186. A system with the open loop transfer function G(s) =	K
     	s(s + 2)(s2 + 2s + 2)

     
     is connected in a negative feedback configuration with a feedback gain of unity. For the closed loop system to be marginally stable, the value of K is ____

1. 1. 2

   
   
   

   2. 3

   
   
   

   3. 4

   
   
   

   4. 5

   
   
   

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1. View Hint View Answer Discuss in Forum

   Characteristic equation is, 1 + G(s)H(s) = 0
   ⇒ 1 + G(s) = 0 (∴ H(s) = 1)
   

   ⇒ 1 +	k	= 0
   	s(s + 2)(s2 + 2s + 2)

   
   ⇒ s⁴ + 4s³ + 6s² + 4s + k = 0
   Constructing Routh- array, we have
   
   For the closed loop system to be marginally stable, 20 – 4k = 0
   ⇒ k = 5

   Correct Option: D

   Characteristic equation is, 1 + G(s)H(s) = 0
   ⇒ 1 + G(s) = 0 (∴ H(s) = 1)
   

   ⇒ 1 +	k	= 0
   	s(s + 2)(s2 + 2s + 2)

   
   ⇒ s⁴ + 4s³ + 6s² + 4s + k = 0
   Constructing Routh- array, we have
   
   For the closed loop system to be marginally stable, 20 – 4k = 0
   ⇒ k = 5

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187. The system ẋ = Ax + Bu with A =		-1			2		, B =		0		is
     		0			2				1

     
     

1. 1. stable and controllable

   
   
   

   2. stable but uncontrollable

   
   
   

   3. unstable but controllable

   
   
   

   4. unstable and uncontrollable

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given : ẋ = Ax + Bu ;
   

   A =		-1			2		, B =		0
   		0			2				1

   
   For stability, characteristic equation
   
   ∴ M = (s + 1) × (s – 2) = 0
   ⇒ (s – 2) (s + 1) = 0
   ⇒ s = 2, s = – 1
   So one root in right hand side of s-plane, so system is unstable.
   

   For controllability : B =		0
   		1

   
   

   AB =		-1			2				0		=		2
   		0			2				1				2

   
   
   = 0 – 2 = – 2 ≠ 0
   So controllable.
   

   Correct Option: C

   Given : ẋ = Ax + Bu ;
   

   A =		-1			2		, B =		0
   		0			2				1

   
   For stability, characteristic equation
   
   ∴ M = (s + 1) × (s – 2) = 0
   ⇒ (s – 2) (s + 1) = 0
   ⇒ s = 2, s = – 1
   So one root in right hand side of s-plane, so system is unstable.
   

   For controllability : B =		0
   		1

   
   

   AB =		-1			2				0		=		2
   		0			2				1				2

   
   
   = 0 – 2 = – 2 ≠ 0
   So controllable.
   

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188. The characteristic equation of a closed-loop system is s(s + 1)(s + 3) + k(s + 2) = 0, k > 0. Which of the following statements is true?

1. 1. Its roots are always real

   
   
   

   2. It cannot have a breakaway point in the range – 1 < Re[s] < 0

   
   
   

   3. Two of its roots tend to infinity along the asymptotes Re[s] = – 1

   
   
   

   4. It may have complex roots in the right half plane

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given : s(s + 1) (s + 3) + k(s + 2) = 0; k > 0
   

   ⇒ 1 +	k(s + 2)	= 0
   	s(s + 1)(s + 3)

   
   But 1 + G(s) H(s) = 0
   

   ∴ G(s) H(s ) =	k(s + 2)
   	s(s + 1)(s + 3)

   
   Roots s = 0, s = – 1, s = – 3 (poles); s = – 2 (zero)
   It has real pole or zeros.
   

   φA =	2q + 1	× 180°
   	n - m

   
   
   At s = 0, Asymptotes
   

   φ0 =	(2*0 + 1)180	= 90°
   	2

   
   

   At s = 1, φ1 =	(2*1 + 1)180	= 270°
   	2

   
   Centroid,
   

   (-σA) =	∑ real parts of poles - ∑ real poles of zeros
   	number of poles - number of zeros

   
   ∴ Re[s] = – 1

   Correct Option: C

   Given : s(s + 1) (s + 3) + k(s + 2) = 0; k > 0
   

   ⇒ 1 +	k(s + 2)	= 0
   	s(s + 1)(s + 3)

   
   But 1 + G(s) H(s) = 0
   

   ∴ G(s) H(s ) =	k(s + 2)
   	s(s + 1)(s + 3)

   
   Roots s = 0, s = – 1, s = – 3 (poles); s = – 2 (zero)
   It has real pole or zeros.
   

   φA =	2q + 1	× 180°
   	n - m

   
   
   At s = 0, Asymptotes
   

   φ0 =	(2*0 + 1)180	= 90°
   	2

   
   

   At s = 1, φ1 =	(2*1 + 1)180	= 270°
   	2

   
   Centroid,
   

   (-σA) =	∑ real parts of poles - ∑ real poles of zeros
   	number of poles - number of zeros

   
   ∴ Re[s] = – 1

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189. The measurement system shown in the figure uses three sub-systems in cascade whose gains are
     

     specified as G1,G2 and	1
     	G3

     
     The relative small errors associated with each respective subsystem G₁ , G₂ and G₃ are ε₁ , ε₂ and ε₃. The error associated with the output is
     

1. 1. ε1 + ε2 +	1
      	ε3

   
   
   

   2. 	ε1 . ε2
      	ε3

   
   
   

   3. ε₁ + ε₂ - ε₃

   
   
   

   4. ε₁ + ε₂ + ε₃
      

   
   
   

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1. View Hint View Answer Discuss in Forum

   The relative error of product or division of different quantities is equal to the sum of relative errors of individual quantities.

   Correct Option: D

   The relative error of product or division of different quantities is equal to the sum of relative errors of individual quantities.

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190. The polar plot of an open loop stable system is shown below. The closed loop system is
     
     

1. 1. always stable

   
   
   

   2. marginally stable

   
   
   

   3. unstable with one pole on the RH s-plane

   
   
   

   4. unstable with two poles on the RH s-plane
      

   
   
   

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1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: D

   NA

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