**Relative Stability Analysis MCQ**

**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

**Answer:** b

**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)

**Answer:** c

**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 s ^{5}+s^{4}+2s^{3}+2s^{2}+11s+10=0.**

a) Stable

b) Marginally stable

c) Unstable

d) None of the mentioned

**Answer:** b

**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.

**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

**Answer:** c

**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. s ^{3}+10s^{2}+18s+K.**

a) K>16 and K<9

b) K<16

c) 9<K<16

d) K<9

**Answer:** c

**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

**Answer:** d

**Explanation:** Using Routh array, for stability k<6.

**7. The characteristic equation of a feedback control system is s ^{3}+Ks^{2}+9s+18. When the system is marginally stable, the frequency of the sustained oscillation:**

a) 1

b) 1.414

c) 1.732

d) 3

**Answer:** d

**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, s ^{4}+3s^{3}+5s^{2}+6s+k+10=0. The condition for stability is**

a) K>5

b) -10<K

c) K>-4

d) -10<K<-4

**Answer:** d

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

**9. The polynomial s ^{4}+Ks^{3}+s^{2}+s+1=0 the range of K for stability is _____________**

a) K>5

b) -10<K

c) K>-4

d) K-1>0

**Answer:** d

**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 by3s ^{4}+10s^{3}+5s^{2}+2=0. This system is:**

a) Stable

b) Marginally stable

c) Unstable

d) Linear

**Answer:** c

**Explanation:** There is missing coefficient so system is unstable.