CONTROL SYSTEM (EE0704) STUDY NOTES · Control System Engineering – By: I.J.Nagrath, M.Gopal Automatic Control Systems – By Benjamin C.Kuo Control Systems Engineering – By S.N.Sivanandam

CONTROL SYSTEM

(EE0704)

STUDY NOTES

M.S.Narkhede

Lecturer in Electrical Engineering

GOVERNMENT POLYTECHNIC, NASHIK (An Autonomous Institute of Government of Maharashtra)

Samangaon Road,

Nashik Road, Nashik - 422 101

Phone : (0253) 2461221, Fax : (0253) 2450236

Website: www.gpnashik.com

E-mail : [email protected]

http://msnarkhede.blog.com

2

Preface

Dear students,

It gives me immense pleasure in bringing these study notes for you. In fact this is my first

attempt to provide computerized notes to students in this format. I have also given you

Switchgear & Protection Notes, but those were in power point presentation form. These

notes are prepared by referring following books.

Control System Engineering – By: I.J.Nagrath, M.Gopal

Automatic Control Systems – By Benjamin C.Kuo

Control Systems Engineering – By S.N.Sivanandam

These notes cover your first three modules i.e. Basis of Control system, Time response of

Control system & Concept of stability in all for 55 marks. For next modules you can refer my

OHP transparencies.

I must thank our honorable Principal Dr. R.S.Naidu for his constant encouragement in

providing latest study material to students in good readable form. Thanks are also due to my

colleagues Prof. Mrs. S.S.Umare & Prof.Mrs.D.R.Kirtane for their kind support.

Thanks are due to Prof. D.D.Lulekar, HOD of Electrical Engg. Department for providing the

infrastructure required for preparing these notes.

Hope these notes will be useful to you in studying Control System.

M.S.Narkhede

LEE, GP Nashik

Thursday, 12th

February 2009.


3

CONTROL SYSTEMS

An average human being is capable of performing a wide range of-tasks, including

decision making. Some of these tasks, such as picking up objects, or walking from one point

to another, are normally carried out in a routine fashion. Under certain conditions, some of

these tasks are to be performed in the best possible way. For instance, a marathon runner, not

only must run the distance as quickly as possible, but in doing so, he or she must control the

consumption of energy so that the best result can be achieved. Therefore, we can state in

general that in life there are numerous “objectives” that need to be accomplished, and the

means of achieving the objectives usually involve the need for control systems.

In recent years control systems have assumed an increasingly important role in the

development and advancement of modern civilization and technology. Practically every

aspect of our day-to-day activities is affected by some type of control system. For example, in

the domestic domain, automatic controls in heating and air-conditioning systems regulate the

temperature and humidity of homes and buildings for comfortable living. To achieve

maximum efficiency in energy consumption, many modern heating and air-conditioning

systems in large office and factory buildings are computer controlled.

Control systems are found in abundance in all sectors of industry, such as quality

control of manufactured products, automatic assembly line, machine-tool control, space

technology and weapon systems, computer control, transportation systems, power systems,

robotics, and many others. Even such problems as inventory control, and social and economic

systems control, may be approached from the theory of automatic controls.

Regardless of what type of control system we have, the basic ingredients of the

system can be described by

I. Objectives of control

2. Control system components

3. Results

In block diagram form, the basic relationship between these three basic ingredients is

illustrated in above Fig.

In more scientific terms, these three basic ingredients can be identified with inputs, system

components, and outputs, respectively, as shown in following Fig.

In general, the objective of the control system is to control the outputs c in some prescribed

manner by the inputs u through the elements of the control system. The inputs of the system

are also called the actuating signals, and the outputs are known as the controlled variables.


4

OPEN LOOP SYSFEMS

Above figure illustrates the block diagram of an open-loop control system. A small

input signal r( t) is amplified by an amplifier and the output of this amplifier actuates the

power actuator. The output of this power actuator drives the controlled variable c(t). From the

block diagram it can be seen that the actuating signal or the control action does not change

the output.

An automatic toaster is an example of open-loop control system because it is con-

trolled by a timer. The time required to make a good toast must be estimated by the user, who

is not a part of the system. The quality of toast (output) is determined by the time set by the

user.

CLOSED-LOOP SYSTEMS

Above figure illustrates the block diagram of a basic closed-loop control system in

which the control action (input) is dependent upon the controlled variable (output). The

output van-able c(t) is compared with the reference input r(t). This comparison is done by an

element called error detector. The output of the error detector is the actuating signal e(t).

If the actuating signal is weak, it should be amplified by an amplifier before it is applied to

the power actuator.

An auto pilot mechanism and the air-plane is an example of a closed-loop system. Its

purpose is to maintain a specified air-plane heading despite atmospheric changes. It performs

this task continuously by measuring the actual air-plane heading and automatically adjusting

the air-plane control surfaces, so as to bring the actual air-plane heading into correspondence

with the specified heading. The human pilot is not a part of the control system.


5

DISTINCTION BETWEEN OPEN-LOOP AND CLOSED-LOOP CONTROL SYSTEMS

Merits of Open-loop Systems :

1. Simplest and most economical.

2. Easier to build.

3. Generally not disturbed with unstable operations.

Demerits of Open-loop Systems:

1. Usually inaccurate and unreliable. Hence they are not preferred.

2. Do not adapt to environmental changes or to external disturbances.

Merits of Closed-loop Systems :

1. Accurate due to feedback, leading to faithful reproduction of the input.

2. Reduced sensitivity of the ratio of output to input for variations in system

characteristics.

3. Reduced effects on non-linearities and distortion.

4. Increased bandwidth (The bandwidth of a system is that range of input frequencies

over which the arguments of the system will respond satisfactorily).

Demerits of Closed-loop Systems :

1. Tendency towards oscillation (unstable operation).

2. Reduction in overall gain.

CLASSIFICATION OF CONTROL SYSTEMS:

Feedback control systems are classified as follows.

1) According to the method of analysis and design

These are further classified as linear and nonlinear, time varying or time invariant.

2) According to the types of signal found in the system

These are further classified as continuous-data and discrete-data systems, or

modulated and un modulated systems.

3) According to the main purpose of the system. These may be classified as positional control system , speed control system, temperature

control system etc.


6

EFFECTS OF FEEDBACK:

Feedback has effects on system performance characteristics as stability, bandwidth, overall

gain, impedance, and sensitivity.

Effect of Feedback on Overall Gain :

Feedback could increase the gain of the system in one frequency range but decrease it in

another.

Effect of Feedback on Stability:

Stability is a notion that describes whether the system will be able to follow the input

command. A system is said to be unstable if its output is out of control or increases without

bound.

Feedback can cause a system that is originally stable to become unstable. Certainly, feedback

is a two-edged sword; when it is improperly used, it can be harmful.

Effect of Feedback on Sensitivity: Sensitivity of a system can be decreased by providing a feedback.

Effect of Feedback on External Disturbance or Noise:

All physical control systems are subject to some types of extraneous signals or noise during

operation. Examples of these signals are thermal noise voltage in electronic amplifiers and

brush or commutator noise in electric motors.

The effect of feedback on noise depends greatly on where the noise is introduced into the

system; no general conclusions can be made. However, in many situations, feedback can

reduce the effect of noise on system performance.


7

MATHEMATICAL MODELLING OF LINEAR SYSTEMS

Translational Systems :

Let us consider the above mechanical system. It is simply a mass M attached to a spring

(stiffness k) and a dashpot (viscous friction coefficient f) on which the force F acts.

Displacement x is positive in the direction shown. The zero position is taken to be at the point

where the spring and mass are in static equilibrium.

Let k be the spring constant called as stiffness in N/m.

Let x be the deformation of the spring in meters which is also a displacement of mass M.

When force F in Newtons is applied to a mass M in kg. a reactive force is produced which

acts in a direction opposite to that of acceleration & this force is given by Newton’s third law.

i.e. F =Ma

= M (d2x)/(dt

2 )

Where a is the acceleration in m/s2

X is the displacement of mass in meters.

The spring provides a retarding force which is given by Fs = kx

The viscous force also acts in opposite direction of applied force.

Fr = f.dx/dt

Where, Fr is the viscous friction force in N

F is the coefficient of viscous friction in N/m/s

Therefore,

F= M (d2x)/(dt

2 ) + f.dx/dt + kx --------------------------(1)


8

Rotational Systems :

Consider a rotational mechanical system consisting of a rotatable disc of moment of inertia J

and a shaft of stiffness k. The disc rotates in a viscous medium with viscous friction

coefficient f.

Let T be the applied torque which tends to rotate the disc.

The torque equation obtained from free body diagram is

T- f.(dθ/dt) - kθ = J (d2θ/dt

2)

T = J (d2θ/dt

2) + f.(dθ/dt) + kθ --------------------------(2)

Here θ is the angular displacement in rad.

Electrical systems :

RLC series Circuit :

The resistor, inductor and capacitor are the three basic elements of electrical circuits. These circuits

are analyzed by the application of Kirchhoff’s voltage and current laws.

Let us analyze the L-R-C series circuit shown in above figure. by using Kirchhoff’s voltage

law.

Applied voltage = sum of voltage drops in the circuit.

e = VR + VL + VC

e = Ri + L (di/dt) + (1/c) ∫i.dt

Now, q = ∫i.dt

Therfore ,

E = R.(dq/dt) + L (d2q/dt2 )+ (1/c).q --------------------------(3)


9

RLC Parallel Circuit :

Applying Kirchoff’s current law

C(de/dt) + (1/L) ∫e.dt + e/R = i

In terms of magnetic flux linkage,

Φ = ∫ e.dt

C.(d2Φ/dt2 ) + (1/R) .(dΦ/dt) + 1/L Φ = i

i = C.(d2Φ/dt2 ) + (1/R) .(dΦ/dt) + 1/L Φ --------------------------(4)

So we have got the equations (1), (2), (3) & (4) as follows.

F= M (d2x)/(dt

2 ) + f.dx/dt + kx --------------------------(1)

T = J (d2θ/dt

2) + f.(dθ/dt) + kθ --------------------------(2)

E = R.(dq/dt) + L (d2q/dt2 )+ (1/c).q -------------------------(3)

i = C.(d2Φ/dt2 ) + (1/R) .(dΦ/dt) + 1/L Φ -----------------(4)

Comparing equations (1), (2) & (3) we will get Force- Torque- Voltage analogy.

Similarly comparing equations (1), (2) & (4) we will get Force- Torque- Current analogy as

follows.

Force- Torque- Voltage analogy. Mechanical translational

systems

Mechanical rotational system. Electrical Systems

Force F Torque T Voltage e

Mass M Moment of inertia J Inductance L

Viscous friction coefficient f Viscous friction coefficient f Resistance R

Spring stiffness K Torsional spring stiffness K Reciprocal of capacitance 1/C

Displacement x Angular displacement θ Charge q

Velocity dx/dt Angular velocity dθ/dt Current i

Force- Torque- Current analogy. Mechanical translational

systems

Mechanical

rotational systems

Electrical systems

Force F Torque T Current i

Mass M Moment of inertia J Capacitance C

Viscous friction coefficient f Viscous friction coefficient f Reciprocal of resistance 1/R

Spring stiffness K Torsional spring stiffness K Reciprocal of inductance 1/L

Displacement x Angular displacement θ Magnetic flux linkage Φ

Velocity dx/dt Angular velocity dθ/dt Voltage e


10

BLOCK DIAGRAMS :

Because of its simplicity and versatility, block diagram is often used by control engineers to

portray systems of all types. A block diagram can be used simply to represent the

composition and interconnection of a system. Or, it can be used, together with transfer

functions, to represent the cause-and-effect relationships throughout the system

Block Diagrams of Control Systems :

We shall now define some block diagram elements used frequently in control systems and the

block diagram algebra.

The important block diagram components are as follows.

Block Diagram Algebra :

We can use following rules for reduction of Block Diagrams.


11

Transfer Function :

Transfer function of a system is defined as a ratio of Laplace of output variable to Laplace of

input variable.

Derivation of Transfer function of closed loop system :

Consider a block diagram of a linear feedback control system as shown below.

Let

r(t), R(s) be the reference input.

c(t),c(s) be the output signal (controlled variable)

b(t), B(s) be the feedback signal

ε(t), ε(s) be the actuating signal

e(t),E(s) = R(s) – C(s) = error signal

G(s) = C(s)/ ε(s) = open -loop transfer function or forward path transfer function

M( s) = C(s)/R(s) = closed-loop transfer function


12

H(s) feedback-path transfer function

G(s)H(s) = loop transfer function

The closed-loop transfer function, M(s) = C(s)/R(s), can be expressed as a function of G(s)

and H(s). From above figure we can write

C(s) = G(s) ε(s) ------------------------ (1)

and

B(s) = H(s) C(s) ------------------------ (2)

The actuating signal is written

ε(s) = R(s) - B(s) ------------------------ (3)

Substituting Eq. (3) into Eq. (1) yields

C(s) = G(s)R(s) - G(s)B(s) ------------------------ (4)

Substituting Eq. (2) into Eq. (4) gives

C(s) = G(s)R(s) - G(s)H(s)C(s) ------------------------ (5)

Solving C(s) from the last equation, the closed-loop transfer function of the system is given

by

M(s) = C(s)/ R(s) = G(s)/[1+G(s)H(s)] ------------------------ (6)

SIGNAL FLOW GRAPHS:

A signal flow graph may be regarded as a simplified notation for a block diagram. It

was originally introduced by S. J. Mason as a cause-and-effect representation of linear

systems. In general, besides the difference in the physical appearances of the signal flow

graph and the block diagram, we may regard the signal flow graph to be constrained by more

rigid mathematical relationships, whereas the rules of using the block diagram notation are

far more flexible and less stringent.

A signal flow graph may be defined as a graphical means of portraying the input-output

relationships between the variables of a set of linear algebraic equations.

e.g.

Here y1 & y2 are input & output nodes respectively.

a12 is the gain.

BASIC PROPERTIES OF SIGNAL FLOW GRAPHS

Following are the important properties of the signal flow graph.

1. A signal flow graph applies only to linear systems.

2. The equations based on which a signal flow graph is drawn must be algebraic

equations in the form of effects as functions of causes.

3. Nodes are used to represent variables. Normally, the nodes are arranged from left to

right, following a succession of causes and effects through the system.

4. Signals travel along branches only in the direction described by the arrows of the

branches.

5. The branch directing from node yk to yj, represents the dependence of the variable yj

upon yk but not the reverse.


13

6. A signal yk traveling along a branch between nodes yk and yj, is multiplied by the

gain of the branch, akj so that a signal akj yk is delivered at node.

DEFINITIONS FOR SIGNAL FLOW GRAPHS

Input Node (Source) : An input node is a node that has only outgoing branches. (Example:

node y1 in above figure.

Output Node (Sink) : An output node is a node which has only incoming branches. (Example:

node y2 in above figure.

Path : A path is any collection of a continuous succession of branches traversed in the same

direction.

Forward Path : A forward path is a path that starts at an input node and ends at an output

node and along which no node is traversed more than once.

Loop: A loop is a path that originates and terminates on the same node and along which no

other node is encountered more than once.

Path Gain / Transmittance: The product of the branch gains encountered in traversing a path

is called the path gain.


14

SIGNAL-FLOW-GRAPH ALGEBRA

Based on the properties of the signal flow graph. we can state the following manipulation and

algebra of the signal flow graph.

Consider the following figures

1. The value of the variable represented by a node is equal to the sum of all the signals

entering the node. Therefore, for the signal flow graph of first figure , the value of y1

is equal to the sum of the signals transmitted through all the incoming branches; that

is, y1 = a21y2+a31y3+a41y4+a51y5

2. The value of the variable represented by a node is transmitted through all branches

leaving the node. In the signal flow graph of first figure, we have

y6 = a16y1

y7=a17y1

y8=a18y1

3. Parallel branches in the same direction connecting two nodes can be replaced by a

single branch with gain equal to the sum of the gains of the parallel branches. An

example of this case is illustrated in Fig 3.


15

4. A series connection of unidirectional branches, as shown last figure , can be replaced

by a single branch with lain equal to the product of the branch gains.

5. Signal flow graph of a feedback control system.

Figure 1 shows the signal flow graph of a feedback control system whose block

diagram is given in Figure 2.

Figure (1)

Figure (2)

Therefore, the signal flow graph may be regarded as a simplified notation for the

block diagram. Writing the equations for the signals at the nodes E(s) and C(s) we have

ε(s) = R(s)—H(s)C(s)

and

C(s) = G(s) ε(s)

The closed-loop transfer function is obtained from these two equations,

C(s)/R(s) = G(s) / [1+G(s)H(s)]

MASON’S GAIN FORMULA FOR SIGNAL FLOW GRAPHS

Given a signal flow graph or a block diagram, it is usually a tedious task to solve for its input-

output relationships by analytical means. Fortunately, there is a general Mason’s gain formula

available which allows the determination of the input-output relationship of a signal flow

graph by mere inspection.

The general gain formula is,


16


17

TIME RESPONSE OF CONTROL SYSTEM

Introduction :

Since time is used as an independent variable in most control systems, it is usually of

interest to evaluate the state and output responses with respect to time, or simply, the time

response. In the analysis problem, a reference input signal is applied to a system, and the

performance of the system is evaluated by studying the system response in the time domain.

For instance, if the objective of the control system is to have the output variable track the

input signal, starting at some initial condition, it is necessary to compare the input and the

output response as functions of time. Therefore, in most control system problems the final

evaluation of the performance of the system is based on the time responses.

The time response of a control system is usually divided into two parts: the transient

response and the steady-state response. Let c(t) denote a time response; then, in general, it

may be written as

c(t) = ct(t) + Css(t)

Where,

ct(t) = transient response

css(t) = steady state response

In network analysis it is sometimes useful to define steady state as a condition when

the response has reached a constant value with respect to the independent variable. In control

systems studies, however, it is more appropriate to define steady-state as the fixed response

when time reaches infinity.

Therefore, a sine wave is considered as a steady-state response because its behavior is

fixed for any time interval, as when time approaches infinity.

Similarly, the ramp function c( t) = t is a steady-state response, although it increases

with time.

Transient response is defined as the part of the response that goes to zero as time becomes

very large.

All control systems exhibit transient phenomenon to some extent before a steady state is

reached. Since inertia, mass, and inductance cannot be avoided entirely in physical systems,

the responses of a typical control system cannot follow sudden changes in the input

instantaneously, and transients are usually observed. Therefore, the control of the transient

response is necessarily important.

The steady-state response of a control system is also very important, since when compared

with the input, it gives an indication of the final accuracy of the system.

When the steady-state response of the output does not agree with the steady-state of the input

exactly, the system is said to have a steady-state error.

The study of a control system in the time domain essentially involves the evaluation of the

transient and the steady-state responses of the system.

TYPICAL TEST SIGNALS FOR THE TIME RESPONSE OF CONTROL SYSTEMS

To facilitate the time-domain analysis, the following deterministic test signals are often used.

Step Input Function : The step input function represents an instantaneous change in the

reference input variables For example, if the input is the angular position of a mechanical

shaft, the step input represents the sudden rotation of the shaft. The mathematical

representation of a step function is


18

where R is a constant. Or

r(t) =Rus(t)

where , us(t) is the unit step function. The step function is not defined at t=0.

It shown graphically as below.

Ramp Input Function : In the case of the ramp function, the signal is considered to have a constant

change in value with respect to time. Mathematically, a ramp function is represented by,

or simply,

The ramp function is shown in following figure. If the input variable is of the form of the

angular displacement of a shaft, the ramp- input represents the constant-speed rotation of the

shaft.


19

Parabolic Input Function : The mathematical representation of a parabolic input function is

or simply

The graphical representation of the parabolic function is shown in following figure.


20

TlME-DOMAlN PERNORMANCE OF CONTROL SYATEMS-

TRANSIENT RESPONSE

The transient portion of the time response is that part which goes to zero as time becomes

large. The transient performance of a control system is usually characterized by the use of a

unit step input. Typical performance criteria that are used to characterize the transient

response to a unit step input include overshoot, delay time, rise time and settling time.

Following figure illustrates a typical unit step response of a linear control system. The above-

mentioned criteria are defined with respect to the step response:

1. Maximum overshoot : The maximum overshoot is defined as the largest deviation of

the output over the step input during the transient state. The amount of maximum

overshoot is also used as a measure of the relative stability of the system. The

maximum overshoot is often represented as a percentage of the final value of the step

response; that is,

2. Delay time : The delay time Td is defined as the time required for the step response to

reach 50 percent of its final value.

3. Rise time : The rise time Tr is defined as the time required for the step response to rise

from 10 percent to 90 percent of its final value

4. Settling time :The settling time Td is defined as the time required for the step response

to decrease and stay within a specified percentage of its final value. A frequently used

figure is 5 percent.


21

STABILITY OF CONTROL SYSTEM

Roughly speaking, stability in a system implies that small changes in the system input, in

initial conditions or in system parameters, do not result in large changes in system output.

Stability is a very important characteristic of the transient performance of a system. Almost

every working system is designed to be stable.

A linear time-invariant system is stable if the following two notions of system stability at

satisfied:

(i) When the system is excited by a bounded input, the output is bounded.

(ii) In the absence of the input, the output tends towards zero (the equilibrium state of the

system) irrespective of initial conditions.

The stability of a system is governed by roots of the characteristic equations. Following

figure shows the various locations of roots of characteristic equation & corresponding

impulse responses.


22

From above it is clear that if the roots of characteristic equation lie in left half of the s plane

the system is stable. This can be shown by following figure.

From above we can summarize the relation between the transient response and the

characteristic equation roots as follows.

1. When all the roots of the characteristic equation are found in the left of the s plane,

the system responses due to the initial conditions will decrease to zero as time

approaches infinity.

2. If one or more pairs of simple roots are located on the imaginary axis of the s plane,

but there are no roots in the right half of the s plane, the responses due to initial

conditions will be undamped sinusoidal oscillations.

3. If one or more roots are found in the right half of the s plane, the responses will

increase in magnitude as time increases.

For analysis and design purposes stability is classified into absolute stability and relative

stability. Absolute stability refers top the condition of stable or unstable. It is a yes or no

condition. Once the system is stable then it is determined how stable it is? This degree of

stability gives relative stability.

METHODS OF DETERMINING STABILITY UNEAR CONTROL SYSTEMS Although the stability of linear time-invariant systems may be checked by investi-

gating the impulse response, or by finding the roots of the characteristic equation, these

criteria are difficult to implement in practice. For instance, the impulse response is obtained

by taking the inverse Laplace transform of the transfer function, which is not always a simple

task. The solving of the roots of a high-order polynomial can only be carried out by a digital

computer. The methods outlined below are frequently used for the stability studies of linear

time-invariant systems.

1) Routh - Hurwitz criterion:

It is an algebraic method that provides information on the absolute stability of a linear

time-invariant system. The criterion tests whether any roots of the characteristic equation lie

in the right half of the s-plane. The number of roots that lie on the imaginary axis and in the

right half of the s-plane are also indicated.


23

2) Nyquist criterion:

It is a semi graphical method that gives information on the difference between the

number of poles and zeros of the closed-loop transfer function by observing the behavior of

the Nyquist plot of the loop transfer function. The poles of the closed-loop transfer function

are the roots of the characteristic equation. This method requires that we know the relative

location of the zeros of the closed-loop transfer function.

3) Root locus plot:

It represents a diagram of loci of the characteristic equation roots when a certain

system parameter varies. When the root loci lie in the right half of the s-plane, the closed-

loop system is unstable.

4)Bode diagram:

The Bode plot of the loop transfer function G(s)H(s) may be used to determine the

stability of the closed-loop system. However, the method can be used only if G(s)H(s) has no

poles and zeros in the right-half s-plane.

5)Lyapunov’s stability criterion:

It is a method of determining the stability of nonlinear systems, although it can also be

applied to linear systems. The stability of the system is determined by checking on the

properties of the Lyapunov function of the system.

HURWITZ CRITERION:

Consider that the characteristic equation of a linear time-invariant system is of the

form

where all the coefficients are real numbers.

The Hurwitz determinants of above equation are given by,

Hurwitz criterion states that , all roots of above characteristic equation lie in left half of the s

plane if Hurwitz determinants , Dk where k= 1,2,3….n are all positive.


24

ROUTH HURWITZ CRITERION: The first step in the simplification of the Routh-Hurwitz criterion is to arrange the polynomial

coefficients into two rows. The first row consists of the first, third, fifth,... coefficients, and

the second row consists of the second, the fourth, sixth,... coefficients, as shown in the

following tabulation:

The next step is to form the following array of numbers by the indicated operations (the

example shown is for a sixth-order system):

The array of numbers and operations given above is known as The Routh tabulation or the

Routh array. The first column of a0 & a1 on the left side is used for identification purpose.

Once the Routh tabulation has been comp1eted, the last step in the Routh— Hurwitz criterion

is to investigate the signs of the numbers in the above first column of the tabulation. The

following conclusions are drawn.

The roots of the polynomial are all in the left half of the s-plane if all the elements of the first

column of the Routh tabulation are of the same sign. If there are changes of signs in the

elements of the first column, the number of sign changes indicates the number of roots with

positive real parts i.e lying in right hand side of s plane.

It means if there is a sign change in first column the system is unstable & the number of sign

changes indicates the number of roots lying on the right hand side of the s plane.


25

Examples on Routh- Hurwitz criterion:

1) Comment on the stability of the system, whose characteristic equation is given as

, by applying Routh -Hurwitz criterion.

Solution :

Applying Routh -Hurwitz criterion we get Rouths tabulation as,

Since there are two sign changes in the first column of the tabulation, the system has two

roots located in the right half of the s plane & is unstable.

2) Comment on the stability of the system, whose characteristic equation is given as

, by applying Routh -Hurwitz criterion.

Solution :

Applying Routh -Hurwitz criterion we get Rouths tabulation as,

Since there are two changes in sign in the first column, the equation has two roots in the right

half of the s-plane & is unstable.


26

ROOT LOCUS TECHNIQUE :

Consider a design problem in which the designer is required to achieve the desired

performance for a system by adjusting the location of its closed-loop poles in the s-plane by

varying one or more system parameters. The Routh’s criterion, obviously does not help much

in such problems.

For determining the location of the closed-loop poles, we may use technique of factoring the

characteristic polynomial and determining its roots, since the closed-loop poles are the roots

of the characteristic equation. This technique is very laborious when the degree of the

characteristic polynomial is three or higher. Furthermore, repeated calculations are required

as a system parameter is varied for adjustments.

A simple technique, known as the root locus techniques, for finding the roots of the

characteristic equation, introduced by W.R. Evans, is extensively used in control engineering

practice. This technique provides a graphical method of plotting the locus of the roots in the s-plane as a given system parameter is varied over the complete range of values ( may be from zero to

infinity). The roots corresponding to a particular value of the system parameter can then be located on

the locus or the value of the parameter for a desired root location can be determined from the locus.

The root locus is a powerful technique as it brings into focus the complete dynamic response of the

system and further, being a graphical technique, an approximate root locus sketch can be made

quickly and the designer can easily visualize the effects of varying various system parameters on root locations. Also in the design of control systems it is often necessary to investigate the performance of a

system when one or more parameters of the system varies over a given range. This technique

is useful in such case.

A characteristic equation can be written as,

where, K is the parameter considered to vary between -∞ and +∞. The coefficients

a1,a2….an,b1,b2…..bm are assumed to be fixed. These coefficients can be real or complex,

although our main interest here is in real coefficients.

Now based on variation of K following categories are defined.

1) Root Loci : The portion of the root loci when K assumes positive values that is

2) Complimentary root loci : The portion of the root loci when K assumes negative

values, that is ,

3) Root contours : loci of roots when more than one parameter varies.

4) Complete root loci : It refers to the combination of the root loci and the

complementary root loci.

ROOT LOCUS CONCEPT: Consider a transfer function of a system as

Here the characteristic equation of the system is

This second order system under consideration is always stable for positive values of a and K

but its dynamic behavior is controlled by the roots of above equation.


27

The roots are given by,

From above it is seen that as any of the parameter (a or K) varies, the roots of the

characteristic equation change. Let us vary K by keeping a constant.

Following conditions are obtained.

the roots are real and distinct, when K=0, the two roots are s1=0, s2= -a i.e.

they coincide with the open loop pole of the system.

(2) K=a2/4, the roots are real and equal in value. i.e.s1=s2= - a/2

(3) a2/4<K<∞, the roots are complex with real part = - a/2 i.e. unvarying real part.

The root locus with varying K is plotted in following figure.

1) The root locus plot has two branches starting at the two open-loop poles (s = 0, and s =-a)

for K = 0,

2) As K is increased from 0 to a2/4, the roots move towards the point (- a/2, 0) from opposite

directions. Both the roots lie on the negative real axis which corresponds to an over damped

system. The two roots meet at s = -a/2 for K=a2/4. This point corresponds to a critically

damped system. As K is increased further (K> a2/4), the roots break away from the real axis,

become complex conjugate and since the real part of both the roots remains fixed at - a/2, the

roots move along the line σ =- a/2 and the system becomes under damped.

3)For K> a2/4, the real parts of the roots are fixed, therefore the settling time is nearly

constant.

----------------------------------------------------------------------------------------------------------------

Example on Steady state error: 1) If a 10 volts reference supply is used to regulate a 100 volts supply & if H is a constant

equal to 0.1,calculate the error signal when output voltage is exactly 100volts.

Solution:

The error signal ε(t) is given by following formula.

ε(t) = r(t)-b(t)

i.e. ε(s) = R(s) – H(s)C(s)

= 10-(0.1)x(100)

=0

____________________Best of Luck_________________________

CONTROL SYSTEM (EE0704) STUDY NOTES · Control System Engineering – By: I.J.Nagrath, M.Gopal Automatic Control Systems – By Benjamin C.Kuo Control Systems Engineering – By S.N.Sivanandam

Documents