Question

For the control system shown in the Figure, the transfer function of a plant,
\[ G(s)=\frac{1}{(s+1)(s+2)} \]
is connected in cascade with a compensator
\[ C(s)=K(s+\alpha), \]
where \(K\) and \(\alpha\) are positive real valued constants. The compensator and plant are placed in the forward path of a unity negative feedback system with input \(R(s)\) and output \(Y(s)\).
Which of the following pairs \((K,\alpha)\) represent the correct values for the closed loop system to have poles at \(\left(-3\pm j\sqrt{5}\right)\)?

• A. \(2, 3\)
• B. \(3, 4\)
• C. \(2, 4\)
• D. \(3, 3\)

Hint:

Form the characteristic equation (s+1)(s+2)+K(s+alpha)=0 and match its sum and product of roots to the given complex pole pair.

Answer 1

The Correct Option is B

Step 1: Write the open loop transfer function.
The forward path has the compensator in cascade with the plant, so the open loop transfer function is
\[ L(s)=C(s)G(s)=\frac{K(s+\alpha)}{(s+1)(s+2)} \]

Step 2: Write the closed loop characteristic equation.
For a unity negative feedback system, the characteristic equation is \(1+L(s)=0\), that is
\[ (s+1)(s+2)+K(s+\alpha)=0 \]

Step 3: Expand the equation.
\[ (s+1)(s+2)=s^2+3s+2 \]
So the characteristic equation becomes
\[ s^2+3s+2+Ks+K\alpha=0 \] \[ s^2+(3+K)s+(2+K\alpha)=0 \]

Step 4: Use the desired pole locations to get sum and product of roots.
The desired closed loop poles are \(-3+j\sqrt5\) and \(-3-j\sqrt5\).
Sum of roots:
\[ (-3+j\sqrt5)+(-3-j\sqrt5)=-6 \]
Product of roots:
\[ (-3+j\sqrt5)(-3-j\sqrt5)=(-3)^2-(j\sqrt5)^2=9-(-5)=9+5=14 \]

Step 5: Match with the characteristic equation coefficients.
For a quadratic \(s^2+bs+c=0\) with these roots, sum of roots \(=-b\) and product of roots \(=c\). Comparing with \(s^2+(3+K)s+(2+K\alpha)=0\):
\[ -(3+K)=-6\quad\Rightarrow\quad 3+K=6\quad\Rightarrow\quad K=3 \]
\[ 2+K\alpha=14\quad\Rightarrow\quad K\alpha=12 \]

Step 6: Solve for alpha.
\[ \alpha=\frac{12}{K}=\frac{12}{3}=4 \]

Step 7: Analyze the options.

(A) 2, 3: Gives \(K=2\), which does not satisfy \(3+K=6\). Incorrect.

(B) 3, 4: Matches both \(K=3\) and \(\alpha=4\) found above. Correct.

(C) 2, 4: The value \(K=2\) fails the sum condition. Incorrect.

(D) 3, 3: \(K=3\) is right, but \(\alpha=3\) does not satisfy \(K\alpha=12\), since \(3\times3=9\neq12\). Incorrect.

Step 8: Final conclusion.
The correct pair is \[ \boxed{(K,\alpha)=(3,4)} \]