# Steady error control system sinusoidal disturbance

Consider the control system closed-loop represented such a block diagram below:

Given that: $$\G_C(s) = K\$$ and $$\G(s) = 1/s\$$

a) Determine the values of $$\ K\$$ which the closed-system loop is stable.

b) Suppose that the disturbance $$\ d(s)\$$ be sinusoidal with amplitude A and frequency $$\ w\$$. The exactly value of $$\ w\$$ is unknown, but know itself $$\ 0 \leq w \leq 10 \$$ rad/s

Is it possible to choose a value of $$\ K\$$ such that, in steady state, the amplitude of the output value is less than or equal to 1% of the value A?

If your answer is "YES", so compute the value of $$\ K\$$ that guarantees this attenuation.

If your answer is "NO", so show that there is no value of $$\ K\$$ that guarantees this attenuation.

However my question is very similar, i have some doubts about the letter b)

$$\ \dfrac{y(s)}{d(s)}=\dfrac{1}{1+KG(s)}\Rightarrow \bigg|\dfrac{y(s)}{d(s)}\bigg|_{s=j\omega}=0.1\$$

$$\\frac{1}{|1+K\frac{1}{j10}|} = 0.1\Rightarrow 10 = \sqrt{1+\frac{K^2}{100}}\Rightarrow 100 = 1+\frac{K^2}{100}\implies \,\,\,\,\boxed{K = 10\sqrt{99}}\$$

Is this correct?

• What's the signal entering the G(s) block? – Chu Jun 5 '19 at 16:50
• @Chu $G(s) = \frac{1}{s}$ – miguel747 Jun 5 '19 at 17:12
• No, what’s the signal at the output of the 2nd summing junction and the input to G(s)? – Chu Jun 5 '19 at 20:38

$$y(s)/d(s) = G(s)/(1+KG(s))$$
• Huge details about the position of disturbance (thank you) and I realize that the value of $|y(s)/d(s)| = 0.01A$ in terms of amplitude $A$ but it is another detail. – miguel747 Jun 5 '19 at 17:37