LT P C 3 2 0 4 OBJECTIVES - Jeppiaar Institute of Technology · NPTEL Video Lecture Notes on...

REGULATION: 2017 ACADEMIC YEAR: 2018-2019

JIT-JEPPIAAR/EEE/Ms.J.ANGELIN/II YR /SEM / 04/IC 8451/CS/UNIT1-5/QB KEYS/VER 1.0

6.1

IC8451 CONTROL SYSTEMS

OBJECTIVES:

➢ To understand the use of transfer function models for analysis physical systems and

introduce the control system components.

➢ To provide adequate knowledge in the time response of systems and steady state error

analysis.

➢ To accord basic knowledge in obtaining the open loop and closed–loop frequency

responses of systems.

➢ To introduce stability analysis and design of compensators.

➢ To introduce state variable representation of physical systems and study the effect of state

feedback.

UNIT I SYSTEMS AND THEIR REPRESENTATION (9)

Basic elements in control systems – Open and closed loop systems – Electrical analogy of

mechanical and thermal systems – Transfer function – Synchros – AC and DC servomotors –

Block diagram reduction techniques – Signal flow graphs.

UNIT II TIME RESPONSE (9)

Time response – Time domain specifications – Types of test input – I and II order system response

– Error coefficients – Generalized error series – Steady state error – Root locus construction-

Effects of P, PI, PID modes of feedback control –Time response analysis.

UNIT III FREQUENCY RESPONSE (9)

Frequency response – Bode plot – Polar plot – Determination of closed loop response from open

loop response - Correlation between frequency domain and time domain specifications.

UNIT IV STABILITY AND COMPENSATOR DESIGN (9)

Characteristics equation – Routh Hurwitz criterion – Nyquist stability criterion-

Performance criteria - Effect of Lag, lead and lag-lead compensation on frequency response–

Design of Lag, lead, Lag/Lead compensator design using bode plots.

UNIT V STATE VARIABLE ANALYSIS (9)

Concept of state variables – State models for linear and time invariant Systems – Solution of state

and output equation in controllable canonical form – Concepts of controllability and observability.

TOTAL (L:45+T:30): 75 PERIODS

COURSE OUTCOMES

At the end of the course, the student should have the:

➢ Ability to develop various representations of system based on the knowledge of

➢ Mathematics, Science and Engineering fundamentals.

➢ Ability to do time domain and frequency domain analysis of various models of linear

➢ system.

➢ Ability to interpret characteristics of the system to develop mathematical model.

➢ Ability to design appropriate compensator for the given specifications.

➢ Ability to come out with solution for complex control problem.

➢ Ability to understand use of PID controller in closed loop system.

LT P C

3 2 0 4



6.2

TEXT BOOKS

1. Nagrath, I.J. and Gopal, M., “Control Systems Engineering”, New Age International

Publishers, 2017.

2. Benjamin C. Kuo, “Automatic Control Systems”, Wiley, 2014.

REFERENCES

1. Katsuhiko Ogata, “Modern Control Engineering”, Pearson, 2015.

2. Richard C.Dorf and Bishop, R.H., “Modern Control Systems”, Pearson Education,2009.

3. John J.D., Azzo Constantine, H. and Hoopis Stuart, N Sheldon, “Linear Control System

Analysis and Design with MATLAB”, CRC Taylor& Francis Reprint 2009.

4. Rames C.Panda and T. Thyagarajan, “An Introduction to Process Modelling

Identification and Control of Engineers”, Narosa Publishing House, 2017.

5. M.Gopal, “Control System: Principle and design”, McGraw Hill Education, 2012.

6. NPTEL Video Lecture Notes on “Control Engineering “by Prof. S. D. Agashe, IIT

Bombay.



6.3

SUBJECT CODE: IC 8451 YEAR /SEMESTER: II/04

SUB NAME: CONTROL SYSTEMS SUB HANDLER: ANGELIN J

UNIT I SYSTEMS AND THEIR REPRESENTATION

Basic elements in control systems – Open and closed loop systems – Electrical analogy of mechanical and

thermal systems – Transfer function – Synchros – AC and DC servomotors – Block diagram reduction

techniques – Signal flow graphs.

PART * A

Q.No. Questions

1.

What is control systems. (May/June 2013) (BTL1)

A system when the output quantity is controlled by varying the input quantity, then the system is

called control system. The output quantity is called controlled variable or response and input

quantity is called command signal or excitation.

2.

Define transfer function. (Nov/Dec 2010,13) (BTL1)

The transfer function of a system is defined as the ratio of Laplace transform of output to Laplace

transform of input with zero initial conditions.

3.

What is feedback? What type of feedback is employed in control system? (Nov/Dec 2013)

(BTL2)

the feedback is a control action in which the output is sampled, and a proportional signal is given

to input for automatic correction of any changes in desired output. Negative feedback is employed

in control system.

4.

What are the characteristics of negative feedback? (May/June 2014) (BTL3)

The characteristics of negative feedback are as follows

a. Accuracy in tracking steady state value.

b. Rejection of disturbance signals.

c. Low sensitivity to parameter variations.

d. Reduction in gain at the expense of better stability.

5.

Differentiate open loop and closed loop system. (Nov/Dec 2010,14, April/May 2010) (BTL4)

S.NO Open loop Closed loop

1. Inaccurate and unreliable Accurate and reliable

2. Simple and economical Complex and costly



6.4

3. Changes in output due to

external disturbances are

not corrected automatically

Changes in output due to

external disturbances are

corrected automatically

4. They are generally stable Great efforts are needed to

design a stable system

6. What is effect of positive feedback on stability? (May/June 2015) (BTL2)

The positive feedback increases the error signal and drives the output to instability. The positive

feedback is used in minor loops in control systems to amplify certain internal signals or parameters.

7.

List the advantages of closed loop system. (Nov/Dec 2015, May/June 2012) (BTL1)

a. Closed loop systems are accurate and reliable.

b. Changes in output due to external disturbances are corrected automatically.

8.

What is Block diagram? What are its basic components? (May 17, Nov/Dec 2015, Nov 11)

(BTL2)

A block diagram of a system is a pictorial representation of the functions performed by each

components of the system and shows the flow of signals. The basic elements of the block diagram

are blocks, branch points and summing points.

9.

Define open loops and closed loop control system. (April/May2011, Nov/Dec 11) (BTL1)

The control systems in which the output quantity has no effect upon the input quantity are called

open loop control system. This means that the output is not feedback to the input for correction.

The control systems in which the output quantity has an effect upon the input quantity in order to

maintain the desired output value are called closed loop control system.

10.

What are the advantages of open loop control system? (Nov/Dec 2012) (BTL1)

a. Simple and economical

b. Stable system

11.

What are the properties of signal flow graph? (May/June 2012) (BTL1)

The basic properties of signal flow graph are

a. Signal flow graph is applicable to linear systems.

b. It consists of nodes and branches. A node is a point representing a variable or signal.

A branch indicates functional dependence of one signal to other.

c. A node adds the signals of all incoming branches and transmits this sum to all other

branches.

d. Signals travel along branches only in the marked direction and when it travels it gets

multiplied by the gain or transmittance of the branch.

e. The algebraic equations must be in the form of cause and effect relationship.

12.

Why negative feedback is preferred in control system? (May 17, Nov/Dec 2016) (BTL2)

The negative feedback results in better stability in steady state and rejects any disturbance signals.

It also has low sensitivity to parameter variations. Hence negative feedback is preferred in closed

loop control system.



6.5

13.

What are the differences between synchro transmitter and synchro control transformer?

(Nov/Dec 2016) (BTL4)

a. Rotor of synchro transmitter is of dumb bell shape and the rotor of control transformer is

cylindrical.

b. The rotor winding of synchro transmitter is excited by an AC voltage. In control

transformer, the induced emf in the rotor is used as an output signal (error signal).

14.

Give practical example of open loop systems. (Dec 11) (BTL2)

The practical examples of open loop control systems are, sprinkler used to water a lawn, automatic

toaster, traffic light controller, automatic door opening and closing system.

15.

Give practical example of closed loop systems. (Dec 11) (BTL2)

The practical examples of closed loop control systems are, human being, home heating system,

speed control systems, ship stabilization system, missile launching system, voltage stabilizer,

temperature control systems.

16.

Name three functional components used in control system. (May 03, Dec 14) (BTL2)

The three functional components used in control systems are:

a. Error detectors

b. Controller

c. Feedback

17.

State Mason’s Gain formula. (APRIL/MAY 15 and Repeated question, will be there in all

AU question papers) (BTL1)

18.

Write the analogous electrical elements in force voltage analogy for the elements of

mechanical translational system. ( BTL3)

a. Force-voltage e.

b. Velocity v-current i.

c. Displacement x-charge q.

d. Frictional coefficient B-Resistance R.

e. Mass M- Inductance L.



6.6

f. f. Stiffness K-Inverse of capacitance 1/C.

19.

Write the analogous electrical elements in force current analogy for the elements of

mechanical translational system. ( BTL3)

a. Force-current I.

b. Velocity v-voltage v.

c. Displacement x-flux φ.

d. Frictional coefficient B-conductance 1/R.

e. Mass M- capacitance C.

f. f. Stiffness K-Inverse of inductance 1/L.

20.

Write the force balance equation of an ideal mass, dashpot, spring element. (BTL2)

a. F = M d2x/dt2 for mass element.

b. F = B dx/dt for dash pot element.

c. F = kx for spring element.

21. a. What are the disadvantages of block diagram representation? (Nov 18) (BTL2)

b. a. It is a tedious method of calculating transfer function.

c. b. overall gain of the system cannot be computed.

PART * B

1.

Write the differential equations governing the mechanical translational system shown in the

figure. Draw the electrical equivalent analogy circuits. BTL3 (13M)

Answer: Page 1.29-Nagoor kani

➢ Draw the free body diagram (2M)

1. Mechanical translational system can be obtained using three basic elements Mass

(M), Spring (K), Damper (B).

2. F = M d2x/dt2 for mass element.

3. F = B dx/dt for dash pot element.

4. F = kx for spring element.

➢ Write the differential equations (2M)

𝑀1𝑑2𝑥1

𝑑𝑡2 + 𝐵1𝑑𝑥1

𝑑𝑡+ 𝐵

𝑑

𝑑𝑡(𝑥1 − 𝑥) + 𝐾1𝑥1 + 𝑘(𝑥1 − 𝑥) = 0

𝑀2𝑑2𝑥

𝑑𝑡2 + 𝐵2𝑑𝑥

𝑑𝑡+ 𝐵

𝑑

𝑑𝑡(𝑥 − 𝑥1) + 𝑘(𝑥 − 𝑥1) = 𝑓(𝑡)



6.7

➢ Write the rule for force voltage and force current analogy (3M)

1. Force voltage: M→L ; K→1/C ; B→R; velocity→I

2. Force current: M→C; K→1/L; B→1/R

➢ Draw the electrical circuit equivalent to mechanical system (3M)

➢ Write the electrical equations using KVL and KCL.(3M)

2.

Derive the transfer function of AC servo motor in control system. BTL4 (13M)

Answer: Page 55-Notes

➢ Definition of Servomotor :- (2M)

1. Motor used for feedback control systems are called servomotors. It is also called

automatic control system.

2. Converts electrical system into angular motion.

3. 2 types: DC servomotor and AC servomotor.

➢ Definition of AC Servomotor :- (3M)

1. Motor which runs at zero speed as its base speed is driven by error signal with AC

supply is called servomotor.

2. Used in closed loop servo systems, high speed instrument servos, and low power

applications.

3. Speed control done using armature voltage control and field control.

➢ Operating principle of AC Servomotor (4M)

Salient features include rugged construction, Reliable in operation, Light Weight and No

Slip rings.

➢ Transfer function derivation (4M)

Ө(s)/Vc2(s)=Km/s(1+stm)

3.

Compare open and closed loop systems. BTL4 (13M)


➢ Definition of open and closed loop systems (3M)

Open loop: any physical system that does not automatically correct the variation in its

output is called open loop system.



6.8

Closed loop: A system in which control action is somehow dependent on the output. The

output is given to feedback element and compared with the reference signal.

➢ Block diagram of open and closed loop systems (4M)

➢ Advantages of open and closed loop systems (2M)

Open loop:

1. Simple and economical.

2. Easier to construct.

3. Less Stable.

Closed loop:

1. Accurate.

2. More stable.

3. Less noise affected.

➢ Disadvantages of open and closed loop systems (2M)

open loop

1. Inaccurate.

2. Unreliable.

3. Cannot correct automatically.



6.9

Closed loop:

1. Complex and costly.

2. Oscillatory response.

3. Overall gain reduced.

➢ Applications of open and closed loop systems (2M)

Open loop:

1. Electric switch.

2. Automatic washing machine.

3. Electric toaster.

Closed loop:

1. Traffic light control.

2. Liquid level control.

3. Temperature level control.

4.

Write the differential equations governing the mechanical rotational system as shown in the

figure. Draw the both electrical analogous circuits.399(Apr/May 2017) BTL3 (13M)

Answer: Page1.46-Nagoor kani


1. Mechanical Rotational system can be obtained using three basic elements Moment of

Inertia (J), Spring (K), Damper (B).

2. F = M d2θ/dt2 for mass element.

3. F = B d θ /dt for dash pot element.

4. F = k θ for spring element.

➢ write differential equations (2M)



6.10

𝐽1𝑑2𝜃1

𝑑𝑡2 + 𝐵1𝑑(𝜃1−𝜃2)

𝑑𝑡+ 𝐾1(𝜃1 − 𝜃2) = 𝑇

𝐽2𝑑2𝜃2

𝑑𝑡2 + 𝐵1𝑑(𝜃2−𝜃1)

𝑑𝑡+ 𝐾1(𝜃2 − 𝜃1) + 𝐵2

𝑑(𝜃2−𝜃3)

𝑑𝑡= 0

𝐽3𝑑2𝜃3

𝑑𝑡2 + 𝐵2𝑑(𝜃3−𝜃2)

𝑑𝑡+ 𝐾2(𝜃3 − 𝜃2) = 0

➢ Write the rule for torque voltage and torque current analogy (3M)

Torque-voltage rule:

1. Angular Velocity v-current i.

2. Angular Displacement x-charge q.

3. Frictional coefficient F-Resistance R.

4. Mass J- Inductance L.

5. Stiffness K-Inverse of capacitance 1/C.

Torque -current rule:

1. Angular Velocity v-voltage v.

2. Angular Displacement x-flux φ.

3. Frictional coefficient F conductance 1/R.

4. Mass J- capacitance C.

5. Stiffness K-Inverse of inductance 1/L.


5.

Differentiate DC and AC servo motors (or) With neat diagrams, explain the working of AC

and DC servo motors.(Nov/Dec 2014) BTL4 (13M)

Answer: notes.

➢ Definition of Servomotor (3M)

1. Motors are used for feedback control systems are called servomotors.it is also called

automatic control system.

2. Converts electrical system into angular motion.

3. 2 types: DC servomotor and AC servomotor.

➢ Definition of AC Servomotor & DC Servomotor, Block diagram of AC & DC

Servomotor (4M)

1. Motors which runs at zero speed as its base speed is driven by error signal with AC supply

is called servomotor.



6.11

2. Used in closed loop servo systems, high speed instrument servos, and low power

applications.

3. Speed control done using armature voltage control and field control.

➢ Operating principle of AC & DC Servomotor (4M)

Salient features include rugged construction, Reliable in operation, Light Weight and No

Slip rings.

➢ Applications of AC & DC Servomotor (2M)

DC:

1. Large power applications.

2. Robotics.

AC

1. Solar Tracking System.

2. Antenna Positioning.

3. Robotic Vehicle.

6.

Convert the given block diagram shown in fig. to signal flow graph and determine the closed

loop transfer function C(s)/R(s). And verify by block diagram reduction approach. BTL4

(13M)

Answer: Page 1.56-Nagoor Kani

➢ Draw the signal flow graph.

➢ Give the numbering to input, output, summing point and branching point (2M)

➢ Apply block diagram reduction rules to find transfer function (6M)



6.12

1. Reduce series blocks.

2. Reduce parallel blocks

3. Reduce minor feedback loops.

4. Shift summing point to left and take-off point to right.

5. Splitting the summing points.

➢ Apply Masons gain formula (2M)

➢ Find transfer function (3m)

C(s)/R(s)= G1G2G3G4/(1+G3G4H1+G2G3H2+G1G2G3G4)

7.

Derive the transfer function of an armature controlled DC servomotor (or) Obtain the

mathematical model of an armature controlled DC motor. (Nov/Dec 2016) BTL3 (13M)


➢ Explain speed control techniques (2M)

a)Armature control

b)Field control

➢ Draw the equivalent circuit diagram (2M)



6.13

➢ Derive the differential equation (4M)

𝑉𝑎 = 𝑖𝑎𝑅𝑎 + 𝐿𝑎𝑑𝑖𝑎

𝑑𝑡+ 𝑒𝑏

𝑇 = 𝐽𝑑2𝜃

𝑑𝑡2 + 𝐵𝑑𝜃

𝑑𝑡

➢ Take Laplace transform (2M)

𝑖𝑎(𝑠)𝑅𝑎 + 𝐿𝑎𝑠𝑖𝑎(𝑠) + 𝐸𝑏(𝑠) = 𝑉𝑎(𝑠)

𝐽𝑠2𝜃(𝑠) + 𝐵𝑠𝜃(𝑠) = 𝑇(𝑠)

➢ Find transfer function (3M)

8. Obtain the transfer function using Mason’s Gain formula for the system given. (Apr/May

2017). BTL3 (13M)

s( )

Va s( )

K m

s R a L a s+( ) J s b+( ) K b K m+



6.14


➢ Give the numbering to all nodes(2M)

➢ Find No.of forward paths and individual loop gain(3M)

No of forward path ,K=6

Forward path gain:

1. P1=G2G4G6

2. P2=G3G5G7

3. P3=G1G2G7

4. P4=G3G8G6

5. P5=-G1G3G7G8H1

6. P6=-G1G2G6G8H2

➢ Calculate the value of delta(3M)

Δ=1+G4H1+G5H2+-G1G2H1H2+G4G4H1H2

➢ Apply Masons gain formula(2M)



6.15

➢ Find transfer function(3M)

C(s)/R(s)= -G1G3G7G8H1-G1G2G6G8H2/(1+G4H1+G5H2-G1H8H1H2+G4G5H1H2)

9.

Write the differential equation for the electric circuit as shown in fig. hence Find Y2(s)/F(s).

(Nov/Dec 2016). BTL2 (13M)

Answer: Page1.10-Nagoor Kani

➢ Draw the free body diagram of mass M1 &M2 (3M)

➢ Write differential equations in terms of displacement (4M)

𝑀1𝑑2𝑦1

𝑑𝑡2 + 𝐵1𝑑𝑦1

𝑑𝑡+ 𝑘1𝑦1 + 𝑘2(𝑦1 − 𝑦2) = 𝑓(𝑡)

𝑀2𝑑2𝑦2

𝑑𝑡2 + 𝐾2(𝑦2 − 𝑦1) = 0




6.16

1. M1s2y1(S)+Bsy1(s)+K1y1(s)+K2(y1(s)-y2(s))-F(s)

2. M2 s2Y2(s)+K2(y2(s)-y1(s))=0

➢ Rearrange the equation and find transfer function (3M)

➢ Y2(s)/F(s)=K2/[M1s2+Bs+(K1+K2)] [M2s

2+K2]-k22

10.

Obtain the electrical analogy of the mechanical systems shown in the following Fig(Nov 18)

Answer: Page1.101-Nagoor Kani


1. Mechanical translational system can be obtained using three basic elements Mass

(M), Spring (K), Damper (B).




➢ Write the differential equations (2M)

𝑀1𝑑2𝑥1

𝑑𝑡2 + 𝐵1𝑑𝑥1

𝑑𝑡+ 𝐵2

𝑑

𝑑𝑡(𝑥1 − 𝑥2) + 𝐵3

𝑑

𝑑𝑡(𝑥1 − 𝑥3) + 𝐾1𝑥1 = 𝑓(𝑡)

𝑀2𝑑2𝑥2

𝑑𝑡2+ 𝐵2

𝑑(𝑥2 − 𝑥1)

𝑑𝑡+ 𝐵

𝑑

𝑑𝑡(𝑥2 − 𝑥3) = 0

𝑀3𝑑2𝑥3

𝑑𝑡2+ 𝐵3

𝑑(𝑥3 − 𝑥1)

𝑑𝑡+ 𝐵

𝑑

𝑑𝑡(𝑥3 − 𝑥2) = 0

➢ Write the rule for force voltage and force current analogy (3M)

1.Force voltage: M→L ; K→1/C ; B→R; velocity→I

2.Force current: M→C; K→1/L; B→1/R




6.17

➢ Write the electrical equations using KVL and KCL(3M)

PART * C

1.

What are the basic elements of mechanical rotational and translational systems? Write its

torque balance and force balance equations. (May/June 2016). BTL3 (15M)


➢ Elements of mechanical system(2M)

1. Mass(M), Dashpot(B), Spring(K)

2. Moment of inertia (J), Dashpot(F), Spring(K)

➢ Draw the free body diagram(2M)

➢ Apply Newton’s second law(1M)

The second law states that the acceleration of an object is dependent upon two variables -

the net force acting upon the object and the mass of the object.

➢ Write differential equations(2M)

1. F(t)= M d2θ/dt2 + B dθ/dt+kθ

2. T(t)= M d2θ/dt2 + B dθ/dt+ kθ

➢ Force balance equation of translational system(4M)




➢ Force balance equation of rotational system(4M)

1. Mechanical Rotational system can be obtained using three basic elements Moment of

Inertia (J), Spring (K), Damper (B).

2. F = M d2θ/dt2 for mass element.

3. F = B d θ /dt for dash pot element.

4. F = k θ for spring element.

Torque-voltage rule:

1. Angular Velocity v-current i.

2. Angular Displacement x-charge q.

3. Frictional coefficient F-Resistance R.

4. Mass J- Inductance L.

5. Stiffness K-Inverse of capacitance 1/C.

Torque -current rule:

1. Angular Velocity v-voltage v.



6.18

2. Angular Displacement x-flux φ.

3. Frictional coefficient F conductance 1/R.

4. Mass J- capacitance C.

2.

For a non-unity negative feedback control system whose open loop transfer function is G(s)

and feedback path transfer function is H(s), obtain the control ratio using Mason’s gain

formula. (Nov/Dec 2015). BTL6 (15M)


➢ Draw the closed loop system(2M)

➢ Choose G(s) and H(s) value(2M)

G(s) = any value.

H(s)=1

➢ Apply Masons gain formula(5M)

➢ Find the transfer function(6M)



6.19

➢ G(s) = output/input.

3.

Derive the transfer function of an RLC series circuit. (Nov/Dec 2015). BTL5 (15M)


➢ Draw the RLC Circuit(2M)

➢ Write KVL equations to input and output side (3M)

1. Ri+ Ldi/dt + 1/Cʃi.dt = Vi

2. 1/Cʃi.dt=Vo

➢ Take Laplace transform of differential equations(4M)

Ri(s)+L s i(s) + 1/Cs i(s)= v(s)

➢ Rearrange the Laplace equations and find transfer function(6M)

G(s)=Vo/Vi

4.

With a neat diagram, derive the transfer function of a field-controlled control system.

(Nov/Dec 2015) BTL5 (15M)


➢ Explain speed control techniques(2M)

a) Armature control b) Field control

➢ Draw the equivalent circuit diagram(2M)



6.20

➢ Derive the differential equation (3M)

𝑉𝑓 = 𝑖𝑓𝑅𝑓 + 𝐿𝑓

𝑑𝑖𝑓

𝑑𝑡+ 𝑒𝑏 𝑇 = 𝐽

𝑑2𝜃

𝑑𝑡2+ 𝐵

𝑑𝜃

𝑑𝑡


𝑖𝑓(𝑠)𝑅𝑓 + 𝐿𝑓𝑠𝑖𝑓(𝑠) + 𝐸𝑏(𝑠) = 𝑉𝑓(𝑠)

𝐽𝑠2𝜃(𝑠) + 𝐵𝑠𝜃(𝑠) = 𝑇(𝑠)


UNIT II TIME RESPONSE

Time response – Time domain specifications – Types of test input – I and II order system response – Error

coefficients – Generalized error series – Steady state error – Root locus construction- Effects of P, PI,

PID modes of feedback control –Time response analysis.

PART * A

Q.No. Questions

1.

State the various standard test inputs used in control system. (Dec 05, May 11, 14) (BTL1)

The various standard test inputs used in control system are step ramp, parabolic, impulse and

sinusoidal.

2. Define peak overshoot. (April/May 17) (BTL1)

It is defined as the ratio of the maximum peak value to final value, where maximum peak value is

measured from final value.

3. What is steady state error? (May 04,06,07,11, Dec 11, 15) (BTL1)

s( )

V f s( )

K m

s J s b+( ) L f s R f+( )



6.21

The difference between the desired output i.e. reference input and the actual output of the system

is called steady state error which is denoted as ess.

4.

What is rise time? (Dec 04,08, Nov 11, May 14) (BTL1)

It is the time required for the response to rise from 10% to 90% of the final value for over damped

systems and 0 to 100% of the final value for under damped systems. The rise time is reciprocal of

the slope of the response at the instant, the response is equal to 50% of the final value.

5. What is settling time? (May 03, 05, 10, Dec 14) (BTL1)

The settling time is defined as the time required for the response to decrease and stay within

specified percentage of its final value (within tolerance band)

6. List out any four time-domain specification. (Dec 03,08, 15) (BTL1)

The various time domain specifications are delay time, rise time, peak time, peak overshoot, settling

time and steady state error.

7.

Mention the characteristics of PI controller. (Dec 03, May 07, 13, 14) (BTL2)

The PI controller has following characteristics

➢ It increases order of the system.

➢ It increases TYPE of the system.

➢ Design of Ki must be proper to maintain stability of system. So, it makes system r relatively

less stable.

➢ Steady state error reduces tremendously for same type of inputs.

8.

Define transient response and steady state response of a system. (May 10) (BTL1)

The output variation during the time it takes to achieve its final steady value is called transient

response.

Steady state response is that part of the time response which remains after complete transient

response vanishes from the system output.

9.

How a control system is classified depending on the value of damping? (May 11) (BTL2)

➢ Under damped system (0< ᵷ < 1 ) ➢ Critically damped system (ᵷ = 1)

➢ Over damped system (ᵷ > 1)

➢ Un-damped system (ᵷ = 0)

10

Why derivative controller is not used in the control system? (May 11, May 15) (BTL2)

The derivative controller acts only during transient period when the error varies with time and does

not produce any corrective action for a constant error as derivative of a constant error is zero.

Hence the derivative controller is never used alone but always used along with some other type of

controller.

11 What is the effect of adding a pole to a second order system? (May 04, 08) (BTL2)

The second order system is generally stable. If a pole is added to it, it becomes third order due to

which it becomes less stable in nature. It increases peak overshoot and settling time.



6.22

12

State the advantages of generalized error coefficients. (May 06) (BTL1)

➢ It gives variation of error as a function of time.

➢ It uses any input other than the standard input.

➢ As variation of error as a function of time is available, the design of the system becomes

easy and optimum.

13

Define adaptive control system. (Dec 06) (BTL4)

Adaptive control system has a property or an automatic control system having a property of

automatically changing the characteristics, system parameters and / or structure of the controller

during the normal operation with the aim of maintaining a given standard of performance under

varying inputs and varying arbitrary, conditions of operation is called as adaptive or self-adjusting

system.

14 What is the function of controller? (BTL2)

A controller accepts error as its input and manipulates the error according to the requirement of the

system and gives output to the plant or the process to be controlled.

15

What is type 0 and type 1 system? (May 2015) (BTL2)

The value of N in the denominator polynomial of loop transfer function decides the type number

of the system

➢ N- Number of poles at origin.

➢ N = 0, then the system is type 0 system,

➢ N = 1, the system is type 1 system.

➢ N = 2, then the system is type 2 system and so on.

16

What is the effect of PI controller on the system performance? (Dec 14) (BTL2)

The PI controller increases the order of the system by one, which results in reducing, the steady

state error. But the system becomes less stable than the original system.

17 Define peak time. (Dec 09) (BTL1)

The time at which the peak overshoot occurs in the time response of a second order system is called

a peak time.

18

What will happen to damping factor and natural frequency of oscillations if gain K of a

second order system is increased? (May 07) (BTL3)

As the value of gain K increases the natural frequency of oscillations and the value of damping

ratio decreases. Due to this system becomes more oscillatory in nature.

19

How can the maximum overshoot of a system be decreased without affecting the steady state

error? (BTL3)

With the use of PD i.e. proportional plus derivative controller, it can be observed that transient

response and value of damping ratio increases without affecting steady state error. As damping



6.23

ratio increases, the maximum overshoot decreases. So, using PD controller it is possible to decrease

maximum overshoot without affecting the steady state error.

20

Why derivative is not used in control systems? (BTL 5)

The derivative controller produces a control action based on the rate of change of error signal and

it does not produce corrective measures for any constant error. Hence derivative controller is not

used in control system.

PART * B

1.

Determine the response of the system and find time domain specifications of the given figure.

ζ =0.5 (May/June 2016). BTL3 (13M)

Answer: Page 2.26-Nagoor kani.

➢ Choose G(s) and H(s) (2M)

1. G(s)=16/s(s+0.8)

2. H(s) = Ks+1.


1. G(s)H(s)= G(s)/1+G(s)H(s)

2. G(s) H(s)= 16/s2+(0.8+k) s+16.

➢ Compare with standard second order expression(2M)

1. G(s) =ɷn2/ s2+2ℇɷn s+ ɷn2.

➢ Find ζ and wn (2M)

1. wn 2=16

wn =sqrt (16) = 4

2. 2 ζ wn =(0.8+K) =>K=0.2

Final transfer function:



6.24

3. 16/ s2 + 4s+16.

➢ Find response & time domain specifications using formulas(5M)

1. Formula for Rise time: -

Rise time = 0.64s

2. Formula for Peak time: -

Peak time = 0.907s

3. Formula for Settling time: -

Settling time = 1.5s

4. Formula for Peak overshoot: -

Peak overshoot =16.3%

2.

A unity feedback system has 10/s2(s+2). find the steady state error and generalized error

coefficients for r (t) =6t4+5t3+4t2+2t+3. (May/June 2016). BTL3 (13M)


➢ Choose G(s) and H(s) (1M)

1. G(s)=10/s2(s+2)

2. H(s)=1


G(s)H(s)= G(s)/1+G(s)H(s)

➢ Apply r(t) in S domain to the system (5M)

1. E(s) = R(s)/ 1+G(s) H(s).

2. R(s) =3/s+ 2/s2 + 4/s3 + 5/s4 + 6/s5.



6.25

3. G(s)=10/s2(s+2)

➢ Find ess using final value theorem in S domain (5M)

3.

Derive the time domain specifications of a second order system. (May/June 2016). BTL1

(13M)

Answer: Page 2.17- Nagoor Kani

➢ Rise time definition and its formula (3M)

The time required for the response to rise from 0% to 100% of its final value. This is

applicable for the under-damped systems.

➢ Peak time definition and its formula (3M)

The time required for the response to reach the peak value for the first time. It is denoted

by tp. At t=tp, the first derivate of the response is zero.

➢ Settling time definition and its formula (3M)

The time required for the response to reach the steady state and stay within the specified

tolerance bands around the final value. The settling time is denoted by ts.

➢ Maximum overshoot definition and its formula (4M)

Peak overshoot Mp is defined as the deviation of the response at peak time from the final

value of response. It is also called the maximum overshoot



6.26

4.

For a unity feedback control system, the open loop transfer function is given by

G(s) = 10(s+2)/s2(s+1). Find the position, velocity, and acceleration error coefficients and find

the steady state error when the input is R(s) =3/s-2/s2+1/3s3. (May/June 2016). BTL3

(13M)


➢ Choose G(s) and H(s) & Find transfer function (3M)

➢ Find Kp, Kv, Ka (5M)

1. Kp=α

2. Kv= α

3. Ka=20

➢ Apply r(t) in S domain to the system (5M)

1. E(s) = R(s)/ 1+G(s) H(s).

2. R(s) =3/s-2/s2+1/3s3

3. G(s)=10(s+2) /s2(s+1)

➢ Find ess using final value theorem in S domain (5M)

ess = 1/60.

5.

With a neat diagram explain the effect of PD controller in detail. (May/June 2016).

Answer: Page 2.62-Nagoor Kani BTL5 (13M)

➢ Transfer function of P & D Controller (3M)

U(s)/E(s) = Kp (1+Td S)

➢ Block diagram of PD Controller (3M)

➢ Compare with standard second order expression (3M)

Loop transfer function= Gc(s)G(s) / 1+ Gc(s)G(s)H(s)

= Kp (1+Td s) * {ɷn2/ s2+2ℇɷn s+ ɷn2.}



6.27

= ɷn2 (Kp+ Kd s) / (s2+ (2ℇɷn + Kdɷn2) s +Kp ɷn2)

➢ Justify the results(4M)

1. Addition of zero Increases the peak overshoot and reduces the rise time.

2. Increased damping reduces the peak overshoot.

6.

Write explanatory notes on PI and PD controllers. (May/June 2016) BTL5 (13M)

Answer: Page 2.54 & 2.56-Nagoor Kani

➢ Transfer function of PI & PD Controller (3M)

1. U(s)/E(s) = Kp (1+Td S)

2. U(s)/E(s) = Kp (1+TiS)/TiS



1. Loop transfer function PD= Gc(s)G(s) / 1+ Gc(s)G(s)H(s)

= Kp (1+Td s) * {ɷn2/ s2+2ℇɷn s+ ɷn2.}


2. Loop transfer function PI= Gc(s)G(s) / 1+ Gc(s)G(s)H(s)

= Kp (1+Ti s / Ti s) * {ɷn2/ s2+2ℇɷn s+ ɷn2.}

= Kiɷn2(10+ Ti s) / (s3+ (2ℇɷns2 + Kpɷn2 s +Kiɷn2)


PD controller:



PI controller:

1. Increase in order results in less stability

2. Increase in the type reduces the steady state error.



6.28

7.

Derive the expressions for the unit step response of a second order of the underdamped and

undamped systems. (Nov/Dec 2015). BTL1 (13M)

Answer: Page 2.10 & 2.11-Nagoor kani

➢ Draw block diagram of second order system(2M)


1. G(s)H(s)= G(s)/1+G(s)H(s)

2. G(s) =ɷn2/ s2+2ℇɷn s+ ɷn2.

➢ Apply the input signal (3M)

R(s)= 1/s

➢ Use partial fraction method find the response of the system(6M)

8.

The open loop transfer function of a unity feedback system is given by G(s) =1/s (1+s). The

input to the system is described by r (t)= 4+6t.Find the generalized error coefficients and

steady state error. (Nov/Dec 2015). BTL3 (13M)


➢ Choose G(s) and H(s) & Find transfer function(2M)

R(s) =4/s+6/t2

G(s)=1/s(s+1)

H(s)=1

➢ Apply r(t) to find ess using initial value theorem or final value theorem(6M)

( ) ( )

+−

−−=

−tety n

tn 2

21sin

1

11



6.29

➢ Find C0, C1, C2 from static error constants(5M)

1. C0=1/1+kp

2. C1=1/Kv

3. C3=1/ka

9.

(i)Develop an Expression to find steady state error of closed Loop system BTL 4 (6M)

Answer: Page.2.48-Nagoor kani

➢ Choose G(s) and H(s) & Find transfer function (3M)

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

2. e(t)= LT-1(E(s))

G(s)=transfer function

H(s)= feedback

➢ Apply r(t) to find ess using initial value theorem or final value theorem (3M)

1. Ess= t→∞ LT (e(t))

(ii) A unity feedback system has the forward transfer function

For the input r (t) = 1 + 5t, formulate the minimum value of K so that the steady state error

is < 0.1. (Use final value theorem). BTL 4 (7M)



1. G(s)=Ks/(s+1)2.

2. H(s)=1



6.30

3. R(s)=1/s+5/s2

➢ Apply r(t) to find ess using final value theorem (3M)

10.

The open loop transfer function of a unity feedback system is given by G(s)=40/(s(0.2s+1)).

Determine the steady state error using error series approach for the input

r(t)=3t+4t2.).(May/June 2016). BTL3 (13M)



1. G(s)=40/s(0.2s+1)

2. R(s)=3/s2+4/s4

3. H(s)=1


➢ Find C0, C1, C2 from static error constants(5M)

1. C0=1/1+kp

2. C1=1/Kv

3. C3=1/ka

11.

A unity feedback control system has the open loop transfer function G(s)=K/(s+A) (s+2). Find

the values of K and A, so that the damping ratio is 0.707 and the peak time for unit step

response is 1.8s. (Nov/Dec 2015). BTL2 (13M)



1. G(s)= K/ (s +A) (s+2)

2. ꜫ=0.7

3. TPJ=1.8



6.31

➢ Compare with standard expression(2M)

G(s) =ɷn2/ s2+2ℇɷn s+ ɷn2= K/(s+A) (s+2)

➢ Find Wn, also find K, A by using peak time(9M)

1. 2ℇɷn=(A+2)

2. Tp=1.8

π/wn(sqrt(1-ℇ2)) =1.8 (calculate Wn)

3. Substitute the value in eq.1 and find the value of A.

13.

Obtain the impulse and step responses of the following unity feedback control system with

open loop transfer function. G(s) =6/(s(s+5)). (Nov/Dec 2015). BTL2 (13M)

Answer: Page 2.81--Nagoor kani


1. C(s)/R(s)=G(s)/1+G(s)

2. C(s)=R(s)* 6/s(s+5)

3. Step response R(s)=1/s

C(s)= 1/s * 6/s(s+5)

4. Impulse response R(s)=1

C(s)= 1 * (6/s(s+5))

➢ Apply input signal R(s), Find response of the system(5M)

The time domain response c(t) is obtained by taking inverse Laplace transform of C(s)

➢ Find derivative of response of system is response of impulse system(6M)

The time domain response c(t) is obtained by taking inverse Laplace transform of C(s)

14.

Sketch the root locus of the system.

G(s)=K(s+3)/s(s+1) (s+2) (s+4). (Apr/May 2017). BTL 6 (13M)


➢ Find poles and zeros (2M)

1.Poles=0, -1, -2, -4 (n=4)

2.Zeros= -3 (m=1)

➢ Centroid and angle of asymptotes (3M)



6.32

1.centroid=sum of poles-sum of zeros/ n-m.

2. angle of asymptotes = +/- 1800 (2q+1)/ n-m.

➢ Find breakaway points (4M)

1.G(s)/ 1+G(s)

2. s(s+1) (s+2) (s+4) + K(s+3) =0

➢ Find intersection of real axis with imaginary axis (4M)

1.s(s+1) (s+2) (s+4) + K(s+3) =0

2.s→ jw

3. equate the real and imaginary parts.

15.

Sketch the root locus of the system whose open loop transfer function is G(S) = K / S (S+2)

(S+4). Find the value of K so that the damping ratio of the closed loop system is 0.5.

BTL 6 (13M)



1.Poles=0, -2, -4 (n=3)

2.Zeros= 0 (m=0)





1.G(s)/ 1+G(s)

2. s(s+2) (s+4) + K=0


1.s(s+2) (s+4) + K=0

2.s→ jw

3. equate the real and imaginary parts.

PART * C

1.

Explain the effect of P, PI and PID controllers on the system performances. (Nov/Dec 2015).

BTL5 (15M)


➢ Transfer function of P, I & D Controller (5M)



6.33


PD controller:

U(s)/E(s) = Kp (1+Td S)

Loop transfer function = Gc(s)G(s) / 1+ Gc(s)G(s)H(s)

= Kp (1+Td s) * {ɷn2/ s2+2ℇɷn s+ ɷn2.}


PI controller:

U(s)/E(s) = Kp (1+TiS)/TiS

Loop transfer function PI= Gc(s)G(s) / 1+ Gc(s)G(s)H(s)

= Kp (1+Ti s / Ti s) * {ɷn2/ s2+2ℇɷn s+ ɷn2.}

= Kiɷn2(10+ Ti s) / (s3+ (2ℇɷns2 + Kpɷn2 s +Kiɷn2)

PID controller:

U(s)/E(s) = Kp (1+ 1/Tis +Tds)

Loop transfer function PID = Gc(s)G(S)/1+ Gc(s)G(s)H(s)

= Kp (1+ 1/Tis +Tds) * {ɷn2/ s2+2ℇɷn s+ ɷn2.}

➢ Block diagram of P, PI, PID Controller (5M)

➢ Justify the results (3M)

PD controller:



PI controller:

1. Increase in order results in less stability

2. Increase in the type reduces the steady state error

PID controller:

1. Proportional controller stabilizes the gain but introduces the steady state error.



6.34

2. The integral controller eliminates the error.

3. The derivative controller reduces the rate of change of error.

2.

A unity feedback system has the forward transfer function G(s) =K1 (2s+1)/s (5s+1) (1+s)2

The input r(t)=1+6t is applied to the system. Determine the minimum value of K1, if the

steady state error is to be less than 0.1. (Apr/May 2017). BTL3 (15M)



1. G(s) =K1 (2s+1)/s (5s+1) (1+s)2

2. R(s)=1/s+6/s2


1. Ess=0.1.

➢ Find C0, C1, and C2 from static error constants (5M).

1. C0=1/1+kp

2. C1=1/Kv

3. C3=1/ka

➢ Find value of K(2M)

1. K should be greater than 60

3.

Determine the open loop transfer function for a unit feedback control system with unit

impulse response given by

C (t) = −te −t + 2e−t for (t>0) BTL4 (15M)

Answer: page 71- notes

➢ Laplace transform: (3M)

1. C(s)/R(s)=G(s)/1+G(s)

2. C(s)= -1/(s+1) +2/(s+1)

➢ Response of impulse input: (5M)

R(s)=1



6.35


➢ Justify the results (3M)

4.

(i)A certain unity negative feedback control system has the following open loop transfer

function

Derive an expression for the output variable as a function of time when the input applied is

unit step. What is the percentage overshoot, peak time and rise time? BTL 6 (7M)

Answer: Notes

➢ Response of impulse input: (2M)

➢ C(s)/R(s)=G(s)/1+G(s)

➢ C(s)=1* 10(s+10) / s(s+2)

➢ R(s)=1/s

➢ Laplace transform: (2M)

The time domain response c(t) is obtained by taking inverse Laplace Transform of C(s)

c(t)=LT-1 {C(s)}


G(s)= {ɷn2/ s2+2ℇɷn s+ ɷn2.}

➢ wn=3.162 rad/sec

➢ ℇ=0.316

➢ Tp= 1.047 sec



6.36

➢ Tr=0.63 sec

➢ Peak overshoot: 4.2144 units

(ii)For unity feedback system having

Determine the type of system, error coefficients and the steady state error. BTL6 (8M)

r (t) = 1 + 3t + (t2/2).

Answer: Notes

➢ Steady state error constants: (4M)

1. E(s) = R(s)/ 1+G(s) H(s).

2. R(s) =1/s + 3/s2+ ½ 1/s5

3. G(s)=5(s+1)/(s2(s+3) (s+10))

4. Kp= lt s→0 G(s)

5. Ki=Lt s→0 s G(s)

6. Ka=Lt s→0 s2G(s)

➢ Steady state error:(4M)

1. Ess=1/1+Kp

2. Ess= 1/Kv

3. Ess=1/Ka

5.

Explain how PI, PD and PID compensation will improve the time response of a system with

a neat block diagram and derivation. BTL 6 (15M)

Answer: Notes

➢ Transfer function of P & D Controller(4M)



6.37

U(s)/E(s) = Kp (1+Td S)



Loop transfer function = Gc(s)G(s) / 1+ Gc(s)G(s)H(s)

= Kp (1+Td s) * {ɷn2/ s2+2ℇɷn s+ ɷn2.}



PD controller:


2. Increased damping reduces the peak overshoot

6.

Derive the time response analysis of a first order system for step and ramp input.

(15M) BTL 6

Answer: Notes

➢ Response of step and ramp input: (5M)

1. Closed loop transfer function=> C(s)/R(s)= 1/1+Ts

2. R(s)=1/s

3. C(s)=(1/T)/s(s+1/T)

➢ Inverse Laplace transform: (5M)

C(t)=LT-1{C(s)}


G(s)=1/1+Ts

7

Sketch the root locus of the system having G(s) =K(s+9)/s(s2+4s+11). (Apr/May 2017).

BTL 5 (15M)



1.Poles=0, -2+j2.64, -2-j2.64 (n=3)

2.Zeros= -9 (m=1)




6.38




1.G(s)/ 1+G(s)

2. s(s2+4s+11) + K(s+9) =0

➢ Angle of departure(2M)

1.θd = 180-(angle subtended by poles to P2) + (angle subtended by zeros to P2)


1. s(s2+4s+11) + K(s+9) =0

2.s→ jw

3. equate the real and imaginary parts

8

A unity feedback control system has an open loop transfer function

G(S) = K / s (s2+4s+13). Sketch the root locus. BTL 4 (15M)



1. Poles=0, -2+j3, -2-j3 (n=3)

2. Zeros= -9 (m=1)





1.G(s)/ 1+G(s)

2. s(s2+4s+13) + K=0


1.θd = 180-(angle subtended by poles to P2) +( angle subtended by zeros to P2)


1. s(s2+4s+13) + K=0

2. s→ jw

3. Equate the real and imaginary parts

9.

A unity feedback control system has an open loop transfer function

G(S) = K(s+3) / s (s+1) (s+2) (s+4). Sketch the root locus. (nov/dec 2018) BTL 4 (15M)



1. Poles=0, -1, -2, -4 (n=4)

2. Zeros= -3 (m=1)





6.39



1.G(s)/ 1+G(s)

2. s (s+1) (s+2) (s+4) +k(s+3) =0


1.θd = 180-(angle subtended by poles to P2) +( angle subtended by zeros to P2)


1. s (s+1) (s+2) (s+4) +k(s+3) =0

2. s→ jw

3. Equate the real and imaginary parts

UNIT III FREQUENCY RESPONSE

Frequency response – Bode plot – Polar plot – Determination of closed loop response from open loop

response - Correlation between frequency domain and time domain specifications.

PART * A

Q.No. Questions

1.

Define gain Margin. (May 03,11,13, Dec 06,08,09,11,12,14) (BTL1)

The gain margin (G.M.) is defined as the margin in gain allowable by which gain can be increased

till system reaches on the verge of instability. Mathematically it can be defined as reciprocal of the

magnitude of the G(jw)H(jw) measured at phase crossover frequency.

2.

What is meant by frequency response? (April/May 17) (BTL1)

The magnitude and phase angle of sinusoidal transfer function of a system are real function of

frequency ω, and so they are called frequency response.

3.

Define phase margin. (Dec 03,06,08,09,10,11,12, May 03,11,12,13) (BTL1)

The amount of additional phase lags which can be introduced in the system till the system reaches

on the verge of instability is called phase margin.

4.

State any four-frequency domain specification. (Nov 08,10,15 May 07,11) (BTL1)

➢ Resonant Peak

➢ Resonant frequency

➢ Cutoff region

➢ Phase Margin

➢ Gain Margin

➢ Phase cross over frequency

➢ Gain cross over frequency



6.40

5.

What are the advantages of Bode plot? (May 06, Dec 09,10) (BTL2)

➢ It shows both low and high frequency characteristics of transfer function in single

diagram.

➢ The plots can be easily constructed using some valid approximations.

➢ Relative stability of system can be studied by calculating G.M. and P.M. from the

ode plot.

➢ The various other frequency domain specifications like cut-off frequency,

bandwidth etc. can be determined.

➢ Data for constructing complicated polar and Nyquist plots can be easily obtained

from Bode plot.

➢ Transfer function of system can be obtained from the bode plot.

6.

What is gain crossover frequency and phase crossover frequency? (Dec 10, May 06,11, 14)

(BTL1)

➢ Gain crossover frequency (Wgc): The frequency at which magnitude of G(jw)H(jw) is unity

is called gain crossover frequency.

➢ Phase crossover frequency (Wpc): The frequency at which phase angle of G(jw)H(jw) is -

180 deg. is called phase crossover frequency.

7.

What is meant by corner frequency in frequency response analysis? (May 05) (BTL2)

A frequency up to which the magnitude contribution of a factor is negligible and can be neglected

is called its corner frequency. It is the frequency at which low frequency and high frequency

asymptotes intersect each other. At the corner frequency, a change in the slope of a magnitude plot

occurs. Frequency range and the number of points is chosen automatically.

8.

What is the use of Nichol’s chart in control system? (Dec 14, May 15) (BTL2)

➢ Nichol’s chart used to find closed loop frequency response from open loop frequency

response.

➢ The frequency domain specifications can be determined from Nichols chart.

➢ The gain of the system can be adjusted to satisfy the given specifications.

9. Define BIBO stability. (Dec 14) (BTL1)

A linear relaxed system is said to have BIBO stability if every bounded (finite) input results in a

bounded (finite) output.

10.

What are the characteristics of phase lead network? (May 15) (BTL2)

➢ Increases system bandwidth which usually correlates to reduce rise and settling times and a

susceptibility to high frequency noise.

➢ The phase of the forward path transfer function in the vicinity of the zero-gain crossover

frequency. This increases the phase margin of the closed loop system and hence the relative

stability.

11.

What is the basis for selection of a compensator for a system? (May 15) (BTL2)

When the system is to be redesigned to meet the required specifications, it is necessary to alter the

system by adding an external device to it. The system must provide,

➢ Attenuation in the high frequency range to give a system enough phase margin.



6.41

➢ Large bandwidth, short rise time and less settling time.

12.

Define M & N circles. (BTL1)

➢ The magnitude of closed loop transfer function with unity feedback can be shown to be for

every value of M. These circles are called M circles.

➢ If the phase of closed loop transfer function with unity feedback is α, then tan α will be in

the form of circles for every value of α; these circles are called N circles.

13.

How the resonant peak (Mr), resonant frequency (wr), and band width are determined from

Nichols chart? (BTL2)

The resonant peak is given by the value of µ. Contour which is tangent to G (jw) locus. The resonant

frequency is given by the frequency of G (jw) at the tangent point. The bandwidth is given by

frequency corresponding to the intersection point of G(jw) and –3dB M-contour.

14.

What is meant by Cut of frequency? (BTL1)

it is denoted by ωb. the frequency at which the magnitude of the closed loop response is 3 dB down

from its zero-frequency value is called cut-off frequency.

15.

What is meant by resonant peak? (BTL1)

Resonant peak (Mr): It is the maximum value of magnitude of the closed loop frequency

response.

16.

What is meant by resonant frequency? (BTL1)

Resonant frequency (ωr): The frequency at which resonant peak Mr occurs in closed loop frequency

response is called resonant frequency.

PART * B

1.

Define all frequency domain specifications of a second order control system after plotting the

response. (Apr/May 2017). BTL1 (13M)


➢ Definition (5M) and Derivations of frequency domain frequency specifications: (8M)

1. Resonant peak:

It is the peak (maximum) value of the magnitude of T(jw)T(jw). It is

denoted by Mr.

2. Resonant Frequency:

It is the frequency at which the magnitude of the frequency response has peak

value for the first time. It is denoted by ωr.



6.42

3. Cut off rate:

the slope of the log magnitude curve near the cut off frequency is called cut off rate

.

4. Band width:

The bandwidth is the range of frequencies in which the gain of the system is more

than -3dB.the frequency at which the gain is -3dB is called cut off frequency.

𝑤𝑏 = 𝑤𝑛[1 − 2𝜀2 + √2 − 4𝜀2 + 4𝜀4]1/2

5. Gain margin:

The values of gain to be added to the system in order to bring the system to the verge

of instability.

1. Kg= 1/|G(jw)| Wpc

2. Kg in dB= 20 log Kg

6. Phase margin:

The additional phase lag to be added at the gain cross over frequency in order to bring

my system to the verge of instability.

1. µ=1800+ɸgc

2. ɸgc = angle(G(jw)

2.

Sketch the polar plot for a system whose open loop transfer function is 4/(s+2) (s+4). Find

gain margin and phase margin. (Apr/May 2017). BTL3 (13M)

Answer: Page.3.54- Nagoor kani

➢ Convert G(s) into time constant form (3M)

1. G(s)=4/ (0.5s+1) (0.25s+1)

➢ S=jw (2M)

1. G(jw)=4/(0.5jw+1) (0.25jw+1)

➢ Find magnitude and phase angle(2M)

1. Magnitude = |G(jw)|

= 4/sqrt[(1+0.25w2) (1+0.625w2)]

2. Phase angle = angle of G(jw)

=-tan-1 (0.5)- tan-1 (0.25)

➢ Draw polar plot(4M)

➢ Find PM & GM (2M)

1. Gain margin:



6.43


of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

2. Phase margin:



µ=1800+ɸgc ɸgc = angle(G(jw))

3.

Consider a unity feedback open loop transfer function G(s) =100/s (1+0.1s) (1+0.2s). Draw

the bode plot and find gain margin and stability of the system. (Nov/Dec 2016). BTL3 (13M)



1. 100/s(1+0.1s) (1+0.2s)

➢ S= jw (2M)

1. G(jw)= 100/jw(1+0.1jw) (1+0.2jw)




= -90-tan-1 (0.1)- tan-1 (0.2)

➢ Draw bode plot(4M)

➢ Find PM & GM(2M)

1. Gain margin:


of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log (Kg)

2. Phase margin:




4. List any four frequency domain specifications. (May/June 2016). BTL1 (13M)




6.44

➢ Definition (5M) and Derivations of frequency domain frequency specifications: (8M)

1. Resonant peak:


denoted by Mr.




3. Cut off rate:


.

4. Band width:



𝑤𝑏 = 𝑤𝑛[1 − 2𝜀2 + √2 − 4𝜀2 + 4𝜀4]1/2

5.

Draw the bode magnitude and phase plot for the unity feedback system with G(s) =40/s

(1+0.1s) and hence determine phase margin and gain margin. (May/June 2016). BTL3

Answer: Page 3.32-Nagoor Kani (13M)


1. 100/s(1+0.1s)

➢ S= jw (2M)

2. G(jw)= 40/jw(1+0.1jw)




= -90-tan-1 (0.1w)

➢ Draw bode plot(4M)

➢ Find PM & GM(2M)

1. Gain margin:



6.45


of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

2. Phase margin:



µ=1800+ɸgc ɸgc = angle(G(jw)).

6.

Sketch the bode plot for the following transfer function and determine the value of K for the

gain cross over frequency of 5 rad/sec G(s) =ks2/ (1+0.2s) (1+0.02s). (Apr/May 2017). BTL3



1. 100/s(1+0.1s) (1+0.2s)

➢ S= jw (2M)

2. G(jw)= K(jw)2/(1+0.2jw) (1+0.02jw)




= 180-tan-1 (0.2w)- tan-1 (0.02w)

➢ Draw bode plot (4M)


3. Gain margin:


of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

4. Phase margin:






6.46

➢ Find the value of K

7.

Sketch the polar plot for the following transfer function and determine the gain and phase

margin. G(s)=1/s(s+1) (1+2s). (Apr/May 2017) BTL3 (13M)



➢ G(s)=1/s(s+1) (1+2s)

➢ S=jw (2M)

G(jw)1/jw(jw+1) (2jw+1)


Magnitude = |G(jw)|

Phase angle = angle of G(jw)

=90-tan-1 (w)- tan-1 (2w)



3. Gain margin:


of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

4. Phase margin:




8.

Obtain the relationship between any three frequency domain specifications in terms of time

domain specifications. (Nov/Dec 2016). BTL4 (13M)

Answer: Page.3.7-Nagoor Kani

➢ Calculate time domain specifications such as maximum overshoot, natural frequency,

damping ratio (4M)

1. Rise time definition



6.47

The time required for the response to rise from 0% to 100% of its final value. This is

applicable for the under-damped systems.

2. Peak time definition

The time required for the response to reach the peak value for the first time. It is denoted

by tp. At t=tp, the first derivate of the response is zero.

3. Settling time definition and its formula

The time required for the response to reach the steady state and stay within the specified

tolerance bands around the final value. The settling time is denoted by ts.

4. Maximum overshoot definition and its formula

Peak overshoot Mp is defined as the deviation of the response at peak time from the final

value of response. It is also called the maximum overshoot

➢ Calculate frequency domain specifications such as resonant peak, damping ratio,

resonant frequency (4M)

5. Resonant peak:


denoted by Mr.






6.48

7. Cut off rate:


.

8. Band width:



𝑤𝑏 = 𝑤𝑛[1 − 2𝜀2 + √2 − 4𝜀2 + 4𝜀4]1/2

9. Gain margin:


of instability.

1. Kg= 1/|G(jw)| Wpc

2. Kg in dB= 20 log Kg

10. Phase margin:



µ=1800+ɸgc

ɸgc = angle(G(jw)

➢ By using various values of damping ratio compare the results (5M)

9.

The closed loop transfer function of a system C(s)/R(s)=81/s2+7s+81 calculate the values of

resonant frequency, resonant peak and band width. (Nov/Dec 2015). BTL3 (13M)

➢ Resonant peak:

It is the peak (maximum) value of the magnitude of T(jw). It is

denoted by Mr.

Ans:

➢ Resonant Frequency:


value for the first time. It is denoted by ωr

wr= wn* sqrt(1-2ꜫ2)

Ans:



6.49

➢ Band Width:



𝑤𝑏 = 𝑤𝑛[1 − 2𝜀2 + √2 − 4𝜀2 + 4𝜀4]1/2

Ans:

10.

Explain how open loop response can be obtained from closed loop response. BTL 2 (13M)


➢ Find magnitude of G(s)

1. The magnitude of closed loop transfer function with unity feedback can be shown to be for

every value of M. These circles are called M circles.

2. If the phase of closed loop transfer function with unity feedback is α, then tan α will be in

the form of circles for every value of α; these circles are called N circles.

➢ Convert into in the form of circle equation

➢ Find phase angle of G(s) and Find α

PART * C

1.


margin. G(s)=1/s2(s+1) (1+2s). (Apr/May 2017) BTL3 (13M)



➢ G(s)= 1/s2(s+1) (1+2s).

➢ S=jw (2M)

G(jw)=1/(jw)2(jw+1) (2jw+1)


Magnitude = |G(jw)|


=90-tan-1 (w)- tan-1 (2w)





6.50

5. Gain margin: - Kg=0


of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

6. Phase margin: PM= -900




2.


margin. G(s)=(1+0.2s) (1 +0.025s)/s3(0.005s+1) (1+0.001s). (Apr/May 2017) BTL3

(13M)



➢ G(s)=(1+0.2s) (1 +0.025s)/s3(0.005s+1) (1+0.001s).

➢ S=jw (2M)

G(jw)=(1+0.2jw) (1 +0.025jw)/(jw)3(0.005jw+1) (1+0.001jw).


Magnitude = |G(jw)|


=90-tan-1 (w)- tan-1 (2w)





of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

8. Phase margin: PM= -770





6.51


3.


margin. G(s)=1/s(s+1)2. (Apr/May 2017) BTL3 (13M)



➢ G(s)=1/s(s+1)2

➢ S=jw (2M)

G(s)=1/jw(jw+1)2.


Magnitude = |G(jw)|


=90-tan-1 (w)- tan-1 (2w)





of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

10. Phase margin: PM= 210




5.

Sketch the Nyquist plot for the system whose loop transfer function is

G(s)H(s)=K(s+2)/[s(s+3) (s+6)] and find the range of K for stability of closed loop. Check your

answer with Routh’s Criterion. Nov/Dec 2018 BTL 5 15M


➢ Put K=1


➢ G(s)H(s)=(s+2)/[s(s+3) (s+6)]



6.52

➢ S=jw (2M)

G(s)H(s)=(s+2)/[s(s+3) (s+6)].


Magnitude = |G(jw)|






of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

12. Phase margin: PM= 210




6

Sketch the bode plot for the following transfer function G(s)=75(1+0.2s)/s(s2+16s+100).

(Apr/May 2017). BTL3



1. G(s)=75(1+0.2s)/s(s2+16s+100).

2. S= jw (2M)

3. G(jw)= 75(1+0.2jw)/jw(jw2+16jw+100)




= tan-1 (0.2w)-90- tan-1 ((0.16w)/(1-0.01w2))



Gain margin: - Kg = infinity



6.53


of instability.

Kg= 1/|G(jw)| Wpc

Kg in dB= 20 log Kg

5. Phase margin=920




7

Sketch the bode plot for the following transfer function G(s)=20/s(1+3s) (1+4s). obtain the

gain cross over and phase cross over frequency. BTL 4 (15M)



1. G(s)=20/s(1+3s) (1+4s).

2. S= jw (2M)

3.G(jw)=20/jw(1+3jw) (1+4jw).




= -tan-1 (3w)-90- tan-1 (4w)


➢ Find Wgc and Wpc (4M)

Wgc=1.1 rad/sec

Wpc=0.25 rad/sec



6.54

UNIT IV STABILITY AND COMPENSATOR DESIGN

Characteristics equation – Routh Hurwitz criterion – Nyquist stability criterion- Performance criteria -

Effect of Lag, lead and lag-lead compensation on frequency response–Design of Lag, lead, Lag/Lead

compensator design using bode plots.

PART * A

Q.No. Questions

1. What is characteristics equation. (APRIL/MAY 17) (BTL1)

The denominator polynomial of C(s) / R(s) is the characteristics equation of the system.

2.

State what is stability in control system (BTL1)

A linear relaxed system is said to have BIBIO stability if every bounded input result in abounded

output.

3.

What will be the nature of impulse response when the roots of characteristic equation are

lying on imaginary axis? (BTL2)

If the root of characteristic equation lies on imaginary axis the nature of impulse response is

oscillatory.

4.

What is the relationship between Stability and coefficient of characteristic polynomial?

(BTL3)

If the coefficient of characteristic polynomial is negative or zero, then some of the roots lie on the

negative half of the S-plane. Hence the system is unstable. If the coefficients of the characteristic

polynomial are positive and if no coefficient is zero, then there is a possibility of the system to be

stable provided all the roots are lying on the left half of the S-plane

5.

Write Routh stability criterion? (BTL2)

Routh criterion states that the necessary and enough condition for stability is that all the elements

in the first column of the Routh array is positive. If this condition is not met, the system is unstable

and the number of sign changes in the elements of the first column of Routh array corresponds to

the number of roots of characteristic equation in the right half of the S-plane.

6.

What is limitedly stable system? (BTL2)

For a bounded input signal if the output has constant amplitude oscillations, then the system may

be stable or unstable under some limited constraints such a system is called limitedly stable

system.

7.

In Routh array what conclusion you can make when there is a row of all zeros? (BTL2)

All zero rows in the Routh array indicate the existence of an even polynomial as a factor of the

given characteristic equation. The even polynomial may have roots on imaginary axis



6.55

8.

What is a principle of argument? (BTL2)

The principles of arguments state that, Let F(S) are analytic function and if an arbitrary closed

contour in a clockwise direction is chosen in the S-plane so that F(S) is analytic at every point of

the contour. Then the corresponding F(S) plane contour mapped in the F(S) plane will encircle

the origin N times in the anti-clockwise direction, where N is the difference between number of

poles and zeros of F(S) that are encircled by the chosen closed contour in the S plane.

9.

How the roots of characteristic are related to stability? (BTL1)

If the root of characteristic equation has positive real part, then the impulse response of the system

is not bounded. Hence the system will be unstable. If the root has negative real parts, then the

impulse response is bounded. Hence the system will be stable.

10.

What is the necessary condition for stability? (BTL1)

The necessary condition for stability is that all the coefficients of the characteristic polynomial be

positive. The necessary and enough condition for stability is that all the elements in the first

column of the Routh array should be positive.

11.

State the requirements for BIBO Stability. (BTL2)

The requirements of the BIBO stability is that the absolute integral of the impulse response of the

system should take only the finite value.

12.

What is auxiliary polynomial? (BTL2)

In the construction of Routh array a row of all zero indicates the existence of an even polynomial

as a factor of given characteristic equation. In an even polynomial the exponents of S are even

integers or zero only. This even polynomial factor is called auxiliary polynomial. The coefficients

of auxiliary polynomial are given by the elements of the row just above the row of all zeros.

13.

Why compensation is required in feedback control systems? (BTL4)

In feedback control systems compensation is required in the following situations:

➢ When the system is unstable, then compensation is required to stabilize the system and to

meet the desired performance.

➢ When the system is stable, compensation is provided to obtain desired performance.

14.

When lag, lead, lag-lead compensation is employed? (BTL4)

➢ Lag compensation is employed for a stable system for improvement in steady state

performance.

➢ Lead compensation is employed for stable/unstable system for improvement in transient

state performance.



6.56

➢ Lag-lead compensation is employed for stable/unstable system for improvement in both

steady state and transient state performance.

15.

Discuss the effect of adding a pole to open loop transfer function of a system. (BTL2)

The addition of pole to open loop transfer function of a system will reduce the steady state error.

The closer the pole to origin lesser will be the steady state error. Thus, the steady state performance

of the system is improved. Also, the addition of pole will increase the order of the system, which

in turn makes the system less stable than the original system.

16.

Discuss the effect of adding a zero to open loop transfer function of a system. (BTL2)

The addition of zero to open loop transfer function of a system will improve the transient response.

The addition of zero reduces the rise time. If the zero is introduced close to origin the peak

overshoot will be large. If the zero is introduced far away from the origin in the left half of s plane,

then the effect of zero on the transient response will be negligible.

17.

What are the characteristics of lag compensator? (BTL2)

The lag compensator improves the steady state performance, reduces the bandwidth and increases

the rise time (which results in slower transient response). If the pole introduced by the

compensator is not cancelled by a zero in the system, then the lag compensator increases the order

of the system by one.

18.

Write the characteristics of lead compensator. (BTL2)

The lead compensation increases the bandwidth and improves the speed of response. It also

reduces the peak overshoot. If the pole introduced by the compensator is not cancelled by a zero

in the system, then lead compensation increases the order of the system by one. When the given

system is stable/unstable and improvement in transient state response then lead compensation is

employed.

19.

What is the transfer function of lag, lead and lag-lead compensator? (BTL2)

Gc(s)=(s+1/T)/(s+1/Tβ)

Gc(s)=(s+1/T)/(s+1/Tα)

Gc(s)=(s+1/T1) (s+1/T2)/(s+1/T1β) (s+1/ T2α)

20.

State about lead-lag compensator. (APRIL/MAY 17) (BTL2)

The lead-lag compensation is a design procedure in which a lead-lag compensator is introduced

in the system to meet the desired specifications.



6.57

21

Define Nyquist stability criterion. (May 17, 13, Dec 15) (BTL1)

If the Nyquist plot of the open loop tf G(s) H(s) corresponding to the Nyquist contour in the s-

plane encircles the critical point -1+j0 in the counter clockwise direction as many times as the

number of right half s-plane poles of G(s)H(s), the closed loop system is stable.

PART * B

1.

Draw the Nyquist plot for the system whose open loop transfer function is G(S)H(S) =K/S

(S+2) (S+10). Determine the range of K for which closed loop system is stable. BTL3 (13M)



1.G(s)H(s)= K/s(0.5s+1) (0.1s+1)

➢ Select contour regions (3M)

1.System has one pole at origin.

2.Select S-plane enclosing entire right half except the origin.

➢ Apply four sections and draw the F(s) contour (4m)

1.Four sections C1, C2, C3 and C4.

➢ Finally draw all contour regions in single Nyquist plot (4m)

1. Stability:

-0.00417k = -1

K=240

This contour passes through the point -1+j0

2. K<240:

The closed loop system is stable

3. K>240

The closed loop system is unstable.

2.

Construct Nyquist plot for a feedback control system whose open loop transfer function is

given by G(S)H(S) =K(1+s)2/ s3. Comment on the stability of open loop and closed loop

transfer function. BTL3 (13M)



1. G(s)H(s)= K(1+s)2/ s3

2. G(jw)H(jw)= K(1+jw)2/(jw)3




6.58

1. System has three poles at origin.


➢ Apply four sections and draw the F(s) contour (4m)


➢ Finally draw all contour regions in single Nyquist plot (4m)

1. Stability:

-2k = -1

K=0.5

This contour passes through the point -1+j0

2. K<0.5:

The closed loop system is Unstable

3. K>0.5

The closed loop system is stable.

3.

Sketch the Nyquist plot for a system with the open loop transfer function G(S)H(S) =(1+4S)

/s2 (1+S) (2S+1). Determine the stability of the system. BTL3 (13M)



3. G(s)H(s)= (1+4S) /s2 (1+S) (2S+1)

4. G(jw)H(jw)= (1+4jw)/jw2(1+jw) (2jw+1)


1. System has two poles at origin.


➢ Apply four sections and draw the F(s) contour (4M)


➢ Finally draw all contour regions in single Nyquist plot (4M)

1.closed loop system is unstable

2.Two poles of closed loop system are lying on the right half of s plane.

4.

Construct Routh array and determine the stability of the system represented by the

characteristics equation S5+S4+2S3+2S2+3S+5=0.Comment on the location of the roots of

characteristic equation. BTL2 (13M)



6.59


➢ Form Routh array from Characteristic equation (5M)

➢ Check the first column of the Routh array (5M)

1. If it is positive, system is stable

2. If it is not positive, system is unstable

➢ Sign changes in first column (3M)

1. All are negative→all roots on left half

2. Sign changes→ corresponds to roots in right half

5.


characteristics equation S7+9S6+24S4+24S3+24S2+23S+15=0comment on the location of the

roots of characteristic equation. BTL2 (13M)




1.If it is positive, system is stable

2.If it is not positive, system is unstable


1.All are negative→all roots on left half

2.Sign changes→ corresponds to roots in right half

6.

The open loop transfer function of a unity feedback control system is G(s)=K/[s(s+1) (s+2)].

Design a suitable Lag-lead compensator to meet the following specifications: static velocity

error constant Kv=10 sec-1, phase Margin = 50 degree and gain margin >= 10db.

BTL 5 (13M)

Answer Page.5.61-Nagoor kani

Bode plot: 5M

Procedure: 4M

Compensated bode plot:4M

➢ Find the value of K using velocity error constant

➢ Convert G(s) into time constant form

➢ S=jw



6.60

➢ Find magnitude and phase angle

➢ Draw bode plot

➢ Find PM & GM

➢ If the PM is not matched with the given specifications, lag lead compensator is

employed

➢ Using design procedure of lag- lead compensator find compensator transfer

function

Find overall transfer function

7.

Consider the unity feedback whose open loop transfer function is G(s)=K/[s(0.1s+1)

(0.2s+1)] system to be compensated to meet the following specifications: static velocity

error constant = 30 sec-1, phase margin>=50 degrees. Band width (wb)=12 rad /sec.

BTL 6 (13M)


Bode plot: 5M

Procedure: 4M




➢ S=jw


➢ Draw bode plot

➢ Find PM & GM

➢ If the PM is not matched with the given specifications, lead compensator is

employed

➢ Using design procedure of lead compensator find compensator transfer function

➢ Find overall transfer function.



6.61

8.


characteristics equation S4+8S3+18S2+16S+5=0.Comment on the location of the roots of

characteristic equation. BTL2 (13M)









9.


characteristics equation S6+2S5+8S4+12S3+20S2+16S+16=0.Comment on the location of the

roots of characteristic equation. BTL2 (13M)









10.


characteristics equation S6+S5+3S4+3S3+3S2+2S+1=0.Comment on the location of the roots

of characteristic equation. BTL2 (13M)






6.62






PART * C

1.

A unity feedback system has an OLTF G(s) = K / s(s+0.5). Design a Lead-Lag compensator

is to meet the following specifications. ζ= 0.5, Wn=5 rad/sec, velocity error constant, Kv =

80. BTL3 (15M)

Answer Page.5.61-Nagoor kani

Bode plot: 5M

Procedure: 4M




➢ S=jw


➢ Draw bode plot

➢ Find PM & GM


employed


function

➢ Find overall transfer function

2.

Consider the unity feedback system whose OLTF is G(s) = K / s (s+3) (s+6). Design a lag-

lead compensator to meet the following specifications. (i) Velocity error constant, Kv = 80,

(ii) P.M 35º BTL 3 (15M)

Answer: Page 5.55- Nagoor kani

Bode plot: 5M



6.63

Procedure: 4M




➢ S=jw


➢ Draw bode plot

➢ Find PM & GM


employed


function

➢ Find overall transfer function

3.

Draw the frequency response of lag compensator with necessary expressions. BTL 3 (13M)


➢ G(s), s can be replaced by jw (4M)

1. 𝐺𝑐(𝑠) =(𝑠+1/𝑇)

(𝑠+1

𝛽𝑇)

2. 𝐺𝑐(𝑠) =(𝑗𝑤+1/𝑇)

(𝑗𝑤+1

𝛽𝑇)



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➢ Find magnitude and Phase angle. (4M)

𝐺𝑐(𝑗𝑤) =1 + 𝑗𝑤𝑇

1 + 𝑗𝑤𝛽𝑇=

√1 + (𝑤𝑇)2 < 𝑡𝑎𝑛−1𝑤𝑇

√1 + (𝑤𝛽𝑇)2 < 𝑡𝑎𝑛−1𝑤𝛽𝑇

➢ Draw the graph between frequency versus phase angle (5M)

➢ Draw the graph between frequency versus magnitude

4.

Draw the frequency response of lead compensator with necessary expressions. BTL4 (15M)

Answer: Page.5.28-Nagoor Kani




6.65

1. 𝐺𝑐(𝑠) =(𝑠+1/𝑇)

(𝑠+1

𝛼𝑇)

2. 𝐺𝑐(𝑠) =(𝑗𝑤+1/𝑇)

(𝑗𝑤+1

𝛼𝑇)

➢ Find magnitude and Phase angle

𝐺𝑐(𝑗𝑤) =1 + 𝑗𝑤𝑇

1 + 𝑗𝑤𝛼𝑇=

√1 + (𝑤𝑇)2 < 𝑡𝑎𝑛−1𝑤𝑇

√1 + (𝑤𝛼𝑇)2 < 𝑡𝑎𝑛−1𝑤𝛼𝑇

➢ Draw the graph between frequency versus phase angle

➢ Draw the graph between frequency versus magnitude



6.66

5

Draw the frequency response of lead lag compensator with necessary expressions.

BTL4(15M)

Answer: Page 5.28 – Nagoor Kani


3. 𝐺𝑐(𝑠) =(𝑠+1/𝑇1)

(𝑠+1

𝛽𝑇1)

(𝑠+1/𝑇2)

(𝑠+1

𝛼𝑇2)

4. 𝐺𝐺𝑐(𝑠) =(𝑗𝑤+1/𝑇1)

(𝑗𝑤+1

𝛽𝑇1)

(𝑗𝑤+1/𝑇2)

(𝑗𝑤+1

𝛼𝑇2)

➢ Find magnitude and Phase angle. (4M)

𝐺𝑐(𝑗𝑤) = 1 + 𝑗𝑤𝑇1

1 + 𝑗𝑤𝛽𝑇1

1 + 𝑗𝑤𝑇2

1 + 𝑗𝑤𝛼𝑇2

=√1 + (𝑤𝑇1)2 < 𝑡𝑎𝑛−1𝑤𝑇1

√1 + (𝑤𝛽𝑇1)2 < 𝑡𝑎𝑛−1𝑤𝛽𝑇1

√1 + (𝑤𝑇2)2 < 𝑡𝑎𝑛−1𝑤𝑇2

√1 + (𝑤𝛼𝑇2)2 < 𝑡𝑎𝑛−1𝑤𝛼𝑇2

➢ Draw the graph between frequency versus phase angle (5M)

➢ Draw the graph between frequency versus magnitude (2M)



6.67

UNIT –V STATE VARIABLE ANALYSIS

Concept of state variables – State models for linear and time invariant Systems – Solution of state and

output equation in controllable canonical form – Concepts of controllability and observability.

PART * A

Q.No. Questions

1.

What are the advantages of state space analysis? (BTL2)

➢ It can be applied to non-linear as well as time varying systems.

➢ Any type of input can be considered for designing the system.

➢ It can be conveniently applied to multiple input multiple output systems.

➢ The state variables selected need not necessarily be the physical quantities of the system.

2. What are phase variables? (BTL1)

The phase variables are defined as the state variables which are obtained from one of the system

variables and its derivatives.

3.

Define state variable. (BTL1)

The state of a dynamical system is a minimal set of variables (known as state variables) such that

the knowledge of these variables at t-t0 together with the knowledge of the inputs for t > t0,

completely determines the behavior of the system for t > t0.

4.

Write the general form of state variable matrix. (BTL1)

The most general state-space representation of a linear system with m inputs, p outputs and n state

variables are written in the following form:

Ẋ= AX + BU

Y = CX + DU

Where = state vector of order n X1.

U = input vector of order n X1.

A=System matrix of order n Xn.

B=Input matrix of order n Xm

C =output matrix of order p Xn

D = transmission matrix of order p Xm.

5.

What is the necessary condition to be satisfied for design using state feedback? (BTL2)

The state feedback design requires arbitrary pole placements to achieve the desire performance.

The necessary and enough condition to be satisfied for arbitrary pole placement is that the system

is completely state controllable.

6.

What is controllability? (BTL1)

A system is said to be completely state controllable if it is possible to transfer the system state

from any initial state X(t0) at any other desired state X(t), in specified finite time by a control

vector U(t).

7. What is observability? (BTL1)

A system is said to be completely observable if every state X(t) can be completely identified by

measurements of the output Y(t) over a finite time interval.

8. What is similarity transformation? (BTL2)

The process of transforming a square matrix A to another similar matrix B by a transformation



6.68

P- |AP| = B is called similarity transformation. The matrix P is called transformation matrix.

9.

What is meant by diagonalization? (BTL2)

The process of converting the system matrix A into a diagonal matrix by a similarity

transformation using the modal matrix M is called diagonalization.

10.

What is modal matrix? (BTL2)

The modal matrix is a matrix used to diagonalize the system matrix. It is also called

diagonalization matrix.

If A = system matrix

M = Modal matrix

And M-1=inverse of modal matrix

Then M-1AM will be a diagonalized system matrix

11.

How the modal matrix is determined? (BTL2)

The modal matrix M can be formed from eigenvectors. Let m1, m2, m3 …. mn be the eigen

vectors of the nth order system. Now the modal matrix M is obtained by arranging all the

eigenvectors column wise as shown below.

Modal matrix, M = [m1, m2, m3 …. mn].

12.

What is the need for controllability test? (BTL2)

The controllability test is necessary to find the usefulness of a state variable. If the state variables

are controllable then by controlling (i.e. varying) the state variables the desired outputs of the

system are achieved.

13.

What is the need for observability test? (BTL2)

The observability test is necessary to find whether the state variables are measurable or not. If the

state variables are measurable then the state of the system can be determined by practical

measurements of the state variables.

14.

State the condition for controllability by Gilbert’s method. (BTL2)

Case (i) when the Eigen values are distinct

Consider the canonical form of state model shown below which is obtained by using the

transformation X=MZ.

Ẋ= ΛZ + U

Y=Z + DU

Where, Λ = M-1AM; = CM, = M-1B and M = Modal matrix.

In this case the necessary and enough condition for complete controllability is that, the matrix

must have no row with all zeros. If any row of the matrix is zero, then the corresponding state

variable is uncontrollable.

Case (ii) when Eigen values have multiplicity

In this case the state modal can be converted to Jordan canonical form shown below:

Z= JZ + U

Y=Z + DU Where, J = M-1AM

In this case the system is completely controllable, if the elements of any row of that correspond

to the last row of each Jordan block are not all zero.



6.69

15.

State the condition for observability by Gilbert’s method. (BTL2)

Consider the transformed canonical or Jordan canonical form of the state model shown below:

which is obtained by using the transformation, X =MZ

Z= ΛZ + U

Y=Z + DU (Or)

Z= JZ + U

Y=Z + DU where =CM and M=modal matrix.

The necessary and enough condition for complete observability is that none of the columns of the

matrix be zero. If any of the column has all zeros, then the corresponding state variable is not

observable.

16.

State the duality between controllability and observability. (BTL2)

The concept of controllability and observability are dual concepts and it is proposed by Kalman

as principle of duality. The principle of duality states that a system is completely state controllable

if and only if its dual system is completely state controllable if and only if its dual system is

completely observable or vice versa.

17. Define servo mechanism. (BTL 1)

The servomechanism is a feedback control system, in which the output is mechanical position or

time derivatives of positions.

18.

what is state diagram? Draw the block diagram representation of state model.

The pictorial representation of the state model of the system is called state diagram. The state

diagram of the system can be either in block diagram or in signal flow graph form.

19. What is state vector? (BTL 1)

The state vector is a (n x1) column matrix whose elements are state variables of the system.it is

denoted by X(t).

20. Write the properties of state transition matrix.



6.70

PART * B

1.

Determine the state model of armature-controlled DC motor. BTL1 (13M)

Answer: Page 2.26-Nagoor kani.

➢ Draw the electrical circuit (3M)

➢ Write differential equations in terms of displacement (3M)

𝑉𝑎 = 𝑖𝑎𝑅𝑎 + 𝐿𝑎𝑑𝑖𝑎

𝑑𝑡+ 𝑒𝑏

𝑇 = 𝐽𝑑2𝜃

𝑑𝑡2 + 𝐵𝑑𝜃

𝑑𝑡

➢ Apply the state variables to differential equation (2M)

𝑥1 = 𝑖𝑎 ; 𝑥2 =𝑑𝜃

𝑑𝑡= 𝜔 ; 𝑥3 = 𝜔

➢ Rearrange the equation and derive state model (2M)

)()(.5

)()()(.4

)()()0(.3

)()(.2

)0(.1

020112

1

ktt

tttttt

txtx

tt

I

k=

−=−−

−=

−=

=

−



6.71

𝒙𝟏 = −𝑅𝑎

𝐿𝑎

𝑥1 −

𝐾𝑏

𝐿𝑎𝑥2 +

1

𝐿𝑎𝑢

𝒙𝟐 = −𝐾𝑡

𝐽

𝑥1 −

𝐵

𝐽𝑥2

𝒙𝟑 = 𝑥2

➢ Draw the block diagram for state equation and output equation (3M)

2.

Construct the state model for a system characterized by the differential equation

d3y/dt3+6d2y/dt2+11dy/dt+6y+u=0.Give the block diagram representation of the state model.

BTL3 (13M)

Answer: Page: 6.26-Nagoor kani

➢ Convert differential equations in to dot variables (2M)

𝑥1 = 𝑦

𝑥2 = 𝑑𝑦

𝑑𝑡= 𝑥1 𝑥2 =

𝑑2𝑦

𝑑𝑡2 = 𝑥2

➢ Apply the state variables to dot variables (2M)

1. 𝑥1 = 𝑥2

2. 𝑥2 = 𝑥3 3. 𝑥3 = − 6𝑥1 − 11𝑥2 − 6𝑥3 − 𝑢



3.

Obtain the state model of the system whose open loop transfer function is given as,

Y(s)/U(s) =10/s3+4s2+2s+1. BTL3 (13M)


➢ Use inspection, cascade, signal flow graph method (1M)

➢ Take inverse Laplace transform & Convert differential equations in to dot variables (2M)






6.72

4.

A feedback system has a closed loop transfer function Y(s)/U(s) = 10(s+4)/s(s+1) (s+3),

construct block diagram representation of each state model. BTL3 (13M)







5.

Determine the canonical state model of the system, whose transfer function is

T(s) = 2(s+5)/(s+1) (s+3) (s+4). BTL4 (13M)







6.

Compute state transition matrix A = 0 -1.

-2 -3 BTL3 (13M)


➢ Find eigen values by using |SI -A |=0 (4M)

➢ Find inverse matrix of |SI -A | (4M)

➢ Use partial fraction method to find A, B, C (2M)

➢ Find inverse Laplace of φ(s) to get φ(t) (3M)

7. Consider the state model given below, verify the system is observable and controllable.

(Apr/May 2015) BTL4 (13M)



6.73


➢ From the state model identify A, B, C Matrix (2M)

➢ Use Gilberts method or Kalman’s method (2M)

➢ In Gilberts Method Find Ḃ and C. The value of Ḃ does not contain zero value then the

system is completely controllable. The value of C does not contain zero value then the

system is completely observable. (5M)

➢ In Kalman’s method Find Qo and Qc. The value of Qo does not contain zero value then

the system is completely controllable. The value of Qo does not contain zero value then

the system is completely observable. (4M)

8.

Test the controllability and observability by using any one method of the given state space

representation model. (Apr/May 2015) BTL4 (13M)

Answer: Refer Notes



➢ In Gilberts method Find Ḃ and C. The value of Ḃ does not contain zero value then the






9.

Consider the following differential equation, obtain the state model in diagonal form.

(nov/dec 2015) BTL 3 (13M)



6.74

Answer: Refer Notes



➢ Draw the block diagram for state equation and output equation(5M)

10.

Consider the system defined by

Check controllability and observability. (nov/dec 2015) BTL3 (13M)

Answer: Refer Notes








the system is completely observable. (4M).

PART * C

1.

Test the controllability of the following state model by using both the methods.

(nov/dec 2016) BTL4 (15M)

Answer: Refer Notes



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➢ The system is completely observable.

2.

Verify the system is completely controllable and observable. (nov/dec 2016)

BTL4 (15M)

Answer: Refer Notes









3.

Check the controllability of the following state model. (nov/dec 2015) BTL4 (15M)





6.76








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