Ind the optimal control optimal states and optimal

Question

Exam - Optimal Control

1. Find the external of the functional

J = ₁∫² x^·2(t)/2t³ dt

With x(1) = 1 And x(2) = 10.

2. Find the Optimal control, optimal states, and optimal co-states for the problem:

min_u J = ½ ₀∫⁵ u²(t)dt

s.t. x^··(t) = u(t)

x(0) = 2, x(5) = 0, x^·(0) = 2, x^·(5) = 0

u is unconstrained.

3. Given

x^·₁ = x₂

x·₂ = -Ω²x₁ + u   Ω > 0, constant

y = x₁

If u(t) = 1  t > 0, x₁(0) = 0, x₂(0) = 1

Find y(t), t > 0.

4. min_x,y J(x, y) = xy

s.t. 4x² + y² = 4

Consider

Min_{t_f} J = ₀∫^t_f dt

x^··(t) = u(t)

x(0) = x₀, x·(0) = v₀

x(t_f) = x·(t_f) = 0

t_f FREE

|u| ≤ 1

(a) Derive A formula for tf as a function of x₀ and v₀.

(b) Write a matlab script that outputs

x₁''(t) vs t

x2'' (t) vs t

u'' (t) vs t

When the input is the initial state X₀, v₀.

Turn in plots for the case x₀ = 10 And v₀ = 10 And demonstrate that your simulation works.

For various other initial conditions.

Hint: Show that the switching curve can be written as s_w = x₁ + ½ x₂|x₂|.