bv_cvxbook_extra_exercises

We consider ve norms frobenius norm xs y 1t f j s

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Unformatted text preview: f (t) is obtained from a single rocket engine on the spacecraft, with a given maximum thrust; an attitude control system rotates the spacecraft to achieve any desired direction of thrust. The thrust force is therefore characterized by the constraint f (t) 2 ≤ F max . The fuel use rate is proportional to the thrust force magnitude, so the total fuel use is T td γ f ( t) 0 2 dt, where γ > 0 is the fuel consumption coefficient. The thrust force is discretized in time, i.e., it is constant over consecutive time periods of length h > 0, with f (t) = fk for t ∈ [(k − 1)h, kh), for k = 1, . . . , K , where T td = Kh. Therefore we have vk+1 = vk + (1/m)fk − ge3 , pk+1 = pk + (1/2)(vk + vk+1 ), where pk denotes p((k −1)h), and vk denotes p((k −1)h). We will work with this discrete-time model. ˙ For simplicity, we will impose the glide slope constraint only at the times t = 0, h, 2h, . . . , Kh. 123 (a) Minimum fuel descent. Explain how to find the thrust profile f1 , . . . , fK that minimizes fuel consumption, given the touchdown time T td = Kh and discretization time h. (b) Minimum time desce...
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