Control engineering a guide for beginners

Control Engineering
A guide for beginners

Manfred Schleicher
Frank Blasinger

Preface
This work is intended to be of practical assistance in control engineering technology. It will help
you to select and set up a suitable controller for various applications. It describes the different
types of controller and the options for setting them up. The explanations and definitions are provid-
ed without using advanced mathematics, and are mainly applied to temperature-control loops.
In this new and revised edition, Chapters 3 and 5 have been extensively updated.
We wish to thank our colleagues for their valuable support in writing this book.

Fulda, January 2003

Manfred Schleicher Frank Blasinger

JUMO GmbH & Co. KG, Fulda, Germany
Copying is permitted with source citation!

3rd Edition

Part number: 00323761
Book number: FAS 525
Printing date: 02.04
ISBN: 3-935742-01-0

Inhalt
1 Basic concepts ............................................................................ 7
1.1 Introduction .................................................................................................. 7
1.2 Concepts and designations ........................................................................ 7
1.3 Operation and control .................................................................................. 7
1.4 The control action ...................................................................................... 11
1.5 Construction of controllers ....................................................................... 12
1.6 Analog and digital controllers ................................................................... 18
1.6.1 Signal types .................................................................................................. 18
1.6.2 Fundamental differences .............................................................................. 20
1.7 Manipulating devices ................................................................................. 23
1.8 Other methods of achieving constant values .......................................... 25
1.8.1 Utilizing physical effects ............................................................................... 25
1.8.2 Constructional measures ............................................................................. 25
1.8.3 Maintaining constant values by operation ................................................... 26
1.9 Main areas of control engineering ............................................................ 27
1.10 Tasks of the control engineer .................................................................... 28

2 The process ................................................................................ 29
2.1 Dynamic action of technical systems ...................................................... 29
2.2 Processes with self-limitation ................................................................... 32
2.3 Processes without self-limitation ............................................................. 33
2.4 Processes with dead time ......................................................................... 35
2.5 Processes with delay ................................................................................. 37
2.5.1 Processes with one delay (first-order processes) ........................................ 38
2.5.2 Processes with two delays (second-order processes) ................................. 39
2.5.3 Processes with several delays (higher-order processes) ............................. 41
2.6 Recording the step response .................................................................... 41
2.7 Characteristic values of processes .......................................................... 43
2.8 Transfer coefficient and working point .................................................... 43

Inhalt
3 Continuous controllers .............................................................. 45
3.1 Introduction ................................................................................................ 45
3.2 P controller ................................................................................................. 45
3.2.1 The proportional band .................................................................................. 47
3.2.2 Permanent deviation and working point ...................................................... 49
3.2.3 Controllers with dynamic action ................................................................... 52
3.3 I controller ................................................................................................... 53
3.4 PI controller ................................................................................................ 54
3.5 PD controller ............................................................................................... 57
3.5.1 The practical D component - the DT1 element ............................................ 60
3.6 PID controller .............................................................................................. 61
3.6.1 Block diagram of the PID controller ............................................................. 62

4 Control loops with continuous controllers .............................. 63
4.1 Operating methods for control loops with continuous controllers ....... 63
4.2 Stable and unstable behavior of the control loop ................................... 64
4.3 Setpoint and disturbance response of the control loop ......................... 65
4.3.1 Setpoint response of the control loop .......................................................... 66
4.3.2 Disturbance response .................................................................................. 67
4.4 Which controller is best suited for which process? ................................ 68
4.5 Optimization ................................................................................................ 69
4.5.1 The measure of control quality ..................................................................... 70
4.5.2 Adjustment by the oscillation method .......................................................... 71
4.5.3 Adjustment according to the transfer function or process step response ... 72
4.5.4 Adjustment according to the rate of rise ...................................................... 75
4.5.5 Adjustment without knowledge of the process ............................................ 76
4.5.6 Checking the controller settings .................................................................. 77

Inhalt
5 Switching controllers ................................................................ 79
5.1 Discontinuous and quasi-continuous controllers ................................... 79
5.2 The discontinuous controller .................................................................... 80
5.2.1 The process variable in first-order processes .............................................. 81
5.2.2 The process variable in higher-order processes .......................................... 83
5.2.3 The process variable in processes without self-limitation ........................... 85
5.3 Quasi-continuous controllers: the proportional controller .................... 86
5.4 Quasi-continuous controllers: the controller with dynamic action ....... 89
5.4.1 Special features of the switching stages ..................................................... 90
5.4.2 Comments on discontinuous and quasi-continuous
controllers with one output .......................................................................... 90
5.5 Controller with two outputs: the 3-state controller ................................ 91
5.5.1 Discontinuous controller with two outputs ................................................... 91
5.5.2 Quasi-continuous controller with two outputs,
as a proportional controller .......................................................................... 93
5.5.3 Quasi-continuous controller with two outputs and dynamic action ............ 94
5.5.4 Comments on controllers with two outputs ................................................. 94
5.6 The modulating controller ......................................................................... 95
5.7 Continuous controller with integral motor actuator driver .................... 98

6 Improved control quality through special controls .............. 101
6.1 Base load .................................................................................................. 101
6.2 Power switching ....................................................................................... 103
6.3 Switched disturbance correction ........................................................... 104
6.4 Switched auxiliary process variable correction .................................... 107
6.5 Coarse/fine control .................................................................................. 107
6.6 Cascade control ....................................................................................... 108
6.7 Ratio control ............................................................................................. 110
6.8 Multi-component control ......................................................................... 111

Inhalt
7 Special controller functions .................................................... 113
7.1 Control station / manual mode ............................................................... 113
7.2 Ramp function .......................................................................................... 114
7.3 Limiting the manipulating variable ......................................................... 114
7.4 Program controller ................................................................................... 115
7.5 Self-optimization ...................................................................................... 116
7.6 Parameter/structure switching ............................................................... 118
7.7 Fuzzy logic ................................................................................................ 118

8 Standards, symbols, literature references ............................ 121

1 Basic concepts
1.1 Introduction
Automatic control is becoming more and more important in this age of automation. In manufactur-
ing processes it ensures that certain parameters, such as temperature, pressure, speed or voltage,
take up specific constant values recognized as the optimum, or are maintained in a particular rela-
tionship to other variables. In other words, the duty of control engineering is to bring these param-
eters to certain pre-defined values (setpoints), and to maintain them constant against all disturbing
influences. However, this apparently simple duty involves a large number of problems which are
not obvious at first glance.
Modern control engineering has links with almost every technical area. Its spectrum of application
ranges from electrical engineering, through drives, mechanical engineering, right up to manufactur-
ing processes. Any attempt to explain control engineering by referring to specialized rules for each
area would mean that the control engineer has to have a thorough knowledge of each special field
in which he has to provide control. This is simply not possible with the current state of technology.
However, it is obvious that there are certain common concepts behind these specialized tasks. It
soon becomes clear, for example, that there are similar features in controlling a drive and in pres-
sure and temperature control: these features can be described by using a standard procedure. The
fundamental laws of control engineering apply to all control circuits, irrespective of the different
forms of equipment and instruments involved.
A practical engineer, trying to gain a better understanding of control engineering, may consult vari-
ous books on the subject. These books usually suggest that a more detailed knowledge of control
engineering is not possible, without extensive mathematical knowledge. This impression is com-
pletely wrong. It is found again and again that, provided sufficient effort is made in presentation, a
clear understanding can be achieved, even in the case of relationships which appear to demand an
extensive mathematical knowledge.
The real requirement in solving control tasks is not a knowledge of many formulae or mathematical
methods, but a clear grasp of the effective relationships in the control circuit.

1.2 Concepts and designations
Today, thanks to increasing standardization, we have definite concepts and designations for use in
control engineering. German designations are laid down in the well-known DIN Standard 19 226
(Control Engineering, Definitions and Terms). These concepts are now widely accepted in Germany.
International harmonization of the designations then led to DIN Standard 19 221 (Symbols in con-
trol engineering), which permits the use of most of the designations laid down in the previous stan-
dard. This book keeps mainly to the definitions and concepts given in DIN 19 226.

1.3 Operation and control
In many processes, a physical variable such as temperature, pressure or voltage has to take up a
specified value, and maintain it as accurately as possible. A simple example is a furnace whose
temperature has to be maintained constant. If the energy supply, e.g. electrical power, can be var-
ied, it is possible to use this facility to obtain different furnace temperatures (Fig. 1). Assuming that
external conditions do not change, there will be a definite temperature corresponding to each value
of the energy supply. Specific furnace temperatures can be obtained by suitable regulation of the
electrical supply.

However, if the external conditions were to change, the temperature will differ from the anticipated
value. There are many different kinds of such disturbances or changes, which may be introduced
into the process at different points. They can be due to variations in external temperature or in the

JUMO, FAS 525, Edition 02.04 7

1 Basic concepts

Fig. 1: Operation and control

8 JUMO, FAS 525, Edition 02.04

1 Basic concepts
heating current, or caused by the furnace door opening. This type of temperature control takes no
account of the actual furnace temperature, and a deviation from the required value may not be no-
ticed by the operator.
Some form of control is necessary if the furnace temperature has to maintain its value in spite of
changes in external conditions, or non-constant disturbances which cannot be predicted. In its
simplest form the control may just be a thermometer which measures and indicates the actual fur-
nace temperature. The operator can now read the furnace temperature, and make appropriate ad-
justments to the energy supply, in the event of a temperature deviation (Fig. 1).
The energy supply is now no longer pre-determined, but is linked to the furnace temperature. This
measure has converted furnace operation into furnace control, with the operator acting as the con-
troller.
Control involves a comparison of the actual value with the desired value or setpoint. Any deviation
from the setpoint leads to a change to the energy supply. The energy input is no longer fixed, as is
the case with simple operation, but depends on the actual process value attained. We refer to this
as a closed control loop (Fig. 2)
If the connection to the temperature probe is broken, the control loop is open-circuited. Because
there is no feedback of the process value, an open control loop can only be used for operation.

Fig. 2: The closed control loop
The control loop has the following control parameters (the abbreviations conform to DIN 19 226):
Process variable (process value, PV) x: the process value is the control loop variable which is
measured for the purpose of control and which is fed into the controller. The aim is that it should al-
ways be made equal to the desired value through the action of the control (example: actual furnace
temperature).
Desired value (setpoint, SP) w: the predetermined value at which the process variable has to be
maintained through the action of the control (example: desired furnace temperature). It is a param-
eter which is not influenced by the control action, and is provided from outside the control loop.
Control difference (deviation) e: difference between desired value and process variable e = w - x
(example: difference between required and actual furnace temperature).


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Disturbance z: an effect whose variation exerts an unfavorable influence on the process value (in-
fluence on the controlled variable through external effects).
Controller output YR: it represents the input variable of the manipulating device (the manipulator
or actuator).
Manipulating variable y: a variable through which the process value can be influenced in the re-
quired way (e.g. heating power of the furnace). It forms the output of the control system and, at the
same time, the input of the process.
Manipulation range Yh: the range within which the manipulating variable can be adjusted.
Control loop: connection of the output of the process to the input of the controller, and of the con-
troller output to the process input, thus forming a closed loop.
It consists of controller, manipulator and process.
The physical units involved can differ widely:
process value, setpoint, disturbance and deviation usually have the same physical units such as
°C, bar, volts, r.p.m., depth in metres etc. The manipulating variable may be proportional to a heat-
ing current in amps or gas flow in m3/min, or is often a pressure expressed in bar. The manipulation
range depends on the maximum and minimum values of the manipulating variable and is therefore
expressed in the same units.


1 Basic concepts
1.4 The control action
The basic task of the controller is to measure and prepare the process value PV, and compare it
with the setpoint SP; as a result it produces the appropriate manipulating variable MV. The control-
ler has to perform this action in a way which compensates for the dynamic characteristics of the
controlled process. This means that the process value PV should reach the setpoint SP as rapidly
as possible, and then fluctuate as little as possible about it.
The action of the controller on the control loop is characterized by the following parameters:
- the overshoot: Xo ,
- the approach time: Ta, the time taken for the process value PV to reach the
new setpoint SP for the first time,
- the stabilization time: Ts,
- and also agreed tolerance limits ± ∆x (see Fig. 3)

Fig. 3: Criteria for control action
The controller is said to have “stabilized” when the process is operating with a constant manipulat-
ing variable MV, and the process value PV is moving within the agreed tolerance band ± ∆x.
In the ideal case the overshoot is zero. In most cases this cannot be combined with a short stabili-
zation time. In certain processes, e.g. speed controls, rapid stabilization is important, and a slight
overshoot beyond the setpoint can be tolerated. Other processes, such as plastics production ma-
chinery, are sensitive to a temperature overshoot, since this can quite easily damage the tool or the
product.


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1.5 Construction of controllers
The choice of a suitable controller depends essentially on its application. This concerns both its
mechanical features and its electrical characteristics. There is a wide range of different designs and
arrangements, so only a few will be discussed here. The discussion is limited to electronic control-
lers, and excludes mechanical and pneumatic control systems. The user, who is faced with choos-
ing a controller for his particular application, will be shown initially which types are available. The
listing is not intended to be comprehensive.

Mechanical variations:
- Compact controllers (process controllers) contain all the necessary components (e.g. display,
keypad, input for setpoint etc.) and are mounted in a case which includes a power supply. The
housing usually has one of the standard case sizes, 48mm x 48mm, 48mm x 96mm,
96mm x 96mm or 72mm x 144mm.
- Surface-mounting controllers are usually installed inside control cabinets and mounted on a
DIN-rail or the like. Indicating devices such as process value display or relay status LEDs are not
usually provided, as the operator does not normally have access to these controllers.
- Rack-mounting controllers are intended for use in 19-inch racks. They are only fitted with a
front panel and do not have a complete housing.
- Card-mounted controllers consist of a microprocessor with suitable peripherals, and are used
in various housing formats. They are frequently found in large-scale installations in conjunction
with central process control systems and PLCs. These controllers again have no operating or in-
dicating devices, since they receive their process data via an interface from the central control
room through software programs.

Functional distinctions
The terms that are used here are covered and explained in more detail in later chapters (see Fig. 4).
- Continuous controllers
(usually referred to as proportional or analog controllers)
Controllers which receive a continuous (analog) input signal, and produce a controller output
signal that is also continuous (analog). The manipulating signal can take on any value within the
manipulation range. They usually produce output signals in the range 0 — 20mA, 4 — 20mA or
0 — 10V. They are used to control valve drives or thyristor units.

- Discontinuous controllers
2-state controllers (single-setpoint controllers) with one switching output are controllers that pro-
duce a discontinuous output for a continuous input signal. They can only switch the manipulating
variable on and off, and are used, for instance, in temperature-control systems, where it is only
necessary to switch the heating or cooling on or off.
3-state controllers (double-setpoint controllers) have two switching control outputs. They are sim-
ilar to 2-state controllers but have two outputs for manipulating variables. These controllers are
used for applications such as heating/cooling, humidifying/dehumidifying etc.


1 Basic concepts
- Quasi-continuous controllers
Quasi-continuous controllers with one switching output are controllers that achieve a quasi-
continuous action. The average value of the controller output over a defined time interval shows
approximately the same time-dependent variation as a continuous controller. Applications are, for
instance, temperature control (heating or cooling), where improved control-loop performance is re-
quired. In practice, quasi-continuous controllers with one switching output are also described as 2-
state controllers.
Quasi-continuous controllers with two switching outputs can steer a process in opposing di-
rections (for example, heating/cooling or humidifying/dehumidifying). These controllers also
achieve a quasi-continuous action, by pulsing the switched outputs. In practice, all controllers that
use two outputs to steer a process in opposing directions are referred to as 3-state controllers.
Here the outputs need not necessarily be switched, but can be continuous.

- Modulating controllers
Modulating controllers have two switching outputs and are specially designed for motorized actua-
tors which are used, for instance, to drive a valve to the open and closed positions.

- Actuating controllers
Actuating controllers are also used for motorized actuators and again have two switching outputs.
They differ from modulating controllers by requiring feedback of the actuator position (stroke re-
transmission).


1 Basic concepts

Fig. 4: Difference in controller functions


1 Basic concepts
All these types of controller (apart from the discontinuous controller) can be implemented with dif-
ferent forms of dynamic response. This is often referred to as the “controller structure”. The terms
used are P, PI, PD or PID controllers (see Fig. 5).
Different setpoint arrangements
The setpoint can be set manually on the controller by means of a potentiometer, or by using keys
to input digital values. The setpoint is indicated in either analog form (pointer of a setpoint knob), or
digitally as a numerical value.
Another possibility is the use of an external setpoint. The setpoint is then fed in as an electrical sig-
nal (e.g. 0 — 20mA) from some external device. As well as these analog signals, it is also possible
to use digital signals for setting the setpoint. The signals are fed into the controller through a digital
interface and can be derived from another digital instrument, or from a computer linked to the con-
troller. If this external setpoint operates according to a fixed time sequence (program), this is also
referred to as program or sequence control.


1 Basic concepts

Fig. 5: Typical step responses
Evaluation of the process variable
The process variable must be available as an electrical signal. Its form depends on the sensor used
and on the processing of this signal. One possibility is to connect the transducer signal (sensor,
probe) directly to the controller input. The controller must then be capable of processing this signal;
in many temperature probes the output signal is not a linear function of the temperature, and the
controller must have a suitable linearization facility.


1 Basic concepts
The other possibility is the use of a transmitter.
The transmitter converts the sensor signal into a standard signal (0 — 20mA, 0 — 10V) and usually
also linearizes the signal. In this case the controller need only have an input for standard signals.
The process value is normally displayed on the controller. This can be in the form of a digital dis-
play (numerical indication), which has the advantage of being readable from a longer distance. The
advantage of the analog display (pointer movement) is that trends such as rising or falling of the
process variable are clearly visible, as well as the position within the control range.

Fig. 6: Example for external connections to a controller

In many cases the process value requires further processing, e.g. for a recorder or for remote indi-
cation. Most controllers provide a process value output where the process variable is given out as
a standard signal.
In order to signal movements of the process variable above or below certain values, the controllers
are provided with so-called limit comparators (limit value or alarm contacts), which provide a signal
if the process value infringes set limits. This signal can then be used to trigger alarms or similar
equipment.


1 Basic concepts
1.6 Analog and digital controllers

1.6.1 Signal types
Technical systems can be classified according to the type of signals at their inputs and outputs.
The signals differ in their technical nature. In control systems we often find temperature, pressure,
current or voltage as signal carriers which, at the same time, determine the units of measurement.
The signals can be divided into different types, depending on their range of values and variation
with time.

Fig. 7: Various signal forms


1 Basic concepts
Analog signals
Analog signals have the greatest number of possible signal levels. The measuring device converts
the process variable PV, for example a temperature, into a signal corresponding to this tempera-
ture. Each temperature value corresponds to a value of the electrical signal. If the temperature now
varies continuously, the signal will also vary continuously. We call this a value-continuous signal.
The essential element in defining analog signals is that such signals pass continuously through a
full range of values.
The time course is also continuous; at every instant the signal value corresponds to the tempera-
ture at this instant. It is therefore also a time-continuous signal (see Fig. 7a). In an application where
the measuring device operates through a channel selection switch in which the contact arm is ro-
tating continuously, the measured signal is only sampled at certain discrete times. The signal is
then no longer time-continuous, but time-discrete (see Fig. 7b). On the other hand, the measure-
ment remains value-continuous, since the measured signal is fully reproduced at each sampling in-
stant.
Digital signals
Digital signals belong to the group of discrete signals. Here the individual signal levels are repre-
sented by numerals (digitally). This means that discrete signals can only take up a limited number
of values. The variation of such discrete signals with time always appears as a series of steps.
A simple example of a system with discrete signals is the control system of a passenger lift or ele-
vator, which can only take up discrete values for the height. This type of signal appears in control
systems using computers, or digital controllers. The important feature here is that the analog sig-
nals can only be converted into digital signals by discretization of the signal level. There are no
longer any intermediate values. However, assuming that the conversion takes place at an effective-
ly unlimited speed, it is still possible to have a time-continuous signal (see Fig. 7c). In practice, the
technical methods available limit the conversion to a time-discrete form. In other words, the ana-
log/digital converter, used in digital control, only carries out the conversion process at discrete time
intervals (sampling time). From the analog signal we obtain a result which is both value-discrete
and time-discrete (see Fig. 7d).
It is quite evident that conversion of analog to digital signals in this way leads to a loss of informa-
tion about the measured signal.
Binary signals
In their simplest form the signals can only have two states, and are therefore called binary signals.
The control engineer is already familiar with this type of signal. The two states are normally de-
scribed as “0” and “1”. Every switch used to turn a voltage on and off produces a binary signal as
its output variable. Binary signals are also referred to as logic values and are assigned the values
“true” and “false”. Virtually all digital circuits in electrical engineering work with this type of logic
signals. Microprocessors and computers are built up from such elements, which only recognize
these two signal states (see Fig. 7e).
3-state signals
Signals with the next higher information content after binary signals are 3-state signals (sometimes
called tri-state signals). They are often used in connection with motors. Essentially, a motor can
have three operating states. The motor can be stationary, or it can rotate clockwise or anticlock-
wise. Corresponding elements with a 3-state action are frequently found in control engineering,
and are of great interest. Each of the three signal levels can have any desired value; in certain cas-
es each signal level can be a positive signal, or the magnitude of the positive and negative signals
can be different (see Fig. 7f).


1 Basic concepts
1.6.2 Fundamental differences
A controller produces a relationship between the process variable PV and the setpoint SP, and de-
rives from it the manipulating variable MV. There are a number of ways to carry out this task: me-
chanical, pneumatic, electrical, mathematical. The mechanical controller, for example, alters a sig-
nal through a lever system, the electronic controller through operational amplifiers. With the intro-
duction of more powerful and low-cost microprocessors, another type of electrical controller has
cornered the market in recent years, the microprocessor controller (digital controller). The mea-
surement signal is no longer processed in an operational amplifier, but is now calculated using a
microprocessor. The different structures found in these digital controllers can be described directly
in mathematical terms.
The term “digital” means that the input variable, the process value, must initially be digitized, i.e.
converted into a numerical value, as described in Chapter 1.6.1, before the signal can be pro-
cessed by the microprocessor. The calculated output signal (the manipulating variable) then has to
be converted back to an analog signal, by a digital to analog converter, to control the process, or
alternatively, fed directly to a digital actuator. There is very little functional difference between digi-
tal and analog controllers, so this is not covered in-depth in the context of this book.
Use of a digital display is, in itself, not an adequate criterion for calling an instrument a digital con-
troller. There are instruments which work on analog principles, but which have a digital display.
They do not have an internal microprocessor to calculate the signals, and are therefore still referred
to as analog controllers.

Fig. 8: Principle of analog and digital controllers


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Fig. 9: Arrangement of analog and digital controllers


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Advantages and disadvantages of digital controllers
Analog controllers are built up from operational amplifiers. The control parameters are set by
means of potentiometers, trimmers or solder links. Controller structure and characteristics are
largely predetermined by the design and construction. They are used where there is no requirement
for very high accuracy, and where the required features of the controller, such as its dynamic ac-
tion, are already known at the planning stage. Because of its speed of reaction, the analog control-
ler has clear advantages in extremely fast control loops.
In digital controllers a microprocessor converts all analog inputs into numerical values, and uses
them to calculate the manipulating variable. This has certain advantages compared with analog
processing:
- increased accuracy of control, depending on the measurement signal and the technology used
(e.g. A/D converter). Unlike components which are affected by tolerances and drift, the mathe-
matical relationships used have a constant accuracy and are unaffected by ageing, variations in
components and temperature effects.
- high flexibility in the structure and characteristics of the controller. Instead of having to adjust
parameters or unsolder components, as in analog controllers, a digital controller can be modi-
fied by simply programming a new linearization, controller structure etc. by inputting numerical
values
- facility for data transfer. There is often a need to modify or store information about process sta-
tus variables, or pass it on for different uses, and this is very simple to achieve using digital
technology. Remote setting of parameters through data systems, such as process management
systems via a digital interface, is also quite simple.
- control parameters can be optimized automatically, under certain conditions.
Digital controllers also have disadvantages compared with controllers operating on analog princi-
ples. The digital display, normally standard with digital controllers, makes it more difficult to identify
trends in process values. Digital instruments are more sensitive to electromagnetic interference.
The processor needs a certain time to calculate parameters and to carry out other tasks, so that
process values can only be read in at certain time intervals. The time interval between two succes-
sive readings of the process variable is referred to as the sampling time, and the term “sampling
controller” is often used. Typical values of the sampling time in compact controllers are in the range
50 — 500msec. There are no technical reasons why controllers with sampling times less than
1 msec could not be built. If the process is relatively slow compared with the sampling time, the
behavior of a digital controller is similar to that of an analog controller, since the sampling action is
no longer noticeable.


1 Basic concepts
1.7 Manipulating devices
The purpose of the manipulating device is to influence the process variable. Its main task is to reg-
ulate a mass or energy flow. Mass flows may have either gaseous or liquid state, e.g. natural gas,
steam, fuel oil etc.

Fig. 10: Overview of different manipulators


1 Basic concepts

Fig. 11: Overview of different actuators
Energy flows often take the form of electrical energy. The energy supply can be varied discontinu-
ously through contacts, relays or contactors, or continuously by means of variable transformers,
variable resistors or thyristor units.


1 Basic concepts
The manipulating device is frequently operated by an actuator where the controller cannot operate
it directly, for instance, if it cannot provide sufficient power, or where the output of the controller is
in the wrong energy form for driving the manipulator. The controller then operates either a mechan-
ical-pneumatic or electrically powered driver. For example, the relays built into switching control-
lers can normally only handle currents up to 5A; external contactors or solid-state relays are then
used to control the higher power required by the process.
Table 1 gives a brief overview of the various manipulators/drivers and their operation from suitable
controllers.

Controller type Operated manipulators/drivers
Continuous controllers Adjustable resistor
Thyristor unit
Valves, flaps, slides
Speed-controlled motors
2-state controllers Contact
Relay, contactor, solenoid valve
Solid-state relay for heating, cooling etc.
3-state controllers (switching) Heating, cooling, relays etc.
Modulating controllers Actuating motors (AC, DC, 3-phase etc.)
Table 1: Controller types and manipulators/drivers

1.8 Other methods of achieving constant values
Automatic control, i.e. measurement of the process variable PV, comparison with the setpoint SP,
and production of the manipulating variable MV, is not the only possible way of ensuring that a pa-
rameter is kept constant. There are several other methods of achieving this, which often offer a
more cost-effective solution, as an alternative to automatic control.

1.8.1 Utilizing physical effects
There are a number of physical values which remain constant over a wide range even when sub-
jected to varying external influences. They include, for example, the melting point of a substance.
While ice is melting, the temperature remains constant at 0°C. Physical effects like this are suc-
cessfully used in many measurements, particularly in the laboratory. In this way, a temperature can
be maintained constant to a high degree of accuracy, without the expense of sophisticated control
equipment.

1.8.2 Constructional measures
To some extent, parameters can be held constant through suitable constructional features. For ex-
ample, a constant liquid level can be maintained in a container or tank, in spite of variations in the
inflow rate, just by providing an overflow (see Fig.12a). Another example is a swimming pool, where
the water level can be maintained constant by providing an overflow all round the pool.


1 Basic concepts

Fig. 12: Methods of achieving constant values

1.8.3 Maintaining constant values by operation
As already discussed in Chapter 1.3, “Operation and control”, a parameter can be kept constant
by suitable operation. An example could be to maintain a constant furnace temperature. Assuming
a constant voltage, i.e. a steady power supply to an electrically heated furnace, the setting of an
energy regulator can be varied to provide different furnace temperatures. By noting these tempera-
tures, i.e. by producing a temperature scale and attaching this to the energy regulator, we can then
set any desired furnace temperature. As the adjustment is made by hand, we refer to this as man-
ual operation. The input parameter in this form of temperature control is the setting of the energy
regulator, the output variable is the furnace temperature, which can be displayed on a suitable indi-
cating instrument (see Fig. 1).


1 Basic concepts
Adjustment of the input parameters need not take place manually, but can be automated: this is
then called automatic operation. As an example, take the control of a mixing process. The task
consists of producing a flow Q2 which is proportional to an externally determined flow Q1 in order
to achieve a particular mixture ratio (see Fig. 12b). Here the flow Q1 is determined as the input vari-
able, and is applied to the operating equipment. The output of the operating equipment operates a
manipulator which changes the flow Q2.
From this it is clear that a process variable can also be kept constant by simple operation. Howev-
er, it should be borne in mind that operation has considerable disadvantages compared with auto-
matic control. If the process is subjected to a disturbance, or there is a change in the transfer char-
acteristic of the manipulating device, there can be undesirable changes in output, even with a fixed
transfer action between input and output variables.

1.9 Main areas of control engineering
Today, control engineering has applications in almost every area of technology. In Chapter 1.1 we
have already seen that these different applications have certain common features, which can be
described through a standard procedure. A number of main application groups have evolved as a
result of differing process variables, stabilization rates, types of machinery and equipment, and
certain special features of the application field.

! industrial process control
! drive control (speed control)
! control of electrical variables
! positional control
! course control

Fig. 13: Main areas of control engineering
Industrial process control
This heading covers the control of temperature, pressure, flow, level etc. in many different industrial
applications. If we look at the criterion “stabilization time”, this can have an order of magnitude
ranging from milliseconds, e.g. in pressure control, up to several hours in the case of temperature
control of larger installations (industrial furnaces).
Drive control (speed control)
This group includes speed control of motors on different machines and installations, such as in
plastics manufacture, paper production or textile machinery. Specially designed controllers are
normally used for these applications, since they have to remain stable during fast disturbances in
the range of tenths of seconds.
Control of electrical variables
This refers to stabilization of electrical parameters, e.g. voltage, current, power or even frequency.
This type of equipment is used in power generation or to stabilize characteristic values in supply
networks. Here again there are very fast disturbances, in the range of tenths of seconds or even
shorter.


1 Basic concepts
Position control
This involves the positioning of tools, workpieces or complete assemblies, either in two or three di-
mensions. Examples include a milling machine and the positioning of guns on ships and tanks.
Once again, stabilization at the setpoint must be very rapid and very accurate.
Course control
The control of the course of ships or planes. Here the controller has to satisfy special demands,
such as high processing speed and operational safety, combined with low weight.

1.10 Tasks of the control engineer
So far we have discussed various concepts and designations, the differences between operation
and control and the various forms of controllers and manipulators. We can now summarize the
tasks a control engineer has to face in practice.
The most important tasks for a control engineer are as follows:

! Determining the process variable
! Checking whether automatic control
offers significant advantages
! Determining the measurement site
! Assessing the disturbances
! Selecting the manipulator
! Selecting a suitable controller
! Installation of the controller
in accordance with applicable regulations
! Starting up, adjusting parameters, optimizing
Fig. 14: Tasks of a control engineer
By control engineer, we don’t mean specialist engineers and technicians from universities or re-
search departments, who work in the laboratory developing controllers, control algorithms or spe-
cial control circuits. Specialists such as these require a much more extensive knowledge. Instead
we are addressing people working on site who may have to optimize an unsatisfactory control loop
or convert from manual operation to automatic control, or those involved in the design of a control
loop for a new installation. In most cases these operations can be tackled without using advanced
mathematics. All that is really needed is a basic understanding, pragmatic rules and knowledge
gained from past experience.
As a general principle for planning a control system, it should be borne in mind that when high-
performance demands are placed on a controller, the costs will increase considerably.


2 The process
2.1 Dynamic action of technical systems
The process is the element of a system which has to be controlled in accordance with the applica-
tion duty. In practice, the process represents either an installation or a manufacturing process
which requires controlling. Normally, the process covers a number of elements within a system.
The input is the manipulating variable y received from the control device. The output is represented
by the process variable x. As well as these two variables there are the disturbances z which affect
the process to some extent, through external influences or process-dependent variations.
An example of a process is a gas-fired furnace (see Fig. 15). At the start of the process is the valve,
which has as its input the manipulating variable of the controller. The valve controls the gas flow to
the burner. The burner produces heat energy by burning the gas, which brings the charge up to a
higher temperature. If the temperature in the charge is measured (process value), this also forms
part of the process. The final component of the process here is the sensor, which has the job of
converting the temperature into an electrical signal. Disturbances here are all the variables in the
process which, when they change, result in a different temperature for the same valve setting.
Example: If the manipulating variable is just large enough to give the required temperature in the
charge, and a disturbance occurs due to a fall in outside ambient temperature, then, if the manipu-
lating variable is not changed, the temperature in the charge will also be lower.

Fig. 15: Input and output variables of a process
When designing a control loop, it is important to know how the process responds when there is a
change in one of the influencing variables mentioned above. On the one hand, it is of interest to
know the new process value reached when stable conditions have been attained, following such
changes. On the other hand, it is also important to find out how the process value varied with time
during the transition to the new steady-state value. A knowledge of the characteristics determined
by the process is essential and can help to avoid difficulties later on, when designing the process.


2 The process
Although processes have many different technical arrangements, they can be broadly categorized
by the following features:
- with and without self-limitation,
- with and without dead time or timing elements,
- linear or non-linear.
In most cases, however, a combination of individual characteristics will be present.
An accurate characterization and detailed knowledge of the process is a prerequisite for the design
of controls and for the optimum solution of a control task. It is not possible to select suitable con-
trollers and adjust their parameters, without knowing exactly how the process behaves. The de-
scription of the dynamic action is important to achieve the objective of control engineering, i.e. to
control the dynamic behavior of technical dynamic systems and to impose a specific transient re-
sponse on the technical system.
Static characteristic
The static behavior of a technical system can be described by considering the output signal in rela-
tion to the input signal. In other words, by determining the value of the output signal for different in-
put signals. With an electrical or electronic system, for instance, a voltage from a voltage source
can be applied to the input, and the corresponding output voltage determined. When considering
the static behavior of control loop elements, it is of no importance how a particular control element
reaches its final state. The only comparison made is limited to the values of the input and output
signals at the end of the stabilization or settling time.
When measuring static characteristics, it is interesting to know, amongst other things, whether the
particular control loop element exhibits a linear behavior, i.e. whether the output variable of the
control element follows the input proportionally. If this is not the case, an attempt is made to deter-
mine the exact functional relationship. Many control loop elements used in practice exhibit a linear
behavior over a limited range. With special regard to the process, this means that when the manip-
ulating variable MV is doubled, the process value PV also doubles; PV increases and decreases
equally with MV.
An example of a transfer element with a linear characteristic is an RC network. The output voltage
U2 follows the applied voltage U1 with a certain dynamic action, but the individual final values are
proportional to the applied voltages (see Fig. 16). This can be expressed by stating that the pro-
cess gain of a linear process is constant, as a change in the input value always results in the same
change in the output value.
However, if we now look at an electrically heated furnace, we find that this is in fact a non-linear
process. From Fig. 16 it is clear that a change in heater power from 500 to 1000W produces a larg-
er temperature increase than a change in power from 2000 to 2500W. Unlike the behavior of an RC
network, the furnace temperature does not increase to the same extent as the power supplied, as
the heat losses due to radiation become more pronounced at higher temperatures. The power
must therefore be increased to compensate for the energy losses. The transfer coefficient or pro-
cess gain of this type of system is not constant, but decreases with increasing process values. This
is covered in more detail in Chapter 2.8.


2 The process

Fig. 16: Linear and non-linear characteristics
Dynamic characteristic
The dynamic response of the process is decisive for characterizing the control loop. The dynamic
characteristic describes the variation in the output signal of the transfer element (the process)
when the input signal varies with time. In theory, it is possible for the output variable to change im-
mediately and to the same extent as the input variable changes. However, in many cases, the sys-
tem responds with a certain delay.

z

y x
t

t t
z

Process
y x

Fig. 17: Step response of a process with self-limitation


2 The process
The simplest way of establishing the behavior of the output signal is to record the variation of the
process value PV with time, after a step change in the manipulating variable MV. This “step re-
sponse” is determined by applying a step change to the input of the process, and recording the
variation of PV with time. The step change need not necessarily be from 0 to 100%; step changes
over smaller ranges can be applied, e.g. from 30 to 50%. The dynamic behavior of processes can
be clearly predicted from this type of step response, which will be discussed in more detail in
Chapter 2.6.

2.2 Processes with self-limitation
Processes with self-limitation respond to a change in the manipulating variable or to a disturbance
by moving to a new stable process value. This type of process can dissipate the energy supplied
and achieve a fresh equilibrium.
A classic example is a furnace where, as the heating power is increased, the temperature rises until
a new equilibrium temperature is reached, at which the heat lost is equal to the heat supplied.
However, in a furnace, it takes some time to achieve the new equilibrium following a step change in
the manipulating variable. In processes without delays, the process value immediately follows the
manipulating variable. The step response of such a process then has the form shown in Fig. 18.

Fig. 18: Process without delay; P process
In this type of process with self-limitation, the process value PV is proportional to the manipulating
variable MV, i.e. PV increases to the same extent as MV. Such processes are often called propor-
tional processes or P processes. The relationship between process value x and manipulating vari-
able y is given by:

∆x = K S • ∆y


2 The process
The factor KS is known as the process gain (transfer coefficient). The relationship will be discussed
in more detail in Chapter 2.8.
Examples of proportional processes are:
- mechanical gearing without slip
- mechanical transmission by lever
- transistor (collector current Ic follows the base current IB with virtually no delay)

2.3 Processes without self-limitation
A process without self-limitation responds to a change in the manipulating variable or to a distur-
bance by a permanent constant change in the process value. This type of process is found in the
course control of an aircraft, where a change in the manipulating variable (rudder deviation) pro-
duces an increase in the process value deviation (course deviation) which is proportional to time. In
other words, the course deviation continually increases with time (see Fig. 19).

Fig. 19: Process without self-limitation; I process
Because of this integrating effect, such processes are also called integral processes or I process-
es. In this type of process, the process value increases proportionally with time as a result of a step
change ∆y in the manipulating variable. If the change in MV is doubled, the process value will also
double after a certain time.
If ∆y is constant, the following relationship applies:

∆x = K IS • ∆y • t

KIS is called the transfer coefficient of the process without self-limitation. The process value now
increases proportionally with both the manipulating variable change ∆y, as in a process with self-
limitation, and also with time t.


2 The process
Additional examples of processes without self-limitation are:
- an electric motor driving a threaded spindle
- the liquid level in a tank (see Fig. 20)

Fig. 20: Liquid level in a tank; I process
Probably the best known example of a process without self-limitation is a liquid container with an
inflow and an outflow. The outlet valve, which here represents the disturbance, is assumed to be
closed initially. If the inlet valve is now opened to a fixed position, the liquid level (h) in the container
will rise steadily at a uniform rate with time.
The level in the container rises faster as the inflow rate increases. The water level will continue to
rise until the container overflows. In this case, the process does not self-stabilize. Taking the effect
of outflow into consideration, no new equilibrium is reached after a disturbance (except when in-
flow = outflow), unlike the case of a process with self-limitation.
In general, processes without self-limitation are more difficult to control than those with self-limita-
tion, as they do not stabilize. The reason is, that following an overshoot due to an excessive
change in MV by the controller, the excessive PV cannot be reduced by process self-limitation.
Take a case where the rudder is moved too far when making a course adjustment, this can only be
corrected by applying an opposing MV. An excessive change in MV could cause the process value
to swing back below the desired setpoint, which is why control of such a process is more difficult.


2 The process
2.4 Processes with dead time
In processes with a pure dead time the process only responds after a certain time has elapsed, the
dead time Tt. Similarly, the response of the process value is delayed when the manipulating vari-
able changes back (see Fig. 21).

Fig. 21: Process with dead time; Tt process
A typical example here is a belt conveyor, where there is a certain time delay before a change in the
chute feed rate is recorded at the measurement location (see Fig. 22).
Systems like this, which are affected by a dead time, are called Tt processes. The relationship be-
tween process value x and manipulating variable y is as follows:

∆x = K S • ∆y

but delayed by the dead time Tt.


2 The process

Fig. 22: Example of a process with dead time; belt conveyor
Another example is a pressure control system with long gas lines. Because the gas is compress-
ible, it takes a certain time for a pressure change to propagate. By contrast, liquid-filled pipelines
have virtually no dead time, since any pressure change is propagated at the speed of sound. Relay
switching times and actuator stroke times also introduce delays, so that such elements in the con-
trol loop frequently give rise to dead times in the process.
Dead times pose a serious problem in control engineering, since the effect of a change in manipu-
lating variable is only reproduced in the process variable after the dead time has elapsed. If the
change in manipulating variable was too large, there is a time interval before this is noticed and
acted on by reducing the manipulating variable. However, if this process input is then too small, it
has to be increased once more, again after the dead time has elapsed, and so the sequence con-
tinues. Systems affected by dead time always have a tendency to oscillate. In addition, dead times
can only really be compensated for by the use of very complex controller designs. When designing
and constructing a process, it is very important that dead times are avoided wherever possible. In
many cases this can be achieved by a suitable arrangement of the sensor and the application point
of the manipulating variable. Thermal and flow resistances should be avoided or kept to a mini-
mum. Always try to mount the sensor at a suitable location in the process where it will read the av-
erage value of the process conditions, avoiding dead spaces, thermal resistances, friction etc.
Dead times can occur in processes with and without self-limitation.


2 The process
2.5 Processes with delay
In many processes there is a delay in propagation of a disturbance, even when no dead time is
present. Unlike the case explained above, the change does not appear to its full extent after the
dead time has elapsed, but varies continuously, even following a step change in the disturbing in-
fluence.
Continuing with the example of a furnace, and looking closely at the internal temperature propaga-
tion:
If there is a sudden change in heating power, the energy must first of all heat up the heating ele-
ment, the furnace material and other parts of the furnace until a probe inside the furnace can regis-
ter the change in temperature. The temperature therefore rises slowly at first until the temperature
disturbance has propagated and there is a constant flow of energy. The temperature then contin-
ues to rise. Over a period of time the temperature of the heating element and the probe come clos-
er and closer together; the temperature increases at a lower rate and approaches a final value (see
Fig. 23).

Fig. 23: Processes with delay


2 The process
As an analogy, consider two pressure vessels which are connected by a throttle valve. In this case,
the air must flow into the first vessel initially, and build up a pressure there, before it can flow into
the second vessel. Eventually, the pressure in the first vessel reaches the supply pressure, and no
more air can flow into it. As the pressures in the two vessels slowly come into line with each other,
the pressure equalization rate between the two vessels becomes slower and slower, i.e. the pres-
sure in the second vessel rises more and more slowly. Following a step change in the manipulating
variable (in this case the supply line pressure) the process value (here the pressure in the second
vessel) will take the following course: a very slow rise to begin with until a certain pressure has built
up in the first vessel, followed by a steady rise and then finally an asymptotic or gradual approach
to the final value.
The transfer function of this type of system is determined by the number of energy stores available
which are separated from each other by resistances. This concept can also be used when referring
to the number of delays or time elements present in a process.
Such processes can be represented mathematically by an equation (exponential function) which
has an exponential term for each energy store. Because of this relationship, these processes are
designated as first-order, second-order, third-order processes, and so on.
The systems may be processes with or without self-limitation, which can also be affected by dead
time.

2.5.1 Processes with one delay (first-order processes)
In a process with one delay, i.e. with one available energy store, a step change in MV causes the
PV to change immediately without delay and at a certain initial rate of change: PV then approaches
the final value more and more slowly (see Fig. 24).

Fig. 24: First-order process; PT1 process


2 The process

Example:
R A typical example of a first-order
process is the charge or discharge
Uin Uout
of a capacitor through a resistor.
The plot of the process variable
(capacitor voltage) follows a typi-
cal exponential function.
-t
Uout = Uin (1 - e RC )

For a step change ∆y the relationship is as follows:

-t
⎛ ----⎞
-
∆x = K S • ∆y • ⎜ 1 – e ⎟
T
⎜ ⎟
⎝ ⎠
The term in brackets shows that a step change in MV does not produce a corresponding immedi-
ate change in PV. Instead PV slowly approaches the final value in a characteristic manner. As the
time t increases (large value of t/T), the value of the expression in the brackets tends towards 1, so
that for the final value, ∆x = KS • ∆y.
As shown in Fig. 24, after a time t = T (time constant), the PV has reached 63% of the final value.
After a time t = 5 T, the PV has almost reached 100% of the final value.
Such processes are also referred to as T1 processes. If it is a process with self-limitation, it is re-
ferred to as a PT1 process; a process without self-limitation is an IT1 process. Processes with one
delay (first-order) occur very frequently. Examples are:
- heating and cooling of a hot water tank
- filling a container with air or gas via a throttle valve or a small bore pipe

2.5.2 Processes with two delays (second-order processes)
In a process with two delays there must be two storage elements, connected together by a resis-
tance. Such processes can be characterized by specifying the transfer coefficient KS and the time
constants T1 and T2. Here, in contrast to a first order process, the step response of the process
value starts with a horizontal phase and also has a point of inflection (see Fig. 25).
The course of the step response cannot be drawn by simply combining T1 and T2. For a step
change ∆y and for T1 not equal to T2 , the relationship is as follows:

–t –t
⎛ -----
- -----⎞
-
⎜ T1 T1 T2 T 2⎟
∆x = K S • ∆y • ⎜ 1 – ---------------- e + ---------------- e ⎟
- -
⎜ T1 – T2 T1 – T2 ⎟
⎝ ⎠


2 The process

Fig. 25: Second-order process; PT2 process
Such a process is normally called a PT2 process. As already discussed, second-order processes
always have a point of inflection, where the radius of curvature changes from a left-hand to a right-
hand curve. First-order processes do not have this point of inflection.
Typical examples of this type of action are:
- filling two containers in series with air or gas through restrictors (see Fig. 23)
- charging up two RC networks in series
- temperature rise in a heated hot-water tank, where the thermometer is mounted in a pocket.


2 The process
2.5.3 Processes with several delays (higher-order processes)
If there are more than two storage elements, the process has a correspondingly higher order.
Interestingly enough, the transfer function characteristic of a higher-order process shows very little
change from that of a second-order process. The rise of the curve does, however, become increas-
ingly steeper and more delayed, until, with an infinite number of time delay elements, it approaches
a pure dead time (see Fig. 26).
The order of a process is an important characteristic, particularly when describing it mathematical-
ly. In practice, almost every process is made up of a large number of widely differing energy stor-
age elements, such as protective fittings, filling materials for temperature probes, dead spaces in
manometers, etc. As a result, it is quite impossible to give an accurate mathematical description of
an actual process.

Process value
y x

∆y

infinite order
t0 t t0 t

Fig. 26: Processes with several delays
In practice, the exact order of the process is not as important as might appear at first glance. Of
much greater significance are the longest delay times, which determine the nature of the process.
As the order of the process increases, it becomes more and more difficult to control, since it ap-
proximates more and more to a system with dead time. A combination with a pure dead time is
also possible, when the controllability deteriorates even more. Controllability is improved when
there are significant differences between the time constants of the individual process elements.
The worst case occurs when the time constants have the same value.

2.6 Recording the step response
The step response of a process, i.e. the course of the process value PV following a step change in
manipulating variable MV can be characterized by two time values:
- the delay time Tu, and the
- response time Tg
If these times are known, a quick estimate of the controllability of a process can be made, and the
control parameters determined in a simple way, as explained later. The order of the process is ig-
nored when using this approach, where it is assumed that any process is made up of a dead time
Tu and a first-order process with a time constant Tg.
To determine such a transfer function and the resulting delay and response times, a recorder is
connected to the transducer (sensor) and the manipulating variable (e.g. heating current) changed
suddenly. Obviously, the change in MV should be limited to a value such that the new setpoint can
be reached without damaging the system. The course of the process value is recorded, a tangent
is drawn to the curve at the point of inflection, and Tu and Tg are determined as shown in Fig. 27.


2 The process

y

Dy

t
x Tg

inflection tangent

inflection point

Dx
Dt
t
Tu

Fig. 27: Determining the delay time and response time
The ratio of delay time to response time gives information about the character of the process and
its controllability:
Tg/Tu more than 10: process easy to control
Tg/Tu between 10 and 3: process can be controlled
Tg/Tu less than 3: process difficult to control
As the ratio of response time Tg to delay time Tu reduces, there is an increasing delay before the
change in manipulating variable is communicated to the controller, and the controllability is pro-
gressively reduced. As explained in Chapter 2.5.3, a low Tg/Tu ratio corresponds to a steep gradi-
ent on the graph, representing a higher-order process which is difficult to control because of its
tendency to overshoot.
Fast processes with Tg/Tu less than 3 are comparatively rare in furnaces, for example, since the
temperature disturbance propagates relatively slowly and continuously through the furnace materi-
al. One exception is the type of furnace where the heating acts directly on the charge. The situation
is quite different with pressure control: opening an air-lock can lead to a sudden drop in pressure,
to which the controller must respond with an equally fast increase in supply pressure. Pressure
equalization in the system takes place just as quickly, so that the entire control process is complet-
ed within a short space of time. In these processes the dead times are long in relation to the delay
times. Certain chemical processes (reactions, neutralization) can sometimes proceed very quickly.
As well as the delay and response times, another important characteristic of the process can be
determined, the maximum rate of rise Vmax. It is obtained from the slope of the tangent at the point
of inflection (see Fig. 27):


2 The process

change in process variable ∆x
V max = ------------------------------------------------------------ = -----
- -
unit time ∆t
As shown later in the section on optimization, the maximum rate of rise is also referred to when
setting control parameters.
DIN Specification 19 226 refers to the start-up value A instead of the rate of rise. This start-up val-
ue is the reciprocal of the maximum rate of rise of the process value PV for a sudden change in the
manipulating variable from 0 — 100%:

1
A = ------------
-
V max

2.7 Characteristic values of processes
The delay times and response times (standard values) of some typical processes are shown in
Table 2 below.

Process Type of process Delay time Tu Response time Tg
variable
Temperature small electrically-heated oven 0.5 — 1 min 5 — 15min
large electrically-heated furnace 1 — 5 min 10 — 20min
large gas-fired reheating furnace 0.2 — 5 min 3 — 60min
autoclave 0.5 — 0.7 min 10 — 20min
high-pressure autoclave 12 —15 min 200 — 300min
injection molding machine 0.5 — 3 min 3 — 300min
extruder 1 — 6 min 5 — 60min
packaging machine 0.5 — 4 min 3 — 40min
Pressure drum boiler, gas or oil-fired 0sec 150sec
drum boiler, solid fuel-fired 0 — 2min 2.5 — 5min
Flow gas pipelines 0 — 5sec 0.2 — 10sec
liquid pipelines 0sec 0sec
Table 2: Delay times and response times (standard values) for some processes
The values given in the table should be taken as average values and serve only as a rough guide.
For practical applications, the values of delay time and response time should be determined by
carrying out a step response test.

2.8 Transfer coefficient and working point
Previous sections have dealt mainly with the dynamic characteristic of the process (course of the
step response), i.e. its behavior with respect to time. Chapter 2.1 has already mentioned the static
characteristic, and described the final values for various manipulating variables. No account is tak-
en of changes in the process value with respect to time.
The transfer coefficient is given by the ratio of output to input value, in this case the ratio of the
change in process variable to the change in manipulating variable.

K S = ----------------------------------------------------------------------- = -----
- -
change in manipulating variable ∆y


2 The process
The transfer coefficient KS is also called the process gain.
In many cases the process gain KS is not constant over the entire range of the process variable, as
the following case will explain: in a furnace, a small increase in heating power is sufficient to pro-
duce a large increase in temperature at the lower end of the temperature range; at the upper end of
the temperature range, however, a much larger change in energy flow is required to achieve the
same effect (see Fig. 28).

Temperature °C

Dy
300
Dx
AP2
Dy
200
Dx
AP1
100 WP: working point

Heating power kW
05 10 15
ð KS depends on the working point
Dx 100 °C °C
WP1: K S = = = 20
Dy 5 kW kW

Dx 30 °C °C
WP2: K S = = = 6
Dy 5 kW kW

Fig. 28: Process gain and working point
The reason for this is that the furnace used in the example represents a non-linear process. In ad-
dition to temperature processes, such processes also include processes where the friction is pro-
portional to speed, relationships between motor power and speed, etc.
The delay time and response times in non-linear processes also depend on the working point. Pro-
cess gain KS, delay time, response time and other such values must be referred to a working point,
i.e. to a pair of values of MV and PV, at which they have been evaluated. In general, they are not
valid for other working points; in non-linear processes, a new set of values must be determined for
each different working point, since the controller setting depends on the process parameters. In
such processes the action of the controller is only optimized at the working point of the process for
which the values were evaluated. If this is changed, for example, if a different process temperature
is required, the controller has to be re-tuned to achieve optimum control.
Generally, the working point should lie in the range from middle to upper third of the transfer func-
tion at full power. A working point in the lower third is less satisfactory, because of the large excess
power. Although the desired value (setpoint) is reached more rapidly in this case, the controllability
is made worse. A working point in the upper part of the characteristic is also unsatisfactory, due to
the lack of reserve power and resultant slow stabilization, and is also unsatisfactory from the point
of view of disturbances.


3 Continuous controllers
3.1 Introduction
After discussing processes in Chapter 2, we now turn to the second important element of the con-
trol loop, the controller. The controller has already been described as the element which makes the
comparison between process variable PV and setpoint SP, and which, depending on the control
deviation, produces the manipulating variable MV. The output of a continuous controller carries a
continuous or analog signal, either a voltage or a current, which can take up all intermediate values
between a start value and an end value.
The other form of controller is the discontinuous or quasi-continuous controller in which the manip-
ulating variable can only be switched on or off.
Continuous controllers offer advantages for certain control systems since their action on the pro-
cess can be continuously modified to meet demands imposed by process events. Common indus-
try standard output signals for continuous controllers are: 0 — 10V, 0 — 20mA, 4 — 20mA. On a
continuous controller with a 0 — 20 mA output, 10% manipulating variable corresponds to an out-
put of 2mA, 80% corresponds to 16mA, and 100% equals 20mA.
As discussed in Chapter 1, continuous controllers are used to operate actuators, such as thyristor
units, regulating valves etc. which need a continuous signal.

3.2 P controller
In a P controller the control deviation is produced by forming the difference between the process
variable PV and the selected setpoint SP; this is then amplified to give the manipulating variable
MV, which operates a suitable actuator (see Fig. 29).

Process value (x)
Control
deviation
e = (w - x)
Amplifier
(Kp)
Manipulating
variable (y)
Setpoint (w)

Fig. 29: Operating principle of a P controller
The control deviation signal has to be amplified, since it is too small and cannot be used directly as
the manipulating variable. The gain (Kp) of a P controller must be adjustable, so that the controller
can be matched to the process.
The continuous output signal is directly proportional to the control deviation, and follows the same
course; it is merely amplified by a certain factor. A step change in the deviation e, caused for exam-
ple by a sudden change in setpoint, results in a step change in manipulating variable (see Fig. 30).



P controller
e Step response

e = (w - x)

t
y

y = KP • (w - x)

t0 t

Fig. 30: Step response of a P controller
The step response of a P controller is shown in Fig. 30.
In other words, in a P controller the manipulating variable changes to the same extent as the devi-
ation, though amplified by a factor. A P controller can be represented mathematically by the follow-
ing controller equation:

y = KP • ( w – x )

The factor KP is called the proportionality factor or transfer coefficient of the P controller and corre-
sponds to the control amplification or gain. It should not be confused with the process gain KS of
the process.
So, in an application where the user has set a KP of 10 %/°C, a P controller will produce a manipu-
lating variable of 50 % in response to a control difference of 5 °C.
Another example would be a P controller for the regulation of a pressure, with a KP set to 4 %/bar.
In this case, a control difference of 20 bar will produce a manipulating variable of 80 %.


3.2.1 The proportional band
Looking at the controller equation, it follows that, in a P controller, any value of deviation would
produce a corresponding value of manipulating variable. However, this is not possible in practice,
as the manipulating variable is limited for technical reasons, so that the proportional relationship
between manipulating variable and control deviation only exists over a certain range of values.

The X P band
Heater power
kW
Manipulating variable MV
% Setpoint
w
XP
50 100

25 50

50 100 150 200 T / °C

Different controller gains through different X P bands

MV
% X P1 = 50 °C
w XP2 = 150 °C
X P1
100 X P3 = 250 °C
80 XP2

50

50 100 150 200 250 300 T / °C

Fig. 31: The position of the proportional band
The top half of Fig. 31 shows the characteristic of a P controller, which is controlling an electrically
heated furnace, with a selected setpoint w = 150°C.
The following relationship could conceivably apply to this furnace
The manipulating variable is only proportional to the deviation over the range from 100 to 150°C,
i.e. for a deviation of 50°C from the intended setpoint of 150°C. Accordingly, the manipulating vari-
able reaches its maximum and minimum values at these values of deviation, and the highest and
lowest heater power is applied respectively. No further changes are possible, even if the deviation
increases.
This range is called the proportional band XP . Only within this band is the manipulating variable
proportional to the deviation. The gain of the controller can be matched to the process by altering
the XP band. If a narrower XP band is chosen, a small deviation is sufficient to travel through the full
manipulating range, i.e. the gain increases as XP is reduced.


The relationship between the proportional band and the gain or proportionality factor of the con-
troller is given by the following formula:

1
X P = ----- • 100%
-
KP

Within the proportional band XP , the controller travels through the full manipulating range yH, so
that KP can be determined as follows:

y H max. manipulating range
K P = ----- = -------------------------------------------------------
- -
XP proportional band
The unit of the proportionality factor KP is the unit of the manipulating variable divided by the unit
of the process variable. In practice, the proportional band XP is often more useful than the propor-
tionality factor KP and it is XP rather than KP that is most often set on the controller. It is specified in
the same unit as the process variable (°C, V, bar etc.). In the above example of furnace control, the
XP band has a value 50°C. The advantage of using XP is that the value of deviation, which produc-
es 100% manipulating variable, is immediately evident. In temperature controllers, it is of particular
interest to know the operating temperature corresponding to 100% manipulating variable. Fig. 31
shows an example of different XP bands.
An example
An electric furnace is to be controlled by a digital controller. The manipulating variable is to be
100% for a deviation of 10°C. A proportional band XP = 10 is therefore set on the controller.
Until now, for reasons of clarity, we have only considered the falling characteristic (inverse operat-
ing sense), in other words, as the process variable increases, the manipulating variable decreases,
until the setpoint is reached. In addition, the position of the XP band has been shown to one side of
and below the setpoint.
However, the XP band may be symmetrical about the setpoint or above it (see Fig. 32). In addition,
controllers with a rising characteristic (direct operating sense) are used for certain processes. For
instance, the manipulating variable in a cooling process must decrease as the process value in-
creases.



Fig. 32: Position of the proportional band about the setpoint
The advantages of XP bands which are symmetrical or asymmetrical about the setpoint will be dis-
cussed in more detail under 3.2.2.

3.2.2 Permanent deviation and working point
A P controller only produces a manipulating variable when there is a control deviation, as we al-
ready know from the controller equation. This means that the manipulating variable becomes zero
when the process variable reaches the setpoint. This can be very useful in certain processes, such
as level control. However, in our example of the furnace, it means that heating power is no longer
applied when the control deviation is zero. As a consequence, the temperature in the furnace falls.
Now there is a deviation, which the controller then amplifies to produce the manipulating variable;
the larger the deviation, the larger the manipulating variable of the controller. The deviation now


takes up a value such that the resulting manipulating variable is just sufficient to maintain the pro-
cess variable at a constant value.
A P controller always has a permanent control deviation or offset
This permanent deviation can be made smaller by reducing the proportional band XP . At first
glance, this might seem to be the optimal solution. However, in practice, all control loops become
unstable if the value of XP falls below a critical value - the process variable starts to oscillate.
If the static characteristic of the process is known, the resulting control deviation can be found di-
rectly. Fig. 33 shows the characteristic of a P controller with an XP band of 100°C. A setpoint of
200°C is to be held by the controller. The process characteristic of the furnace shows that a manip-
ulating variable of 50% is required for a setpoint of 200°C. However, the controller produces zero
manipulating variable at 200°C. The temperature will fall, and, as the deviation increases, the con-
troller will deliver a higher manipulating variable, corresponding to the XP band. A temperature will
be reached here, at which the controller produces the exact value of manipulating variable required
to maintain that temperature. The temperature reached, and the corresponding manipulating vari-
able, can be read off from the point of intersection of the controller characteristic and the static
process characteristic: in this case, a temperature of 150°C with a manipulating variable of 40%.

Permanent control deviation
y/%
Controller characteristic
Setpoint w
100

50 X P = 100 °C
40

100 150 200 300 400 T / °C
T / °C
Static process characteristic
400

200

25 50 75 100 y/%

y/% Working point correction
W
100

WP
50

50 100 150 200 250 300 T / °C

Fig. 33: Permanent deviation and working point correction


It is clear that in a furnace, for instance, a certain level of power must be supplied in order to reach
and maintain a particular setpoint. So it makes no sense to set the manipulating variable to zero
when there is no control deviation. The manipulating variable is usually set to a specific percentage
value for a control difference of 0. This is called working point correction, and can be adjusted on
the controller, normally over the range of 0 — 100%. This means that with a correction of 50%, the
controller would produce a manipulating variable of 50% for zero control deviation. In the example
given, see Fig. 33, this would lead to the setpoint w = 200°C being reached and held. We can see
that the proportional band exhibits a falling characteristic that is symmetrical about the setpoint. If
the process actually requires the manipulating variable set at the working point, as in our example,
the control operates without deviation.
Setting the working point in practice
In practice, the process characteristic of a process is not usually known. However, the working
point correction can be determined by manually controlling the process variable at the setpoint val-
ue that the controller is to hold later. The manipulating variable required for this is also the value for
the working point correction.
Example
In a furnace where a setpoint of 200°C is to be tracked, the controller would be set to manual
mode and the manipulating variable slowly increased by hand, allowing adequate time after the
change for the end temperature to be reached. A certain value of manipulating variable will be de-
termined, for example 50%, which is sufficient for a process variable of 200°C. This manipulating
variable is then fed in as the value for the working point correction.
After feeding in the value for the working point correction, the controller will only operate without
control difference at the particular setpoint for which the working point correction was made. Fur-
thermore, the external conditions must not change. If other disturbances did affect the process,
(for example, a fall in the temperature outside a furnace), a control difference would be set once
again, although this time the value would be smaller.
We can summarize the main points about the control deviation of a P controller as follows
(controller with falling characteristic, process with self-limitation):
Without working point WP
- The process variable remains in a steady state below the setpoint.

With working point WP (see Fig. 33)
- below the working point (in this case 0 — 50% manipulating variable)
process variable is above the setpoint
- at the working point (in this case 50% manipulating variable)
process variable = setpoint
- above the working point (in this case 50 — 100% manipulating variable)
process variable is below the setpoint

In a P controller, the output signal has the same time course as the control deviation, and because
of this it responds to disturbances very rapidly. It is not suitable for processes with a pure dead
time, as these start to oscillate due to the P controller. On processes with self-limitation, it is not
possible to control exactly at the setpoint; a permanent deviation is always present, which can be
significantly reduced by introducing a working point correction.


3.2.3 Controllers with dynamic action
As we saw in the previous chapter, the P controller simply responds to the magnitude of the devia-
tion and amplifies it. As far as the controller is concerned, it is unimportant whether the deviation
occurs very quickly or is present over a long period.
When a large disturbance occurs, the initial response of a machine operator is to increase the ma-
nipulating variable, and then keep on changing it until the process variable reaches the setpoint.
He would consider not only the magnitude of the deviation, but also its behavior with time (dynam-
ic action).
Of course, there are control components that behave in the same way as the machine operator
mentioned above:
- The D component responds to changes in the process variable. For example, if there is 20% re-
duction in the supply voltage of an electric furnace, the furnace temperature will fall. This D
component responds to the fall in temperature by producing a manipulating variable. In this
case, the manipulating variable is proportional to the rate of change of furnace temperature, and
helps to control the process variable at the setpoint.
- The I component responds to the duration of the deviation. It summates the deviation applied to
its input over a period of time. If this controller is used on a furnace, for example, it will slowly in-
crease the heating power until the furnace temperature reaches the required setpoint.
In the past, dynamic action was achieved in analog controllers by feeding part of the manipulating
variable back to the controller input, via timing circuits. The feedback changes the input signal (the
real control deviation) so that the controller receives a simulated deviation signal that is modified by
a time-dependent factor. In this way, using a D component, a sudden change in process variable,
for example, can be made to have exactly the same initial effect as a much larger control deviation.
In this connection, because of this reverse coupling, we often talk about feedback. In modern mi-
croprocessor controllers, the manipulating variable is not produced via feedback, but derived
mathematically direct from the setpoint and process variable.
We will avoid using the term feedback in this book, as far as possible.
The components described above are often combined with a P component to give PI, PD or PID
controllers.


3.3 I controller
An I controller (integral controller) integrates the deviation signal applied to its input over a period of
time. The longer there is a deviation on the controller, the larger the manipulating variable of the
I controller becomes. How quickly the controller builds up its manipulating variable depends firstly
on the setting of the I component, and secondly on the magnitude of the deviation.
The manipulating variable changes as long as there is a deviation. Thus, over a period of time, even
small deviations can change the manipulating variable to such an extent that the process variable
corresponds to the required setpoint.
In principle, an I controller can fully stabilize after a sufficiently long period of time, i.e. setpoint =
process variable. The deviation is then zero and there is no further increase in manipulating vari-
able.
Unlike the P controller, the I controller does not have a permanent control deviation
The step response of the I controller shows the course of the manipulating variable over time, fol-
lowing a step change in the control difference (see Fig. 34).

Fig. 34: Step response of an I controller
For a constant control deviation ∆e, the equation of the I controller is as follows:

1
∆y = --- • ∆e • t
-
TI

Here TI is the integral time of the I controller and t the duration of the deviation. It is clear that the
change in manipulating variable y is proportional not only to the change in process variable, but
also to the time t.


If the control deviation is varying, then:

1 s
TI
- ∫
y = --- e • dt • --
K
-

The integral time of the I controller can also be evaluated from the step response (see Fig. 34):

∆e • ∆t
T I = -----------------
∆y
If the process variable is below the setpoint on an I controller with a negative operating sense, as
used, for example, in heating applications, the I controller continually builds up its manipulating
variable. When the process variable reaches the setpoint, we now have the possibility that the ma-
nipulating variable is too large, because of delays in the process. The process variable will again in-
crease slightly; however, the manipulating variable is now reduced, because of the sign reversal of
the process variable (now above the setpoint).
It is precisely this relationship that leads to a certain disadvantage of the I controller
If the manipulating variable builds up too quickly, the control signal which arises is too large, and
too high a process variable is reached. Now the process variable is above the setpoint and the sign
of the deviation is reversed, i.e. the control signal decreases again. If the decrease is too sudden, a
lower process value is arrived at, and so on. In other words, with an I controller, oscillations about
the setpoint can occur quite frequently. This is especially the case if the I component is too strong,
i.e. when the selected integral time TI is too short. The exception to this is the zero-order process
where, because there are no energy storage possibilities, the process variable follows the manipu-
lating variable immediately, without any delay; the control loop forms a system which is not capa-
ble of oscillation.
To develop a feel for the effect of the integral time TI, it can be defined as follows: The integral
time TI is the time that the integral controller needs to produce its constant control difference at its
output (without considering sign). Imagine a P controller for a furnace, where the response time TI
is set at 60sec and the control difference is constant at 2°C. The controller requires a time TI =
60sec for a 2% increase in manipulating variable, if the control difference remains unchanged at
2°C.
Summarizing the main points, the I controller removes the control deviation completely, in contrast
to the P controller.
An I controller is not stable when operating on a process without self-limitation, and is therefore un-
suitable for control of liquid levels, for example. On processes with long time constants, the I com-
ponent must be set very low, so that the process variable does not tend to oscillate. With this small
I component, the I controller works much too slowly. For this reason, it is not particularly suitable
for processes with long time constants (e.g. temperature control systems). The I type of controller
is frequently used for pressure regulation, and in such a case Tn is set to a very low value.

3.4 PI controller
As we have found in the I controller, it takes a relatively long time (depending on Ti ) before the con-
troller has built up its manipulating variable. Conversely, the P controller responds immediately to
control differences by immediately changing its manipulating variable, but is unable to completely
remove the control difference. This would seem to suggest combining a P controller with an I con-
troller. The result is a PI controller. Such a combination can combine the advantage of the P con-
troller, the rapid response to a control deviation, with the advantage of the I controller, the exact
control at the setpoint.


We can obtain the step response of a PI controller simply by superimposing the step responses of
a P and an I controller, as shown in Fig. 35.

e

De

y t0 t

I controller

y t

PI controller

S P controller
Tn t

Fig. 35: Step response of a PI controller
If the diagonally rising straight line of the PI manipulating variable is projected back to its point of
intersection S with the time axis, it intercepts a length of time there. With a PI controller, this corre-
sponds to the reset time Tn .
For a control deviation e = ∆e = constant, we obtain the following equation for the PI controller:

∆y = ----- • 100% • ⎛ ∆e + ----- • ∆e • t⎞ = ----- • 100% • ∆e • ⎛ 1 + ----- • t⎞
1 1 1 1
- - - -
XP ⎝ Tn ⎠ X ⎝ Tn ⎠
P
The reset time is a measure of the extent to which the duration of the control deviation affects the
control function. A long reset time means that the I component has little influence, and vice versa.
From the equation above, it is evident that the real amplification of the I component is the factor

1 1
----- • 100% • -----
- -
XP Tn

With a PI controller, therefore, a change in proportional band XP also causes a change in the inte-
gral action. If the proportional gain of a PI controller is increased by reducing XP , the integral action
will also be increased, so the controller will make a faster integration of the control difference.
It is also possible to interpret Tn as the time interval required for the I component to produce the
same manipulating variable y (for a given deviation), as that already produced by the P component


(following a step change). The formula given above is only valid when the deviation remains con-
stant during the time interval t. If this is not the case, the relationship is as follows:

1 1 1
XP
-
XP
-
Tn
- ∫
∆y = ----- • 100% • e + ----- • 100% • ----- • e • dt

As mentioned earlier, a PI controller can, in principle, be built up by combining a P controller and an
I controller. With a sudden deviation, the manipulating variable is initially formed by the P compo-
nent (see Fig. 36). Because of the changed manipulating variable, the process variable moves to-
wards the setpoint, i.e. the deviation is reduced, and with it the manipulating variable produced by
the P controller. Now the manipulating variable produced by the I component ensures exact con-
trol. Whereas the P component of the manipulating variable steadily decreases as the setpoint is
approached, the I component continues to build up. Here, however, the increase is also smaller,
because of the reducing deviation, until finally, when the setpoint is reached, nothing more is add-
ed to the current manipulating variable. When the system has stabilized, the manipulating variable
of the PI controller is produced solely by the I component.

T / °C

400 Setpoint w
300
200
100

t
T / °C

400
300
200
100

t

y/%

P component I component
100
50 % power required
50

t

Fig. 36: Formation of the manipulating variable in a PI controller


Summarizing the main points:
In a PI controller, the P component causes the manipulating variable to respond immediately to the
control deviation. The PI controller is therefore much faster than an I controller. The I component in-
tegrates the control deviation at the output of the controller, so that the closed control loop acts to
reduce the remaining deviation.

3.5 PD controller
If a large disturbance occurs in a control loop which is being controlled manually, bringing with it a
change in the manipulating variable, the operator will try to cushion the effect of the disturbance by
making a large initial adjustment of the actuator. He then quickly reduces the adjustment, so that
the new equilibrium of the control loop can be approached gradually. A controller which responds
in a similar way to the above operator is the PD controller: it consists of a P component with a
known proportional action, and a D component with a derivative action. This D component re-
sponds not to the magnitude or duration of the control deviation, but to the rate of change of the
process variable. Fig. 37 shows how such a PD controller builds up its manipulating variable.
Fig. 37 explains how the PD controller works. If a new setpoint is applied, the manipulating variable
is increased by the P component; this component of the manipulating variable is always propor-
tional to the deviation. The process variable responds to the increased manipulating variable, for
example, a furnace temperature rises. As soon as the process variable changes, the D component
starts to take effect: while the process variable increases, the D component forms a negative ma-
nipulating variable, which is subtracted from the manipulating variable of the P component, finally
producing the manipulating variable at the controller output. When the process variable is tracking
the setpoint, the D component “brakes”, thus preventing the manipulating variable overshooting
above the setpoint.
If the process variable has reached its maximum value after an overshoot above the setpoint, and
is now reducing, the D component gives out a positive manipulating variable. In this case, the D
component counteracts the change in process variable.
The D component only intervenes in the process when there is a change in process variable. The
size of the manipulating variable of the D component depends on the rate of change of the process
variable, that is on the magnitude of ∆x/∆t (see the gradient triangle in Fig. 37). In addition, the ef-
fect can be changed at the controller via the time Td (derivative time), which we will get to know in
this chapter. A pure D controller is not suitable for control, as it does not intervene in the process
when there is a constant deviation, or when the process variable remains constant.



T / °C

400 Setpoint w
300

200
100
t

T / °C

400 Process value X
300

200
Dx
100 Dt t

yp /%
P component
100

t

-100

yD /%

D component
100

t

-100

Fig. 37: Formation of the manipulating variable in a PD controller


Fig. 38 shows the ramp function response for a PD controller, where we can imagine the increasing
control deviation resulting from a falling process variable.

Fig. 38: Response of a PD controller to a ramp function
From Fig. 38 we can see that there is a noticeable manipulating variable from the D component at
the start of the ramp function, since this manipulating variable is proportional to the rate of change
of the process value. The P component needs a certain time, namely the derivative time Td , to
reach the same value manipulating variable as the D component has built up. The derivative time is
obtained by projecting the diagonally rising line back to its point of intersection S with the time
axis.
Mathematically, the rate of change v is obtained from the change in control deviation “de” per unit
time “dt”:

de
v = ----
-
dt
For the PD controller, this leads to the following control equation:

y = ----- • 100% • ⎛ e + T v • ---- ⎞
1 de
- -
XP ⎝ dt ⎠

In principle, the D component has the following effects:
As soon as the process variable changes, the D component counteracts this change.
For a controller with an inverse operating sense (i.e. for heating), this means for example:


- If the process variable reduces as a result of a disturbance in the process, the D component
forms a positive manipulating variable, which counteracts the reduction in the process variable.
- If the process variable increases as a result of a disturbance in the process, the D component
forms a negative manipulating variable, which counteracts the increase in the process variable.

3.5.1 The practical D component - the DT1 element
In principle, we could also consider the step response of a PD controller in the same way as previ-
ously for P and PI controllers. Now, however, the rate of change at a step is infinitely large. In theo-
ry, the D signal derived from a step would therefore be an infinitely high and infinitely narrow spike
(see Fig. 39). Theoretically, this means that the manipulating variable has to take up an infinitely
high value for an infinitely short time, and then return immediately to the value produced by the P
component. This is simply not possible, for both electrical and mechanical reasons. Furthermore,
such a short pulse would scarcely influence the process. In practice, the immediate decay is pre-
vented by forming the D component through a DT1 element. This element consists of a D compo-
nent, which we have already met in this chapter, in series with a T1 element. The T1 element be-
haves like a first-order process with a transfer coefficient of 1.
Fig. 39 shows the step response of the “practical” D component. T1 is the time constant of the T1
element. In practice, this time constant is set at Td /4, and when Td is changed, the time constant is
changed by the same ratio. The derivative time Td can be determined from the step response of
the “practical” D component, on the basis of the ratio T1 = Td /4.
T1 is specified by the manufacturer, and cannot be altered by the user.

e

De

t
y
Narrow spike Theory

t
y yh
Practice

Td
De
T1

T1 t

Fig. 39: Step response of a DT1 element


Summarizing the main points:
A pure D controller has no practical importance since it takes no account of a permanent deviation,
and simply responds to the rate of change of the process variable. By comparison, the PD control-
ler is very widely used. The D component ensures a fast response to disturbances, whereas its
“braking behavior” also stabilizes the control loop. The D component is not suitable for processes
with pulsating variables, such as pressure and flow control.
The main application for the PD controller is where tools or products are prone to damage if the
setpoint is exceeded. This applies particularly to plastics processing machines. However, PD con-
trollers, like the P controller, always have a permanent deviation, when controlling processes with
self-limitation.

3.6 PID controller
We have seen earlier that the combination of a D component or an I component with a P controller
offered certain advantages in each case. Now it seems logical to combine all three structures, re-
sulting in the PID controller.
With this controller, the XP , Tn , Td parameters are adjusted for the P, I and D action. These three
components can be seen in the step response of a PID controller (see Fig. 40).

e

De

t0 t
y D component
I component

Td /4 KP • De P component

Tn t

Fig. 40: Step response of a PID controller
According to DIN 19 225, such a controller obeys the following controller equation:

∆y = K P • ⎛ e + ----- e • dt + T d • ---- ⎞
1 de
⎝ Tn
-∫ -
dt ⎠

(ideal PID controller)
As already discussed in the previous section, the individual parameters (KP , Td , Tn ) have different
effects on the individual components.


larger XP (corresponds to smaller KP ): corresponds to smaller P component
larger Tn : corresponds to reduced I component
larger Td : corresponds to increased D component

On some controllers with PID action, Td and Tn cannot be adjusted separately. Practical experi-
ence has shown that optimum performance is obtained with a ratio Td = Tn / 4 to 5. This ratio is fre-
quently a fixed setting on the controller, and only one parameter can be varied (usually Tn ).
We can summarize by noting that the PID controller brings together the best characteristics of the
P, I and D controllers. The P component responds with a suitable manipulating variable when a de-
viation occurs. The D component counteracts changes in the process variable, and increases the
stability of the control loop. The permanent deviation is removed by the I component. The PID type
of controller is used for most applications.

3.6.1 Block diagram of the PID controller

Fig. 41: Block diagram of the PID controller
As we have already seen in this chapter, from the controller equations for the PI, PD and PID con-
trollers, the I and D actions of a PID controller are influenced not only by the adjustment of the Tn
and Td parameters, but also by the proportional gain with XP . If the proportional gain of a PID con-
troller is doubled (by halving XP), the controller not only has double the proportional action, but the
I and D components are also increased to double the value.
An example
The PID controller shown in Fig. 41 has settings Tn = 10sec and XP = 100 (the D component should
be disregarded in this example). The control deviation is 2.
When KP and XP are given as percentage values, the P component has a gain of:

1 ⎛ K P = ----- • 100 %⎞
1
-
⎝ XP ⎠
The control deviation is thus offered directly to the I component. We already know from Chapter 3.3
“I controller”, that an I controller requires a time equal to Tn to fully reproduce the input signal at its
output (percentage values). The I component would thus require 10sec before it has increased its
manipulating variable by 2%. XP is now set to 50, so that the gain of the P component is 2.
Now the control difference is first amplified by a factor of 2, before it is offered to the I component.
The I component now increases its manipulating variable by 4% every 10 seconds. The effect of
the I component was also amplified by a factor of 2.
Changing the proportional gain in a PID controller
changes the I and D action to the same extent


4 Control loops with continuous controllers
4.1 Operating methods for control loops with continuous controllers
The previous chapters dealt with the individual elements of a control loop, the process and the
controller. Now we consider the interaction between these two elements in the closed control loop.
Amongst other things, the stable and unstable behavior of a control loop should be examined, to-
gether with its response to setpoint changes and disturbances. In the section on “Optimization”,
we will come across the various criteria for adjusting the controller to the process.
We also often refer to the static and dynamic behavior of the control loop. The static behavior of a
control loop characterizes its steady state on completion of all dynamic transient effects, i.e. its
state long after any earlier disturbance or setpoint change. The dynamic behavior, on the other
hand, shows the behavior of the control loop during changes, i.e. the transition from one state of
rest to another. We have already discussed this kind of dynamic behavior in Chapter 2 “The pro-
cess”.
When a controller is connected to a process, we expect the process variable to follow a course like
that shown in Fig. 42.

Fig. 42: Transition of the process variable in the closed control loop


- After the control loop is closed, the process variable (x) should reach and hold the predeter-
mined setpoint (w) as quickly as possible, without appreciable overshoot. In this context, the
run-up to a new setpoint value is also called the setpoint response.
- After the start-up phase, the process variable should maintain a steady value without any appre-
ciable fluctuations, i.e. the controller should have a stable effect on the process.
- If a disturbance occurs in the process, the controller should again be able to control it with the
minimum possible overshoot, and in a relatively short response time. This means that the con-
troller should also exhibit a good disturbance response.

4.2 Stable and unstable behavior of the control loop
After the end of the start-up phase, the process variable should take up the steady value, predeter-
mined by the setpoint, and enter stable operation. However, it could happen that the control loop
becomes unstable, and that the manipulating variable and process variable perform periodic oscil-
lations. Under certain circumstances, this could result in the amplitude of these oscillations not re-
maining constant, but instead increasing steadily, until it fluctuates periodically between upper and
lower limit values. Fig. 43 shows the two cases of an unstable control loop.
Here, we often talk about the self-oscillation of a control loop. Such unstable behavior is mostly
caused by low noise levels present in the control loop, which introduce a certain restlessness into
the loop. Self-oscillation is largely independent of the construction of the control loop, whether it
be mechanical, hydraulic or electrical, and only occurs when the returning oscillations have a larger
amplitude than those sent out, and are in phase with them.

Fig. 43: The unstable control loop
If certain operating conditions, (e.g. new controller settings), are changed in a control loop that is in
stable operation, there is always a possibility of the control loop becoming unstable. However, in
practical control engineering, the stability of the control loop is an obvious requirement. We can
generalize by stating that stable operation can be achieved in practice by choosing a sufficiently
low gain in the control loop and a sufficiently long controller time constant.


4.3 Setpoint and disturbance response of the control loop
As already mentioned, there are basically two cases which result in a change in the process vari-
able. When describing the behavior of a process in the control loop, we use the terms setpoint re-
sponse or disturbance response, depending on the cause of the change:
Setpoint response
The setpoint has been adjusted and the process has reached a new equilibrium.
Disturbance response
An external disturbance affects the process and alters the previous equilibrium, until a stable pro-
cess value has developed once again.
The setpoint response thus corresponds to the behavior of the control loop, following a change in
setpoint. The disturbance response determines the response to external changes, such as the in-
troduction of a cold charge into a furnace. In a control loop, the setpoint and disturbance respons-
es are usually not identical. One of the reasons for this is that they act on different timing elements
or at various intervention points in the control loop.
In many cases, only one of the two types of process response is important.
When a motor subjected to continuously variable shaft loading still has to maintain a constant
speed, it is clearly only the disturbance response which is of importance. Conversely, in the case of
a furnace, where the charge has to be brought to different temperatures over a period of time, in
accordance with a specific setpoint profile, the setpoint response is of more interest.
The purpose of control is to influence the process in the desired manner, i.e. to change the setpoint
or disturbance response. It is impossible to satisfactorily correct both forms of response in the
same way. A decision must therefore be made whether to optimize the control for disturbance re-
sponse or setpoint response. More about this in the section on “Optimization”.


4.3.1 Setpoint response of the control loop
As already explained, the main objective in a control loop with a good setpoint response is that,
when the setpoint is changed, the process variable should reach the new setpoint value as quickly
as possible and with minimal overshoot. Overshoot can be prevented by a different controller set-
ting, but only at the expense of the stabilization time (see Chapter 4.1, Fig. 42). After closing the
control loop, it takes a certain time for the process variable to reach the setpoint value predeter-
mined at the controller. This approach to the setpoint can be made either gradually (creep) or in an
oscillatory manner (see Fig. 44).
Which particular control loop response is considered most important varies from one case to an-
other, and depends on the process to be controlled.

Fig. 44: Approach to the setpoint


4.3.2 Disturbance response
When the start-up phase is complete and the control loop is stable, the controller now has the task
of suppressing the influence of disturbances, as far as possible. When a disturbance does occur, it
always results in a temporary control deviation, which is only corrected after a certain time. To
achieve good control quality, the maximum overshoot, the permanent control deviation and the
stabilization time should be as small as possible (see Chapter 1.4, Fig. 3). As the size of distur-
bances of the characteristics in a control loop normally has to be accepted as given, good control
quality can only be achieved by a suitable choice of controller type and an appropriate optimiza-
tion.
The disturbances can act at different points in the process. Depending on the point of application
of the disturbance, its effect on the dynamic transition of the process variable will differ. Fig. 45
shows the course of a disturbance step response of the process, when a disturbance acts at the
beginning, in the middle and at the end of the process.

Fig. 45: Disturbance step response of a process


4.4 Which controller is best suited for which process?
After selecting a suitable controller according to type, dimensions etc. (see Chapter 1.5), the prob-
lem now arises of deciding which dynamic response should be employed to control a particular
process. With modern microprocessor controllers, the price differentials between P, PI and PID
controllers have been eroded. Hence it is no longer crucial nowadays, whether a control task can
still be solved with just a P controller.
Regarding dynamic action, the following general points can be made:
P controllers have a permanent deviation, which can be removed by the introduction of an I com-
ponent. However, there is an increased tendency to overshoot, because of this I component, and
the control becomes a little more sluggish. Accurate stable control of processes affected by delays
can be achieved by a P controller, but only in conjunction with an I component. With a dead time,
an I component is always required, since a P controller, used by itself, leads to oscillations. An I
controller is not suitable for processes without self-limitation.
The D component enables the controller to respond more quickly. However, with strongly pulsating
process variables, such as pressure control etc., this leads to instabilities. Controllers with a D
component are very suitable for slow processes, such as those found in temperature control.
Where a permanent deviation is unacceptable, the PI or PID controller is used.
The relationship between process order and controller structure is as follows:
For processes without self-limitation or dead time (zero-order), a P controller is adequate. Howev-
er, even in apparently delay-free processes, the gain of a P controller cannot be increased indefi-
nitely, as the control loop would otherwise become unstable, because of the small dead times that
are always present. Thus, an I component is always required for accurate control at the setpoint.
For first-order processes with small dead times, a PI controller is very suitable.
Second-order and higher-order processes (with delays and dead times) require a PID controller.
When very high standards are demanded, cascade control should be used, which will be dis-
cussed in more detail in Chapter 6. Third-order and fourth-order processes can sometimes be con-
trolled satisfactorily with PID controllers, but in most cases this can only be achieved with cascade
control.
On processes without self-limitation, the manipulating variable must be reduced to zero after the
setpoint has been reached. Thus, they cannot be controlled by an I controller, since the manipulat-
ing variable is only reduced by an overshoot of the process variable. For higher-order processes
without self-limitation, a PI or PID controller is suitable.
Summarizing the selection criteria results in the following tables:
Permanent deviation No permanent deviation
P PD PI PID
Temperature simple process simple process suitable highly suitable
for low demands for low demands
Pressure mostly unsuitable mostly unsuitable highly suitable; for pro- suitable, if process val-
cesses with long delay ue pulses not too much
time I controller as well
Flow unsuitable unsuitable suitable, but I controller suitable
frequently better
Level with short dead time suitable suitable highly suitable
suitable
Conveyor unsuitable because of unsuitable suitable, but I controller nearly no advantages
dead time mostly best compared with PI

Table 3: Selection of the controller type for controlling the most important process variables


Process Controller structure
P PD PI PID
pure dead time unsuitable unsuitable very suitable, or
pure I controller
first-order with suitable if deviation is suitable if deviation is highly suitable highly suitable
short dead time acceptable acceptable
second-order with deviation mostly too deviation mostly too not as good as PID highly suitable
short dead time high for necessary XP high for necessary XP
higher-order unsuitable unsuitable not as good as PID highly suitable
without self-limitation suitable suitable suitable suitable
with delay

Table 4: Suitable controller types for the widest range of processes

4.5 Optimization
Controller optimization (or “tuning”) means the adjustment of the controller to a given process. The
control parameters (XP , Tn , Td ) have to be selected such that the most favorable control action of
the control loop is achieved, under the given operating conditions. However, this optimum action
can be defined in different ways, e.g. as a rapid attainment of the setpoint with a small overshoot,
or a somewhat longer stabilization time with no overshoot.
Of course, as well as very vague phrases like “stabilization without oscillation as far as possible”,
control engineering has more precise descriptions, such as examining the area enclosed by the os-
cillations and other criteria. However, these adjustment criteria are more suitable for comparing in-
dividual controllers and settings under special conditions (laboratory conditions). For the practical
engineer working on the installation, the amount of time taken up and the practicability on site are
of greater significance.
The formulae and control settings given in this chapter are empirical values from very different
sources. They refer to certain idealized processes and may not always apply to a specific case.
However, anyone with a knowledge of the various adjustment parameters, on a PID controller, for
example, should be able to adjust the control action to satisfy the relevant demands.
Apart from the mathematical derivation of the process parameters and the controller data derived
from them, there are various empirical methods. One method consists of periodically changing the
manipulating variable and investigating how the process variable follows these changes. If this test
is carried out for a range of oscillation frequencies of the setpoint, the amplitude and phase shift of
the resulting process variable fluctuations can be used to determine the frequency response curve
of the process. From this it is possible to derive the control parameters. Such test methods are
very expensive, involve increased mathematical treatment, and are not suitable for practical use.
Other controller settings are based on empirical values, obtained in part from lengthy investiga-
tions. Such methods of selecting controller settings (especially the Ziegler and Nichols method and
that of Chien, Hrones and Reswick) will be discussed in more detail later.


4.5.1 The measure of control quality
Standard text book instructions for controller optimization are usually based on step changes in,
for example, a disturbance or the setpoint. Disturbances are usually assumed to act at the start of
the process.

Disturbance change

yz = 10 % of yH

x
X max
A1 A3 ∆x = ± 1 % of w
w

A4
A2
Ts

t0 t

Setpoint change
x
X max
A2
A4
∆x = ± 1 % of w
w1

A1 A3
w0
Ts

t0
t

Fig. 46: The measure of control quality
This type of disturbance is also the most important one, as it frequently occurs in normal operation,
testing is very feasible and because of its clear mathematical analysis. Fig. 46 shows that for a step
change disturbance, the overshoot amplitude Xo and the stabilization time Ts offer a measure of
quality. For a more exact definition of the stabilization time, we have to establish when the control


action is regarded as complete. It is convenient to regard stabilization after a disturbance as being
complete, when the control difference remains within ±1% of the setpoint w. For expediency, the
size of the disturbance is taken as 10% of yH .
In addition to the overshoot amplitude and the stabilization time, for mathematical analysis, the
area of the control error is also used as a measure of control quality (see Fig. 46).
Linear control area (linear optimum): [A]min = A1 - A2 + A3 ...
Magnitude control area (magnitude optimum): [A]min = | A1 | + | A2 | + | A3 | + ...

Squared control area (squared optimum): [A]min = A12 + A22 + A32 + ...
Without doubt, quite apart from any other considerations, one controller setting can be said to ex-
hibit better control quality than another, if the resulting overshoot amplitudes are smaller and the
stabilization time is shorter. Some tests indicate, however, that within certain limits it is possible to
have a small overshoot at the expense of a longer stabilization time, and vice versa. For the given
control error area, there is a definite controller setting at which the areas are at a minimum.
As mentioned several times previously, differing levels of importance are attached to the various
measures of control quality, depending on the type of process variable and the purpose of the in-
stallation (see also Chapter 4.3 “Setpoint and disturbance response of the control loop”).

4.5.2 Adjustment by the oscillation method
In the oscillation (or limit cycle) method, devised by Ziegler and Nichols, the control parameters are
adjusted until the stability limit is reached, and the control loop formed by the controller and the
process starts to oscillate, i.e. the process variable performs periodic oscillations about the set-
point. The controller setting values can be determined from the parameters found from this test.
The procedure can only be used in processes that can actually be made unstable and where an
overshoot does not cause danger. The process variable is made to oscillate by initially reducing the
controller gain to its minimum value, i.e. by setting the proportional band to its maximum value.
The controller must be operating as a pure P controller; for this reason, the I component (Tn ) and
the D component (Td ) are switched off. Then the proportional band XP is reduced until the process
variable performs undamped oscillations of constant amplitude.
This test produces:
- the critical proportional band XPc , and
- the oscillation time Tc of the process variable (see Fig. 47)



X P < XPc X P > XPc
x x
w w

t t
X P = XPc
x
w

Tc

t

Fig. 47: Oscillation method after Ziegler and Nichols
The controller can then be set to the following values:
Controller structure
P XP = XPc / 0.5
PI XP = XPc / 0.45
Tn = 0.85 · Tc
PID XP = XPc / 0.6
Tn = 0.5 · Tc
Td = 0.12 · Tc

Table 5: Adjustment formulae based on the oscillation method
Without doubt, the advantage of this process is that the control parameters can be studied under
operational conditions, as long as the adjustments described succeed in achieving oscillations
about the setpoint. There is no need to open the control loop. Recorder data is easily evaluated;
with slow processes, the values can even be determined by observing the process variable and us-
ing a stopwatch. The disadvantage of this method is that it can only be used on processes which
can be made unstable, as mentioned above.
The Ziegler and Nichols adjustment rules apply mainly to processes with short dead times and with
a ratio Tg /Tu greater than 3.

4.5.3 Adjustment according to the transfer function or process step response
Another method of determining the parameters involves measuring process-related parameters by
recording the step response, as already described in Chapter 2.6. It is also suitable for processes
which cannot be made to oscillate. However, it does require opening the control loop, for instance,
by switching the controller over to manual mode in order to exert a direct influence on the manipu-
lating variable. If possible, the step change in manipulating variable should be made when the pro-
cess variable is close to the setpoint.


A method that can be used to determine the control parameters when the process parameters are
known has been developed by Chien, Hrones, and Reswick (CHR). This approximation method
yields good control parameters, not only for disturbances, but also for setpoint changes, and is
suitable for processes with PTn structure (with n equal to 2 or greater).
The step response is used to determine the delay time Tu, the response time Tg and the process
transfer coefficient KS (see Fig. 46).

K S = ----------------------------------------------------------------------- = -----
- -
change in manipulating variable ∆y

y

Dy

t

x Tg

Inflection tangent

Dx Point of inflection

t
Tu

Fig. 48: Adjustment according to the step response
The values found are applied using the following setting rules:
Controller structure Setpoint Disturbance
P XP = 3.3 · KS · (Tu/Tg ) · 100% XP = 3.3 · KS · (Tu/Tg ) · 100%
PI XP = 2.86 · KS · (Tu/Tg ) · 100% XP = 1.66 · KS · (Tu/Tg ) · 100%
Tn = 1.2 · Tg Tn = 4 · Tu
PID XP = 1.66 · KS · (Tu/Tg ) · 100% XP = 1.05 · KS · (Tu/Tg ) · 100%
Tn = 1 · Tg Tn = 2.4 · Tu
Td = 0.5 · Tu Td = 0.42 · Tu

Table 6: Formulae for adjustment according to the step response
Example:
Tn, Td and XP have to be determined for a temperature control process. The future operating range
is at 200°C. The heater power can be continuously adjusted using a variable transformer, and the
maximum output is 4kW. The disturbance response parameters for a PID structure have to be eval-
uated.


First the heater power is set to give a temperature close to the future working point, for example
180°C at 60% heater power. Now the heater power is suddenly increased to 80% and the varia-
tion in temperature recorded. The inflection tangent is then drawn, giving Tu as 1 min and Tg as 10
min. If it is difficult to determine the point of inflection, the change in manipulating variable must be
increased, i.e. by starting the test at a lower heater power and ending at a higher heater power. The
final temperature in the case illustrated here is 210°C.
This gives the following values:

∆x 210 °C – 180 °C 30 °C
K S = ----- = ----------------------------------- = ----------- = 1.5 °C/%
- - -
∆y 80 % – 60 % 20 %
Using the values obtained for Tu and Tg , the parameters are calculated as follows:

T n ≈ 2.4 • T u ≈ 2.4 • 1 ≈ 2.4 min ≈ 144 sec

T d ≈ 0.42 • T u ≈ 0.42 • 1 ≈ 0.42 min ≈ 25 sec

Tu °C 1 min
X P ≈ 1.05 • K S • ----- ≈ 1.05 • 1.5 ---- • -------------- • 100 % ≈ 15,75 °C
- - -
Tg % 10 min

We should not overlook a certain disadvantage of this process. In practice, the graph very rarely
shows a very clear point of inflection. Hence, drawing the tangent at the point of inflection can lead
to errors in determining the values of Tu and Tg , which may or may not be significant. The method
illustrated is still very useful for forming a first impression of the controller settings. Other criteria
can then be used to tune the settings.


4.5.4 Adjustment according to the rate of rise
In some cases, there can be difficulties in determining the response time Tg when using the meth-
ods described above. Very often, the manipulating variable can only be set to either 0 or 100%.
Operating the process continuously at 100% manipulating variable can be highly destructive.
A more constructive alternative is to avoid determining Tg , and instead to evaluate the maximum
rate of rise Vmax . To do this, the manipulating variable is suddenly set to 100% and the output of
the process observed (see Fig. 49). The process variable will only start to change after a certain
time, following the change in manipulating variable. The rate of change will increase continuously
until the point of inflection is reached. At the point of inflection, the process variable approaches its
final value more and more slowly. Using this method, it is necessary to wait until the point of inflec-
tion is reached, and then set the manipulating variable back to 0% again. It is important to remem-
ber that, especially in processes with long delays, such as furnaces, the process variable can con-
tinue to increase considerably, even after the heating has been switched off.

Fig. 49: Adjustment according to the rate of rise
The tangent at the point of inflection is now drawn, and Vmax determined from the gradient triangle.
Using the delay time determined in a similar manner from the step response, the controller settings
can be implemented in accordance with the table which follows later.
The method described yields even better values if the controller and any manipulating device that
might be present allow the manipulating variable to be set to any value. In this case, the step
change in manipulating variable should be made close to the setpoint required later:
Example:
The future setpoint value of a furnace is 300°C. The existing controller is set to manual mode and
the manipulating variable manually increased until the furnace temperature reaches 280°C; this
temperature is reached at, say, 60% manipulating variable. Now the manipulating variable is sud-
denly set to 100%, and the point of inflection awaited. To apply the adjustment according to the


rate of rise, for this example, we also need the height of the step change in manipulating variable
∆y (40%) and the manipulation range yH (in this case 100%, as the controller can be set to 100%
manipulating variable) for use in the formulae which follow later.

Controller structure 100% step change in MV Any changes in MV
P XP = Vmax · Tu XP = Vmax · Tu · yH/∆y
PI XP = 1.2 · Vmax · Tu XP = 1.2 · Vmax · Tu · yH/∆y
Tn = 3.3 · Tu Tn = 3.3 · Tu
PID XP = 0.83 · Vmax · Tu XP = 0.83 · Vmax · Tu · yH/∆y
Tn = 2 · Tu Tn = 2 · Tu
Td = 0.5 · Tu Td = 0.5 · Tu
PD XP = 0.83 · Vmax · Tu XP = 0.83 · Vmax · Tu · yH/∆y
Td = 0.25 · Tu Td = 0.25 · Tu

Table 7: Formulae for adjustment according to the slew-rate response,
for processes with self-limitation

4.5.5 Adjustment without knowledge of the process
Occasionally, a controller has to be adjusted to a process where it is simply not possible to record
a transfer function or to open the control loop. If the process is not overly slow, the controller is ini-
tially set to a pure P structure with the largest proportional band possible, so that pure P action is
achieved.
The setpoint is set close to the future operating point, and the process value indication on the con-
troller is observed. After some time, the process value will stabilize at a value quite some way from
the setpoint. This is because of the low gain through the large proportional band setting. XP is now
reduced, as a result of which the deviation from the setpoint becomes smaller and smaller. As XP is
further reduced, a point is eventually reached at which the process value starts to oscillate periodi-
cally. There is no point in reducing XP any further, as it would only increase the amplitude of these
oscillations. These oscillations are not usually symmetrical about the setpoint; their mean value is
either above or below the setpoint. The reason for this, as we have already established, is the con-
tinued presence of the permanent deviation that a P controller produces.
The proportional band is now increased once again, until the process value becomes stable.
Next, the I component is added (PI structure), and the reset time Tn is reduced step by step. The
process variable slowly approaches the setpoint, as a result of the I component. Reducing Tn still
further accelerates the approach, but also leads to oscillations. We now apply a disturbance to the
process, either by changing the setpoint or an external disturbance. The approach to the new set-
point is monitored. If the process value overshoots, we have to increase Tn . If the approach is only
very slow, the reset time setting can be reduced still further.
The D component can be activated next, if required, (PID structure), by setting Td to a value of ap-
proximately Tn /4.5.
The procedure described above is a widely used practical method, suitable for simple processes.


4.5.6 Checking the controller settings
We cannot expect the control loop to achieve optimum performance with the initial parameter set-
tings. Some readjustment will usually be required, particularly on processes that are difficult to
control, with a Tg / Tu ratio less than 3. The step response of the process variable to a setpoint
change clearly shows any mismatch of the control parameters. The resulting transient response
can be used to draw conclusions about any necessary corrections. Alternatively, an external dis-
turbance can be applied to the process, for example, by opening a furnace door, and then analyz-
ing the effects of the disturbance. A recorder is used to monitor the process variable, and the con-
troller setting adjusted if necessary (see Fig. 50).
Increasing the proportional band XP - corresponding to a reduction in controller gain - leads to a
more stable transient response. Without an I component, a permanent deviation can be detected.
Reducing XP reduces the deviation, but a further reduction in proportional band eventually leads to
undamped oscillations. Setting the controller setting just below self-oscillation, by setting a small
XP , leads to a small deviation, but this is not the optimum setting, as in this case the control loop is
only very lightly damped. As a consequence, even small disturbances cause the process variable
to oscillate.
The I component reduces the permanent deviation in accordance with the reset time Tn . If the I
component is too low (Tn too large), the visible effect is that the process variable only creeps grad-
ually towards the setpoint. A larger I component (Tn small) acts like an excessive control gain, and
makes the control loop unstable, resulting in oscillations.
A large derivative time Td has an initial stabilizing effect, but, with a pulsating process variable, it
can also make the control loop unstable.
Fig. 50 indicates possible incorrect settings. It uses as an example the setpoint response of a third-
order process with a PID controller.
When optimizing a controller, only one parameter should be adjusted at a time, then the effect of
this change awaited before changing further parameters. Furthermore, we have to consider wheth-
er the controller should be optimized for disturbance response or setpoint response.
It is found, for example, that a “tight” controller setting with a high controller gain may indeed give
a fast approach to the setpoint, but the control loop is poorly damped because of the high gain.
This could mean that a short duration disturbance produces oscillation. In other words, a lower
controller gain slows down the approach to the setpoint somewhat, but makes the entire control
loop more stable.



x x

w w

X P too large Tn , Td too small

t t

x x

w w

X P too small Tn , Td too large

t t

x

w

optimum adjustment

t

Fig. 50: Indications of possible incorrect adjustments


5 Switching controllers
5.1 Discontinuous and quasi-continuous controllers
With the continuous controllers described previously, with P, PD, I, PI and PID actions, the manipu-
lating variable y can take on any value between the limits y = 0 and y = yH. In this way, the control-
ler is always able to keep the process variable equal to the setpoint w.
In contrast to continuous controllers, discontinuous and quasi-continuous controllers do not have
a continuous output signal, but one that can only have the state ON or OFF. The outputs from such
controllers are frequently implemented as relays, but voltage and current outputs are also com-
mon. However, unlike the continuous controller, these are binary signals that can only have a value
of 0 or the maximum value. These signals can be used to control devices such as solid-state re-
lays.

continuous
continuous fine graduation controller
of manipulating variable w y
controller
( 0 – 100 %)
-x

comparator
coarse graduation with hysteresis
discontinuous w y
of manipulating variable
controller
( 0 or 100 %)
-x

continuous switching
fine graduation controller stage
quasi-continuous w yR y
of manipulating variable
controller
( 0 – 100 %)
-x

Fig. 51: Continuous, discontinuous and quasi-continuous controllers
In addition to these controller types with binary outputs, there are also 3-state and multi-state con-
trollers, where the manipulating variable output can have 3 or more levels. A tri-state controller
would, for instance, be used for heating and cooling tasks, or humidification and dehumidification.
It might be assumed that controllers with outputs which can only be in the ON or OFF state would
only produce an unsatisfactory control action. But surprisingly enough, satisfactory results for the
intended purposes can be achieved in many control processes, particularly with quasi-continuous
controllers. Discontinuous and quasi-continuous controllers are very widely used, because of the
simple construction of the output stage and the actuators that are required, resulting in lower
costs. They are found universally in those areas of process control where the processes are rela-
tively slow and can be readily controlled with switching actuators.
The simplest controller with a binary output is the discontinuous controller, which is effectively a li-
mit switch that simply switches the manipulating variable on or off, depending on whether the pro-
cess variable goes below or above a predetermined setpoint. A simple example of such a control-
ler is the two-state bimetallic temperature controller in an electric iron, or a refrigerator thermostat.
Quasi-continuous controllers can be put together, for example, by adding a switching stage to the
output of a continuous controller (see Fig. 51), thus converting the continuous output signal into a
switching sequence. P, PD, I, PI and PID actions can also be implemented for these controllers
(Fig. 51) and the foregoing remarks about continuous controllers are also applicable.


5.2 The discontinuous controller
The discontinuous controller has only 2 switching states, i.e. the output signal is switched on and
off, depending on whether the process variable goes below or above a predetermined limit or set-
point. These devices are also often used as limit monitors, which initiate an alarm message when a
setpoint is exceeded.
A simple example of a mechanical discontinuous controller is, as previously explained, the bimetal-
lic switch of an electric iron, which switches the heating element off when the set temperature is re-
ached and switches it on again when the temperature falls by a fixed switching differential (hystere-
sis). There are other examples in the field of electronic controllers. For example, a resistance ther-
mometer (Pt 100), whose electronic circuitry switches heating on if the temperature falls below a
certain value, say 5°C, to provide frost protection for an installation. In this case, the resistance
thermometer together with the necessary electronic circuitry takes the place of the bimetallic
switch.

Fig. 52: Characteristic of a discontinuous controller
The discontinuous controller shown here supplies 100% power to the process until the setpoint is
reached. If the process variable rises above the setpoint, the power is taken back to 0%. Apart
from the hysteresis, we see that the discontinuous controller corresponds to a continuous control-
ler with no proportional band (XP = 0) and therefore “infinite” gain.


5.2.1 The process variable in first-order processes
If we connect a discontinuous controller, such as a rod thermostat, to a first-order process (e.g. a
thermostatic bath with water circulation, warmed by an immersion heater), we find that the course
of the process variable and manipulating variable is as shown in Fig. 53. In theory, the controller
should switch off the energy when the setpoint is reached, the process variable would fall imme-
diately and once again go below the setpoint. The controller would immediately switch on again,
and so on. Because an idealized first-order process has no delay time, the relay would switch on
and off continuously, and would be destroyed in a very short time.
For this reason, a discontinuous controller usually incorporates a switching differential XSd (also
known as hysteresis) about the setpoint, within which the switch status does not change. In prac-
tice, the switching differential is often to one side of the setpoint, either below (for example with
heating) or above (for example with cooling). Fig. 53 shows a case where the switching differential
lies below the setpoint. The switch-off point of the controller is the setpoint w. In practice, as the
process is not ideal (it has some delay time), the higher and lower values of the process variable do
not coincide exactly with the switching edges of the differential (XSd).

Fig. 53: Discontinuous controller in a first-order process


What matters however, is that the controller only switches when the process variable has moved
outside the differential band that has been set. The process variable continually fluctuates, at least
between the values Xhi and Xlo. The fluctuation band of the process variable is therefore influenced
by the switching differential.
In a process with delay, the discontinuous controller can only maintain the process variable con-
stant between the values Xhi and Xlo. The on-off switching is due to the manipulating variable being
too large to maintain the process variable constant when it is switched on, and too small when it is
switched off. In a large number of control tasks, where the process variable only needs to be main-
tained approximately constant, these fluctuations are not a problem. An example of this is a dome-
stic electric oven, where it does not matter if the actual temperature fluctuates between 196°C and
204°C for a baking temperature of 200 °C.
If these continuous fluctuations of the process variable do cause problems, they can be minimized
to a limited extent by selecting a smaller switching differential Xsd. This automatically leads to
more switching operations per unit time, i.e. the switching frequency increases. This is not always
desirable, as it affects the life of the controller relay.
It can be shown (mathematical details are not entered into here) that the following relationship
exists between the switching frequency (fsw) and the parameters T, Xmax and XSd :

1 1 X max 1 X max
f sw = ----------- = - • ------------ • -
- - - - valid for x ≈ ------------
-
T osc 4 X Sd T 2

fsw : switching frequency
Tosc : period of oscillation
Xmax : max. process variable reached with the controller output permanently switched on
XSd : switching differential
T : time constant of the first-order process
We can see from this relationship that the shorter the time constant (T), the higher the switching
frequency. A control process with short time constants will therefore produce a high switching fre-
quency, which would contribute to rapid wear of the switching stage of the controller. For this rea-
son, a discontinuous controller is unsuitable for this type of process.


5.2.2 The process variable in higher-order processes
In a process with delay, we have seen that under ideal conditions the fluctuation band is determi-
ned only by the switching differential XSd of the controller. The process itself has no effect here. In
a process with several delays, which can be described as delay time, response time and transfer
coefficient, this is no longer the case. As soon as there are any delays the process variable will
continue to rise or fall after switch-off and will only return after reaching a maximum. Fig. 54 shows
how the process variable overshoots the response threshold of the relay when the manipulating va-
riable is switched on and off.

Fig. 54: Discontinuous controller in a higher-order process
This produces an overshoot of the process variable, with limits given by the values Xhi and Xlo. This
means that the process variable fluctuates even when the controller has zero switching differential,
as the process only reacts to the change in manipulating variable after the end of the delay time.
Once again, take the electrically heated furnace as an example. If the energy supply is switched off
when the setpoint is reached, the temperature still continues to rise. The reason is that the tempe-
rature in the furnace only permeates slowly, and when the setpoint is reached, the heater rod is al-
ready at a higher temperature than that reported by the sensor. The rod and furnace material conti-
nue to supply additional heat. Similarly, when the heating is switched on again, heating-up is rather
sluggish and initially the temperature continues to fall a little further after switch-on.


The more powerful the heater, the greater is the temperature difference between the heater rod and
the sensor during heating-up, because of the process delay, and the process variable will overs-
hoot the setpoint even more during heating-up. We use the term excess power in this connection,
meaning the percentage by which the maximum power of a furnace is greater than the power re-
quired to approach a setpoint.
Example: A furnace which requires a manipulating variable of 2kW on average to stabilize at a set-
point of 200°C, but has a 4 kW continuous output rating, has an excess power of 100% at the wor-
king point of 200°C.
This means that the higher the excess power, the wider is the fluctuation band ∆x of the process
variable about the setpoint.
Now the present (but unwanted) fluctuation band of the process variable can be estimated for the
case where 100% excess power is available:
It is assumed that the switching differential XSd = 0

Tu X max
∆x = X max • -----
- valid for x ≈ ------------
-
Tg 2

As we can see, the fluctuation band is dependent not only on Xmax (with a linear process this is
proportional to the excess power) but also on the ratio Tu/Tg , whose reciprocals we are already fa-
miliar with from Chapter 2, and which give a measure of how good the controllability of a process
is. The shorter the delay time in comparison with the response time, the narrower is the fluctuation
band. The formula given for the fluctuation band ∆x is valid for XSd = 0. If there is a switching diffe-
rential, this is also added to the fluctuation band.
This gives us the formula:

Tu
∆x = X max • ----- + X Sd
-
Tg

The formula for the period of oscillation is: Tosc = 4Tu (valid for XSd = 0)
If a switching differential XSd has been set, then the period of oscillation is slightly longer. From this
we can derive the maximum switching frequency, which can be used to predict the expected con-
tact life:

1
f osc = -------
-
4T u


5.2.3 The process variable in processes without self-limitation
Because the step responses of an integrating process are linear, the behavior of a discontinuous
controller is easy to describe and calculate. Here again the process value fluctuates between the
given limits Xhi and Xlo (Fig. 55). In an ideal process without delay time Tu, the limit values are equal
to the switching differential XSd.

Fig. 55: Discontinuous controller in a process without self-limitation
The switching frequency fsw is given by:

Kp • yH
1
f sw = ----------- = -------------------
-
T osc 2 • X Sd

Kp : proportionality factor of the process
yH : maximum value of the manipulating variable
An example of such an application is a discontinuous controller used as a limit switch for level con-
trol of a water tank. The tank is used as a storage reservoir, from which water is drawn to meet de-
mand or into which a constant amount flows.
Summarizing, we can say that the discontinuous controller offers the advantage of simple con-
struction and few parameters which have to be set. The disadvantage is the fluctuation of the pro-
cess variable about the setpoint. In non-linear processes these fluctuations can be wider in the lo-
wer operating range than in the upper, because the process has excess power here. Approaching
the setpoint in the lower operating range will often result in wider fluctuations than in the upper
operating range. The area of application for such discontinuous controllers is limited to applicati-
ons where precise control is not required. In practice, these controllers are implemented through
mechanical thermostats, level switches etc. If an electronic controller with a sensor is used, the
controller is almost always provided with a dynamic action.


5.3 Quasi-continuous controllers: the proportional controller
As we have already seen, a quasi-continuous controller consists of a continuous controller and a
switching stage. If this controller is operated purely as a proportional controller, then the characte-
ristics which we have already met in Chapter 3.2.1 “The proportional band” apply equally here.

y
XP
100 %

w x

Fig. 56 : Proportional band of a quasi-continuous proportional controller
The quasi-continuous controller whose characteristic is shown in Fig. 56 always gives out a 100%
manipulating variable, as long as the process value lies below the proportional band. As the pro-
cess value enters the proportional band and approaches the setpoint, so the manipulating variable
becomes progressively lower.
How can a controller with a switched output provide a virtually constant energy supply i.e. stepless
dosage?
In the end it is immaterial whether a furnace is operated at 50% heating power all the time or at
100% heating power for only half the time. The quasi-continuous controller changes the switch-on
ratio or ON-time ratio (also known as duty-cycle) of the output signal instead of changing the size
of the output signal. An ON-time ratio of 1 corresponds to 100% of the manipulating variable, 0.25
corresponds to 25% of the manipulating variable, and so on.
The ON-time ratio, or duty-cycle R is defined as follows:

T on
R = -----------------------
-
T on + T off

Ton = ON time
Toff = OFF time
Multiplying the ratio R by 100 gives the relative ON-time in % of R, which corresponds to the mani-
pulating variable in %.

R(%) = y = R • 100%

With a quasi-continuous controller the characteristic of the process (especially the time constants)
exerts a strong influence on the course of the process variable. In a process where a disturbance is
transmitted relatively slowly (a process with long time constants) and where energy can be stored,
there is a smoothing effect on any pulses. With a suitable switching frequency, the use of a quasi-
continuous controller with these processes achieves a similar result to that achieved using a conti-
nuous controller.


The situation is different with a very fast process, where there is hardly any smoothing of the con-
stantly changing flow of energy, and the process variable fluctuates accordingly. Hence quasi-con-
tinuous controllers are preferably used where the process is comparatively slow, and are especially
popular in temperature control systems.

Fig. 57: Power control
The definition of ON-time ratio (or duty-cycle) means the ratio of the switch-on time of a controller
output to the sum of the switch-on and switch-off times, e.g. an ON-time ratio of 0.25 means that
the power supply is switched on for 25% of the total time. It gives no information on the actual du-
ration of the periods during which the switching cycles take place.
For this reason, the so-called cycle time (Cy) is defined, which fixes this time period. It represents
the period during which switching on and off takes place once, i.e. it is equal to the sum of the
switch-on and switch-off times (Fig. 57). The switching frequency is the reciprocal of the cycle time.
Fig. 57 shows the same ON-time ratio (R = 0.25) for different cycle times.


For a given ON-time ratio of 0.25 and a cycle time of Cy = 20 sec, this means that the energy sup-
ply is switched on for 5 seconds and switched off for 15 seconds. If the cycle time is 10 sec, the
energy supply is switched on for 2.5 seconds and switched off for 7.5 seconds. In both cases, the
power supplied is 25 %, but with a finer dosage with Cy = 10 sec. The fluctuations of the process
variable are smaller in the second case.
Theoretically, the ON-time of the controller is given by the following relationship:

y • Cy
T on = ---------------
-
100 %
Ton = ON time
y = manipulating variable in %
Cy = cycle time
This means that a shorter cycle time results in a finer dosage of the energy supply. On the other
hand, there is increased switching of the actuating device (relay or contactor). The switching fre-
quency can easily be determined from the cycle time.
Example:
The cycle time of a controller used for temperature control is Cy = 20 seconds. The relay used has
a contact life of 1 million switching operations. The value given for Cy results in 3 switching opera-
tions per minute, i.e. 180 per hour. For 1 million operations, this gives a life of 5555 hours = 231
days. Based on an operating time of 8 hours per day, this represents approx. 690 days. Assuming
around 230 working days per year we arrive at an operating life of approx. 3 years.
Generally, the cycle time is selected so that the control process is able to smooth out the energy
bursts supplied, to eliminate periodic fluctuations of the process variable as far as possible. At the
same time, the number of switching operations must always be taken into account. With a micro-
processor controller however, the value set for the cycle time Cy is not held constant over the who-
le of its working range. A detailed discussion of this point is rather complicated and would be too
advanced at this stage. If it is possible to operate a switching P controller in manual mode, the in-
fluence on Cy can be observed by direct input of a manipulating variable.
When Cy is matched to the dynamic action of the process, the behavior of a quasi-continuous con-
troller (as a proportional controller with dynamic action) can definitely be comparable with that of a
continuous controller, which also explains its name. With quasi-continuous controllers the different
manipulating variables are the result of a variation of the ON-time ratio, but there is no discernible
difference in the course of the process variable when compared to that of a continuous controller.


5.4 Quasi-continuous controllers: the controller with dynamic action
A quasi-continuous controller, operated as a pure proportional controller and with Cy suitably mat-
ched, shows almost the same behavior in a process as does a continuous controller with P action.
Although it reacts very quickly to changes in the control deviation, it cannot reduce the control de-
viation to zero, which is also the case with a proportional controller. A quasi-continuous controller
can also be configured as a PID controller, which means that it slows down as the setpoint is ap-
proached and stabilizes accurately at the setpoint.
A quasi-continuous controller (and also a P controller) can be pictured as a combination of a conti-
nuous controller and a switching stage connected to the output. The continuous controller calcula-
tes its manipulating varaible from the course of the process variable deviation and controls the
switching stage accordingly. The switching stage calculates the relative ON-time of the switching
stage output. The output of the switching stage is pulsed in accordance with the ON-time ratio and
the value set for Cy.

Fig. 58: Quasi-continuous controller with dynamic action,
as a continuous controller with a switching stage
Example: The continuous controller produces a manipulating variable of 50%. Likewise, for the
switching stage, 50% manipulating variable means an ON-time of 50%. Let us assume that the va-
lue set for Cy is 10 seconds, then the switching stage will turn the input on and off every 5 seconds.


5.4.1 Special features of the switching stages
As already described, the switching stage calculates the relative ON-time of the switching stage
output on the basis of the manipulating variable of the continuous controller. The output of the
switching stage is pulsed in accordance with the relative ON-time and the value set for Cy.
Using analog technology, an additional D action stems from the technical implementation of the
switching stage. If the continuous controller preceding the switching stage is operated as a propor-
tional controller, then combining it with the switching stage results in a PD action. This means that
if the analog controller referred to has a pure P action (i.e. only a proportional band XP can be set),
then the quasi-continuous controller, obtained by connecting a switching stage to its output, has
an additional D action which is not adjustable. As this characteristic proved to be very useful, it has
also been retained in the microprocessor controllers. The value of the derivative component is nor-
mally set by the manufacturer and therefore cannot be adjusted.
The following table shows the control configurations of a quasi-continuous controller and the actu-
al control actions:

Set control action Actual control action
P PD
PD PDD
I PI
PI PID
PID PD/PID
Table 8: Control configurations of a quasi-continuous controller, and actual control actions
Example: A quasi-continuous controller configured as a proportional controller actually has PD ac-
tion, because of the switching stage connected to its output.
If the continuous controller has a PID structure, the existing switching controller has a P, an I and
two D components. PD/PID controllers of this type are widely used with switched temperature
controllers, as they offer the best start-up action for this application.
Depending on the controller structure, various setting parameters are obtained for the quasi-conti-
nuous controller:

PD PDD PI PID PD/PID
XP XP - XP XP
Cy Cy Cy Cy Cy
- - Tn Tn Tn
- Td - - Td
Table 9: Setting parameters for differing dynamic actions

5.4.2 Comments on discontinuous and quasi-continuous
controllers with one output
In practice, discontinuous and quasi-continuous controllers are often brought together under the
concept of the 2-state controller, on the basis that the output of the controller can only assume two
conditions, either on or off. These controllers are configured at the instrument by defining the con-
trol type, namely 2-state controller. If the proportional band XP is now set to 0, we have a disconti-
nuous controller. If a proportional band XP greater than 0 is selected, a quasi-continuous controller
is obtained, on which the appropriate control parameters (Tn, Td and Cy) can be set.


5.5 Controller with two outputs: the 3-state controller

5.5.1 Discontinuous controller with two outputs
A discontinuous controller with two outputs can be thought of in simple terms as a combination of
two discontinuous controllers, each with one output, but linked together, one below the other. They
can be used for heating, for example, when below the setpoint and for cooling when above the set-
point. Other applications are, for instance, humidification and dehumidification in a climatic cabi-
net, etc. Each output of the controller is assigned a manipulating variable. E.g. the first controller
output would often be used for heating and the second output for cooling. All parameters associa-
ted with the “heating controller” are identified by an Index1 and those associated with the “cooling
controller” by Index2.
If the process variable varies within a fixed interval about the setpoint – the contact spacing XSh –
then neither output is active (Fig. 59). This contact spacing is necessary to prevent continual swit-
ching between the two manipulating variables e.g. heating and cooling registers, when the process
variable is unsteady.
As well as the contact spacing, discontinuous controllers with two outputs also have a hysteresis
for each of the heating and cooling contacts, which are normally indicated by the switching diffe-
rentials XSd1 and XSd2 . These two parameters eliminate any contact “chatter”, when the process
variable moves from the heating zone or cooling zone into the contact spacing.
With regard to the switching differentials XSd1 and XSd2, and the related switching frequency and
control quality in connection with the process characteristics, the same considerations apply as
for a discontinuous controller with only one output.
With this controller, the control accuracy which can be achieved is limited by the switching hystere-
sis values and the contact spacing (Fig. 59).



Discontinuous controller with 2 outputs

X Sd2
Cooling

w X Sh

Heating
X Sd1

Switching differential (X Sd1 , X Sd2 ) and contact spacing (X Sh )

Characteristic
y

100 %
X Sd1

XSh

w x

X Sd2

- 100 %

Fig. 59: Characteristic of a discontinuous controller with two outputs


5.5.2 Quasi-continuous controller with two outputs, as a proportional controller
Even a quasi-continuous 3-state controller, where each of the two outputs is operated by a propor-
tional controller, can be thought of in simple terms as a combination of two quasi-continuous con-
trollers linked together, one below the other. The switching differential does not apply here, but a
contact spacing can be set.
Fig. 60 shows the characteristic of a quasi-continuous 3-state controller used to control an climatic
cabinet.

Fig. 60: Characteristic of a quasi-continuous controller with two outputs,
as a proportional controller
As shown in Fig. 60, the two proportional bands XP1 and XP2 can be adjusted independently for a
quasi-continuous controller with two outputs. This is necessary, because in general the process
gain is different for the two manipulating variables, as the heating register influences the process
differently from the cooling (through a fan, for example).
The way in which this controller works is described below: The process value in the cabinet is
26°C. Now the control is switched on:
Heating: The heating relay is energized and the heating heats up with 100% manipulating variable,
whereupon the process value increases. The heating manipulating variable continually reduces
from a process value above 27°C (on reaching the proportional band), the heating relay starts to
pulse and the switch-on times become progressively shorter. The control deviation and hence the
manipulating variable become smaller, until a manipulating variable is obtained which is just suf-
ficient to maintain the process value. Now the process value is below the contact spacing, and we
obtain a positive manipulating variable (for example 28.5°C and 25% manipulating variable).
Cooling: Now the ambient temperature increases (disturbance), whereupon the inner chamber of
the cabinet is heated. The process value increases – on entering the contact spacing (29°C) the
manipulating variable is 0%, and there is neither heating nor cooling. The cooling relay only starts
to pulse above a temperature of 31°C (the manipulating variable becomes negative). Likewise, the
control deviation reaches a value such that the manipulating variable produced is just sufficient to
maintain the resulting process value.


5.5.3 Quasi-continuous controller with two outputs and dynamic action
With a quasi-continuous controller with two outputs and dynamic action, it is additionally possible
to set a Tn and a Td for each of the two manipulating variables. With this controller, the three con-
trol components (P, I and D) are only all effective together outside the contact spacing. If a process
enters the contact spacing, the P component is not effective. In the contact spacing, only the I and
D components are effective, and in principle the process variable can therefore be stabilized exact-
ly.
Because there is no P component in the contact spacing, a large contact spacing has an adverse
effect on the dynamic response, as the controller is slow in the contact spacing. The size of the
contact spacing chosen should be no larger than necessary, but no unwanted changeover bet-
ween the manipulating variables should occur, either when approaching the setpoint (possible
overswing of the process variable above the setpoint) or when stabilizing at the setpoint (fluctuati-
ons of the process variable about the setpoint).
Too small a contact spacing can lead to a pointless waste of energy in an installation
Table 10 shows the setting parameters of a quasi-continuous controller with two outputs and dyna-
mic action:

set effective
Discontinuous XP1 = 0 – – – XSh Xd1; Xd2
XP2 = 0
Quasi-continuous P PD XP1; XP2 – – Cy1; Cy2 XSh –
PI PID XP1; XP2 Tn1; Tn2 – Cy1; Cy2 XSh –
PID PD/PID XP1; XP2 Tn1; Tn2 Td1; Td2 Cy1; Cy2 XSh –
PD PDD XP1; XP2 – Td1; Td2 Cy1; Cy2 XSh –
I PI – Tn1; Tn2 – Cy1; Cy2 XSh –
Table 10: Setting parameters for a quasi-continuous controller,
with two outputs and dynamic action

5.5.4 Comments on controllers with two outputs
In Chapter 5.5 we met controllers where the two outputs could influence a process variable in two
directions. The controllers described always had two outputs of the same type (discontinuous or
quasi-continuous). In practice, it may turn out that the outputs are different. It may well be, for ex-
ample, that a controller has to provide a discontinuous output for cooling and a quasi-continuous
output for heating. Such a controller cannot be classified under any of Chapters 5.5.1, 5.5.2 or
5.5.3.
To limit the number of names, in practice all controllers with two outputs which can influence a pro-
cess variable in two directions are referred to as...
. . . 3-state controllers
irrespective of whether the outputs are discontinuous, quasi-continuous or continuous. As an ex-
ample, mention should be made of a controller which operates a thyristor-controlled power unit
and a refrigerator unit and thus maintains constant temperature in a climatic cabinet. The two
plants require two controller outputs – but the controller must provide a continuous output for hea-
ting and a switched output for cooling.


5.6 The modulating controller
Actuators (actuator drives) have three operating conditions: opening, holding, closing.
Electrical drives in particular are widely used for this application, where a motor controlled for
clockwise or anticlockwise rotation drives a worm gear to operate a valve, throttle, variable trans-
former or similar device. Both DC and 3-phase motors are used, with single phase motors used for
smaller drives, switched by contactors or relays (see Chapter 1.7).
These drives stand out from the controller applications already discussed in one particular way.
When the heating is switched on in a furnace, it operates immediately at full power, and when it is
switched off, the supply of power stops immediately. In contrast, actuators require a certain time to
reach the maximum manipulating variable (valve opening etc.). In addition, an electrical actuator
holds the position it has reached, even when there is no signal from the controller. The actuator
can, for example, stay at 60 % open, even though it is not operated by the controller at this time.
The controller must take these properties into account.
Modulating controllers are used for this type of actuator drive.
The modulating controller consists of a continuous controller (P or PD) and a switching element. If
we regard the valve position as the manipulating variable, the combination of modulating controller
and regulating valve exhibit PI or PID action.
To understand the operation of the modulating controller take a look at Fig. 61:

Fig. 61: The modulating controller, with a regulating valve in the control loop
The modulating controller shown controls the temperature in a furnace via a regulating valve in the
gas flow. The switching stage provides two relay outputs which drive the valve open and closed
over the range 0 to 100%.
The controller, the switching element and the regulating valve must now be thought of as a single
unit. The modulating controller (meaning here the continuous controller and the switching element)
can be configured for PI or PID action. If a control deviation occurs, the valve will exhibit the corre-
sponding PI or PID action. If then, for example, PI action is set on the controller, then the combined
modulating controller and regulating valve (with the valve opening as manipulating variable) will
have PI action.
Example: The modulating controller of Fig. 61 was configured as a PI controller. The proportional
band XP was set to 25°C, and the reset time Tn to 120 seconds. The associated regulating valve
has an actuator stroke time Ty (the time required by the actuator to travel from 0 to 100% or from
100 to 0% manipulating variable) of 60 seconds. Fig. 63 shows the step response of the system.



Fig. 62: Step response of the modulating controller and regulating valve system
Fig. 62 shows the step response of a PI controller with the parameters XP = 25°C and Tn = 120sec.
The step change in control deviation occurs at t = 60 sec and amounts to 10°C. The modulating
controller has the following settings: XP = 25°C, Tn = 120sec, and the actuator stroke time Ty =
60seconds. By operating via the “Open” output of the modulating controller, the PI action of the
combined modulating controller and regulating valve will be implemented. The opening of the valve
will, of course, lag behind the manipulating variable of a PI controller, as it has a stroke time of 60
seconds.
The modulating controller receives no indication of the exact position of the valve. It assumes that
the valve opens and closes at exactly the same speed. The modulating controller calculates the
time for which the “Open” contact must be closed, until, in theory, the valve position corresponds
to the manipulating variable of the corresponding PI controller. For this to work, the modulating
controller must have knowledge of the actuator stroke time.
The modulating controller also has its contact spacing set so as to lie symmetrically about the set-
point. Within the contact spacing, no control operation occurs on the actuator, which means that if
the process variable enters the contact spacing, the valve will remain in its old position.
With modulating controllers, a minimum pulse duration TMmin can be taken into account. This may
be necessary because of minimum switch-on times of the actuator drive (e.g. play in the gears).
With a microprocessor controller, however, it is at least the sampling time or cycle time of the con-
troller. The minimum pulse duration TMmin can be set directly on many controllers.
The minimum pulse duration has a direct influence on the positioning accuracy of the actuator, and
consequently on the expected control accuracy.
The following relationship generally applies for linear processes:

T Mmin
∆x = X Max • ---------------
-
Ty

∆x : control accuracy
XMax : maximum process value
TMmin : minimum pulse length
Ty : actuator stroke time


It is important that the contact spacing XSh of a modulating controller is not set smaller than the
control accuracy ∆x, calculated from the minimum pulse duration. Choosing a contact spacing
smaller than this value will result in permanent fluctuations of the process variable as the actuator
continually changes over from clockwise to anticlockwise rotation, making excessive demands on
the actuator.
If the correct contact spacing is chosen, the true control deviation will be smaller than the set con-
tact spacing, because the final pulse runs the actuator into the contact spacing and thereby redu-
ces the control deviation.
As the actuator drive has the same characteristic for clockwise and anticlockwise rotation, there is
only one setting each for XP, Tn and Td . The setting parameters are then as follows:

Dynamic action PI PID
Setting parameters XP XP
Tn Tn
- Td
Ty Ty
XSh XSh
Table 11: Setting parameters with the modulating controller


5.7 Continuous controller with integral motor actuator driver
A “continuous controller with integral motor actuator driver” or, for short, an actuating controller, is
much more suitable for operating a motorized actuator than is a modulating controller. It forms a
type of cascade structure (see Chapter 6.6), and consists of a continuous controller and a subordi-
nate actuating controller (Fig. 63).

Fig. 63: The actuating controller with a regulating valve in the control loop
The continuous controller outputs the manipulating variable, based on the course of the control de-
viation and the parameters set on the controller (Fig. 63). The usual control structures, i.e. P, PI, I,
PD, PID can be set for the continuous controller. The duty of the actuating controller is now to re-
gulate this manipulating variable on the regulating valve. The actuating controller operates the ac-
tuator via two switching outputs, and receives an actuator retransmission signal (usually a stan-
dard signal 0/4 — 20mA, 0/2 — 10V etc.), which feeds the actuator position back to the controller.
Example: The continuous controller determines a manipulating variable of 20% from the course of
the control deviation. The actuating controller now controls the valve at 20% opening. The valve
provides a 0 — 10V actuator retransmission signal that corresponds to 0 — 100% opening of the
valve. If the actuating controller has controlled the valve to 20% opening, the actuator retransmis-
sion signal would be 2V.
An actuator stroke time must also be fed into the actuating controller, which the controller then
uses to optimize its control parameters.
Where a motor is being operated which has an appreciable overrun (poor braking action), juddering
of the actuator motor can be avoided by increasing the contact spacing (XSh).
Advantages of the actuating controller in comparison with the modulating controller:
Unlike the modulating controller, the actuating controller offers the advantage of a subordinate
controller structure. If a control deviation occurs, the actuating controller ensures that the motor is
driven directly to a new position. This is achieved by comparing the actuator position with the ma-
nipulating variable (yR) of the continuous controller.
A modulating controller does not receive an actuator retransmission signal and must always assu-
me a linear actuator action. If the actuator has a non-linearity, or play is present in the actuator me-
chanism, this assumption will only be an approximation.


An actuating controller offers the following setting parameters for the corresponding control action:

Dynamic PD PDD PI PID PD/PID
action
Setting XP XP - XP XP
parameters - - Tn Tn Tn
- Td - - Td
Ty Ty Ty Ty Ty
XSh XSh XSh XSh XSh
Table 12: Setting parameters with the actuating controller

Fig. 64: Example of an application for an actuating controller
Example:
The actuating controller described is used to control the outflow temperature of a heating system.
The main element is a mixer valve whose chambers “C” and “W” for cold and warm water are lin-
ked through piping to the water return pipe. The mix temperature is measured by a Pt100 and can
be varied by adjusting the position of the slider “S”, which is operated by an actuator motor. The
input variable of the actuator motor is in the form of switching pulses for opening and closing the
outflow opening.


6 Improved control quality through special controls
So far, we have only considered single-loop control circuits, where controller and process form a
closed signal loop. However, when using such single-loop control circuits, there are limits to the
control quality which can be achieved in certain control processes. It is possible to go beyond the
control quality limits imposed by the single-loop control circuit by using multi-loop control circuits,
or by switching auxiliary variables on and off. To some extent, relatively simple solutions can lead
to considerable improvements in control quality.

6.1 Base load
With a base load setting, the controller only influences part of the total manipulating variable, and a
fixed proportion is continuously supplied to the process (combination of control and operation). It
could then be the case that, for example in an electrically heated furnace, one section of the heat-
ing elements is controlled by the controller, whereas another section is supplied at full supply volt-
age (see Fig. 65).

K1
Furnace
R1
N

R2
L1

x 100 % power

with base load

without base load

t

Fig. 65: Base load setting
Essentially, base load setting offers the following advantages:
- The actuator, e.g. a thyristor controller, can be more compact and less expensive, as it only
needs to control low power.
- Load fluctuations on the supply network, as a result of power consumption in bursts caused by
the switching controller, are similarly reduced.
- If the controller fails, the process can still be operated with the base load component of the total
power.


Against the advantages shown, there are also a number of disadvantages:
- The dynamic control action is impaired, especially with regard to disturbances. As the controller
now no longer provides the full manipulating variable, the cooling curve, for instance, is not only
shifted by the amount of the base load setting, but is also clearly flatter (see Fig. 65). If, for any
reason, the power requirement suddenly becomes less than the base load setting, the controller
is helpless in this situation, as it cannot reduce the manipulating variable below the value of the
base load.
- In addition, the base load setting must also be matched to the setpoint. If the setpoint is
changed downwards, for instance, the excess power could suddenly be too large; with an
upward change, the excess power could be too small. In such cases, the base load should be
changed at the same time as the setpoint.


6.2 Power switching
If a process is being operated with different setpoints or working points, it is better to switch the
applied power rather than use a base load. In an electrically heated furnace, this can be achieved
by switching part of the heating elements through a limit value switch (limit comparator as an ad-
vance contact), in order to facilitate furnace operation at full power in the upper temperature range
(see Fig. 66).
This gives the following advantages:
- At any one time, the process can be operated in the upper third of the characteristic valid at that
time (see Fig. 66). In this way, the excess power at small values of the process variable can be
minimized.
- The dynamic control action is rather better here when compared with the base load method, as
in this case the control power can be reduced to zero (after falling below the changeover point).
There can be a disadvantage with this circuit if it operates with a setpoint close to the changeover
point, as the process has two different values of process gain here.

Fig. 66: Power switching


6.3 Switched disturbance correction
The effect of a disturbance can often be predicted within certain limits. For example, opening a fur-
nace door leads to a fall in temperature of 30°C. Instead of first waiting for the process to respond
to this disturbance and then for the controller to take corrective action, the disturbance can be re-
sponded to directly. To do this, the furnace door is fitted with a position switch that increases the
manipulating variable (e.g. heating power) by several percent when the furnace door opens.
This principle is known as switched disturbance correction. It is useful when the cause and effect
of a disturbance are known, and where the disturbance occurs frequently and reproducibly. The
disturbance is quickly compensated by the rapid response made possible without time delays
caused by the controller and process.
We will now look at three different possibilities of switched disturbance correction:
Maintaining the disturbance constant
The effect of the disturbance on the process variable is eliminated by maintaining the disturbance
constant by means of an auxiliary control loop (see Fig 67 a). Maintaining the disturbance constant
should only be used when suitable technology is available to measure disturbances and maintain
them constant.
An example of this is the temperature control of a gas-fired annealing furnace. Here, the main dis-
turbance, gas pressure, can be maintained constant by an in-line pressure controller, which at the
same time can also reduce the higher supply pressure to the lower burner pressure. The block dia-
gram of this method can be applied to our own example:
The controller has the job of bringing the process variable x of the process (the temperature of the
annealing furnace) to the setpoint w, by giving out the manipulating variable y. If the disturbance z
(the gas pressure) is not maintained constant, then, when the gas pressure fluctuates, the control-
ler has to change its output repeatedly, if it is to hold the same setpoint. The auxiliary controller (the
pressure controller) now maintains a constant gas pressure, so that this disturbance no longer in-
fluences the annealing furnace.



a) Maintaining the disturbance constant
z

Auxiliary
controller

w
y
Controller Process
x

b) Additive switched disturbance correction

z
y
z
w
y yz
Controller Process
x

c) Multiplicative switched disturbance correction

z
0—100%

z
KP
w
Controller Process
x

Fig. 67: Forms of switched disturbance correction
Additive/multiplicative switched disturbance correction
With both these methods, when the disturbance changes, the manipulating variable y of the con-
troller is manipulated to counteract the effect of the disturbance (see Fig. 67 b, c).
With additive switched disturbance correction (Fig. 67 b) the manipulating variable (y) is in-
creased by an amount proportional to the disturbance. In other words, this type of switched distur-
bance correction takes into account any offset shifts in the process. Controllers that allow such a
switched disturbance correction to be implemented (compact controllers), normally provide an in-
put for the switching signal. A signal proportional to the disturbance is applied to the controller in-
put, which influences the manipulating variable in accordance with the setting. To illustrate this, we
can take the example above where the furnace door is opened. When the door is opened, the ma-
nipulating variable is increased by a fixed amount.


A multiplicative switched disturbance correction exerts an influence on the controller gain KP . As
the measured disturbance changes its value, so the value of KP set at the controller is changed in
the same ratio, in the range 0 — 100% (see Fig. 67 c). This method is suitable for use in processes
where the manipulating signal (controller output) must be changed to the same extent as any dis-
turbance which may occur.

Fig. 68: Neutralization plant
As an example, a neutralization plant could be quoted, in which alkaline waste water is neutralized
with acid (see Fig. 68). The process variable is the pH value, which should be in the neutral range.
The controller exerts an influence on the pH value by changing the inflow of acid (y). First of all, let
us consider how the plant operates without multiplicative switched disturbance correction. As-
sume that the controller has stabilized at a defined flow rate with, say, 30% manipulating variable.
Now, the disturbance (flow) changes, and the quantity of waste water per unit time is now twice as
large. The pH value will now increase, and the controller will increase its manipulating variable until
the process variable reaches the setpoint again. This will be the case with 60% manipulating vari-
able (double the quantity of acid). We can see that the manipulating variable must be kept propor-
tional to the disturbance to maintain the same setpoint, other conditions remaining unchanged.
This can be achieved by measuring the disturbance (flow) and applying multiplicative switching.
The disturbance is scaled at the controller over the range from zero to the maximum disturbance
value which could occur; the controller now changes its proportional action to the same extent,
over the range 0 — 100%.
If we now look at our example again:
Assume here that the controller has stabilized again with, say, 30% manipulating variable. Now the
disturbance (flow) changes to twice the value. Likewise, through the multiplicative switched distur-
bance correction, the proportional gain (that corresponds to the overall gain, see also Fig. 41) is set
to double its value. The manipulating variable of the controller immediately increases to 60% and
there are no larger control deviations.
The examples of switched disturbance correction shown here apply to discontinuous controllers
with 2-state action and continuous controllers. The relationships for 3-state, modulating and actu-
ating controllers are more complex, and will not be discussed here.


6.4 Switched auxiliary process variable correction
Where a disturbance cannot be measured or localized, it is possible to derive an auxiliary process
variable Xaux from the process, where Xaux has a shorter time delay than the main process variable
x, and apply it to the controller input, after suitable conversion (see Fig. 69). In this way, the distur-
bances at the process input (e.g. supply disturbances) are quickly reported to the controller.
However, Xaux is normally applied through an adaptive timing element, so that the process variable
is not distorted under stabilized conditions. With this arrangement, two control loops, each with its
own complete signal path, are coupled together. It should be noted that the control loop can possi-
bly become unstable as a result of overly strong switching of the auxiliary process variable and an
unsuitable controller setting.

Fig. 69: Switched auxiliary process variable correction

6.5 Coarse/fine control
Two control loops in series are used to maintain some parameter of a mass flow or energy flow
constant. The residual deviation from the first controller, the coarse controller (C1), is corrected by
the second, fine controller (C2) – see Fig. 70.

Fig. 70: Coarse/fine control


Here again we can use as an example a pH control system for neutralizing industrial waste water.
Because of the large variations in inflow normally present, and the changing composition, it is often
appropriate to connect two control loops in series, so that the variations in pH value are maintained
within the permissible tolerances.

6.6 Cascade control
Cascade control can significantly improve the control quality. This applies in particular to the dy-
namic action of the control loop, in other words, the transition of the process variable following set-
point changes or disturbances. Processes with a Tg / Tu ratio less than 2 or 3 can be difficult to
control with a simple control system; because of the relatively long delay time, the controller does
not become aware of how it should respond until a very late stage. We therefore try to split the con-
trol loop into several partial loops (usually two), which are controlled separately. Control of these
partial loops is much easier, as each has only a fraction of the overall delay time. This arrangement
is also known as multi-loop or networked control.
Fig. 71 shows the block diagram for cascade control.

Fig. 71: Cascade control
An auxiliary process variable xaux is derived from the process and applied to the input of an auxilia-
ry controller, the output of which controls the manipulating variable y. The setpoint w1 of the auxil-
iary controller is determined by the manipulating variable of the main controller, such that the pro-
cess variable reaches the set value. The auxiliary control loop can be set to respond more rapidly,
and quickly eliminates all disturbances at the input to the process.
The subordinate auxiliary controller is constructed in the same way as an ordinary controller. How-
ever, it must have an input for an electrical setpoint signal, as its setpoint is set by the supervisory
controller. In other respects, it must be matched to the demands of its duty, with regard to input,
output etc. The auxiliary controller has the job of changing the auxiliary process variable very
quickly, in proportion to the manipulating variable of the main controller; hence P or PD controllers
are normally used for this application, or also, less frequently, a PI controller. The master controller,
set for setpoint response, is usually a PI or PID controller.


For cascade control, it is important that the subordinate loop is at least 2 — 3 times faster than the
outer loop, as otherwise the overall control loop will tend to oscillate. One advantage of cascade
control is that the dynamic response of the control loop is much improved. Another advantage is
that the controllers are much easier to adjust. The master controller is switched to manual mode,
and the slave controller is optimized. Then the master controller is optimized, with the slave con-
troller kept in automatic mode.
An example of cascade control is the temperature control of a furnace heated by a gas burner (see
Fig. 72).

Fig. 72: Cascade temperature control for a burner
The master controller outputs a manipulating variable y1 in the range 0 — 100%, on the basis of
the control difference applied to it. The slave controller now receives this manipulating variable as
its setpoint, but only after the signal is normalized: on the basis of the normalization, the setpoint of
the slave controller (w1) amounts to 0 — “maximum gas flow”, corresponding to 0 — 100% manip-
ulating variable of the master controller. With its manipulating variable, the master controller practi-
cally presets the desired gas quantity per unit time. The slave controller has the job of controlling
the gas flow accurately. The slave controller now takes over part of the timing elements and cor-
rects disturbances at the input to the process, for example, fluctuations in the gas pressure. The
control action is improved on this basis, and, in certain cases, processes can only be controlled by
introducing cascade control.


6.7 Ratio control
Ratio controllers are used in burner controls (control of the gas/air mixture ratio), analytical tech-
niques (mixing of reagents) and in process engineering (preparation of mixtures). These controllers
have two process value inputs. The ratio of the two input variables is the real process variable. The
value required for this ratio is set as the setpoint, directly at the controller.
A ratio controller is frequently used as a slave controller. Here, the controller has the task of control-
ling the quantities of two substances in such a way that the mixing ratio stays constant when differ-
ing total quantities of the mixture are required. With this kind of slave control, there are two set-
points: the mixing ratio and the total quantity. Accordingly, two controllers are used, one of which
controls the total quantity of the mixture per unit time, whilst the other influences the mixing ratio,
by adjusting the dosage of the separate components. As the total quantity per unit time is the ulti-
mately decisive setpoint, this controller is designated as master controller, whilst the subordinate
controller controls the substance mixing ratio to meet the requirements of the master controller.

Fig. 73: Ratio control
An example of this is the mixture control shown in Fig. 73: two substances have to be mixed in a
fixed ratio to each other, whilst the demand for the quantity of the mixture fluctuates according to
production requirements. Two control circuits are required for this, one to control the total quantity
of both substances after mixing, the other to control the mixing ratio. In controlling the total quanti-
ty, it is sufficient to influence only one component, since the other is made to follow according to
the set ratio. However, the mixing ratio is controlled independently of the master controller, so that
the master controller and its associated valve have been fitted purely to control the air flow and
hence the total quantity. Without the master controller, only the mixing ratio remains constant,
whereas the total quantity of the two substances is disregarded.
A ratio controller is a standard controller whose input stage has two inputs to suit this modified
specification. With regard to the dynamic action, all the variations of the standard controller could


conceivably be used. Because of the nature of the process, the controllers are usually continuous,
modulating or actuating controllers, with PI or PID action. With microprocessor controllers, func-
tions such as ratio control can usually be configured directly.

6.8 Multi-component control
In a multi-component control system, various process-dependent variables produce the process
value for the controller and determine the control deviation, as in a steam/feed water control, for in-
stance (see Fig. 74).

Fig. 74: Multi-component control
In this case, the individual process values can each be allocated a different weighting factor, so
that they affect the control deviation to different extents; the main process variable is normally allo-
cated the highest weighting factor.
In the example given, the following relationship might apply:

x = x 1 + 2b ( x 2 – ( 2ax 3 + c ) )

x3 = steam flow
x2 = feedwater flow
x1 = water level
If we now consider the relationship in the above formula without the constants, we have:

x = x1 + ( x2 – x3 )


We can see that when the steam draw-off x3 is equal to the cold water supplied x2, the expression
in brackets becomes zero, and the process value x now depends only on the water level x1. Such a
multi-component controller must have suitable number of inputs, and, if necessary, a facility for
combining the individual signals via certain computations. In the example shown, there are three
components, hence this circuit has been given the name “3-component control”.


7 Special controller functions
This chapter introduces a number of additional functions, i.e properties which are specific to par-
ticular controllers and are required for certain control tasks. With modern microprocessor-based in-
struments, such options can be implemented using suitable software.

7.1 Control station / manual mode
With a control or auto-manual station, the manipulating variable can be altered directly. The control
loop is interrupted so that the process variable has no significance. The controller now acts purely
as an operating device (see Fig. 75).

Fig. 75: Use of a control station
This feature is useful where valves or actuators have to be fully opened or closed for cleaning, as
this would not otherwise be possible. Control stations are also invaluable for trials and test runs,
when the manipulating variable has to be operated in manual mode, i.e. without automatic control.
They can be integrated in the controller or arranged as a separate instrument.
Nowadays, with many microprocessor controllers, the function of a control station is provided by
the manual mode setting. If a controller is switched from automatic mode (where the controller is
trying to control the process value at the setpoint) to manual mode, the control function is disabled.
Now the actuator, the thyristor power regulator, the cooling etc., can be controlled by setting the
manipulating variable manually. Manual operation is possible with all types of controller.
In manual mode, the operator could set a value for the actuator very different to the value of the
current manipulating variable set by the controller. However, such an extreme change could have a
destructive effect on the actuator. This problem is overcome by the provision of bumpless transfer
from automatic to manual mode, where the manipulating variable remains at its current value, and
can then be changed manually.
In case of a broken probe, caused, for example, by a cable fault or mechanical damage to the sen-
sor, automatic control is no longer possible, and the controller switches off the energy supply for
safety reasons. In certain processes this could destroy the batch or cause a lengthy loss of pro-
duction, owing to the long warm-up times or similar such conditions. Here, controllers offering
manual operation have the advantage that, in such cases, the process can still be operated manu-
ally, albeit with reduced accuracy, and brought to completion.


7.2 Ramp function
Ramp functions are often used in processes with delicate products or to protect heating elements,
as they ensure that the process variable approaches the setpoint slowly. In processes with large
excess power, it also makes sense to limit the rate of rise of the process variable. With this func-
tion, the user sets a maximum rate of change (e.g. in °C/min, °C/hr etc.) at which a ramp setpoint
travels towards the main setpoint. The process variable continuously tracks this changing value
(see Fig. 76).

Fig. 76: Diagram of the ramp function
When the ramp function is activated, the ramp setpoint is normally made equal to the current pro-
cess value, and then altered towards the main setpoint at the set gradient. Once the setpoint is
reached, the ramp function is terminated, and the instrument controls at the set value until, for in-
stance, the main setpoint is changed. If it does change, the newly activated value will once again
be approached by a ramp. In this way, both rising and falling ramps can be implemented.

7.3 Limiting the manipulating variable
A manipulating variable limit can be used to limit the controller output signal at either a maximum
or a minimum value. One application is where the actuating device fitted (e.g. pump, electric heat-
ing etc.) is over-sized; it avoids excess power and its associated problems, such as the process
variable overshooting the setpoint. Further, a minimum manipulating variable limit can be a wise
precaution in the control of gas burners, for instance. Setting a minimum manipulating variable
(e.g. 3%) avoids the gas supply being interrupted and thus the burner going out. The controller
then only gives out a manipulating variable within the range of the set minimum or maximum val-
ues.
In a continuous controller with a 0 — 10V output signal and a 5 — 95% range limitation, the output
signal is restricted to the range 0.5 — 9.5V.


7.4 Program controller
So far, we have always assumed that the process variable has to be maintained constant at a fixed
setpoint value. Such controls are also called fixed setpoint controls. In comparison, there are also a
number of manufacturing processes where the setpoint does not represent a fixed value for the
control system. Instead it represents a parameter which varies with time, i.e. a specific profile for
the variation of the setpoint with time is required. Controllers used for this application are called
program or profile controllers. They are often encountered as temperature controls, e.g. in anneal-
ing furnaces for metallic materials, in ceramic kilns, in chemical engineering for the batch manufac-
ture of products, in climatic chambers etc.
The selectable program time is normally of the order of minutes, hours or even days. Fig. 77 shows
such a profile for the course of the setpoint. Conditioned by the process, the course of the process
variable does not normally show the same sharp transitions as the setpoint profile.

w/x Setpoint
Process value
200 °C
Rise too steep

150 °C

100 °C

50 °C

2h 4h 6h 8h 10 h t

Fig. 77: Diagram of a setpoint profile
The setpoint profiles are usually predetermined by an external programming device or a computer;
they are fed into the controller via the interface or a separate setpoint input. As the process variable
has to follow the continuously varying setpoint, the arrangement is also called follower control.
The rise of the setpoint profile must not exceed the rise of the process characteristic, otherwise the
process variable no longer follows the program profile, but rises at the maximum rate set by the
process characteristic (see Fig. 77). This is particularly important in non-linear processes such as
temperature controls, where the process gain decreases with higher process values. This can
sometimes be avoided by programming the instrument so that the program run is terminated as
soon as a certain deviation occurs, i.e. the process variable can no longer follow the program.


7.5 Self-optimization
Optimum adjustment of a controller to the process can be a time-consuming affair, particularly if
the process concerned is rather slow. Furthermore, as we saw in Chapter 2.8, the optimal values
for XP , Td and Tn are dependent on the working point. So it is quite likely that several sets of con-
trol parameters will have to be found for one process.
It seems an obvious step to let the controller itself make automatic adjustments to the control pa-
rameters. With microprocessor controllers, the facility for automatic controller setting (self-optimi-
zation) is available.
Basically, the choice lies between controllers which determine the control parameters simply on
the basis of user requirements, and adaptive controllers, where the settings are continually
checked and changed.
The methods used for self-optimization are usually built into the controller as software functional
blocks, and identify the process on the basis of the response of the process variable to step
changes in manipulating variable, and from this determine the best control parameters. The values
determined for XP , Td, Tn etc. can be refined by the user (see Fig. 78).

Fig. 78: Operating principle of self-optimization
In principle, a self-optimization can be arranged, for instance, to take place about the setpoint: if
the controller has stabilized the process variable at the setpoint, the self-optimization can then be
started, and the controller outputs 100% and 0% manipulating variable alternately. The controller
determines the best parameters by examining the oscillations of the process variable about the
setpoint, and then automatically accepts the parameters. With this method, it should be ensured
that no damage can be caused when the process variable exceeds the setpoint.


Fig. 79 shows another type of self-optimization: in this case, the self-optimization process is start-
ed when the process value is below the setpoint. The controller determines the delay time from the
initial response of the process variable. The controller calculates a switching level based jointly on
the delay time and the gradient of the response of the process variable, and if this level is exceed-
ed it sets the manipulating variable to 0%. With a linear process, interposing the switching level
prevents the process variable from overshooting the setpoint. With non-linear processes, the over-
shoot is not completely prevented, but is at least reduced. In all, the controller outputs 100% ma-
nipulating variable twice, interrupted by a 0% manipulating variable output. Afterwards, the con-
troller accepts the optimized parameters and controls accurately at the setpoint.

Fig. 79: Fluctuation of the process variable about the switching line
Manufacturers normally assume a process with self-limitation and without dead time elements as
the basis for determining the control parameters. The closer the actual process corresponds to this
model, the more effective is the self-optimization.


7.6 Parameter/structure switching
By switching between parameter sets, it is possible to operate the controller in a control process
which requires different settings as the conditions vary. A set may comprise various parameters,
such as dynamic response, XP bands, cycle time Cy etc. Some controllers even allow switching be-
tween complete controller structures, such as switching from PD to PID structure.
Switching between sets is initiated either via the controller’s logic inputs or via programmed values
which depend on the control deviation or various setpoint ranges (see Fig. 80).

Fig. 80: Schematic diagram of parameter set switching
Parameter set switching of this type finds application, for instance, when there are repeated start-
ups during automatic operation, where the setpoint must be reached in minimum time, and where
the process variable x must not overshoot the setpoint. This is not normally possible, however, with
the parameters used for stable operation. As an example, depending on the control deviation, the
controller switches from P action with a somewhat smaller XP band to PI action before the setpoint
is reached. Another application where switching can help is when running different charges in an
industrial furnace (annealing furnace). The control action of the process changes according to the
loading (half charge, full charge), and the controller must adapt to each individual case. Various pa-
rameter sets can be allocated, based on certain preliminary tests. Moreover, this special function
can be used to advantage where different working points are operated over the full range of the
characteristic, and where a non-linear process is concerned, in which variable margins of excess
power are to be expected.

7.7 Fuzzy logic
Fuzzy logic was developed in the USA by Lotfi A. Zadeh in 1965. This mathematical method is
based on the fuzzy set theory, which represents an extension of Boolean algebra. This technology
has undergone continuous development, and its advantages were quickly recognized in Japan,
where they began to apply the technology to a multitude of control tasks (fuzzy control). This appli-
cation of fuzzy logic to automation and control engineering represents a logical extension of tradi-
tional control technology.


A detailed coverage of this technology is outside the scope of this book; only its application in
combination with a PID controller will be illustrated. It should be mentioned briefly that fuzzy logic
deals with the subjective uncertainty of expressions, such as “temperature too high”, “pressure too
low” or “humidity too low”, as well as decision-making based on such linguistic variables. In this
way, it reconstructs the imprecise concepts of human thought.
The following illustrates how fuzzy logic and a PID control algorithm work together to improve the
disturbance and setpoint responses of a control loop. Fig. 81 shows the simplified block diagram
of a PID controller combined with a fuzzy module. The input variables of the fuzzy controller are the
control deviation and the time derivative of the process variable, as well as information on whether
the controller should operate for setpoint or disturbance response. The output manipulating vari-
able of the fuzzy controller is weighted by a parameter Fc1 and added to the manipulating variable
of the PID controller. In this way, the manipulating variable acting on the process is made up of the
manipulating variable of the PID controller and that of the fuzzy controller. A second output from
the fuzzy controller is a control output which influences the PID control parameters according to
parameter Fc2. Thus, with just two parameters, Fc1 and Fc2, the fuzzy controller can be matched
to the process.

Fig. 81: Simplified block diagram of a PID controller with fuzzy module
The fuzzy controller incorporates linguistic control blocks based on “IF-THEN” rules. These linguis-
tic rules determine the transient response of the fuzzy controller to setpoint changes and distur-
bances.
The combination of a fuzzy controller in parallel with a conventional controller offers several advan-
tages:
- With a non-linear process behavior or on higher-order processes, the fuzzy controller can com-
pensate for specific inaccuracies of the PID controller by supportive intervention under critical
operating conditions.
This is exemplified by systems whose behavior changes over the operating time. The fuzzy control-
ler is less sensitive to process parameter changes than the controller with its fixed parameter set-
tings. This means that variable process characteristics are dealt with more effectively by the com-
bination of fuzzy logic and the PID controller.


- For setpoint response, the integral fuzzy controllers increase the damping factor of the control
loop in certain conditions. This means that any tendencies to oscillate during the run-up to the
setpoint are reduced and overshoots are damped considerably.
- The problem of optimizing the controller specifically for setpoint or disturbance response is
lessened by the fuzzy components, as fuzzy control combines the control theory of both optimi-
zation criteria.
- When disturbances occur, the fuzzy controller responds more dynamically than the PID control-
ler and hence stabilizes the disturbance more rapidly. The criterion of control effectiveness is
improved, i.e. the resulting control error area is smaller than with stabilization by the PID control-
ler alone (see Fig. 82).

Fig. 82: Disturbance response of a third-order process, using a controller with and without
fuzzy module
We can summarize by saying that fuzzy logic, in its application as fuzzy control in automation and
control engineering, is highly regarded today, despite initial problems in accepting this technology.
The fuzzy concept has proved to be a powerful tool, particularly in complex control tasks. Tradi-
tional control technology will certainly not be replaced by fuzzy logic, but technological processes
will emerge that can be controlled more economically and more safely using fuzzy logic. The com-
bination of the fuzzy controller with the PID controller is a sensible example of this.
From this we can record that the fuzzy controller in general, as well as the combination of a tradi-
tional conventional controller and fuzzy module, bring about increased flexibility, so that the con-
troller can be better matched to different processes and control objectives.


8 Standards, symbols, literature references
Standards
Standards and guidelines are very important in engineering, including control engineering. They lay
down concepts and designations, component values, dimensions, as well as important numerical
values and numerical ranges.
From the large number of published standards and guidelines, the following list below has been
limited to those which are useful for the basic principles of control engineering and have been used
in the preparation of this book.

DIN 19 221 1993-05 Messen, Steuern, Regeln; Formelzeichen der
Regelungs- und Steuerungstechnik
(Measurement, operation, control; symbols in control engineering)
DIN 19 225 1981-12 Messen, Steuern, Regeln; Bennenung und Einteilung von Reglern
(Measurement, operation, control; control engineering concepts)
DIN 19 226 (1-5) Regelungs- und Steuerungstechnik;
Begriffe und allgemeine Grundlagen
(Control Engineering; Definitions and Terms)
DIN 19 236 1977-01 Messen, Steuern, Regeln; Optimierung, Begriffe
(Measurement, operation, control; optimization, definitions)

Guideline VDI/VDE 2189 Beschreibung und Untersuchung von
Sheet 1 Zwei- und Mehrpunktreglern ohne Rückführung
(Description and analysis of two-state and multi-state
controllers without feedback)
Guideline VDI/VDE 2189 Beschreibung und Untersuchung von
Sheet 2 Zwei- und Dreipunktreglern mit Rückführung
(Description and analysis of two-state and three-state
controllers with feedback)
Guideline VDI/VDE 2189 Beschreibung und Untersuchung von Dreipunkt-Schrittreglern
Sheet 3 (Description and analysis of modulating controllers)
Guideline VDI/VDE 2189 Beschreibung und Untersuchung digital arbeitender
Sheet 4 Kompaktregler
(Description and analysis of digital compact controllers)
Guideline VDI/VDE 2190 Beschreibung und Untersuchung stetiger Regel-
Sheet 1 geräte Grundlagen
(Description and analysis of continuous control
instrument fundamentals)

Symbols

Cy cycle time, Ton switch-on time,
quasi-continuous controller quasi-continuous controller
e deviation Tosc oscillation time,
discontinuous controller
fsw switching frequency, Ts stabilization time,
discontinuous controller quasi-continuous controller
KI integral coefficient of the controller Tt dead time


8 Standards, symbols, literature references
KIS transfer coefficient of process TI integral time
without self-limitation
KS transfer coefficient (process gain) Ty stroke time
of the process
KP proportionality factor of the controller Vmax maximum rate of rise
R ON-time ratio, w setpoint (SP)
quasi-continuous controller
R(%) ON-time ratio, x process variable, process value (PV)
quasi-continuous controller in%
t time (continuous) XMax maximum process value (process variable)
T time constant of first-order process Xo overshoot
Ta approach time XP proportional band of the controller
TC oscillation time at XPc XPc critical XP , at which the control loop
with P controller oscillates uniformly
Td derivative time of the controller XSd switching differential,
discontinuous controller
Tg response time XSh contact spacing
TMmin minimum pulse duration, y manipulating variable (MV)
modulating and actuating controllers
Tn reset time of the controller yH manipulation range
Toff switch-off time, z disturbance
quasi-continuous controller
Tu delay time

References
- E. SAMAL / W. BECKER ‘Grundriss der Praktischen Regelungstechnik 18’
Basics of practical control engineering 18 (Oldenburg)
- LENZ /OBERST / KOEGST ‘Grundlagen der Steuerungs- und Regelungstechnik’
Fundamentals of control engineering (Hüthig Verlag)
- P. BUSCH ‘Elementare Regelungstechnik’
Elementary control engineering (Vogel Verlag)
- BÖTTLE / BOY / GROTHUSMANN ‘Elektrische Mess- und Regeltechnik’
Electrical measurement and control engineering (Vogel Verlag)
- D. WEBER ‘Regelungstechnik- Wirkunsweise und Einsatz elektronischer Regler’
Control engineering - operation and application of electronic controllers (Expert Verlag)
- D. WEBER ‘Elektronische Regler- Grundlagen, Bauformen und Einstellkriterien’
Electronic controllers - fundamentals, types and adjustment criteria (JUMO)
- JUMO GmbH & Co. KG ‘Betriebsanleitung- Universeller Programmregler
JUMO DICON 401/501’
Operating Instructions - universal profile controller JUMO DICON 401/501


Index
A
actuating controller 98
actuator retransmission signal 98
actuator stroke time 95–96, 98
actuators 24–25
analog and digital controllers 18
approach time 11
auxiliary control loop 104
auxiliary controller 108
auxiliary process variable 107

B
base load 101
base load setting 101
behavior
- dynamic 63
- static 63

C
cascade control 108
characteristic 47
- falling 48
- rising 48
Chien, Hrones and Reswick 69
coarse controller 107
coarse/fine control 107
constant deviation 57
construction of controllers 12
contact spacing 91, 96
continuous controller 12, 45
control deviation 9
control difference 9
control error area 71
- linear 71
control loop 10
- closed 9–10
- multi-loop 101
- single loop 101
control quality 101, 108
control station 113
controller
- continuous 12, 45
- discontinuous 12, 79, 90
- dynamic action 89, 94
- switching 12
controller adjustment
- according to the rate of rise 75
- according to the transfer function 72

123

Index
- by the oscillation method 71
controller gain 71
controller output 10
controller setting 66
controller with dynamic action 85
course control 28
cycle time 87–88

D
D component 60
delay time 41
derivative time 57
digital and analog controllers 18
discontinuous controller 12
disturbance 10, 104
disturbance response 63, 65
drive control 27
duty-cycle 86
dynamic action 52
dynamic characteristic 31
dynamic response 68

F
fine controller 107
fluctuation band 82, 84
fuzzy controller 119
fuzzy logic 118
fuzzy module 119

G
gain 45

H
hysteresis 80, 91

I
I controller 53, 62
I processes 33
inflection tangent 41, 74

M
manipulating devices 23
manipulating variable 10, 45

124

Index
manipulation range 10
manual mode 113
master controller 108
modulating controller 95
multi-state controller 79

O
ON-time ratio 86, 88
ON-time, relative 86
operation 26
oscillation method 71
overshoot 11, 67

P
PD controller 57
period of oscillation 82, 84
PI controller 54, 60
PID action 62
PID controller 61–62
point of inflection 74
position control 28
power control 87
power switching 103
process gain 30
process step response 72
process transfer coefficient 73
process value (process variable) 9
process with self-limitation 31
processes 43
- first-order 39
- higher-order 41
- pure dead time 35
- second-order 40
- stable 54
- unstable 50
- with and without dead time 35
- with and without self-limitation 34
- with delay 38
- with one delay 38
- with two delays 39
- without self-limitation 33, 85
proportional band 47
proportionality factor 46, 48
PT1 process 39
PT2 process 40

125

Index
Q
quasi-continuous controllers 13

R
ramp function 114
rate of change 60
rate of rise Vmax 75
rate of rise vmax 42
ratio control 110
reset time 55
response time 41–42

S
self-optimization 116
setpoint 64
setpoint (desired value) 9
setpoint response 63, 65
signal types 18
signals 18
- 3-state 19
- analog 19
- binary 19
- digital 15
slave controller 110
speed control 27
stabilization time 11
start-up value 43
static characteristic 30
static process characteristic 50
step response 16, 31–32, 41
switched disturbance correction 104
- additive 105
- multiplicative 105
switching differential 81–82, 84, 91
switching frequency 82, 86

T
three-state controller 12, 25, 79, 91
tolerance limits 11
transfer coefficient 30, 33, 43
transfer function 38, 72
Tt processes 35

126

Index
W
working point 43, 49

X
Xp band 47
- asymmetrical 49
- symmetrical 48

Z
Ziegler and Nichols 69

127

measurement • control • recording

JUMO GmbH & Co. KG · 36039 Fulda, Germany · Phone +49 661 6003-396
Fax +49 661 6003-500 · e-mail: manfred.schleicher@jumo.net · Internet: www.jumo.net

02.04/00323761 ISBN 3-935742-01-0

Control engineering a guide for beginners

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