Question

Cosider a unity negative feedbackk system, with open-loop transfer function (a is positive constant).
If the system is stable for K > -1, then which of the following is correct ?

$G\left(s\right)=\frac{K(s^2+1)}{(s+1)(s+a)}$

• A. a < 2
• B. 0 < a < 1
• C. a > 1
• D. a > 0

Answer 1

The correct option is C a > 1

Since the system is unity feedback H(s) = 1
Characteristice equation : 1 + G(s)H(s) = 0

$1+\frac{K(s^2+1)}{(s+1)(s+a)}=0$
$s^2+(a+1)s+a+K{s}^2+K=0$
$(1+K)s^2+(a+1)s+(k+a)=0$
using Routh's tabular form,

$\begin{array}{ccc}{s}^2& (1+K)& (a+K)\\ s^1& (a+1)& \\ s^0& (a+K)& \end{array}$

For stability of the system
$(1+K)>0\Rightarrow K>-1$
$a+1>0\Rightarrow a>(-1)$
$a+K>0\Rightarrow a>(-K)\Rightarrow a>1\because K>(-1)$

So, a > 1 is the correct answer.