

spi= +

and,

( ) ( )e

s y

s K s K G sss

r

p is=

+ +

=→ο

lim.

. . 1

0

2.16

Thus, with proportional plus integral (PI) control, for the servo system speed transfer function the Steady State Error is zero. Dynamic Response The effect of feedback upon the dynamic response of the servo control system velocity controller can be seen from a consideration of the closed-loop transfer function. From Figure 2.16 it is possible to write

( ) ( ) ( ) ( )( ) ( )y s

K s G s y s

K s G sr

ω =+

1

11

2.17 Recall that the speed transfer function is, from Equation 2.9,

( )G sG

Ts11

1=

+

2.18 With proportional control, the closed-loop transfer function is obtained by combining Equations 2.17 and 2.18 to give

( ) ( )y sk G

Ts k Gy s

p

prω =

+ +

1

11

2.19

Page 2-26


or

( ) ( )y sG

T sy sc

crω =

+1

1

1

11

2.20a

Where the closed-loop gain is Gc11 given by:

GK G

k Gcp

p11

1

11=

+

2.20b and the closed-loop constant is Tc11 given by:-

TT

k Gcp

11 1 1=

+

2.20c From Equation 2.20c it can be seen that the closed-loop speed of response can be increased by reducing the time constant Tc11

. This in turn is achieved by

increasing the proportional gain kp. If the system controlled by a proportional plus integral controller, the closed-loop system is given by

( ) ( ) ( )y sk k s G

Ts k G s k Gy s

i p

p irω =

+

+ + +

12

1 11( )

2.21

By comparing the denominator of Equation 2.21 with the standard expression for the denominator of a second order transfer function:-

( ) ( )y ss s

y sn

n nr=

+ +ω

ξω ω

2

2 22

2.22 it is possible to show that

ω nik G

T2 1=

2.23 and

Page 2-27


Page 2-28

21 1ξω n

pk G

T=

+

Thus by use of Equation 2.23, it is possible to achieve a desired increase in system transient response performance in terms of a second order closed-loop response. This is done by selecting ki and kp to give desired values of ωn and ξ 2.2.7 Controller Design: Angular Position Control Figure 2.17 represents the possible block diagram configuration for feedback control of angular position.

Figure 2.17 Feedback Control of Angular Position Notice that the control system has two feedback loops. An inner loop feeds back a proportion, kv, of the system velocity, while an outer (position) loop feeds back the sensed output position yθ(s). The role of the inner velocity loop is to improve transient performance of the overall system. This can be seen by considering the overall closed-loop transfer function with proportional control, such that ; ( )K s k p=

( ) ( )y sk G G y s

s T s k G k G Gp r

v p

θ =+ + +

1 22

1 11( )

2

2.24


Again this can be compared with the standard second order Equation (Equation 2.22) and the following results obtained:

ω

ξω

np

nv

k G G

T

k G

T

2 1 2

12 1

=

= +

2.25

By selecting kp and kv appropriately it is possible to obtain the desired dynamic performance, as specified by ωn and ξ. Note that when kv=0 (i.e. there is no velocity feedback) it is not possible to specify the damping factor; this can lead to very oscillatory behaviour when the system proportional gain is increased. 2.2.8 Controller Design: Disturbance Rejection

Figure 2.18 Velocity Control System Consider the velocity control system discussed in Section 2.2.6, but with the servo-system model extended as indicated by Equation 2.5a to include the effect of the generator load. Figure 2.18 shows this situation. The load disturbance transfer function is (from the second term on the right hand side of Equation 2.5a):

Page 2-29


( ) ( )G sG

Tsli=+ 1

2.26 where G k . l l

'= The closed-loop equation for the system, including the influence of the generator load is, from Figure 2.18, given by

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )y sG s K s

G s K sy s

G s

G s K sv srω =

+−

+1

1 11 1l

l

2.27 Proportional Compensation: If proportional control is applied, then K(s) = kp and if kp is large then the effect of the load change upon the output will be small. In fact the larger kp is the smaller the effect of the load change upon yω. Integral Compensation: If integral plus proportional control is applied, then if a load is applied the integral term will integrate any non-zero error until the effect of the load is removed. This can be seen by writing the closed-loop equation for Figure 2.18 with proportional and integral control:-

( ) ( ) ( ) ( )y s

k sk G

Ts k G s k Gy s

sG v s

Ts G k s k Gi p

p ir

p iω =

+

+ + +−

+ + +( )

( )1

21 1

21 11 1

l l

2.28

The numerator of the load disturbance term contains a term s (i.e. a zero at the origin) which indicates that for constant load voltages νl(s), the effect upon yω

(s) will be zero in the steady state.

Page 2-30


Feed Forward Compensation: From the previous paragraphs it is seen that proportional control reduces the effect of load changes and integral control action removes the steady state effect of loads. There is, however, a way of reducing the effect of load changes even more. This involves feeding a signal proportional to the load demand into control action. This is called Feed Forward control and is shown in block diagram form in Figure 2.19

Figure 2.19 Feed Forward Control The idea of Feed Forward control is to take a proportion of the load voltage vι and after passing it through a suitable controller Kf(s), add it to the motor input voltage, v, such that it compensates for the effect of the load upon the speed, yω. By correctly selecting Kf(s) it is possible to completely compensate for the influence of the load voltage vi. This is done by selecting Kf(s) such that,

( )K sG s

G sf = l ( )

( )1

2.29 From the equations defining Gl(s) and G1(s), (Equations 2.26 and 2.29 respectively) the feed forward controller required to exactly cancel the load disturbance is a constant Kf, given by

KG

Gf = l

1

2.30

Page 2-31


Thus by calculating Kf according to Equation 2.30 it is possible to exactly cancel the effect of changing load upon the speed signal. In practice, Kf is often selected experimentally, to approximately remove disturbances and combined with a proportional plus integral controller which removes the remainder of the load effects. 2.3 Advanced Principles Of Control: Non-Linear System Elements The treatment of the servo-control problem thus far has considered the system to be linear. In a practical servo-system, however, a number of non-linearities occur. The most frequently occurring forms of non-linearity are incorporated into the servo-system in a block of simulated non-linearities. The non-linear elements can be connected in series with the servo-motor in order to systematically investigate the influence which non-linearities have upon practical system performance. 2.3.1 Amplifier Saturation

Figure 2.20 Saturation In a practical electronic amplifier for a servo-motor drive there are maximum and minimum output voltages which cannot be exceeded. These maximum and minimum values are due to the limitation imposed by the values of the amplifiers. For example, if the power supply to an amplifier provides ±15 V,

Page 2-32


then the amplifier output cannot exceed these limits, no matter what the gain of the amplifier. This feature is termed 'Saturation' and is illustrated in Figure 2.20. The saturation amplifier works normally with a specified linear gain relationship between the input voltage, vi, and the output voltage, vo, for inputs in the range -vmin and vmax. Beyond these limits, the output voltage, vo, is constant at either vmax. or vmin. The servo motor drive amplifier saturates at ± 10V, but in order to show separately the effects of saturation the non-linear element block incorporates a saturation element (Figure 2.21).

Figure 2.21 Saturation Element The saturation block is switched into the circuit using the enabling switch. With the saturation disabled the input signal passes through the saturation block unmodified. The gain of the saturation amplifier is unity and the voltage at which the amplifier saturates is controlled by a calibrated 'level control'.

Page 2-33


2.3.2 Amplifier Dead-Zone A further feature of a practical amplifier is the dead-zone (or dead-band as its is sometimes called), whereby the amplifier output is zero until the input exceeds a certain level at which the internal losses are overcome, i.e. mechanical losses such as 'stiction'. Hereafter, the amplifier behaves normally. Figure 2.22 shows a typical dead-zone amplifier characteristic.

Figure 2.22 Typical Dead-Zone Characteristic Amplifier dead-zone characteristics are inherent in motors in which a certain (minimum) amount of input is required in order to turn the motor against friction and other mechanical losses. Once the motor begins to turn, the amplifier and motor respond in the normal linear way. The servo motor amplifier has a small dead-zone, but in order to show separately the effects of dead-zone the non-linear element block incorporates a dead-zone element (Figure 2.23).

Page 2-34


Figure 2.23 Dead-Zone Controls

The dead-zone block is switched into circuit using the enable switch. With the dead-zone disabled the input signal passes directly through the dead-zone block unmodified. The gain of the dead-zone element is the linear region is unity, and the dead-zone width and location can be controlled by 'width' control and 'location' control (Figure 2.23). 2.3.3 Anti-Dead-Zone (Inverse Dead-Zone)

Figure 2.24 Inverse Non-Linearity Anti-Dead-Zone Characteristic

Page 2-35


One way in which the non-linear characteristics can be compensated for is by using an inverse of the non-linearity characteristics. In the specific case of a dead-zone non-linearity, the corresponding inverse non-linearity is the anti-dead-zone characteristic shown in Figure 2.24. By selecting the anti-dead-zone levels vap and -van to correspond to the dead-zone levels vdp and vdn the two non-linearity cancel exactly. In order that the effects of anti-dead-zone can be demonstrated the non-linear element block incorporates an anti-dead-zone element (Figure 2.25).

Figure 2.25 Anti-dead-Zone Block The anti-dead-zone block is switched into circuit using the enable switch. With the anti-dead-zone disabled the input signal passes directly through the anti-dead-zone block unmodified. The gain of the anti-dead-zone element in the linear region is unity and the anti-dead-zone 'width' and 'location' can be adjusted by the 'width' control and the 'location' control . These are shown in Figure 2.25. 2.3.4 Hysteresis (Backlash) A common and yet unwelcome form of non-linearity in mechanical drives is hysteresis or backlash. This form of non-linearity is caused by worn or poor tolerance mechanical couplings (usually gearboxes) in which the two elements of the coupling separate and temporarily lose contact as the

Page 2-36


direction of movement changes. This can be illustrated with reference to Figure 2.26, in which the worn or incorrectly meshed gears temporarily lose contact during a change in direction of the driving gear. As a result the driven (or output) gear remains stationary until the driving (or input) gear has traversed and made contact again with the driven gear. The region where no contact exists is termed the 'backlash gap'.

Figure 2.26 Hysteresis or ‘Backlash’

Page 2-37


Figure 2.27 Input/Output Characteristic of a Hysteresis/Backlash Device The input/output characteristic of a hysteresis/backlash device is shown in Figure 2.27. Notice that the hysteresis is a 'directional' non-linearity in that the output signal depends upon the direction of change of the input signal and (during the backlash gap) the post direction of change. The servo-system gearbox has been selected to have a small hysteresis characteristic, such that backlash in the servo-system should not be a problem. However, in order to show the effects of hysteresis, the non-linear element block incorporates a hysteresis element to add realism to the system. The hysteresis block is switched into the circuit using the enable switch. With the hysteresis disabled the input signal passes directly through the hysteresis block unmodified. The magnitude of the hysteresis is adjusted by the 'backlash' control, as shown in Figure 2.28.

Page 2-38


Figure 2.28 Backlash Gap (Hysteresis) Control 2.3.5 Composite Non-Linearities The phenomena of dead-zone, saturation and hysteresis often, unhappily, occur together in a system. The combined effects of these non-linearities can be introduced with the non-linear blocks by switching in the desired combination of non-linearities. For example, a saturating non-linearity with dead-zone can be produced by enabling these blocks and adjusting the controls appropriately. Care should be taken to ensure that the composite non-linearity is practically reasonable. For example, the dead-zone width should always be less than the level at which saturation occurs. Used together with the servo-system motor the non-linear blocks enable the demonstration of important limitations to control system design caused by non-linearity.

Page 2-39


SECTION 3.0 DIGITAL CONTROL TECHNIQUES In this Section we consider basic ideas and methods used in digital control techniques and outline the digital form of the algorithms which may be applied to the CE110 Servo Trainer. 3.1 Fundamental Digital Control Principles Microprocessors and computers have become increasingly important tools for the engineer in recent years for design, data analysis and other routine purposes. However, it is in the field of system control that these devices have had the most significant impact on most branches of science and engineering. The speed and flexibility of operation enables them to be programmed for a much wider range of eventualities than their equivalent analogue circuits. Software may be written to generate control functions based on the error between actual and demanded values of the variables such that the optimum transient response and steady state condition is attained. As with any system which requires accurate control, whether digital or analogue, the system must include some method of measuring the relevant physical parameters and then be able to respond to any changes so detected. In a computer controlled system, the transducer signals are converted into the required digital format and then fed to the input port of the computer.

Figure 3.1 Under software control the computer then interprets and compares this data with a programmed demand value held in memory and uses the result to affect it's response, as shown in Figure 3.1.

Page 3-1


There are, however, disadvantages in using digital techniques in control applications. These mainly arise because of the periodic sampling of the data and also the subsequent update of the output signal. Nevertheless, provided the normal precautions on sample rate selection are followed, digital control can produce excellent results. In the sub-sections which follow we will illustrate how common digital control algorithms are obtained and provide guidance on such issues as sample rate selection. 3.1.1 Representation of a Digital Controller The schematic diagram of Figure 3.1 can be re-drawn for control studies in the form shown in Figure 3.2 In Figure 3.2 the analogue to digital converters (ADC) are represented by sampling switches which close at ΔT second intervals. The sample interval, ΔT, is determined by the control system designer/programmer and selected

such that the sampling frequency

=

TΔ

1fs Hz is at least twice the desired

bandwidth of the control system. In Figure 3.2 the output signals y(t) and the reference signal yr(t) (assumed here to be generated externally), are sampled by the ADC system to become the sampled signals at the sample interval j, YJ and YRJ. The control signal at sample interval j, UJ, is output to the system via the digital to analogue converter (DAC). The DAC is represented by a sampler with a hold mechanism which holds the voltage on the output of the DAC until it is updated at the next sampled interval. In this way the controller algorithm output UJ is converted to the control signal u(t).

Page 3-2


Figure 3.2

Page 3-3


3.2 Software Implementation of a Three Term Controller The output of a three-term controller may be written as,

u t e t K K e t d t Kde t

d tp i d( ) ( ). ( ). .( )

= + +

Where, Kp, Ki and Kd are the coefficients of the proportional, integral and differential terms respectively. Varying each of these terms will directly affect the response of the controller and so careful selection is important. If any of these coefficients were to be set to zero, then the whole of the respective term will be removed from the overall control function. From the previous section it is clear that there are three possible control strategies that may require programming on the microcomputer.

a. Proportional only b. Proportional and Integral c. Proportional, Integral and Derivative.

Each one shall now be considered in turn, and developed into a flow chart as the first step in preparing a digital control program. In the following sections, each physical parameter is represented in the way it may be written into a computer program. This is not intended to be an alternative to the symbols used in Section 2, but instead a practical application of them. The symbols used are,

YRJ Reference Signal at Sample Interval j YJ System Output at Sample Interval j EJ - Error Signal at Sample Interval j UPJ - Proportional Term of Control Signal at Sample Interval j UIJ - Integral Term of Control Signal at Sample Interval j UDJ - Differential Term Control Signal at Sample Interval j

Page 3-4


KP - Proportional Constant KI - Integral Constant KD - Differential Constant UJ - Combined Three-Term Controller Output at Sample Interval j

3.2.1 Proportional Control This is the simplest form of control and requires the computer to multiply the error signal by a constant value, KP.

Figure 3-3

From Figure 3-3, the control equation for the computer program can be used to express the Proportional constant as,

UPJ EJ KP= × Where,

EJ YRJ YJ= −

The flowchart shown in Figure 3.4 illustrates how such a procedure would be implemented on a microcomputer. It shows a simple implementation of the control loop whereby the computer outputs the control signal to the DAC and then simply waits for the sample interval to end before commencing the control actions again.

Page 3-5


Figure 3-4

Page 3-6


This form of controller implementation is very wasteful of computer time and would only be used in the simplest of microprocessor implementations. In all other situations the computer would be interrupt driven. In this mode of operation the computer would be performing some main task (such as updating the computer display) and this task would be interrupted every ΔT seconds in order to perform the control task for that sample interval. Upon completion of the control task the computer would resume its main task again. 3.2.2 Proportional and Integral Control The additional control function is the integration term. If the process of integration and its meaning is examined in discrete time format, then,

dt T→ Δ This may be graphically represented as shown in Figure 3.5. In the form of a mathematical series this becomes,

∞

++=0

etc......TΔ)1(fTΔ)0(fdt)t(f

or,

f t d t f t T( ) ( )= Δ From Figure 3.5 the discrete approximation to the integral term is a summation of all the errors up to the present sample interval. Thus, at the jth sample interval, the time is

J T∗ Δ such that;

=

−

=

+∗=→TJΔ

0

J

0N

1J

0N

TΔENTΔEJTΔENdt)t(E

Page 3-7


Figure 3-5

If we define the integral component of the controller output at time step J to be UIJ, then we can write as follows

UIJ EN TN

J

== Δ

0

or ( )UIJ EJ T UI J= ∗ + −Δ 1

Where UI(J-1) is the integral controller component at the previous (j-1)th sample interval. A flow chart for a PI controller is shown in Figure 3.6. Note that normally the digital PI controller would have protective software to prevent it overflowing or under flowing numerically. This "anti-wind up" software is present in all commercial implementations.

Page 3-8


Figure 3-6

Page 3-9


3.2.3 Proportional, Integral and Derivative Control The additional function for this mode of control is the derivative of the error signal,

dt

de.KDED =

Again, in discrete time intervals the equation becomes,

TΔ

EΔ.KDED =

This is shown graphically in Figure 3.7.

Figure 3.7

Page 3-10


Figure 3.8

Page 3-11


Page 3-12

The sample interval ΔT is constant, because the sampling periods are equal, and so it can be included in the value of KD. This eliminates the need for division in a machine code program and saves computation time generally. The change in error at the jth sample interval, EDJ, is given by;

EDJ EJ E J= − −( )1

where E J( − 1) is the error at the previous time step. The control function for derivative control may be expressed as,

UDJ KD EJ E J= × − −( ( 1))

The total PID output expression becomes,

UJ UPJ UIJ UDJ= + + A flowchart to implement the PID algorithm is given in Figure 3.8. 3.3 Implementation of Computer Control In a practical software package, the above procedure would need to be embedded within a much larger program so that essential facilities such as displaying the current input/output/scaling parameters, allowing them to be amended as required, data logging, and so on, are included.


SECTION 4.0 EXPERIMENTATION 4.1 Introduction The experiments described in this section are designed to provide full practical support to the theory given in Sections 2 and 3 of this manual. These experiments, when used in conjunction with the theory, may be considered as a self-contained course in practical control principles and applications. Additionally, once the basic principles have been investigated and understood, the equipment may be easily configured to illustrate a wider range of control topics. This may be necessary to comply with the experimental requirements of a particular syllabus. In each experiment it is assumed that the CE110 Servo Trainer is used in conjunction with the CE120 Controller. For any other combination it will be necessary to modify the instructions provided. It is recommended that each student is supplied at the beginning of the experimental session with a photocopy, or similar, of the relevant experiment. Accordingly, TecQuipment Ltd give their permission for any part of this manual to be copied provided that it is for internal college use only. On completion, the results, graphs and conclusions can then be compared and commented upon against the typical results provided in Section 5. The experimental connection diagrams are given for each experiment to both reduce setting up time as well as simplifying the presentation. This will not only increase the proportion of each laboratory period spent performing the experiments but will also provide a better understanding of what is being achieved by each configuration. It is, however, important that care is taken to identify the correct sockets before a connection is made to achieve the required circuit and performance. It is recommended that the experiments are completed in the order given since the performances of the later assignments are to be compared with the earlier, more basic, ones.

Page 4-1


The blank experimentation circuit diagram provided in Appendix 1 is to allow users to develop their own test circuits. May we suggest that you photocopy the original outline drawings of the CE110/120 and then add the required connection leads. In this way the original may be used to produce an indefinite number of copies. The CE110 Servo Trainer/CE120 Controller combination has been designed to provide a totally self-contained control system, with all devices and facilities required to assemble and investigate a wide range of control situations. However, the experiments provided may be additionally enhanced by the use of commonly available laboratory equipment, such as oscilloscopes and XY/Yt recorders. In the experiments provided, where a transient response is required to be analysed, the use of an optional Yt Chart Recorder has been recommended. Any additional instruments should be suitably connected to the experimental circuits provided - adapters are provided to change from the 2mm connection format used throughout the CE Range to either a 4mm or BNC format. In many cases it may be found convenient to use the Digital Section of the CE120 Controller (and the software supplied) to monitor system performance. By connecting the A-D inputs (up to eight are available) to the relevant points in the analogue control systems, facilities are readily available via a computer to not only acquire and display data but also to 'save' it for later consideration. Throughout the experiments the user will be also be able to produce graphical hard copy of each experiment via a printer.

IMPORTANT

The performance of this equipment, as with any other scientific instrument, is dependent upon it being connected a reliable and stable voltage mains supply. The Serial Number Plate, mounted at the rear of the unit, defines the correct power supply requirements. Should the power supply vary during usage, for whatever reason, it must be anticipated that the performance of the equipment will be affected and the quality of the results impaired. In extreme cases it may be necessary to consider the use of a voltage stabilising device. TecQuipment can accept no responsibility for damage caused to equipment which is connected to an unsuitable supply voltage.

Page 4-2


4.2 Experiment 1: Basic Tests and Transducer Calibration Object: The object of this experiment is to calibrate the circuits of the Servo Trainer, namely the input actuator (the motor circuit) and also the output sensors (the speed and angular position sensors) Apparatus: CE110 Servo Trainer CE120 Controller

IMPORTANT Access is gained to the inertial load of the CE110 Servo Trainer, by a door to

the rear left of the front panel. When operating the equipment you should ensure that the selected inertial load is firmly secured by the knurled nut provided and that the access door is firmly closed. The access door has a

micro-switch which prevents the motor turning when the door is open. It is important therefore when closing the door to ensure the door is firmly shut

and the micro-switch is engaged.

Procedure Part 1 Motor Calibration Characteristic Connect the equipment as shown in Figure E1.1 Initial Control Settings: CE110 Clutch disengaged (i.e. position shaft not connected). Rear access panel firmly closed (check micro-switch contact is made) Smallest inertial load installed. (No additional discs). CE120 Potentiometer in the centre position and reading 0 V. Slowly increase the potentiometer voltage (turning the potentiometer control clockwise) until the motor just starts to turn. This is the size of the positive dead-zone for the motor drive amplifier, enter it into the second row of the Table (E1.1) provided. Increase the potentiometer to 1 V, record the corresponding motor speed from the speed display on the CE110 front panel.

Page 4-3


Figure E1.1 Enter your results in Table E1.1. Increase the potentiometer voltage in 1V steps to 10V and record the corresponding speed in Table E1.1. Repeat the procedure with negative voltages. Repeat the above procedure with the clutch engaged, and complete Table E1.2. Avoid running the Servo Trainer at high speed for prolonged periods with the clutch engaged, as this may cause excessive wear of the gearbox. Plot the results from Table E1.1 and Table E1.2.

Page 4-4


Motor Drive Voltage (V) (Positive)

Motor Speed (rpm)

Motor Drive Voltage (V) (Negative)

Motor Speed (rpm)

0 0 0 0

Dead-Zone Size= 0 Dead-Zone Size= 0

1 -1 2 -2 3 -3 4 -4 5 -5 6 -6 7 -7 8 -8 9 -9 10 -10

Table E1.1 Motor Drive Calibration (Clutch Disengaged)


Motor Speed (rpm)


Motor Speed (rpm)

0 0 0 0

Dead-Zone Size= 0 Dead-Zone Size= 0

2 -2 3 -3 4 -4 5 -5 6 -6 7 -7 8 -8 9 -9 10 -10

Table E1.2 Motor Drive Calibration (Clutch Engaged)

Page 4-5


Part 2: Speed Sensor Setting Connect the equipment as shown in Figure E1.2

Figure E1.2

Page 4-6


Initial Control Settings: CE110 Clutch disengaged Rear Access panel firmly closed Smallest Inertial load installed. (No additional discs). CE120 Potentiometer in the centre position and reading 0 V. Slowly increase the potentiometer voltage until the speed sensor reads 1 V. Enter the corresponding speed reading in Table E1.3. Repeat the process in steps of 1 V for positive and negative speed sensor readings in the range –9 V to +9 V. Plot your results.

Motor Speed (rpm)

(Positive)

Speed Sensor Output

(V)

Motor Speed (rpm)

(Negative)

Speed Sensor Output

(V)

1 -1

2 -2

3 -3

4 -4

5 -5

6 -6

7 -7

8 -8

9 -9

Table E1.3 Speed Sensor Calibration

Page 4-7


Part 3: Angular Position Transducer Calibration Connect the equipment as shown in Figure E1.3

Figure E1.3

Page 4-8


Initial Control Settings: CE110 Clutch Engaged: Rear Access panel firmly closed Smallest inertial load installed CE120 Potentiometer in the centred position and reading 0V output. Increase the potentiometer voltage slowly until the output shaft begins to turn. Measure the angular position sensor output at angular increments of 30° starting at -150° and enter your results in Table E1.4 (Hint: with the output shaft turning at a slow but steady speed, disconnect the potentiometer from the motor drive input and position the output shaft at the desired angle by manually making and breaking the connection to the motor drive). Plot your results.

Indicated Angle (°) Position Sensor Output (V)

-150

-120

-90

-60

-30

0

30

60

90

120

150

Table E1.4 Output Shaft Angular Position Sensor Calibration

Conclusions: In completing this experiment you will have familiarised yourself with the Servo Trainer's main functions and measured their characteristics. You should comment on these characteristics (e.g. are they linear?) and discuss why the motor drive characteristic differs with the clutch engaged and disengaged.

Page 4-9


4.3 Experiment 2: Response Calculating and Measurements Object:

The object of this experiment is to determine the gain, G1 and time constant, T, of the servo-motor transfer function with differing inertial loads where the servo motor transfer function is given by

y

v

G

Tsω =

+1

1

and yω = the speed sensor output voltage v = the motor drive input voltage Apparatus: CE110 Servo Trainer CE120 Controller Chart Recorder Procedure: Part 1 Motor Drive Input to Speed Sensor Output Gain Characteristic The steady state gain relating the motor drive input voltage to the speed sensor output voltage may be calculated by combining the results of Parts 1 and 2 of Experiment 1. Alternatively, the characteristic may be measured directly as detailed in the following procedure. Connect the equipment as shown in Figure E2.1 (do not make the dotted connection) Initial Control Settings: CE110 Clutch disengaged

Rear Access panel firmly closed Smallest inertial load mounted. (No additional discs).

CE120 Potentiometer in the centre position and reading 0 V.

Page 4-10


Figure E2.1

Increase the potentiometer voltage in steps of 1 V to 9 V, recording the corresponding speed sensor output (to do this disconnect the potentiometer/voltmeter connection and make the dotted connection), in Table E2.1.

Page 4-11



Speed Sensor Output

(V)


Speed Sensor Output

(V) 1 -1 2 -2 3 -3 4 -4 5 -5 6 -6 7 -7 8 -8 9 -9

Table E2.1 Motor Drive Voltage/Speed Sensor Characteristics (Clutch

Disengaged

Repeat the process for voltages – 1 V to –9 V. Repeat the procedure with the clutch engaged and enter the results in Table E2.2. Plot the results to obtain the required characteristics and measure the slope in order to obtain the steady state gain G.


Speed Sensor Output

(V)


Speed Sensor Output

(V) Dead-Zone Size= 0 Dead-Zone Size=

0

2 -2 3 -3 4 -4 5 -5 6 -6 7 -7 8 -8 9 -9 10 -10

Table E2.2. Motor Drive Voltage/Speed Sensor Characteristics (Clutch Engaged)

Page 4-12


Part 2 Measurement of Time Constant Connect the equipment as shown in Figure E2.2

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Figure E2.2

Page 4-13


Initial Control Settings: CE110 Clutch disengaged

Rear access panel firmly closed No additional inertial loads mounted

CE120 Potentiometer output set to 5 V. Function Generator set to square wave with frequency of 0.05 Hz and level 1 V

The square wave from the function generator applies a step change of 1 V in either direction about the operating input of 5 V. The transitions in the square wave signal provide step changes in the input. The output of the speed sensor will therefore be a series of step responses. Connect the output of the speed sensor to a chart recorder and plot the step response (suggested chart speed 10mm/second or faster). Repeat the above procedure with each of the inertial loads installed. From the step responses calculate the time constant T of the servo-motor transfer function. Conclusions: Comment on the shape of the motor drive voltage to speed sensor output voltage characteristic. Discuss why the time constant for various inertial loads increases as the size of the load increases.

Page 4-14


4.4 Experiment 3: Proportional Control of Servo Trainer Speed Object: The object of this experiment is to implement a proportional controller of the Servo Trainer speed and to investigate the closed transient response, and the steady state errors. Apparatus: CE110 Servo Trainer CE120 Controller

Chart Recorder Procedure: Part 1: Steady State Errors Connect the equipment as shown in Figure E3.1, this has the corresponding block diagram shown in Figure E3.2. Initial Controller Settings: CE110 Clutch Disengaged

Large inertial load installed Rear Access door firmly closed.

CE120 Potentiometer turned fully anti-clockwise (i.e. set to 0V output) PID Controller: Proportional gain set to 10 and switched in, Derivative and Integral blocks switched out. In this part of the experiment we seek to verify that the steady state error, ess, for a constant reference signal, yr, is given by:-

ey

k Gssr

p

=+1 1

E3.1

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±10V

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±10V

Figure E3.1

Page 4-16


Figure E3.2 First investigate whether the steady state error is proportional to the reference signal, yr. Increase the reference speed, as given by the potentiometer output, in steps of 1 V from 2 V to 10 V and record the corresponding errors signals in Table E3.1. Use Equation E3.1, the value of kp (10) and G1 calculated in Experiment 2 (use G1=1 if you have not done Experiment 2) to calculate the theoretical values of ess for the various values of yr and enter your results in Table E3.1 in the column provided.

Potentiometer Setting

(Reference Speed yr)

(V)

Measured Steady State Error Signal

(V)

Theoretical Steady State Error Signal

(V)

2 3 4 5 6 7 8 9 10

Table E3.1 Steady State Error For Various Reference Speeds

Investigate whether the steady state error is inversely proportional to the controller gain kp. Set the potentiometer to give a reference speed signal, yr, of 5V. Vary the controller gain from 1 to 10 in steps of 1 and record the

Page 4-17


corresponding error signal readings in Table E3.2. Use Equation E3.1 to calculate the theoretical values of the error for each kp value and enter the results in Table E3.2

Potentiometer

Controller Gain kp

Measured Steady State Error Signal

(V)


(V) 1 2 3 4 5 6 7 8 9 10

Table E3.2 Steady State Error for Various Controller Gains

Page 4-18


Part 2: Transient Response Connect the equipment as shown in Figure E3.3, this corresponds to the block diagram of Figure E3.4.

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To ChartRecorder

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30:1

±10V±10V�

0-10V�

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+ve

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±10V

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±10V

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Figure E3.3

Page 4-19


Figure E3.4 Initial Controller Settings: CE110 Clutch disengaged Large inertial load installed Rear access door firmly closed CE120 Potentiometer set to 5V. Function generator set to square wave, frequency of 0.05Hz, offset 0V, level 1V. PID Controller Proportional controller kp=1, integral and derivative blocks switched out. In this part of the experiment we investigate how the transient response of the Servo Trainer is affected by the proportional controller gain kp. Use the square wave output to generate a series of step changes in reference speed and plot the corresponding speed response using the chart recorder (suggested time base 10mm/sec) for proportional gains of kp=0.5, 1,2,4. Calculate the closed-loop time constants, Tc11

, from the graph and compare the

results with the theoretical values obtained using the equation.

TT

k Gclp

l=

+1 1

E3.2 Enter the results in Table E3.3.

Page 4-20


Gain kp Measured Closed Loop Time Constant

(sec)

Theoretical Closed Loop Time Constant

(sec)

0.5 1 2 4

Table E3.3 Comparison Of Measured Closed Loop Time Constants With Theoretical Values

Note for the large inertial load the following approximate values may be used; T=1.5 sec and G1=1

Conclusions: Discuss the steady state error results and in particular give reasons for any sufficient differences between the measured and theoretical values of steady state errors. Discuss the step response results and the differences between the measured and theoretical closed-loop time constants.

Page 4-21


4.5 Experiment 4: Proportional plus Integral Control of Servo Trainer Speed

Object: The object of this experiment is to investigate the effect of proportional plus integral control upon the servo-motor speed control loop in terms of steady state errors, disturbance rejection and transient response. Apparatus: CE110 Servo Trainer CE120 Controller

Chart Recorder Procedure Part 1: Effect of Integral Action on Steady State Errors Connect the equipment as shown in Figure E4.1, this corresponds to the block diagram shown in Figure E4.2 Initial Control Settings: CE110 Clutch Disengaged

Rear access door firmly closed Largest Inertial Load Installed

CE120 Potentiometer turned fully anti-clockwise (i.e. set to 0V input). PID Controller: Proportional gain kp set to 1, integral gain ki set to 0.1 and switched out. Differential gain switched out. Function Generator: Select offset zero, level zero DC,

Page 4-22


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Error Signal

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30:1

±10V±10V�

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0

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±10V

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64

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±10V

Figure E4.1

Page 4-23


Figure E4.2 Slowly increase the potentiometer output voltage to 4V, and observe the steady state error. (for kp=1 this should be approximately 2V). Observe the error signal as integral action takes effect, as follows:- with ki=0.1, press the integrator reset button and switch the integrator into the controller. (Note: it is most important to press the reset button each time an integrator is switched into a circuit. Failure to do so can cause unpredictable results). Observe the speed slowly increase and the error signal slowly decrease to zero as the integrator output increases so as to cancel the error. Switch the integrator out of the circuit. Repeat the above procedure for ki=0.5, 1, 2, 4, 6, and 10. Note that as ki is increased the error is reduced to zero more rapidly until a point is reached when the error overshoots zero, and oscillates before settling to zero. The oscillations became more pronounced as the ki is increased. Part 2: Selection of Integral and Proportional Controller Gains In Part 1 it was demonstrated that integral action will remove steady state error, and also that the speed at which it does so is influenced by the size of the integral gain, ki. In this part the effect of ki upon the speed of response and the experimental choice of kp and ki for optimum transient response are investigated.

Page 4-24


Use the equipment connections for Part 1, specifically:- Connect the equipment as shown in Figure E4.1 and note that this corresponds to the block diagram of Figure E4.2. Initial Control Settings CE110 Clutch disengaged Rear access door firmly closed Largest inertial load installed (all discs). CE120 Potentiometer set to 5V PID controller: Proportional gain kp set to 1, integral gain ki set to 3,

differential gain switched out. Function generator: Select square wave, frequency 0.05Hz, offset zero, level 1V

The square wave generator signal provides a series of step changes in the reference signal which can be used to investigate the step response of the servo-speed control system. Investigate the effect of proportional gain upon the control system step response by plotting the response for values of kp=1, 0.1, and 0.01 (suggested chart recorder speed 2mm/sec). Comment on the shape of the results in terms of speed of response and amount of overshoot. Investigate the effect of integral gain upon the control system step response by setting kp= 1, and plotting the step response for values of ki= 0.5, 1, 5, and 10. Comment on the shape of the resulting step responses in terms of speed of response and amount of overshoot.

Page 4-25


Page 4-26

Part 3: Selection of Proportional and Integral Gains for Specific Transient Response

Use the equipment connections for Part 1. Specifically:- Connect the equipment as shown in Figure E4.1 and note that this corresponds to the block diagram of Figure E4.2 Initial Control Settings CE110 Clutch disengaged Rear access door firmly closed Largest inertial load installed CE120 Potentiometer set to 5V PID Controller : see following procedure. Function generator: Select square wave, frequency 0.05Hz Select the proportional and integral gains to give a closed-loop transient response with desired undamped natural frequency ωn, and compare results with standard plots for second order step responses and discuss possible reasons for differences between the observed responses and the standard second order plots.

Damping Factor

ξ

ki

=

T

Gnω 2

1

kp

−=

1

n

G

1Tξω2

1

0.8

0.6

0.4

Table T4.1 Proportional and Integral Gain Calculations For Desired Natural

Frequency fn = 1

4H z And Various Damping Factors


Page 4-27

Natural Frequency

fn

Natural Angular Frequency ω πn nf= 2

ki

=

T

Gnω 2

1

kp

−=

1

n

G

1Tξω2

1

2 π

1

3

3

π2

1

4

2

π

1

5

5

π2

Table E4.2 Proportional And Integral Gain Calculations For Various

Desired Natural Frequencies and a damping factor of 0.5

Calculate the gains ki, kp, using the equations:-

kT

Gin=

ω 2

1

E4.1

kT

Gpn= −2 1

1

ξω

E4.2

Use the values of open loop gain G1, and time constant T obtained in Experiment 2 (if you have not done Experiment 2, then use G1=1 and T=1.5 sec (for the largest inertial load).

With a desired undamped natural frequency of fn = 1

4Hz (or ωn = π

2 rads/sec)

calculate the values of ki and kp for responses with damping factors of ξ= 1, 0.8, 0.6, and 0.4. Enter your calculated gains in Table E4.1 and plot the step response for each set of gains (suggested chart recorder time base settings 2mm/sec). [Note: it may be necessary to approximate the theoretical gains which you calculate to the nearest available gain setting on the CE120 unit]. Compare the results with standard plots for second order step responses. Discuss possible reasons for differences between the plots.


Now select a damping factor ξ=0.5 and calculate the values of ki and kp for

step responses with natural frequencies fn =1

2

1

3

1

4

1

5, , , and and enter

your calculated gains in Table T4.2. Plot the step responses for each of the set of gains (suggested chart recorder time base settings 2mm/sec).

Note: It may be necessary to approximate the theoretical gains which you

calculate to the nearest available gain setting on the CE120 unit

Conclusions: Discuss the results in the manner mentioned in the experiments. Could it happen that the design method for ki, kp leads to a negative value of kp? How would this be implemented and what does it mean?

Page 4-28


4.6 Experiment 5: Disturbance Cancellation and Feed Forward Control Object: The object of this experiment is to investigate the ability of a proportional, and proportional plus integral controller to reject disturbances caused by the load generator and to show how feed forward control can assist in this respect. Apparatus CE110 Servo Trainer CE120 Controller Chart Recorder Procedure Part 1: Disturbance Rejection Properties of Proportional plus Integral

Controllers Connect the equipment as shown in Figure E5.1 (do not connect the dotted connection). The corresponding block diagram is shown in Figure E5.2 Initial Controller Settings CE110 Clutch Disengaged Rear Access door firmly closed Largest Inertial load Installed (all discs) CE120 Potentiometer set to 6V PID Controller: Proportional Gain kp=1, integral action switched out differential action switched out Investigate the effect of the disturbance applied by the load generator on a proportional speed controller. With a proportional gain of kp=1 measure the error signal. Now apply a 10V load demand on the generator input (do this by making the dotted connection in Figure E5.3) and note the new error signal. Record your results in Table T5.1. Repeat the above exercise for kp= 4, 7, and 10. Note that the change in the error signal should decrease as kp is increased.

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Figure E5.1

Page 4-30


Figure E5.2

kp Error Signal (No Load)

V

Error Signal (10V Load)

V

Change In Error (Column 3-Column 2)

1

4

7

10

Table T5.1 Influence Of Load Disturbances rpm Steady State Error For

Various kp

Investigate the effect of the load generator upon a proportional plus integral controller. Set a proportional gain, kp, of 1 and an integral gain, ki, of 1. Press the reset button and switch in the integral control term. Notice steady state error go to zero as the integral action comes into effect. Connect the speed sensor output to the chart recorder (suggested time base 2mm/second) and plot the speed sensor output as the 10V input is applied to the generator load input. Repeat the above exercise for kp=ki=4, 7, 10. Comment on the disturbance rejection properties for the various gain settings. Use the voltmeter to measure the output of the PID controller block with and without the +10V signal applied to the generator load input. Enter your results in the first row of Table T5.2.

Page 4-31


Part 2 Feed Forward Control for Disturbance Rejection. Connect the equipment as shown in Figure E5.3 (do not connect the dotted connection). This connection diagram corresponds to the control system block diagram shown in Figure E5.4. Initial Control Settings CE110 Clutch Disengaged Rear Access door firmly closed Largest inertial load installed CE120 Potentiometer set to 5V.

PID Controller: Proportional gain kp=1, Integral Gain ki=1, Differential block switched out.

Investigate the effect of the feed forward controller gain upon the load disturbance rejection properties of the speed control system. With a feed forward gain of kf=0.1, and the speed sensor output connected to the chart recorder (suggested chart speed 2mm/sec), make the dotted connection. This will apply the 10V load demand to the generator input and to the feed forward controller block. Observe the plot of the speed response and measure the output of the PID controller block before and after the application of the load. Enter the results in Table T5.2 Repeat the above procedure for kf=0.2, 0.3, and 0.4.

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Block Output

kf

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Figure E5.3

Page 4-33


Figure E5.4

Feed Forward Gain

kf

PI Controller Output With No

Load (V)

PI Controller Output With 10V

Load (V)

Change In Controller Output (V)

No feed forward

(kf=0)

0.1

0.2

0.3

0.4

Table T5.2 Controller Output for Various Levels of Feed Forward Gain

Page 4-34


Page 4-35

For each gain value calculate the change in the output of the PID controller block as the load is applied. Enter the results in Table T5.2 and comment on the plots of load rejection response and the contents of the table. Select a suitable feed forward gain which cancels the influence of loads. Conclusions Discuss the relative load disturbance rejection properties of a) proportional, b) proportional plus integral c) proportional plus integral plus feed forward.


4.7 Experiment 6: Angular Position Control: Proportional Control Object: The object of this experiment is to investigate the angular position control performance and response of the Servo Trainer under proportional control. Apparatus CE110 Servo Trainer CE120 Controller Chart Recorder Procedure Part 1: Basic Tests and Dead-Zone Compensation Connect the equipment as shown in Figure E6.1 (do not make the dotted connection) this corresponds to the block diagram shown in Figure E6.2. Initial Control System CE110 Clutch Engaged

Rear Access Door firmly closed Largest inertial load installed

Reference position dial set to zero, All non-linear blocks switched out of circuit

CE120 Proportional gain kp=1 Investigate the effect of the motor amplifier dead-zone on the accuracy of the position control system. Slowly turn the reference position dial clockwise until the output shaft just begins to move. Note the angle (θ1) on the reference position and turn the dial anti-clockwise, and note the angle (θ2) at which the output dial begins to move. The difference between (θ1-θ2) between these two angles is the effective dead-zone width of the controller. Note the difference in Table E6.1 Repeat above procedure for kp=2, 4, 6, 8, 10.

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alignment

OutputAngleDial

ReferenceAngle Dial

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Figure E6.1

Page 4-37


Figure E6.2

Proportional Gain kp

Width Of Output Angle Dead-Zone (°)

1

2

4

6

8

10

Table E6.1 Effective Dead-Zone: In Output Angle For Proportional

Controller

Notice that the dead-zone decreases in width as the controller gain is increased. Investigate next the compensation for dead-zone using anti-dead-zone. Set the anti-dead-zone block alignment control (see Figure E6.1), and the level to zero. Select a proportional gain of kp=1, and switch on the anti-dead-zone block. Slowly increase the anti-dead-zone level to 1V and turn the reference position potentiometer. Note that the effective dead-zone in the output angle is much reduced and the error signal is correspondingly reduced. Increase the anti-dead-zone level to 1.5V (this is approximately the width of the amplifier dead-zone with the clutch engaged and as measured in Experiment

Page 4-38


1). Note that the dead-zone is now almost completely removed. Increase the anti-dead-zone level to 2V. The Servo Trainer will now start 'humming' due to high frequency instability due to excessive anti-dead-zone. Decrease the anti-dead-zone to 1V. Discuss why excessive anti-dead-zone may cause high frequency oscillations. Part 2: Servo Trainer Step Response Connect the equipment as shown in Figure E6.1. This corresponds to the block diagram of Figure E6.2 (do not make the dotted connection). Initial Control Setting CE110 Clutch engaged Rear access door firmly closed Largest inertial load installed (all discs) Reference position dial set to zero Anti-dead-zone set to 1 V and switched on CE120 Potentiometer set to 1 V Proportional gain kp =1 Investigate the position controller step response for various levels of proportional gain. Introduce a step change in reference position using the potentiometer (to do this make the dotted connection in Figure E6.1). Plot the angular position output signal using a chart recorder (suggested chart speed 5 mm/sec). Repeat for kp= 2, 4, and 8 and discuss the relative performance of the controller with the various gains. Conclusions Comment on the use of anti-dead-zone as a method of cancelling amplifier dead-zone.

Page 4-39


4.8 Experiment 7: Angular Position Control: Velocity Feedback

Object: The object of this experiment is to investigate the use of velocity feedback as a means of improving the transient response of an angular position control system. Apparatus CE110 Servo Trainer CE120 Controller Chart Recorder Procedure Connect the equipment as shown in Figure E7.1 (do not make the dotted connection). Note that this corresponds to the block diagram shown in Figure E7.2

Initial Control Settings CE110 Clutch Engaged Rear Access Door firmly closed Largest inertial load installed (all discs) Reference position dial set to zero CE120 Proportional gain kp1=1, kp2=5 Velocity feedback gain kv=0.01 Function generator: select square wave, frequency 0.1Hz, offset zero level 1V. Investigate the effect of very large proportional gains on the system transient and steady state error response. Plot the square wave response as the proportional gain is increased from kp (= kp1kp2)= 5 to 100 in the following steps: kp= 5, 10, 20, 40, 80, 100. Note the corresponding error signal in Table E7.1. The chart speed suggested for the plotter is 5 mm/sec.

Page 4-40


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kP1 kP2

kvError Signal

G M

30:1

±10V±10V�

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0

+ve

-ve

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6

±10V

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+ve-ve

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Figure E7.1

Page 4-41


Figure E7.2

Proportional Gains

kp = kp1 kp2

Error Signal

(V)

5

10

20

40

80

100

Table E7.1 Steady State Errors For Verifying Proportional Gains

Note that the steady state error becomes rather small, even without anti-dead-zone compensation, when the proportional gain becomes very large (e.g. greater than 80). However, the transient response becomes more oscillatory. Thus, while the controller designer would like to have very large kp for good steady state performance, the associated transient performance is not acceptable. Investigate how the transient response can be improved for a high gain proportional controller by the addition of velocity feedback. Make the dotted connection in Figure E7.1. This completes the velocity feedback loop shown in Figure E7.2. Set the proportional gain kp (= kp1kp2) to 80 and the velocity

Page 4-42


feedback gain kv=1. Plot the square wave response of the controller for kv=1, 3, 5, 8, and 10. Note the reduction in oscillatory behaviour in the step responses which are formed by the square wave transitions (suggested chart speed 5 mm/seconds) Investigate how the inertial load influences the position controller response. Open the rear access panel and remove the inertial load. Close the access panel firmly to ensure the micro-safety switch re-engages. Plot the square wave responses again with kp= 80 and kv= 10, 5, 3, 1, and 0. Discuss the responses. Conclusions Explain why velocity feedback has a beneficial effect on the system transient response.

Page 4-43


4.9 Experiment 8: Angular position control and the influence of non-linearity

Object: The object of this experiment is to illustrate the practical effects of non-linearity upon the performance of a position control Servo Trainer using the non-linear blocks provided. Apparatus: CE110 Servo Trainer CE120 Controller Chart Recorder Procedure: Part 1: Amplifier Saturation Connect the equipment as shown in Figure E8.1 (this corresponds to the block diagram of Figure E8.2) Initial Control Settings: CE110 Clutch Engaged Rear Access Door Firmly Closed Large inertia load installed (all three discs) Reference signal set to 0 degrees Saturation non-linearity switched on and set to 10 V, all other non-linear elements switched out. CE120 Proportional gain kp1= 8, kp2= 10 Velocity feedback gain kv= 0.01 Function generator: select square wave, Frequency 0.05 Hz offset zero. Adjust the level control to give a +1 V (step). This gives a 1V pulse for ten seconds, then no pulse for ten seconds while the system recovers.

Page 4-44


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kV

To ChartRecorder

To ChartRecorder

Saturation Level Hysteresis Level

kP2kP1

G M

30:1

±10V±10V�

0-10V�

0

+ve

-ve

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2

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8

6

±10V

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±10V

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+ve-ve

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6 108

64

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±10V

Figure E8.1

Page 4-45


Figure E8.2

Investigate the influence of amplifier saturation level upon the Servo Trainer step response. With the saturation level set to 10 V, plot the input to the motor drive amplifier and the output position step response using the square wave input from the function generator (suggested chart speed 5mm/sec). Repeat the above procedure for saturation levels of 8 V, 5 V, 3 V. What happens when the saturation level is reduced to 1 V and why? Investigate the effect of input step amplitude on the Servo Trainer response when saturation is present. Set the saturation level to 4 V and plot the response of the servomotor input and the angular position response for square wave input levels of 0.5 V, 2 V and 5 V. Comment on the response characteristics. Part 2: Hysteresis Connect the equipment as shown in Figure E8.3, this connects the hysteresis block between the angular position output and the controller. In this way it simulates backlash in the gearbox. Initial Control Settings CE110 As for Part 1 except saturation non-linearity is now switched out. The hysteresis block switched on and its control set to 0. All other non-linear elements switched out. CE120 As for Part 1 except kp1=1 (giving a total gain of 1 x 10 =10)

Page 4-46


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To ChartRecorder

To ChartRecorder

Saturation Level Hysteresis Level

kP2kP1

G M

30:1

±10V±10V�

0-10V�

0

+ve

-ve

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±10V

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±10V

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+ve-ve

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6 108

64

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±10V

Figure E8.3

Page 4-47


Investigate the effect of hysteresis/backlash non-linearity upon the Servo Trainer step response. With the hysteresis level set to 0V, plot the input to the motor drive amplifier and the output position step response (suggested chart speed 5mm/sec) using the square wave input from the function generator. Repeat the above procedure for hysteresis levels of 1V, 2V, 3V, 4V and 7V. Explain what happens when the hysteresis level is increased and why? Conclusions Discuss the ways in which non-linearity adversely affects system response.

Page 4-48


4.10 Experiment 9: Non-linear System Characteristics Object: The object of this experiment is to examine the characteristics of the non-linear elements and to investigate their influence upon typical signals. Apparatus: CE110 Servo Trainer CE120 Controller Chart Recorder Procedure Part 1: Non-Linear System Characteristics (static)

Connect the equipment as shown in Figure E9.1 (do not make the dotted connection). Initial Control Setting: CE110 Rear Access panel firmly closed. All non-linear elements switched out CE120 Potentiometer turned fully anti-clockwise. Investigate the dead-zone characteristic for a dead-zone width of 2V. Switch the dead-zone on, adjust the dead-zone offset to zero and set the dead-zone level at 2V. Use the CE120 voltmeter to set the potentiometer to -10V (disconnect the existing connection to the voltmeter and put the dotted connection in place to do this). Note the corresponding non-linear block output in Table E9.1. Increase the potentiometer voltage to -9V and repeat the above procedure up to +10V. Plot the resulting characteristic.

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Anti-deadzoneLevel

Anti-deadzoneOffset

DeadzoneLevel

DeadzoneOffset

SaturationLevel

HysteresisWidth

G M

30:1

±10V±10V�

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0

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±10V

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Figure E9.1

Page 4-50


Dead-Zone Input Voltage Dead-Zone Output Voltage

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

+1 +2 +3 +4 +5 +6 +7 +8 +9 +10

Table E9.1 Dead-Zone Characteristics for a Dead-Zone Width of 2V

Investigate the anti-dead-zone characterised for an anti-dead-zone width of 2V. Switch off the dead-zone. Switch on the anti-dead-zone, adjust the offset to zero and set the level to 2V. Set the potentiometer to -8V and note the corresponding non-linear block output in Table E9.2. Repeat the above procedure in potentiometer voltage steps of 1V to +8V. Plot the resulting characteristic.

Page 4-51


Anti-Dead-Zone Input Voltage

Anti-Dead-Zone Output Voltage

-8 -7 -6 -5 -4 -3 -2 -1 0

+1 +2 +3 +4 +5 +6 +7 +8

Table E9.2 Anti Dead-Zone Characteristics for an

Anti-Dead-Zone Width of 2V Investigate the saturation characteristic for a saturation of 5V. Switch off the anti-dead-zone, switch on the saturation characteristics and set the level to 5V. Set the potentiometer to -10V and note the corresponding non-linear block output in Table E9.3. Repeat the above procedure in potentiometer voltage steps of 1V to +10V. Plot the resulting characteristic.

Page 4-52


Saturation Input Voltage Saturation Output Voltage

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

+1 +2 +3 +4 +5 +6 +7 +8 +9 +10

Table E9.3 Saturation Characteristic for a Saturation Level of 5V

Investigate the hysteresis characteristic for a hysteresis width of 1V. Switch off the saturation characteristic. Switch on the hysteresis characteristic and set the width to 1V. Starting from a potentiometer setting of 0V, decrease the potentiometer voltage to -5V (Note - the hysteresis is a directional non-linearity so that it is important in which direction a voltage is approached. Therefore do not go past a voltage and then go back to it as this will alter the measured characteristic. Make sure that you approach the specified voltage from the specified direction). Record the output voltage in Table E9.4. Increase the potentiometer voltage in steps of 1V to +5V, noting the output voltage at each stage and on no account decreasing the voltage during the steps. Enter the results in Table E9.4

Page 4-53


Next start to decrease the potentiometer voltage in steps of 1V to -5V and note the results in Table E9.3. Plot the characteristic. Hysteresis Input

Voltage (Increasing)

Hysteresis Output Voltage

Hysteresis Input Voltage

(Decreasing)


-5 5 -4 4 -3 3 -2 2 -1 1 0 0

+1 -1 +2 -2 +3 -3 +4 -4 +5 -5

Table E9.4 Hysteresis Characteristic for a Hysteresis Width of 1V

Page 4-54


Part 2: Non-linear Characteristic (Dynamic)

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Figure E9.2

Page 4-55


Page 4-56

Connect the equipment as shown in Figure E9.2 Initial Control Setting CE110 Rear Access Panel firmly closed All non-linear elements switched out. CE120 Function generator: select sine wave, Frequency 0.02, offset zero, level 5V. Investigate the anti-dead-zone characteristic's influence on a sine wave. Switch the anti-dead-zone on, adjust the offset to zero and set the level of anti-dead-zone to 1.5V. Plot the anti-dead-zone block response using the chart-recorder. (suggested time-base, 1mm/sec) Investigate the dead-zone characteristics influence on a sine wave. Switch the anti-dead-zone off. Switch the dead-zone on, adjust the offset to zero and set the level to 2V. Plot the dead-zone block response using the chart-recorder (suggested time-base, 1mm/sec) Investigate the hysteresis characteristics influence on a sine wave. Switch the saturation off. Switch the hysteresis on and adjust the hysteresis width to 2V. Plot the hysteresis block response using the chart recorder. (suggested time-base, 1mm/sec). Conclusions: Discuss how the various non-linearities would affect a control system.


SECTION 5.0 RESULTS AND COMMENTS Note: These results are typical only – actual results may differ slightly. 5.1 Experiment 1: Results and Comments Your results should comprise the following completed tables and plotted calibration curves. a) Motor drive calibration (clutch disengaged and engaged) b) Speed sensor calibration c) Output shaft angular position sensor calibration Part 1: Motor Calibration Characteristic

Motor Drive Voltage, V, (positive)

Motor Speed (rpm)

Motor Drive Voltage ,V, (Negative)

Motor Speed (rpm)

0 0 0 0

0.50

(enter dead-zone size)

0 -0.50

(enter dead-zone size)

0

1 130 -1 -130

2 325 -2 -320

3 525 -3 -525

4 726 -4 -723

5 929 -5 -926

6 1145 -6 -1127

7 1328 -7 -1330

8 1530 -8 -1533

9 1740 -9 -1735

10 1943 -10 -1951

Table E.1.1 Motor Drive Calibration (Clutch Disengaged)

Page 5-1



Motor Speed (rpm)


Motor Speed (rpm)

0 0 0 0

1.60

(enter dead-zone

width)

0 -1.60

(enter dead-zone

width)

0

2 145 -2 -140

3 360 -3 -365

4 540 -4 -535

5 745 -5 -740

6 940 -6 -915

7 1140 -7 -1110

8 1335 -8 -1300

9 1520 -9 -1490

10 1700 -10 -1675

Table E1.2 Motor Drive Calibration (Clutch Engaged)

The completed Tables E1.1 and E1.2 give typical calibration data required. Figure E5.1.1 shows the corresponding plots. Note the dead-zone is increased with the clutch engaged because of the increased static friction in the gearbox. The slope of the characteristic with the clutch engaged is nominally unaltered since the rotating friction coefficient is low for the gearbox. From Figure E5.1.1 the shape of the linear region gives the motor drive sensitivity as

k rpm volt rev s volt

k rev volt

i

i

= =

=

1550

8

1550

60 8

3 2

/.

/ .

. / sec.

where we are using one revolution as the angular unit.

Page 5-2


Figure E5.1.1. Motor Calibration Curve

Page 5-3


Page 5-4

Part 2: Speed Sensor Calibration The completed Table E1.3 gives a typical set of results. Figure E5.1.2. shows the corresponding plots.

Motor Drive Speed (rpm)

Speed Sensor Input (V)

Motor Speed (rpm)

Speed Sensor Output (V)

198 1 -202 -1

410 2 -405 -2

605 3 -601 -3

805 4 -796 -4

1003 5 -1000 -5

1198 6 -1197 -6

1402 7 -1400 -7

1597 8 -1500 -8

1806 9 1800 -9

Table E1.3 Speed Sensor Calibration

From Figure E5.1.2 the speed sensor sensibility, kω1 is given by the slope

kV

rpmV rev

volt rev

ω = =

=

6

1200

6 60

1200

0 3

./ .se

. / sec.

c

Figure E5.1.2 Speed Sensor Calibration


Page 5-5

Part 3: Angular Position Transducer Calibration The completed Table E1.4 shows typical results for this characteristic. Figure E5.1.3 shows the corresponding plot.

Indicated Angle (°)

Position Sensor Output (V)

-150 -8.50

-120 -6.85

-90 -5.0

-60 -3.4

-30 -1.74

0 0

30 1.65

60 3.3

90 4.95

120 6.6

150 8.3

Table E1.4 Output Shaft Angular Position Sensor Calibration

From Figure E5.1.3, the slope of the characteristic gives the transducer sensitivity as:-

Angular Position Sensor Sensitivity = K Vθ = =5 5

1000 055

.. / degree

Alternatively, and because the speed and motor sensitivity are measured with respect to revolutions per second, the angle transducer sensitivity can be written as

kθ = 20 volts/revolution.

The sensitivity of the amplifier/motor and the associated sensors can now be depicted in schematic form:- where kω = 0.3 volt sec/rev ki = 3.2 rev/sec volt kθ = 20 volts/rev


Figure E5.1.3 Angle Position Sensor Calibration

Page 5-6


5.2 Experiment 2: Results and Comments Your results should comprise; a) Completed Tables E2.1, E2.2 for the steady state motor drive voltage to

speed sensor output. b) The step responses of the Servo Trainer with various inertial loads. Part 1: Motor Drive Input to Speed Sensor Output Gain Characteristic The completed Tables E2.1 and E2.2 give typical calibration data for this characteristic. Figure E5.2.1 is a plot of these characteristics from which the gain G1 is obtained as the slope. Thus clutch disengaged:

GV

V1

6 1

61= ≈

.

clutch engaged:

GV

V1

6 2

61= ≈

.


Speed Sensor Input

(V)


Speed Sensor Output

(V)

1 0.7 -1 -0.6

2 1.65 -2 -1.65

3 2.68 -3 -2.65

4 3.65 -4 -3.67

5 4.66 -5 -4.70

6 5.67 -6 -5.74

7 6.68 -7 -6.66

8 7.70 -8 -7.75

9 8.80 -9 -8.70

Table: E2.1 Motor Drive Voltage/ Speed Sensor Characteristic (clutch disengaged)

Page 5-7



Speed Sensor Input

(V)


Speed Sensor Output

(V)

1.6 Dead-Zone value

0 -1.5 Dead-Zone value

0

2 0.7 -2 -0.75 3 1.85 -3 -1.70 4 2.75 -4 -2.80 5 3.80 -5 -3.75 6 4.70 -6 -4.73 7 5.70 -7 -5.62 8 6.65 -8 -6.55 9 7.55 -9 -7.5 10 8.20 -10 -8.42

Table E2.2 Motor Drive Voltage/Speed Sensor Characteristic (clutch engaged)

Figure E5.2.1

Page 5-8


Part 2: Measurement of Time Constant

Figure E5.2.2 Step Response Tests With Various Inertial Loads (Chart Speed 10mm/sec)

Page 5-9


Figure E5.2.2 shows typical step response tests for various inertial load. The step amplitude was 2V from an initial steady level of 4V. Using the construction for time constant calculation given in Section 2.2.5 the following typical results were obtained from Figure E5.2.2

Large inertial load T ≈ 1.5 sec Medium inertial load T ≈ 1.0 sec Small inertial load T ≈ 0.5 sec Note: The measurements are approximate, and typical of those found with the Servo Trainer. The time constant increases with inertial load because, from Equation 2.5b (Section 2.2.2), the time constant is directly proportional to the inertia. The inertia of a disc is approximately proportional to its thickness, hence, the increase in T is approximately proportional to the thickness of the inertial loads.

Page 5-10


5.3 Experiment 3: Results and Comments Your results should comprise the following a) Completed Tables E3.1, E3.2; b) Measured closed-loop step responses with proportional controller

gains kp = 0.5, 1, 2, 4; c) A comparison of measured closed-loop time constants with those

predicted by theory.

Potentiometer Output (reference Speed yr)

(V)

Measured Steady Speed Error Signal

(V)


(V)

2 0.22 0.18

3 0.32 0.27

4 0.42 0.36

5 0.5 0.45

6 0.59 0.54

7 0.68 0.64

8 0.77 0.77

9 0.86 0.86

10 0.95 0.95

Table E3.1 Steady State Error for Various Reference Speeds

Proportional Controller Gain kp

Measured Steady State Error Signal (V)

Theoretical Steady State Error Signal (V)

1 2.69 2.5

2 2.34 1.67

3 1.53 1.25

4 1.23 1

5 1.01 0.83

6 0.86 0.71

7 0.73 0.62

8 0.64 0.55

9 0.56 0.5

10 0.51 0.45

Table E3.2 Steady State Errors for Various Controller Gains

Page 5-11


Part 1: Steady State Errors The completed Tables E3.1, E3.2 give typical results for steady state errors. The theoretical values of steady state error ess where is calculated using

ey

k Gssr

p=

+1 1

and G1 =1

Notice that the difference between the measured and theoretical steady state errors decreases as the value of yr increases. This is because the small dead-zone in the Servo Trainer introduces errors which become smaller as the reference signal becomes very much larger than the dead-zone width. Part 2: Transient Response Figure E5.3.1 shows typical step responses of the proportional speed controller with the gains kp = 0.5, 1,2,4. Using the construction outlined in Section 2 the experimental closed-loop time constants are tabulated in Table E3.3, along with the theoretical values from the Equation shown in the experiment.

Figure E5.3.1 Transient Response of Servo Trainer under Proportional Control of Speed (Chart Speed 10mm/sec)

Page 5-12


Gain kp Measured Closed Loop

Time Constant (sec)

Theoretical Closed Loop Time Constant

(sec)

0.5 1.0 1.00 1 0.75 0.75 2 0.48 0.50 4 0.34 0.30

Table E3.3 Comparison of Measured Closed Loop Time Constants with Theoretical Values

Given the inaccuracy associated with measuring time constants from graphs the correspondence between actual and theoretical time constants is good. The actual and theoretical values will start to diverge at far higher values of kp, since under high gain conditions the drive amplifier saturates under transient conditions.

Page 5-13


5.4 Experiment 4: Results and Comments Your results should comprise: a) A set of step response plots for ki=3 and various kp settings b) A set of step response plots for kp=1 and various ki settings completed

Tables E4.1 and E4.2 c) A set of step response plots corresponding to the gain setting in

Tables E4.1 and E4.2

Damping Factor ξ ki

=

T

Gnω 2

1

kp

=−

2 1

1

ξωnT

G

1 3.7 3.7

0.8 3.7 2.8

0.6 3.7 1.8

0.4 3.7 0.9

Table E4.1 Proportional and Integral Gain Calculations for a Closed Loop Desired Natural Frequency fn=¼hz and Various Damping Factors

Natural Frequency fn

Natural Angular Frequency ωn

ki

=

T

Gnω 2

1

kp

=−

2 1

1

ξωnT

G

1

2 π 14.8 3.7

1

3

2

3

π 6.6 2.1

1

4

π2

3.7 1.4

1

5

2

5

π 2.4 0.9

Table E4.2 Proportional And Integral Gains. Calculation for a Closed Loop

Damping Factor of ξ=0.5 and Various Natural Frequencies

Page 5-14


Part 1: Effect of Integral Action on Steady State Error Your observation should be that integral action removes steady state error. The rate at which it removes the steady state error increases as ki increases, with eventually oscillations beginning to appear. Part 2: Selection of Integral and Proportional Controller Gains.

Figure E5.4.1. P+I Controller With Various Proportional Gains (Chart Speed 2mm/sec)

Figure E5.4.1 shows typical responses as the proportional gain is altered. The damping is altered, but the natural frequency (or speed of response) does not change. Figure E5.4.2 shows typical responses as the integral gain is altered. The damping and the speed of response are effected by changes in ki.

Page 5-15


Figure E5.4.2. P+I Controller With Various Integral Gains (Chart Speed 2mm/sec)

Page 5-16


Part 3: Selection of Proportional and Integral gains for specified Transient Response

Figure E5.4.3 P+I Controller Design For Various Damping Factors (Chart Speed 2mm/sec)

Page 5-17


Figure E5.4.4 P+I Controller Design For Various Natural Frequencies

(Chart Speed 2mm/sec)

The completed Tables E4.1 and E4.2 give the theoretical gains required. Note that the smallest value of kp is 0.005 which is smaller than the smallest available on the CE120. Approximate this value with kp = 0.01. Likewise other values are approximately to the nearest decimal place. Figures E5.4.3, E5.4.4 show typical step response results associated with these gain settings. Note that the results correspond approximately with the standard second order response which would be expected. The exception is the plot for fn=1/2 which is more oscillatory than the corresponding standard response. This is because in representing the servo-transfer function as first order, the dynamics of the motor armature circuit were neglected (see Section 2.2.2). At fast response rates this is no longer a valid assumption.

Page 5-18


The proportional gain kp is given by:

kT

Gpn=

−2 1

1

ξ ω

Thus for ξ sufficiently small, the required value of kp will become negative. This means that the proportional control action provides a positive feedback component. This can be realised in the CE120 by making up a P + I controller using the separate amplifier blocks rather than the PID block.

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5.5 Experiment 5: Results and Comments Your results should comprise:- a) A completed Table E5.1 b) A set of plots showing the speed sensor output for various integral

and proportional gains when a 10V load demand is applied to the load generator.

c) A completed Table E5.2 d) A set of plots showing the speed sensor output for various feed

forward controller gains when a 10V load demand is applied to the load generator.

Part 1: Disturbance Rejection Properties of Proportional Plus Integral

Controllers Table E5.1 shows typical results for the influence of proportional gain upon disturbance rejection. As kp is increased the relative effect of a load disturbance on the control system is reduced but not completely removed.

kp Error Signal No Load,

V

Error Signal 10V Load

V

Change in Error (Column 3 - Column 2)

1 3.21 3.64 0.43

4 1.5 1.92 0.42

7 0.82 1.12 0.3

10 0.59 09.82 0.23

Table E5.1 Influence of Load Disturbance Upon Steady State Error For

Various kp

Page 5-20


Figure E5.1 Plot of Speed Response to 10V Load Disturbance applied to Generator for Various Proportional & Integral Controller Gains

Figure E5.1 shows plots of the speed response during the application and removal of a 10V input to the generator load control input. Note that the integral action removes the steady state value of the disturbance, and that the transient effect is reduced as the controller gains are increased.

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Part 2 Feed Forward Control for Disturbance Rejection

Feed Forward Gain, kf

PI Controller Output With No Level Input (V)

PI Controller Output With 10V

Load Input (V)

Change In Controller Output (V)

No Feed Forward (kf=0)

5.5 8.1 2.6

0.1 5.5 7.1 1.6

0.2 5.5 6.1 0.6

0.3 5.5 5.1 -0.4

0.4 5.5 4.1 -1.4

Table E5.2 Controller For Various Levels of Feed Back Gain

Figure E5.2 Plot of Speed Responses to 10V Load Disturbances applied to

Generator for Various Feed Forward Gain Values

Page 5-22


Table E5.2 shows typical results for the effect of feed forward on the PI controller contribution to the control system with a load disturbance. Notice that as the feed forward gain is increased, the change in the PI control output during the application of a load reduces. A level of feed forward gain (kf = 0.3) is deduced at which the PI controller output changes hardly at all. At this level of kf the feed forward controller is almost completely cancelling the effect of the load disturbance. Increasing kf beyond this level over compensates for the load and the PI controller output will reduce correspondingly. (e.g. kf = 0.4). Figure E5.2. shows plots of the corresponding transient response of the speed output during load application with various levels of kf. Note that with kf = 0.3 the speed is almost unaffected by the load disturbance. Your results should show that the proportional plus integral controller with feedback is the best way of removing the effects of load disturbances. The additional comment should be made that the load may affect the system differently at different operating points. For example, reduce the reference speed potentiometer voltage from 6V to 4V and apply feed forward. You will find kf should be 0.2 to achieve cancellation of the load disturbance. This is because the load imposed by the generator is speed dependent. In industry the feed forward gain would be adjusted according to the operating speed of the system.

Page 5-23


5.6 Experiment 6: Results and Comments Your results should comprise: a) A table (E6.1) of effective dead-zone widths for various controller

gains. b) A set of step response plots for the position input for various

controller gains.

Proportional Gain

kp

Width of Output Angle Dead-Zone (°)

1 35

2 10

4 7

6 4

8 3

10 2

Table E6.1 Table of Effective Dead-Zone in Output Angle for Proportional

Controller

Part 1: Basic tests and Dead-Zone Compensation A typical set of results for dead-zone widths are contained in Table E6.1. The use of the anti-dead-zone should reduce the dead-zone width and eventually cancel it completely. If excessive anti-dead-zone is used it over compensates for the amplifier dead-zone and introduces a very high gain element in the motor drive characteristic. Most systems will become unstable for very high feedback gains. The Servo Trainer is no exception and oscillates with an amplitude equal to that of the excess anti-dead-zone. Part 2: Servo Trainer Step Response Figure E5.6.1 shows typical step responses for the Servo Trainer position controller under proportional control, kp. The response becomes faster as the

Page 5-24


value of kp is increased, and more oscillatory. The tendency for excess anti-dead-zone to cause oscillations increases as kp is increased.

Figure E5.6.1 Step Response of Position Control System with Proportional Controller. (Chart Speed 5mm/sec)

In general anti-dead-zone is used to remove the majority of large amounts of dead-zone. The remainder is removed by increasing the proportional gain. In the experiment, the use of kp=10 alone decreased the dead-zone to 4°, combined with a little anti-dead-zone the final closed-loop dead-zone was very small. Increasing kp, however, makes the system more oscillatory (see Figure E5.6.1.). This can be compensated for using velocity feedback (see Experiment 7).

Page 5-25


5.7 Experiment 7: Results and Conclusions Your results should comprise: a) A completed Table E7.1 b) Step/square wave responses with various proportional controller

gains. c) Step/square wave responses with various velocity feedback gains,

with and without inertial loading.

Proportional Gain kp=kp1 kp2

Steady State Error (V)

5 -0.15

10 -0.03

20 -0.03

40 -0.02

80 -0.01

100 -0.00

Table E7.1 Steady State Errors for varying Proportional Gains

The completed Table E7.1 shows typical results for the steady state error experiment. Note that for Experiment 2 an error of 0.05 corresponds to an angle of 1°. Figure E5.7.1 shows the influence of increasing the proportional gain. Note response becomes very oscillatory for high gains. It would become even more oscillatory if it was not for the fact that the servo-amplifier is being driven into saturation by the highly amplified error signal. The conclusion should be that high gain is good for removing steady state error without recourse to anti-dead-zone, but that it leads to a very oscillatory response.

Page 5-26


Figure E5.7.1 Position Controller with various values of Proportional Gain

(Chart Speed 5mm/sec)

Page 5-27


Figure E5.7.2 Position Controller with various levels of Velocity Feedback

(kp=80) (Chart Speed 5mm/sec)

Page 5-28


Figure E5.7.3 Position Controller with various levels of Velocity Feedback and minimum Inertial Loading

Figure E5.7.2 shows the influence of velocity feedback. Notice that the velocity feedback reduces the oscillatory behaviour as it is increased. This is confirmed by Equations 2.25 (Section 2) which show that proportional and velocity feedback together lead to a second order response in which kp

Page 5-29


controls the natural undamped frequency, ωn, and (for a fixed ωn), kv controls the damping faster, e.g.

ω

ξω

np

nv

k G G

T

k G

T

2 1 2

12 1

=

= +

E5.7

Using the actuator/sensor sensitivities calculated in Experiment 1, it is possible to determine appropriate values of G1, and G2. Likewise, the time constant T evaluated in Experiment 2 may be used, to find theoretical values for ωn, and ξ from Equation E5.7. However, with the high gains used in the experiment the amplifiers are saturating such that the linear analysis calculations (Equation E5.7) will be only approximate. See the next experiment for confirmation of this. Figure E5.7.3 show the influence of reducing the motor inertial load. The response is much faster and requires less velocity feedback to reduce the oscillations. Velocity feedback in a position control system is useful because it feeds back the rate of change of the output position. Thus it gives some prediction of large changes in the angle and supplies a compensating signal. The net result is a less oscillatory response.

Page 5-30


5.8. Experiment 8: Results and Conclusions

Figure E5.8.1. Servo system Step Response With Various Levels Of Amplifier Saturation (Chart Speed 5mm/sec)

Your results should comprise:- a) Plots of the Servo Trainer square wave/step response with various

levels of amplifier saturation and for various reference input levels. b) Plots of the Servo Trainer response with various levels of hysteresis.

Page 5-31


Figure E5.8.2. Servo Trainer Step Response With Saturating Amplifier And Various Input Amplitudes (Chart Speed 5mm/sec)

Part 1: Amplifier Saturation Figure E5.8.1 shows the typical effects of saturation upon the Servo Trainer response. As the saturation level is decreased, the controller spends more time in saturation mode. This limits the speed of output response and causes the step response to differ significantly from the linear response. When the saturation level is at 1V, it is below the amplifier dead-zone. Therefore no signal gets through to the motor and the controller fails to function. As the size of the input demand increases, the amplifier spends more time in saturation such that for long periods the output is 'slewing' at a rate

Page 5-32


determined solely by the saturation level. Figure E5.8.2 shows typical results illustrating the longer periods in saturation as the input amplitude increases.

Mo

tor

Am

pli

fier

Inp

ut

Po

siti

on

An

gle

Ou

tpu

t

Hy

ster

esis

=0

Hy

ster

esis

=3

Hy

ster

esis

=7

Figure E5.8.3 Servo Trainer Step Response with various levels of Hysteresis (kp = 10, kv=0.01) (Chart Speed 5mm/sec)

Page 5-33


Part 2: Hysteresis Figure E5.8.3 shows typical results for various levels of hysteresis . The system oscillates uniformly as soon as you add hysteresis, but the important point is that the response period increases (frequency becomes lower) as you increase the hysteresis. This is because the hysteresis delays the feedback signal so the system takes longer to respond. Note that the need to move across a backlash region at each step input makes the output more oscillatory. In the limit a certain width of backlash is achieved that the Servo Trainer inertia builds up a sufficient amount of momentum as it crosses the backlash region to make it oscillate about zero. Generally, amplifier saturation is a fact of life which is not necessarily bad. It does not necessarily destabilise the closed-loop system, but it sets an upper limit on the speed at which the output can move between two levels. This is called the 'slew rate' of the system and must be kept in mind when designing the linear part of the system. Hysteresis or backlash is to be avoided at all costs since it limits the accuracy of control. Also it causes the system to oscillate (at worst) and fail at best. Mechanical design changes are usually the only remedy. Dead-Zone is likewise undesirable, but its effect is reduced by high controller gain, anti-dead-zone or dither. These last two remedies cause some high frequency oscillations in the system and are to be handled with care to prevent mechanical damage or wear.

Page 5-34


5.9 Experiment 9: Results and Comments


-10 -7.9

-9 -7.0

-8 -5.9

-7 -4.9

-6 -3.9

-5 -2.9

-4 -1.9

-3 -0.8

-2 0

-1 0

0 0

+1 0

+2 0

+3 1.1

+4 2.1

+5 3.1

+6 4.1

+7 5.1

+8 6

+9 7

+10 8

Table E9.1 Dead-Zone Characteristics for Dead-Zone Width of 2V

Page 5-35


Anti-Dead-Zone Input

Voltage Anti-Dead-Zone Output

Voltage

-8 -10

-7 -9.0

-6 -8.1

-5 -7.03

-4 -6

-3 -5

-2 -3.85

-1 -2.85

0 -

+1 2.8

+2 3.9

+3 5

+4 6

+5 7.1

+6 8.1

+7 9

+8 10.1

Table E9.2 Anti-Dead-Zone Characteristics for Anti Dead-Zone Width of 2V

Page 5-36



-10 -5

-9 -5

-8 -5

-7 -5

-6 -5

-5 -5

-4 -4.0

-3 -3

-2 -2

-1 -1

0 0

+1 +1

+2 +2

+3 +3

+4 +4

+5 +5

+6 +5

+7 +5

+8 +5

+9 +5

+10 +5

Table E9.3 Saturation Characteristics for Saturation Level 5V

Page 5-37



(Increasing)



(decreasing)


-5 -4.4 5 4.2

-4 -4.3 4 4.2

-3 -4.1 3 3.8

-2 -3.1 2 2.8

-1 -2.2 1 1.78

0 -1.2 0 0.8

1 -0.2 -1 -0.2

2 0.9 -2 -1.2

3 2.2 -3 -2.4

4 3.3 -4 -3.8

5 4.2 -5 -4.9

Table E9.4 Hysteresis Characteristics for Hysteresis Width of 1V

Figure E5.9.1. Dead-Zone Characteristics for 2V Width Dead-Zone

Page 5-38


Figure E5.9.2 Anti-Dead-Zone Characteristics for 2V Width Anti-Dead-

Zone

Page 5-39


Figure E5.9.3 Saturation Characteristics for 5V Level of Saturation

Page 5-40


Figure E5.9.4 Hysteresis Characteristics for a Hysteresis Width of 1V Your results should consist of a) Tables of input/output characteristics for the various non-linear

elements. b) Plots of the input/output characteristics. c) Chart Recorder plots of the response of the non-linear elements to a

signal wave input. Part 1: Non-Linear System Characteristics (Static) Tables E9.1 to E9.4 show typical results for non-linear element characteristics. Figures E5.9 to E5.9.4 show the corresponding plots. Note that you may have problems obtaining a reasonable hysteresis plot because of the need to proceed in one and only one direction when completing Table E9.4.

Page 5-41


Figure E5.9.5. Plot of Non-Linear Responses to Sine Wave Inputs (Chart Speed 1mm/sec). Sine Wave Amplitude: 4.5V, Frequency 0.02 Hz.

Page 5-42


Part 2: Non-Linear System Characteristics (Dynamic) Figure E5.9.5 shows typical sine wave responses of the four non-linear elements. Note the following facts which can be important in assessing the influence of these non-linearities on a dynamical system: a) Anti-dead-zone: This gives a very high gain region about the zero

point. In normal use this would be employed to cancel the effect of dead-zone. However, used on its own or in excess it will lead to rapid low amplitude oscillations in a feedback loop since it creates an infinite feedback gain at low amplitudes.

b) Dead-Zone: This gives a zero gain region around the zero point. In

normal situations it gives a reduced control system accuracy. c) Saturation: This sets an upper limit on the maximum and minimum

output of an amplifier. In normal use it sets a limit on the maximum rate at which a system can operate.

d) Hysteresis: This gives a gain reduction and phase lag between input

and output. It is this phase lag (see Figure E5.9.5) which gives rise to instability when hysteresis occurs in a feedback system.

Page 5-43


Page 5-44


APPENDIX 1. BLANK EXPERIMENT CIRCUIT DIAGRAM

Page A1-1


Page A1-2

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