eee iv control systems [10es43] notes

Control Systems 10ES43

Department of EEE, SJBIT Page 1

10ES43 CONTROL SYSTEMS (Common to EC/TC/EE/IT/BM/ML)

Subject Code:10ES43 IA Marks: 25

No. of Lecture Hrs./Week : 04 Exam Hours : 03

Total No. of Lecture Hrs.:52 Exam Marks : 100

PART A

UNIT 1:

Modeling of Systems: Introduction to Control Systems, Types of control systems, Effect of

feedback systems, Differential equations of physical systems Mechanical systems-

Friction, Translational systems (Mechanical accelerometer, Levered systems excluded),

Rotational systems, Gear trains. Electrical systems, Analogous systems. 6 Hours

UNIT 2:

Block diagrams and signal flow graphs: Transfer functions, Block diagrams, Signal Flow

graphs (Statevariable formulation excluded). 7 Hours

UNIT 3:

Time Response of feed back control systems: Standard test signals, Unit step response of

First and second order systems, Time response specifications, Time response specifications

of second order systems, steady state errors and error constants. 7Hours

UNIT 4:

Stability analysis: Concepts of stability, Necessary conditions for Stability, Routh-Hurwitz

stability criterion, Relative stability analysis; Special cases of RH criterion. 6 Hours

PART B

UNIT 5:

Root Locus Techniques: Introduction, basic properties of root loci, Construction of root

loci. 6 Hours

UNIT 6:

Stability analysis in frequency domain: Introduction, Mathematical preliminaries, Nyquist

Stability criterion, (Inverse polar plots excluded), Assessment of relative stability using

Nyquist criterion, (Systems with transportation lag excluded). 7Hours



UNIT 7:

Frequency domain analysis: Correlation between time and frequency response, Bode plots,

All pass and minimum phase systems, Experimental determination of transfer functions,

Assessment of relative stability using Bode Plots. 7 Hours

UNIT 8:

Introduction to State variable analysis: Concepts of state, state variable and state models for

electrical systems, Solution of state equations. 6 Hours

TEXT BOOK :

1. Control Systems Engineering, I. J. Nagarath and M.Gopal, New Age International (P)

Limited, 4 Edition 2005

2 Modern Control Engineering, K. Ogata, PHI, 5th Edition, 2010.



Table of contents Sl .no Contents Page number

1

UNIT 1:Modeling of Systems:

Introduction to Control Systems

5 to 12

Types of control systems, Effect of feedback systems,

Differential equations of physical systems Mechanical

systems- Friction

Translational systems (Mechanical accelerometer,

Levered systems excluded)

Rotational systems, Gear trains. Electrical systems,

Analogous systems.

2

UNIT 2: Block diagrams and signal flow graphs

Transfer functions

13 to 23 Block diagram

Signal Flow graphs (Statevariable formulation excluded).

3

UNIT 3: Time Response of feed back control systems:

Standard test signals

24 to 46

Unit step response of First and second order systems,

Time response specifications

Time response specifications of second order systems,

steady

state errors and error constants.

4

UNIT 4: Stability analysis:

Concepts of stability

47 to 65 Necessary conditions for Stability

Routh-Hurwitz stability criterion

Relative stability analysis



Special cases of RH criterion.

5

UNIT 5: Root Locus Techniques:

Introduction

66 to 84 basic properties of root loci

Construction of root loci.

6

UNIT 6: Stability analysis in frequency domain:

Introduction

85 to 102

Mathematical preliminaries

Nyquist Stability criterion, (Inverse polar plots excluded)

Assessment of relative stability using Nyquist criterion,

7

UNIT 7: Frequency domain analysis:

Correlation between time and frequency response

103 to 118

Bode plots

All pass and minimum phase systems

Experimental determination of transfer functions

Assessment of relative stability using Bode Plots.

8

UNIT 8: Introduction to State variable analysis:

Concepts of state

119 to 127 state variable and state models for electrical systems

Solution of state equations



UNIT-1

Modeling of Systems

Introduction to control systems

A system is an arrangement of or a combination of different physical components connected

or related in such a manner so as to form an entire unit to attain a certain objective

Control system is an arrangement of different physical elements connected in such a

manner so as to regulate, director command itself to achieve a certain objective

Requirements of good control system are accuracy, sensitivity, noise, stability, bandwidth,

speed, oscillations

Types of control systems

A system in which the control action is totally independent of the output of the system is

called as open loop system

Example: Automatic hand driver, automatic washing machine, bread toaster, electric lift,

traffic signals, coffee server, theatre lamp etc.

A system in which the control action is somehow dependent on the output is called as

closed loop system

The elements of closed loop system are command, reference input, error detector, control

element controlled system and feedback element



Elements of closed loop system are:

1. Command : The command is the externally produced input and independent of the

feedback control system.

2. Reference Input Element: It is used to produce the standard signals proportional to the

command.

3. Error Detector : The error detector receives the measured signal and compare it with

reference input. The difference of two signals produces error signal.

4. Control Element : This regulates the output according to the signal obtained from error

detector.

5. Controlled System : This represents what we are controlling by feedback loop.

6. Feedback Element : This element feedback the output to the error detector for

comparison with the reference input.

Example: Automatic electric iron, servo voltage stabilizer, sun-seeker solar system, water

level controller, human perspiration system.

Feedback system is that in which part of output is feeded back to input

In feedback system corrective action starts only after the output has been affected

Advantages of closed loop system:

1. Accuracy is very high as any error arising is corrected.

2. It senses changes -in output due to environmental or parametric change, internal

disturbance etc. and corrects the same.

3. Reduce effect of non-linearities.

4. High bandwidth.

5. Facilitates automation.

Disadvantages

1. Complicated in design and maintenance costlier.

2. System may become unstable.



Advantages of open loop system:

1. They are simple in construction and design.

2. They are economic.

3. Easy for maintenance.

4. Not much problem of stability.

5. Convenient to use when output is difficult to measure.

Disadvantages of open loop system

1. Inaccurate and unreliable because accuracy is dependent on accuracy of calibration.

2. Inaccurate results are obtained with parameter variations, internal disturbances.

3. To maintain quality and accuracy, recalibration of controller is necessary from time to

time.

Feed forward system is that in which the corrective action is initiated without waiting for

the effect of disturbance to show up in the output

System having multiple inputs and multiple outputs is known as multiple output (MIMO)

control system

A servomechanism is a power amplifying feedback control system in which the controlled

variable is mechanical position or its time derivative such as velocity, acceleration

A regulator or regulating control system is a feedback control system in which the

reference input remains constant for long periods/entire intervals of operation

An adaptive control system is one that continuously and automatically measures the

dynamic characteristics of the plant.

The system which follows the principle of superposition and proportionality is called a

linear system.

The motion take place along a straight line is known as translational motion.

Rotational motion of a body is the motion about a fixed axis.

The elements of rotational system are inertia (J), damping coefficient (B) and torsional

stiffness (K).

Mechanical system



For every mechanical system, there is analogous electrical system.

forces resisting motion in any given direction is zero.

For mechanical network, analogous electrical network can be obtained by using f-v and f-i

analogy.

Force-voltage analogy: In this method force is analogous to voltage.

Similarly, displacement n charge q.

Force-current analogy: In this method force is analogous to current.

Mechanical rotational system:

(a) Force-voltage analogy:

(b) Force-current analogy :

Mechanical coupling:

Laplace transform of signal x (t) is denoted by X (s)

Example problems:

Q 1. Draw the f-1 analogous mechanical system for the electrical circuit of fig. below:



Ans. f (t) is analogous to e (t)

f (t) is analogous to R.

f (t) frictional force f is analogous to r.

Spring constant K is analogous to reciprocal

Mass M is analogous to inductance L

Q. 2. Draw the mathematical model of the following

system and obtain the transfer function.



Ans. Using torque-voltage analogy, we have

The equation is

Q. 3. Write force-current analogous quantities.

Ans. Force analogous to current



Q. 4. Draw the mathematical model of the following

system and obtain the transfer function.



Ans. Writting nodal equation

At node x



UNIT-2

Block diagrams and signal flow graphs

Transfer function

The ratio of laplace transform of the output to the laplace transform of input under the

assumption of zero initial conditions is defined as the transfer function Of system. It is

denoted by G(s).

Importance : Transfer function is highly important because of following reasons

1. It is used to give the gain of given blocksystem.

2. The system poles/zeros can be found from transfer function.

3. Stability can be determined from characteristic equation.

4. The system differential equation can be obtained from transfer function by replacing.

s-variable with linear differential operator

Significance of Transfer Function

Where C(s) is laplace transform of output

R(s) is laplace transform of input.

Transfer function gives the gain of the given block system.

Properties of Transfer Function

1. The transfer function is independent of the inputs to the system.

2. The transfer function of a system is the laplace transform of its impulse response for zero

initial conditions.

3. The system poles/zeros can be found out from transfer function.

4. The transfer function is defined only for linear invariant systems. It is not defined for

non-linear systems.

Limitations of transfer function are listed below

1 Transfer function is valid only for linear time invariant system.

2 It does not take into account the initial conditions initial conditions loose its significance.

3 It does not give any idea about how the present output is progressing.



Poles are the v

make the transfer function value as infinity

make the transfer function value as zero

The characteristic equation can be obtained by equating the denominator polynomial of the

transfer function to zero

The number of poles at the origin defines the type of system

Block diagram algebra

Block diagram gives a pictorial representation of a control system by way of short handing

the transfer function Signal flow graph further shortens the representation of a control

system by eliminating summing symbol take-off point and block This elimination is

A pictorial representation of the relationship between input and output of a system is termed

as block diagram.

The direction of flow of signal from one block to other is indicated by an arrow.

The point in a block diagram at which signal can be added or subtracted is termed as

summing point.

Gain is the ratio of laplace transform of output to laplace transform of input .

Blocks in series are algebraically combined by multiplication.

The lines drawn between the blocks to indicate the connections between the blocks are

termed as branches.

The point from which a signal is taken for the feedback purpose is called as take-off point.

The order of summing point can be changed if two or more summing points are in series.

Signal Flow Graphs



A signal flow grow is a pictorial representation of a system and it displays graphically, the

transmission of signal in system

Node: It is a point from where branches originate or terminate or pass through.

Branch : It is connecting link between two nodes.

Path : The time traced by connecting two or more node is called path.

Loop : It is a path that originates and terminates on same node and along which node other

node is traversed more than once.

Mason s gain formula is used to find the gain of signal flow graph

where i = Number of forward path

= Gain of ith forward path

= System determinant

= 1 (sum of all individual loops) + (sum of all gain products of two non-

touching loops) - (sum of all gain product of three non-

The gain associated with each branch is called branch transmittance

The independent and dependent variable of a control system are represented by small

circles as nodes.

The relationship between nodes is represented by drawing a line between two nodes Such

Example problems:

Q. 1. Find out the for the system shown in the following block diagram. Dec/Jan

2010



Ans. First draw the signal flow graph.

Step I Obtain total number of forward paths

There is only one forward path

Step II. Total number of single loop

There are two loops. Thus

Step III. Value of

As there is one forward path which touch all the loops

Step IV. Obtain transfer function

Q. 2. The transfer function of a system is .

Calculate the phase shift at

Ans.

Hence here is phase shift of zero corresponding to



Q. 3. Laplace transform of a function f (t) is given by

Find out the initial and final values of f (t).

Ans.

Applying final value theorem

Q. 4. Find out the inverse Laplace transform of the function .

Ans.

The term can be factorized as, (s + 2) (s + 3)

Using partial fraction expansion



Q. 5. Represent the following set of equations by a signal June/July

2011

flow graph and determine the overall gain relating .

Ans. Given equations are :

required signal flow graph is

Step I. Obtain the total number of forward paths

Step II. Obtain the number of single loops



Step lll. Obtain the number of two non-touching loop

Step IV. Number of three non-touching loops

--no

Step V. Find the value of

Overall transform function

Q. 6. Simplify the block diagram in fig and obtain the transfer June/July

2010

function relating C(s) and R(s).

Ans.



There are no loops.

Q. 21. From the block diagram shown in the figure below draw the corresponding

signal flow graph and evaluate close loop transfer function relating

Dec/Jan 2006

the output and input.

Ans. Required signal flow graph is:

s gain formula:

Individual loop,

Non-touching loops = 0



Q. 7. The transfer function of a system is given by

Determine the state model in canonical form using parallel decomposition method.

Ans.

Using partial fractions



State diagram representation in diagonal canonical form by parallel decomposition

Q. 8. Determine the transfer function of the system given in fig.

Ans.

eee iv control systems [10es43] notes

Documents

order systems

electrical systems

levered systems

rotational systems

modeling of systems

analogous systems

types of control systems

control systems engineering