# Control Systems Questions and Answers – Relative Stability Analysis

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This set of Control Systems Multiple Choice Questions & Answers (MCQs) focuses on “Relative Stability Analysis”.

1. A system with unity feedback having open loop transfer function as G(s) = K(s+1)/s3+as2+2s+1. What values of ‘K’ and ’a’ should be chosen so that the system oscillates ?
a) K =2, a =1
b) K =2, a =0.75
c) K =4, a =1
d) K =4, a =0.75

Explanation: Solving Routh Hurwitz table whenever row of zero occurs, the roots are located symmetrically on the imaginary axis then the system response oscillates, a =1+K/2+K. If K =2 is consider then a =0.75.

2. The open loop transfer functions with unity feedback are given below for different systems.
Among these systems the unstable system is
a) G(s) =2/s+2
b) G(s) =2/s(s+2)
c) G(s) =2/(s+2)s^2
d) G(s) =2(s+1)/s(s+2)

Explanation: 1+2/s^2(s+2) =0. The coefficient of‘s’ is missing. Hence the system is unstable.

3. Determine the stability of closed loop control system whose characteristic equation is
s5+s4+2s3+2s2+11s+10=0.
a) Stable
b) Marginally stable
c) Unstable
d) None of the mentioned

Explanation: By Routh array s =0 and s =+j. It is having a pair of conjugate root lying on imaginary axis. System is marginally stable.
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4. Determine the condition for the stability of unity feedback control system whose open loop transfer function is given by
G(s) = 2e-st/s(s+2)
a) T >1
b) T <0
c) T <1
d) T >0

Explanation: G(s) =2(1-sT)/s(s+2)
By Routh array analysis, for stable system, all the elements of first column need to be positive T<1.

5.Determine the value of K such that roots of characteristic equation given below lies to the left of the line s = -1. s3+10s2+18s+K.
a) K>16 and K<9
b) K<16
c) 9<K<16
d) K<9

Explanation: In Routh array analysis the first column must be positive and after solving K<16 and K>9.

6. Consider a negative feedback system where G(s) =1/(s+1) and H(s) =K/s(s+2). The closed loop system is stable for
a) K>6
b) 0<K<2
c) 8<K<14
d) 0<K<6

Explanation: Using Routh array, for stability k < 6 and k > 0.

7. The characteristic equation of a feedback control system is s3+Ks2+9s+18. When the system is marginally stable, the frequency of the sustained oscillation:
a) 1
b) 1.414
c) 1.732
d) 3

Explanation: Solve using Routh array and for the system to be marginally stable, K = -2. Polynomial for sustained oscillation w = 3 rad/s.

8. Consider a characteristic equation, s4+3s3+5s2+6s+k+10=0. The condition for stability is
a) K>5
b) -10<K
c) K>-4
d) -10<K<-4

Explanation: Solve Roth array for the system stable, -10<K<-4.

9. The polynomial s4+Ks3+s2+s+1=0 the range of K for stability is _____________
a) K>5
b) -10<K
c) K>-4
d) K-1>0

Explanation: Solving using Routh array we get K-1>0 and is always negative for K>1.

10. The characteristic equation of a system is given by3s4+10s3+5s2+2=0. This system is:
a) Stable
b) Marginally stable
c) Unstable
d) Linear

Explanation: There is missing coefficient so system is unstable.

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