### GATE 2005 ECE Control Systems - Complete Video Solutions with Answers

GATE 2005

1. A linear system is equivalently represented by two sets of state equations
Ẋ = AX + BU   and   Ẇ = CW + DU.
The eigen values of the representations are also computed as |λ| and |μ|. Which one of the following statements is TRUE?

Solution :

2. Which one of the following polar diagrams corresponds to a lag network?

3. Despite the percentage of negative feedback, control systems still have problems of Instability because the
a) components used have non-linearity
b) dynamic equations of the sub-systems are not know exactly
c) mathematical analysis involves approximations
d) system has large negative phase angle at high frequencies

4. The polar diagram of a conditionally stable system for open loop gain K = 1 is shown in figure. The open loop transfer function of the system is known to be stable. The closed loop system is stable for

5. In the derivation of expression for peak percent overshoot, which one of the following conditions is NOT required?
a) system is linear and time invariant
b) system transfer function has a pair of complex conjugate poles and no zeros
c) there is no transportation delay in the system
d) the system has zero initial conditions

6. A ramp input applied to an unity feedback system results in 5% steady state error. The Type number and zero frequency gain of the system are respectively
a) 1 and 20
b) 0 and 20
c) 0 and 1/20
d) 1 and 1/20

7. A double integrator plant, G(s) = K/S2, H(s) = 1 is to be compensated to achieve the damping ratio ξ = 0.5, and an undamped natural frequency ωn = 5 rad/sec. Which one of the following compensator Gc(s) will be suitable?

8. A unity feedback system is given as G(s) = k(1 – s) / [s(s + 3)]. Indicate the correct root locus diagram.

The open loop transfer function of a unity feedback is given by G(s) = 3e–2s / [s(s + 2)].
9. The Gain and Phase cross over frequencies in rad/sec are, respectively
a) 0.932 and 1.26
b) 0.632 and 0.485
c) 0.485 and 0.632
d) 1.26 and 0.632

10. based on the above results, the Gain and Phase Margins of the system will be
a) –7.09 dB and 87.5o
b) 7.09 dB and 87.5o
c) 7.09 dB and –87.5o
d) –7.09 dB and –87.5o

### GATE 2003 ECE Control Systems - Complete Video Solutions with Answers

GATE 2003

1. The Laplace transform of i(t) is given by I(s) = 2 / [s(1 + s)]. As t → ∞, the value of i(t) tends to
a) 0
b) 1
c) 2
d) ∞

2. Figure shows the Nyquist plot of the open loop transfer function G(s)H(s) of a system. If G(s)H(s) has one right hand pole, the closed loop system is

a) always stable
b) unstable with one closed loop right hand pole
c) unstable with two closed loop right hand poles
d) unstable with three closed loop right hand poles

3. A PD controller is used to compensate a system. Compared to the uncompensated system, the compensated system has
a) a higher type number
b) reduced damping
c) higher noise amplification
d) larger transient overshoot

4. The driving point impedance Z(s) of a network has the pole-zero locations as shown in figure. If Z(0) = 3, then Z(s) is

5. The signal flow graph of a system is shown in figure. The transfer function C(s)/R(s) of the system is

6. The root locus of the system G(s)H(s) = K / [s(s + 2)(s + 3)] has the break away point located at
a) (-0.5, 0)
b) (-2.548, 0)
c) (-4, 0)
d) (-0.748, 0)

7. The approximate Bode magnitude plot of a minimum phase system is shown in figure. The transfer function of the system is

8. A second order system has the transfer function C(s)/R(s) = 4 / [s2 + 4s + 4] with r(t) as the unit step function, the response c(t) of the system is represented by

9. The gain margin and the phase margin of a feedback system with G(s)H(s) = s/(s + 100)3 are
a) 0 dB, 0o
b) ∞, ∞
c) ∞, 0o
d) 88.5 dB, ∞

10. The zero input response of a system given by the state space equation

### GATE 2004 ECE Control Systems - Complete Video Solutions with Answers

GATE 2004

1. The gain margin for the system with open loop transfer function G(s)H(s) = 2(1 + s)/s2, is
a) ∞
b) 0
c) 1
d) - ∞

2. Given G(s)H(s) = K /[s(s + 1)(s + 3)], the point of intersection of the asymptotes of the root loci with the real axis is
a) - 4
b) 1.33
c) - 1.33
d) 4

3. Consider the Bode magnitude plot shown in figure. The transfer function H(s) is

4. The transfer function H(s) of an RLC circuit is given as 106/[s2+20s+106], then the Q – factor of this circuit is
a) 25
b) 50
c) 100
d) 5000

5. For the circuit shown in figure, the initial conditions are zero. Its transfer function Vo(t)/Vi(t) is

6. A system has poles at 0.01 Hz, 1 Hz and 80 Hz; zeros at 5 Hz, 100 Hz and 200 Hz. The approximate phase of the system response at 20 Hz is
a) - 90o
b) 0o
c) 90o
d) - 180o

7. Consider the signal flow graph shown in figure. The gain X5/X1 is

8. The open loop transfer function of a unity feedback system is G(s) = K/[s(s2 + s + 2)(s + 3)]. The range of K for which the system is stable is
a) 21/44 > K > 0
b) 13 > K > 0
c) 21/4 < K < ∞
d) - 6 < K < ∞

9. for the polynomial R(s) = s5 + s4 + 2s3 + 2s2 + 3s + 15, the number of roots which lie in the Right Half of the S-plane is
a) 4
b) 2
c) 3
d) 1

10. The state variable equations of a system are:
1 = -3x1 -x2 + u
2 = 2x1
y = x1 + u
Then the system is
a) controllable but not observable
b) observable but not controllable
c) neither controllable nor observable
d) controllable and observable

11. The state transition matrix eAt for the given matrix A, is

### GATE 2006 ECE Control Systems - Complete Video Solutions with Answers

GATE 2006

1. The open loop transfer function of a unity gain feedback control system is given by G(s) = K/[(s + 1)(s + 2)]. The gain margin of the system in dB is given by
a) 0
b) 1
c) 20
d) ∞

2. In the system shown below, x(t) = (sint).u(t). In steady state, the response y(t) will be

3. Consider two transfer functions G1(s) = 1 / (s2 + as + b) and G2(s) = s / (s2 + as + b). The 3 dB bandwidths of their frequency responses are, respectively

Solution :

4. The unit step response of a system starting from rest is given by c(t) = 1 – e2t for t ≥ 0. The transfer function of the system is
a) 1 / (1 + 2s)
b) 2 / (2 + s)
c) 1 / (2 + s)
d) 2s / (1 + 2s)

5. The Nyquist plot of G(jω)H(jω) for a closed loop control system, passes through ( –1, j0) point on the GH – plane. The gain margin of the system in dB is equal to
a) infinite
b) greater than zero
c) less than zero
d) zero

6. The positive values of "K" and "a" so that the system shown in the figure below oscillates at a frequency of 2 rad/sec respectively are

a) 1, 0.75
b) 2, 0.75
c) 1, 1
d) 2, 2

7. The unit impulse response of a system is h(t) = et, t ≥ 0. For this system, the steady state value of the output for unit step input is equal to
a) –1
b) 0
c) 1
d) ∞

8. The transfer function of a phase lead compensator is given by Gc(s) = (1 + 3Ts)/(1 + Ts) where T ≥ 0. The maximum phase shift provided by such a compensator is
a) π/2
b) π/3
c) π/4
d) π/6

9. A linear system is described by the following state equation, then the state transition matrix of the system is

Consider a unity gain feedback control system whose open loop transfer function is G(s) = (as + 1)/s2.
10. The value of "a" so that the system has a phase margin equal to π/4 is approximately equal to
a) 2.40
b) 1.40
c) 0.84
d) 0.74

11. With the value of "a" set for phase margin of π/4, the value of unit impulse response of the open loop system at t = 1 second is equal to
a) 3.40
b) 2.40
c) 1.84
d) 1.74

### GATE 2007 ECE Control Systems - Complete Video Solutions with Answers

GATE 2007

1. If the closed loop transfer function of a control system is given as T(s) = (s – 5)/[(s + 2)(s + 3)], then it is
a) an unstable system
b) an uncontrollable system
c) a minimum phase system
d) a non-minimum phase system

2. If the Laplace Transform of a signal y(t) is Y(s) = 1/[s(s – 1)], then its final value is:
a) – 1
b) 0
c) 1
d) unbounded

3. A control system with a PD controller is shown in the figure.
If the velocity error constant Kv = 1000 and the damping ratio ξ = 0.5, then the values of Kp and KD are
a) Kp = 100, KD = 0.09
b) Kp = 100, KD = 0.9
c) Kp = 10, KD = 0.09
d) Kp = 10, KD = 0.9

4. The transfer function of a plant is T(s) = 5/[(s + 5)(s2 + s + 1)]. The second order approximation of T(s) using dominant pole concept is

5. The open loop transfer function of a plant is given as G(s) = 1/(s2 – 1). If the plant is operated in a unity feedback configuration, then the lead compensator that can stabilize this control system is

6. A unity feedback control system has an open loop transfer function G(s) = – K/[s(s2 + 7s + 12)]. The gain k for which s = –1+j1 will lie on the root locus of this system is
a) 4
b) 5.5
c) 6.5
d) 10

7. The asymptotic Bode plot of a transfer function is as shown in figure. The transfer function G(s) corresponding to this Bode plot is

8. The state space representation of a separately excited DC servo motor dynamics is given as
where ω is the speed of the motor, ia is the armature current and u is the armature voltage. The transfer function of the motor is

9. The eigen value and eigen vector pairs (λi, vi) for the system are

Solution :

10. The system matrix A is