An aircraft is in level flight at airspeed v(t) m/s with thrustT(t) N at cruising altitude. Suppose that at v0 = 250 m/s, theaerodynamic drag experienced by the aircraft at this altitudeis:

Fd(v) = 0.25v 2 . (1)

Then, an extremely simplified model relating v(t) to T(t)is:

mvË™(t) + Fd(v(t)) = T(t), (2)

where m = 25000 kg. Assume v(t) is always positive.

Question 1. Linearize (2) at v0 = 250 m/s, and an appropriatenominal thrust T0. That is, create a new linear system model ˙δv +aδv = bδT that is accurate for small perturbations δv(t) = v(t) −v0, δT = T(t) − T0.

Question 2. Suppose we want the aircraft to fly at v0 = 250 m/s.However, precise modelling of drag is very difficult. Suppose inreality the dynamics are slightly different:

mvË™(t) + 0.23v(t) 2 = T(t). (3)

To control v(t) to be close to the reference 250m/s, you tryusing a proportional controller to control the thrust:

T(t) = T0 + k(v0 − v(t)), (4) with k = 100.

Here T0 is the value calculated in Question 1, but thecontroller is applied to the true dynamics (3). Find the finalfinal value of v in: (a) open loop (i.e. just apply T(t) = T0), and(b) closed loop (using (4)). You may need to solve a quadraticequation. Recall that v(t) is assumed to be positive. Which one isbetter?