Question

In a closed loop system, the process to be controlled is an integrating process with transfer function $10/s$. the controller proposed to be used is an integral controller with transfer function $1/{\tau }_{i}s$. When a step change is applied in set point to such a closed loop system. The output of the closed loop system will exhibit

• A. Overdamped response
• B. unstable response
• C. Underdamped response
• D. undamped response

Answer 1

The correct option is D undamped response

$G\left(s\right)=\frac{10}{s}:G_C\left(s\right)=\frac{1}{{\tau }_{i}s}$

Closed loop Transfer function = $\frac{G_C\left(s\right)G\left(s\right)}{1+G_C\left(s\right)G\left(s\right)}=\frac{\frac{10}{s}\times \frac{1}{{\tau }_{i}s}}{1+\frac{1}{{\tau }_{i}s}\times \frac{10}{s}}$

$=\frac{10/{\tau }_i}{s^2+10/{\tau }_i}$

the root are $s^2+10/{\tau }_i=0$

$S=\pm j\sqrt{\frac{10}{{\tau }_i}}(\therefore \text{roots are purely imaginary}$