1.

INTRODUCTION TO THE AUTOMATIC CONTROL

1.1. The essence and object of the automatic control
It could be imagined that the man has thought of simplification of its work
from the beginning of the ages - how to get the water into its house in a more simple
way, how it were possible to move more goods at once, how to make other works
more quickly, could it be possible to make these jobs so that they could be done
without assistance. The objective of it was the transformation from the work to the
mechanised work and so has the technology developed during centuries – first the
methods were found how to lease the force /energy) requiring works to the machines
and thereafter the possibilities to leave the information processing to the machines as
well. Although the transformation of the energy level to the machines has happened
already long time ago, the processing of the information by machines
has taken place on a quite primitive level until the development on the computer
technology. The development of the computer technology has speeded up the
development of many other branches of sciences, the development control technology
included which in turn enabled to entrust to the machines activities requiring more
decisiveness.
For the description of such systems, where the right to decide is trusted to the
machines the term automated system was applied. This term is inherited from the
Greek, where the word automatos means self-acting in English. By different literature
sources, the systems are divided differently by the level of automation, by the grade of
automation [1], for instance, that which is described in the able 1.1
Table 1.1. Classification of system by the grade of automation
Grade Replaced function Example
A (0) Missing Hand working tools, spinning reel as
example
A (1) Energy Drill
A (2) Skilfulness Lathe with feeding
A (3) Diligence Automatic lathe with open control loop
and repeated production cycle
A (4) Decision Lathe with closed control cycle and
computer control
Such a division is provisional; therefore, it is suitable sometimes to use simpler on the
control-based division. The control is an information processing process, expressed
purposeful rearrangement of some action. As an action technical, as well biological
and social processes could serve. [2]. Focusing on the technical processes one my
distinguish in them
hand control, corresponding to the automation level(0);
automated control, matching A (1) to A(3);

1
 and automatic control, matching grades A(3) and higher.

As technical systems, a set of equipment functioning together for fulfilment of the

objective of the work is meant .The originality of the work is described by the
physical properties of the equipment, which could be described by mathematical
equations, after solution of which is it possible to analyse the system. If the system
does not comply with required conditions then is it necessary to operate the system in
such a way that it will comply with the requirements. In case if the system is able to
apply such impacts without man involvement, one has to deal with automatic control
system

1.2. Recitals
The automatic control systems are divided in to two groups – open loop systems and
closed loop systems. In the open loop systems, preliminary defined mathematical
models control the process, without controlling the satisfaction of the results of
controlled process to the required ones. Such a control suits simple systems, so as the
undesirable influences or disturbances are affecting the results of the process. The
process noise arises from the physical patterns influencing the process, affecting its
behaviour. The measurement noise arises from the measurement errors of the
measuring devices or sensors. In general, the first of them is a random value, which is
not possible to forecast, however the measurement error could be taken into account
preliminary and influence the system in such a way that in the control action
(influence of the control to the system) the arising error will be compensated. Such
control is called the compensation of disturbances.

a)
n
s Con- u Sys- y s input variable
trol tem

b) u control action

n n disturbance
s e u Sys- y
Con-trol
tem e error or deviation
±

y output variable

Figure 1.1. Automatic control systems a) open loop system, b) closed loop system
In the closed loop systems, the process is controlled inspecting the compliance of the
results with the given criteria or the feedback about the results is functioning. The
process is controlled by the error or in dependence on the difference of the actual and
desired result. The controlled process is called control object, in general, and its res-
ult - output y. The device, which is forming the control action, is called control
device or also regulator. The system is influenced by input variables, which are in-
herited form outside of the system. These inputs are set point s, which determines
what is required from the system and the disturbances n which are disturbing the pro-
cess functioning

2
In the Figure 1.1. an automatic control system consisting from blocks and directed
influences between blocks is shown. The blocks are called transfer links and the
influences – signals. If to examine one individual block, then the following relation
holds:
y =W ⋅s . (1.1)
The quantity W is called transfer function and it
describes the dependance of output variable on the
input variable..

Figure 1.2. Transfer link

Example 1.1.
If to apply a force to the spring, then as a result of it the
length of the spring is changing. Expressing it based on
the equation (1.1)
x =W ⋅F .

From the Hook’s law the following expression is

F x known:
F =k⋅x

and from these expressions it is possible to conclude,

that
1
W=
k

Figure 1.3. A spring .

In the block diagrams or flow charts three types of signal processing elements are
present (addition link, branching link and multiplication link) and three types of main
or simple structures (series structure, parallel structure and feedback loop) [3].
.
s1 y For the adding link the following mathematical relation
holds
+ -
s2 y=s 1−s 2 . (1.2)

Figure 1.4. Addition

link

s s

3
 s1
Figure 1.5.
y
X Branching link
Mathematical presentation of the multiplication link is
s2
the following
y=s 1⋅s2 . (1.3)
Figure .1.6.
Multiplication link

s y For the series structure the followings holds

W1 W2
y=W 1⋅W 2⋅s . (1.4)
Figure 1.7. Series structure

W1 The output of the parallel structure is

s y expressed as

±
y=W 1±W 2⋅s (1.5)
W2

Figure 1.8. Parallel structure

Feedback structure is presented

mathematically s y
W1
W1 ±
y= ⋅s (1.6)
1∓W 1⋅W 2
W2

Figure 1.9. Feedback structure

The transfer functions are considered more detailed in the next chapter.
The transfer functions are used for the description of the dependence of one output
variable on the one input variable. In case of more complicated systems, where it is
needed to follow multiple outputs depending on multiple inputs simultaneously or
multilinked systems (?), the calculation using transfer functions becomes
complicated. The calculation becomes easier if to describe the system by means of the
state equations (look also division 2.3).

4
ẋ= A⋅x B⋅u
y=C⋅xD⋅u , (1.7)

where
A – state matrix of the system;
B – input matrix;
C – putpu matrix;
D – disturbance’s matrix;
x – state vector;
u – input vector;
y – output vector.
As a state of the system the aggregated whole of all state variables of the system is
called. For instance, while describing the state of a spring we have there two variable
quantities – the force and the shift. By the force, we have to deal with input variable,
by the shift – with output variable. At the same time there could be more state
variables which are presented precisely or approximately, yet the question about their
suitability to be considered as output variables is determined by the following criteria:
as output variable such state variable could be, which could be precisely measured
and dos not belong into input variables simultaneously. These criteria could be
applied also to the transfer functions:
D

x' x y
B ∫ C
s

Figure 1.10. Graphical presentation of state equations

1.3. The components of an automatic control system

Although the automatic control systems could be divided into control objects and
control devices in general, it is not possible to determine, which devices are used in
the system. Also it necessary to know the so-called background components. to
understand the influence of possible disturbance.
Therefore, beginning the description from the control object or device, or from the
process, which is to be controlled, one of the components will be the controllable
device, electric motor, for example. This device is influenced by control actions and
also by distrubances, as a result of which the output of the control object or the result
of the process changes. To control the result it has to be measured, therefore also the
measuring device or sensor, to which, dependent on the structural individualities,
systematic or random errors are influencing, belongs to the control object. A random
error could be influence of the temperature, presence of outer fields etc. System errors
are contingent upon the accuracy of the measuring device.
If the signal of the measuring device are not suitable for the control device, digital
computer cannot process analogous signals, for instance, the it will be required to add

5
 to the measurement device a signal converter, which changes the physical properties
of the signal and/or makes the quantification of the signal. In this energy conversion
process the errors similar to the measurement devices appear.
As a centre of the automatic control system the control device could be considered,
which is replacing the man, deciding about the output of the system, in case if one has
to deal with feedback system, or, in case of open system, takes into consider all
possible deviations. In addition, to this part of the system, different disturbances are
influencing and similarly to the adjustment of the sensor signal to the control device
could be needed to adjust the control signal to the control object. Which means to
have another signal converter in the system.

n1 n2 n3
y
s e Regu- u Actuat Pro- Meas
± lator or cess uring

Figure 1.11. Automatic control system

Although, it could not be left out of attention, that the set value, with which the order
for the control system is submitted has certain error. However, the most complications
are introduced to the system by the time. Namely, for the processing of the signal in
each link takes certain time as well as the motion of the signal in the connection lines
between links which is a serious problem in large factories. Therefore, the control
device regulates mainly the system state that has already passed.

1.3.1. Regulator
Regulator is so-called crucial element of the automatic control system, which forms
the control action based on deviation. Therefore, the adding or comparing link before
the regulator belongs to the control device. Also belongs to this association the
element for producing the set value, for which a simple switch could be serve in most
simple cases (to switch the process on or off), or potentiometer by sliding contact of
which the set value could be changed smoothly.
Real implementation of the regulator depends directly on the tasks and requirements
for the regulation quality. This could be implemented by analogue computers, digital
computers, and relays or by some other means. The regulator could be an individual
device or could be incorporated into the actuator.
It is most simple to represent a temperature regulator of the relay action

Bimetal Such regulator could be used in the

electric radiators. Into the feeder line
of a radiator, a bi-metal element will
be placed, which opens and closes the
contacts depending on the temperature.
It is possible due to the different linear
Contacts
expansion coefficients of the metals
used in the bi-metal element, which
determines the binding of the element
Feeder lines

6
 Figure 1.12. Bimetal-regulator
From the incorporated regulators the
controllers could be mentioned, where
the regulators are implemented by
software, or the microprocessor
decides based on the information
obtained from the measuring device
about the suitable control action. The
LOGO! Controller could be put into
operation similarly to the bi-metal
regulator. .
Figure 1.13. Siemens LOGO!Controller

1.3.2. Actuator
Actuator is that part of the control system, which
amplifies the control action and converts it acceptable
for the controlled device. Regulator, represented in the
Figure 1.12. is an actuator at the same time, but it is only
possible by small power control systems In case of large
power control systems the converters must be used, so
as the regulators are constructed as small power devices,
to provide the minimal dimensions and reduction of the
losses. For instance, if the output signal of the regulator
lies in-between 0…5 V and the rated voltage of the
controlled electric motor is 400 V, then for their match
an actuator or a (semiconductor) converter has to be
used, which has to compose the internal variables
(opening angels of the thyristors) from the control
action, on the bases of which the supply voltage of the
motor will be formed.
The regulator could be incorporated into the converter,
what mist the margin between control device and control
object

Figure 1.14. ABB ACS-800 converter

1.3.3. Process
If generalised, a process is a device,
which is subject to control in general.
This is the only part of the automatic
control system, the parameters of
which could not be changed; therefore,
it forms the base of the control system.
and the system will be built around it.

7
Parameters of all other parts of the
system could be changed or they
would be replaced. For instance, let
the process be regulation of the speed
of an asynchronous machine, to which
different regulators (continuous,
discrete, etc.), actuators (frequency
converter, rheostat etc), and measuring
devices (velocity transducer,
incremental encoder etc) could be
chosen. By selection of these elements,
one has to decide how they will match
mutually and with the machine. The
machine will be replaced if it becomes
clear that it does not match the given
task and in this case, all elements will
be reselected.

1.3.4. Measuring device.

The measuring device is used for obtaining information about the system parameters.
All parameters required for successful control are subject of measuring. For example,
for the control of the electric motor speed it is enough to measure the speed and
supply voltage only, however, the result will be better if there will be measured the
machine current as well, and it will be regulated. Not always is the measurement of
input necessary, for instance, if it is constant or is changing according to certain
regularity. In addition, it could not always be possible to measure the input, especially
by some disturbances.
The measuring device is the most widespread element of the automatic control
systems, so as there are many quantities, which could be and are to be measured. For
instance, in case of an electric drive the following sensors could be possible: - Hall
transducer and magnetic resistances for the measurement of the magnetic field of the
rotor air gap. tacho-generator or velocity transducer for the measurement of the speed,
resolver, optical incremental sensor. The two last suit the determination of the rotor
shaft position. For the measurement of the current a current transformer will be
suitable, The there will be another shunt and Hall sensor, dynamometer for the
measuring of the torque which is usually replaced by the measurement of the current,
so as the torque is a product of the current. For the measurement of the voltage
transformers, voltage dividers etc are suitable. The question is not about hot or what
to measure, but about the purposefulness of the measurement. This problems arises
with the processing duration, so as a certain time will be required for the data
collection, evaluation and decision or the formation of the control action, therefore, by
the time of formation of the control action the system could be in emergency state
already.
The measuring devices could be abandoned at all, in case if the output of the control
object will be precisely determined in each time moment. One of those control objects
could be the step-motor, which has precisely determined rotor position, if the motor is
not overloaded.

8
1.3.4.1. Transducers of speed and position
Tacho-generators represent a direct current machine, which transforms the rotation
speed or mechanical signal into voltage or an electrical signal. [4]. It is described by
the following equation
u t =k m⋅t  , (1.8)
where km – a machine constant, characterising construction and excitation of the
machine.
Main shortcoming of the tacho-generator is the wear of brushes, therefore they require
frequent maintenance, and also it is impossible to determine with it the position of the
shaft.
There exist asynchronous and synchronous tacho generators, however, they are not
widespread
Resolver represents a high frequency
dynamotor, the primary winding of which is
settled on the shaft of the rotating machine A
and the secondary side has two windings,
which are mutually shifted by 90 degrees.
With the rotation of the primary winding a
signal of variable amplitude is induced on
the secondary side, which, after the filtering
of the currier frequency gives a sinus-signal B
A and a co sinus-signal B. Based on the
instantaneous values of these signals it is
possible to determine the position of the
shaft and from that, after differentiation - the
speed. The advantage of this device is the
absence of brushes [5].
.
Figure 1.16. Principal diagram of
a resolver
Main element of an optical incremental transducer (Figure 1.17) is the incremental
disc, which is sat on the rotating shaft. The disc is divided into several sectors, light
penetrable, and light impervious (1024, 2048 or 4097 light penetrable sectors). These
sectors are exposed with LED light sources mainly; the light spot is directed through
condenser and filter. The condenser concentrates the light spot and the filter also has
light penetrating and light impervious sectors, the width of which corresponds to the
width of the disc sectors. On the other side of the disc four photocells are placed,
which are shifted in relation to each other be the quarter of the sectors width and
which are pair wise switched back-to-back. As a result of it, a situation arises where
by the rotation of the incremental disc a current is generated in the photocells what
corresponds to the sinus and cosines signals, where the period of the signal
corresponds to two sectors width (one penetrating and another impervious), but not to
one shaft rotation. Therefore, the frequency of the signals obtained is a product of the
number of sectors and of the rotation speed of the shaft. But - with the tangent it is
possible to determine the position of te shaft inside of one period, therefore the given

9
transducer has to count how many periods is passed during this turn, or - it has to
increment. To enable the intermediate control of the calculation results the
incremental disc has a reference point, which is monitored by the fifth photocell,
which fixes, that one turn of the shaft has occurred [5].
There exist another optical position transducer, called absolute transducer or encoder.
In this case, the disc is not divided into ribbed arches but into zones (similar to the
chess board). The arrangement of the zones depends on the way of the encoding and
on the number of bits used in the encoding. As a disadvantages of this transducer are
accuracy (the zones could not be placer with the same density as could be placed
sectors) and large amount of data (for each bit there exists one photocell [5].

LED Condensor
Filter

Incremental Photocell
disc

Reference

a) b)
Figure 1.17. Optical incremental transducer a) construction b) functioning principle

1.3.4.2. Current transducers

Current transformers are the most widespread
devices for measurement of the current. So one
has to deal with a transformer, although with
transformers of special type, they are suitable for
measuring in the alternating current systems only.
The advantage of current transformers is heir
simplicity that is expressed in their construction.
The primary winding of the transformer is a cord
or cable, the current of which is measured, and
secondary winding is placed around tha cable and
on its terminal an emf, proportional to the current,
will be induced Figure 1.18. Current
transformer
Transducer with Hall’s sensor is settable for
operation both by direct current as well by
alternating current. The current to be measured
crosses the transducer’s winding and induces in it
a magnetic flux, which crosses the Hall’s sensor.
So as the magnetic flux is proportional to the
magnetic flux. and the voltage on the Hall’s sensor
Figure 1.19.Hall’s transducer terminal is proportional to the magnetic flux, than

10
 as a result we have a voltage signal proportional to
the current [5].

1.4. Control methods

The classical control method belongs to older and therefore simpler systems. It is
used by systems with one input and one output mainly, In this case one has to deal
with linear systems which could be described by linear differential equations and
which could be analysed by Lap lace’ transformation and frequency methods. These
systems could in turn be divided into stabilisation, regulating and following systems.
Stabilisation systems are used in case if it is desired to keep the output on certain
given level. Regulating systems are applied in case if the output will be changed by
some certain principle. Programmed control systems belong to this category.
Following systems change their output in dependence on their input, the changes of
which are not planned beforehand.
Example 1.2.

Uvar T°
230V AC

U(T°)

Figure 1.21. Automatic control system with a classical control method

Required room temperature is set on the thermostat, Depending on the difference of
the actual and desired room temperatures the regulator builds up, by the mean of the
electronics of supply block, a voltage suitable for supply cable, which in turn
determines the heating intensity.
The temperature is measured by a transducer, which transforms the temperature into
voltage signal, which is directed to the regulator.

Modern control methods are used by multicoupled systems, i.e. by systems with
multiple inputs and outputs, where multiple outputs are regulated simultaneously.
Example 1.3.
U, I, f
α
ω
T, p
Boiler
Water Turbine Generator
x1
Air
CO2

Fuel x2

x3

11
 Figure1.22. Automatic control system with modern control method
In the figure 1.22, an electricity production complex is represented in a simplified
way. Into the combustion chamber air and fuel are sent, emerging energy in the
combustion of which is transferred to the water steam, which rotates the turbine In
this connection the generator also will rotate and electricity will be produced. To hold
the network voltage and frequency by changing load stable, the load current will be
monitored. Therefore the rotation speed of the generator has to be regulated, which is
made or by changing the angels of turbine blades or changing of the steam energy.
The latter could be changed by changing the pressure and temperature of the steam,
which is made by boilers pump (not shown in the figure) or changing the combustion
process adding fuel and air to it as required. During the combustion process, the
exhaust gases are monitored, to provide the optimal combustion process and maximal
efficiency.
Description of such systems by means of transfer functions would be complicated;
therefore, state equations in digital form are mainly used for the description of modern
control systems.
The highest of control methods are the intellectual methods, which are based on the
intuitive estimations of the programmer, on the fuzzy logic or expert evaluation, for
instance. These methods are applied when one has to deal with the uncertainty of the
control object. In this case, the state variables are not estimated quantitatively but
qualitatively (large, small, for example) and these variables will be related to the
conditional clauses IF-THEN according to the algorithm. To such systems self-
adjusting, self-programming, self-learning and self-organising systems belong [2]

1.5. The purpose of the control

Proceeding from the given above the purpose of the control could be formulated as
follows [2]:
The main purpose of the control is to form a set of control actions, in result of which
the system will behave as desired. This set of actions has to provide to he system a
control corresponding to the quality requirements (reaction speed, overshoot,
attenuation) of the system. Thereby, one has to consider the physical conditions
(power limits of the equipment) and subjective decisions (danger to the personnel).
Solution of this problem means the synthesis or creation of the automatic control
system. Therefore, the object under study will be separated from the surrounding
media, and its parameters and structure will be determinate. This is called
identification of the object. Quit often it occurs notional, so as in the project stage
the does not exist a real object where from a mathematical model imitating the
behaviour of a real object could be composed. If the real object does exist then it is
possible to identify it investigating the reactions of the object or transients produced
by standardized input signals. . That reaction could be also called as signal echoes.
Mathematical equations or models could describe the shape of the signal echo.
How well the model behaves, depends on that how exactly is the control object
modelled. The more complicated is the object the greater is the rate of uncertainty.
Therefore, as the uncertainty is described by the evaluations, the uncertainties are

12
arising into model, in turn complicating the solution of the problem. In case if the
model is composed without the presence of any indeterminacy, the model is called
fully determinate. Those systems could be easily controlled by open loop, however, in
practice, such systems are rather exclusions, therefore most of real systems operate on
the feedback principle, to diminish the system with the application of measuring
devices and the uncertainty of the model with it.
Most of the mathematical models and of the systems are dynamic, i.e. the state of the
system depends on the previous one. For instance, the movement of a link of the serial
cinematic robot is influenced by the movements of all other links. Those systems
could be called inertial systems or systems with after-effect. These systems are
described by the means of differential or difference equations.
As next stage, the objective of the control will be set. Therefore, an output variable
will be determined which will be controlled, deflecting torque, angle velocity or
position for instance, in case of an electric motor, and for instance, the torque will be
hold constant or will be changed according to certain law.
In the run of the solution of a control problem the permissible set signals and control
actions. These signals must be determined beforehand due to the limitation formulated
in the task.
As last stage, the measure of the control quality must be determined. For this purpose
a set of different criteria serve, based on which it is possible to compare different
systems, executing the same tasks. Also it is possible based on this measure to
optimise the system, evaluate the system operation and reduce the influence of
disturbances to the output.

13
2. DESCRIPTION OF AUTOMATIC CONTRL SYSTEMS
For sufficient control of an automatic control system the dynamic properties of both -
as of the process as well of the system, must be known. The dynamic properties of a
component or of a system could be determined by calculations or experimentally.

2.1. Description of systems by differential equations

Example2.1.
i(t) For the resistor Ohm’s law holds
u t = R⋅i t  .
u(t) R
So as the resistance is depending on the temperature,
then more precisely will it be
u t = R i , t ⋅i t  .
Fidure2.1.A resistor
Example 2.2.
i(t) Voltage equation of a condenser
t t
C 1
u(t) u t =
C
∫ i  d =U 0 C1 ∫ i  d  .
−∞ 0

Figure 2.2. A condenser

Example 2.3.
i(t) The voltage drop of a winding is expressed as follows
di t 
u(t) R u t = R⋅i t L⋅ .
dt
L

Figure 2.3. A coil

Example 2.4.
Let it be a dripping buckle with water hanged on the
spring [3]. Proceeding from the Newton’s II law and
Hook’s law for this system could be written
d
[m t ⋅ẋ ]k  x ⋅x=m t ⋅g
dt
m t ⋅ẍ ṁ t ⋅ẋk  x ⋅x=m t ⋅g
Joonis 2.4. Lekkiv anum

Example 2.5.
For a RLC circuit the Kirchof-s second law R L
and corresponding differential equations
could be applied. us(t) uy(t)
i(t) C

14
 d i t 
u s t =R⋅i t L⋅ u y t 
dt
d u t 
i t =C⋅ y Figure 2.5. RLC-C-circuit
dt
d u y t  d 2 u y t 
u s t =RC⋅ LC⋅ u y t 
dt dt 2

Example 2.6.
x
Based on Newton\s II law the following is
valid> m F
Fh
2
d x dx
m⋅ 2 =F −⋅m⋅g⋅sgn 
dt dt
Figure 2.6. Linear motion
Example 2.7.
Newton’s II law holds for the rotating
movement also Th
J
d
J =T −T h  . T
dt
ω
The difference here is, that frictional
moment always works against of the drive
moment
Figure 2.7. Rotational motion
R L
Example 2.8.
J, ω, Tk
From the theory of electrical drives it is us(t) M
i(t) km
known the fundamental equation of the
direct current electric motor:
d i t 
u s t =R⋅i t L⋅ k m⋅t 
dt
d t  Figure 2.8. A DC drive
J⋅ =k m⋅i t −T k t 
dt

Assuming, that inertial moment and the load moment of the system are constant in
time (the frictional moment is negligible compared to the load moment), then it is
possible to replace mechanical equation into voltage equation:
2
J⋅L d t  J⋅R d t  R L d T k t 
u s t = ⋅ 2
 ⋅ k m t  ⋅T k t  ⋅ .
km dt km dt km km dt

15
2.2. Description of systems by the mean of transfer functions
2.2.1. Laplace transformation
Solution of differential equations without support of mathematical software is
complicated, therefore there are various solution techniques are developed for them.
One of the oldest is the operator method or Laplace transformation.
∞
F  p = ℒ { f t  }=∫ f t ⋅e− p⋅t , (2.1)
0

where F(p) – operator representation of the differential equation

ƒ (t) – differential equation
p – complex variable or operator
Here one has to deal with a mathematical manipulation, in the run of which the time
axis of the process will be turned around or switched over to the frequency axis. As a
result of it processes that had occurred in the beginning of the time axis or fast
processes are imaged at the end of frequency axis and slow processes at the beginning
of frequency axis. The mathematical background of such manipulation is not easy to
understand, however, solution of differential equation in operator form is possible not
knowing the integration of differential equations, so as the differential equations will
be replaced with simple algebraic functions.
The execution of transformations is comparatively easy, however, the inverse
transformation or returning into time space is a complicated mathematical procedure,
therefore for the execution of the transformations tables (Appendix 1) are used, if
there do not exist ready software solutions at hand.
Transformation of a differential

ℒ
{ dn
dt } n
n f t  = p ⋅F  p  (2.2)

And on an integral
1
ℒ {∫ f t  }= ⋅F  p  . (2.3)
p

Example 2.9.
The representation function of the differential equation of the Example 2.9.
di t 
u t = R⋅i t L⋅ ⊶ u  p=R⋅i  p  p⋅L⋅i  p
dt
Example 2.10.
The representation function of the differential equation of the Example 2.10
2
J⋅L d t  J⋅R d t  R L d T t 
u s t = ⋅ 2  ⋅ k m t  ⋅T k  ⋅ k ⊶
km dt km dt km km dt
J⋅L 2 J⋅R R L
u s  p = ⋅ p ⋅p  ⋅ p ⋅p k m⋅ p  ⋅T k  p  ⋅p⋅T k  p 
km km km km

16
2.2.2. The transfer function
The transfer function describes the relation of the output
s W y variable of a transfer link to the input variable of it.
y
Figure 2.9. The transfer link W=
s .
(2.4)
The transfer function will be transformed from the representation function due to its
simplicity. One of the advantages of the transfer function is its reversibility
−1 s
W =
y . (2.5)

Example2.11.
In the Example 2.3 a RL cit was presented and in the Example 2.9 the representation
function of it was given. Determining the supply voltage being as input variable and
the current as output variable the corresponding transfer function could be found as
follows
u  p =R⋅i  p  p⋅L⋅i  p =[ R  p⋅L ]⋅i  p 
1
i  p 1 R
W  p = = =
u  p  R  p⋅L L .
1 p⋅
R
The transfer function found is a little bit unhandy, it is recommended to do some more
transformations
1 L
=K and =T ,
R R

where K – transfer ratio;

T – time constant of the process
i  p K
W  p = =
u  p  1  p⋅T
The transfer ratio describes the gain of given link and it is always a value with a
1 A
dimension. (In given case  = V ), the unit of which determines the unit of the
transfer function
The time constant of the process describes the temporal properties of the transient
processes occurring in the transfer link. This is also a value with a dimension, the unit
of which is a second. So as the transfer function determined the transfer function unit,
the it follows from here, that the denominator of the function must be without
dimension, from where it follows in turn that operator p also is a value with a
dimension, the unit of which is s-1.
If to consider the current as input variable and voltage as output variable, then the
corresponding transfer function will be

17
 u p 1 p⋅T
W  p = =R p⋅L= .
i  p K
The result is reciprocal of the first determined transfer function, which reduces
essentially the amount of calculations in the analysis and synthesis of automatic
control systems
Example 2.12.
In the Example 2.5 a RLC circuit was considered the output variable was the
condensers voltage and input variable the supply voltage. Corresponding to this
transfer function will be determined as follows
2
d u t  d u y t 
u s t =RC⋅ y LC⋅ u y t  ⊶ u s  p =RC⋅p⋅u y  p LC⋅p 2⋅u y  p u y  p 
dt dt 2

u y  p 1
W  p = = .
u s  p  LC⋅p 2RC⋅p1
L
Substituting RC =T 2 and R =T 1 the transfer function will have the form

uy p K
W  p = =
u s  p  T 1 T 2⋅p 2T 2⋅p 1 .

Example 2.13.
In the Example 2.8 the representation function of a DC drive was brought
J⋅L 2 J⋅R R L
u s  p = ⋅ p⋅p  ⋅ p ⋅p k m⋅ p  ⋅T k  p ⋅p⋅T k  p .
km km km km

In given function there are three non-parametric values us, ω and Tk. The transfer
function presumes two non-parametric variables only. So as the electrical drive is
controlled by voltage. then it could be considered as input variable. However, load
moment is a random value which is not possible to control, and therefore could be
considered rather as a disturbance then a control action or output variable. The
velocity thereby is easy to control and eve easier to measure (with a tachogenerator)
Therefore it suits for an output variable. Thus, we have to deal with one output value
and two input values (disturbance also is an input variable), hence – two transfer
functions could be expressed:
 p 1
W j  p= =
u s  p J⋅L 2 J⋅R
⋅p  ⋅pk m ,
km km
R L⋅p
−
 p km
W h  p= = .
T k  p  J⋅L 2 J⋅R
⋅p  ⋅pk m
km km

1 J⋅R L
After arrangement and making substitutions k =K , 2 =T 2 and R =T 1 we will
m km
have in result:

18
  p  K
W j  p = =
u s  p  T 1 T 2⋅p T 2⋅p 1 ,
2

RL⋅p
−
 p k 2m .
W h  p= =
T k  p T 1 T 2⋅p2 T 2⋅p1

These are called transfer function and disturbance transfer functions correspondingly.
Each of them describes the dependence of the output on the different input signal and
as one can see the transfer functions are different, and as a prevision it could be
mentioned that if the drive is adjusted on the ideal execution of the control then the
reduction of the disturbances influence is less effective and the same holds to the
opposite case. We shall return to this problem in the following chapters.
If to compare the transfer functions of the RLC circuit and of the DC drive, one could
be convinced that mathematically we are dealing with similar systems, although their
physical backgrounds are different. From this one my conclude, that systems behave
similarly, whereas they could be described similarly and controlled similarly or,
typical solutions could applied to them and they could be named standard links.
Standard links are considered in Chapter 2.5.

2.2.3. The Bode diagram

Bode diagram is otherwise called logarithmic amplitude-phase frequency
characteristic. Here one has to deal with graphical representation of the transfer
function, characterising the dependence of system properties on the frequency. It is
widely used in the acoustics and by the calculations of filters, thou as it is possible to
see on them whether the given object by given frequencies gives a damping or
amplification of the signal, and which will be the phase shift of the signal. In the
automatic control we don’t deal with continually changing frequency processes,
however, for the analysis of the systems the Bode diagram is useful, so as it is easy to
determine about system stability on the diagram (see also Chapter 3.1)
There are two different representation forms of Bode diagram, but historically it has
developed here in Estonia the use of the representation, where decibel (dB) is taken as
measure of logarithmic amplitude, which is used as a relative gain. Also the
application of per units enables to join different objects graphically.
While describing the relation of the input and output power of an object the
logarithmic mode of the description is used which simplifies their representation as
graphics. Mathematically it is expressed as
Pv
L =log (2.6)
Ps

and the unit is 1 bell, that means the relation of powers 10:1. So as quite often we
have to deal with smaller relations, the as a unit decibel is used
Pv
L =10⋅log (2.7)
Ps

The unit (decibell was at the beginning related to the power or „squared values“), then
for the linear values based on in the electricity known relation P ~ I2 holds

19
 Pv I 2v I
L =10⋅lo g =10⋅lo g 2 =20⋅lo g v =20⋅lo g K . (2.8)
Ps Is Is

The transformation of the transfer function to the logarithmic form is made by the
condition that p = jω
lo g W  p =lo g W  j =lo g K  e j =lo g K  j   , (2.10)
However, if the Bode diagram is constructed by hand, then it will be solved by
asymptotes
Example 2.9.
Let be given a transfer function
K⋅1 p⋅T 1 
W  p = ,
1 p⋅T 2

From which the frequency function will be expressed

K⋅1 j ⋅T 1 
W  j = .
1 j ⋅T 2

The frequency function could be divided into two separate functions

W 1  j =1  j ⋅T 1 and W 2  j = K

1  j ⋅T .
2

For this functions the corner frequencies will be determined:

1 1
1 = =
T 1 and 2 T 2 .

Proceeding from the corner frequencies the asymptotes of both of functions

determinate
W 1  j =1T 1 j ∣ =1
≪ 1
1 ,
W 1  j =1T 1 j ∣ =T 1  j 
≫ 1

K
W 2  j = ∣ =K
1 j ⋅T 2 ≪  2

K K .
W 2  j = ∣ =  j −1
1 j ⋅T 2 ≫  T 2 2

The value of the power of the frequency variable is showing the inclination of the
asymptote. In case of positive power the asymptote proceeds up and by negative
values it proceeds down. The value of the power determines the value of the
inclination (slop), or how many amplitude decades (20 dB) will the amplitude change
during one frequency decade (10x). Also by the value of power are phase shifts
determined. – the sign shows the direction of the phase shift and its value shows how
many 90-degree phase shifts occurs.
Let it be given for the construction of a diagram K = 2, T1 = 2 s and T2 = 0,5 s.
With it, the corner frequencies are

20
 1 1
1 = =0,5 s−1 and 1 = =2 s−1 .
2s 0,5 s

The asymptotes are expressed as:

W 1.1 j =1 , W 1.2 j =2 s  j  , W 2.1 j =2 and W 2.2 j =1 s−1  j −1 .

The logarithms of the gains are

L 1.1=20⋅log 1=0 and L 2.1=20⋅log 2=6,02 .

21
 2

7
6
3 5

E 9 A
F G

I J K
8 B H L

ω1 ω2

Figure 2.10. The asymptotes of the Bode diagram

By the construction of the Bode diagram one proceeds from the following preceding
calculations
Step 1 The gain of the first transfer function is 0 dB until the first corner function
Step 2 After the corner frequency the gain will increase with the rate 20 dB for
one decade
Step 3 The gain of the other transfer function is 6,02 dB until the next corner
frequency
Step 4 After the corner frequency the gain will decrease with the speed 20 dB for
a decade
Step 5 The gains of different functions could be arithmetically added, so the gain
of the system is 6,02 dB until the first corner frequency
Step 6 After first corner frequency the gain of the first function will increase
also, and therefore the gain of the whole system with a speed 20 dB for a
decade
Step 7 After the second corner frequency the gain of the second function starts to
decrease and so as the velocities of the increase and of the decrease are
equal, the gain of the system does not increase nor decrease..
Step 8 At low frequencies the phase shift of the first subsystem is missing
Step 9 In the first subsystem at the corner frequency the phase shift occurs in the
positive direction, for construction of which a point will be marked, the
coordinates of it are corner frequency and half of the final value of the
phase shift (in given case 45 degrees). Through this pint a line will be
drown, which begins in the point with coordinates initial phase angel and
0.7 decade length before the corner frequency and ends in the point the

22
 coordinates of which are final phase angel and 0.7 of the decade length
after the corner frequency.
Step A The first subsystem does not have any more corner frequencies, therefore
it holds the phase angel obtained at the end of the first step
Step B The second subsystem also begins with a 0-degree phase shift.
Step C In the given subsystem the phase shift occurs in the negative direction,
which will be constructed similarly to the step 9
Step D In this system also the other corner frequencies are missing
Step E So as the logarithmic systems could be added, then for the sake of
simplification of the mathematics the phase-frequency characteristic line
of the second subsystem will be shifted by 90-degrees upwards.
Step F The phase shifting processes are partly overlapping and a vertical line
gives the lower limit of it by frequency.
Step G Similar to the step F, describes the determination of the upper limit
Step H Adding two systems we will get as a result overall phase shift equal zero
to the first phase shifting process
Step I The first subsystem produces a positive phase shift until the lower limit of
overlapping
Step J Between the limits of overlapping the slopes of both processes are equal
but opposite signs, therefore the system keeps the phase shift obtained by
the previous step.
Step K After the upper limit of overlapping the whole system is influenced by the
second subsystem, producing negative phase shift
Step L So as the system does not have more corner frequencies; the system holds
the phase shift obtained by the previous step

Figures 2.11. The calculated Bode diagram

23
.2.4. Structure diagrams
In case if the system appears to be too complicated, it will be presented consisting
from separate transfer functions, which could simplify the understanding of the
processes occurring in the system.
Example 2.10.

km

U - I Tm Td φ
1/R km 1 1
1+p - pJ ω 2π

Tk

Figure 2.12. Structure diagram of a separately exited DC motor

The first transfer function described transformation of the voltage signal into a current
signal, which in turn will be transformed into machine torque. The difference of the
machine torque and load torque is a dynamic moment, by division of it by inertial
torque and integrating, one will have the speed signal. Integrating the speed signal in
turn one gets as a result the inductors position [6]
And so as we have to deal with simple structures, the transfer function of the control
could be expressed by the following
1
R 1
⋅k m⋅
L pJ
1 p
 p  R
W j  p= = ,
U  p 1
R 1
1 ⋅k m⋅ ⋅k
L pJ m
1 p
R
Which, after arrangement and substitutions reduces to the form
 p  K
W j  p = =
U  p  T 1 T 2⋅p T 2⋅p 1 .
2

The same transfer function was determined in the Example 2.8.

In case if the automatic control system is not presented by simple structure diagrams,
then, for the determination of the transfer function of this system the following
possibilities exist:
transform the structure diagram into simple schemes
represent the structure diagram with open loops
to use the Mason's equation
The mentioned above solution methods are analysed by determination of the transfer
function of the automatic control system presented in the figure 2.13. [2].

24
 W6

s + y
W1 W2 W3 W4
- - +
W8

W7

W5

Figure 2.13. Structure diagram of an automatic control system

2.2.4.1. Determination of the transfer function by transformations

For the determination of the transfer function of a structure diagram by
transformations one has to begin from the middle of the structure diagram, where it is
possible to determine a simple part of the structure diagram, for which it is possible to
determine unique one input and one output. In the first chapter simpler structure
diagrams were given.
As a middle point of the automatic control system presented in the figure 2.13 could
be considered the transfer link W3, which builds up a parallel structure with the link
s1 was mot an adding
W8, if there y link in the input of sthe
1
y
link W3. It follows from which,
W W
that given scheme does not have simple structures and the structure mast be
transformed. Transformed could be distribution and addition links from the input to
the output of the link and contrariwise, thereby the values of the signals 1/Wmust remain
unchanged after
s transformation
1 s1

Figure 2.14. Transformation of a distribution link from the input to the output.

s1 y s1 y
W W

y y

Figure 2.15. Transformation of a distribution link from the output to the input

25
 s1 y s1 y
W W

W
s2
s2
Figure 2.16.Transformation of an addition link form the input to the output

s1 y s1 y
W W

1/W

s2 s2

Figure 2.17. Transformation of an addition link from the output to input.

Example 2.11.
Preceding from these transformation rules the addition linkWW3 could be transformed
from the input to the output, as a result W
ofe1what the structure
e2
diagram
W3 will have
W6the
form:
s y
W1 W2 W3 W4
- -
W8

W7

W5

Figure 2.18. The transformed diagram after first step

After the transformation in the structure diagram two simple parts of structure arise,
which could be replaced by equivalent transfer links, the transfer function of which
are expressed
W e1=W 3W 8

and
W e2=W 3⋅W 6

and the structure diagram will take the form (Figure 2.19), where a new part of
structure diagram arises, the transfer function of which is as follows:
W e3=W e1⋅W 2=W 3 W 8 ⋅W 2 .

26
 We2
We3
s y
W1 W2 We1 W4
- -

W7

W5

Figures 2.19. The transformed diagram after second step.

We2

s y
W1 We3 W4
- -
W7

W5

Figure 2.20. The transformed diagram after 3rd step

The following stem will be the transformation of the addition link We3 from the output
to input

We4
1/ We2
We3
s y
W1 We3 W4
- -
We5 W7

W5

Figure 2.21.The transformed diagram after step 4

After the transformation two following simple structure arose, the transfer functions
of which are expressed as
W e2 W 3⋅W 6
W e4 = =
W e3 W 3W 8⋅W 2

and
W e3 W 3 W 8 ⋅W 2
W e5 = = .
1W e3⋅W 7 1 W 3W 8 ⋅W 2⋅W 7
We7 We4
We6
s y
W1 We5 W4
27
-
W5
 Figure 2.22.The transformed diagram after step 5
The equivalent transfer link of the arising serial structure
W 3W 8⋅W 2⋅W 4
W e6 =W e5⋅W 4=
1W 3W 8⋅W 2⋅W 7

builds with the link We4 a feedback structure the equivalent of which will be
W 3 W 8⋅W 2⋅W 4
W e6 1W 3W 8 ⋅W 2⋅W 7
W e7 = = =
1−W e4⋅W e6 W 3⋅W 6 W 3W 8 ⋅W 2⋅W 4
1− ⋅
W 3 W 8⋅W 2 1W 3W 8 ⋅W 2⋅W 7
W 3W 8 ⋅W 2⋅W 4
=
1W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8−W 3⋅W 4⋅W 6
.

We8
s y
W1 We7
-
W5

Figure 2.23. The transformed structure after step 7

Next to last transformation will be the transformation of the serial structure
W 3W 8 ⋅W 2⋅W 4⋅W 1
W e8 =W 1⋅W e7 = ,
1 W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8−W 3⋅W 4⋅W 6

that forms for the last transformation a feedback structure of the forward transfer
The result of the last transformation is the transfer function of the whole system
W 3W 8⋅W 2⋅W 4⋅W 1
W e8 1W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8−W 3⋅W 4⋅W 6
W= = =
1W e8⋅W 5 W 3W 8 ⋅W 2⋅W 4⋅W 1
1 ⋅W
1 W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8−W 3⋅W 4⋅W 6 5

W 1⋅W 2⋅W 3⋅W4 W 1⋅W 2⋅W 4⋅W 8

=
1W 1⋅W 2⋅W 3⋅W 4⋅W 5W 1⋅W 2⋅W 4⋅W 5⋅W 8W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8−W 3⋅W 4⋅W 6
.

28
2.2.4.2. Determination of the transfer function of an open loop structure
diagram
Example 2.12
This method is efficient for the verification of the transfer function of an automatic
y
control system that is determined by other methods. Switching off the distribution
points of diagram represented in figure 2.13 an open
W6 loop structure diagram presented
in figure 2.24 is obtained.
s x1 x2 + x3 x4 y
W1 W2 W3 W4
- - +

W5 W7 W8

x4 x2
y

Figure 2.24. Structure diagram with open loops

To simplify the calculations, intermediate variables xi. are added to the system. By the
replacement of the intermediate variables it must be taken care, that the number of
them is enough for description of the system after splitting of the distribution points.
The intermediate variables could be considered as state variables. In the following
each intermediate variable, except inputs, will be described as equations via other
state variables
y=W 4⋅x 4
x 4 =x 3W 8⋅x 2
x 3=W 3⋅ x 2W 6⋅y  .
x 2=W 2⋅ x1−W 7⋅x 4 
x 1=W 1⋅s−W 5⋅y 

To express the output signal from the input signal the simultaneous equations are
solved using replacement method.
x 2=W 2⋅[ W 1⋅ s−W 5⋅y−W 7⋅x 4 ] =W 1⋅W 2⋅s−W 2⋅W 5⋅y−W 2⋅W 7⋅x 4

x 3=W 3⋅[ W 1⋅W 2⋅s−W 2⋅W 5⋅y−W 2⋅W 7⋅x 4W 6⋅y ] =
=W 1⋅W 2⋅W 3⋅s−W 2⋅W 3⋅W 5⋅y−W 2⋅W 3⋅W 7⋅x 4W 3⋅W 6⋅y
x 4 =W 1⋅W 2⋅W 3⋅s−W 2⋅W 3⋅W 5⋅y−W 2⋅W 3⋅W 7⋅x 4 W 3⋅W 6⋅y
W 8⋅[ W 1⋅W 2⋅s−W 2⋅W 5⋅y−W 2⋅W 7⋅x 4 ] ⇒
W 1⋅W 2⋅W 3W 1⋅W 2⋅W 8 ⋅s−W 1⋅W 2⋅W 3⋅W 5−W 3⋅W 6W 1⋅W 2⋅W 5⋅W 8 ⋅y
x 4=
1 W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8

W 4⋅W 1⋅W 2⋅W 3W 1⋅W 2⋅W 8⋅s−W 1⋅W 2⋅W 3⋅W 5−W 3⋅W 6 W 1⋅W 2⋅W 5⋅W 8 ⋅y
y= ⇒
1W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8

29
 W 1⋅W 2⋅W 3⋅W4 W 1⋅W 2⋅W 4⋅W 8
W=
1W 1⋅W 2⋅W 3⋅W 4⋅W 5W 1⋅W 2⋅W 4⋅W 5⋅W 8 W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8−W 3⋅W 4⋅W 6
.

2.2.4.3. Determination of the transfer function by Mason’s formula

For the transformation of the structures of the automatic control systems the graph
theory, which is a branch of discrete mathematics while dealing with the topology of
system elements, could be efficiently used. From this field stems the S.J. Mason’s
formula, the simplified form of which could be used for the simplification of
structural diagrams that do not have hermit loops. As hermit loop are those feedback
parts loops of structure diagram named, which does not have common elements.
Simplified Manson’s formula is used by the structure diagram presented in the
figure 2.13, where all feedback loops contain elements which are involved into some
other feedback loop or where the hermit loops are absent or the following formula
could be used for the simplification of structure diagrams: [2]
W oj
W= , (2.11)
1∓ W ki

where Woj – transfer function of the j-th forward chain

Wki – transfer function of the i-th opened feedback loop.
If the feedback is negative, then Wki has a „+“-sign, in case of positive feedback the
sign is „-“.
Example 2.13.
From the structure diagram represented in figure 2.13 two forward chains could be
noticed:
W o 1=W 1⋅W 2⋅W 3⋅W 4
W o 2=W 1⋅W 2⋅W 4⋅W 8 .

Also five feedback loops could be noticed here, the open loop transfer functions of
which express as follows:
W k 1=W 1⋅W 2⋅W 3⋅W 4⋅W 5
W k 2 =W 1⋅W 2⋅W 4⋅W 5⋅W 8
W k 3=W 2⋅W 3⋅W 7
W k 4=W 2⋅W 7⋅W 8
W k 3=−W 3⋅W 4⋅W 6 .

Replacing the found transfer functions into the formula (2.11) the result will be:
W 1⋅W 2⋅W 3⋅W4 W 1⋅W 2⋅W 4⋅W 8
W=
1W 1⋅W 2⋅W 3⋅W 4⋅W 5W 1⋅W 2⋅W 4⋅W 5⋅W 8 W 2⋅W 3⋅W 7W 2⋅W 7⋅W 8−W 3⋅W 4⋅W 6
.
The Masons full formula is used in cases when one has to deal with hermit loops.
According to the definition of the hermit loops given above, in the figure 2.25 a
corresponding structure of an automatic control system is presented. For
simplification of this structure the formula [7] is used:

30
  W o j⋅ j
W= , (2.12)


where Δ – determinant of the automatic control system

Δj – determinant of the j-th forward chain.
The determinant of the automatic control system is calculated as follows:
=1− W k i  W ej⋅W ek∣ j≠k − W ej⋅W ek⋅W el∣ j≠ k≠l ... , (2.13)
where Wex – transfer function of an open hermit loop.
The determinant of the forward chain is used similar to the system determinant to
assign remaining untouched loops after deletion of the i-th forward chain.
Example 2.14.

W8

s -
W1 W2 W4 W5 W7
- - y

W3 W6

Figure 2.25. An automatic control system with hermit loops

The given structure diagram has three feedback loops, two of which don’t have
common elements or, one has to deal with two hermit loops.
From the structure diagram the transfer function of the forward chain
W o 1=W 1⋅W 2⋅W 4⋅W 5⋅W 7

and three transfer functions of open feedback structures

W k1=−W 1⋅W 2⋅W 4⋅W 5⋅W 7⋅W 8
W k2 =−W 2⋅W 3 ,
W k3 =−W 5⋅W 6

two of which are hermit structures

W e1=−W 2⋅W 3
W e2=−W 5⋅W 6 .

To assign the determinant of the structure, the found transfer functions will be placed
into equation (2.13)
=1W 1⋅W 2⋅W 4⋅W 5⋅W 7⋅W 8W 2⋅W 3W 5⋅W 6W 2⋅W 3⋅W 5⋅W 6 .

By deletion of the forward chain all loops will be broken, therefore its determinant
equals to 1. Placing the results into equation (2.12) one gets as result the transfer
function of the system:
W 1⋅W 2⋅W 4⋅W 5⋅W 7⋅1
W= .
1W 1⋅W 2⋅W 4⋅W 5⋅W 7⋅W 8W 2⋅W 3 W 5⋅W 6 W 2⋅W 3⋅W 5⋅W 6

31
2.3. Description of the systems by state equations
In the division 1.2 state equations, which were presented mathematically with
equation (1.7), were considered briefly.
ẋ= A⋅x B⋅u
y=C⋅xD⋅u .
It is possible to compose the state equations for the continuous systems proceeding
from the differential equations or structure diagrams

2.3.1. Determination of the state equations from the differential

equations
Example 2.15.
Assigning state equation to the DC drive with controlled speed, familiar from the
Example 2.8, the differential equation of the drive is written first
d i t 
u s t =R⋅i t L⋅ k m⋅t 
dt
d t  .
J⋅ =k m⋅i t −T k t 
dt

After arrangement the equations take the following form

d i t  R k 1
=− ⋅i t − m ⋅t  ⋅u s t 
dt L L L
k ,
d t  m 1
= ⋅i t − ⋅T k t 
dt J J
which, if represented in matrix form, is

[ ] [ ][ ] [ ][
d i t  R km 1
− − 0
dt
d t 
dt
=
km
J
L
0
L
⋅ i t  
t 
L
0 −
]
⋅ u s t 
1 T k t  .
J

  x  u
ẋ A B

By a drive with speed control usually both as current as well speed are monitored. The
reasons for the monitoring of the current are more detailed considered by the
synthesis of automatic control systems. If to proceed from the state equation and
output variables the output equation could be expressed as follows:

[] [] [] [] [] .

i t  = 1 0 ⋅ i t   0
t  0 1 t  0
0 ⋅ u s t 
0 T k t 
y C x D u

Example 2.16.
Applying position control to the drive described in the previous example, the
differential equation will present itself in following

32
d i t  R k 1
=− ⋅i t − m ⋅t  ⋅u s t 
dt L L L
d t  k m 1
= ⋅i t − ⋅T k t 
dt J J
d t 
=t 
dt
and the state equations will take the following form

[ ] [ ][ ] [ ]
d i t 
R k 1
dt − − m 0 0
L L i t  L
d t 
dt
d t 
= km
J
0 0
⋅ t  
t 
0 −
J 
u t 
1 ⋅ s
T k t  [ ]
0 1 0 0 0 u
dt  x 
 A B
ẋ

[ ] [ ][ ] [ ][
i t  1 0 0 i t  0 0

t
  0 0 1 t  
  0 0 
u t 
t  = 0 1 0 ⋅ t   0 0 ⋅ s
T k t  . ]
C D u
y x

2.3.2. Determination of the state equations from the structure

diagrams
Example 2.17.

k7 k6

s1 x1 x2 x3 - x4
k1 k2 k3 k4 k5
- - T2p T3p+ T4p+ - T5p y

k8

k10
s2
k9

Figure 2.26. An automatic control system.

Using the principle of an structure diagram with open loops, one may written
k5
y= ⋅ x −k ⋅s 
T 5⋅p 4 10 2
k4
x 4= ⋅ x −k ⋅y 
T 4⋅p1 3 6
k3
x 3= ⋅[ x k ⋅ x −k ⋅x  ]
T 3⋅p1 2 7 1 8 4

33
 k2
x 2= ⋅ x −k ⋅x 
T 2⋅p 1 8 4
x 1=k 1⋅ s 1−k 9⋅y  .

By the arrangement of the equations the operator variable will be transferred to the
left side of the equality sign and the state variables ordered by their indices. So as in
the last equation the operator variable is missing, it will placed in other equations
k5 k 5⋅k 10
p y= ⋅x 4 − ⋅s 2
T5 T5
k 4⋅k 6 1 k4
p x 4=− ⋅y− ⋅x 4 ⋅x 3
T4 T4 T4
k 1⋅k 3⋅k 7⋅k 9 k 3⋅k 7⋅k 8 1 k3 k 1⋅k 3⋅k 7
p x 3=− ⋅y− ⋅x 4− ⋅x 3 ⋅x 2 ⋅s1
T3 T3 T3 T3 T3
k 1⋅k 2⋅k 9 k ⋅k k ⋅k
p x 2=− ⋅y− 2 8⋅x 4 1 2⋅s1 .
T2 T2 T2

Representing the matrixes in general form

[ ][ ][ ] [ ][ ]
py a 11 a12 a 13 a 14 y b 11 b12
p x4 a a 22 a 23 a 24 x 4 b b 22 s1
= 21 ⋅  21 ⋅
p x3 a 31 a32 a 33 a 34 x 3 b 31 b32 s 2
p x2 a 41 a 42 a 43 a 44 x 2 b 41 b 42

    u
ẋ A x B

[]
y
y ] =[
[
y C
x
x3
x2
d 11 d 12]⋅ s 1
c 11 c12 c 13 c14 ]⋅ 4  [
D s2 []
 u
x

and applying the comparison principle of constants it could be expressed as follows

k5 k 4⋅k 6 1 k4 k 1⋅k 3⋅k 7⋅k 9
a 12= a 21=− a 22=− a 23= a31=−
T5 T4 T4 T4 T3
k ⋅k ⋅k 1 k3 k ⋅k ⋅k k ⋅k
a 32=− 3 7 8 a 33=− a 34= a 41=− 1 2 9 a 42=− 2 8
T3 T3 T3 T2 T2
k 5⋅k 10 k 1⋅k 3⋅k 7 k 1⋅k 2
b 12=− b 31= b 41= c 11=1
T5 T3 T2

The unrevealed constants equal to zero.

2.4. Description of systems by characteristic curves

If it is impossible to use a mathematical approach for the description of the system,
one has to deal with an improving device with incomplete data for it, for instance, or
with a black box, the characteristic curves of the device or process will be studied- A
characteristic curve is the graphical description of the process. Two kinds of

34
characteristic curves are distinguished – dynamic and static. Dynamic characteristic
curve characterises the transient processes or transition of the output variable of the
device from one value to another, and this in the time domain. Those characteristics
are also called transient characteristics. Static characteristics describe the dependence
of the system output on the all variables, the disturbances included, by the condition
that transient processes in the device are terminated. Which of characteristic to use for
the study of the device depends on the objective of the study. The dynamic
characteristic describes fast processes and they are used if the quality and the duration
of the transient process are in importance, by an automatic miller, for instance. The
static characteristics are in interest if the parameters of the transient process do not
influence essentially the operation of the device, usual water pump, for example.

2.4.1. Dynamic characteristic curve

Dynamic characteristic curve represents the response of the device or of the process to
the input signal. It could be named as signal echo. So as the input signals could be
different, (different amplitude, different time progress of the signal, etc) then for the
comparison of different devices amongst, the following input functions are used: [3]:
unit jump
unit impulse;
linear function;
cosine function.
The time development of unit jump
describes the characteristic curve in the
figure 2.27. Mathematical representation of
this function is the following

t =0 , kui t 0 . (2.14)

1 , kui t ≥0
In case if one has to deal with a real device,
the mathematical description of what is
sought, then a signal will be applied to the Joonis 2.27. Ühikhüpe
input of the device, corresponding to the unit
jump, and output signal will be recorded.
After the termination of transient processes Joonis 2.28. Hüppekaja
the mathematical presentation of the device
will be assessed by the recorded curve, from
the figure 2.28, for example, could be read
t
−
T
h t =K⋅1−e 
If one has to deal with engineering design of
a device which is wanted to simulate, then
the differential equation describing the
device will be solved by the condition,
where the input signal is permanently 1 and
time will be let run to infinity.

35

Joonis 2.29. Ühikimpulss

The progress of a unit impulse in time is
described with the characteristic curve in
figure 2.29. Mathematical description of this
function is the following:

t = 0 , kui t ≠0 . (2.15)

∞ , kui t =0
Also is this function called Dirac’s function.
So as the infinity is an indefinite value, then
in practice the impulse is realised so that its
area equals to 1 whereas its duration
approaches to zero. Thereby the amplitude
must not exceed the permissible for the
device, maximum signal, for instance, if the
isolation of the connection cord is intended
for the use up to 1 kV, then there must not
be applied the voltage of higher amplitude.
As for the jump echo it is possible for the
impulse echo determine the characteristic
equation, for example, from the figure 2.30.
t
K −
w t = ⋅e T .
T
As in the figure 2.28 as well in the figure 2.30 the same device is described, to which
different input signals were given. If one has to do with the same system, the should
there exist a mathematical relation between those two signals, which is
h t =∫ w t dt (2.16)
and these relations are graphically presented in the figure 2.31.

Aegruum Sagedusruum

δ(t) Protsess w(t) Laplace W(p)

1
∫ ∫ p

σ(t) Protsess h(t) Laplace H(p)

Figure 2.31. Graphical relation of the unit impulse and of the jump
The progress of the linear function in time is described in the figure 2.32.
Mathematical presentation of this function is
R t =K 1⋅t . (2.17)
The alteration of the output of a device produced by linear function is indicated by f
(t), for instance, the expression of the echo of a linear function: is

36
 −t
T
f t = K⋅t −T T ⋅e 
In the fig8ure 2.33 the signal echo of the system investigated with two previous
functions is presented, from which it is possible to determine the time constant of the
system. If to extend the tangent to the ordinate axis, then through the intersection
point it is possible to determine the gain of the device.

Joonis 2.32. Lineaarfunktsioon Joonis 2.33. Signaalikaja

The figure 2.34 describes the progress of the cosine function in time. Mathematical
description of this function is
s t =s⋅cos ⋅t  . (2.18)
Te cosine function
2.34. is used for investigation of time properties of a 2.35.
deviceSignaalikaja
or process.
Joonis Koosinusfunktsioon Joonis
Thereby, the transient processes following the application of the input are not in
interest, but the change of the amplitude and the phase shift, so as in the evolved state
the frequency of the output signal of the device equals to the frequency of the input
signal. The signal echo of the cosine function of the studied device is given in the
figure 2.35.

37
2.4.2. Static characteristic
Static characteristic describes the system in the
evolved state that could be expressed from the
operator form of an automatic control system
presented in the figure 2.36. [3], [8]
W R⋅W P WP
y  p = ⋅s  p  ⋅n  p  .
1 ∓W R⋅W P 1 ∓W R⋅W P
Joonis 2.36.
(2.19) Automaatjuhtimissüsteem
The equation (2.19) describes the output value of a
closed system, however, for an open system the
equation will have another form
y  p =W R⋅W P⋅s  pW P⋅n p  . (2.20)
To determine the evolved value of the output variable the given above equation will
be solved (depending on the system) as follows
y st t =lim p⋅y  p , (2.21)
p0

which will be resolved for each different set and disturbance value. For the real
existing systems the set value will be applied to the system, evaluate and if possible,
modify the disturbance, and the output variable will be measured. As it follows, the
result could be presented graphically

Figure 2.37. Static characteristic

From this graph it is easy to read the value of the output variable by a given set value
and given disturbance value. Also it gives estimation about the change of the output
value by changing of the disturbance. Formerly, the linearization of those graphs
around the operation point was used, now these characteristics are used less. In the
use are remained some concepts from the static characteristics
Static or evolved output value is the value that the output variable takes if from the
beginning of the transient process has passed infinite time. It could be calculated by
the equation (2.21).

38
The permanent control error is the difference between the desired and realized
values of the output variable. Mathematically it could be expressed as
x d st=lim  K⋅s  p− y  p , (2.22)
p 0

where K – planned gain of the system.

The efficiency of the regulator in holding the system in a stable operation describes
the ability of the regulator to reduce the influence of disturbances on the operation of
the system. The evaluation of the efficiency could be expressed from the equations
(2.19) and (2.20), where the open system descries the system without regulator ( WR =
1; s(p) = 0) and the closed system describes the system with regulator (s(p) = 0). Their
ratio is the rate of the efficiency
y reg 1
= . (2.23)
y 1∓W R⋅W P

From the equation (2.23) might be concluded, that in the system with closed loop the
influence of the disturbance will be decreased by 1-WRWP times

2.5. Standard links

In the division 2.2.2 it was concluded, that if to control voltage of a RCL circuit and
the speed of an DC motor by the input voltage, then both systems behave similarly an
it is possible to describe them by similar mathematical equations. Thus, if there exist
mathematically similar control objects, then. consequently there exist mathematically
similar control devices, from where it follows in turn, that if for one control object
matches one control device, then for another similar object could match a control
device which is similar to the control device of the first control object. In other words
- one has to do with standard solutions and the used links of the automatic control
systems are therefore named standard links.

2.5.1. Proportional link

Proportional link is also named an amplification link or
a link without inertia, or for short P/link. The output
signal of o P/link changes simultaneously with input
jump signal and also jump-wise and without delay. The
difference of the signal levels is due to the gain only
Joonis 2.38. P-lüli
The differential equation: y t = K⋅s t  (2.24)
The transfer function W  p =K (2.25)
The impulse echo w t = K⋅ t  (2.26)
The jump echo: h t =K⋅ t  (2.27)

39
 a) b)
Figure 2.39. A P-link K=1. a) The jump echo, b) the Bode diagram

2.5.2. Integration link

The integrating link is also called an astatic link and I-
link. The output signal of an ideal integrating link
increases (or decreases) continuously with a constant
speed if xs ≠ 0 and is constant. The speed is determined
by the value of the jump in the input. By a real
integrating link (described by IT1-loink) growth ratio
of the output signal is zero in first moment and grows
slowly to the final speed.
The differential equation v̇ t =K⋅u t  (2.28)
K
The transfer function: W  p = p (2.29)

The impulse echo: w t = K⋅ t  (2.30)

The jump echo h t =K⋅t (2.31)

a) b)
Figure 2.41. An I-link, K=1. a) The jump echo, b) the Bode diagram

40
2.5.3. Differentiation link
The another name of the differentiation link is D-link.
The output signal of an ideal differentiation link is a
hyper-short impulse of infinite great amplitude. For a
real differentiation link (described as DT1-link) The
output signal grows very rapidly to a final value and
diminishes thereafter gradually with reducing speed to
zero
The differential equation y t = K⋅ṡ t  (2.32)
The transfer function: W  p =K⋅p (2.33)
d
The impulse echo w t = K⋅  t  (2.34)
dt

The jump echo

: h t =K⋅t  (2.35)

a) b)
Figure 2.43. D-link K=1. a) Jump echo, b) Bode diagram

2.5.4. Integration link with time constant

Integration link with a time constant or short IT1
describes a real integration link, which, different from
the ideal one das distortions in the output.
Differential equation T⋅ÿ t  ẏ t =K⋅s t  (2.36)
Joonis 2.44. IT1-lüli
K
Transfer function: W  p = (2.37)
p⋅1T⋅p
−t
Impulse echo: w t = K⋅1−e T
 (2.38)

−t
Jump echo h t =K⋅t −T T⋅e T
 (2.39)

41

a) b)
 Figure 2.45. IT1-llinkK=1, T=1 s. a) jump echo, b) Bode diagram

2.5.5. Differentiation link with time constant

Differentiation link with a time constant or short DT1
describes a real differentiation link that, different from
the ideal one has distortions in its output

Joonis 2.46. DT1-lüli

Differential equation: T⋅ẏ t  y t =K⋅ṡ t  (2.40)
K⋅p
Transfer function: W  p = (2.41)
1 T⋅p
t
K K −
Impulse echo: w t = t − 2⋅e T (2.42)
T T
t
K −
Jump echo: h t = ⋅e T (2.43)
T

a) b)

Figure 2.47. DT1-link K=1, T=1 s. a) jump echo, b) Bode diagram

2.5.6. Delay link

Delay link behaves like P-link, but reacts to the input
with certain delay. The delay link is noticed as PTh-link.
Notion! Bode diagram is built with MathCAD software

Joonis 2.48. PTh-lüli

42
Differential equation y t = K⋅s t −T h  (2.44)
Transfer function: W  p =K⋅e−T ⋅p h
(2.45)
Impulse echo w t = K⋅ t −T h  (2.46)
Jump echo: h t =K⋅ t−T h  (2.47)
20

10

W 0

10

20
0.01 0.1 1 10 100
wi

qi

3
0.01 0.1 1 10 100
wi

a) b)

Figure 2.49. PTh-link K=1, Th=1 s. a) jump echo, b) Bode diagram

2.5.7. Aperiodic link

Aperiodic link is also called inertial link, relaxation link
and PT1-link. The output signal starts to change
immediately, first with maximal speed, thereafter with
gradually reducing speed until it reaches maximal value
after (3...5)T. The transient characteristic represents an Joonis 2.50. PT1-lüli
exponent curve
Differential equation T⋅ẏ t  y t =K⋅s t  (2.48)
K
Transfer function W  p = (2.49)
1 T⋅p
t
−
Impulse echo: w t = K⋅e T (2.50)

−t
Jump echo h t =K⋅1−e T
 (2.51)

43

a) b)
 Figure 2.51. PT1-linki K=1, T=1 s. a) jump echo, b) Bode diagram

2.5.8. Oscillator link

The oscillator link could be named as PT2-link, therefore, that it has two time
constants already. For better understanding of the operation of the oscillator link one
has to introduce some new quantities. From the examples 2.12 and 2.13 the following
transfer function is known
K
W  p = , (2.52)
1T 2 pT 1 T 2 p2
From which, while substituting
1
0= , (2.53)
T 1T 2
where ω0 – the characteristic of the object or radian frequency of un-damped
oscillation
1
=
2T 1 , (2.54)

where α – damping factor

it is possible to express the equation (2.52) in other form
K⋅20
W  p = . (2.55)
202 ⋅p  p 2

Also it is possible to express with the damping factor and radian frequency of
characteristic other parameters characterising the system
s=  20− 2 , (2.56)
where ωs – radian frequency of damping oscillation
2
T= , (2.57)
s

where T – period of damping oscillations


=
0
, (2.58)
2

where χ – damping factor 1 .8

K(1+exp(-αt))
1 .6

1 .4

1 .2
h (t )

0 .8

0 .6
T
0 .4
44
0 .2
K(1-exp(-αt))
0
0 1 2 3 4 5 6 7 8 9 10
A eg
 Figure 2.52. An example of the transient characteristic of an oscillation link
Depending on the damping factor three kinds of oscillation links are distinguished
Damping factor χ > 1

K, T1, T2
s y

Figure 2.53. PT1-link

Differential equation T 1⋅T 2⋅ÿ t T 1T 2 ⋅ẏ t  y t =K⋅st  (2.59)
K
Transfer function: W  p = (2.60)
1T 1⋅p⋅1T 2⋅p
t t
K −
T
−
T
Impulse echo: w t = ⋅e −e  1 2
(2.61)
T 1−T 2
t t
1 −
T
−
T
Jump echo h t =K⋅[1− ⋅T 1⋅e −T 2⋅e ] 1 2
(2.62)
T 1−T 2

45
a) b)
 Figure 2.54. PT1-link K=1, T1=0,3 s, T2= 2,4 s. a) jump echo, b) Bode diagram
Damping factor χ = 1

K, T, T
s y

Figure 2.55. PT1-link

2
Differential equation T ⋅ÿ t 2⋅T⋅ẏ t  y t =K⋅s t  (2.63)
K
Transfer function: W  p = 2 (2.64)
1T⋅p 
t
K −
Impulse echo: w t = 2⋅t⋅e T (2.65)
T
t
t −
Jump echo: h t =K⋅[1−1 ⋅e T ] (2.66)
T

a) b)
Figure 2.56. PT1-link K=1, T1=0,3 s, T2= 0,3 s. a) jump eco, b) Bode diagram
Damping factor 0 < χ < 1
K, χ, ω0
s y

Figure 2.57. PT1-link

1 2⋅
Differential equation 2
⋅ÿ t  ⋅ẏ t  y t = K⋅s t  (2.67)
0 0

46
 K
W  p =
Transfer function: 1 2 2⋅ (2.68)
⋅p  ⋅p1
20 0
0
⋅sin   1 − ⋅0⋅t 
−⋅0⋅t 2
Impulse echo w t = K⋅ ⋅e (2.69)
 1− 2

Jump echo

h t =K⋅{1−e−⋅ ⋅t⋅[cos   1− 2⋅0⋅t 
0
⋅sin   1− 2⋅0⋅t ]} (2.70)
 1 −2

a) b)
-1
Figure 2.58. PT1-link K=1, χ=0...1, ω0=5 s . a) Jump echo, b) Bode diagram

2.5.9. Proportional-integration link

PI-link combines proportional and integration links.
This link is used in regulators mainly
1
Differential equation: ẏ t = K R⋅ ⋅s t  ṡ t  (2.71)
TI
Joonis 2.59. PI-lüli
1T I⋅p
Transfer function: W  p =K R⋅ (2.72)
T I⋅p
1
Impulse echo w t = K R⋅ t  ⋅t  (2.73)
TI

1
Jump echo h t =K R⋅1 ⋅t  (2.74)
TI

For the connection of the parameters of a PI-link with gain of an I-link the following
equation is valid
KR
K I= . (2.75)
TI

47

a) b)
 Figure 2.60. PI-link K=1, TI=1 s. a) jump echo, b) Bode diagram

2.5.10. Proportional-differentiation link

PD-link combines proportional link and differentiation
link. Given link is used in regulators mainly
Differential equation: y t = K R⋅s t T D⋅ṡ t  (2.76)
Transfer function W  p =K R⋅1T D⋅p (2.77)
Joonis 2.59. PI-lüli
d
Impulse echo: w t = K R⋅ t T D⋅ ⋅t  (2.78)
dt

Jump echo: h t =K R⋅ t T D⋅ t  (2.79)

To connect the parameters of a PD-link with the gain of a D-link the following
relation holds:
K I =K R⋅T D . (2.80)

a) b)
Figure 2.62. PD-link K=1, TD=1 s. a) jump echo, b) Bode diagram

2.5.11. Proportional-integration- differentiation link

PID-link combines all three main elements – P-link, I-
link and D-link. PID-regulators are most widespread
control devices, whereas PI- and PD-regulators are
realised with PID-regulator, where the gain of the D- or
I link equals zero. Joonis 2.63. PID-lüli

48
 1
Differential equation ẏ t = K R⋅ ⋅s t  ṡ t T D⋅s̈ t  (2.81)
TI
2
1T I⋅p T I⋅T D⋅p
Transfer function: W  p =K R⋅ (2.82)
T I⋅p

1 d
Impulse echo w t = K R⋅ t  ⋅t T D⋅ ⋅t  (2.83)
TI dt

1
Jump echo: h t =K R⋅ t  ⋅t T D⋅ t  (2.84)
TI

Joonis 2.64. PID-lüli K=1, TI= 50 s, TD=0.5 s. a) hüppekaja, b) Bode diagramm

Exercises to the chapter 2.

Exercise 1.
Express the dependence of the output
voltage of the system presented in
figure 2.65 on the input voltage with a
differential equation. Transform the
found equation into operator form and
determine the transfer function of the
system. Repeat the same by the Joonis 2.65. RC-ahel
condition that output variable is the
current.

Exercise 2.
Determine the transfer function of the system represented in figure 2.66.

R1 R2
C
us(t) L uv(t)

Figure 2.66. RLC-circuit

Exercise 3.

sFind the transfer

W function of theWstructure diagram represented
W in figure 2.67
W y
1 2 4 5
-

W3
49
 Figure 2.67.Structure diagram of an automatic control system
Exercise 4.
Find the transfer function of the structure diagram represented in figure 2.68

s y
-

Figure 2.68.Structure diagram of an automatic control system

3. ANALYSIS OF THE AUTOMATIC CONTROL

SYSTEMS
It is essential by the analysis of automatic control systems to determine the main
properties of the system. These are the controllability and observability, but also
stability and the quality of dynamics. A system is controllable, if during a finite time
interval ta ≤ t ≤ tl exist such a control action u (t), that takes the system from given
initial state into the final state or sufficiently close to it. The system is fully
controllable, if there exist a control action with which is would be possible to bring
the mapping point of a system to the zero point of the coordinates. The system is

50
partly controllable if the control action does not influence all of state variables or if a
part of state variables does not affect the system output.

R. Kalman has formulated the criteria of controllability depending on the of the

order of the differential equation of the controllable part of the system. The system is
fully controllable, if the order of the differential equation of the controllable part of
the system equals to the order of its state matrix (matrix A). The system is partly
controllable, if the order of the differential equation is less than the order of the state
equation, and the system is not controllable if the order of the differential equation is
zero.

The observability of the system characterises the dependence between is outputs and
state variables, whereby the system could be or fully or partly observable. By Kalman,
a system is observable, if the order of the differential equation of its observable part
equals to the order of the state equation. System is observable, if the order of its
differential equation is less of the order of its state equation, and not observable if the
order of its differential equation is zero

The theory, considering the stability of the system is based on the investigation of the
stability problems of the solutions of differential equations, known from mathematics.
By the classical methods of automatic control, the system stability is controlled by the
transfer function and characteristic equation.

As the static accuracy of the system the correlation of the system, output to the
control input in steady-state operation is meant. The divergence of the output from the
value determined by the control input is called steady-state operation error. In the
process of the calculation of the steady-state operation, error the quality of the
transient process is compared with the desired quality index [2].

3.1. Stability of the system

The dynamics of an automatic control system is characterised by the transient process

following the violation of its equilibrate or steady state. Deviation from the equilibrate
state is induced by the change if the system has input signal(s). A linear automatic
control system could be stable or unstable.

The system is stable, if after the transient process the steady state of the automatic
control system recovers within a finite time interval. Thereby the stable system could

51
equilibrate in a new state, if the reason of the deviation preserves, or in the previous
state if the reason of the deviation elapses, or when the system is invariant in relation
to the given input signal.

The system s unstable, if during the transient process the equilibrate state does not
restore. By the deviation from the equilibrate state the oscillation of the state variables
with growing amplitude might occur, or the state variables diverge monotonously
from the steady state.

a) b)

Figure 3.1.Transient characteristics of a stable system a) balancing in a new state b)

balancing in previous state

a) b)

52
 Figure 3.2. Transient characteristics of an unstable system a) the variable sets to
oscillate with growing amplitude b) the variable diverges monotonously from the
equilibrate state

3.1.1. Stability criteria

3.1.1.1. Determination of the stability by the transfer function

If the transfer function is given in form [2]

b 0⋅p nb 1⋅p n−1 ...b n−1⋅p b n

W  p = , (3.1)
a 0⋅p ma 1⋅p m−1 ...a m−1⋅p a m

then the equation

m m−1
a 0⋅p a 1⋅p ...a m−1⋅pa m=0 (3.2)

is called characteristic equation of the transfer function (3.1), based on the solutions
of which it is possible to decide about the stability of this system described and
evaluate other quantities characterising the transient process.

The solutions of the equation (3.2) could be expressed in the form

p i=ℜ[ p ]± j⋅ℑ[ p ] , (3.3)

where R – real part of the solution

I – imaginary part of the solution

And knowing, that the solution of the differential equation, describing the system

dn d n −1 dm d m−1
b 0⋅ n
⋅y t b 1⋅ n−1
⋅y t ...b n⋅y t =a 0⋅ m
⋅s t a 1⋅ m−1⋅s t ...a n⋅s t 
dt dt dt dt
(3.4)

could be presented in the form

y t = y c t  y d t  , (3.5)

where yc(t) – the solution describing forced motion of the variable

yd (t) – the solution describing free motion or transient process of the variable

53
Placing the solutions of the characteristic equation (3.3) into the general form of the
free motion

y d t = C i⋅e p ⋅t
i
(3.6)

the last, in case of real solution, expresses as follows

y d t = C i⋅e ℜ [ p]⋅t

i
(3.7)

and, in case of complex solutions, as follows:

y d t = C i⋅e ℜ [ p]⋅t⋅sin ℑ[ p]i⋅ti  .

i
(3.8)

It is known from the previous division, that the system is stable if the state variable
equilibrates or in a new or in the previous state or, the free movement of the system
grows down to zero. Mathematically presented

lim y d t =0 , (3.9)

t ∞

on the bases of which it is, possible to conclude from the equations (3.7) and (3.8),
that for each solution of the characteristic equation is valid

ℜ[ p ]i 0 (3.10)

Alternatively, the system is stable if the real parts of all solutions of the characteristic
equation are negative

Example 3.1.

Transfer function of the PT1-block is

K
W=
1T⋅p

And its characteristic function

1T⋅p=0 .

The solution of the characteristic function is

1
p =−
T ,

On the ground of which it could be confirmed, that the PT 1-block is stable by any
values of its parameters.

54
Example 3.2.

The transfer function of the I-block is

K
W=
p

And its characteristic equation

p =0 .

On this basis it is possible to confirm, that the I-block is substantially unstable

For easier survey, the solutions are presented graphically on the complex plane, which
is called p-plane. Using the facilities of the mathematical software, it is easily possible
to add there the axes of damping and characteristic frequencies. The damping rate
could be read by the radiuses from the zero of the p-plane, the values of which
correspond to the sine of the angle between the radius and the imaginary axe. Fur
reading the radian frequency one has to read the distance of the point from the zero
point.

Example 3.3.

The transfer function of the PT3-block is given in the form

100
W= 3 2 .
0.004⋅p 0.06⋅p 0.3⋅p1

Determining the poles of the transfer function (zeros of the characteristic equation)
and locating the results on the p-plane one gets as result the pole diagram presented in
the figure 3.3.

55
 Figure 3.3. Pole-zero diagram

So as one has to do here with the characteristic equation of third order, one has to deal
with three poles also, which are presented by crosses on the diagram. One of which is
placed on the real axe, other describe conjugate complex values, that enables to
conclude that one has to do with an oscillating process with damping ratio 0.5 and
radian frequency 5 s-1. However, system itself is stable, so as real parts of all poles are
negative.

3.1.1.2. Routh’s stability criterion

E. J. Routh has formulated the criteria of system stability, by simultaneous

implementation of which it is possible to assert, that the system is stable [2], [9]:

The necessary but not sufficient condition for the system stability is that all
coefficients of the characteristic equations should have the same sign (positive) or
ai > 0. If required, the whole characteristic equation has to be multiplied by -1;

Necessary and sufficient condition for the system stability is that all Routh’s
coefficients should be positive or Ri > 0.

The general form of the characteristic equation was given with equation (3.2)

a 0⋅p m a 1⋅p m−1...a m−1⋅pa m=0

On the assumption of the general form of the characteristic equation, the following
algorithm determines the Routh’s coefficients

56
 R 0=a 0 a2 a4  0
R 1=a 1 a3 a5  0
R0 R0 R0
R 2=a 2− ⋅a a ' 4 =a 4− ⋅a a ' 6=a 6−
⋅a  0
R1 3 R1 5 R1 7
(3.11)
R1 R1 R1
R 3=a 3− ⋅a ' 4 a ' 5=a 5− ⋅a ' 6 a ' 7=a 7− ⋅a ' 8  0
R2 R2 R2
⋮ ⋮ ⋮ ⋱ ⋮
R m=a m 0 0 0 0

Example 3.4.

Let be given differential equation describing a system.

s3 s2 s 1 1 1
0.045 ⋅y t 2.78 ⋅ÿ t −78.6 ⋅ẏ t −897 ⋅y t =0.8 ⋅s t 0.027 ⋅n t  .
m m m m V T

Based on the Lap lace transformation, handled in the chapter two, the characteristic
equation could be expressed as

s3 3 s2 s 1
0.045 ⋅p 2.78 ⋅p 2−78.6 ⋅p−897 =0 ,
m m m m

Which will be at first studied with the Routh's first criterion

a 00  0.0450 The condition met;

a 10  2.780 The condition is met

a 20  −78.60 The condition is not met!

On this bases on might conclude that the system is not stable

3.1.1.3. Hurwitz’s stability criterion

A. Hurwitz simplified the Routh’s criterion and formulated it as follows: [2], [9]:

Necessary but not sufficient condition for the stability of a system is, that all
coefficients of the characteristic equation should be with the same sign (positive)
or ai > 0. If required the whole characteristic equation must be multiplied by –1.;

Necessary and sufficient condition for the system stability is that Hurwitz’s
determinant and its diagonal minors must be positive.

Hurwitz’s determinant is composed from the coefficients of the characteristic

equation as follows:

57
 [ ]
a1 a3 a5 a7  0
a0 a2 a4 a6  0
0 a1 a3 a5  0
= 0 (3.12)
0 a0 a2 a4  0
⋮ ⋮ ⋮ ⋮ ⋱ ⋮
0 0 0 0 0 am

In addition, the diagonal minors correspondingly

1=[ a 1 ] 0 , (3.13)

2 =
[ ] a1 a 3
a0 a 2
0 , (3.14)

[ ]
a1 a 3 a 5
3= a 0 a 2 a 4 0 etc.. (3.15)
0 a1 a3

Example 3.5.

Let the system given in the Example 3.ǎ be upgraded with a PD-regulator, after what

the differential equation will have the following form (in condition that the
disturbances are not considered)

s3 s2 s 1
0.045 ⋅y t 2.78 ⋅ÿ t 61.2 ⋅ẏ t 234 ⋅y t =0 ,
m m m m

Which could be considered now as characteristic equation and investigate it with the
Hurvitz’a first criterion

a 00  0.0450 The condition met

a 10  2.780 The condition is met

a 20  61.20 The condition is met

a 30  234 0 The condition is met

Based on this, one might study the system based on the second criterion

=
[ ] a1 a 3
a0 a 2
0  125.140 The condition is met

58
1=[a 1]0  2.780 The condition is met

Herewith it might be concluded, that the system is stable

3.1.1.4. Nyquist’s stability criterion

Nyquist’s stability criterion belongs to the frequency methods, where the frequency
characteristic curves of switched-off feedback or open loop systems are investigated,
that enables the stability of systems with delay blocks to be assessed, what has not
been possible by Routh nor by Hurwitz. However, the mathematical presentation of
the Nyquist general criterion is complicated and therefore one special case of this
criterion is more often applied, on the bases of which it is possible to evaluate the
stability of predominately used automatic control systems, however, the results should
be taken with some reservation. The special case is called the Nyquist’s left-hand rule,
that could be applied to the system with negative feedback and it is formulated as
follows: [2], [3], [6], [8]:

The necessary but not sufficient stability condition of a closed system is, that the
radian frequency curve of the open system does not embrace the point (-1, j0) on
the complex plane.

It means, that when moving on the radian frequency curve from the frequency 0 to ∞
direction, the Nyquist’s point (-1, j0) remains to the left from the radian frequency
curve. So, as here one has to do with sufficient condition only, then it is recommended
to check the result with some other method.

To obtain the equation of the radian frequency curve the following transformation will
be made in the transfer function:

p= j  , (3.16)

As a result of what the equation of the radian frequency curve could be presented in
generic form:

b 0⋅ j nb 1⋅ j n−1...b n−1⋅ j b n

W  j = . (3.17)
a 0⋅ j ma 1⋅ j m−1...a m−1⋅ j a m

In addition, the transfer function of the open loop cold is determined according to the
generic structure of the automatic control system, presented in the figure 3.4.

59
W 0=W R⋅W P , (3.18)

n
s e WR u WP y
±

Figure 3.4. Generic structure of an automatic control system

That also could be presented in the form of equation (3.17), which in turn is presented
as

W 0=ℜ[W 0  j ] j⋅ℑ[W 0  j ] , (3.19)

On the bases of what the Nyquist’s diagram on the complex plane will be composed.

Example 3.6.

In the figure 3.5. A structure consisting of three PT1-blocks is represented

K, T1 1, T2
s y
-
1, T3

Figure 3.5.Automatic control system

Transfer function of the opened structure of witch is expressed as

K
W 0  j =
10.1 s⋅j ⋅10.5 s⋅j ⋅10.2 s⋅j  .

On this bases the Nyquist’s diagram (figure 3.6) for different gains will be composed,
thereby the red line means (K = 5), that one has to deal with a stable system, By green
line (K = 12,5) one has to do with a conditionally stability or boundary conditions that
are to be controlled more precisely (to be remembered, all equations for calculation
are adducted to the real system). The blue line (K = 20) describes an unstable system;
so as the Nyquist’s point is embraced by the radian frequency curve

a) b)
60
Figure 3.6. Nyquist’s diagram. a) full diagram, b) blow-up around the Nyquist’s point

For the confirmation of the conclusion in the figure l 3.7 the step response of the
closed system is given

a) b) c)

Figure 3.7. Step response of the closed system a) K = 5, b) K = 13,5, c) K = 20.

The results approve the stability properties stated by Nyquist, whereby in the figure
3.7.b) on has to deal with a stable system, but the quality of the transient process does
not meet the requirements

3.1.1.5. Stability criteria on the basis of Bode diagram

This method is based on the Nyquist’s stability criterion, the results of which are
presented on the Bode diagram and concluded from it formulation is [2], [3], [6], [8]:

Necessary but not sufficient stability condition of closed system is, that the
ordinate φ(ω) or phase angel at the cutting frequency ωL of the frequency axe on
the Bode diagram of an open system is less than |180°|. by absolute value.S

61
As cutting frequency is the frequency considered, by which the ordinate L(ω) of the
frequency axe of the logarithmic amplitude equals zero.

ωL

Figure 3.8. Bode diagram of an unstable system

3.1.1.6. Other methods

With the development of the computer technology a number of different stability

determination method are remained in history, more widely known from them might
have been Mihhailov’s hodograph and Nichols’s diagram.

Mihhailov has proceeded from the closed characteristic equation, that was solved by
frequency method and the values of which were placed on the complex axes. Proceed
from that Mihhailov has formulated a stability criterion of its own name [2]:

System is stable, when by variation of the frequency ω = 0...∞ the hodograph (a

curve on the plane describing the solutions of the function) crosses on the

62
 complex plane in the positive direction in series n quadrants, where n is the order
of the differential equation (characteristic equation) of the system in consider.

10

-1 0

-2 0

-3 0
Im

-4 0

-5 0

-6 0

-7 0
-1 0 0 -8 0 -6 0 -4 0 -2 0 0 20
Re

Figure 3.9. Mihhailov’s hodograph of a stable 3-rd order system

Nichols’s diagram could be considered as a Nyquist’s diagram, represented in another

form of this diagram, which for the Nyquist’s stability criterion is valid.

Figure 3.10. Nichols’s diagram of a stable system

63
3.1.2. Structural stability

Structural stable are called systems which could be convert stable by changing their
parameters, whereby the structure of the system and character of blocks do not need
to be changed. System, which could not be stabilised by changing its parameters (time
constants or gains) are called structurally unstable [2]

. Example 3.7.

Let it be given the characteristic equation of a feedback serial connection of one PT1
and two I-blocks

T 1⋅p 3 p 2 K 1⋅K I 2⋅K I 3=0 ,

Which when investigating by Hurwitz could be expressed as

a 1⋅0−a 0⋅a 30

Alternatively, given system is unstable by all possible values of its parameters, i.e. the
system is structurally unstable.

Example 3.8.

Let it be given the characteristic equation of a feedback system of three PT1 blocks

T 1⋅T 2⋅T 3⋅p 3T 1⋅T 2T 2⋅T 3T 3⋅T 1 ⋅p 2T 1⋅T 2⋅T 3 ⋅p K 1⋅K 2⋅K 3 =0

And by Hurwitz

a 1⋅a 2−a 0⋅a 3=?

Or, if the system is unstable by some values of its parameters, then it is possible to
stabilise it changing its parameters, not changing structure of the system, or one has to
do with structurally stable system

3.1.3. Determination of parametric stability domains

After stability control, it is often required to analyse the impact of some parameters to
the system properties. For instance, it is required to explicate how to impose certain
parameters to stabilise an unstable system. In addition, it could be needed to study
how much could be changed the gain of the system or of a part of the system for
system remaining stable.

64
From the above it is known, that the coefficients of the characteristic equation of the
system are determined by the parameters of the automatic control system, i.e. transfer
rates and time constants. In turn, the solutions of the characteristic equation are
determined by the coefficients of the equations. Thus, there exist steady dependence
between solutions of the characteristic equation and system parameters.

Representing the solutions of the characteristic equation on the complex plane and if
the solutions are located left to the imaginary axe, then the system is stable. Shifting
of some element to the right from the imaginary axe makes system unstable. Hence,
on the complex plane of the solutions the imaginary axe is the margin between stable
and unstable domains.

Similarly could the margin curve of a stability characterising domain be determined in

the coordinates of the characteristic equation’s coefficients. While changing of all
coordinates, it proceeds, in accordance with the order of the system, in the n-
dimensional space. It could be determined a surface in this space, that is called D-
surface. In one side from the surface is the space where the system is stable, in
another side the space where the system is unstable. The D-margin is determined by
the values of the coefficients of the characteristic equation, which correspond to the
stability margin. If some of the characteristic equation’s coefficients of a stable
system should exceed the crucial value matching the stability margin, the system turns
out unstable.

D-margin could be determined for the parameters that contain in the coefficients of
the system. Usually the D-margin is drawn for one or two parameters.

Example 3.9.

Let the following equation describe the dependence between the control voltage of the
magnets of a monorail s (t), the air gap between the train and the rail x (t); and the
asperity of the path n (t):

0.445⋅y t 1.72⋅ÿ t −288.3⋅ẏ t −2241⋅y t =0.5⋅s t − K n⋅n t 

Appling the Hurwitz criterion it could be concluded, that one has to do with an
unstable system

a 0=0.4450 Condition met

65
a 1=1.720 Condition met

a 2=−288.30 Condition net

Therefore, as the parameters of this system could not be changed, then one has to do
with structurally unstable system. In order to make this system usable a stable it will
be upgraded with negative feedback and PD-regulator, after which the system turns to
be structurally stable. Differential equation of the regulator would be

s t =K R⋅ s R t T D⋅s˙R t  ,

But, taking into account that the output f the system is regulator’s input and the input
of the system could be considered as regulator’s output, the differential equation of
the regulator, negative feedback in consider, presents itself in the form

s t =−K R⋅ y t T D⋅ ẏ t  .

New differential equation of the system will be

0.445⋅y t 1.72⋅ÿ t −288.3⋅ẏ t −2241⋅y t =

0.5⋅[−K R⋅ y t T D⋅ẏ t ]−K n⋅n t 

Which after arrangements could be represented in the form

0.445⋅y t 1.72⋅ÿ t −288.3 0.5⋅K R⋅T D ⋅ẏ t 

−22410.5⋅K R ⋅y t =−K n⋅n t  ,

From the characteristic equation of which, in accordance with the Hurwitz’s stability
criterion, could be expressed:

a 0=0.4450 Condition met

a 1=1.720 Condition met

1
a 2=−288.30.5⋅K R⋅T D0 ⇒ T D 576.6⋅ 1. condition
KR

a 3=−22410.5⋅K R 0 ⇒ K R 4482 2. condition

1=[ a 1 ] =[ 1.72 ]0 Condition met

=
[ ]
a1 a 3
a0 a 2
0 ⇒ a 1⋅a 2 −a 0⋅a 30 ⇒ T D 0.25−583⋅
1
KR 3. condition

66
Presenting three appointed conditions graphically it is easy to determine the stability
domain, which is featured in the figure 3.10 as colored domain. If to select the
parameters of the regulator from inside of this domain, one gets a stable system for
sure.

Figure 3.10. Graphical depicture of the D-margins

3.1.4. Some concepts from the stability reserve

In the consideration of the stability margins it became clear, that for all criteria certain
margin could be determined, exceed of which a stable system turns out unstable. Also,
describing an automatic control system by equations one has to consider some errors
due to the following facts

Simplifications are made by composing of the equations

Instead of non-linear equation the linear one are used
System parameters are determined approximately, i.e. with certain error
System parameters could change n time

Therefore, theoretically stable system could proved unstable while implementing in

practice, especially in case if its calculated characteristic curves are too close to the
stability margin. It follows from this, that the parameters of a real system should be
selected so, that the system would have a certain stability reserve. Depending on the
stability criterion different indexes could be used to characterise the stability reserve.

67
For example, by Nyquist’s criterion as amplitude reserve, as well phase reserve is
determined. It could be done using Bode diagram as well.

LR
LR

αR

αR

a) b)

Figure 3.11. Stability reserve for a PT3-element a) by Nyquist b) by Bode (enlarged

image)

The amplitude reserve and phase reserve are considered mainly so as it is possible on
ttheir basis determine the system parameters. For instance, if it is wanted to increase
the amplitude reserve not changing the phase reserve, then one has to change the gain
f the system (remaining –p-block does not impact on the phase). The relevance of
these characterising parameters the following sequence of thoughts could be
explained:

The input signal of the system is sine signal with a frequency by which the gain of the
system is K ≥ 1 and phase shift is exactly 180°. If the output signal-will be directed to
the input of the system by negative feedback, both signals merge, in result of which
the input sign of the system increases, although the set value has not changed, and this

68
process continues with continuing amplification, or the system will “generate” or one
has to do with an unstable process.

3.2. Evaluation of the transient process’s quality

The system is mathematically or theoretically stable if the transien6t process is

damping during a finite time interval. It does not follow from that, any stable process
is suitable for practical application. Practically unsuitable are, for instance, the
systems, where the restoration time of stable state is longer than permissible, or the
amplitudes of the oscillating transient process are too high. In general, the automatic
control structure is applicable, if its transient process is of required quality

Fur the characterisation of the transient process the following main indexes are used:
[2]:

Static or permanent operation error ε;

Duration of the head front tr;
Duration of the transient process ts;
Maximal overregulation σ and corresponding to it time tm;
Vibration or the number of half-waves during the oscillating transient process

Static or permanent operation error ε characterises the accuracy of the system and
is the difference of the settled state of the transient process, and of the steady state
determined by the control action. Systems, static error of which is zero are called
astatic systems.

Duration of the head front tr is considered the time interval what is needed for
changing of the signal response in the interval (0 or 0,1 ... y∞-δ).

Tolerance δ is called the permitted deviation from the steady state determined by the
control action

Duration of the transient process t is called the time periods after which the transient
process has reduced into the tolerance limits

The duration of the heat front and total duration of the transient proceed characterise
the action speed of the system. The last has great importance in those systems, in the

69
operation of which the dynamic processes are essential, launch and breaking
processes, for instance. Increase of the operation speed of systems and truncation of
the transient process duration essentially promotes productivity of equipment.

Figure 3.12. Parameters of the transient process

Maximal over regulation σ is determined by the relation

 ym
=
y∞
. (3.20)

Maximal over regulation determines maximal admissible amplitude of the transient

process, which is stated starting from the operational feature of the system. Let it be
said here, that in most cases the duration of head front and maximal overregulation
are related reciprocally.

The regulating quality could be assessed or experimentally or by calculated signal

response, whereby in the calculations both could be used as analytic as well numerical
and indirect methods.

The grey area in the figure 3.13. characterises the deviation of the output in relation to
the input. In case of an ideal transient characteristic, the grey are would be absent, or
the ouput would follow the input, whereby the measure of difference would be the
gain only. To evaluate the regulating quality in this process the application of the
integral criteria is suitable, in the run of which the differences between ideal transient
characteristic and of the real one are assessed, or simpler, the area of the grey region

70
will be evaluated. For this evaluation the following calculation methods could be
used:

Integral of the absolute error or IAE-criterion

∞
J =∫∣ y t − y ∞∣dt ; (3.21)
0

Integral of the quadratic error or ISE-criterion

∞
2
J =∫∣ y t − y ∞∣ dt ; (3.22)
0

Integral of the time-weighted absolute error or ITAE-criterion

∞
J =∫ t⋅∣ y t − y∞∣ dt ; (3.23)
0

Integral of the time-weighted quadratic error or ITSE-criterion

∞
2
J =∫ t⋅∣ y t − y ∞∣ dt . (3.24)
0

For processes, real transient characteristic of which intersects the ideal transient
characteristic, the quadratic methods should be used, so as the signs of the error is
changing in the process, with, if using absolute methods, could lead to wrong
assessments.

71
 Figure 3.13.Difference of input and output of a PT2-block

3.2.1. Determination of the calculated permanent operation error

The determination of the settled output value and permanent operation error by
calculations is needed for the evaluation of the accuracy of the system under design
even before the simulation, which reduces the scope of work in development. In the
figure 3.14. The general form of an automatic control system is represented. For this
system one can write [3]:

n
s e WR u WP y
±

Figure 3.14. Automatic control system

W R⋅W P WP
y  p = ⋅s  p  ⋅n  p  . (3.25)
1 ∓W R⋅W P 1∓W R⋅W P

For the determination of the settled value the equation (3.25) will be solved as
follows:

y ∞=lim p⋅
p 0 [ W R⋅W P s∞
⋅ 
WP n
⋅ ∞
1∓W R⋅W P p 1∓W R⋅W P p ] . (3.26)

Three solutions of most widespread process-regulator combinations (3.26) are

presented in the table 3.1

Table 3.1. Settled output values of the system and permanent operation error related
to the control input

Regulator Process Settled value Permanent

operation error
K R⋅K P KP 1
PTn y ∞= ⋅s ∞  ⋅n 1K R⋅K P
1 K R⋅K P 1 K R⋅K P ∞
P
1
ITn y ∞=s ∞  ⋅n 0
KR ∞

PTn y ∞=s ∞ 0⋅n ∞ 0

I
ITn Structurally unstable

72
Permanent operation error is given in the operator form as follows;

 p =s  p − y  p  , (3.27)

which, considering the equation (3.25) takes the form

1 WP
 p = ⋅s  p− ⋅n  p . (3.28)
1W R⋅W P 1W R⋅W P

Solving the equation (3.28) by the condition

=lim p⋅
p0 [ 1 s
⋅ ∞−
WP n
⋅ ∞ .
1W R⋅W P p 1W R⋅W P p ] (3.29)

On the basis of these calculations one might conclude, that in the systems containing
one I-block it is possible to reduce the permanent error to zero, Which is not possible
with P-regulator

Exercises to chapter 3.

Exercise 5.

Inspect the stability of the if the transfer function is presented as follows:

 p2 ⋅ p−4.2⋅ p 7.1

W  p = .
 p2.5⋅ p12⋅ p2⋅ p−1

Exercise 6.

Verify the stability of the system by Routh’s criterion if the characteristic equation of
the system is the following

0.0076⋅p 50.051⋅p 40.17⋅p 39.67⋅p 278.1⋅p542=0 .

Exercise 7.

Characteristic equation of a system controlled with two P-regulators is:

T 2⋅p 32⋅T⋅p 2 K⋅K R1 ⋅p K⋅K I⋅K R1⋅K R2=0 .

73
Determine the stability domain by the regulators KR1 and KR2, if the parameters of the
system are K = 1, KI = 0.5 s-1 and T = 2 s.

Exercise 8.

Determine the stability domain of the regulator, if the characteristic equation is given
in the form
2
[T I⋅T P⋅1−K P⋅K R ]⋅p [ T I  K P⋅K R⋅T I −T S ]⋅p K P⋅K R=0 .

Exercise 9.

Appoint the following Joonis 3.15. Siirdekarakteristik

parameters from the
near by graphic

Permanent
operation error;

Maximal over
regulation;

Approximate IAE-
criterion;

Approximate ISE-
criterion.

74