7.

SUBJECT DETAILS

7.6. LINEAR CONTROL SYSTEMS

7.6.1 Objective and Relevance

7.6.2 Outcome

7.6.3 Prerequisites

7.6.4 Syllabus

i.JNTU

ii.GATE

iii.IES

7.6.5 Suggested Books

7.6.6 Websites

7.6.7 Experts’ Details

7.6.8 Journals

7.6.9 Findings and Developments

7.6.10 Session Plan

i. Theory

ii.Tutorial

7.6.11 Student Seminar Topics

7.6.12 Question Bank

i.JNTU

ii.GATE

iii.IES
7.6.1 OBJECTIVE AND RELEVANCE

In this course it is aimed to introduce to the students the principles and

applications of control systems in everyday life. The basic concepts of block
diagram reduction, time domain analysis solutions to time invariant systems
and also deals with the different aspects of stability analysis of systems in
frequency domain and time domain.
The main objective of this course is to give the student a complete
understanding of control systems that play a vital role in the advances in
engineering and science. In addition to its extreme importance in space
vehicle systems, missile guidance systems, robotic systems and the like,
automated control has become an important and integral part of modern
manufacturing and industrial processes. The advances in theory and practice
of automatic control provide the means for attaining optimal performance of
dynamic systems, improving productivity, relieving the drudgery of many
routine repetitive manual operations and the time saving of various
operations of the systems. This course is introduced to enable the student to
understand the process of simulation and modeling of the control systems.

7.6.2 OUTCOME

After going through this course the student gets a thorough knowledge on
open loop and closed loop control systems, concept of feedback in control
systems, mathematical modeling and transfer function derivations of
translational and rotational systems, Transfer functions of synchros, AC and
DC servo motors, Transfer function representation through block diagram
algebra and signal flow graphs, time response analysis of different ordered
systems through their characteristic equation and time domain specifications
, stability analysis of control systems in S -domain through R-H criteria and
root locus techniques, frequency response analysis through bode diagrams,
with which he/she can able to apply the above conceptual things to real world
electrical and electronics problems and applications.

7.6.3 PREREQUISITES

Basic knowledge on network theory, Fourier and Laplace transforms

elementary matrix algebra and basic electrical machines theory.
7.6.4.1 JNTU SYLLABUS

UNIT-I
OBJECTIVE
The objective of this unit is to give an idea on various control systems, the
importance of feedback in control systems and the methodology of obtaining
mathematical model of any physical system.

SYLLABUS
INTRODUCTION:
Concepts of control systems, open loop and closed loop control systems and their
differences, different examples of control systems, classification of control sys tems,
feedback characteristics, and effects of feedback. Mathematical models, differential
equations, impulse response and transfer functions, translational and rotational
mechanical systems.

UNIT-II
OBJECTIVE
The objective of this unit is to deal with the constructional and operational features
of control system components and the various tools available for the reduction of
complex systems into simpler ones.

SYLLABUS
TRANSFER FUNCTION REPRESENTATION:
Transfer function of DC servo motor, AC servo moto r, synchro transmitter and
receiver, block diagram representation of systems considering electrical systems as
examples, block diagram algebra, representation by signal flow graph, reduction
using Mason’s gain formula.

UNIT-III
OBJECTIVE
The objective of this unit is to deal with the transient and steady state analysis of
various systems for different types of test signals and the study of proportional
derivative and proportional integral systems.

SYLLABUS
TIME RESPONSE ANALYSIS:
Standard test signals, time response of first order systems, characteristic equation of
feedback control systems, transient response of second order systems, time domain
specifications, steady state response, steady state errors and error constants, effects
of proportional derivative, proportional integral systems.
UNIT-IV
OBJECTIVE
The objective of this unit is to deal with the concept of stability, various stability
criterions for determining the stability of a given system and the graphical
technique available for the analysis and design of control systems.

SYLLABUS
STABILITY ANALYSIS IN S-DOMAIN:
The concept of stability, Routh's stability criterion, qualitative stability and conditional stability,
limitations of Routh's stability
Root Locus Technique:
The root locus concept, construction of root loci-effects of adding poles and zeros to G(s)H(s) on
the root loci.

UNIT-V
OBJECTIVE
The objective of this unit is to deal with the analysis of control systems using
frequency response plots and determination of transfer function from Bode plots.

SYLLABUS
FREQUENCY RESPONSE ANALYSIS:
Introduction frequency domain specifications, Bode diagrams, determination of
frequency domain specifications and transfer function from the Bode diagram phase
margin and gain margin-stability analysis from Bode plots.

7.6.4.2 GATE SYLLABUS

UNIT –I
Principles of feedback

UNIT-II
Transfer function, block diagram

UNIT-III
Response for linear time invariant systems, steady state errors

UNIT-IV
Routh stability criterion, root locus

UNIT-V
Bode plots
7.6.4.3 IES SYLLABUS

UNIT -I
Mathematical modelling of physical system

UNIT -II
Control systems components: Electromechanical components block diagrams
and signal flow graphs and the reduction.

UNIT-III
Time domain and frequency domain analysis of linear dynamical system.
Errors for different type of inputs

UNIT-IV
Stability analysis using Routh -Hurwitz array, root locus

UNIT-V
Bode plot, estimation of gain and phase margin

7.6.5 SUGGESTED BOOKS

TEXT BOOKS
T1 Control Systems theory and applications, S.K Bhattacharya, Pearson.
T2 Control Systems,N.C.Jagan,BS Publications

REFERENCE BOOKS
R1 Control Systems, Anand Kumar, PHI, 2008.
R2 Control Systems Engineering,S.Palani,Tata -McGraw-Hill.
R3 Control Systems, Dhanesh N.Manik, Cengage Learning.
R4 Control Systems Engineering, I. J. Nagrath and M. Gopal, 2nd Edn.,New Age
International (P) Ltd.
R5 Control Systems by N.K Sinha, 3 rd Edn.,New Age International (P) Ltd,
1998.

7.4.6 WEB SITES

1. www.contro.eng.com.ac.uk
2. melot.ee.usyd.edu.au
3. ocw.mit.edu
4. regpro.mechatronic.uni
5. www.control.utoronto.ca
6. www.controlengg.com
7. www.insrumentation.com
 8. www.ecpsystems.com
9. www.ieeecss.org

7.4.7 EXPERTS’ DETAILS

INTERNATIONAL
1. Dr.Rodolphe Sepulehre,
Professor in Systems and Modelling,
University of Toronto,
email:r.sepulehre@ulg.ac.be.

2. Dr.W.M.Wonham,
Systems Control Group,
Department of Electrical and Computer Engineering,
University of Toronto,
email : wonham@control.toronto.edu.

3. Mr. S.M. Joshi,

NASA, Longley Research Center,
Hampton, USA,
email :smjoshi@lage.nasa.gov.

4. Mr.Sameer S. Saab,
Department of Electrical and Computer Engineering,
Lebanese American University,
Byblous,Lebanan,
email :ssaab@lau.edu.lb.

NATIONAL
1. Dr. Sanjay P. Bhat,
Department of Aero Space Engineering,
IIT, Bombay,
email :bhat@aero.iitb.ac.in.

2. Dr. Jagadish Kumar,

Department of EEE,
IIT,Madras, Chennai,
email : vjk@iitm.ac.in.

REGIONAL
1. Dr. T. Lingareddy,
Professor and HoD,
Electrical and Electronics Engineering Department,
Chaitanya Bharathi Institute of Technology,
Gandipet, Hyderabad,
 Ph.: 0841-3232486 (Off), 0841-3233909 (Res).

2. Dr.S.Partha Sarathy,
Chief Consultant,
Algologic Research and Solutions,
Secunderabad,
email : drpartha@gmail.com.

7.6.8 JOURNALS

INTERNATIONAL
1. IEEE Transactions on Automatic Control
2. IEEE Control Systems Magazine
3. Instrumentation and Control

NATIONAL
1. Journal of Institution of Engineers (Electrical Enggneering)
2. Journal of Systems Society of India
3. Journal of Instrument Society of India

7.2.9 FINDINGS AND DEVELOPMENTS

1. Sufficient Conditions for Closed Loop Asymptotic Controllability and

Stabilization by Smooth Time-varying Feedback Integrator, J. Tsinias, IEEE
Transactions on Automatic Control System, Vol. 53, No 8, October 2008.
2. Robust Adaptive Control of Feedback Linearizable MIMO Non-linear
Systems with Prescribed Performance, C.P. Bechliou Lis and G A Rovithakis,
IEEE Transactions on Automatic Control System, Vol. 53, No 8, October
2008.
3. Input to State Stability of Time Delay Systems: A Link with Exponentia l
Stability, P. Pipe and M.Dambrise, Transactions on Automatic Control
System, Vol 53, No. 6, July 2008.
4. A Scenario Approach to Robost Control Design, Giuseppe C.Cda fire and
Marco. C. Campi, IEEE Transactions on Automatic Control System, Vol.51,
No.5, May 2006.
5. Non Blocking Supervisory Control of State Tree Structures, Chuan. Ma and
W.M.Wonham, IEEE Transactions on Automatic Control System, Vol.51,
No.5, May 2006.
7.1.10 SESSION PLAN

i. THEORY

Sl. Topics in JNTU Lecture Suggested

Modules and Sub modules Remarks
No. syllabus No. Books
UNIT-I
1 Concepts of control Definition T1-Ch1, T2-Ch1 GATE
systems Examples of day to day L1 R1-Ch1, R2-Ch1
application of control systems R3-Ch1
2 Open loop and closed Open and closed loop definition T1-Ch1, T2-Ch1 GATE
loop control systems Comparison between open and L2 R1-Ch1, R2-Ch1 IES
and their difference closed loop R3-Ch1
Examples of open loop and closed T1-Ch1, T2-Ch1 GATE
loop control systems L3 R1-Ch1, R2-Ch1 IES
R3-Ch1
3 Different examples of Different examples of control T1-Ch1, R1-Ch1
control systems systems R2-Ch1, R3-Ch1
L4
Classification of Classification of control systems
control systems
4 Feedback Feedback definition and T1-Ch1, T2-Ch3 GATE
characteristics importance R1-Ch1, R2-Ch4 IES
L5
Effects of feed back Effect of feedback on gain
Effect of feedback on stability
Reduction of parametric variations T1-Ch1, T2-Ch3 GATE
by use of feedback R2-Ch4, R3-Ch7 IES
Sensitivity definition
Sensitivity of system to parametric
L6
changes for open loop and closed
loop system
Effect of feedback on external
disturbances
Control over system dynamics by T2-Ch3,R1-Ch1 GATE
feedback R2-Ch4 IES
Effect of feedback on system L7
dynamics
Effect of feedback on band width
5 Mathematical models Transfer function, definition T1-Ch1, T2-Ch2 GATE
Differential equations Concept of poles and zeros R1-Ch3, R2-Ch2 IES
Impulse response and Transfer function from differential L8 R3-Ch2
transfer functions equations of physical systems
Impulse response model
6 Translational and Significance of mathematical T1-Ch1, T2-Ch2 GATE
rotational mechanical modelling R1-Ch3, R2-Ch2 IES
systems Modelling of mechanical systems L9 R3-Ch2
using translational and rotational
elements
Modelling of electrical systems T1-Ch4, T2-Ch2 GATE
L10
R2-Ch2, R3-Ch2 IES
Analogy between mechanical T2-Ch2, R2-Ch2 GATE
translational, rotational and L11 R3-Ch2 IES
electrical systems
Sl. Topics in JNTU Modules and Sub modules Lecture Suggested Books Remarks
No. syllabus No.
UNIT-II
7 Transfer functions of Basic principle and operation T1-Ch4, T2-Ch4
DC servomotor Features of field and armature R1-Ch4, R2-Ch2
controlled DC servomotor L12
Characteristics and applications of
DC servomotor
8 AC servomotor Construction principle and T2-Ch4,R2-Ch2
operation
Torque-speed characteristics L13
Features and application of AC
servomotor
9 Synchro transmitter Synchro control transformer T2-Ch4
L14
and receiver Applications
10 Block diagram Block diagram representation of T1-Ch4, T2-Ch2 GATE
representation of RC circuits, DC motor armature R2-Ch2, R3-Ch2 IES
systems considering and field control L15
electrical systems as
examples
11 Block diagram Block diagram of closed loop T1-Ch4, T2-Ch2 GATE
algebra system R2-Ch2, R3-Ch2 IES
L16
Block diagram of SISO and
MIMO systems
Rules of block diagram algebra T1-Ch3, T2-Ch2 GATE
L17
Problems R2-Ch2 IES
12 Representation of Properties and terminology of T1-Ch4, T2-Ch2 GATE
signal flow graph, signal flow graph R2-Ch2, R3-Ch2 IES
reduction through Methods to obtain signal flow L18
Mason’s gain graph from system equations and
formulae block diagram and vice versa
Derivation of Mason’s gain T1-Ch3, T2-Ch2 GATE
formulae R2-Ch2 IES
L19
Comparison of block diagram and
SFG methods
Problems T1-Ch3, T2-Ch2 GATE
L20
R2-Ch2, R3-Ch5 IES
UNIT-III
13 Standard test signals Definition and classification T1-Ch7, T2-Ch5 GATE
Standard test signals R1-Ch4, R2-Ch5 IES
Step L21
Ramp
Parabolic
14 Time response of 1st Time response of 1st order systems T1-Ch7, T2-Ch5 GATE
L22
order systems with step, ramp, parabolic inputs R1-Ch4, R2-Ch5 IES
15 Characteristic Characteristic equation of feedback T1-Ch7, T2-Ch5 GATE
equation of feedback control system L23 R1-Ch4, R2-Ch5 IES
control system
Sl. Topics in JNTU Lecture
Modules and Sub modules Suggested Books Remarks
No. syllabus No.
16 Transient response of Derivation of unit step response T2-Ch4, T2-Ch5 GATE
L24
2nd order systems of 2nd order systems R1-Ch7, R2-Ch5 IES
 Effect of zeta on 2nd order system R3-Ch4
performance
17 Time domain Derivation of peak time, peak T2-Ch4, T2-Ch5 GATE
specifications overshoot, settling time and rise L25 R1-Ch7, R2-Ch5 IES
time R3-Ch4
18 Steady state response Derivation of steady state error T2-Ch5, R1-Ch7 GATE
Steady state errors and Effect of input (type and R2-Ch5, R3-Ch7 IES
error constants magnitude) on steady state error
L26
Analysis of type 0,1 & 2 systems
Disadvantages of steady state
error coefficients method
Problems T2-Ch5, R1-Ch7 GATE
L27
R2-Ch5, R3-Ch7 IES
19 Effects of proportional PD and PI controllers R1-Ch10, T2-Ch5 GATE
derivative and Transfer function R3-Ch9 IES
L28
proportional integral Purpose
systems Realization
UNIT-IV
20 The concepts of BIBO Stability T2-Ch6, R1-Ch6 GATE
L29
stability Absolute and relative stability R2-Ch6, R3-Ch6 IES
21 Routh stability Hurwitz criterion T2-Ch5, T2-Ch6 GATE
criterion Necessary conditions R1-Ch6, R2-Ch6 IES
Qualitative stability, Routh criterion R3-Ch6
L30
conditional stability Special cases
and limitations of Applications
Routh’s stability Advantages and limitations
Problems T2-Ch6, T2-Ch5 GATE
L31 R1-Ch6, R2-Ch6 IES
R3-Ch6
22 The root locus concept Basic concept of root locus T2-Ch6, T2-Ch7 GATE
construction of root Angle and magnitude criterion L32 R1-Ch8, R2-Ch7 IES
loci R3-Ch6
Rules for construction of root T2-Ch6, T2-Ch7 GATE
locus R1-Ch8, R2-Ch7 IES
L33
Advantages of root locus method R3-Ch6
Problems
Problems T2-Ch6, T2-Ch7 GATE
L34 R1-Ch8, R2-Ch7 IES
R3-Ch6
23 Effects of adding poles Effects of adding open loop poles T2-Ch10, R1-Ch8 GATE
and zeros to G(s)H(s) and zeros L35 R3-Ch4 IES
on the root loci
Sl. Topics in JNTU Lecture
Modules and Sub modules Suggested Books Remarks
No. syllabus No.
UNIT-V
24 Introduction Advantages and limitations of T2-Ch8, R1-Ch9 GATE
L36
frequency response methods R2-Ch8 IES
25 Frequency domain Co-relation between time domain T1-Ch9, T2-Ch8 GATE
specifications and frequency domain for 2nd R1-Ch8 IES
L37
order systems
Problems
26 Bode diagrams Bode plots of standard factors of T1-Ch9, T2-Ch8 GATE
L38
G(jw) H(jw) R1-Ch8, R3-Ch10 IES
Steps to sketch Bode plot T1-Ch9, T2-Ch8 GATE
L39
Advantages of Bode plot R1-Ch8, R2-Ch8 IES
 Problems R3-Ch10
Problems T1-Ch9, T2-Ch8 GATE
L40 R1-Ch8, R2-Ch8 IES
R3-Ch10
27 Determination of Concept of phase and gain T1-Ch9, T2-Ch9 GATE
frequency domain margin R1-Ch8, R3-Ch10 IES
L41
specifications and Calculation and improvement of
transfer function from G.M and P.M.
Bode diagrams Relative stability analysis from T1-Ch9, T2-Ch9 GATE
Phase margin and gain G.M and P.M R1-Ch8, R3-Ch10 IES
L42
margin Calculation of transfer function
Stability analysis from from magnitude
Bode plots Problems T1-Ch9, T2-Ch9 GATE
L43
R1-Ch8, R3-Ch10 IES

ii. TUTORIAL

S. No Topics scheduled Salient topics to be discussed

1 Concepts of control systems Discussion of various examples of
Open loop and closed loop control system and their control systems in day to day
difference life Classification of control
Different examples of control systems systems
Classification of control systems
2 Feedback characteristics of control systems Transfer functions of various electrical
Mathematical models Differential equations networks
Impulse response and transfer functions
3 Translational and rotational mechanical systems Determination of transfer function of
Block diagram representation of systems considering translational and rotational
electrical systems as examples mechanical systems
Representation of complex systems by
block diagram
4 Block diagram algebra Problems on block diagram algebra
Representation of signal flow graph and Mason's gain formula
Reduction through Mason’s gain formulae
5 Transfer functions of DC servomotor and AC servomotor Review on mathematical models of DC
Synchro transmitter and receiver servomotor and AC
servomotor
6 Standard test signals Transient response of 2nd order system
Time response of 1st order systems for various standard test
Characteristic equation of feedback control system signals
Transient response of 2nd order systems
7 Time domain specifications Problems on time domain
Steady state response specifications
Steady state errors and error constants Steady state errors and error constants
Effects of proportional derivative and proportional integral
systems
8 The concepts of stability Problems on Routh stability criterion
Routh stability criterion
Qualitative and conditional stability
9 Root locus concept Problems on construction of root locus
Construction of root locii for various systems
Effects of adding poles and Zeros to G(s)H(s) on root locii
10 Frequency response analysis Problems on frequency domain
Introduction specifications
Frequency domain specifications Construction of Bode plot for various
Bode diagrams systems
11 Determination of frequency domain specifications and Problems on determination of transfer
transfer function from Bode diagrams function from Bode plots
Phase margin and gain margin Problems on phase margin and gain
Stability analysis from Bode plot margin

7.6.11 STUDENT SEMINAR TOPICS

1. Nano Positioning for Probe-based Data Storage, Abu Sebastian, Vol. 28, No. 4, IEEE
Control Systems Magazine, August 08.
2. Sensing and Control in Optical Drives, Amir H. Chaghajerdi, Vol. 28, No. 3, IEEE
Control Systems Magazine, June 08.
3. Functional Electrical Stimulation, Closed-loop Control of Induced Muscle Contractions,
C.L. Lynch and M.R. Popovic, Vol. 98, No. 2, IEEE Control Systems Magazine, April
08.
4. Coordinating Control for an Agricultural Vehicle with Individual Wheel Speeds and
Steering Angles, Robo Armwrestler, Chul-Goo Kang, Vol. 28, No. 5, IEEE Control
Systems Magazine, October 08.
5. Control of Variable Speed Wind Turbines,K.E.Johnson,Lucy Y.Rao,Mark.J.Balas and
Lee J.Fingersh,Vol.26,No.3,IEEE Control System Magzine,June 2006.
6. Robust Tuning of PI and PID Controllers, Birgitta Kristiansson and Bengt
Lennartson,Vol.26, No.1, IEEE Control System Magzine, February 2006.

7.6.12 QUESTION BANK

UNIT-I

1. a) Explain the feedback effect on parameter variation.

b) Find the transfer function of the following system show in figure.

(Dec 14,May/June 13,Sep 08)

1. a) Explain the various types of control systems with suitable examples.

2. a) Derive the relavent expressions to establish the effect of feedbackon

sensitivity ,signal to noise ratio and rate of response
 b) For the electrical systems shown in the figure P1(C)for the given input i 1 ,find
v 3 in terms of i 1

(Nov/Dec 12)
3. For the geared system shown below in Figure 3, find the transfer function relating the
angular displacement θL to the input torque T1, where J1, J2, J3 refer to the inertia of
the gears and corresponding shafts. N1, N2, N3, and N4 refer to the number of teeth on
each gear wheel.

(May/June 2012)

4. Derive the differential equation relating the position y (t) and the force f (t) as shown
in figure 5.Determine the transfer function y/f

(May/June 2012)

5. A load having moment of inertia J and frictional coefficient B is driven through

a gear as shown in Figure 10, write the differential equations and find transfer
function θ2(s)/T(s)
 (May/June 2012)
6. (a) Distinguish between open loop and closed loop systems. Explain merits and
demerits of open loop and closed loop systems.
(b) With suitable example explain the classification of control systems.
(May/June 2012)
7. i. Explain the effect of feedback on stability.
ii. Explain the temperature control system concepts using open loop as well as
closed loop system.
(Apr 11)
8. i. Explain temperature control system with neat block diagram.
ii. Human being is an example of closed loop system. Justify your answer.
(Dec 10)

9. i. Explain sensitivity?
ii. Determine the sensitivity of the closed loop system shown in fi gure (b) at
w= 1 rad/sec w.r.t
a. forward path transfer function
b. feedback path transfer function.
(Dec 10)

10. Explain the following terms :

i. Impulse response
ii. Rotational mechanical systems
iii. Translational systems
iv. Sensitivity. (Dec 10)

11. i. How can you control the system dynamics by using feedback?
ii. What is a mathematical model of a physical system? Explain briefly. (Dec 10)

12. i. Explain the traffic control system concepts using open loop as well as closed
loop system.
ii. Why is negative feedback invariably preferred in closed loop systems?
(May 10)
13 .Define and explain the following terms
i. Characteristic equation
ii. Order of a transfer function
iii. Type of a transfer function
iv. Poles and zeros of a transfer function. (May 10, 09)

14. i. Illustrate at least three applications of feedback control systems?

ii. Explain translatory and rotary elements of mechanical systems? (May 10)

15. i. Write the important differences between open loop and closed loop systems
with suitable examples.
ii. Obtain the transfer function of the following system. Shown in figure.

(May 10)
16. i. Find the transfer function of the following system show in figure.

ii. Derive the transfer function of the following network figure by assuming
R 1 =5Mohms and R 2 = 5Mohms, C 1 =0.1uF and C 2 =0.1uF.

(May 09, Sep 08)

17.i. Explain the difference between systems with feedback and without feedback.
ii. Explain the advantages of systems with feedback, with suitable examples.
(May 09)

18.i. Explain and derive the relation between impulse response and transfer
function.
C ( s)
ii. Find the transfer function R( s) of a system having differential equation.

(May 09)

19. Explain the following terms:

i. Linear systems and nonlinear systems
ii. Continuous systems and discrete systems
iii. Single input-single output systems (SISO) and multiple input-multiple output
systems (MIMO)
iv. Static systems and dynamic systems. (May 09)

20. i Explain the linearizing effect of feedback.

ii. The dynamic behaviour of the system is described by the equation
, where ‘e’ is the input and ‘C’ is the output. Determine the
transfer function of the system. (May 09)

21. i. Explain the effect of feedback on bandwidth?

ii. Explain reduction of parameter variations by use of feedback systems?
(May 09)

22. Define and explain the following terms

i. Characteristic equation
ii. Order of a transfer function
iii. Type of a transfer function
iv. Poles and zeros of a transfer function. (May 09)

23.i. Find the transfer function for the following mechanical system: Shown in
figure.

ii. Explain the limitations of closed loop system over open loop system.
 (Sep 08)

24. i.Obtain the transfer function for the following network Figure.

ii. Explain the effects of disturbance signals by use of feedback. (Sep 08)

26. i. Define transfer function and what are its limitations?

ii. Find the transfer function of the following system.Shown in figure.

(May 08)

27. i. Explain the traffic control system concepts using open loop as well as closed
loop system.
ii. Why is negative feedback invariably preferred in closed loop systems?
(May 08)

28. i. Explain the effect of feedback on stabili ty.

ii. Explain the temperature control system concepts usi ng open loop as well as
closed loop system. (Sep 08)

29. i. Write the important differences between open loop and closed loop systems
with suitable examples.
X 0 (s)
ii. Obtain the transfer function X 1 (s) of the following system. Shown in figure.
 (May 08)

30. i. For the mechanical system Figure given, write down the differential
equations of motion and hence determine the Y2(s)/F(s).

ii. Describe the analogy between electrical and mechanical systems. (May 08)

31. i. Explain, with example, the use of control system concepts to engineering and
non engineering fields
ii. For the electrical network given below, derive the transfer function

(May 08, 05)

32. i. Explain the basic components of control systems?

ii. Find the transfer function for the system given figure.

Where, M is the mass of the system.

K is the spring deflection
B is the coefficient of viscous damping. (May 08)

33. For the mechanical system given in Figure.

 Obtain the following:
i. Mathematical model
X 1 ( s) X ( s)
and 2
ii. The transfer functions. F ( s) F ( s) (May 08, 06)

34. i llustrate atleast three applications of feedback control systems?

ii. Explain translatory and rotary elements of mechanical systems? (Sep 07)

35. i. Obtain the transfer function of the following system and draw its analogous
electrical circuit. Figure.

ii. Explain the advantages and features of transfer function. (Sep, May 07)

36. i. Explain the linearizing effect of feedback.

ii. The dynamic behaviour of the system is described by the equation, dC / dt +
10C = 40e, where ‘e’ is the input and ‘C’ is the output. Determine the
transfer function of the system. (Sep 07)

37 .i. Explain regenerative feedback?

ii. Determine the sensitivity of the closed loop transfer function T(s) =
C(s)/R(s) to variations in arameter K at w = 5 rad/sec. Assume the normal
value of K is

(Sep 07)
38.i. Explain the differences between open-loop and closed-loop systems.
ii. Determine the Transfer Function of the electrical network. Figure
 (May 07)

39. Explain the effect of feedback on noise to signal ratio. (May 07)

40.i. Explain the effect of feedback on the stability of a closed loop system?
ii. Explain the effect of feedback on the sens itivity of a closed loop system?
(May 07, 05)

41. Define the following terms.

i. Concept of the system
ii. Control system. (May 07)

42.i. Explain the classification of control systems?

ii Find the transfer function relating displacement ‘y’ and ‘x’ for the following
system. Shown in figure.

(May 07)
43.i. What is feedback? Explain the effects of feedback?
ii. What is the sensitivity function and explain it with respect to open loop and
closed loop systems?
(May 07)
44.i. Derive the transfer function for the following rotational mechanical systems.
Shown in figure.

ii. List out the limitation of open loop systems over closed loop systems.
(May 07)
45. Explain the operation of ordinary traffic signal, which control automobile
traffic at roadway intersections. Why are they open loop control systems?
How can traffic be controlled more effectively?
(Apr 06)

46. Explain the concept of multivariable control systems. (Apr 06)

47 Find the transfer function X(s)/F(s) of the system figure given below.

(Apr 06)
48. i. Find the transfer function X(s)/F(s) of the system given below:

ii. Define transfer function and determine the transfer function of RLC series
circuit if voltage across the capacitor is output variable and input is voltage
source V(s). (Apr 06, 04)

49. i. Define transfer function and what are its limitations.

ii. Obtain the transfer function for the following electrical network.

(Apr 06, 04)

50. By means of relevant diagrams explain the working principles of a practical

closed loop system.
(Apr 06)
51. Distinguish between:
i. Linear and non linear system
ii. Single variable and multivariable control systems
iii. Regenerative and degenerative feeds back control systems.
Give an example for each of the above. (Apr 05)
52. Define system and explain about various types of control systems with
examples and their advantages.
(Apr 05)

53. i. Explain about various types of control syste ms with examples briefly.
ii. Explain the differences between open loop and closed loop system. (Apr 05)

54. i. For the mechanical system given below, derive an expression for the transfer
function

(Nov 03)

(Apr 04, 03)

55. Derive the transfer function of the following network shown in fig. :

(Apr 04, Nov 03)

56. Discuss the effects of feedback on system dynamics by unit feedback and
regenerative feedback. Give suitable examples (Apr 04)

57. Define system and explain about various types of control systems with
examples and their advantages. (Apr 04)

58. Find the transfer function of a circuit given below

(Nov 03)

59. i. Discuss the advantages of providing feedback to an open -loop control

system.
ii. Find the transfer functions Y 1 (S)/F(S) and Y 2 (S)/F(S) for the system shown
in Fig.
 (Nov 03)

60. For the electrical system shown below draw the signal flow graph and hence
find the gain by Mason’s gain formula.

(Nov 03)

61. Explain the open loop control system with practical example (Nov 03)

62.i. Explain the concepts of control systems.

ii. Obtain the transfer function for the following electrical network.

(Nov 03)

63. For the mechanical system given below, derive an expression for the transfer
function

(Nov 03)

64. i. Explain the terms (a) time response (b) frequency response and (c) transfer
function.
ii. Obtain the transfer function for the following electrical system.
 iii. What are the different types of standard test input signals used in testing the
response of a control system? Which type of input signal is widely used?
Why? (Apr 03)

65. Derive the transfer function of a closed loop system (Apr 03)

66. Draw the analogous (use force-voltage analogy) electrical system for the
mechanical system. Shown in fig.

(Apr 03)
67. i. Explain the following terms with suitable examples. (Jan 03)
a. Open loop
b. Feed back
c. Signal flow graphs.
ii. Obtain the transfer function x(s)/Î(s) for the electromechanical system shown
in figure.
Assuming that the coil has a back emf and the coil current i 2 produces a
force
Fc = k 2 i 2 on the mass M.

68. i. Obtain the mathematical model for the mec hanical system shown in figur
 ii. Draw the force-voltage and force-current electrical analogous circuits for the
system show in fig.
iii. Verify the result by writing mesh and node equations. (Jan 03)

69. What do you understand by the terms “Open -loop system”, “closed loop-
system”, “Manually control system” and automatic control system? Give one
practical example of each type with proper diagram and exp lanation.(IES 93)

UNIT-II
1. a) Derive the transfer function of field controlled dc servo motor.
b) The block diagram of a speed control system is shown below in figure2.
Determine its transfer function.

(Dec 14)

1. a) Determine the transfer function of the block diagram shown in figure 2.

b) Determine the closed loop transfer function for the signal flow graph shown in the figure 3.
 (May/June13)

2. Find the closed loop transfer function of the system whose block diagram is
given in the fig.P2 using block diagram reduction techniques and verify the
result using signal flow graph technique

(Nov/Dec12)

3. (a) Find the transfer function of the system shown in Figure 2.

(b) Find the transfer function of a AC servo motor.

(May/June12)

4. (a) Explain the Armature voltage controlled DC servomotor and obtain its transfer
function.
(b) Obtain the overall transfer function for the block diagram in Figure 4.
 (May/June12)

5. Using block diagram reduction techniques, find the closed loop transfer function
of the system whose block diagram is given in Figure 9 and verify the result using
signal flow graph technique

(May/June12)

6. Using block diagram reduction techniques, find the closed loop transfer function of
the system whose block diagram is given in Figure 11 and verify the result using
signal flow graph technique.

(May/June12)

7. i. Determine the transfer function C(s)/ R(s) for the following block diagram.
 ii. Explain the properties of signal flow graphs. (Apr 11)

8. i. Reduce the given block diagram and hence obtain the transfer function
C(s)/R(s).

ii. Explain the need of Mason's gain formula for any system reduction. (Dec 10)

9. i. The signal flow graph shown in figure has one forward path and two isolated
loops. Determine the overall transfer function relating x and x

ii. Explain the differences between AC s ervomotor and DC servomotor. (Dec 10)

10. i. Reduce the given block diagram and hence obtain the transfer function
C(s)/R(s)

ii. Explain synchro with neat sketch. (Dec 10)

11. i. Reduce the given block diagram and hence obtain the transfer function
C(s)/R(s)
 ii. Explain the practical applications of servomotors. (Dec 10)

12. i. Reduce the given block diagram and hence obtain the transfer function
C(s)/R(s)

ii. Explain synchro transmitter. (May 10)

13. i. Reduce the given block diagram and hence obtain the transfer function
C(s)/R(s)

ii. Explain synchro transmitter. (May 10)

14. i. Reduce the given block diagram and hence obtain the transfer function
C(s)/R(s).
 ii. Explain DC servomotor and its features.

(May 10)
15. Reduce the given block diagram and hence obtain the transfer functions:

i.
ii.
iii.

(May 10)

16. Derive the transfer function for the field controlled d.c. motor with neat
sketch and explain the advantages of field controlled d.c. motor over
armature controlled d.c. motor. (May 09)

17. i. The signal flow graph shown in figure has one forward path and two non -
touching loops. Determine the overall transmittance relating x 6 and x 1 .

ii. Explain the advantages of Mason’s gain form ula. (May 09)

18. Represent the following set of equation by a signal flow graph and determine
the overall gain relating x 5 and x 1 . (May 09)
x 2 = ax 1 + fx 2
x 3 = bx 2 + ex 4
 x 4 = cx 3 + hx 5
x 5 = dx 4 + gx 2
19. i. Reduce the given block diagram and hence obtain the transfer function

ii. Explain the need of Mason’s gain formula for any system reduction. (May 09,
08)

20. i. Derive the transfer function of an a.c. servomotor and draw its
characteristics.
ii. Explain the Synchro error detector with circuit diagram. (May 09, 08, 07, 03)

21. i. With a neat sketch explain the construction and principle of working of the
synchro transmitter and receiver.
ii. Derive the transfer function for the synchro transmitter receiver.(May 09, 04)

22. i. Explain the disadvantages and advantages of block diagram reduction process
over signal flow graph.
ii. Explain the rules of block diagram reduction. (May 09, 07)

23. i. Derive the transfer function of a field controlled d.c. Servomotor and develop
the block diagram. Clearly state the assumptions made in the derivation.
ii. What are the effects of feedback on the performance of a system? Briefly
explain. (Sep 08)

24. i. Determine the overall transfer function relating C and R for the system
whose block diagram is given.

ii. Explain the properties of block diagrams. (Sep 08, 07)

25. i. Explain the advantages of signal flow graph over block diagram reduction
process?
ii. Expalin the following terms related to signal flow graph:
a. Node b. Branch c. Forward path gain d. Loop gain. (Sep 08)

26. i. Obtain the output of the system given below. Figure. (Sep 08)

ii. Determine the overall transfer function from the signal flow graph given in
figure.

27. i. Determine the transfer function for the following block diagram

ii. Explain the properties of signal flow graphs. (Sep 08, 07)
28. i. Determine the transfer function for the following block diagram

ii. Define various terms involved in signal flow graphs. (Sep 08, May 07)
29. i. Reduce the given block diagram and hence obtain the transfer function .
 ii. Explain the need of signal flow graph representation for any system. (May
08)

30. i. Reduce the given block diagram and hence obtain the transfer function

ii. Explain the working principle of synchro receiver with neat sketch. (May 08)

31. i. State and explain mason’s gain formula for the signal flow graph.
ii. What are differences between block diagram reduction and signal flow graph
reduction?
(May 08, Sep 07)
32. Find the transfer function matrix for the two input and output system shown
in the given Figure.

(May 08)
33. i Explain how the potentiometers are used as error seensing devices . Give a
typical application of it with single line diagram.
ii. Discuss the effect of disturbance signal of the speed control system for a
gasolineengine as shown in Figure assuming K=10. (May 08, 06, 04)
34. Explain the procedure for deriving the transfer function and derive the
transfer function for servo.
(Sep 07)
35. Derive the Transfer Function for the field controlled d.c. servomotor with
neat sketch. (May 07)

36. Derive the Transfer Function for a.c. servomotor. Explain about torque -speed
characteristics of a.c. servomotor. (May 07, 06, 04)

37. i.
For the system in the above figure, obtain transfer function
a. C/R b. C/D
ii. Verify the above transfer function using signal flow graph. (May 07)

38. i. Draw the signal flow graph for the system of equations given below and
obtain the overall transfer function using mason’s rule
X2 = X1 - X6
X3 = G1X2 - H2X4 - H3X5
X4 = G2X3 - H4G6
X\ = G 5 X 4
X6 = G4X5
ii. Simplify the Figure 1 of the system given below by block diagram reduction
technique and determine the transfer function of the system.
 (May 07, Apr 04)
39. Reduce the following Figure using block diagram reduction technique find
C/R and verify the transfer function by applying mason’s gain formula.

(May 07)
40. i. For the system represented in the given Figure obtain transfer function (Sep
06, Dec 04)
a. C/R 1 b. C/R 2

ii. Write down the signal flow equations and draw the signal flow graph for the
above system.

41. i. What is feedback and explain about the Reduction of parameter variation by
use of feedback.
ii. Consider the feedback control system shown in figure and the normal value
of process parameter K is 1. Evaluate the sensitivity of transfer function T(s )
= C(s)/R(s) to variations in parameter K.
 (Sep 06)
42. Derive the transfer function of a field controlled dc Servomotor and develop
the lock diagram. Clearly state the assumptions made in the derivation.
(Apr 06)

43. Following figure shows a voltage control device. The gain factors of the
amplifier and generator are 1.5 amp/volt input and 80V/amp field current
respectively.

(Apr 06)

i. If the output voltage were to be 250 volts on no load how much should be the
referece voltage with a feedback potentiometer setting at 0.2?
ii. If the feedback is open how much will be the input voltage for the same
output voltage?
iii. What will be the improvement in the performance with feedback if a load
current of 20A is supplied by the generator. Assume -ve feedback in the
above cases
iv. What happens if the feedback point set ting is increased?

44. i. Explain the effect of feedback on noise to signal ratio (Apr 06)
ii. With the help of sketches, explain the construction and working principle of
a Synchro transmitter.

45. i. Deive the transfer function of the Figure given, using block diagram
reduction technique
 ii. Derive the transfer function of the following system shown in fig.

(Apr 06)

46 Discuss Mason’s gain formula. Obtain the overall transfer function C/R from
the signal flow graph shown.

(Apr 06)
47. i.Derive the transfer function of an a.c. servomotor and draw its characteristics.
ii. Explain the Synchro error detector with circuit diagram. (Dec 04)
iii. Write down the signal flow equations and draw the signal flow graph for t he
above system.

48. i. Derive the transfer function of an DC servomotor and draw its

characteristics. (Dec 04)
ii. Explain the Synchro error detector with circuit diagram.

49. Simplify the block diagram of the syst em given below by block diagram
reduction technique and determine the transfer function of the system.
 (Dec 04)

50. i. Explain the traffic control system concepts using o pen loop as well as closed
loop system.
ii. Determine the overall transfer function from the signal flow graph given
below:

(Apr 04)

51. Write notes on the following:

i. Modems
ii. Synchro transmitter and receiver. (Apr 04)

52. Write notes on the following:

i. Field controlled d.c. servomotor
ii. Armature controlled d.c. servomotor. (Apr 04)

53. i. Derive the transfer function of D.C.servomotor with armature voltage

control.
ii. Draw the connection diagram of a Synchro pair used as a position indicator?
(Apr 04)

54. i.Derive the Transfer Function for the field controlled d.c. servomotor with
neat sketch. (Apr 04)
ii. Draw the connection diagram of a synchro pair used as a position indicator.

55. i. Why a conventional induction motor is not suitable for servo application s.
ii. Derive the transfer function for the a.c. servomotor with neat diagram.
(Nov 03)

56. The rotor of a a.c. servo motor is built with high resistance i.e. low X/R ratio
why? What will happen if we use the rotor of an ordinary 3 phase induction
motor having high X/R ratio as the rotor of an a.c. servo motor? Determine
 the transfer function q(s)/V c (s) for an a.c. servo motor where q is the angular
position of shaft in radians and V c (s) is the control phase voltage. (Nov 03)

57. Reduce the following block diagram using block diagram reduction technique
and compute C(s)/R(s).

(Nov 03)
58. i. Define signal flow graph and how do you construct signal flow graph from
equations. (Nov 03)
ii. Draw the signal flow graph for the system shown below and hence find gain
from Mason’s gain formula.

59. Explain the principle of operation of a.c. and d.c. tachometer. Hence derive
the transfer functions for the same. (Apr 03)

60. i. Describe the operation of synchro as an error detector. Derive the necessary
Transfer function.
ii. Discuss the effects of negative feedback on the following
a. Gain of the system
b. Band width
c. Sensitiveness to parameter variation. (Apr 03)

61. For the following block diagram draw the signal flow graph and hence
determine C/R using Mason’s gain formula.

(Apr 03)

62. i. What are the advantages of signal blow graph over block diagram ?
ii. Simplify the block diagram shown below using block diagram r eduction
techniques. Obtain the closed loop transfer function C(s)/R(s). (Apr 03)
 iii. Construct signal flow graph for the above system and verify the result
obtained in part ii. Using Masons’ gain formula.

63. i.What is servomotor?

ii. What are the characteristic of servo motors?
ii. Derive transfer function of a DC motor when it is operated i n field control
mode. Assume necessary data. (Jan 03)

64. i. What is the difference between ac servomotor and two phase induction
motor? (Jan 03)
ii. What is tachogenerator?

65. An ac servo motor has both windings excited with 115 V a.c. It has a stall
torque of 2 lb-ft. Its coefficient of viscous friction is 0.2 lb ft. sec.
i. Find it’s no load speed.
ii. It is connected to a constant load of 0.9 lb ft and coefficient of viscous
friction of 0.05 lb-ft.sec. through a gear pass with a ratio of 4. Find the
speed at which the motor will run. (Jan 03)

66. i. With aid of neat sketch, explain the working principle of synchro transmitter.
ii. Derive the transfer function of synchro receiver. (Jan 03)

67. How does a two-phase servomotor differ from a Normal Induction motor?
Find its transfer function and explain how the motor constants can be
estimated. (Jan 03)

68. Find the outputs C 1 and C 2 of the system. (IES 97)

C (S)
R (S)
69. Give basic properties of SFG. Find the transfer function for a system
whose signal flow graph is shown below: (IES 96)
70. State and explain Mason’s gain formulae. Hence find the transfe r function
C (S)
R (S)
for the system whose signal flow graph is

(IES 94)

71. Draw a signal flow graph and evaluate the closed loop transfer function of a
system whose block is given by

(IES 94)
72. State Mason’s Gain Formulae. Hence otherwise. Find I and E C in terms of the
input varaibles. E(0) i 1 (0) and e c (0) for the network whose signal flow graph
is shown in fig.

(IES 94)

73. A servo mechanism is used to control the angular position q 0 of a mass

through a command signal. The M.I. of moving parts referred to the load shaft
is 200 Kg-m 2 and the motor torque at the load is 6.88 x 10 .4 N-m/rad of error.
 The damping torque coefficient referred to the load shaft is 5 x 10 .3 N-
m/rad/sec.
i. Find the time response of the servo mechanism to a step input of 1 rad and
determine the frequency of transient oscillation, the time to rise to the peak
overshoot and the value of the peak overshoot.
ii. Determine the steady-state error when the command sign al is a constant
angular velocity of 1 revolution/minute.
iii. Determine the steady-state error which exists when a steady torque of 1200
N-m is applied at load shaft (IES 94)

74. For a two winding transformer (mutually coupled circuit) equations relating
the terminal voltages are given by
V 1 = (R 1 + L 1 P) i 1 - Mpi 2
-V 2 = (R 2 + L 2 P) i 2 - Mpi 2 Where P is a differential operator. Rearrange the
above equations in appropriate cause -effect from to obtain the signal flow
graph as shown in figure.

Using mason gain formulae, prove that i 1 is given by

(R 1  L 2 P) U 1  Mpv 2
P (L 1 L 2  M 2 )  (R 1 L 2  R 2 L1 ) p  R 1 R 2
2
(IES 93)

75. Use block diagram reduction methods to obtain the equivalent transfer
function from R to C.

(IES 93)

76. What is a signal flow graph. Give its properties for the signal flow graph
shown in the figure below. Find, using Mason’s gain formulas, the transfer
function c(s)/r(s). (IES 92)
77. Consider the system shown in figure : (IES 92)

i. In the absence of derivate feedback (a=0) determine the damping factor and
natural frequency. Also determine the steady state error resulting from a unit
ramp.
ii. Determine the derivative feedback. Constant a, which will increase the
damping factor of system to 0.7. What is the steady state error to unit ramp
with this setting of the derivative feed back constant?

UNIT-III
1. a) What is the beneficial effect of derivative error compensation on important performance
indices of a type-1 control system? Elaborate.

b) A unity negative feedback control system has the plant

Determine its peak overshoot and settling time due to unit step input. Determine the
range of K for which the settling time is less than 1sec. (Dec 14)

1 . a) Explain the standard test signals that are used in the time-domain analysis.
b) A unity feedback control system has an open loop transfer function
G(s)=K/s(s+10).Determine the gain ‘K’ so that the system will have a damping ratio of
0.5. For this value of ‘K’, determine the settling time, peak overshoot and time to peak
overshoot for a unit-step unit.
(May/June 13)

2. a) A unity feedback control system has the forward transfer function

G(s)=25/s(s+6).find the rise time ,peak time and maximum over shoot for
unit step input.
b) Find the error constants and steady state error for the velocity input r(t)=2t
and step input of 2 units. The system is described by G(s) H(s) =10/s(s+5)
(Nov/Dec12)

3. a) For the system shown in Figure 1, determine K1, K2, and `a' such that the system
will have a steady state gain of 1.0, a damping ratio & _ = 0 .6, !n = 5.0.
b) A unity feedback control system has the forward transfer function, G(s) =25/s(s+6). Find
the rise time, peak time and the maximum over shoot for unit step input.
 (May/June 12)

4. Consider the system shown in Figure 6.

(a) The damping ratio of this system is 0.158 and the undamped natural frequency is
3.16rad/sec. To improve the relative stability, we employ tachometer feedback Figure 7.
(b) Shows such a tachometer-feedback system. Determine the value of Kh so that the
damping ratio of the system is 0.5. Draw unit-step response curves of both the original
and tachometer-feedback systems. Also draw the error-versus-time curves for the unit-
ramp response of both systems.

(May/June 12)
5. (a) Consider the differential equation system given by Ϋ + 3ý+ 2y = 0; y (0) =0.1 ý (0) =
0.05.Obtain the response y (t), subjected to the given initial condition.
(b) Consider a unity-feedback control system whose open-loop transfer function is G(s) =
K/s(Js+B) . Discuss the effects of varying the values of K and B on the steady-state error
in unit-ramp response.
(May/June 12)

6. (a) Explain error constants Kp , Kv, Ka for type 1 system.

(b) Given the open loop transfer function of a unity feedback system as G(s) =10/s(0.1s+1) ,
find Kp , Kv, Ka.
(May/June 12)

7. i. State how the type of a control system is determined? How it effect the
steady-state error of the system?

ii. A unity feed-back system has . Determine

a. Type of the system?
b. All the error coefficients?
c. Error for ramp input with magnitude. (Apr 11, Dec 10)

8. i. Define the following terms:

a. Steady-state error b. Settling time
c. Peak overshoot d. type and order of a control system.
ii. Sketch the transient response of a second order system and derive the
expression for rise time and peak overshoot? (Dec, May 10)

9. i. Explain the significance of generalized erro r series?

ii. For a system , find the value of K to
limit the steady state error to 10 when the input to the system is
r(t)=(1+10t+40)/2 t 2 . (Dec, May 10)

10. i. Explain error constants K p , K v , K a for type-1 system?

ii. A unity feed back system has an open loop transfer function
. Determine its damping ratio, peak overshoot and time
required to reach the peak output. Now a derivative component having T.F.
of s/10 is introduced in the system. Discuss its effect on the values obtained
above?
(Dec 10)
11. i.Define type and order of a control system and hence find the type and order
of the following systems?

a.

b.
 c.

d.
ii. The unit step response of a second order linear system with zero initial state
is given by c(t) = 1 + 1:25e –6t Sin(8t – tan –1 1:333). Determine the damping
ratio, un damped natural frequency of oscillations and peak overshoot?
(May 10)

12. i. Derive the static error constants and list the disadvantages?
ii. Find step, ramp and parabolic error coefficients and their corresponding
steady-state error for unity feed-back system having the T.F

. (May 10)

13. i. Explain the significance of generalized error series?

ii. For a system G(s) H(s) = Find the value of K to limit the steady
state error to 10 when the input to the system is r(t)=1+10t+40/2 t 2 .
(May 09, 07, Sep 07)

14. i. Explain error constants K p , K v and K a for type-1 system?

ii. A unity feed back system has an open loop transfer function
Determine its damping ratio, peak overshoot and time required to reach the
peak output. Now a derivative component having T.F. of s/10 is introduced in
the system. Discuss its e”ect on the values obtained above?
(May 09, Sep 07)
15. i. Derive the static error constants and list the disadvantages?
ii. Find step, ramp and parabolic error coefficients and their corresponding
14( s  3)
G( s) 
steady-state error for unity feed-back system having the T.F . s( s  5)(s 2  3s  2)

(May 09)
16. For a unity feed back system having G(s) = K/s(2+sT) find the following
i. The factor by which the gain K should be multiplied to increase the damping
ratio from 0.15 to 0.6
ii. The factor by which the time constant sh ould be multiplied to reduce the
damping ratio from 0.8 to 0.4. (May 09)

17. i. Establish the relation relation between ζ and M p for a step response of a
second order system?
d2y dy
4  8 y  8x
ii. A system is given by differential equation dt 2 where y=output x=
dt ,
input. Determine all the time domain specifications and obtain output
response for unit step input ? (May 09)
18. i. What are the types of controllers that are used in closed loop system?
Explain them?
ii. The response of a system subjected to a unit step input is c (t) = 1 + 0.2e -60t -
1.2e -10t Obtain the expression for the closed loop transfer function? Also
determine the Undamped natural frequency and damping ratio of the system?
(May 09, Sep 08)

19. i. Define the following systems and sketch their output wave forms for an unit
step i/p
a. Under damped system
b. Undamped system
c. Over damped system
d. Critically damped system
ii. For a second order system ζ= 0.6, w n = 5rad/sec. Find the values of w d , T r ,
T p , T s and M p . (May 09)

20. i. Define time constant and explain its importance.

ii. A unit feedback system is characterized by an open -loop transfer function
G(s) = K/s(s+5). Determine the gain K so that the system will have a
damping ratio of 0.5. For this value of K determine settling time, peak
overshoot and times to peak overshoot for a unit -step input. (Sep 08)

21. i. What are the different time domain specifications of a dynamical system?
Explain important specifications of a second ordered system to unit step
input.
ii. The open loop transfer function of a unity feedback system is given by G(s) =
K/s(Ts+1), where K and T are positive constants. By what factor should the
amplifier gain be reduced so that the peak overshoot of unit -step response of
the system is reduced from 75% to 25%? (Sep 08)

22. i. Explain the important time? response specification of a standard second

ordered system to a unit step input.
ii. Derive expressions for time domain specifications of a standard second
ordered system to a step input. (Sep 08)
23. The overall T.F. is a unity feed back control system is given by

i. Find K p , K v , K a
ii. Determine the steady state error if the input is r(t)=1 + t + t 2 . (Sep 08)

24. i. What is the difference between type and order of a control system? Explain
each with an example?
ii. The figure shows PD controller used for the system. Determine the value of
T d so that the system will be critically damped? Calculate it’s settling time?
 (Sep 08)
25. i. Explain about various test signals used in control system?
ii. Measurement conducted on a servomechanism shows the system response to
be C(t) = 1 + 0.2e -60t -1.2e -10t , when subjected to a unit step input. Obtain
the expression for closed loop T.F., the damping ratio and undam ped natural
frequency of oscillations? (Sep 08, 07)

26. Consider the system shown in Figure. Determine the value of k such that the
damping ratio ζ is 0.5. Then obtain the rise time, t r , peak time t p , maximum
overshoot M p , and settling time ts in the unit-step response.

(Sep 08, May 07)

27. i. What is meant by time response? Explain about
a. Steady- state response b. Transient response
ii. A unity feed-back system is characterized by an open loop T.F G(s)
=k/s(s+10). Determine the gain K so that the system will have a damping
ratio of 0.5. For this value of K, determine T s , T p and M p for a unit step
input. (May 08)

28. i. What are generalized error constants? State the advantages of gene ralized
error coefficients?
ii. For a first order system, find out the output of the system when the input
applied to the system is unit ramp input? Sketch the r(t) and c(t) and show
the steady state error. (May 08)

29. i. How steady state error of a control system is determined? How it can be
reduced?
ii. Determine the error coefficients and static error for G(s) = 1/s(s+1)(s+10) ,
H(s) = s + 2. (May 08)

30. i. Derive the expression for rise time, peak time peak overshoot and settling
time of second order system subjected to a step input.
 ii. A unity feedback control system has a loop transfer function.
10
G( s) 
s( s  2)
Find the rise time, percentage overshoot, peak time and settling time for a
step input of 12 units.
(May 08)
31. i. Determine whether the largest time constant of the characteristic equation
given below is greater than, less than, or equal to 1.0 sec. s 3 + 4s 2 +6s+4=0
ii. Figure. is a block diagram of a space-vehicle attitude-control system.
Assuming the time constant T of the controller to b e 3 sec., and the ratio K/J
to be 2/9 rad 2 /sec 2 , find the damping ratio and natural frequency of the
system.

(May 08)
32. i. Define type and order of a control system and hence find the type and order
of the following systems?
100 200
a. G(s) H(s) = s( s  4s  200)
2
b. G(s) H(s) = s ( s  10s  200)
2 2

4( s  10s  100)
2
200
s ( s  3) s 2  2 s  10) (1  01s )(1  0.5s )
c. G(s) H(s) = d. G(s) H(s) =
ii. The unit step response of a second order linear system with zero initial state
is given by c(t) = 1 + 1.25e -6t Sin(8t - tan -1 1.333). Determine the damping
ratio, undamped natural frequency of oscillations and peak overshoot?
(May 08)

33. i. For an under damped second order system, define various time domain
specifications?
ii. The forward path T.F. of a unity feed back control system is given by G(s)
=2/s(s+3). Obtain the expression for unit step response of the system?
(Sep, May 07)

34. The open-loop transfer function of a servo system with unity feedback isG(s )
= 10/s(0.1s+1). Evaluate the static error constants ( K P , Ku, and Ka) for the
system. Obtain the steady-state error of the system when subjected to an
input given by the polynomial (May 07, 06, 05)

35. Consider a system shown in Figure, employing proportional plus error -rate
control. Determine the value of the error -rate factor Ke so that the damping
 ratio is 0.5. Determine the values of settling time, maximum overshoot when
subjected to with and without error-rate control a unit step input.

(May 07, 06)

36. For a unity feedback system G(s) = . Determine the characteristic
equation of the system. Hence calculate the undamped frequency of
oscillations, damped frequency of oscillations, damping ratio, peak
overshoot, and time required to reach the peak output, settling time. A unit
step input is applied to the system. (May 07)

37. The open loop transfer function of a c ontrol system with unity feedback is
given by G(s)= 100/s(s+01s) . Determine the steady state error of the system
when the input is 10+10t+4t 2 . (May 07)

38. For the feedback control system shown in figure. It is required that:
i. The steady-state error due to a unit-ramp function input is equal to 1.5.
ii. The dominant roots of the characteristic equation of the third -order system
are at?1+j1 and ?1-j1. Find the third-order open-loop transfer function G(s)
so that the foregoing two conditions are satisfied.

(May 07)
39. i. Why derivative controller is not used in control systems? What is the effect
of PI controller on the system performance?
ii. The system shown in figure uses a rate feed back controller. Determine the
tachometer constant K t so as to obtain the damping ratio as 0.5. Calculate the
corresponding w d , T p , T s and M p .

(May 07)
40. i. Define the following terms:
a. Steady-state error
b. Settling time
c. Peak overshoot
d. type and order of a control system.
ii. Sketch the transient response of a second order system and derive the
expression for rise time and peak overshoot? (May 07)

41. A second order servo has unity feedback G(s) = 500/s(s+5 ). Sketch the
transient response for unity step input, and calculate peak overshoot, settling
time, peak time. (Sep 06)

42. A feedback system employing output-rate damping is shown in Figure:

(Sep 06)
i. In the absence of derivative feedback (K 0 =0), determine the damping factor
and natural frequency of the system. What is the steady state error resulting
from unit-ramp input?
ii. Determine the derivate feedback constant K 0 , which will increase the
damping factor of the system to 0.6. What is the steady-state error to unit-
ramp input with this setting of the derivative feedback constant?
iii. Illustrate how the steady-state error of the system with derivative feedback to
unit-ramp input can be reduced to same value as in part (a), while the
damping factor is maintained at 0.6.

43. The open loop transfer function of a unity feedback control system is
K
G(s) 
s (1  Ts )
i By what factor should the amplifier gain “K” be multiplied in order to
increase the damping ratio from 0.2 to 0.8?
ii. By what factor should “K” be multiplid so that the maximum overshoot for a
step input decreases from 60% and 10%. (Apr 06)

44. i. Explain error constants K p , K v and K a for type I system.

ii. Given the open-loop transfer function of a unity feedback system as
Gs  
100
S 0.1s  1
Find K p , K v and K a . (Apr 06, Jan 03)
 C (S ) n2
 2
45. i. Consider the closed-loop system given by R(s) s  2n s  n
2

Determine the values of ζ and w n so that the system responds to a step input
with approximately 5% overshoot and with a settling time of 2 sec. (Use the
2% criterion)
ii. What are the time response specifications? Explain each of them. (Apr 06)

46. i. Consider the system shown in Figure. Determine the values of K and k such
that the system has a damping ratio of 0.7 and an undamped natural
frequency of 4 rad/sec.

ii. Consider the unit-step response of a unity-feedback control system whose

open-loop transfer function is G(s) =1/s(s+1). Obtain the rise time, peak
time, maximum overshoot, and setting time, peak time, maximum overshoot,
and setting time. (Apr 06, 05)
47. A Unity feedback system has forward transfer function G(s)=20/(s+1).
Determine and compare the response of the open and closed loop systems for
unit step input.Suppose now that parameter variation occurring during
operating conditions causes G(s) to modi fy toG(s) =20/(s+0.4). What will be
the effect on unit-step response of open and closed loop systems? Comment
upon the sensitivity of the two systems to parameter variations. (Apr 06)

48. i. Find the steady-state error to

a. a unit step input b. a unit ramp input and
ii. a unit parabolic input (r = l /2t 2 ) for a unity feedback systems that have the
10
G (s) 
following forward transfer functions. s (s  4)(s 2  3s  12 )
2

iii. The open loop transfer function of a servo system with unity feedback is
given G(s)=500/s(1+0.1s) Evaluate the error series for the above system and
determine the steady state error when the input is r(t) = 1 + 2t + t 2 .(Apr 05)

49. i. What are the time response specifications? Explain each of them.
ii. For a negative feedback control system having forward path transfer function
G(s)=k/s(s+6) and H(s) = 1. Determine the value of gain k for the system to
have damping ratio of 0.8. For this value of gain k, determine the complete
time response specifications. (Nov 04)
50. An open loop transfer function of unity feedback System is given by
G(s)=25/s(s+5) Determine the damping factor, undamped natural frequency,
damped natural frequency and time response for a unit step input. (Apr 04)

51. i. What are the different time domain specifications of a dynamical system?
Explain important specifications of a second ordered system to unit step
input.
ii. The open loop transfer function of a unity feedback system is given by G(s) =
K/s(Ts+1), where K and T are positive constants. By what factor should the
amplifier gain be reduced so that the peak overshoot of unit -step response of
the system is reduced from 75% to 25%? (Apr 04)

52. i. Explain the Routh’s criterion to determine the stability of a dynamical

system and give its limitations.
ii. Find the step, ramp and parabolic error coefficients and their corresponding
steady state errors for unit feedback control syste m having the transfer
14s  3
G(s) 
ss  5
 s 2  2s  5 

function   (Apr 04)

53. i. A unity feedback system has a forward path transfer function G(s) =9/s(s+1) Find the
value of damping ratio, undamped natural frequency of the system, percentage overshoot,
peak time and settling time.
ii. Measurements conducted on servomechanism show the system response to be
When subjected to a unit-step unit. Obtain the expression for the closed -loop
transfer function. (Apr 04)

54. i. Explain error constants Kp, KV and Ka for types-II system.

ii. A unity feed-back system is characterized by the open-loop transfer function
G s  
1
s0.5s  10.2s  1
Determine the steady-state errors for unit-step, unit-ramp and unit-
acceleration input. Also find the damping ratio and natural frequency of the
dominant roots. (Apr 04)

55. i. Derive the expression for rise time, peak time peak overshoot and settling
time of second order system subjected to a step input.
ii. A unit feedback control system has a loop transfer function. G(s)=10/s(s+2)
Find the rise time, percentage overshoot, peak time and sett ling time for a
step input of 12 units. (Apr 04)

56. i. The open loop transfer function of a control system with unity feedback is
given by G(s)=100/s(s+0.1s). Determine the steady state error of the system
2
when the input is 10+10t+4t .
 ii. A unit feedback system has an open loop transfer function
G(s)=k/(s 2 +4s+5)(s+2).Use RH test to determine the range of positive values
of K for which the system is stable. (Apr 04)

57. A unity feedback system with G(s)=k/s(s+1) 2 is desired to limit the steady
state error to a value not exceeding 0.25 to unit ramp input.
i. find the value of K and also
ii. Find the requirement value of K for stability. (Apr 04)

58. The open loop transfer function of a control system unit feedback is given by
150
G(s) 
s 1  0.25 s 
i. Evaluate the generalized error series for the system.
ii. Determine the steady state error for an input r(t) = (1+5) u(t). (Apr 04)

59. i. For an underdamped second order system, define various time domain specifications.
ii. A unity feedback control system has the forward transfer function: G(s)=25/s(s+6)
Find the rise time, peak time and the maximum over shoot for unit step input.
(Apr 04)
2
60. Determine different error co-efficient for a system having G(s)=k/s(s +2s+5) and.
H s  
10
s4
Determine the steady state error if input is r(t) = 5 + 10t + . Assume K = 20. (Apr 04)

61. i. Obtain the unit-impulse response and the unit-step response of a unity-feedback system
2s  1
Gs   2
whose open-loop transfer function is s
ii. A unity feedback system is characterized by the open loop transfer function
Gs  
1
s0.5s  10.2s  1 . Determine the damping ratio and natural frequency of the dominant
roots (Apr 04)

62. The control system having unity feedback has G(s)=20/s(s+1)(1+4s) . Determine
i. Static error co-efficient.
2
ii. Steady state error if input = r(t) = 2+4t+t /2. (Apr 04)

63. For a unit feedback system G(s) =36/s(s+0.72). Determine the characteristic
equation of the system. Hence calculate the undamped frequency of
oscillations, damped frequency of oscillations, damping ratio, peak
overshoot, time required to reach the peak output, settling time. A unit step
input is applied to the system. (Apr 04)

64. i. Explain the concept of stability of a control system and explain a method to
determine the stability of a dynamical system.
 ii. A unity feedback control system is characterized by the open-loop transfer function
K s  13
Gs  
ss  3s  7
a. Using the Routh’s criterion, determine the range of values of K for the
system to be stable.
b. Check if for K=1, all the roots of the characteristic equation of the above
system have damping factor greater than 0.5. (Apr 04)

65. A unity feedback control system has G(s) =100/s(s+5).If it is subjected to

unity step input. Determine
i. Damped frequency of oscillation.
ii. Maximum peak overshoot
iii. Time to reach for first overshoot
iv. Settling time
v. Output response (Nov 03)

66. i. For the given system shown in the figure below, find damping factor and
natural frequency when:
a. K D = 0; b. K D = 1. (Apr 03)

ii. Determine the damping ratio and undamped natural frequency of oscillatory
roots and percentage of peak over shoot for a unit step given by
G s  
1
s 1  0.5s 1  0.2 s  and the system is unity feedback type.

67. A system oscillates with frequency w, if it has poles at s = ±jw and no poles
in the right half s-plane.Determine the values of ‘k’ and ‘a’, s o that the
system shown in fig oscillates at a frequency of 2rad/sec. (Apr 03)

68. i. For the system shown in the figure below, determine K 1 , K 2 , and a such that
the system will have a steady state gai n of 1.0,a damping ratio, d=
0.6,w n =5.0.
 ii. A unity feedback control system has the forward transfer function, find the
rise time, peak time and the maximum over shoot for unit step input. (Apr 03)

69. i.Explain the terms :

a. Proportional control;
b. Integral control; and
c. Proportional plus derivative control.

Show that the steady error for unit step input is zero. (Apr 03)

70. i. Find the step, ramp and parabolic error coefficients and their corresponding
steady state errors for unity feedback control system having transfer function:
14 s  3
G s  
ss  5  s  2s  5 
2

G s  
25
s s  6 
ii. A unity feedback control system has the forward transfer function:
Find the rise time, peak time and the maximum over shoot for unit step input.
(Jan 03)

71. In a closed loop control system, the open loop transfer function G(s)=K/s 2 and
feedback transfer function H(s) = as+b. If k = 20, find the values of a,b so
that overshoot is 16% and the time constant is 0.1sec. Also determine the
steady state error, if the input to the system is a unit ramp. (Jan 03)

72. The open-loop transfer function of a Servo system with a unity feedback is
G(s)=4/s(1+0.1s) Determine the dynamic error for a input r(t) = 1+2t+t 2 by using the
dynamic error coefficients. (Jan 03)

73. The open loop transfer function of a servo system wit h unity feedback is
given by G(s)=500/s(1+0.1s).Evaluate the error constants K P , K v and K a .
Determine the steady state error when the input is r(t) = 1+2t+t 2 . (Jan 03)
74. i. The open loop transfer function of a servo system with unity feed back is
given by G(s)=500/s(1+0.1s).Evaluate the error series for the above system and
determine the steady state error when the inputs
2
r(t) = 1+2t+t .
ii. The open loop transfer function of a unity feed back control system is given by
Gs  
K

 
S  2S  3 s 2  6S  25 Applying
Routh Hurwitz criterion discuss the
stability of the closed loop system as a function of K.
(Jan 03)

75. i. Define: (i) rise time (ii) settling time (iii) delay time (iv)over shoot for a
second order system.
ii. The open loop transfer function of a unit feed back control system is given by
G s  
A
S1  ST 

By what factors should the amplifier gain A be multiplied so that:

a. Damping ratio is increased from 0.2 to 0.4.
b. The over shoot of unit step response is reduced from 80% to 40%. (Apr 02)

76. i. Define the following time domain specifications

a. Rise time
b. Peak time
c. Setting time
d. Peak over shoot and
e. Steady state error
G s  
K

ii. For unity feedback system having forward path transfer function STs  1
,
determine the time domain specifications for units step input, with K = 100
and T = 0.1 sec.
iii. Derive the expressions for steady state error coefficients for unity feedback
system with
a. Step input b. Velocity input and c. Acceleration input. (Apr 02)

77. Sketch the time domain response C(t) of typical under damped 2 nd system to
a step input r(t). In this sketch indicate the time domain specifications.
(IES 96)

78. An open loop transfer function of unity feedback s ystem is given by

G(s)=5/s 2 (s+5) Determine the damping factor, undamped natural frequency,
damped natural frequency and time response for a unit step input. (IES 95)
 C (S ) S 2  35  4

79. A circuit has the following transfer function , R (S ) S  45  4 .Find C(t) when
2

r(t) is a unit step. State if the circuit is undamped, underdamped, critically

damped or overdamped. (IES 02)

UNIT-IV

1. The open loop transfer function of a feedback system is G(s) H(s) =K/s(s+4) (s2+4s+20).
Draw the root locus and investigate the stability of the system.
(Dec 14)

1. The open loop transfer function of a unity feedback system is given by

G(s)=K/(s+2)(s+4)(s2+6s+25) Using R-H criterion discuss the stability of the closed-
loop system as a function of ‘K’. Determine the values of ‘K’ which will cause
sustained oscillations in the closed-loop system. What are the corresponding oscillation
frequencies? (May/June 13)

2. A unity feedback control system has an open loop transfer function G(s) = K/s 2 (s+2)
Sketch the Root-Locus plot and show that the system is unstable for all values of‘K’.
(May/June 13)

3. a) sketch the root locus plot for the control system with a forward transfer
function G(s)=k(s+2)/s 2 +2s+3 and H(s)=1
b) A unity feedback system has the forward transfer function G(s)
=k(s+2)/s(s+3)(s+7).using R -H criterion ,find the range of K for which the
closed loop system is stable. (Nov/Dec 12)

4. Sketch the root-locus diagram of a control system whose loop transfer function is
G(s)H(s) = K/s(s+4)(s+10) . Using the diagram or otherwise find the values of gain at
breakaway points and at point of intersection of the loci with the imaginary axis.
(May/June 12)
5. A unity feedback control system has an open loop transfer functionG(s) =
K(1+0:2s)(1+0:025s)/s3(1+0:001s)(1+0:004s) Sketch the complete root locus for -
K<œ.Indicate the crossing points of the loci on the j! axis and the corresponding values
of K at these points. Also indicate the range of K for which system closed loop system is
stable. (May/June 12)
6. (a) Determine the range of value of k for the system to be stable which is characterized
by the equation s3 + 3Ks2 + (k+2)s+4 = 0.
(b) Explain how Routh Hurwitz criterion can be used to determine the absolute
stability of a system. (May/June 12)

7. The open loop transfer function of a feedback control system with unity feedback
is G(s) = K/(s+10)n . Sketch the root loci of the characteristic equation of the closed
loop system for -œ< K < œ, with
(a) n=3
(b) n=4. Show all important information on the root loci. (May/June 12)

8. i. Define the the following terms

a. absolute stability
b. marginal stability
c. conditional stability
ii. By means of RH criterion determine the stability of the system represented
by the characteristic equation S 4 + 2S 3 + 8S 2 + 4S + 3 = 0
iii. State the advantages of RH Stability criterion? (Apr 11)

9. i. Define the following terms

a. Stable system
b. Critically stable system
c. Conditionally stable system.
ii. For the system having characteristic equation 2S 4 + 4S 2 + 1 = 0, find the
following
a. the no. of roots in the left half of s-plane
b. the no. of roots in the right half of s -plane
c. the no. of roots on the imaginary axis.
Use the RH stability criterion (Dec 10)

10. i.Show that that the break-away and break-in points, if any, on the real axis
for the root locus for where N(s) and D(s) are rational
polynomials in S, can be obtained by solving the equation .

ii. Check whether the points (-1+ j) and (-3+ j) lie on the root locus of a system
given by . Use the angle condition. (Dec 10)

11. Sketch the root locus plot of a unity feed back system whose open loop T.F is
.
(Dec 10)

12. The open loop T.F. of a control system is given by .

Sketch the root locus plot and determine
 i. the break-away points
ii. The angle of departure from complex poles
iii. the stability condition. (Dec 10)

13. i. Sketch the root locus plot of a unity feed - back system whose open loop T.F
is
ii. What is break-away and break-in point? How to determine them?
iii. From the given root locus plot how can you determine the gai n margin and
phase margin for the specified gain value `K'. (May 10)

14. Sketch the root locus plot for the system having .(May 10)

15. Using RH stability criterion determine the st ability of the following system.
i.Its loop t.f. has poles at s=0, s= -1, s= -3 and zero at s= -5, gain K of forward
path is 10
ii. It is a type-1system with an error constant of 10 sec-1 and poles at s= -3 and
s=-6. (May 10)
16. Determine the values of k and b, so that the system whose open transfer

function is oscillates at a frequency of oscillations of 2

rad/sec. Assume unity feed back. (May 10)

17. Sketch the root locus plot of a unity feed back system whose open loop T.F is
K ( s  9)

G(s) = s( s 2  4s  11)

(May 09, Sep 08)

18. i. The open loop t.f. of a unity feed -back system is given by G(s) =

Find the restriction on K so that the closed loop system is

absolutely stable?
Ke  s

ii. A feed-back system has an an open loop t.f of G(s)H(s) =. s( s  5s  9)

2

Determine by the use of the RH criterion, the max. value of K for the closed
loop system to be stable? (May 09, Sep 07)

19. i. Sketch the root locus plot for a unity feedback system whose open loop T.F.
is given by
ii. What is break-away and break-in point? How to determine them?
iii. From the given root locus plot how can you determine the gain margin and
phase margin for the specified gain value ‘K’. (May 09)

20. i. State and explain Routh Hurwitz stability criterion

ii. Construct Routh array and determine the stability of the system whose
characteristic equation is S 6 + 2S 5 + 8S 4 + 12S 3 + 20S 2 +S16 = 0. Also
 determine the no.of roots lying on right half of s -plane, left half of s-plane
and on imaginary axis? (May 09)

21. Sketch the root locus plot for the systems whose open loop transfer function
is given by Determine
i. the gain margin
ii. the phase margin for K=6. (May 09)

22. i. What is root locus plot? Explain with suitable example?

ii. What are the features of root locus plot?
iii. Check whether the point s= -3 + j 5 lies on the root locus of a system having

. Determine the corresponding value. (May 09)

23. Explain the stability of the system from the root locus plot in the following
situations with suitable examples?
i. addition of open loop poles
ii. addition of open loop zeros. (May 09)

24. i. Show that the Routh?s stability criterion and Hurwitz stability criterion are
equivalent.
ii. Consider a unity-feedback control system whose open-loop transfer function
K
is G(s) = s( Js  B) . Discuss the effects that varying the values of K and B has
on the steady-state error in unit-ramp response. (Sep 08)

25. i. The open loop transfer function of a control system with unity feedback is
9
G(s) = (1  s)(1  2s)(1  3s)
. Show that the system is stable.
ii. A unity feedback system is characterized by the open loop transfer function
1
G(s) = s(0.5s  1)(0.2s  1)
. Determine the steady state errors for unit step, unit
ramp and unit acceleration input. (Sep 08)

26. i. The open loop Transfer function for a unity feedback system is given by G(s)
K
= s(1  s  T1 )(1  sT2 )
Find the necessary conditions for the system to be stable
using Routh-Hurwitz method.
100K
ii. The open loop transfer function of a unity feedback system is G(s) = s( s  10)
.
Find the static error constants and the steady state error of the system when
subjected to 10 an input given by the polynomial .r(t) = P o + P 1 t + P 2 2t 2
(Sep 08)
27. i. Find the Number of roots with positive, Negative and Zero real parts for a
following polynomial using Routh’s Hurwitz criterion s 4 + 6s 3 - 31s 2 + 80s -
100=0.
ii. System Oscillates with a frequ ency W if it has poles at s=+ jw and no poles
in the right half of the s-plane. Determine the values of K and a for the
characteristics equation s 3 + as 2 +2s+1+K(s+1) =0 at a frequency of 2rad/sec.
(Sep 08)

28. Sketch the root locus plot of a unity feedback system whose open loop T.F is
K ( s 2  2s  2)
G(s) = ( s  2)(s  3)(s  4)

(Sep 08)
29. Determine the values of k and b, so that the system whose open transfer
K ( s  1)
function is G(s) = s 3  bs 2  3s  1 oscillates at a frequency of oscillations of
2rad/sec. Assume unity feed back.
(Sep 08)
30. i. Explain the RH stability Criterion ?
ii. The open loop transfer function of a unity feed back control system is given

by G(s) = . Apply RH stability criterion, determine the value of

K in terms of T\ and T 2 for the system to be stable? (Sep 08, 07)

31. i. Define the term root locus and state the rule for finding out the root locus on
the real axis?
ii. Calculate the angle of asymptotes and the centroid for the system having
K ( s  3)
G(s)H(s) = s( s  2)(s  4)(s  5) .
K
iii. For G(s)H(s) = s(s  1)(s  3) , find the intersection point of the root locus with the
jw - axis? (May 08)

32. i. What are the necessary conditions to have all the roots of the characteristic
Equation in the left half of s-plane?
ii. What are the difficulties in RH stability cri terion? Explain, how you can over
come them? (May 08)

33. i. Explain how Routh Hurwitz criterion can be used to determine the absolute
stability of a system.
ii. For the feedback control system shown in Figure. it is required that :
a. the steady-state error due to a unit-ramp function input be equal to 1.5.
b. the dominant roots of the characteristic equation of the third -order system
are at ?1+j1 and ?1-j1. Find the third-order open-loop transfer function G(s)
so that the foregoing two conditions are satisfied.
 (May 08)
34. i. Explain how Routh Hurwitz criterion can be used to determine the absolute
stability of a system.
ii. For the feedback control system shown in Figure. it is required that :
i. the steady-state error due to a unit-ramp function input be equal to 1.5.
ii. the dominant roots of the characteristic equation of the third -order system
are at ?1+j1 and ?1-j1. Find the third-order open-loop transfer function G(s)
so that the foregoing two conditions are s atisfied.
(May 08, 07)
35. i. Find the roots of the characteristic equation for systems whose open -loop
transfer functions are given below. Locate the roots in the s -plane and
indicate the stability of each system.
1
a. G(s) H(s) = ( s  2)(s  4)
5( s  3)
b. G(s) H(s) = s( s  3)(s  8)

Ke  s

ii. A feedback system has an open-loop transfer function of G(s)H(s)= s( s  5s  9)

2

. Determine the use of Routh criterion, the maximum value of K for the
closed-loop system to be stable. (May 08)

36. Using RH stability criterion determine the stability of the foll owing systems.
i. Its loop t.f. has poles at s=0, s= -1, s= -3 and zero at s= -5, gain K of forward
path is 10
ii. It is a type-1system with an error constant of 10 sec-1 and poles at s= -3 and
s=-6. (Sep 07)

37. i. Open loop T.F. of a unity feedback system is G(s) = (Sep 07)
a. Prove that break-away and break-in points will exist only when |a| > |b|
ii. Prove that the complex points on the root locus form a circle with center ( -a,

0) and radius .
38. i. Show that the breakaway and break-in points, if any, on the real axis for the
root locus for G(s)H(s)= where N(s) and D(s) are rational polynomials

in s, can be obtained by solving the equation

ii. By a step by step procedure draw the root locus diagram for a unity negative

feedback system with open loop transfer function G(s)= . Mark all the
 salient points on the diagram. Is the system stable for all the values of K?
(May 07, 06)

39. The open-loop transfer function of a unity feedback control system is given

by G(s)= By applying the Routh criterion, discuss the

stability of the closed-loop system as a function of K. Determine the values
of K, which will cause sustained oscillations in the closed -loop system. What
are the corresponding oscillation frequencies? (May 07)

40. i. What are root loci? Summarize the steps that are used as rules for
constructing the root locus.
ii. Draw the root locus of a system having open loop transfer function G(s)H(s)=
K
s ( s  4)(s 2  4s  20)
. Indicate the salient points of root locus. (May 07)
41. A unity feedback system has an open loop transfer function G(s)=
. Use RH test to determine the range of positive values of K for which the
system is stable. (May 07)

42. The open loop transfer function of a unity feedback system is G(s) =

. Sketch the complete root locus and determine the value of K

i. For the system to be stable
ii. For the system to be marginally stable and hence the frequency of
oscillation
iii. To provide critical damping
iv. To give an e”ecting damping factor 0.5. (May 07)

43. The open loop T.F. of a control system is given by G(s)H(s) =

Sketch the root locus plot and determine
a. the break-away points
b. The angle of departure from complex poles
c. the stability condition. (May 07)
44. For a unity feedback system having forward path transfer function G(S) =

Determine
a. The range of values of K
b. Marginal value of K
c. Frequency of sustained oscillations. (May 07)
45. Sketch the root locus plot a unity feed back system with an open loop transfer function
K
G( s) 
s( s  2)(s  4) .
Find the range K for which the system has damped oscillatory response.
Explain the procedures for constructing root locus (Sep 06)
46. i. The characteristic equation of a feedback control system is s 3 + (K + 0.5) s 2
+ 4Ks + 50 = 0. Using R-H criterion determine the value of K for which the
system is stable.
ii. Determine whether the largest time constant of the characteristic equation
given below is greater than, less than, or equal to 1.0 sec. s 3 + 4s 2 + 6s + 4 =
0 (Apr 06)

47. The characteristic equation of a control system is given by s 4 + 20s 3 + 15s 2

+ 2s + k = 0 use Routh -Hurwitz criterion to find the value of K for which the
system will be marginally stable and the frequency of the corresponding
sustained Oscillations. (Apr 06)
K ( s  0.5)
48. A unity feedback system has a plant G(s) = s( s  1)( s 2  2s  2) sketch the root
locus and find the roots when ζ = 0.5 (Apr 06)

K
G (s) 
49. A unity feedback system has the plant transfer function . s(s 2  24s  144)
Find the angle of departure from the complex pole and the gain when the
poles are at imaginary axis. (Apr 06)

50. i. Find the angle of arrival and the angle of departure at the complex zeros and
complex poles for the root locus of a system with open -loop transfer function
K (s 2  1)
G (s) H (s) 
s(s 2  4s  8)
ii. Draw the root locus diagram for a feedback system with open -loop transfer
K (s  5)
G (s) 
function s(s  3)
following systematically the rules for the construction
of root locus. Show that the root locus in the complex plane is a circle.
(Apr 05)

51. i. Determine the Breakaway points of the system which have hte open loop
transfer function
G(S) H(S) = k (s+4) / (s 2 + 2s + 4)
ii. Derive the magnitude and phase angle. (Apr 05)

K (s  3)
G (s)H(s) 
52. For the function (s  1)(s  2)
prove that part of root locus is circular.
Find the center, and radius of the circle. What are the breakaway points?
(Apr 05)

53. i. Determine the breakaway points of the system which have the open loop transfer function
K s  4
Gs H s   2
s  2s  4 
ii. Derive the magnitude and angle criteria for stability. (Nov 04)
54. i. Determine the range of value of k for the system to be stable and is
characterized by the equation
3 2
s + 3Ks + (K+2)s+4 = 0.
Gs  
K
ii. A unity feedback system has an open loop transfer function s  2 s 2
 4 s  8

. Use Routh’s test to determine the range of positive values of ‘K’ for which
the system is stable. (Nov 04)
K
G(s) 
55. A unit feedback control system given by s1  0.1s1  0.2s
i. Sketch the root locus diagram of the system.
ii. Determine the limiting value of gain K for stability
iii. Determine the value of gain at which the system is critically damped.
(May 04)

56. Calculate the values of K and w for the point in s -plane at which the root
locus of intersects the imaginary axis. (May 04)

Gs H s  
K
57. A unity feedback system has an open loop transfer function 
ss  3 s 2  2s  2 .
Sketch the root locus as “K” varied from 0 to infinity. (May 04)

58. The open loop transfer function of a unity feedback system control system is
K s  2
Gs  
given by 
ss  3s  4 s 2  2s  2 
i. Sketch the root locus diagram as a function of K.
ii. Determine the value of K which makes the relative damping ratio of the
closed loop complex poles equal to 0.707. (May 04)

59. i. Discuss the rules of construction of root loci.

ii. A unity feedback control system has the follow ing open loop transfer function
K ( s  3)
G( s) 
S ( s 2  2s  2)(s  5)(s  6)
Sketch the root locus diagram. Calculate the value of K
corresponding to the damping ratio of 0.42. (Jan 03)

60. Sketch the root locus plot of a unity feedback system with an open -loop
K
G( s) 
transfer function S ( s  2)(s  4) (Jan 03)

Find the range of values of K for which the system has damped oscillatory
response. What is the greatest value of K which can be used before
continuous oscillations occur? Also determine the frequency of continuous
oscillations.
Determine the value of K so that the dominant pair of complex poles of the
system has a damping ratio of 0.5.
61. The characteristic polynomial of a system is q(s) = 2s 5 + s 4 + 4s 3 + 2s 2 + 2s
+ 1. The system is
i. stable
ii. unstable
iii. marginally stable
iv. oscillatory (GATE 02)

62. Consider the feedback control system as shown

i. Find the transfer function of the system and its characteristic equation
ii. Use the R-H criterion to determine the ranges of k for which the system is
stable (GATE 01)

63. The loop transfer function of a feed back control system is given by G(s)H(s)
k (s  1)
= s(1  Ts)(2s  1) , k>0, using R-H criterion determine the region of k -T plane in
which the closed loop system is stable
(GATE 99)
64. The loop transfer system .of a single loop control system is given by
s 4  20s 3  15s 2  2s  k =0
i. Determine the range of k for the system to be stable
ii. Can the system be marginally stable? if so, find the required value of k and
the frequency of sustained oscillation
(GATE 98)
k ( s  3)
A system having an open loop transfer function G(s) = s(s  2s  2) is used in the
2
57.
control system with unity feedback. Using R -H criterion find the range of k
for which the feed back system is stable.
(GATE 96)
58. State the properties of Hurwitz Polynomial. (GATE 93)

59. Test the following Polynominals for Hurwitz Property.

F(S) = S 4 + S 3 + 5S 2 + 3S + 4 (IES 94)

60. By means of Routh’s Criteria determine the stability of the following system.
S 5 + S 4 + 3S 3 + 9S 2 + 16S + 10 = 0
S 6 + 3S 5 + 5S 4 + 9S 3 + 8S 2 + 6S + 4 = 0 (IES 93)

61. Which of the following statements about the equation below, for R -H
criterion is true 4s  s  3s  5s  10 =0
4 3 2
 i. it has only one root on the imaginary axis
ii. it has two roots in the right half of the s -plane
iii. the system is stable
iv. the system is unstable (IES 91)

UNIT-V

1. Sketch the Bode plot for the transfer function G(s) =ke-0.1s/s (1+s) (1+0.01s) and determine
the system gain k for the gain cross over frequency to be 5 rad/sec.
(Dec 14)

1. Define phase margin and gain margin and sketch the bode plot for the following
Transfer function G(s) H(s) =K s2 / (1+0.25s) (1+0.025s)
(May/June13)

2. The open loop transfer function of a unity feedback control system is G(s)
=k/s(1+0.1s)(1+s)
a) Determine the value of the k so that the resonance peak M P of the system
isequal to 1.4
b) Determine the value of k so that the gain margin of the system is 20db
c) Determine the value of the k so that the phase margin of the system is 60 0
(Nov/Dec12)

3. A unity feedback control system has the transfer function G(s) = K/s(s+a)
(a) Find the value of `K' and `a' to satisfy the frequency domain specifications of
Mr=1.04 and! r = 11.55 rad/sec.
(b) Evaluate the settling time and bandwidth of the system for the values of K
and a determined in part (a). (May/June12)

4. Sketch the Bode plot for the following transfer function and determine the system
gain K for the gain cross over frequency !c to be 5 rad/sec. G(s) = Ks2/(1+0:2s)(1+0:02s)
(May/June12)
5. The block diagram representation of a second-order type 0 system is shown in
Figure 8. Derive the frequency domain characteristics. (May/June12)

6. (a) Explain the following terms:

i. Frequency response
ii. Phase and gain margins.
(b) Sketch the Bode plot for the following transfer function G(s) = 75(1+0.2s)/s(s2+16s+100)
(May/June12)
7. i. Explain the significance of Bandwidth in the design of linear control
systems.
ii. Show that the error contributed by a simple pole in the Bode magnitude plot
is -3 dB at corner frequency.
iii. The asymptotic plot of a system is shown in figure.

Find the loop transfer function of the system. (Apr 11, Dec 10)

8. i. Define
a. Bandwidth
b. Resonant peak
ii. Explain how stability can be determined from Bode plots
iii. Find resonant peak & resonant frequency given ζ = 0.5. If the damping ratio
is changed to 0.9 find resonant peak & resonant frequency. Comment on the
result. (Dec 10)

9. i. Define
a. Minimum phase tf
b. Non minimum phase tf
ii. Enlist the steps for the construction of Bode plots
iii. Explain the procedure for determination of transfer function from Bode plots.
(Dec 10)

10. i. What do you mean by a critically stable system? How do you find out
whether a given system is critically stable from Bode plots?
ii. Define
a. Gain Margin
b. Phase Margin
iii. Sketch Bode phase angle plot of a system (Dec 10)

11. i. Explain why it is important to conduct frequency domain analysis of linear

control systems.
ii. Sketch the Bode Magnitude plot for the transfer function

Hence find `K' such that gain cross over freq. is 5 rad/sec. (May 10)
12. i. Given If r(t) = 6.8 Sin 4t find the output at steady state.
ii. Write a brief note on log-magnitude Vs phase plots.
iii. In Bode plots if gain cross over frequency is greater than phase cross over
frequency then the system is UNSTABLE. Elaborate. (May 10)

13. i. Explain the term frequency response analysis.

ii. Show that in Bode magnitude plot the slope corresponding to a quadratic
factor is -40 dB/dec.
iii. Explain with the help of examples
a. Minimum phase function
b. Non minimum phase function
c. All pass function. (May 10)

14. i. Define frequency response.

ii. Discuss the advantages & disadvantages of frequency response analysis.
iii. Bring out the correlation between time response & f requency response and
hence show that the correlation exists for the range of damping ratio 0 < ζ<
0:707. (May 10)

15. Sketch the Bode plot s of

15( s  5)
G(s) = s( s 2  16s  100)

Hence find gain cross over frequency. (May 09, 08)

16. i. Explain the term frequency response analysis.

ii. Show that in Bode magnitude plot the slope corresponding to a quadratic
factor is -40 dB/dec.
iii. Explain with the help of examples
a. Minimum phase function
b. Non minimum phase function
c. All pass function. (May 09, 07, Sep 07)

17. i. Explain the significance of Bode plots in stability studies of linear control
systems.
ii. Show that in case of a quadratic factor the phase angle is a function of
frequency w and damping ratio ζ.
iii. The magnitude plot of a system is shown in figure.

Find the open loop transfer function. (May 09)

18. i. Show that Bode plots of a system with transfer function having many factors
can be obtained by adding the Bode plots of individual factors.
ii. In the Bode magnitude plot of a system the slope changed by -40 db / dec at a
frequency w = w 2 . What can be the corresponding factor in the loop transfer
function?
iii. The magnitude plot of a system is shown in figure.

19. A system is given by

i. Draw Bode plots
ii. Find GM & PM
iii. Hence determine the closed loop stability of the system. (May 09)

20. i. Find the angle of arrival and the angle of departure at the complex zeros and
complex poles for the root locus of a system with open -loop transfer function
K ( s 2  1)

G(s)H(s)= s( s  4s  8) .
2

ii. Draw the root locus diagram for a feedback system with open -loop transfer
K ( s  5)
function G(s)= s( s  3)
., following systematically the rules for the construction
of root locus. Show that the root locus in the complex plane is a circl e.
(Sep 08)

21. Define phase margin and gain margin. (Sep 08)

K
22. i. For the function G(s)H(s)= s( s  2)(s  4)
determine the breakaway point and the
value of K for which the root locus crosses the imaginary axis.
ii. Explain the terms with reference to root locus.
a. Asymptotes
b. Centroid
c. Break away point. (Sep 08)

23. i. Derive expression of peak resonance and bandwidth.

ii. Define the following frequency response specifications.
a. Peak Resonance b. Bandwidth c. Phase Margin d. Gain Margin
 (Sep 08)

24. i. Show that the breakaway and break-in points, if any, on the real axis for the
root locus for G(s)H(s)= kN(s)/D(s) , where N(s) and D(s) are rational
polynomials in s, can be obtained by solving the equation dK/dS =0.
ii. By a step by step procedure draw the root locus diagram for a unity neg ative
K ( s  1)
feedback system with open loop transfer function G(s)= s (s  9) . Mark all the
2

salient points on the diagram. Is the system stable for all the values of K?
(Sep 08)

25. i. Find the value of K and a to the following frequency domain specifications
K
M r = 1.04, w r = 11.55 rad/sec. Assume G(s) = s( s  a) unity feed back system.
ii. Sketch the Bode Plot for the following transfer function and determin e in
each case the system gain K for the g ain cross over frequency wc to be 5
Ke 0.1s
rad/sec. G(s) = s ( s  1)(1  0.1s ) (Sep 08)

26. i. With usual notations derive equations for the angle of departure and the
angle of arrival of the root locus from complex poles and zeros.
ii. The characteristic equation of closed -loop system is s 2 + (2+k) s+26=0. Draw
the root locus of the system. Mark the salient points on the diagram. (Sep 08)

27. Write short notes:

i. Frequency domain specifications
ii. Stability analysis from Bode plots . (Sep 08)

28. i. Derive the expressions for resonant peak & resonant frequency and hence
establish the correlation between time response & frequency response.
ii. Given ζ = 0.7 & w n = 10r/s find resonant peak, resonant frequency &
Bandwidth. (Sep 08, May 07)

29. i. Show that for a critically stable system the gain cross over frequency is equal
to phase cross over frequency.
ii. The Gain Margin of a type-1, 2 nd order system is always infinity. Justify.
iii. The Bode plots of a system is shown in figu re.
 iv. From the Bode plots of a unity feed back system, at gain cross over
frequency is found to be .150o &|| at phase cross over frequency is found to
be -12 dBs. Find the stability of the system. (Sep 08)

30. i. State the advantages & limitations of frequency domain analysis

28.5e 0.1s
ii. Sketch the Bode plots of G(s) = s (1  s )(1  0.1s ) Hence find gain cross over
frequency. (May 08)

31. i. Explain gain margin and phase margin. (May 08, 04)
ii. The open loop transfer function of feed back system is G(s)H(s)=k(s+1)/(s-1)
.Comment on stability.

32. i. Define frequency response.

ii. Discuss the advantages & disadvantages of frequency response analysis.
iii. Bring out the correlation between time response & frequency response and
hence show that the correlation exists for the range of damping ratio 0 <ζ<
0.707. (May 08)

33. A unity feedback system has a plant G(s) = K (s+0.5) /s(s+ 1) (s 2 +2s+ 2)
sketch the root locus and find the roots when ζ= 0.5. (May 08)

34. i. Explain the frequency response specifications.

100(0.02s  1)
ii. Draw the Bode Plot for the system having G(s)H(s) = 1)(0.1s)(0.01s  1)
( s  . Find gain and
phase cross over frequency. (May 08)

35. Sketch the Bode Plot for a unity feedback control system with forward path transfer
24
function G(s) = ( s  2)(s  6) Determine the gain margin and phase margin. (May 08)

36. i. The open loop transfer function of a feed back control system is G(s)H(s)=
K (1  2s)
s(1  s )(1  s  s 2 )
. Find the restrictions on K for stability. Find the values of K for
the system to have a gain margin of 3 db. With this value of K, find the phase
cross over frequency and phase margin.
ii. Explain how Bode plot is used to find gain margin and phase margin.
(May 08)

37. i. Determine the breakaway points of the system which have the open loop
K ( s  4)

transfer function G(s)H(s)= ( s 2  2s  4)

ii. Derive the magnitude and angle criteria for stability. (May 08)

38. i. Explain the Relative stability.

 s2
ii. The open loop transfer function of a unity feed back system is G(s)H(s)=( s  1)(s  1)

Comment on the stability. (May 08)

39. i.Write a note on determination of range of ‘K’ for stability using Bode plots.
ii. Define GM & PM and explain how you can determine them from Bode plots.
(Sep, May 07)

40. i. Explain clearly the steps involved in the construction of Bode plots of a
system with loop transfer function consisting of
a an open loop gain K b. one pole at origin c. one quadratic factor
ii. Given G(s) =
Determine the phase angle at 0, 5 & œ frequencies. (Sep 07)

41. i. Define
a. Minimum phase tf b. Non minimum phase tf
ii. Enlist the steps for the construction of Bode plots
iii. Explain the procedure for determination of transfer function from Bode plots.
(Sep 07)

42. i. Explain the concept of phase margin and gain margin.

ii. Draw the Bode Plot for a system having G(s) = Determine:
a. Gain cross over frequency and corresponding phase margin.
b. Phase cross over frequency and corresponding gain margin.
c. Stability of the closed loop system. (May 07)

43. i. Explain the correlation between time and frequency response of a system
ii. Sketch the Bode Plot for a unity feed back system characterized by the open

loop transfer function G(s) = Show that the system is

conditionally stable. Find the range of values of K for which the system is
stable (May 07)

44. i. The open loop transfer function of a unity feed back system is G(s)=

Determine the value of K for the following case:

a. Resonance Peak is required to be equal to 1.58. b. Gain margin of the
system is 21 db.
ii. A certain unity feed back control system is given by the value of K so as to

have .Draw the Bode Plot of the above system. Determine from
the plot of the value of ‘K’ so as to have:
a. Gain margin = 10 db b. Phase margin = 50 0 (May 07)

45. i. Explain the frequency response specifications.

 ii. Draw the Bode Plot for the system having G(s)H(s) = . Find
gain and phase cross over frequency. (May 07)

46. i. Explain why it is important to conduct frequency domain analysis of linear

control systems.
Ks 2
G(s) 
ii. Sketch the Bode Magnitude plot for the transfer function s(1  0.2s)(1  0.02s)

Hence find ‘K’ such that gain cross over freq. is 5 r ad/sec. (May 07)

47. Sketch the Bode plot for a unity feed back system characterized by the open
loop transfer function. Show that the system is conditionally stable. Find the
range of K for which the system is stable. (Sep 06)
K

S (S  3)(s 2  2S  3)
48. Plot the root locus for a system having G(s)H(s) for all
positive values K. Hence obtain gain margin and phase margin. (Apr 06)
200
49. i. For a 2 nd Order system with G(s) = s( s  8) and unit feed back, find various
frequency domain specifications.
ii. Consider a feedback system as shown below figure;

(Apr 06)

Find the value of ‘K’ & ‘a’ to satisfy following frequency domain
specification Mr=1.04, =11.55 rad/sec.

50. i Define phase margin and gain margin

ii. Sketch Bode plot & find value of “K” such that gai n crossover frequency is
Ks 2
G (s) 
5 rad/sec. (1  0.2 s )(1  0.02 s )
(Apr 06)

51. For a 2 nd order system characteristic equation is S 2 +4S+25. Determine the

resonance magnitude, normalized resonance, normalized resonance freq, and
resonance angle. Also find resonance frequency w r (Apr 06)

52. Sketch asymptotic Bode plot & find gain margin & phase margin. By what
factor should ‘K’ be increased or decreased to obtain a gain margin of 40 db.
K 10  (1  a)
G( s) H ( s)  with K  1
s( s  2)(s 2  2s  25)
(Apr 06)
53. Draw the Bode diagram and determine the stability of the closed loop
system with following open-loop transfer function. (Apr 06)

10 (1  0.5s)
G (s) 
54. Sketch the Bode Plot for the following Transfer function s(1  0.1s)(1  0.2s)
Calculate Gain margin and phase margin. (Apr 05)
100 0.02s  1
G(s)H s  
55. Draw the Bode Plot for the system having s  10.1s  10.01s  1 . Find gain
and phase cross over frequency. (May 04)
Ke 0.5S
Gs  
56. Sketch the Bode plot for the Transfer function s2  s 1  0.3s  (May 04)

57. Sketch the Bode plot for the following transfer functions and determine in
each case the system gain K for the gain cross over frequency w c to be 5
Ke 0.1S
G s  
rad/sec. ss  11  0.1s  and the Bode plot for a system having
Gs   , H s   1.
3
s1  0.05s 1  0.2s 
Determine the gain margin, phase margin and stability of closed loop system
2( S  3)
G( S ) H ( S ) 
2( S  1) . (May 04)

58. By analytical method calculate the gain margin in dB of the unity feedback
10
G( s) 
control system with transfer function s( s  1)( s  2) .
(May 04)

59. Consider the closed loop feedback system shown below. Determine the range
of K for which the system is stable.

(May 04)
G s  
1  100s1  s 
60. Sketch the Bode Plot for 1  10s 1  0.1s 
. Assume unity feed back. Obtain
gain margin and phase margin using semi log sheet. (May 04)

61. i. Define peak resonance and band width

2561  0.5s
Gs  
ii. Sketch Bode plot for 
s1  2s s 2  3.2s  64 
(May 04)
100(0.02s  1)
62. Draw the Bode plot for the system having G(s )H(s) = ( s  1)(0.1s  1)(0.01s  1)

Find gain and phase cross over frequency. (Nov 03)

63. Sketch the Bode Plot for the following transfer function
10(1  0.5s)
G(s) = s(1  0.1s)(1  0.2s) Calculate gain margin and phase margin. (Nov 03)

64. Draw the Bode magnitude and phase plots for the following transfer function:
G s  
10

S  s  0.4s  4 
2
 

Also obtain the gain margin and phase margin from the plots. (Jan 03)

65. Sketch the Bode magnitude and phase plots of a closed -loop system which
has the open-loop transfer function G(s) = 2e-sT/ S(1+s)(1+0.5s).Determine
the maximum value of T for the system to be stable. (Jan 03)

66. Draw the Bode magnitude and phase diagrams for the system with the
transfer function
200 (s+0.5)
G(s) = —————————
s(s+10) (s+50)
Also obtain the phase margin and gain margin from the plots. (Jan 03)

67. The gain margin and the phase margin of a feedback system with
s
G( s) H ( s) 
( s  100 ) 3
are
(a) 0 dB,0 0 (b)infinite, infinite (c) infinite,0 0 (d) 88.5dB,infinite
(GATE 03)
1
68. The system with the open loop transfer function G(s) H(s) = 𝑠(𝑠+1) has a gain
margin of
(a) -6 dB (b) 0dB (c) 3.5dB (d) 6dB

(GATE 02)

69. By analytical method calculate the gain margin in dB of the unity feedback
10
control system with transfer function G(s)= 𝑠(𝑠+1)(𝑠+2)
(IES 00)

70. Define stability. Discuss any two methods for finding the stability of a linear
system what are the advantages of Routh criteria of finding stability of a
system over other methods? (IES 00)
 71. The open loop transfer function of unity feedback control system is given by
𝑘
the expression G(s) = (𝑠+2)(𝑠+5) (IES ‘99)

UNIT-VI
52
1. Sketch the nyquist plot for the transfer function G(s)H(s)= (𝑠+2)(𝑠2+2𝑠+5) .find
the relevant stability parameters and discuss its stability. (Nov/Dec12)

2. State and explain Nyquist stability criterion. Draw the Nyquist plot for the open loop
transfer function G(s) = 1/s(1+0:1s)(1+s) and discuss the stability of the closed loop
system. (May/June12)

3. The characteristic equation of a feedback control system is s3+4Ks2+(K+3)s+10=0.

Apply the Nyquist criterion to determine the values of K for a stable closed loop system.
Check the answer by means of the Routh Hurwitz criterion. (May/June12)

4. (a) State and explain the Nyquist stability criterion.

(b) Sketch the Nyquist plot for the transfer function G(s)H(s) = 32/(s+1:5)(s2+2s+5) .Discuss
its stability. (May/June12)

5. (a) Explain the Nyquist criterion for assessing the stability of a closed loop system.
(b) Sketch the polar plot of the transfer function G(s) = 1/(1+T1s)(1+T2s)(1+T3s) . Determine
the frequency at which the polar plot intersects the real and imaginary axis of G(jw) plane.
(May/June12)

6. i. Explain the selection criteria of Nyquist contour in stability analysis of linear

control systems.
ii. Discuss the effect of adding poles& zeros on the stability of a system with
the help of Nyquist plots.
(Apr 11)
7. i. Distinguish between polar plots & Nyquist plots.
ii. Discuss the effect of adding poles & zeros to G(s)H(s) on the shape of
Nyquist plots. (Dec 10)

8. i. Addition of a non zero pole to a transfer function results in further rotation

of the polar plot by –90 0 as w→œ”. Justify with the help of an example
ii. A system is given by . Determine the magnitude & phase
angle at zero & œ frequencies. Hence sketch the polar plot. (Dec 10)

9. i. Compare the Nyquist stability method with other methods & hence br ing out
the advantages of the Nyquist method.
 ii. Relative stability analysis for open loop unstable cannot be carried out by
Nyquist method. Why?
(Dec 10)
10. i. Explain Nyquist stability criterion.
3
ii. With the help of Nyquist plot assess the stabil ity of a system G(s)= 𝑠(𝑠+1)(𝑠+2)
What happens to stability if the numerator of the function is changed from 3
to 30? (Dec 10)

11. i. Draw & explain polar plots for type -0, type-1 & type-2 systems.
ii. Write a note on relation between root loci & Nyquist plots. (May 10)

12. Starting from the “principle of argument” show that the Nyquist plot
encircles the (-1+j0) point (P-Z) times in anticlockwise direction where `P' is
the no. of open loop poles & `Z' is the no. of closed loop poles lying in the
right half of s-plane. (May 10)

13. i. Define Polar plot.

ii. Explain how you can determine relative stability using polar plots.
1
iii. Sketch the polar plot of a system given by G(s)= 𝑠(1+𝑠)(1+2𝑠). If the plot crosses
the real axis determine the corresponding frequency & magnitude.(May 10)

14. i. State Nyquist Stability Criterion.

ii. Explain the use of Nyquist Stability Criterion in the assessment of relative
stability of a system.
iii. Enlist the step-by-step procedure for the construction of Nyq uist plots(May
10)

15. i. Explain Nyquist stability criterion.

3
ii. With the help of Nyquist plot assess the stability of a system G(s) = 𝑠(𝑠+!)(𝑆+2)
what happens to stability if the numerator of the function is changed from 3
to 30? (May 09, Sep 07)

16. i. State & explain “principle of argument”

𝑘
ii. Given𝐺(𝑠) = 𝑠(𝑠+2)(𝑠+10). Sketch Nyquist plot & find range of? K? For
stability. (May 09)

𝑘
17. i. A unity feed back system has 𝐺(𝑠) = .Discuss the effect on
𝑠(1+𝑠𝑇1)(1+𝑠𝑇2)
Nyquist plot when the value of K is
a. low (<critical value)
b. = critical value
c. high ( >critical value)
ii. Pure time delay usually deteriorates the stability. Explain with the help of
Nyquist plots. (May 09)
18. i. Bring out the relevance of relative stability analysis in linear control systems
ii. Discuss the effect of adding one pole & one zero (simultaneously &
separately) to a given transfer function on the polar plot. (May 09)

19. i. What is “Nyquist Contour”?

4𝑠+1
ii. A system is given by G(s) = .Sketch the Nyquist plot & hence
𝑠2 (𝑠+1)(2𝑠+!)
determine the stabilityof the system. (May 09, 07)

20. i. Explain the effect of addition of a pole at the origin on the polar plot of a
given system.
ii. Sketch the polar plot & hence find the frequency at which the plot intersects
0.1
the +ve imaginary axis for the system G(s) = 𝑠2 (1𝑠)(1+0.1𝑠) .Also find the
corresponding magnitude. (May 09)

21. i. Construct the complete Nyquist plot for a unity feed back control system
𝑘
whose open loop transfer function is G(s)H(s) = 𝑠(𝑠2 +2𝑠+2). Find maximum
value of K for which the system is stable.
ii. The open loop transfer function of a unity feed back system is G(s )
1
= 𝑠(1+0.5𝑠)(1+0.1𝑠). Find gain and phase margin. If a phase lag element with
1+2𝑠
transfer function of (1+5𝑠) is added in the forward path, find how much the
gain must be changed to keep the margin same. (Sep, May 08)
𝑘(1+𝑠)
22. i. The open loop transfer function of a feed back system is G(s)H(s)= (1−𝑠)
Comment on stability using Nyquist Plot.
1+0.2𝑠
ii. The transfer function of a phase advance circuit is 1+0.2𝑠 . Find the maximum
phase lag. (Sep 08)
𝑘(𝑠+1)
23. Draw the Nyquist Plot for the open loop system G(s) = 𝑠(𝑠−1) and find its
stability. Also find the phase margin and gain margin . (Sep 08)

24. Sketch the polar (Nyquist) plot on a plain paper for the following transfer
10
function G(s) = 𝑠(1+𝑠)(1+0.005𝑠). (Sep 08)

25. i. Define Polar plot.

ii. Explain how you can determine relative stability using polar plots.
1
iii. Sketch the polar plot of a system given by G(s) = 𝑠(1+𝑠)(1+2𝑠).If the plot
crosses the real axis determine the corresponding frequency & magnitude.
(Sep 08)

26. i. State Nyquist Stability Criterion.

ii. Explain the use of Nyquist Stability Criterion in t he assessment of relative
stability of a system.
 iii. Enlist the step-by-step procedure for the construction of Nyquist plots.
(Sep 08, 07)

27. i. Explain Nyquist stability criterion

ii. A unity feedback control system has an open loop transfer function given by
100
G(s)H(s) = (𝑠+5)(𝑠+2) Draw the Nyquist diagram and determine its stability.
(Sep 08, May 07)

28. Write short notes:

i. Comparision of polar & Nyquist plots
ii. Applications of Nyquist criterion. (May 08)

29. i. With respect to a function q(s) “Every s -plane contour which does not pass
through any singular points of q(s) has a corresponding contour in q(s)
plane” Elaborate.
ii. What is the effect of adding a zero at origin to the to the open loop transfer
function on polar plot?
(May 08)
30. The open loop transfer function of a unity feedback control system is G(s)
10
= (𝑠+1)(𝑠+5) . Draw its polar plot and hence determine its phase ma rgin and gain
margin. (May 08)

31. i. Explain how the type of a system determines the shape of polar plot. (Sep 07)
ii. Write a note on Nyquist criterion for minimum phase & non minimum phase
transfer functions.

32. i. What is “Nyquist Contour”? (Sep 07)

ii. A system is given by
4𝑠+1
G(s) = 𝑠2 (𝑠+1)(2𝑠+1)Sketch the Nyquist plot & hence determine the stability of
the system.
𝑘(1+𝑠)
33. The open loop transfer function of a feed back system is G(s)H(s)= (1−𝑠)
Comment on stability using Nyquist Plot . (May 07)

29(𝑠2 +𝑠+0.1
34. i. Plot the polar plot of 𝐺(𝑠) = 𝑠(𝑠+1)(𝑠+10)
ii. Explain the concept of Nyquist stability criterion. (May 07)

𝑘(𝑠+3)
35. Draw the Nyquist Plot for the open loop system g(s)= and find its
𝑠(𝑠−1)
stability. Also find the phase margin and gain margin (May 07)

36. i. A system has one open loop pole & two closed loop poles in Right Half of s
plane. Show that the Nyquist plot encircles the ( -1+j0) point once in
clockwise direction.
 ii. Addition of poles to the loop transfer function reduces the closed loop
stability of the system. Justify by Nyquist plots. (May 07)

37. i. Explain how polar plots are useful in fi nding the stability of a system
ii. Sketch the Nyquist plot and the stability of the following system 𝑔(𝑠)𝐻(𝑠) =
100
(𝑠+2)(𝑠+4)(𝑠+8)
(Sep 06)

38. Comment on the stability of the system whose open loop transfer function 𝐺(𝑠)𝐻(𝑠) =
1
Also find gain and phase margin (using Nyquist plot). (Sep 06)
𝑠(1+2𝑠)(1+𝑠)

39. Explain how Nyquist contour is selected for stability analysis. (Apr 06)

40. Discuss the stability of the following system using Nyquist stability criterion

(Apr 06, 05)

41. i. Explain what is meant by the Relative stability of a system and the manner in
which this is specified.
ii. Construct the complete Nyquist plot for a unity feed back control system whose open
𝑘
loop transfer function is 𝐺(𝑠)𝐻(𝑠) = 𝑠(𝑠2 +3𝑠−10) Find maximum value of K for which
the system is stable. (Apr 05)
𝑘(1+2𝑠)
42. i. The open loop transfer function of a feed back system is (𝑠)𝐻(𝑠) = (1−𝑠) .
Comment on stability using Nyquist Plot.
1+0.2𝑠
ii. The transfer function of a phase adance circuit is 1+0.2𝑠 . Find the maximum
phase lag. (May 04)
1
43. Sketch the Nyquist Plot for 𝐺(𝑠)𝐻(𝑠) = 𝑠4 (𝑠+𝑝) , 𝑝 > 0 Explain the terms gain
margin and phase margin. (May 04)

44. i. Explain the Polar Plots. (May 04)

ii. The open loop transfer function of a unit feed back control system is given
(1+100𝑠)(𝑠+40)
by G(s)H(s)= 𝑠3 (𝑠+200)(𝑠+1000).Discuss the stability of a closed loop system as a
function of k. Determine values of K which will cause sustained oscillations
in the closed loop system. What are the frequenc ies of oscillations? Use
Nyquist approach.

45. Make a rough sketch of the Nyquist plot for system whose open loop transfer
5
function is G(s)H(s)= 𝑠(1+0.2𝑠)(1+𝑠)
i. Is the above system stable? Explain.
ii. Define gain margin of a system and determine the GM of the system specified
in (i).
iii.Define the PM of a system and indicate how this can be determined from
the Nyquist plot. (May 04)
46. Sketch the complete Nyquist plot for the following loop transfer function
G s  H s  
10

S
2
1  0.25s1  0.5s
And test its stability under closed-loop condition. (Jan 03)

47. i. Explain the Nyquist criterion for accessing the stability of a closed -loop
control system.
100
G( s) H ( s) 
ii. Sketch the Nyquist plot of the transfer function: S ( s  1)(s 2  2s  2)

Discuss its stability. (Jan 03)

G s  
10
48. i. Sketch the polar plot for the transfer function s s  1s  2 

ii. Explain the Nyquist stability criterion. (Jan 03)

3(𝑠+2)
49. Draw the complete Nyquist plot forG(s)H(s)= and discuss stability of
𝑠2 +3𝑠+1)
the system. (Jan 03)
50. The loop transfer function of a single loop control system is given by
100
G(s)H(s) = 𝑠(𝑠+0.01𝑠) 𝑒 −𝑠𝑇 Using the Nyquist criterion ,find the condition for
the closed loop system to be stable. (GATE 98)

51. Explain the meaning and significance of phase and gain margins of a
feedback control systems. How will you obtain the values of these margins
from
i. Polar plots
ii. Bode plots
Illustrate your answer by giving plots for stable and unstable systems
separately. (GATE 94)
1
G( s) 
52. Sketch the Nyquist plots of the given transfer function (1  s )(1  2s )
(IES 00)

53. Sketch the Nyquist plot for the system and comment on the stability or the
1.06
GH 
closed loop systems s ( s  1)(s  2)
(IES 99)
UNIT-VII

1. a) what are the different type of compensators? Explain briefly.

b) show that the lead network and the lag network inserted in cascaded in an
open loop acts as proportional plus derivative control(in the region of small
w) and proportional plus integral control (in the region of large w)
respectively. (Nov/Dec12)

2. A unity feedback system has an open loop transfer function G(s) = K/s(s+2)(s+60) .Design
a Lead-Lag compensator to meet the following specifications:
(a) Phase margin is at least 400
(b) Steady state error for ramp input is 0.04 rad. (May/June 12)

3. The open-loop transfer function of a control system with unity feedback is G(s)
=K/s(0.1s+1)(0.2s+1) . Design a suitable compensator so that the system satisfies the
following performance specifications:
(a) Kv=100; or the steady-state error of the system due to a step ramp function input is 0.01
in magnitude and
(b) Phase margin _40 degrees. (May/June 12)

4. A unity feedback system has an open loop transfer function G(s) = K/s(s+2)(s+60) .Design
a Lead-Lag compensator to meet the following specifications: (May/June 12)

5. The open loop transfer function of the uncompensated system is G(s) = 2500K/s(s+25) .
Design a suitable compensator so that the system satisfies the following performance
specifications:
(a) The phase margin of the system should be greater than 45 degrees.
(b) The steady-state error due to a unit ramp function input should be less than or equal to
0.01 rad/sec. (May/June 12)

6. For the unity feed back control system forward path transfer func tion G(S) =
K/S (S+4) (S+20). Design a lag-lead compensator so that PM > 40 and steady
state error for unit ramp input < 0.04 rad.
(Apr 11, Dec 10)
7. i. Explain the need of lead compensator and obtain the transfer function of lead
lag compensator.
ii. Explain the significance of compensator? (Dec 10)

8. i. What is compensation? What are the different types of compensators?

ii. What is lag-lead compensator, obtain the transfer function of lag -lead
compensator and draw its pole-zero plot?
iii. Explain the different steps to be followed for the design of lag lead
compensator using Bode plot?
 (Dec, May 10)
9. The open loop transfer function of unity feedback system is G(s)=k/s(s+1)
It is desired to have the velocity error constant KV = 12 Sec –1 and phase
margin as 40 0 Design lead compensator to meet the above specifications.
(May 10)

10. Design a lead compensator for unity feed back system whose open loop
𝑘
transfer function 𝐺(𝑠) = 𝑠(𝑠+1)(𝑠+5) to satisfy the following specifications.
i. velocity error constant K V > 50
ii. Phase margin > 20 0 . (May 10)

11. Consider a Unity feedback system with open loop transfer function
100
G(s)= (𝑠+1)(𝑆+2)(𝑠+10) . Design a PID controller. So that the phase margin of the
system is 45 0 at a frequency of 4 rad/sec. and steady state error for unit ramp
input is 0.1. (May 10)

12. Design a lead compensator for unity feed back system whose open loop
transfer function
𝑘
G(S) = 𝑠(𝑠+1)(𝑠+5)to satisfy the following specifications.
i. Velocity error constant K V > 50
ii. Phase margin > 20 0 . (May 09, 08)

13. i. What is compensation? What are the different types of compensators?

ii. What is a lead compensator, obtain the transfer function of lead compensator
and draw pole-zero plot?
iii. Explain the different steps to be followed for the design of lead compensator
using Bode plot?
(May 09, 07, Sep 08, 07)
𝑘
14. For G(s)=. 𝑠(𝑠+2)(𝑠+20) Design a lag compensator given phase margin > 35 0 and
K V < 20. (May 09)
15. i. Explain the need of lead compensator and obtain the transfer function of
lead-lag compensator.
ii. Explain the significance of compensator? (May 09)

16. Design a phase lead compensator for unity feed back system whose open loop
𝑘
transfer function 𝑠(𝑠+1). The system has to satisfy the following specifications.
i. The phase margin of the system > 45 0
ii. Steady state error for a unit ramp input. < 1/15
iii. The gain crossover frequency of the system must be less than 7.5 rad/sec.
(May 09)
17. For the unity feed back control system forward pa th transfer function G(S) =
K/S (S+4) (S+20). Design a l ag-lead compensator so that PM =40 and steady
state error for unit ramp input < 0.04 rad.
(Sep 07)
1+0.2𝑠
18. The transfer function of a phase advance circuit is 1+.2𝑠 Find the maximum
phase lag. (May 07)
1
19. The open loop transfer function of unity feedback is G(S)= 𝑠(𝑠+1)(0.5𝑠+1).Design
a compensator to meet the following specifications. Velocity error constant
K v =5 sec -1 : phase margin=40 0 : gain margin = 10 db (Apr 06)

20. i. Define sensitivity and explain mathematically.

ii. What is a PID controller and derive its transfer function. (Nov 03)
1
21. A unity feedback system has the plant transfer function G p (s) = (𝑠+1)(2𝑠+1)
i. Determine the frequency at which the plant has phase lag of 90 0
ii. An integral controller with transfer function Gc (s) = (k/s) is placed in the
feed forward path of the feedback system. Find the value of k such that the
compensated system has an open loop gain margin of 2.5.
iii. Determine the steady errors of the compensated to unit step and unit ramp
inputs. (GATE 02)
𝑠+1/𝑇
22. The phase-lead network function Gc(s)= 𝑠+1/𝑎𝑇,where a<1 would provide
maqximum phase-lead at a frequency of
1 1
(i) 1/T (ii) a/ (aT) (iii) 𝑇 𝑎 (iv) 𝑎√𝑇 (IES 98)
√
23. Consider a feedback control with open loop transfer function G(s) = k/s(s+1)
Design a series compensator to provide the following specifications:
i. The phase margin of the system must be greater than 45 0 .
ii. When the input to the system is a ramp ,the steady state error of the output in
position should be less than 0.1 degree/deg/sec of the final out velocity
(IES 93)

24. What are compensators explain clearly lag, lead and lag -lead compensators.
How do you proceed to design a lag compensator? (IES 92)
𝑠𝑘1
25. Draw the polar plot of the following transfer function, 𝐺𝐻(𝑠) 𝑠(𝑠+𝑝1)(𝑠+𝑝2)
design a lead compensator. (T1-Ch8)
𝑘1
26. Draw the polar plot of the following transfer function 𝐺𝐻(𝑠) 𝑠(𝑠+𝑝1)(𝑠+𝑝2),
design a lag compensator. (T1-Ch8)
27. Draw the polar plot of the following transfer function, design a
lead compensator. (T1-Ch8)
𝑘1
28. Draw the polar plot of the following transfer function 𝐺𝐻(𝑠) 𝑠(𝑠+𝑝1)(𝑠+𝑝2) ,
design a lag compensator. (T1-Ch8)
29. Draw the polar plot for designing a lag compensator the system. (T1-Ch8)

30. Draw the polar plot for designing a lead compensator the system. (T1-Ch8)

31. Draw the polar plot for designing a lead lag compensator the system.
(T1-Ch8)

32. Consider the open loop transfer function check whether the system is stable
or unstable, if unstable design a appropriate compensator (T1-Ch10)

33. Is the system represented by the characteristic equation, ever conditionally

stable? Why? (T1-Ch10)

34. How would the inclusion of a minor feedback loop with a transfer function
K 2 s,(K 2 >0). Determine the transient and steady performance of the system.
K1/(s(s+a) (T1-Ch10)

35. Outline the design of unity feed back system with a plant given
𝑘1
by 𝐺𝐻(𝑠) 𝑠(𝑠+𝑝1)(𝑠+𝑝2) and the performance specifications Kv=50,(3)The
bandwidth BW of the compensated system must be approximately equal to or
not much greater than that of the uncompensated system because high
frequency ‘noise’ disturbance are present under operating conditions.
(T1-Ch10)

36. Design a compensator which will yield a phase margin of approximately 45 o

1.06
for the system defined by. 𝐺𝐻(𝑠) 𝑠(𝑠+1)(𝑠+2) (T1-Ch10)
37. Design a compensator which will yield a phase margin of 40 o and a velocity
84
constant K v =40 for the system given by 𝐺𝐻(𝑠) 𝑠(𝑠+6)(𝑠+2) (T1-Ch10)

38. What kind of compensation can be used to yield a maximum overshoot of

84
20% for the system defined by 𝐺𝐻(𝑠) 𝑠(𝑠+6)(𝑠+2) (T1-Ch10)
10
39. Design a lead compensator for the given transfer function, 𝐺𝐻(𝑠) 𝑠(𝑠+8)(𝑠+2)
(T1-Ch10)
10
40. Design a lag compensator for the given transfer function , 𝐺𝐻(𝑠) 𝑠(𝑠+8)(𝑠+2)
(T1-Ch10)
41. Design a electronic lead –lag compensator using operational amplifier.
(T1-Ch10)
42. Derive the necessary equations for all possible case f or root locus based
compensator design.
(T1-Ch10)
43. Design a lead- lag compensator for the given transfer function,
C ( s) 2s  0.1

R( s) s 3  0.1s 2  6s  0.1 (T1-Ch10)
44. Design a lag, lead and lead lag compensators for RL. Series circuit
(T1-Ch10)
45. Design a lag, lead and lead lag compensators for RC. Series circuit
(T1-Ch10)
46. Design a lag, lead and lead lag compensators for RLC. Series circuit
(T1-Ch10)
47. Consider a unity feed back system which has an open loop transfer function
(T1-Ch10)
48. A unity feedback system is giv en the transfer function .Design a compensator
such that dominant closed lop poles are located at and the static velocity
error constant equal to 80 o sec -1 , (T1-Ch10)
49. Consider a unity feed back system given by chooses a=1 and determine the
𝑐(𝑠) 2𝑠+0.1
values of K and b if the closed loop poles are located at 𝑅(𝑠) = 𝑠2 +0.1𝑠2 +6𝑠+0.1
(T1-Ch10)

50. The model of aircraft system which has unity feed back is given as Design a
lead lag compensator to reduce the peak overshoot by 20%. (T1-Ch10)
51. Design a lead compensator for a second order unstable system. (T1-Ch10)

52. Describe the briefly the dynamic characteristics of the PI controller, PD

controller and PID controller. (T1-Ch10)
53. Consider a unity feed back system which has an open loop transfer function
𝑐(𝑠) 2𝑠+0.1
= Design a PID controller to reduce the peak ove r shoot.
𝑅(𝑠) 𝑠 +0.1𝑠2 +6𝑠+0.1
2

(T1-Ch10)
UNIT-VIII

1. Write short notes on:

a) Procedure to sketch the polar plot.
b) lead- Compensation.
c) State Transition Matrix and its properties (May/June13)

2. ̇ ⌊0
A system is given by the following vector matrix equation: 𝑋 =
1 0
⌋ 𝑋 + ⌊ ⌋ [𝑢],
−4 −5 1
1
Where the initial conditions are given by 𝑋(0) = [ ] Determine
1
a) state transition matrix
b) zero input response
c) zero state response for u= 1
d) total response (Nov/Dec12)

3. Explain the properties of state transition matrix. A linear time invariant system is described
̇ ⌊ 0 6⌋ 𝑋 + ⌊0⌋ [𝑢]and 𝑌 = [1 0]𝑋, 𝑋(0) = [0]
by the state equation: 𝑋 =
−1 5 1 0
Obtain the state transition matrix. Hence obtain the output response y(t) , t > 0 for a unit
step input. (May/June12)

4. Explain the terms `state' and `state variable'. Prove that the state space representation is not
̇ ⌊ 0 6⌋ 𝑋 + ⌊0⌋ [𝑢] and 𝑌 = [1 0]𝑋, 𝑋(0) = [0]
unique. 𝑋 = (May/June12)
−1 5 1 0

5. A linear time-invariant system is described by the following differential equation:

𝑑 2 𝑦(𝑡) 𝑑𝑦(𝑡)
+2 +y(t)=u(t)
𝑑𝑡 2 𝑑𝑡

Find the state space representation and obtain the complete response to unit step
input and zero initial conditions (May/June12)

6. Consider the network shown in figure and obtain the state variable form ?
 A linear time in variant system is characterized by homogenous state equation.

Compute the solution of homogenous equation, assuming the initial state vector.
(Apr 11)

7. i. Obtain the state model of the system shown in figure.

Consider the state variables as i 1 , i 2 , v

ii. Obtain the state model of a field controlled motor? (Dec, May 10)

8. i. Discuss the properties of state transition matrix.

ii. Determine the canonical state model of system, whose transfer function is
2(𝑠+5)
𝑇(𝑠) = (𝑠+2)(𝑠+3)(𝑠+4).
iii. What are advantages of state space analysis compared to transfer function
analysis? (Dec 10)

9. i. Find a state model (phase variable form) for the system with transfer
𝑦(𝑠) 𝑠+4
function. = 3 2
𝑢(𝑠) 𝑠 +6𝑠 +11𝑠+6
ii. A feedback system is represented by a signal flow graph shown in figure.
a. Construct a state model of the system
b. Diagonalize the Coefficient matrix A of the state model.

(Dec, May 10)

10. i. Obtain the state model of the system whose transfer function is given as.
𝑦(𝑠) 𝑠+10
= 3 2
𝑢(𝑠) 𝑠 +4𝑠 +2𝑠+1
ii. Consider the matrix A compute e At ?

(Dec 10)

𝑠+3
11. i. A control system has a transfer function given byG(s)= . Obtain
(𝑠+1)(𝑠+2)2
the canonical state variable representation.
ii. A system is described by

Find the transfer function? (May 10)

12. The state equation of a linear time invariant system is given by

i. Find the transition matrix ft and the characteristic equation of the system.
ii. Obtain the phase variable form of state model for the given system whose
differential equation is given below.
𝑑 2𝑦(𝑡) 𝑑𝑦(𝑡)
+11 +6y(t)=6u(t) (May 10)
𝑑𝑡 2 𝑑𝑡

13. i. For the given T.F T (s) =

Obtain the state model (phase variable form)?
ii. Construct the state model for a system characterized by the differential
equation. ¨y + 5 ÿ y + 6y = u. (May 09, 07, Sep 07)
14. i. Discuss the significance of state Space Analysis?
ii. Define state variables.
iii. Obtain the state variable representation of an armature controlled D.C
Servomotor? (May 09, 07)

15. i. A control system has a transfer function given by

Obtain the canonical state variable representation.
ii. A system is described by

Find the transfer function? . (May 09, 07, Sep 07)

16. A system described by
- 1 - 4 - 1 0
   
x  - 1 - 6 - 2 x   1 u
- 1 - 2 - 3  1
y = [1 1 1] x
Find the transfer function and construct the signal flow graph.
(May 09, Nov 03)

17. i. A linear tine invariant system is characterized by homogenous state equation.

Compute the solution of homogenous equation, assuming the initial state

vector.
ii. Obtain the state model of armature controlled de motor? (May 09)

18. i. Consider the vector matrix differential equation describe the dynamics of the

system as Determine state transition matrix?

ii. What are the properties of state transition matrix? (May 09)

19. i. Obtain the stat variable model in phase variable form f or the following
̈ ̇= 𝑢(𝑡)
system:𝑦⃛ + 2𝑦 + 3𝑦 + 4𝑦
𝑦(𝑠) 160(𝑠+4)
ii. The closed loop transfer function is given by = 3 2
𝑢(𝑠) 𝑠 +8𝑠 +192𝑠+640
Obtain the state variable model using signal flow graph. (Sep 08)

20. i. Explain properties of state transition matrix

ii. Consider the transfer function
Y(s) / U(s) = (2s 2 + s + 5)/(s 3 + 6s 2 + 11s + 4)
Obtain the state equation by direct decomposition method and also find state
transition matrix.
(Sep 08)
21. i. Write the state equations for the block diagram given figure

ii. For the given plant transfer functions construct the signal flow diagram and
determine the state space model. (Sep, May 08)
22. i. For the given system X = Ax + Bu

where

ii. Given Find the unit step response when,

(Sep 08, May 07)

23. i. The system is represented by the differential equation

Find the transfer from state variable representation.
ii. Consider the RLC network shown in figu re 8b. Write the state variable
representation.

24. i. Discuss the significance of state Space Analysis?

ii. Define state variables.
iii. Obtain the state variable representation of an armature controlled D.C
Servomotor? (Sep 08)

25. i. Obtain the state model of the system shown in figure

Consider the state variables as i 1 , i 2 , v

ii. Obtain the state model of a field controlled motor? (Sep 08, 07)

26. Obtain the two differential state representation for the system with transfer
Y (s) 2

function. U ( s) s 3  6s 2  11S  6 (May 08)
27. Obtain the state model of the system whose transfer function is given as.
Y ( s) 10

U ( s) S 3  4S 2  2S  1

Consider the matrix A compute e At ?

0 1
A
  2  3
(May 08)

28. i. Obtain state space mode for given mechanical system. Figure

(May 08)

ii. Obtain the state equations in canonical form for transfer function given
Y ( s)
 (3s 2  5s  13) /( s  2)(s 2  4s  8)
U ( s)
29. Find the unit step response for the following system with the initial
conditions (May 08)

30. i. Explain properties of state transition matrix (May 08)

ii. Consider the transfer function
Y(s) / U(s) = (2s 2 + s + 5)/(s 3 + 6s 2 + 11s + 4)
Obtain the state equation by direct decomposition method and also find state
transition matrix.

31. i. Define the terms

a. State variable b. State transition matrix.
ii. Obtain the state equation and output equation of the electric network show in
Figure

(May 07, Dec 05, Nov 03)

32. i. Obtain the state variable model in phase variable form for the following
̈ ̇= 𝑢(𝑡)
system: 𝑦⃛ + 2𝑦 + 3𝑦 + 4𝑦
𝑦(𝑠) 160(𝑠+4)
ii. The closed loop transfer function is give n by = 3 2
𝑢(𝑠) 𝑠 +8𝑠 +192𝑠+640
Obtain the state variable model using signal flow graph (May 07)

33. Find the canonical format representation and state transition matrix.

(May 07)
34. Consider the system represented by the differential equation as:

where y is output and u is the input. Obtain the state spac e representation of
the system. (Apr 06)

35. i. For the given system

Obtain Jordan form representation of state equation of A. Also find the

transfer function.
ii.Derive the expression for the transfer function G(s) = Y(s) / U(s) .Given
the state model

(Dec 05)

36. i. Construct the state variable model for the system characteri zed by the
differential equation

ii. Explain properties and significance of state transition matrix. (Dec 05)

37. i. For the given transfer function.Obtain the state model of the system.

1 0
ii. Obtain the state transition matrix (t) given the system matrix 𝐴 = [ ].(Dec 05)
1 1
38. Discuss the advantages and disadvantages of proportional, proportional
derivative, proportional integral and proportional integral derivative control
system (Dec 05)
39. i. For the given system X = AX + BU, Y = CX.

Obtain Jordan form representation of state equation of A. Also find the

trans-fer function.
ii. Derive the expression for the transfer function G(s) = Y(s) / U(s) .Given the
state model
X = AX + BU
Y = CX + D U (Apr 05)
40. i. Reduce the matrix A to diagonal matrix.

ii. Derive the state models for the system described by the differentia l equation
in phase variable form. y +4 y +5 y +2y = 2u+5u (Apr 05)

41. i. Determine the state variable matrix for the circuit shown
ii. A single input-single output system has the matrix equation, find the transfer
function (Apr 05)

42. i. A linear time invariant system is denoted by the differential equation

D 3 y + 3D 2 y + 3Dy + y = U where D = d y/dt
a. Write the state equations
b. Find the state Transition matrix
c. Find the characteristic equation and eigen values of A. (Apr 05, Nov 04)
ii. Obtain state space model for the following system Figure.

43. i. Explain properties of state transition matrix (Nov 04)

ii. Consider the transfer function
Y(s)/U(s) = (2s 2 + s + 5) / (s 3 + 6s 2 + 11s + 4)
Obtain the state equation by direct decomposition method and also find state
transition matrix.

 x1   0 1   x1(t) 
 x    2  3  x 
44. i. Given the state equation  2    2(t) 
Find the State Transmission
1
x0   
Matrix and zero input response for  1
ii. Obtain state variable model in Jordan form for the following system (May 04)
45. i. Obtain the state variable model in phase variable form for the following
system :
ii. The closed loop transfer function is given by
Y s  160s  4 

U s  s 3  8s 2  192s  640
Obtain the state variable model using signal flow graph. (May 04)

46. Obtain the time response of the following sys tem.

and output y = [1 0] x
0 1 0
x  x  1 u
  2  3  
where u(t) is the unit step input and the initial condition x 1 (0), x 2 (0) = 0.
(May 04)

47. Derive the expression for the transfer function from the state model.
x = Ax + Bu
y = Cx + Du (May 04)

48. i. Obtain state variable representation of a field controlled D.C. motor.

 2 1 0
X   0  2 1  x
 0 0  2
ii. Find the state transition matrix for a given system . (May 04)

49. Find the unit step, response of the following system.

 X 1  1 0  X 1  1
X      u
 2  1 1  X 2  1

X T (0) = [1, 0] (May 04)

50. i. Construct the state variable model for the system characterized by the
differential equation.
y + 6y + 11y + 2y = 41 + 1
Also give the block diagram of the model.
ii. Explain properties and significance of state transition matrix. (May 04)
0 1 0
51. i. Given the Matrix 𝐴 = [ 0 0 1]
−6 −11 −5
Write down the characteristic equation and obtain the eigen values. Also
obtain the diagonal matrix.
ii. Explain the advantages of state space model over input-output model.
(May 04)

52. i. Write the state equations for the block diagram given.
 (Nov 03)

ii. For the given plant transfer function construct the signal flow diagram and
determine the state space model.

53. i. Obtain state space model for given mechanical system.

(Nov 03)

ii. Obtain the state equations in canonical form for transfer function given
y(s)/u(s)= (3s 2 +5s+13)/(s+2)(s 2 +4 +8).

54. i. The state equations of a Linear system are as follows.

- 2 0 1  1

   
x   1 - 3 0  x  0 u
 1 1 - 1 1
(Nov 03)
y = [2 1 -1] x

Determine the transfer function y(s)/u(s).

ii. Explain various methods of evaluation of state transition matrix.

55. i. Derive the expression for the transfer function from the state model.
x = Ax + Bu
y = Cx + Du
ii. Obtain state variable representation of an armature controlled D.C. motor.

(Nov 03)
1 2 1
0
56. i. For the given system X = Ax + Bu where 𝐴 = [0 1 3] 𝐵 = [ ]
1
1 1 1
Find the characteristic equation of the system and its roots.
 0 1 𝑥1(𝑡) 0
ii. Given x(t)=[ ][ ] [ ] 𝑢(𝑡) Find the unit step response when, X(0)
−2 −3 𝑥2(𝑡) + 1
1
1
= (Nov 03)

57. i. For the system represented by the following equations, find the transfer
function X(s)/U(s) by signal flow graph technique.
X = x1 + b3 u
𝑥̇ 1= -a 1 x 1 + x 2 + b 2 u
𝑥2̇= -a 2 x 1 + b 1 u (May 03)

58. i. Obtain the solution of a system whose state model is given by X = A X(t) + B
U(t) ; X(O) =X 0 and hence define state Transition matrix.
ii. Obtain the transfer function of a control system whose state model is :
X (t) = A X(t) + B U(t)
Y(t) = CX(t)
0 1 0 0
Where 𝐴 = [0 −1 1 ] 𝐵 = [ 0 ] 𝐶 = [1 0 0] (May 03)
0 −1 −10 10

59. i. Obtain the solution of a system whose state model is given by X = A X(t) + B
U(t) ; X(0) =X 0 and hence define state Transition matrix.
ii. Obtain the transfer function of a control system whose state model is
X t   A X t   B U t  Y t   CX t 


0 1 0
Where A 
0 1 1


0 1  10

0
B
0
 C  1 0 0

10
 (May 03)

60. i. Obtain state variable representation of a field controlled D. C. motor.

ii. Find the state transition matrix for a given system. (May 03)
- 2 1 0 
x   0 - 2 1  x


 0 0 - 2
61. A system is characterized by the following state space equations.

 x1    3 1  x1  0
x    2 0 x   1 u; t  0
 2    2  
x 
y  {1 0}  1 
x 2 
i. Find the Transfer Function
ii. Compute the state transition matrix
 iii. Solve the state equation for a unit step input under zero initial condition
(Jan 03)
62. Write a short on the following:
i. State vector
ii. State transition matrix
(Jan 03)
63. i. Find a state model for the system with transfer function.
Cs  s4

Us  3 2
s  6s  11s  6
ii. Obtain the state space representation of the electrical ne twork shown.

(Jan 03)

0 1 0
64. For the following system determine: 𝑋 = [ ] 𝑋(𝑡) + [ ] 𝑉(𝑡)
−2 −3 1
i. State transition matrix;
ii. State vector x (t). (Jan 03)
0
Assume𝑋(0) = [ ] 𝑎𝑛𝑑 𝑉(𝑡) =1
0

65. For system shown below obtain state variable model: (Jan 03)

d 3 y(t)
 4(t),
66. A control system is described by the differential equation dt 3 where
y(t) is the observed output and u(t) is the input.
i. Describe the system in the state variable form
ii. Calculate the state transition matrix
iii. Is the system controllable (GATE 92)

67. From the following state variable representation, determine the transfer
0 1 0 0
function of the system. 𝑋 = [ 0 0 1 ] 𝑋 + [ 1]U
−40 −44 −44 0
y = [0 1 0]x (IES 02)
68. For the circuit choose a set of state variables and derive the voltage/current
equation necessary for solving the circuit in terms of the choosen state
variables.

(IES 99)

𝑥1 0 1 0 𝑥1 0
69. Show that the system designated by [𝑥2] = [ 0 0 1 ] [𝑥2] + [0] 𝑢 is
𝑥3 −𝑎 −𝑏 −𝑐 𝑥3 1
completely state controllable
(IES 96)