ACS61010 Optimal Control Coursework Spring 2021-22
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ACS61010 Optimal Control
Coursework
Spring 2021-22
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)
m˙ = − T
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˙ = g0 m˙ − D(r,V) + g(r) = g0 d [lnm(t)] − D(r,V) + 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) = CDρ(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 ⊕
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 = 1 s-1 .
[15 marks]
Problem 2. Consider the plant
x˙ 1 = x1 + x2 , x1 (0) = 2,
x˙ 2 = x1 + u, x2 (0) = 2,
and the cost
J(u) = 
[x1 (2) − 4]2 + 
[x2 (2) − 6]2 + 
Z02 x
(t) + x
(t) + u2 (t) dt.
a. Let α = 0 and x1 (2) = 4. Express the optimal control in terms of the optimal costate. Write down the corresponding two-point boundary value problem. By solving this problem in MATLAB (use bvp4c or bvp5c), find the optimal states and costates for β = 0, 0.1, 1, 10, 100. Plot your results in four subplots each showing one state/costate component for all β. For example, subplot(2,2,1) must show x1 for different β .
On the same axes but using dashed lines, plot the optimal states and costates for the case when α = 0 = β , x1 (2) = 4, and x2 (2) = 6. Add a legend, a title, and labels to each subplot. Explain what you observe and why. [25 marks]
b. Let α = 1 = β (free endpoint). Write down the boundary value problem to find the optimal control that guarantees
(x1 (2) − 1)2 + (x2 (2) − 1)2 = 1.
Using MATLAB, find and plot the optimal states, costates, and control. What are the values
of x1 (2) and x2 (2)? Do they satisfy the terminal condition? [25 marks]
2022-05-10