# ELEMENTS OF CONTROL SYSTEMS

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15 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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1

ELEMENTS OF CONTROL
SYSTEMS

International Centre for Water and Energy Systems, PO Box 2623, Abu Dhabi.
UAE.

Keywords:
Systems, Block diagram, Characteristic equation, Characteristic polynomial, Controller,
constitutive relations, Discr
ete time systems, Effort variable, Feedback, Flow variable

Forced Response, Free response, Frequency response, Interconnective constraints, Laplace transform,
Open loop control, Plant, Pole, Sampled data, Signal flow graph, Similarity transformation, SISO,

State
space, State vector, Time invariant systems, Time response, Time
-
varying systems, Transfer function,
Z
-
Transform, Zero

Contents

1. Introduction

2. Sys
tem Modeling

3. Mathematical Models of Dynamical Systems

3.1. Differential Equation Models for Lumped Parameter Systems in Continuous Time Domain

3.2. State Space Description of Lumped Parameter Systems

3.3. Linear Time
-
Invariant Systems

3.4. Discrete
-
Time Systems or Sampled Data Systems

3
.5. Block Diagram Representation and Simplification of Systems:

3.6. Distributed Parameter Systems:

3.7. Deterministic and Stochastic Systems

3.8
. Nonlinear Models and Linearization

3.9. Causal and Non
-
Causal Systems

3.10. Stable and Unstable Systems

3.11. Single
-
Input
-
Single
-
Output (SISO)

and Multiple
-
Input
-
Multiple
-
Output (MIMO) Systems

4. Systems Control

4.1. Open Loop Control

4.2. Feedback Control

4.3. Closed
-
Loop Behavior of Control Systems

4.4. Control Strategies

Glossary

System
: A system is a set of components, physical or otherwise, which are connected in such a manner
as to form and act as an enti
re unit.

Block diagram
: A graphic representation of a system showing the individual elements/subsystems and
their interconnections. Based on certain conventions, block diagrams can be manipulated and
simplified for ease of analysis.

Canonical form
: A cano
nical form is a compact form of the mathematical model that involves
minimal number of parameters.

Characteristic equation
: An algebraic equation that portrays the inherent nature of a linear time
-
invariant dynamical system such as stability. In a rational

transfer function this equation is obtained by
equating the denominator to zero.

Characteristic polynomial
: The denominator of a rational transfer function.

Compensator
: The controller in a control system is sometimes referred to as a compensator.

2

Consti
tutive relations
: The descriptions of the basic physical phenomena and properties of physical
elements. They are also known as material relations.

Continuous time systems
: Systems described in the continuous time domain.

Control signal
: The signal that is
applied to a controlled plant in order to make it respond in a certain
desired way.

Controller
: The device or unit that generates the control signal by considering the error in a control
signal in a control system. A computer may act as a controller in a c
ontrol system.

Discrete time systems
: Sampled data systems or systems described in the discrete time domain.

Effort variable
: A variable in a system whose product with the so
-
called flow variable has the sense of
power (rate of energy). It is also known as

‘across variable’.

Eigenvalue
: The eigenvalue of a matrix
A

is the root of the characteristic equation:
s
I
-
A=0

Feedback
: Feedback is an arrangement by which the actual output of a system is fed back to the input
end for comparison with the desired output.

Flow variable
: A variable in a system whose product with the so
-
called effort variable has the sense of
power (rate of energy). It is also known as ‘through variable’.

Forced Response
: The response of a system due only to the input from outside in the abs
ence of initial
conditions.

Free response
: The response of a system due only to the initial conditions and no other input from
outside.

Frequency response
-
state response of a system to sinusoidal signals of unity amplitude
and variable frequenc
y. This function in the frequency domain is obtained by setting
s = j

in the
system transfer function.

Interconnective constraints
: Conditions arising out of the connections among the elements within a
system that constrain the definition of variables in
a system. They are based on Kirchhoff’s laws in a
generalized setting.

Laplace transform:
A mathematical transformation that converts the calculus of time invariant linear
differential equations into an algebra thereby lending simplicity to the analysis a
nd design of control
systems.

MIMO
: Multiple
-
input
-
multiple
-
output

Open loop control
: Control without feedback

Plant
: The object that is to be controlled.

Pole
: The point in the s
-
plane where the system transfer function attains an infinite value. It is
also a
root of the characteristic equation of the system.

Sampled data
: Signals and information available only at certain sampling instants.

Signal flow graph:
A graphical representation of the interconnections of the subsystems in a system in
which nodes

denote signals and branches represent subsystems.

Similarity transformation
: A transformation in state space that changes the state variable coordinate
system without altering the system properties. The eigenvalues of a matrix remain unaltered under
simil
arity transformation.

SISO
: Single
-
input
-
single
-
output.

State space
: The higher dimensional space in which the dynamics of a system is studied in terms of the
trajectory of the state vector.

State vector
: Vector whose elements are the state variables of a

dynamical system.

Time invariant systems
: Dynamical systems whose properties are time invariant. The parameters of
the model of a time
-
invariant system are constants.

Time response
: The time history of the output of a system.

Time
-
varying systems
: Dynamic
al systems whose properties change in time. The parameters of the
model of a time
-
varying system are independent functions of time.

3

Transfer function
: A mathematical function that characterizes the transfer behavior of a system. It is
the ratio of the Lapl
ace transform of the output in the absence of initial conditions, to the Laplace
transform of the input.

Z
-
Transform:
A mathematical transformation that converts the calculus of time invariant discrete time
dynamical systems into an algebra thereby lending

simplicity to the analysis and design of digital
control systems. The relation
sT
e
z

, with
T

as the sampling period, connects the Laplace and z
-

transforms.

Zero
: The point in the s
-
plane where the system transfer function attains
a zero value.

Summary

This paper presents a perspective of the elements of control systems. Human engineered control
systems form part of automation that is characteristic of our society particularly in the present times.

of certain individual elements assembled and connected in specific
ways to perform functions for which they are intended. Systems are controlled to meet specified needs
and control techniques enhance their performance as control systems. We understand sys
tems for their
behavior by modeling, simulation and analysis. Mathematical models of dynamical systems can be
obtained either in time domain or in frequency domain. A particular model for a system can be
obtained in a chosen form by determining the numeric
al values of the parameters associated with the
model based on input
-
output data. This process is known as system identification. Feedback control
can be designed for a system with a known model with reference to certain performance criteria such as
stabil
-
state accuracy, optimality, disturbance rejection, etc. Controller action can be realized
in a computer that works with sampled signals. In the presence of uncertainties and unknown
disturbances, stochastic estimation and control techniques are

to be applied. When the plant
characteristics vary during the period of operation adaptive control techniques may be used to render
the controller adaptive to the changing conditions. Supported by powerful computational facilities in
the control environme
nt features such as learning and decision making can be incorporated to render
control as intelligent and control systems can be made fully automatic and autonomous. The history of
control dates back to the ancient times but the beginning of an era of theo
ry and practice of automatic
control was made in the 18
th

century following the inception of the governor. Major developments took
place in the 20
th

century.

1. Introduction

Systems are sets of components, physical or otherwise, which are connected in

such a manner as to
form and act as entire units. Control is the effort to make systems act as desired. A process is the action
of a system or alternatively, a system in action.

Humans have created control systems as technical innovations to enhance the

quality and comfort of
their lives. Human engineered control systems are part of automation, which is a feature of our modern
life. They are applied in several aspects of our daily life
-

in heating and air conditioning to control our
living environment an
d in many of our household appliances. They significantly relieve us from the
burden of operation of complex systems and processes and enable us to achieve control with desired
precision. Control systems enable accurate positioning and control of machine t
ools in metal cutting
operations and automate manufacturing processes. They automatically guide and control space
vehicles, aircraft, large sea going vessels, and high
-
speed ground transportation systems. Modern
automation of a plant involves components su
ch as sensors, instruments, computers and application of
techniques of data processing and control. The principles and techniques of automatic control may be
applied in a wide variety of systems in order to enhance the quality of their performance.

4

Contro
l systems are not human inventions; they have naturally evolved in the earth’s living system.
The action of automatic control regulates the conditions necessary for life in almost all living things.
They possess sensing and controlling systems and counter
disturbances. An automatic temperature
control system, for example, makes it possible to maintain the temperature of the human body constant
at the right value despite varying ambient conditions. The human body is a very sophisticated
biochemical processin
g plant in which the consumed food is processed and glands automatically
release the required quantities of chemical substances as and when necessary in the process. The
stability of the human body and its ability to move as desired are due to some very e
ffective motion
control systems. A bird in flight, a fish swimming in water or an animal on the run
-

all are under the
influence of some very efficient control systems that have evolved in them.

The field of automatic control is very well developed. The
established techniques in this field can be
applied to the control of a wide range of systems
-

engineering systems such as machines and complex
plants, natural systems such as biological and ecological systems, and non
-
physical systems such as
economic an
d sociological systems following the understanding of the similarity of the underlying
problems.

Understanding a system for its properties is prerequisite to the creation of a control system for it.
Before attempting to control a system, it is essential
to know how it generally behaves and responds to
external stimuli. Such an understanding is possible with the help of a model. The process of developing
a model is known as modeling.

Physical systems are modeled by applying the phenomenological laws that
govern their behavior. For
example, mechanical systems are described by Newton’s laws and electrical systems by Ohm’s,
Faraday’s and Lenz’s laws. These laws form the basis for the
constitutive properties

of the elements in
a system.

2. System Modeling

P
hysical systems may be regarded as energy manipulating units and modeling them is based on the
distribution and transfer of energy taking place within them. Energy from certain sources enters a
system schematically as shown in Figure 1 and is manipulated w
ithin the system by the various
components and subsystems in accordance with their inherent properties and depending on the manner
in which they are connected inside the system. Energy manipulation phenomena are studied in terms of
a pair of variables whos
e product has the sense of power and thereby the meaning of energy. Some
elements store energy and some convert it onto another form. When an element converts energy into
heat, it is termed as a dissipator. The assignment of the term ‘dissipator’ to such
elements seems to be
prejudiced by their association with heat, a form of energy that is degenerate and vulnerable to loss or
dissipation, although the generated heat may indeed be intended for use, say for heating.

Fig.1: Physical system as an energy ma
nipulator

The energy manipulations in system elements are studied in terms of ‘
effort variables
’ and ‘
flow
variables
’ whose product corresponds to the ‘rate of energy’ or ‘power’ as indicated in Figure 2 in
general. For instance, in an electrical system s
hown in Figure 3, voltage is regarded as an effort
variable and current as the flow variable. Because of the manner in which the effort and flow variables
occur , for instance, as voltage across an element and current through it, they are also termed as

a
cross
’ and ‘
through
’ variables respectively.

5

The elements within a given system may have the property to store or dissipate energy. Energy stores
are classified as
effort stores

and
flow stores
. For example, in electrical systems, inductors accumulate
th
e effort variable (voltage) and capacitors accumulate the flow variable (electric current). Resistors
convert electrical energy into heat and are termed as dissipators.

It is the presence of stores that renders a system ‘dynamic’. Figures 4 and 5 show th
e representations in
fluid and mechanical systems respectively.

Fig.2: Effort and flow variables

Fig.3: A simple electrical system

Fig 4. : A simple fluid system

Fig.5: A simple mechanical system

Mathematical modeling of a system is the process of
obtaining a mathematical description that
adequately describes the aspects of its behavior, which are of interest in the context of a study.
Modeling is by itself a well
-
developed field and there are some general approaches that are applicable
to a wide va
riety of systems. The following are some important approaches to physical system
modeling:

Network methods

Variational methods

Bond graph methods

The

network methods

of system modeling are based on generalization of the methods of electrical
network theo
ry. First, all the elements in the system are described (modeled) by their
constitutive

properties

in terms of storage, dissipation, and conversion by applying the physical laws governing
their behavior. Next, generalized Kirchhoff’s laws are applied to ta
ke into account the connections
among the elements in the system. These give rise to the so
-
called
continuity

and
compatibility

conditions, which constrain the effort and flow variables in accordance with the system configuration.
As a result of these cons
traints, the effort and flow variables of the individual elements in a system
cannot all be assigned independent labels. The variables are bound by the structural configuration of
the system or in other words, the manner in which the individual elements ar
e connected in the system.
Figure 6 shows how the effort variables in a closed loop are constrained, and Figure 7 shows how the
flow variables are constrained. The effort variables in the system of Figure 6 representing a loop are
such that their algebraic

sum is zero. Likewise, the algebraic sum of the flow variables at a junction is
zero. This condition is termed the continuity constraint because this implies continuity, that is, the
inflows and the outflows must be equal at a junction.

Fig.6: Compatibil
ity constraint on effort variables

Fig.7: Continuity constraint on flow variables

6

Graph theoretic methods may be applied as general tools to apply the
interconnectivity constraints
.
These constraints will eliminate the redundancy in the labels chosen to

describe the variables. For
example, in the loop of Figure 6, only one flow variable is to be defined and it applies to all the
components by virtue of the series connection. Furthermore, it is enough if all but one of the effort
variables in the loop are

labeled. The unlabeled variable is naturally determined by the negative sum of
these
n
-
1

variables. Thus application of the interconnectivity constraints brings down the multitude of
the system variables to the appropriate number and mutual relationships.

The resulting equations are
then arranged in the desired form to represent the system model.

The variational methods

of Lagrange and Hamilton avoid explicit formulation of both sets of
interconnectivity constraints. Only one set needs to be directly know
n and the other is complementary
and implicit in these methods. Complex couplings of different energy handling media are particularly
susceptible to the variational approach. In this approach infinitesimal alterations in certain key system
effort or flow a
ccumulation variables, without transgressing the related compatibility or continuity
constraints, are considered as admissible variations. A scalar function known as the variational
indicator has to be zero in a natural configuration. In this approach, var
iational calculus, Hamilton’s
principle and Lagrange’s equation are applied. Lagrange’s equations, which are in terms of certain
energy functions, directly give rise to the differential equations governing the system. This approach is
applicable to composi
te systems containing elements and subsystems belonging to different worlds
-

electrical, mechanical, etc.

Bond graph methods

represent the energetic interactions between systems and their components by
single lines termed as energy bonds. Bond graph repr
esentation is alternative to the network convention
and it is more compact and orderly than the equivalent system graph. It also allows multiport elements
to be modeled explicitly and neatly.

Physical system modeling on the basis of the above approaches c
an be computer aided and software
packages are available for this purpose.

(see Mathematical Models, Physical Laws, Electrical Networks, Graph Theory, Variational methods,
Bond graphs)

3. Mathematical Models of Dynamical Systems

Mathematical models may be

in the form of differential, algebraic or logical equations depending on
the nature of the system
(see
General Models of Dynamic Systems
)
. They are useful in providing an
understanding of the input
-
output behavior and stability studies. They are helpful i
n the analysis or
synthesis of control systems as well as in the simulation studies with the help of analog, digital or
hybrid computers. The mathematical equations are ‘solved’ in devices, computational or otherwise to
display the system behavior. Through

simulation we gain an understanding of the performance of a
system under different situations, without the need to run the actual system.

(see Modeling and Simulation, Computational Methods)

3.1. Differential Equation Models for Lumped Parameter Systems
in Continuous Time Domain

Different classes of differential equations describe different types of dynamical systems. Lumped
parameter systems are described by ordinary differential equations. Lumped linear continuous
-
time
systems are described by linear
differential equations. For instance, the n
-
th order linear differential
equation with the single
input

x

and single
output

y of the general form:

7

m
i
i
i
i
n
i
i
i
i
t
x
dt
d
t
b
t
y
dt
d
t
a
0
0

(1)

satisfies the principle of superposition by virtue of its linear property.
If the coefficients in the above
are constant as in Eq. (2), it represents a linear time invariant system of the form:

m
i
i
i
i
n
i
i
i
i
t
x
dt
d
b
t
y
dt
d
a
0
0

(2)

Working with linear time invariant systems becomes simplified with the help of Laplace transforms.
The

Lapalce transform of a function
]
,
0
[
),
(

t
t
f

is defined as

dt
e
t
f
s
F
st

0
).
(
)
(

(3)

This transformation converts a linear differential equation into an algebraic form in the domain
s

that
represents the complex frequency. Let
1

n
a

without loss of generality. Let the constant initial
conditions be defined as

1
,...,
1
,
0
,
;
1
,...,
1
,
0
,
0
0
0
0

m
k
x
dt
x
d
n
k
y
dt
y
d
k
t
k
k
k
t
k
k

(4)

Then the Laplace transform of Eq. (2) is given by

)]
)
(
(
[
)]
)
(
(
[
0
1
0
1
0
0
1
0
1
0
k
i
k
k
i
i
m
i
i
k
i
k
k
i
i
n
i
i
x
s
s
X
s
b
y
s
s
Y
s
a

(5)

and the transform of the output is

i
n
i
i
m
i
i
k
k
k
i
i
i
n
i
i
n
i
i
k
k
k
i
i
n
i
i
i
m
i
i
i
s
a
x
s
b
s
a
y
s
a
s
X
s
a
s
b
s
Y

0
0
1
0
0
1
0
0
1
0
0
1
0
0
)
(
)
(

(6)

The denominator term

n
i
i
i
s
a
0
is called the
characteristic polynomial
.

The response
Y(s)

consists of two components. The first term is due to the input and therefore it is
referred to as the
forced res
ponse

or the
zero
-
state response
. The set of initial conditions (4) represents
the initial
state

of the system. The coefficient of
X(s)

in the first term of the output expression is
referred as the
transfer function

and it has to be obtained in the absence

of initial conditions as the ratio
of the Laplace transforms of the output to the input. The expression for output in Eq. (6) also contains
another term that depends on the initial conditions only and not on the input. This component of the
system output
is known as the
free response

or the
zero
-
input response
.

8

If a unit impulse function or the Dirac delta function denoted as is considered as the input x(t) =

(t)
,
X(s)

=1, the forced response component in Eq. (6) happens to be equal to the transfer funct
ion itself.
Thus, the transfer function may also be regarded as the (unit) impulse response and in time domain the
unit impulse response is given by the inverse Laplace transform of the system transfer function. The
impulse function is not an ordinary func
tion of time. That is, the value of this function is not
definitively defined at a given time. A unit impulse function

(t)
is indirectly defined by the following
properties:

0
1
)
(
dt
t

(7)

and for any function
]
,
0
[
),
(

t
t
f

continuous at

, defined in the ordinary sense

)
(
)
(
)
(
0

f
dt
t
f
t

(8)

The inverse Laplace transform of Eq. (6) for
y(t)

is obtained be the method of partial fractions as
follows.

Suppose that the characteristic polynomial has
n
1

roots each
equal to

p
1
, n
2

roots each equal to

p
2
,….,n
r

roots each equal to
-
p
r

such that

n
n
r
i
i

1

.
Then

i
n
r
i
i
i
n
i
i
p
s
s
a
)
(
1
0

and the function
Y(s)/X(s)

can be written as

r
i
n
k
k
i
ik
n
i
p
s
c
b
s
X
s
Y
s
F
1
1
)
(
)
(
)
(
)
(

(9)

where
b
n
=
0 unless
m=n
. The coef
ficients are given by

i
i
i
i
p
s
n
i
k
n
k
n
i
ik
s
F
p
s
d
d
k
n
c

)]
(
)
[(
)!
(
1

(10)

These coefficients are also known as the residues of
F(s)

at

p
i

,
i = 1, 2, …r
. Inverse Laplace
transformation of (9) gives:

t
p
k
r
i
n
k
ik
n
i
i
e
t
k
c
t
b
t
f

1
1
1
)!
1
(
)
(
)
(

(11)

where

(t)

is the unit impulse funct
ion, and
b
n

= 0

unless
m=n
.

If a system does not contain dead time elements (delay elements) the transfer function
F(s)

is rational,
that is, a ratio of two polynomials. The roots of the numerator polynomial are referred to as the
zeros

and the roots of
the characteristic polynomial, or that of the denominator are termed the
poles

of the
transfer function. These terms are suggestive of the nature of the function
F(s)

with reference to the
complex frequency variable s. If
F(s)

is viewed as a potential func
tion on the
s
-
plane, the value of the

9

function is zero at the
zeros
. At the points representing the zeros of the characteristic polynomial the
value soars to infinity making the profile of the potential function
F(s)

at these points in the s
-
plane
appear a
s poles. For this reason, the roots of the characteristic polynomial are called the poles. In the s
-
plane, a pole is shown as
x

and a zero as
o
.

The response as

t
, is called the
. The system response as a fu
nction of time
before it reaches the steady state is called the
transient response
. The steady state value can be
determined by applying the final value theorem:
)
(
lim
)
(
lim
)
(
0
s
sF
t
f
f
s
t

, if the limit exists.

Notice that nature of
f(t)

depends on the va
lues of
p
i
, the poles of the transfer function. When a pole is
real the response component due to it is purely exponential. If it is negative, the response decays
asymptotically in time and when it is positive, the response grows. Complex poles appear as c
onjugate
pairs and the response due to such a pair is sinusoidal in nature. If the pair has a negative real part, the
oscillations decay in time and when they have a positive real part, the oscillations grow in amplitude
without limit. Referring to the com
plex s
-
plane, these conditions are interpreted as conditions for
stability for linear time invariant dynamical systems in the following manner. If all the poles of the
transfer function lie inside the left half of the
s
-
plane, the system is asymptotically
stable. If any pole
lies on the imaginary axis, the system is critically stable and if any pole lies in the right half of the
s
-
plane, then the system is unstable. These criteria are illustrated in Figure 8. Routh
-
Hurwitz stability
criteria are used to det
ect the location of the roots, without actually solving the characteristic equation
for its roots. Nyquist criterion ascertains the stability of a closed loop system by examining the transfer
function of the open loop system. A more detailed discussion on
the stability theory of dynamical
systems is given elsewhere
(see
Stability Concepts
)
.

Fig 8. Stability criteria for linear time invariant dynamic systems

The transfer function
F(
j

)

evaluated along the
j

-
axis of the
s
-
plane is of significance as it re
presents
the steady state response of the system to a sinusoidal input of frequency

. It is a complex number
with magnitude representing the amplification/attenuation and a phase angle that is the phase shift
between the input and output signals.

3.2. S
tate Space Description of Lumped Parameter Systems

Differential equations describing a linear time varying systems may be organized in the form of a set of
first order differential equations and written in the form:

u
D
x
C
y
u
B
x
A
x
)
(
)
(
)
(
)
(
t
t
t
t

(12)

whe
re
x

is an
n
-
vector (i.e.,
nx1

matrix) containing the state variables,
u

is an
r
-
vector of inputs and
y

is
a
p
-
vector of outputs.
A
,
B
,
C
, and
D

are respectively
n
x
n, n
x
r, p
x
n,

and
p
x
r

matrices. Often
D

happens
to be a matrix with zeros as its elements so
it is not always shown in the above description. The first
equation is called the state equation and the second is termed as the output equation. If the original
differential equation has constant coefficients, then all these matrices are also constant. Th
is is known
as the state space description. Techniques of handling linear systems in state space are well established
(see
Description and Classification
).

3.3. Linear Time
-
Invariant Systems

10

Laplace transformation applied to the state variable model of

a general linear time invariant system
with lumped parameters in the general state variable form

Du
Cx
y
Bu
Ax
x

(13)

gives

The Laplace transform of the vector of outputs is given by

)
D]U(
B
A)
I
[C(
)
Y(
1
s
s
s

(14)

If we let
D=0

wh
ich is the common case,

)
(
)
(
det
)
(
cof
)
(
s
s
s
s
U
A
I
B
A
I
C
Y

(15)

The denominator term on the right hand side of the above is the characteristic polynomial. The
eigenvalues of the system matrix
A

are the poles. The coefficient of the term
U
(s) is the transfer
fu
nction matrix whose
(i,j)
-
th element happens to be transfer between the
i
-
th input and the
j
-
th output
in the multi
-
input
-
multi
-
output system described by Eq. (15).

Solution of the state equation in time domain is direct by analogy with the first order s
calar equation
dx/dt=ax+bu
.

d
e
e
t
t
t
t
)
(
)
0
(
)
(
0
)
(
Bu
x
x
A
A

(16)

The state variable representation is not unique; it depends on the choice of the set of state variables,
which correspond, to the coordinate system in the n
-
dimensional space, which is referred

to as the state
space. Similarity transformation brings about a change in the state variable description without actually
influencing the properties of the system. Certain state variable representations are termed as
canonical

because they involve minimal

number of system parameters. State variable representation permits
examination of additional properties such as
controllability

and
observability

of a system
(see
System

Characteristics
)
. Figure 9 illustrates these properties.

Fig.9: System controllabi
lity and observability with respect to segregated subsystems

Clearly, the controllable and observable part of the system is reflected in the input
-
output behavior and
the transfer function of the overall system is given by the controllable and observable
part only.

The state variable representation is a more complete description than the transfer function
representation. It presents system behavior both internal and external while the transfer function gives
)
(
)
(
)
(
)
(
)
(
)
0
(
)
(
s
s
s
s
s
s
s
DU
CX
Y
BU
AX
x
X

11

the external (input
-
output) behavior only. The
state space description is very appropriate for finite
dimensional systems, that is, systems described by ordinary differential equations of a finite degree. If
a system has time delays, the resulting delay differential equations cannot be represented easi
ly in state
space form. However, if a state space representation is desired, the delay terms have to be represented,
in some sense of approximation, as finite dimensional elements. Thus the presence of delay terms in a
differential equation gives rise to a
n arbitrary enlargement of the dimension of the state space.

3.4. Discrete
-
Time Systems or Sampled Data Systems

If a system variable (signal)
y
, at any arbitrary instant of time can be varied within known limits
continuously, it is called “continuous”. I
f a signal can take only known discrete amplitude values, then
it is called a “quantized signal”. If a signal is known only at certain discrete instants of time, then it is
known as a discrete
-
time (or discrete) signal. If the signal values are given at un
iformly sampled
instants of time separated by an interval
T
,
T

is referred to as the sampling period. The signal itself is
referred to as ‘sampled’. Systems, in which such signals occur, are called discrete
-
time systems, or
discrete systems or sampled
-
data

systems. In general, if digital computers are employed in control
systems, for instance to act as controllers, only quantized discrete
-
time data is processed. Linear time
invariant discrete time systems are described by difference equations.

,.....
2
,
1
,
0
);
(
....
)
(
)
(
)
(
)
(
)
2
(
....
)
(
)
(
1
0
1
2
1

k
mT
kT
u
b
T
kT
u
b
kT
u
b
nT
kT
y
a
T
nT
kT
y
a
T
nT
kT
y
a
T
kT
y
a
kT
y
m
n
n
n

(17)

They can be studied by applying the methods of
z
-
transform, which for a discrete time signal
f(nT),
n=0,1,2, …
, where
T

is the sampling interval, is defined as

0
)
(
)
(
n
n
z
nT
f
z
F

(18)

z
-
transformation of Eq. (17) gives

n
n
m
m
z
a
z
a
z
a
z
b
z
b
z
b
b
z
U
z
Y

....
1
....
)
(
)
(
2
2
1
1
2
2
1
1
0

(19)

Discrete time systems are described in state space as

)
(
)
(
)
(
)
(
)
(
)
1
(
k
k
k
k
k
k
Du
Cx
y
Bu
Ax
x

(20)

Stability conditions for discrete time systems are discussed with reference to the
z
-
plane. A linear time
invariant discrete
-
time
system described as above is stable if the eigenvalues of the A matrix ( poles of
the transfer function or roots of the characteristic polynomial) lie within the unit circle centered at the
origin of the
z
-
plane.

Strips parallel and symmetric to the real

axis in the
s
-
plane will have to be recognized due to sampling
and the width of these strips is proportional to the sampling frequency
1/T
. The first of these is the
primary strip that is of significance.
The
z
-
transform is related to the Laplace transfo
rm through the

12

relation
sT
e
z

. According to this relation, the entire left half of the
s
-
plane is transformed into the
inside of the unit circle and the right half into the region outside the unit circle in the
z
-
plane. The
j

axis wi
nds itself into the unit circle itself with its various segments in the horizontal strips coinciding
with the same circle periodically.

If continuous time systems are to be discretized, the minimum sampling frequency that is necessary to
preserve the inf
ormation in the sampled signal and to avoid aliasing effects is twice the highest
frequency occurring in the signal spectrum. This criterion is called Shannon’s sampling criterion.
However, one rule of thumb in practice is to select
T

such that

m

T

0.5,
where

m
is the magnitude of
the largest eigenvalue of the system. In actual practice it is desirable to make the sampling interval
much smaller than the value specified by this rule. The result of such a choice is to force all poles to lie
in a small lens

shaped region in the
z
-
plane as shown in Figure 10.

Fig.10: The region of normal operation in the z
-
plane

3.5. Block Diagram Representation and Simplification of Systems:

Systems are denoted as transfer elements or blocks. Transfer elements possess a

unique direction of
action indicated by arrows; their action is not reversible. Every controllable and observable transfer
element has at least one input and at least one output. The output of a transfer element depends only on
its own input but not on th
described by block diagrams with appropriate connections among the blocks. Within each block, the
mathematical description of the transfer element is written; when it is too large and co
mplex to be
accommodated, a symbol denoting the transfer relation is shown. In the case of static nonlinear
elements, the description in the block symbol is either in the form of the functional description of the
nonlinear characteristic or the graph of th
e nonlinear function.

Linear time invariant system models transformed into the
s
-

and
z
-
domains attain simple algebraic
properties as has been already observed above enabling system models to be manipulated algebraically.
For example in the s
-
domain, if
a system
G(s)

is excited by an input signal U(s), the response is given
by
G(s).U(s)
. If two systems
G
1
(s)

and
G
2
(s)

are in cascade, the transfer function of the overall system
is given by their product
G
1
(s)
.
G
2
(s)

as shown in Figure 11.

Fig. 11: Simplif

A system with feedback can be simplified as shown in Figure 12.

Fig. 12: Simplification of systems in a feedback loop

Signal flow diagrams are close in spirit to the block diagrams. In a signal flow graph the signals are
de
noted by nodes and the transfer relations by branches between nodes. A block diagram and its signal
flow graph are shown in Figure 13. In manipulating and simplifying a signal flow graph, Mason’s rule
offers a general procedure:

The overall transfer funct
ion G of a system represented as a signal flow graph is given by

k
k
k
T
G

13

where

k

= the number of forward (from the input end of the graph towards the output end) paths

T
k

= the transfer function of the forward path given by the product of the transfer functi
ons of the

= 1

sum of loop transfer functions + sum of non
-
touching loop transfer functions taken two at a
time

sum of loop transfer functions taken three at a time + ……..

k

=

-

sum of the loop transfer functions touching the k
-
th forward path.

Figure 13: The block diagram and signal flow diagram for a system

Referring to the signal flow graph shown in Figure 13(b) we identify the following:

Transfer function of the forward path =
G
1
G
2
G
3
G
4
G
5

Loop transfer functions:
G
2
H
1
, G
4
H
2
, G
7
H
4
, G
2
G
3
G
4
G
5
G
6
G
7
G
8

Non
-
touching loops taken two at a time:
G
2
H
1
G
4
H
2
, G
2
H
1
G
7
H
4
, G
4
H
2
G
7
H
4

Non
-
touching loops taken three at a time:
G
2
H
1
G
4
H
2
G
7
H
4

=
1

[G
2
H
1

+ G
4
H
2
+ G
7
H
4

+ G
2
G
3
G
4
G
5
G
6
G
7
G
8
] + [G
2
H
1
G
4
H
2

+ G
2
H
1

G
7
H
4
+ G
4
H
2

G
7
H
4
]

[G
2
H
1
G
4
H
2

G
7
H
4
]

1

=
1

G
7
H
4

G = T
k

1
/

3.6. Distributed Parameter Systems:

One can think of systems assembled from several ideal elements such as resistors, capacitors,
inductors, masses, springs, dampers etc. Such systems are called lumped parameter systems. These are
de
scribed by ordinary differential equations. If a system possesses an infinite number of such
infinitesimally small elements that are smoothly distributed, then it becomes a distributed parameter
system (DPS) and such systems are described by partial differ
ential equations. A typical example of
such a system is an electric transmission line. The voltage on such a line is a function of both distance
and time and hence is describable only by a partial differential equation. Distributed parameter systems
are of
ten studied by means of lumped approximations. For example, transmission lines are studied with
the help of the so called T and

approximations.

(see Partial differential equations)

3.7. Deterministic and Stochastic Systems

Uncertainty and disturbances

are usual in real systems. In the deterministic case, the signals and the
mathematical model of a system are known without uncertainty and the time behavior can be
reproduced by repeated experimentation. In the stochastic case this not possible due to unc
ertainty that
exists either in its model parameters or in its signals or in both. The values of the signals or the
variables occurring in the system can only be estimated with the help of the methods of probability and
statistics. The results are presente
d as expected values together with the bounds of error
(see
Probability and Statistics)
.

14

3.8. Nonlinear Models and Linearization

Most natural systems are nonlinear. An important criterion that distinguished nonlinear systems from
linear systems is the pr
inciple of superposition. If this principle holds good as it happens in linear
systems, the sum of all the individual outputs due to several individual inputs, each considered to be
acting alone on the system is equal to the output due to all the inputs ac
ting simultaneously on a
system. Nonlinear systems do not obey the principle of superposition. The response of a linear system
due to a sinusoidal input signal remains sinusoidal with an amplitude modification and phase shift,
whereas a non
-
linear system p
roduces distortion that gives rise to harmonic components of the input
signal frequency in its output. In stochastic situations, the output of a linear system due to Gaussian
random inputs preserves the Gaussian property and in nonlinear systems this is
not the case. Linear
systems are studied by a fairly general framework of techniques. Although there exist some methods to
study nonlinear systems, there is no general methodology, which is universally applicable to nonlinear
systems. For this reason nonli
near systems are in general quite complex.

Nonlinear differential equations are not easy to handle. System operation about an operating point is
often very relevant in practice and an understanding of such a situation is provided by studying the
model of

the system obtained by linearizing it about the chosen operating point. The resulting linear
model is studied with help of the well
-
established techniques for linear systems by considering
deviations about the point and the related signals as sufficiently

small.

Consider the problem of linearizing a function
f(x)

x
o
. Expanding the function in Taylor

f(x)
-

f(
x
o
)

= (df/dx)

(x
-

x
o
)

x =
x
o

which is a linear relationship in the form:

f(x) = m
o

x

The

procedure for linearization for a general nonlinear equation in state space is as follows:

,
,
t
t
t
u
x
f
x

(21)

where

x

(t) = [x
1
(t) ……..x
n
(t)]
T

u

(t) = [u
1
(t) ……..u
r
(t)]
T

and

f
(
x
,
u
) is a vector nonlinear function.

Linearization of this
leads to the linear vector differential equation

)
(
*
)
(
*
)
(
*
t
t
t
Bu
Ax
x

(22)

Where
A

and
B

are Jacobians containing the various partial derivative terms as follows:

15

A

=

u
u
x
x
u
x
u
x
u
x
u
x
n
n
n
n
x
f
x
f
x
f
x
f
,
,
,
,
1
1
1
1

B

=

u
u
x
x
u
x
u
x
u
x
u
x
r
n
n
r
u
f
u
f
u
f
u
f
,
,
,
,
1
1
1
1

(23)

If the system parameters vary with time, it is called a time
-
varying (often referred to as time
-
variable or
non stationary) system. For instance, a rocket represents a time
-
varying system as its mass changes
with time during the course of its flight due t
o the expense of fuel.

3.9. Causal and Non
-
Causal Systems

Causality usually refers to events in time. A causal or nonanticipatory system is one in which the
output
x
o
(t
1
)

at any arbitrary instant
t
1

depends on its input
x
i
(t)

in the past up to and inclu
ding t = t
1
. If
this property does not exist, then the system is non
-
causal. All real systems are causal in their temporal
behavior.

3.10. Stable and Unstable Systems

A system whose response either oscillates within certain finite bounds or grows withou
t bounds is
regarded as unstable. If for every bounded input the output is bounded, the system is said to be I/O
stable (
see
Stability Concepts
). If this is not the case, the system is unstable. Stability in linear time
invariant systems is easily ascertai
ned by applying well
-
established criteria. Stability in a nonlinear
system is quite complex; it depends on the inputs and the point at which the system is operated.
Nonlinear systems can therefore be stabilized by manipulating the input signals acting on t
hem, for
example, sometimes by injecting additional high frequency signals. This is not possible in linear
systems; its stability cannot be altered by external actions; they have to be stabilized by manipulating
their inherent properties, that is, by alter
ing the system parameters. Figuratively therefore we can say
that instability in non
-
linear systems can be cured by medical treatment but in linear systems it requires
surgery.

3.11. Single
-
Input
-
Single
-
Output (SISO) and Multiple
-
Input
-
Multiple
-
Output (M
IMO) Systems

When a system has only one input and one output, it is referred to as a single
-
input
-
single
-
output
(SISO) system. When a system has more than one input or more than one output, it is termed as a
multi
-
input multi
-
output (MIMO) system
(see
Con
trol of linear Multivariable Systems
)
. The various
properties of dynamic systems that are briefly introduced here will be reflected in system models.

In order to obtain a simple mathematical model of an actual relationship in a tractable, but sufficiently

accurate form, the structure as well as the parameters should be identified. System identification can be
accomplished by two approaches. One is based on the physical principles underlying the
phenomenological behavior of the process and the other is call
ed black
-
box modeling in which a
discrete time model is chosen and its parameters are estimated by fitting the input
-
output data. In the

16

former approach, applying the physical laws governing the process, the basic relations are written as
equations represe
nting balance of certain physical entities in the process. Physical system modeling
gives rise to generic models which are native to the continuous
-
time domain and the numerical values
of the parameters in such models can be directly estimated from input
-
o
utput data using the techniques
of identification that are specially developed for continuous
-
time models in the recent decades. In
certain situations, the essential features of the behavior of a system can be quickly obtained without
many details by means

of experiment.

4. Systems Control

4.1. Open Loop Control

To understand the development of control concepts let us consider a SISO system for the sake of
simplicity. The basic action in the control of a system is the application of input (control signa
l). Given
a general understanding of the system response (controlled variable) to inputs, a specific input may be
applied to give rise to the desired response. This is called ‘open loop control’ because of the nature of
the diagram representing such an ac
tion that is shown in Figure 14. The controlled system is also
referred to as the ‘plant’. Open loop control has obvious limitations. For instance, if there is a
disturbance on the output side of the process, control action does not take it into considerat
ion. In order
to remove this limitation, feedback has to be provided.

Figure 14: Open loop control system

4.2. Feedback Control

Figure 15 shows a typical feedback control system. In this system, the actual output is fed back and
compared with the desir
ed response. The resulting error is the basis for the application of a control
signal to the plant. The controller generates the control signal on the basis of the error. If a mechanical
signal has to be applied to the plant, it is generated by an actuator

(not explicitly shown in the figure)
from the output of the controller.

In this arrangement, the control signal takes the actual controlled variable into account including
disturbances if any. The plant is driven (by the control signal) until the error
is reduced. This is the
principle of feedback control in which feedback is negative.

Figure 15: Feedback control system

A comparison would show the following differences between open loop and closed loop control
schemes.

Open loop Control

Open loop ope
ration

The effects of known disturbances alone can be countered. Other disturbances cannot be taken
into account.

As long as the controlled plant is itself stable, the control system cannot become unstable, that
is the controlled variable cannot oscillate

or grow beyond bounds

17

In open loop control the controller is blind to what actually takes place at the output end and goes on
driving the plant in a fixed and predetermined manner.

Close loop control

Closed loop operation using negative feedback

The e
ffects of disturbances are countered by virtue of negative feedback.

Closed loop operation can be unstable even if the plant is stable.

In closed loop or feedback control the controller notices what actually takes place at the output end and
drives the pl
ant in such a way as to obtain the desired output.

There can be two different cases of feedback control. One is to reduce the effect of disturbances.
Certain variables of a process such as the controlled variable should be maintained at given fixed
value
s despite disturbances. Such a control is called set point control, or regulation, or control for
disturbances rejection. The other is tracking, that is, the controlled variable (output) is made to follow,
as closely as possible, the desired command (refer
ence) signal. In both the cases, the controlled
variables (or outputs) should be measured continuously and compared with the respective reference
signals. The resulting error signal has to be made to vanish as much as possible by control action. The
contro
l action involves the use of the error signal itself in generating suitable input signal to drive the
plant. It may be manual or automatic. The steering of a vehicle along a street manually by the driver is
an example of manual control.

4.3. Closed
-
Loop

Behavior of Control Systems

The performance of feedback control systems is assessed in terms of the following aspects of the
closed
-
loop behavior:

Disturbance rejection
: The closed loop system design may be specifically addressed to the rejection
of dis
turbances if the situation specifically warrants. For this purpose it is necessary to characterize the
disturbances for their nature and the point of occurrence in the system. Then, from the point at which
disturbance enters the system, the transfer functi
on of the system may be evaluated towards the output
end. The design of a suitable compensator that yields the desired disturbance rejection properties may
be obtained and inserted in the system.

Tracking behavior
: The tracking behavior is important if th
e output of a system has to follow the
input faithfully in time. This requires that the system has good transient response behavior.

-
state accuracy
: The accuracy with which a feedback control system responds to inputs is
-
stat
e error constants, which are evaluated with reference to inputs in the form
of polynomials in time. The simplest is the zero degree polynomial, or the unit step function and the
other two are the unit ramp and the unit parabola.

Unit step applied at
t=
0:
u(t),
whose Laplace Transform is

1
/s

.

Unit ramp applied at
t=
0:
r(t),
whose Laplace Transform is

1
/s
2

.

Unit parabola applied at
t=
0:
r(t),
whose Laplace Transform is

1
/s
3

.

A feedback system whose overall loop transfer function has
m

poles at the origin

of the
s
-
plane is
known as a type
-
m

system. That is, system type number denotes the number of pure integrating
elements within the feedback loop and as this number increases the steady state behaviour improves

18

provided the system stability does not deteri
orate with the increased number of poles at the origin of
the
s
-
plane.

The steady state error in the case of a type
-
0 system is finite for a step input and becomes infinity for
ramp and parabolic inputs. In the case of a type
-
-
state err
or for step inputs is always
zero but remains finite for ramp inputs and becomes infinite for parabolic inputs. A type
-
2 system has
-
state error for step and ramp inputs but has a finite error for parabolic inputs. The steady
-
state errors are eval
uated in terms of the error constants
(see
Closed Loop Behavior
).

Comparison of the desired and actual output of a system by feedback should ideally provide the error
or disparity information over all time, that is,
past
,
present

and
future

with referenc
e to any point in
time. The aim of most feedback control strategies is to generate a control input to the plant that would
reduce the disparity as far as possible.

Information or knowledge over ‘all time’ including the future is complete and is referred t
o in the
orient as '
trikaalagnaanam
' ( In
Sanskrit

it means

tri=
three
, kaala=
time
, gnanam=
knowledge). The
physical world permits us to know the first two of these 'three times', while the future is left for us only
to ponder. Due to uncertainty in informat
ion and irreversibility of certain physical processes, the
symmetry of time is lost; and the so
-
called ‘
Arrow of Time’

becomes a reality. Nevertheless, efforts
aimed at capturing information as far as possible over ‘all time’ continue by manifesting themse
lves in
the field of
estimation,

in which
smoothing
,
filtering

and
prediction

are concerned with the past,
present and the future respectively.

The controller is the element that implements the desired control strategy; it takes the
error

between
the des
ired and the actual system response and generates the
control input

to the
plant

based on
feedback. The control strategy is the result of consideration of one among a wide variety of techniques
of control.

4.4. Control Strategies

A control strategy is t
he basis on which the control signal is generated from the error signal in a
feedback system. In other words, the controller embodies the control strategy as its characteristic.

One of the simplest forms of controller is a relay, which is a simple, rugge
d, and robust power
amplifier. For any positive/negative value of the error the control signal has its full positive/negative
value. This is known as bang
-
bang control and it leads to a non
-
linear control system. This simple
strategy is in the spirit of th
e advice: ‘use the standard stick to deal even with a small snake’.

To realize a simple linear feedback control strategy, the error itself may be made to act directly as the
control signal to drive the plant. For more rapid action, the error may be amplif
ied to become the
control signal. That is, the control signal
u(t) = K
p

e(t)
, where
K
p

is the amplification factor or
gain

and
e(t)

the error signal. Since this strategy relates u(t) with e(t), it is capable of handling only the ‘present’
with reference to

the instant of time
t
. This is the proportional control action, which is part of a more
general proportional
-
integral
-
derivative (PID) scheme. Presentation of an amplified version of the error
to the plant as control signal makes the plant overact or quic
k
-
acting. Such a strategy drives the plant
harder, and beyond a certain limit may drive it crazy, that is, into instability. It is possible to determine
this limit of stability using the Nyquist criterion in frequency domain. The extent to which one can
pr
ovide amplification in the feedback loop without causing instability, is known as the
gain margin
.

19

The integral strategy

t
I
d
e
K
t
u
0
)
(
)
(

takes the error history from the beginning to the instant of time
t

into consideration in generating th
e
control signal.

The derivative strategy

)
(
)
(
t
u
dt
d
K
t
u
D

probes slightly into the ‘future’ with respect to
t
in generating the control signal.

Thus, the well
-
known PID control strategy may be viewed as an attempt to take into account, the er
ror
information over the ‘three
-
times’ additively together in some way. Therefore, in general, the controller
may be regarded as a dynamic system. The controller is also referred to as the ‘compensator’.

In the more general field of systems engineering, t
he so
-
called
inactive, reactive, interactive

and
proactive

approaches for development are control strategies in a similar spirit. The first approach is
tantamount to open loop control taking nothing into account. The others are feedback
-
based approaches
th
at take into account the past, present and the future respectively in the strategy.

Optimal control
: The system performance may be optimized with respect to the controller parameters
in a chosen structure employing the techniques of optimal control. A qua
dratic functional of the state
and/or input is defined as performance index. This is optimized with respect the control input. Often
the result is converted into a control law in terms of the controller parameters.

The problem of designing a control syst
em for a process, with a given precise model, is
straightforward. The Linear Quadratic Regulator (LQR) problem is a typical example of this class of
problems for optimal control
(see
Design of State Controllers
)
. However, control problems in the real
world

are not so ideal. The problem of control in the presence of noise in measurements is the
stochastic control problem, which is characterized by the application of estimation methods
(see
Control of Stochastic Systems
)
. The Linear Quadratic Gaussian (LQG) p
roblem is for optimal control
in the presence of noise.

(see Optimization)

: Quite often, control problems have to be tackled with no process models readily
served to the designer. Control design has to be carried out on the basis of know
ledge of the process,
which is either developed off
-
line or on
-
line on the basis of available measurements that are usually
subject to uncertainty. This is accomplished by self
-
tuning. Control design may follow a separate
modeling exercise that provides es
timates of an approximate plant model together with the limits of
uncertainty associated with it. The control is then designed to be robust against such uncertainty.
(see

Control
)

If the onus of 'understanding or modeling' the process rests on th
e designer, and if it has to be taken up
while the process is in operation, control techniques will have to be rendered comprehensive by
encompassing some estimation method that is capable of providing on
-
line, on the basis of the
available measurements, a

process model that is adequate for the purpose of control. Predictive ability
is considered to be a desirable feature for a process model for the purpose of control and a class of

20

control techniques based on modeling and prediction are of considerable imp
ortance. Modeling of real
world processes based on the so called ‘black
-
box’ approach, i.e., without the use of physical laws, is
of considerable importance in the fields of control and signal processing. Black
-
box approaches are
motivated by circumstances

in which the methods employing physical principles are either very
complex or surrounded by high uncertainty. These are invariably associated with estimation
-

a process
that is specifically referred to as smoothing, filtering and prediction respectively
according to its focus
on the past, present and future. These techniques support decision and control in a significant way.

In certain situations in practice, a chosen set of controller parameters may not remain valid over the
entire range of operating co
nditions of a plant. The plant dynamics, which is the basis for controller
design, may change thereby necessitating redesign and adaptation of the controller. This is the main
principle of adaptive control, which is illustrated in Figure 16.

Fig.16: An

Robust control
: In reality, despite efforts by identification and parameter estimation, system models
are neither precisely known nor are guaranteed to remain the same under the different conditions of
chniques automatically tune the control action to meet mainly the latter
contingency, the issue of uncertainty is tackled by robust control techniques. Here, the controller is
designed for a nominally specified plant model by taking uncertainties and unmo
delled plant dynamics
such that the resulting control guarantees satisfactory control under the limitations of knowledge of the
plant model
(see
Robust

Control
)
.

Intelligent control
: The term intelligent control cannot be defined precisely as it encompass
es many
unusual features and capabilities that characterize the control as intelligent. An important feature of
intelligent control is the presence of a body of knowledge on various aspects of control coded and made
available in a computer system to aid de
cisions and actions together with a learning capability. Fuzzy
logic control and neural network methods are used in such systems
(see
Fuzzy control Systems, Neural
Control Systems, Expert Control systems
)
.

The fields of systems, control and information
processing are closely related to the science of
cybernetics which attempts to understand the behavior of systems in nature. This understanding leads
to the knowledge towards improving the performance of natural or man
-
years, tech
niques of systems, control and information processing, are handled with less reference to
machines and other man
-
made physical processes, in the general field of ‘Systems Science’.

More detailed presentation of the Elements of Control Systems may be found

under this topic
(see
Introduction to Basic Elements, General Models of Dynamic Systems, System Description in Time
-
Domain, Description in Frequency Domain, Closed
-
loop Behavior
).

Acknowledgements

The author is grateful to Prof. H. Unbehauen for the opp
ortunity to contribute to the EOLSS and for the
helpful suggestions in the preparation of the manuscript.

Bibliography

21

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