Ganti Prasada Rao, *International Centre for Water and Energy Systems, **PO Box 2623**, **Abu Dhabi**. UAE.*

**Keywords: **Systems, Block diagram, Characteristic equation,
Characteristic polynomial, Controller, constitutive relations, Discrete 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**

- Introduction
- System Modeling
- Mathematical Models of Dynamical Systems

3.1 Differential Equation Models for Lumped Parameter Systems in Continuous Time

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

- 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 entire 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 canonical 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.

**Constitutive 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 control 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 absence of initial conditions.

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

**Frequency response**: The steady-state
response of a system to sinusoidal signals of unity amplitude and variable
frequency. This function in the frequency domain is obtained by setting *s =
j**w *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 and 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 similarity 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**: Dynamical systems
whose properties change in time. The parameters of the model of a time-varying
system are independent functions of time.

**Transfer function**: A mathematical
function that characterizes the transfer behavior of a system. It is the ratio
of the

**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 _{} , with *T* as the
sampling period, connects the

**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. Systems are made as collections 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 systems 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 numerical 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 stability,
steady-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 environment 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 theory 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.

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 and 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 tools 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 such 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.

Control 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 processing 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 effective 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 and 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 *constitutive
properties* of the elements in a system.

Physical 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 within 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 whose 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.

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 shown 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 ‘*across*’ and ‘*through*’ variables respectively.

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 the 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 the representations in fluid and mechanical systems respectively.

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 variety 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 theory. 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 take 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 constraints, 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 are 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.

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 known 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 accumulation 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, variational calculus,

*Bond graph methods*
represent the energetic interactions between systems and their components by
single lines termed as energy bonds. Bond graph representation 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 can be computer aided and software packages are available for this purpose.

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

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 in 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)

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:

_{} (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:

_{} (2)

Working with linear time
invariant systems becomes simplified with the help of _{} is defined as

_{} (3)

This transformation converts
a linear differential equation into an algebraic form in the domain *s* that represents the complex frequency.
Let _{} without loss of
generality. Let the constant initial conditions be defined as

_{} (4)

Then the

_{} (5)

and the transform of the output is

_{} (6)

The denominator term _{}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 response*
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 *free response* or the *zero-input response*.

If a unit impulse function or the Dirac delta function
denoted as is considered as the input x(t) =*d(t)*,
*X(s)* =1, the forced response
component in Eq. (6) happens to be equal to the transfer function 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 *d(t) *is indirectly defined by the following
properties:

_{} (7)

and for any function _{} continuous at t,
defined in the ordinary sense

_{} (8)

The inverse *y(t)* is
obtained be the method of partial fractions as follows.

Suppose that the
characteristic polynomial has *n _{1}*
roots each equal to

_{} (9)

where *b _{n}=*0
unless

_{} (10)

These coefficients are also
known as the residues of *F(s)* at *–p _{i}* ,

_{} (11)

where *d(t)*
is the unit impulse function, and *b _{n}
= 0* unless

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 function on the *s*-plane,
the value of the 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 as 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 _{}, is called the *steady state response*. The system
response as a function 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: _{}, if the limit exists.

Notice
that nature of *f(t)* depends on the
values 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
conjugate 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 complex s

The transfer function *F(j**w)* evaluated along the *j**w-*axis of the *s*-plane
is of significance as it represents the steady state response of the system to
a sinusoidal input of frequency *w*. 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.

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:

_{} (12)

where **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. This is known as the state space description. Techniques of
handling linear systems in state space are well established (see *Description and
Classification*).

_{} (13)

gives

_{}

The

_{} (14)

If we let **D=0** which is the common case,

_{} (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 function 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 scalar
equation *dx/dt=ax+bu*.

_{} (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.

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 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 easily 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 an arbitrary enlargement of the dimension of the state space.

If a system variable (signal) *y*, at any arbitrary instant of time can be varied within known
limits continuously, it is called “continuous”. If 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 uniformly
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.

_{} (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

_{} (18)

*z*-transformation of Eq. (17) gives

_{} (19)

Discrete time systems are described in state space as

_{} (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 _{}. 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**w*
axis winds 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 information in the sampled signal and to avoid aliasing effects is
twice the highest frequency occurring in the signal spectrum. This criterion is
called *T*
such that l_{m}
T£
0.5, where l_{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.

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 the loading effect of the following connections. Transfer processes are 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 complex 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 the 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

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

Signal flow diagrams are close in spirit to the block diagrams. In a signal flow graph the signals are denoted 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 function G of a system represented as a signal flow graph is given by

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 functions of the

cascaded elements.

D = 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 + ……..

D* _{k} *= D - sum of the loop transfer functions touching the k-th
forward path.

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}*

*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}*

D = *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}]*

D_{1} = *1 – G _{7}H_{4}*

G = T_{k}*D*_{1}/
*D*

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 described 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 differential 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 often studied by means of lumped approximations. For example, transmission lines are studied with the help of the so called T and P approximations.

(see Partial differential equations)

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 uncertainty 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 presented as expected values together with the bounds of error (see Probability and Statistics).

Most natural systems are nonlinear. An important criterion that distinguished nonlinear systems from linear systems is the principle 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 acting 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 produces 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 nonlinear 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)* about a point *x _{o}*. Expanding the function in

*f(x)- f(x _{o})*

* x =x _{o}*

which is a linear relationship in the form:

*df(x) =
m _{o} *

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

_{} (21)

where

**x ** (t) = [x_{1}(t)
……..x_{n}(t)]^{T}

**u** (t) = [u_{1}(t) …….._{r}^{T}

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

Linearization of this leads to the linear vector differential equation

* _{}* (22)

Where **A** and **B** are Jacobians
containing the various partial derivative terms as follows:

**A** = _{} **B** =_{} (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 to the expense of fuel.

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

A system whose response either oscillates within certain
finite bounds or grows without 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 ascertained 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 them, 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 altering 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.

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
*Control 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 called black-box modeling in which a discrete time model is chosen and its parameters are estimated by fitting the input-output data. In the former approach, applying the physical laws governing the process, the basic relations are written as equations representing 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-output 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.

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 signal). 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 action 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 consideration. In order to remove this limitation, feedback has to be provided.

Figure 15 shows a typical feedback control system. In this system, the actual output is fed back and compared with the desired 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.

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

q Open loop operation

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

q 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

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** *

q Closed loop operation using negative feedback

q The effects of disturbances are countered by virtue of negative feedback.

q 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 plant 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 values 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 (reference) 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 control 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.

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
disturbances 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 function 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 the output of a system has to follow the
input faithfully in time. This requires that the system has good transient
response behavior.

**Steady-state
accuracy**: The
accuracy with which a feedback control system responds to inputs is governed by
the steady-state 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
provided the system stability does not deteriorate 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-1
system the steady-state error for step inputs is always zero but remains finite
for ramp inputs and becomes infinite for parabolic inputs. A type-2 system has
no steady-state error for step and ramp inputs but has a finite error for
parabolic inputs. The steady-state errors are evaluated 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 reference 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 to 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 information 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 themselves 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 desired 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.

A control strategy is the 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, rugged, 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 the 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 amplified to become the control
signal. That is, the control signal *u(t) = K _{p} e(t)*, where

The integral strategy

_{}

takes the error history from the beginning to the instant of
time *t* into consideration in generating the control signal.

The derivative strategy

_{}

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 error 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, the
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 that 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 quadratic 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 system 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) problem
is for optimal control in the presence of noise.

(see Optimization)

**Adaptive control**: 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 knowledge 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 estimates 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
*Adaptive* *Control*)

If the onus of 'understanding or modeling' the process rests on the 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 control techniques based on modeling and prediction are of considerable importance. 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 conditions 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.

**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
operation. While adaptive techniques 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 unmodelled 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 encompasses 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
decisions 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-made processes. In recent years, techniques 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*).

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

Chen C.T.
(1984). *Linear system theory and design*,

*Feedback
Control of Dynamic Systems*, Reading: Addison-Wesley. [This is a widely used
textbook that presents essential principles and design of feedback control
systems]

*Digital
Control of Dynamic Systems*, Reading: Addison-Wesley. [This is a widely used
textbook that presents essential principles and design of digital control
systems]

Gupta M.M.
and Sinha N.K.(Eds.) (1996). *Intelligent Control Systems- Theory and
applications*,

Kuo B.C.
(1982). *Automatic control systems*, Englewood Cliffs: Prentice-Hall.
[This is a widely used textbook that presents essential principles and design
of feedback control systems]