Assignment briefing

Problem 1.  Consider the optimal landing of a rocket booster con-

sidered in Lecture 1. In Problem Set 1, we saw that, if the rocket is

oriented vertically, the model becomes

r˙ = −V

V˙ = − (T + D(r,V)) + g(r)

0

where

❼ r is the radial distance from the Earth’s centre;

❼ V is the speed;

❼ m is the rocket mass;

❼ T ∈ [0,TM ] is the thrust magnitude (control);

❼ D(r,V) is the drag force (a function of r and V);

❼ g(r) is the gravitational acceleration at r;

❼ g0  is the gravitational acceleration at R⊕ (Earth radius);

❼ c is a constant that depends on the engine type.

Assume that the booster’s speed is 1100 km/h when its altitude (the distance to the surface of Earth) is 5 km. The mass of an empty booster is 25,600 kg and it has 2,500 kg of fuel.

a. Show that the optimal control for the minimum-time problem is “bang-bang” . Write down the costate equations and boundary conditions.                                                           [10 marks]

b. Show that the optimal control for the minimum-fuel problem is “bang-bang” . Write down the costate equations and boundary conditions.                                                           [10 marks]

c. The last two equations describing the booster dynamics imply

V˙ =   −  + g(r) =   [lnm(t)] −  + g(r).

Assuming that D(r,V) < mg(r) (the gravity is stronger than the drag), show that the con- sumed fuel is a monotone increasing function of the final time.  (Hint: You need to integrate both sides from 0 to tf  and use V (0) = 0.) What does this observation allow us to conclude about the relation between the solutions of the minimum-time and minimum-fuel problems?

[15 marks]

d. The calculations above suggest that the optimal minimum-time control has the form

T∗ (t) =

t ∈ [0,ts],

t ∈ [ts ,tf ].

Write a MATLAB program that solves the state equations for given ts  and tf . Use this program to find the optimal values of ts  and tf . Plot the optimal state.

When solving the state equations, assume that the drag force is given by

D(r,V) = CD ρ(r)SV2 ,

where

❼ CD  = 0.3 is the drag coefficient;

❼ ρ(r) ≡ 1.225 kg/m3  is the air density (for simplicity we take it constant); ❼ S = 10.75 m2  is the reference area.

The gravitational acceleration is given by

g(r) = g0 R 

where

❼ g0  = 9.8 m/s2  is the acceleration on Earth;

❼ R⊕ = 6,371 km is the Earth radius.

For the selected type of thruster, TM  = 1,375.6 kN and c =  s-1 .

[15 marks]

Problem 2. Consider the plant

x˙1  = x1 + x2 , x˙2  = x1 + u,

and the cost

x1 (0) = 2,

x2 (0) = 2,