A plant with a transfer function $\frac{2}{s (s + 3)}$ is controlled by a PI controller with $K_{p} = 1$ and $k_{i} \geq 0$ in a unity feedback configuaration.The lowest value of $K_{i}$ that ensures zero steady state error for a step change in the referance input is

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Solution

The correct option is A 0
Given, $K_{p} = 1$ , transfer function for PI controller is
$T (s) = K_{p} (1 + \frac{1}{K_{i} s}) = \frac{K_{i} s + 1}{K_{i} s}$
Open loop transfer function
$G (s) = \frac{K_{i} s + 1}{K_{i} s} \times \frac{2}{s (s + 3)} = \frac{2 (K_{i} s + 1)}{K_{i} s^{2} (s + 3)}$
therefore characteristic equation is
$1 + G (s) = 0$
$K_{i} s^{3} + K_{i} s^{2} + 2 K_{i} s + 2 = 0$
Routh array will be

for system to be stable $K_{i} > 0$
and 2 $K_{i} - 2 / 3 > 0$
$\Rightarrow K_{i} > 0 a n d K_{i} > 1 / 3$
$\Rightarrow K_{i} > 1 / 3$

Now $K_{p} \to$ position error constant $= {lim}_{s \to 0} G (s) = \infty$
$∴ e_{s s} = \frac{1}{1 + K_{p}} = 0$
So it is independent of $K_{i}$
therefore lowest value is $K_{i} = 0$

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A plant with a transfer function 2s(s+3) is controlled by a PI controller with Kp=1 and ki≥0 in a unity feedback configuaration.The lowest value of Ki that ensures zero steady state error for a step change in the referance input is

A plant with a transfer function $\frac{2}{s (s + 3)}$ is controlled by a PI controller with $K_{p} = 1$ and $k_{i} \geq 0$ in a unity feedback configuaration.The lowest value of $K_{i}$ that ensures zero steady state error for a step change in the referance input is