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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 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.
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
2
A flywheel or inertial load, which will assist in removing rapid fluctuations in
shaft speed and,
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 the
power supplied
generator and frictional
to 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 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!
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.
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 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 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 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.
Figure 2.6 Typical Response of an Integrating Amplifier supplied with Varying
Input Signal
From Figure 2.6 it can be seen that,
a)
b)
c)
d)
e)
f)
When the input is zero the output remains constant.
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.
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.
If the input itself is damping or changing in any way then the outpu will
follow an integral characteristic, again following the criteria givci in (a) and
(b) above.
When a change in input polarity occurs the output responds in the manner
described above, starting at the instantaneous output value a which the change
occurred.
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 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
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.
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)
b)
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.
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.
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 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 CE120
and then confirming the various system responses.
The Section 2.2 builds upon these fundamental principles and introduces the advanced
topics 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 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.
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
dω
inertia and the shaft acceleration
. Thus
dt
τm = bω + k ι v ι + I
dω
dt
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
di
+ vbemf
dt
2.2
Where v(t) is the motor input voltage
R is the motor armature resistance
L is the armature inductance
i is the armature current
and
v bemf is the motor back emf
The back emf and the motor torque can be written in terms of the motor constant km, thus
v bemf = k m ω
τ m = k mI
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) =
k ' m v( s)
( sI + b)( sL + R) + k
2
m
-
kι ( R + sL)
( sI + b)( sL + R) + k
2
m
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
ω (s) =
k ' m v( s ) k 'ι vι ( s )
Ts + 1
Ts + 1
2.5
Where time constant T is given by
T=
IR
bR + k m2
and
k’m =
km
bR + k m2
k’ι =
kι R
bR = k m2
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
ω (s) =
k 'm
v( s)
(Ts + 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 (s) =
ω ( s)
30 s
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.
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.
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 ki, kω, kθ are the motor, velocity sensor and angle sensor gain constants respectively,
then
ω = ki v
yω = kω ω
yθ = kθ 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) =
G1
v( s)
(Ts + 1)
2.9
Where G1 = ki kω, 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) =
G2
yω ( s )
s
2.10
where
G2 =
kθ
30kω
2.10b
Then the overall transfer function for the servo-control system can be drawr as in Figure
2.14 and written thus:
yθ (s) =
G1G2
v( s)
s (Ts + 1)
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 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 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 removing
discs. 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 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 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 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
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,
ess = lim [e(t)]
t→ oo
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,
ess = lim[s.e( s )]
s→o
2.13
For a constant set speed or reference input y,, the steady state error (froir Figure 2.16) is,
ess = lim
s→o
yr
1 + K ( s ).(G )1 ( s )
2.14
Where K(s) is the controller transfer function and G(s) is the servo systeir transfer
function. If proportional control only is used then,
K(s) = Kp
and,
ess = lim
s→o
yr
1 + K p .G1 ( s )
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) = Kp
Ki
s
ess = lim
s. y r
=0
s + ( K p .s + K i ).G1 ( s )
and,
s→o
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) =
K ( s )G1 ( s ) y r ( s )
1 + K ( s )G1 ( s )
2.17
Recall that the speed transfer function is, from Equation 2.9
G1 (s) =
G1
Ts + 1
2.18
With proportional control, the closed-loop transfer function is obtained by combining
Equations 2.17 and 2.18 to give
yω (s) =
k p G1
Ts + 1 + k p G1
yr (s)
2.19
yω (s) =
Gc11
Tc11 s + 1
y r (s)
2.20a
Where the closed-loop gain is
G
c11
G
c11
given by:
K p G1
=
1 + k p G1
2.20b
and the closed-loop constant is
T
c11
=
T
c11
given by:
T
1 + k p G1
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 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 system
is given by
yω (s) =
(k i + k p s )G1
Ts + (1 = k p G1 ) s + k i G1
2
yr (s)
2.21
By comparing the denominator of Equation 2.21 with the standard expression for the
denominator of a second order transfer function:y(s) =
ω n2
y r ( s)
s 2 + 2ξω n s + ω n2
2.22
it is possible to show that
ω n2 =
k i G1
T
2.23
and
2 ξω n =
1 + k p G1
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 ω n and ξ
2.2.7
Controller Design: Annular 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 closedloop transfer function with proportiona] control, such that K(s) = kp ;
yθ (s) =
k p G1G2 y r ( s )
s 2T + s (1 + k v G1 ) + k p G1G2
2.24
Again this can be compared with the standard second order Equation (Equation 2.22) and
the following results obtained:
ω n2
k p G1G2
=
T
2 ξω n =
1 + k v G1
T
2.25
By selecting kp and kv appropriately it is possible to obtain the desired dynamic
performance, as specified by ω n 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 servosystem 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):
G1 (s) =
Gi
(Ts + 1)
where Gι , = kι .
The closed-loop equation for the system, including the influence of the generator load is,
from Figure 2.18, given by
yω (s) =
G1 ( s ) K ( s )
G1 ( s )
y r ( s) v1 ( s )
1 + G1 ( s ) K ( s )
1 + G1 ( s ) K ( s )
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:yω (s) =
(k i + sk p )G1
Ts + (1 + k p G1 ) s + k i G1
2
y r ( s) -
sG1v1 ( s )
Ts + (1 + G1 k p ) s + k i G1
2
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 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,
Kf (s) =
G1 s
G1 s
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 Kf,
given by
Kf =
G1
G1
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 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 nonlinearities 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, 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’.
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).
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
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 nonlinearity, 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 antidead-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 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'.
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 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.
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 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.