**My answer for (i)**

Given that C =35 in this question.

G(s) = θ(s) / F(s) = 1 / (s^2 - α )

α = (10 + C) / 1000 = 45 / 1000 = 9 / 200

System pole can be found at s

Thus , it tells me that it is a real and district roots and when input signal is applied , output signal from the space booster will be a exponentially growing transient response.

ii. Sketch the impulse response, θ(t).ii. Sketch the impulse response, θ(t).

**My answer for (ii)**

**Question 2**

**My answer for (i)**

I let A = kp + kd s + ki / s and B = 1 / (s^2 - α )

To obtain the closed-loop transfer function with respect to a specific point , we assume that all other inputs are zero. Thus , for θ(s) / θr(s) , we take D(s) = 0 and θ(s)/D(s) , we take θr(s) = 0

Hence, Gr(s) = (AB) / (1 + AB) and Gd(s) = B / (1-AB)

Gr(s) = (kp + kd s + ki / s ) / (s^2 - α + kp + kd s + ki / s )

Gd(s) = 1 / (s^2 - α - kp - kd s - ki / s )

Is my answer correct?