Gain Reduction for Overshoot Specification in a Unity-Feedback System

We study a unity-feedback control system with open-loop transfer function

$$G(s)=\frac{k}{s(1+sT)}, \quad k>0,\; T>0.$$

For a unit-step input, the peak (percent) overshoot depends on the closed-loop damping ratio of the resulting second-order dynamics. In this design problem, we will enforce that the overshoot decreases from $75\%$ to $25\%$ by scaling the amplifier gain $k$ accordingly.

Key relations we will use:

• unity feedback implies the closed-loop characteristic equation is $1+G(s)=0$.
• percent overshoot for underdamped second-order systems is $$\mathrm{PO}=100\,e^{-\frac{\zeta\pi}{\sqrt{1-\zeta^2}}}$$ where $\zeta$ is the damping ratio.
• The given plant/feedback structure yields a standard second-order denominator, from which $\zeta$ can be expressed in terms of $k$ and $T$.

Mermaid overview of the control loop:

Footnotes

1. Percent overshoot and damping ratio relationship (including inverse formula) - Wikipedia “Damping” https://en.wikipedia.org/wiki/Damping - Provides PO = 100 exp(−ζπ/√(1−ζ^2)) and ζ inversion formula. ↩

Step 1: Form the unity-feedback closed-loop transfer function

Unity feedback gives the characteristic equation:

$$1+G(s)=0 \;\Rightarrow\; 1+\frac{k}{s(1+sT)}=0.$$

Multiply by $s(1+sT)$:

$$s(1+sT)+k=0.$$

Expand:

$$s + Ts^2 + k = 0 \;\Rightarrow\; Ts^2 + s + k = 0.$$

Thus the closed-loop transfer function has a denominator proportional to $Ts^2+s+k$.

To match the standard second-order form

$$s^2 + 2\zeta\omega_n s + \omega_n^2,$$

divide by $T$:

$$s^2 + \frac{1}{T}s + \frac{k}{T}=0.$$

Now compare coefficients:

• $2\zeta\omega_n = \frac{1}{T}$
• $\omega_n^2 = \frac{k}{T}$

From $\omega_n^2=\frac{k}{T}$, we get $\omega_n=\sqrt{\frac{k}{T}}$. Then

$$2\zeta\sqrt{\frac{k}{T}}=\frac{1}{T} \;\Rightarrow\; 2\zeta\sqrt{kT}=1 \;\Rightarrow\; \zeta=\frac{1}{2\sqrt{kT}}.$$

Key takeaway:

• damping ratio $\zeta$ is inversely proportional to $\sqrt{k}$: $$\zeta \propto \frac{1}{\sqrt{k}}.$$

This relation lets us compute the ratio $k_2/k_1$ once we find $\zeta_1$ and $\zeta_2$ from the overshoot specs.

Step 2: Relate gain ratio to damping ratio

We derived:

$$\zeta=\frac{1}{2\sqrt{kT}}.$$

With $T$ fixed, this implies:

$$\zeta \propto \frac{1}{\sqrt{k}} \quad\Rightarrow\quad \frac{\zeta_2}{\zeta_1}=\sqrt{\frac{k_1}{k_2}}.$$

Therefore,

$$\frac{k_2}{k_1}=\left(\frac{\zeta_1}{\zeta_2}\right)^2.$$

Insert the computed values:

$$\frac{k_2}{k_1}= \left(\frac{0.215}{0.403}\right)^2 \approx (0.533)^2 \approx 0.284.$$

So the amplifier gain should be reduced to about $0.284$ of its original value, i.e., reduced by a factor:

$$\boxed{0.284 \text{ (approximately)}}$$

Equivalently, it is reduced by about:

$$\boxed{1-0.284 \approx 71.6\%}$$