CONTROL SYSTEMS
UNIT  I
SYSTEMS AND THEIR REPRESENTATION
PART  A (2 MARKS)
1.What is control system?
A system consists of a number of components connected together to perform a specific
function. In a system when the output quantity is controlled by varying the input quantity then
the system is called control system.
2.What are the two major types of control system?
The two major types of control system are open loop and closed loop.
3.Define open loop control system.
The control system 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.
4.Define closed loop control system.
The control system in which the output has an effect upon the input quantity so as to
maintain the desired output value are called closed loop control system.
5.What are the components of feedback control system?
The components of feedback control system are plant, feedback path elements, error
detector and controller.
6.Define transfer function.
The T.F of a system is defined as the ratio of the laplace transform of output to laplace
transform of input with zero initial conditions.
7.What are the basic elements used for modeling mechanical translational system?
Mass, spring and dashpot
8.What are the basic elements used for modeling mechanical rotational system?
Moment of inertia J, dashpot with rotational frictional coefficient B and torsional spring with
stiffness K.
9.Name two types of electrical analogous for mechanical system.
The two types of analogies for the mechanical system are Force voltage and force current
analogy.
10.What is block diagram?
A block diagram of a system is a pictorial representation of the functions performed by each
component of the system and shows the flow of signals. The basic elements of block
diagram are block, branch point and summing point.
11.What is the basis for framing the rules of block diagram reduction technique?
The rules for block diagram reduction technique are framed such that any modification made
on the diagram does not alter the input output relation.
12.What is a signal flow graph?
A signal flow graph is a diagram that represents a set of simultaneous algebraic equations
.By taking L.T the time domain differential equations governing a control system can be
transferred to a set of algebraic equations in sdomain.
13. What is transmittance?
The transmittance is the gain acquired by the signal when it travels from one node to another
node in signal flow graph.
14.What is sink and source?
Source is the input node in the signal flow graph and it has only outgoing branches. Sink is a
output node in the signal flow graph and it has only incoming branches.
15.Define nontouching loop.
The loops are said to be non touching if they do not have common nodes.
16.Write Masons Gain formula.
Masons Gain formula states that the overall gain of the system is
17.Write the analogous electrical elements in force voltage analogy for the elements of
mechanical translational system.
Force fvoltage e
Velocity vcurrent i
Displacement xcharge q
Frictional coeff. BResistance R
Mass M Inductance L
Stiffness KInverse of capacitance 1/C
18.Write the analogous electrical elements in force current analogy for the elements of mechanical translational system.
Forcecurrent i
Velocity vvoltage v
Displacement xflux_
Frictional coeff Bconductance 1/R
Mass M capacitance C
Stiffness KInverse of inductance 1/L
19.Write the force balance equation of M ideal mass element.
F = M d2x /dt2
20. Write the force balance equation of ideal dashpot element.
F = B dx /dt
21. Write the force balance equation of ideal spring element.
F = Kx
22.Distinguish between open loop and closed loop system.
23.What is servomechanism?
The servomechanism is a feedback control system in which the output is mechanical position
(or time derivatives of position velocity and acceleration,)
24.Why is negative feedback invariably preferred in closed loop system?
The negative feedback results in better stability in steady state and rejects any disturbance
signals.
25.What is synchro?
A synchro is a device used to convert an angular motion to an electrical signal or viceversa.
26.What is servomotor?
The motors used in automatic control systems or in servomechanism are called servomotors.
They are used to convert electrical signal into angular motion.
PARTB
1.Write the differential equations governing the Mechanical system shown in fig 1.1.and
determine the transfer function. (16)
2. Determine the transfer function Y2(S)/F(S) of the system shown in fig. (16)
4. Determine the overall transfer function C(S)/R(S) for the system shown in fig. (16)
5. Obtain the closed loop transfer function C(S)/R(S) of the system whose block diagram is
6 . For the system represented by the block diagram shown in fig. Determine C1/R1 and
7. Find the overall gain C(s) / R(s) for the signal flow graph shown below. (16)
8.Find the overall gain of the system whose signal flow graph is shown in fig. (16)
9. Draw a signal flow graph and evaluate the closed loop transfer function of a system whose
block is shown in fig. (16)
10. Write the differential equations governing the mechanical systems shown below. Draw
the forcevoltage and forcecurrent electrical analogous circuits and verify by writing
mesh and node equations. (16)
11. (i) Derive the transfer function for Armature controlled DC motor. (8)
(ii)Derive the transfer function for Field controlled DC motor. (8)
12. (i)Explain DC servo motor. (6)
(ii)Explain the working of AC servomotor in control systems. (10)
13. (i)Explain Synchros and its types. (10)
(ii)Write the rules for block diagram reduction techniques. (6)
UNIT – II
PART A (2 MARKS)
1.What is transient response?
The transient response is the response of the system when the system changes from one
state to another.
2.What is steady state response?
The steady state response is the response of the system when it approaches infinity.
3.What is an order of a system.
The order of a system is the order of the differential equation governing the system. The
order of the system can be obtained from the transfer function of the given system.
4.Define Damping ratio.
Damping ratio is defined as the ratio of actual damping to critical damping.
5.List the time domain specifications.
The time domain specifications are
i.Delay time
ii.Rise time
iii.Peak time
iv.Peak overshoot
6.Define Delay time.
The time taken for response to reach 50% of final value for the very first time is delay time.
7.Define Rise time.
The time taken for response to raise from 0% to 100% for the very first time is rise time.
8.Define peak time.
The time taken for the response to reach the peak value for the first time is peak time.
9.Define peak overshoot.
Peak overshoot is defined as the ratio of maximum peak value measured from the Maximum
value to final value.
10.Define Settling time.
Settling time is defined as the time taken by the response to reach and stay within specified
error.
11.What is the need for a controller?
The controller is provided to modify the error signal for better control action
12.What are the different types of controllers?
Proportional controller
PI controller
PD controller
PID controller
13.What is proportional controller?
It is device that produces a control signal which is proportional to the input error signal.
14.What is PI controller?
It is device that produces a control signal consisting of two terms – one proportional to error
signal and the other proportional to the integral of error signal.
15. What is PD controller?
PD controller is a proportional plus derivative controller which produces an output signal
consisting of two time one proportional to error signal and other proportional to the derivative
of the signal.
16. What is the significance of integral controller and derivative controller in a PID
controller?
The proportional controller stabilizes the gain but produces a steady state error. The integral
control reduces or eliminates the steady state error.
17.Why derivative controller is not used in control systems?
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.
18.Define Steady state error.
The steady state error is defined as the value of error as time tends to infinity.
19.What is the drawback of static coefficients?
The main draw back of static coefficient is that it does not show the variation of error with
time and input should be standard input.
20. What is step signal?
The step signal is a signal whose value changes from zero to A at t= 0 and remains constant
at A for t>0.
21. What is ramp signal?
The ramp signal is a signal whose value increases linearly with time from an initial value of
zero at t=0.the ramp signal resembles a constant velocity.
22.What is a parabolic signal?
The parabolic signal is a signal whose value varies as a square of time from an initial value of
zero at t=0.This parabolic signal represents constant acceleration input to the signal.
23.What are the three constants associated with a steady state error?
Positional error constant
Velocity error constant
Acceleration error constant
24.What are the main advantages of generalized error coefficients?
i) Steady state is function of time.
ii) Steady state can be determined from any type of input
25.What is the disadvantage in proportional controller?
The disadvantage in proportional controller is that it produces a constant steady state error.
26.What is the effect of PD controller on system performance?
The effect of PD controller is to increase the damping ratio of the system and so the peak
overshoot is reduced.
27.What is the effect of PI controller on the system performance?
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.
28.Name the test signals used in control system
the commonly used test input signals in control system are impulse step ramp acceleration
and sinusoidal signals
PART B
1.(a) Derive the expressions and draw the response of first order system for unit
step input. (8)
(b) Draw the response of second order system for critically damped case and when input
is unit step. (8)
2. Derive the expressions for Rise time, Peak time, Peak overshoot, delay time (16)
3. A positional control system with velocity feedback is shown in fig. What is the response of
the system for unit step input. (16)
4. (i) Measurements conducted on a Servomechanism show the system response to be
c(t)=1+0.2 ê60t 1.2 ê –10 t. when subjected to a unit step. Obtain an expression for
closed loop transfer function. (8)
(ii). A positional control system with velocity feedback is shown in fig. What is the response
c(t) to the unit step input. Given that =0.5.and also calculate rise time, peak
time, Maximum overshoot and settling time. (8)
5. (i) A unity feedback control system has an open loop transfer function
G(S)= 10/S(S+2).Find the rise time, percentage over shoot, peak time and settling
time. (8)
(ii) A closed loop servo is represented by the differential equation d2c/dt2 +8 dc/dt = 64 e
Where c is the displacement of the output shaft r is the displacement of the input shaft
and e= rc.Determine undamped natural frequency, damping ratio and percentage
maximum overshoot for unit step input. (8)
6. For a unity feedback control system the open loop transfer function
G(S) = 10(S+2)/ S2 (S+1).Find (a) position, velocity and acceleration error constants.
(b)the steady state error when the input is R(S) where R(S) =3/S –2/S2 +1/3S3 (16)
7. The open loop transfer function of a servo system with unity feed back system is
G(S) = 10/ S(0.1S+1).Evaluate the static error constants of the system. Obtain the steady
state error of the system when subjected to an input given Polynomial r(t) = a0 +a1t +a2 /2 t2
(16)
8. The unity feedback system is characterized by an open loop transfer function is
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 settling time,Peak overshoot and time to Peak overshoot
for a unitstep input. (16)
9.(i) For a servomechanisms with open loop transfer function(S)=10/(S+2)(S+3).What type
of input signal gives constant steady state error and calculate its value. (8)
(ii) Find the static error coefficients for a system whose G(S)H(S)=10/ S(1+S)(1+2S)and
also find the steady state error for r(t)=1+ t + t2/2. (8)
10.(i) Obtain the response of unity feedback system whose open loop transfer function is
G(S) = 4 / S (S+5) and When the input is unit step. (8)
(ii) A unity feedback system has an amplifier with gain KA=10 and gain ratio G(S) = 1 / S
(S+2) in the feed forward Path .A derivative feedback ,H(S)=S KO is introduced as a
minor loop around G(S).Determine the derivative feed back constant ,KO ,so that the
system damping factor is 0.6 (8)
11.(i) Explain P,PI,PID,PD controllers (8)
(ii) Derive the expressions for second order system for under damped case and when the
input is unit step. (8)
UNIT III
PART A (2 MARKS)
1.What is frequency response?
A frequency responses the steady state response of a system when the input to the system
is a sinusoidal signal.
2.List out the different frequency domain specifications?
The frequency domain specification are i)Resonant peak. ii)Resonant frequency.
3.Define –resonant Peak?
The maximum value of the magnitude of closed loop transfer function is called resonant
peak.
4.Define –Resonant frequency?
The frequency at which resonant peak occurs is called resonant frequency.
5.What is bandwidth?
The bandwidth is the range of frequencies for which the system gain is more than 3 dbB. The
bandwidth is a measure of the ability of a feedback system to reproduce the input signal
,noise rejection characteristics and rise time.
6.Define Cutoff rate?
The slope of the logmagnitude curve near the cutoff is called cutoff rate. The cutoff rate
indicates the ability to distinguish the signal from noise.
7.Define –Gain Margin?
The gain margin, kg is defined as the reciprocal of the magnitude of the open loop transfer
function at phase cross over frequency. Gain margin kg = 1 /  G(jw pc) .
8.Define Phase cross over?
The frequency at which, the phase of open loop transfer functions is called phase cross over
frequency wpc.
9. What is phase margin?
The phase margin ,g is the amount of phase lag at the gain cross over frequency required to
bring system to the verge of instability.
11.Define Gain cross over?
The gain cross over frequency wgc is the frequency at which the magnitude of the open loop
transfer function is unity.
12.What is Bode plot?
The Bode plot is the frequency response plot of the transfer function of a system. A Bode plot
consists of two graphs. One is the plot of magnitude of sinusoidal transfer function versus log
w.The other is a plot of the phase angle of a sinusoidal function versus log w.
13.What are the main advantages of Bode plot?
The main advantages are:
i) Multiplication of magnitude can be in to addition.
ii) A simple method for sketching an approximate log curve is available.
iii) It is based on asymptotic approximation. Such approximation is sufficient if
rough information on the frequency response characteristic is needed.
iv) The phase angle curves can be easily drawn if a template for the phase angle
curve of 1+ jw is available.
14.Define Corner frequency?
The frequency at which the two asymptotic meet in a magnitude plot is called corner
frequency.
15.What is Polar plot?
The Polar plot of a Sinusoidal transfer function G(j) is a plot of the magnitude of G(j)
versus the phase angle/argument of G(j) on polar or rectangular coordinates as is varied
from zero to infinity.
16.What is minimum phase system?
The minimum phase systems are systems with minimum phase transfer functions. In
minimum phase transfer functions, all poles and zeros will lie on the left half of splane.
17.What is nonminimum phase transfer function?
A transfer function which has one or more zeros in the right half of splane is known as nonminimum
phase transfer function.
18.What are M circles?
The magnitude of closed loop transfer function with unit feed back can be shown to be in the
for every value if M.These circles are called M circles.
19.What is Nichols chart?
The chart consisting if M & N loci in the log magnitude versus phase diagram is called
Nichols chart.
20.What are two contours of Nichols chart?
Nichols chart of M and N contours, superimposed on ordinary graph. The M contours are the
magnitude of closed loop system in decibels and the N contours are the phase angle locus of
closed loop system.
21.How is the Resonant Peak, resonant frequency, and bandwidth determined from
Nichols chart?
i) The resonant peak is given by the value of .contour which is tangent to G(jw ) locus.
ii) The resonant frequency is given by the frequency of G(jw ) at the tangency point.
iii) The bandwidth is given by frequency corresponding to the intersection point of G(jw ) and
–3dB Mcontour.
22.What are the advantages of Nichols chart?
The advantages are:
i) It is used to find the closed loop frequency response from open loop frequency response.
ii) Frequency domain specifications can be determined from Nichols chart.
iii) The gain of the system can be adjusted to satisfy the given specification
23.What are N circles?
If the phase of closed loop transfer function with unity feedback is , then tan will be in the
form of circles for every value of . These circles are called N circles.
PART B
1. Plot the Bode diagram for the following transfer function and obtain the gain and phase
cross over frequencies. G(S) = 10/ S(1+0.4S) (1+0.1S) (16)
2. The open loop transfer function of a unity feed back system is G(S) = 1/ S(1+S) (1+2S).
Sketch the Polar plot and determine the Gain margin and Phase margin. (16)
3. Sketch the Bode plot and hence find Gain cross over frequency ,Phase cross over
frequency, Gain margin and Phase margin.
G(S) = 0.75(1+0.2S)/ S(1+0.5S) (1+0.1S) (16)
4. Sketch the Bode plot and hence find Gain cross over frequency, Phase cross over
frequency, Gain margin and Phase margin.
G(S) = 10(S+3)/ S(S+2) (S2+4S+100) (16)
5. Sketch the polar plot for the following transfer function .and find Gain cross over frequency
,Phase cross over frequency, Gain margin and Phase margin.
G(S) = 10(S+2)(S+4)/ S (S2 3S+10) (16)
6. Construct the polar plot for the function GH(S) =2(S+1)/ S2. find Gain cross over frequency
,Phase cross over frequency, Gain margin and Phase margin. (16)
7. Plot the Bode diagram for the following transfer function and obtain the gain and phase
cross over frequencies G(S) =KS2 / (1+0.2S) (1+0.02S).Determine the value of K for a
gain cross over frequency of 20 rad/sec. (16)
8. Sketch the polar plot for the following transfer function .and find Gain cross over
frequency, Phase cross over frequency, Gain margin and Phase margin.
G(S) = 400/ S (S+2)(S+10) (16)
9. A unity feed back system has open loop transfer function G(S) = 20/ S (S+2)(S+5).Using
Nichol’s chart. Determine the closed loop frequency response and estimate all the
frequency domain specifications. (16)
10. Sketch the Bode plot and hence find Gain cross over frequency, Phase cross over
frequency, Gain margin and Phase margin.
G(S) = 10(1+0.1S)/ S(1+0.01S) (1+S). (16)
11.Explain in detail about M and N Circles with necessary equations. (16)
UNIT IV
PART A (2 MARKS)
1.What is Nyquist contour?
The contour that encloses entire right half of S plane is called nyquist contour.
2.State Nyquist stability criterion.
If the Nyquist plot of the open loop transfer function G(s) corresponding to the nyquist control
in the Splane encircles the critical point –1+j0 in the counter clockwise direction as many
times as the number of right half Splane poles of G(s),the closed loop system is stable.
3.Define Relative stability.
Relative stability is the degree of closeness of the system, it is an indication of strength or
degree of stability.
4.What are the two segments of Nyquist contour.
i. An finite line segment C1 along the imaginary axis.
ii. An arc C2 of infinite radius.
5.What are root loci?
The path taken by the roots of the open loop transfer function when the loop gain is varied
from 0 to are called root loci.
6.What is a dominant pole?
The dominant pole is a air of complex conjugate pair which decides the transient response of
the system.
7.What are the main significances of root locus?
i. The main root locus technique is used for stability analysis.
ii. Using root locus technique the range of values of K, for as table system can be
determined.
8. What are the effects of adding a zero to a system?
Adding a zero to a system results in pronounced early peak to system response thereby the
peak overshoot increases appreciably.
9. StateMagnitude criterion.
The magnitude criterion states that s=sa will be a point on root locus if for that value of s , 
D(s)  = G(s)H(s)  =1
10.State – Angle criterion.
The Angle criterion states that s=sa will be a point on root locus for that value of s,D(s) =
G(s)H(s) =odd multiple of 180°
11. What is a dominant pole?
The dominant pole is a pair of complex conjugate pair which decides the transient response
of the system.
12.Define BIBO stability.
A linear relaxed system is said to have BIBIO stability if every bounded input results in a
bounded output.
13.What is the necessary condition for stability.
The necessary condition for stability is that all the coefficients of the characteristic polynomial
be positive.
14.What is the necessary and sufficient condition for stability.
The necessary and sufficient condition for stability is that all of the elements in the first
column of the routh array should be positive.
15.What is quadrantal symmetry?
The symmetry of roots with respect to both real and imaginary axis called quadrantal
symmetry.
16.What is limitedly stable system?
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.
PART B
1. (i) Using Routh criterion determine the stability of the system whose characteristics
equation is S4+8S3+18S2+16S+5 =0. (8)
(ii).F(S) = S6 +S52S43S37S24S4 =0.Find the number of roots falling in the RHS plane
and LHS plane. (8)
2. A unity feedback control system has an open loop transfer function
G(S)= K / S (S2+4S+13).Sketch the root locus. (16)
3. 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 (16)
4. A unity feedback control system has an open loop transfer function
G(S)= K (S+9) / S (S2+4S+11).Sketch the root locus. (16)
5. Sketch the root locus of the system whose open loop transfer function is
G(S)= K / S (S+4) (S2+4S+20). (16)
6. A Unity feedback control system has an open loop transfer function
G(S)= K (S+1.5) / S (S+1)(S+5).Sketch the root locus. (16)
7. Draw the Nyquist plot for the system whose open loop transfer function is
G(S)= K / S (S+2)(S+10).Determine the range of k for which closed loop system is stable.
(16)
8. Sketch the Nyquist Plot for a system with the open loop transfer function
G(S) H(S)= K (1+0.5S)(1+S) / (1+10S)(S1). Determine the range of k for which closed
loop system is stable. (16)
9. (i) Determine the range of K for stability of unity feedback system whose open loop
transfer function is G(s) = K / s (s+1)(s+2) (8)
(ii) The open loop transfer function of a unity feed back system is given by
G(s) = K (s+1) / s3+as2+2s+1. Determine the value of K and a so that the system
oscillates at a frequency of 2 rad/sec. (8)
10.(i) 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. (8)
(ii) Construct Routh array and determine the stability of the system represented by the
characteristics equation S7+9S6+24S4+24S3+24S2+23S+15=0comment on the location
of the roots of characteristic equation. (8)
UNIT  V
PART A (2 MARKS)
1.Define Phase lag and phase lead?
A negative phase angle is called phase lag.
A positive phase angle is called phase lead.
2.What are the two types of compensation?
i. Cascade or series compensation
ii. Feedback compensation or parallel compensation
3.What are the three types of compensators?
i. Lag compensator
ii. Lead compensator
iii. LagLead compensator
4.What are the uses of lead compensator?
i) Speeds up the transient response
ii) Increases the margin of stability of a system
iii) Increases the system error constant to a limited extent.
5.What is the use of lag compensator?
Improve the steady state behavior of a system, while nearly preserving its transient
response.
6.When is lag lead compensator is required?
The lag lead compensator is required when both the transient and steady state response of a
system has to be improved.
7.What is a compensator?
A device inserted into the system for the purpose of satisfying the specifications is called as a
compensator.
8.What is Compensation?
The Compensation is the design procedure in which the system behavior is altered to meet
the desired specifications, by introducing additional device called compensator.
9.Why Compensation is necessary in feedback control system?
In feedback control systems compensation is required in the following situations.
i) When the system is absolutely unstable, then compensation is required to
stabilize the system and also to meet the desired performance.
ii) When the system is stable, compensation is provided to obtain the desired
performance.
10.When lag/lead/laglead compensation is employed?
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 transientstate
performance.
LagLead compensation is employed for stable/unstable system for improvement in both
steadystate and transient state performance.
PART B
1. What is compensation? Why it is need for control system? Explain the types of
compensation? What is an importance of compensation? (16)
2. Realise the basic compensators using electrical network and obtain the transfer
function. (16)
3. Design suitable lead compensators for a system unity feedback and having open loop
transfer function G(S)= K/ S(S+1) to meet the specifications.(i) The phase margin of the
system 45º, (ii) Steady state error for a unit ramp input 1/15, (iii) The gain cross over
frequency of the system must be less than 7.5 rad/sec. (16)
4. A unity feed back system has an open loop transfer function G(S)= K/ S(S+1) (0.2S+1).
Design a suitable phase lag compensators to achieve following specifications Kv= 8
and Phase margin 40 deg with usual notation. (16)
5. Explain the procedure for lead compensation and lag compensation (16)
6. Explain the design procedure for lag lead compensation (16)
7. Consider a type 1 unity feed back system with an OLTF G(S) =K/S (S+1) (S+4). The
system is to be compensated to meet the following specifications Kv > 5sec and PM > 43
deg. Design suitable lag compensators. (16)
8. Using Electrical lead network derive the transfer function. (16)
9. Using Electrical lag network derive the transfer function (16)
10. Using Electrical laglead network derive the transfer function (16)
11. Design a lead compensator for a unity feedback system with open loop transfer function
G(S) = K/ S(S+1) (S+5) to satisfy the following specifications (i) Kv > 50 (ii) Phase Margin
is > 20 . (16)
12. Design a lead compensator for G(S) =K / S2 (0.2S+1) to meet the following
Specifications (i)Acceleration ka=10; (ii) P.M=35. (16)
13. Design a Lag compensator for the unity feedback system whose closed loop transfer
function C(s) / R(s) = K / (s (s+4) (s+80) + K) is to meet the following specifications
P.M 33 º. And Kv 30. (16)
14. A unity feedback system has an OLTF G(s) = K / s(s+2)(s+60). Design a LeadLag
compensator is to meet the following specifications.
(i) P.M is atleast 40º , (ii) Steady state error for ramp input 0.04 rad. (16)
