Servo Trainer Theory

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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 intypical 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 practicalinvestigations 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 theanalysis 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 ofa 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 servocontrol 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 fedthrough 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 speedor position of a rotating shaft under varying conditions.



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.

Figure 2.1 Simple Motor & Load System

The load will conventionally consist of two elements,

1 A flywheel or inertial load, which will assist in removing rapid fluctuations in

shaft speed and,2 An electrical generator from which electrical power is removed by a load.

Under equilibrium conditions with a constant shaft speed, we haveElectrical = Mechanical power absorbed by the

power supplied generator and frictionalto motor losses



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 andfrictional 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 tocompensate 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 asystem 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 toincrease or decrease the motor input a certain amount.

c) The manual effort to accomplish the load adjustment, required to achieve the desiredchanges 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 suchoperators 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!



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 andalso 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 andso 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 speedas 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.



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



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 noimmediate 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 theactual 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 ofthe system.

If the Gain of the amplifier was increased, the Steady State Error would be reduced butnot 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 becomeevident by the shaft speed oscillating and the input of the

motor changing rapidly.

Figure 2.4 Proportional Control Amplifier Gain Characteristic



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 constantonce selected. Figure 2.4 shows the characteristic of a typical Proportional Control

Amplifier with the Gain set at different levels, increasing from K 1 to K 5.

In order to maintain a non-zero input to the motor drive, there must always be a non-zeroerror signal at the input to the proportional amplifier.

Hence, on its own Proportional Control cannot maintain the shaft speed at the desiredlevel 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 typicalschematic 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 (K 1 to K 5).

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.



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 outpu willfollow an integral characteristic, again following the criteria givci 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 a which the changeoccurred.

f) The magnitudes achieved at the output are dependent on the magnitude of the

input signal and also the time allowed for the



damping to occur. In a practical integrator, the output signal is also limited bythe 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 isdetermined 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 proportionalamplification 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 shownin 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



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 Setspeed 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 practicethe 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 minimumovershoot, to the Set Level value. In the example shown, the value of Integral Action

chosen is said to achieve Critical Damping.



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 theSet 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 Actionuntil 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 largelevels 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 havea tendency to increase instability.

b) Decreasing the integral action would cause the system to respond more slowlyto 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 waveinput.

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 theDifferentiator 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 Differentiatorwould control the amount of damping provided, both in amplitude and duration.



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 highacceleration 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 suitthe specific application.

This section so far has only dealt with control engineering principles in a very basic way

so that the CEIIO 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 CEIIO and the CE120and then confirming the various system responses.

The Section 2.2 builds upon these fundamental principles and introduces the advancedtopics of mathematical modelling, system tuning and predicting 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 allowscomputers 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, bothquickly 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 theequipment 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 gainingexperience in advance through the use of simulators. Most importantly, they were able to

commit the safety of humans to man the vehicles.



Figure 2.10 Servo Control System: Clutch Disengaged



2.2.2 Servo System Modelling: Speed Control System

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 ( T ) 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 (v/) while the frictional torque can be considered as a torque which is

proportional to the shaft speed (o. The inertial torque is determined by the flywheel

inertia and the shaft accelerationdt

d ω. Thus

τm = bω + k ι vι + Idt

d ω

2.1

Where

b = Friction coefficient of rotating components

k ι = Gain constant of load/generator

I = Inertia of flywheel

The motor electrical circuit is governed by the equation

v(t) = Ri + L dt

di + vbemf

2.2

Where v(t) is the motor input voltageR is the motor armature resistance

L is the armature inductance

i is the armature currentand v bemf is the motor back emf



The back emf and the motor torque can be written in terms of the motor constant k m, thus

v bemf = k m ω

τ m = k m 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 vι (s)

ω (s) =m

m

k R sLb sI

svk 2))((

)('

+++ -

mk R sLb sI

sL Rk 2))((

)(

+++

+ι vι (s)

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

(s) =ω1

)('

+Ts

svk m -1

)('

+Ts

svk ιι

2.5

Where time constant T is given by

T =2

IR

mk bR +

and

k’m =2

m

m

k bR

k

+

k’ι =2

mk bR

R

=

ιk

2.5b

Frequently, we will consider the situation when the servo-control system onl has an

inertial load. In this case v/(s) = 0 and Equation 2.5 simplifies to

(s) =ω )()1(

' sv

Ts

k m

+

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 (s) = s

s

30

)(ω

2.7

Where the constant ‘30’ is associated with the 30:1 reduction in speed through thegearbox. Note that the addition of the gearbox load will also change the gain and timeconstant 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-controlsystem dynamics.

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 produceoutput 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 characteristicwhich 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 inFigures 12.13a, b, and c.

Figure 2.13a Speed vs Motor Input Voltage



Figure 2.13b Sensor Output vs Shaft Speed

Figure 2.13c Sensor Output vs Shaft Position



If k i, k ω, k θ are the motor, velocity sensor and angle sensor gain constants respectively,then

ω = k i v

yω = k ω ω

yθ = k θ θ

2.8

Note that k 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.

yω (s) = )()1(

1 svTs

G

+

2.9

Where G1 = k i k ω, is steady state gain of the transfer function from the input drive

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

In addition, the sensed output shaft position yθ is related to the sensec velocity yω by

yθ (s) = )(2 s y s

Gω

2.10

where

G2 =ω

θ

k

k

30

2.10b

Then the overall transfer function for the servo-control system can be drawr as in Figure2.14 and written thus:

yθ (s) = )()1(

21 svTs s

GG

+

2.11



Figure 2.14 Overall Transfer Function for Servo Control System

Again it should be noted that, because of the increase in loading due to the gearbox, the

value of G1, T will be changed when the clutch is engaged and the gearbox and output

position shaft are connected.

2.2.5 Measurement Of System Characteristics

Motor and Sensor Characteristics

The motor steady state characteristic, and the speed sensor characteristics are obtained byrunning 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) tovarious 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 isknown as drift and is not uncommon in industrial sensors and actuators. The motor

characteristics will change significantly according to operating conditions. Specifically,the gain G,, 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 removingdiscs. This will alter the inertia I and hence (via Equation 2.5b) the system time constant,

T.



System Dynamic Characteristics: Step Response Method.

For a first order system, like the servo-control transfer function for shaft speed, the gain

G, 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 result provided thesystem characteristic is known to be first order.

System Dynamic Characteristics: Direct Calculation

An alternative to step response testing is to measure the system characteristicsindividually and then use Equations 2.5b to calculate the gain and tin 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 ofsystem 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 certainmeasurements (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, intocorrespondence 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 inthe reference speed This requires us to find ways of altering the system dynamic response

vie feedback control. We will consider the steady operating performance separately

below.



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 ss = [e(t)]oot →

lim

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 frequencydomain as,

e ss = lim[ ])(. se so s→

2.13

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

e ss = limo s→ )()).((1 1 sG s K

yr

+

2.14

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

K(s) = K p

and,

ess = limo s→ )(.1 1 sG K

y

p

r

+

2.15



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

K(s) = K p s

Ki

and,

ess = limo s→ )()..(

.

1 sG K s K s

y s

i p

r

++ = 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

yω (s) =)()(1

)()()(

1

1

sG s K

s y sG s K r

+

2.17

Recall that the speed transfer function is, from Equation 2.9

G1 (s) =1

1

+Ts

G

2.18

With proportional control, the closed-loop transfer function is obtained by combining

Equations 2.17 and 2.18 to give

yω (s) =1

1

1 Gk Ts

Gk

p

p

++ yr (s)

2.19



yω (s) = )(1

1

1

1

1 s y

sT

Gr

c

c

+

2.20a

Where the closed-loop gain is G given by:c 11

=Gc 111

1

1 Gk

G K

p

p

+

2.20b

and the closed-loop constant is T given by: c 11

T =c 1111 Gk

T

p+

2.20c

From Equation 2.20c it can be seen that the closed-loop speed of response can beincreased by reducing the time constant T . This in turn is achieved by increasing the

proportional gain k .

If the system controlled by a proportional plus integral controller, the closed loop systemis given by

yω (s) =11

21

)1()(

Gk sGk TsG sk k

i p

pi

+=++ yr (s)

2.21

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

y(s) = )(2 22

2

s y s s

r

nn

n

ωξω

ω

++

2.22

it is possible to show that

2

nω =T

Gk i 1

2.23

and



2 nξω =T

Gk p 11+

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 k i,and k p to give desired values of ω and ξ n

2.2.7 Controller Design: Annular Position Control

Figure 2.17 represents the possible block diagram configuration for feedback control ofangular 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, k v, 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 proportiona] control, such that K(s) = k p ;

yθ (s) =211

2

21

)1(

)(

GGk Gk sT s

s yGGk

pv

r p

+++

2.24



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

ω =2

n

T

GGk p 21

2 nξω =T

Gk v 11+

2.25

By selecting k p and k v appropriately it is possible to obtain the desired dynamic

performance, as specified by ω and ξ · Note that when k n v=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 thegenerator load. Figure 2.18 shows this situation.

The load disturbance transfer function is (from the second term on the right hand side ofEquation 2.5a):



G1 (s) =)1( +Ts

Gi

where Gι , = k ι .

The closed-loop equation for the system, including the influence of the generator load is,

from Figure 2.18, given by

yω (s) = )()()(1

)()(

1

1 s y s K sG

s K sGr

+ - )(

)()(1

)(1

1

1 sv s K sG

sG

+

2.27

Proportional Compensation:

If proportional control is applied, then K(s) = k p and if k p is large then the effect of the

load change upon the output will be small. In fact the larger k p 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 termwill 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 integralcontrol:-

yω (s) = )()1(

)(

11

2

1 s y

Gk sGk Ts

G sk k r

i p

pi

+++

+ -

11

2

11

)1(

)(

Gk sk GTs

sv sG

i p +++

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 vι (s), the effect upon yω (s) will be zero in

the steady state.



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 v and after

passing it through a suitable controller K f (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

K f (s) it is possible to completely compensate for the influence of the load voltage vi. Thisis done by selecting K f (s) such that,

K f (s) = sG

sG

1

1

2.29

From the equations defining Gι (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 K f ,given by

K f =1

1

G

G

2.30



Thus by calculating K, according to Equation 2.30 it is possible to exactly cancel the

effect of changing load upon the speed signal. In practice, K, is often selectedexperimentally, 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 belinear. 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 serieswith 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 minimumvalues are due to the limitation imposed by the values of the amplifiers. For example, if

the power supply to an amplifier provides ±15 V, 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 iscontrolled by a calibrated ‘level control’.



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-zoneamplifier 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 thenormal 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).



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. Thegain 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



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 levelsvdp 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 theanti-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 mechanicalcouplings (usually gearboxes) in which the two elements of the coupling separate and

temporarily lose contact as the 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 (oroutput) 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'.

a) Driving gear turning clockwise

b) Driving gear changes direction and temporarily losses contact with the driven

gear

c) Driving gear turning anti-clockwise with contact to driven gear re-established

Figure 2.26 Hysteresis or ‘Backlash’



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 dependsupon 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 hysteresiselement to add realism to the system.

The hysteresis block is switched into the circuit using the enable switch. With thehysteresis 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.



Figure 2.28 Backlash Gap 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 thenon-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 thecomposite 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 thedemonstration of important limitations to control system design caused by non-linearity.

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Servo Trainer Theory