Calculus of Variations Summer Term 2015 Lecture 14 Universität des Saarlandes 24. Juni 2015 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 1 / 20
Purpose of Lesson Purpose of Lesson: To consider aerospace example c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 2 / 20
Example 14.1 Lauching a rocket Launch a rocket (with one stage) to deliver its payload into Low-Earth Orbit (LEO) at some height h above the Earth s surface. Assumptions: ignore drag, and curvature and rotation of Earth LEO so assume gravitational force at ground and orbit are approximately the same thrust will generate acceleration a, which is predefined by rocket parameters we thrust for some time T, then follow a ballistic trajectory until (hopefully) we reach height h, at zero vertical velocity, and with horizontal velocity matching the required orbital injection speed. c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 3 / 20
Example 14.1 Launching a rocket-2 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 4 / 20
Example 14.1 Launching a rocket-3 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 5 / 20
Example 14.1 Launching a rocket-4 Notation: x = horizontal position y = vertical position u = horizontal velocity v = vertical velocity Initial conditions x(0) = y(0) = u(0) = v(0) = 0. Thrust stops at time T, and then at some later time S, we reach the peak of the trajectory where y(s) = h u(s) = u 0, orbital velocity v(s) = 0 We don t actually care about the final position x(s). c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 6 / 20
Example 14.1 Launching a rocket-5 Control: thrust profile is pre-determined. The only thing we can control (in this problem) is the angle of thrust. Thrust a(t) is constant for our example. Measure the angle of thrust θ(t) relative to horizontal. want to minimize fuel but this is equivalent to minimizing time, e.g., need to get to height h J = t 0 T adt = a 1dt 0 need to get to horizontal velocity u 0 to enter orbit. c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 7 / 20
Constrained equations Example 14.1 Launching a rocket-6 Thrust component: t T Ballistic component: T < t S ẋ = u ẋ = u ẏ = v ẏ = v u = a cos θ u = 0 v = a sin θ g v = g Initial point Initial point: fixed x(0) = y(0) = u(0) = v(0) = 0. x(t ), y(t ), u(t ), v(t ) Final point: free Final point: x(s) free y(s) = h, v(s) = 0, u(s) = u 0 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 8 / 20
1st consider ballistic component Example 14.1 Launching a rocket-7 For t [T, S] we have no control, and ẋ = u ẏ = v u = 0 v = g we can calculate the top of the resulting parabola as and x(t ) and x(s) are free. u(s) = u(t ) v(s) = 0 y(s) = y(t ) + (v(t )) 2 /2g c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 9 / 20
Coordinate transform Example 14.1 Launching a rocket-8 So we can change variables: make the final point t = T, and take variables u, v as before, and z = y + v 2 2g. We can differentiate this and combine with previous results to get the new system DEs u = acosθ v = a sin θ g ż = ẏ + v v/g = v (1 + v/g) = av g sin θ c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 10 / 20
Optimization functional Example 14.1 Launching a rocket-9 Time minimization problem T = T 0 1dt. Including Lagrange multipliers for the 3 system constraints we aim to minimize T 0 (1 + λ u ( u a cos θ) + λ v ( v a sin θ + g) + λ z (ż avg sin θ )) dt subject to u(0) = v(0) = z(0) = 0, θ(0) = free, u(t ) = u 0, v(t ) = free, z(t ) = h, θ(t ) = free. c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 11 / 20
Euler-Lagrange equations Example 14.1 Launching a rocket-10 u : v : z : θ : h u d h dt u = 0 λ u = 0 h v d h dt v = 0 a λ v = λ z g sin θ h z d h dt ż = 0 λ z = 0 h θ d h dt θ = 0 (λ equations give back systems DEs). aλ u sin θ λ v a cos θ λ z av g cos θ = 0 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 12 / 20
Solving the E-L equations Example 14.1 Launching a rocket-11 Take the v equation, and noting that v = a sin θ g λ v = λ z a g sin θ = λ z g ( v + g), λ v = λ z g (v + gt + c) = λ zv g λ zt + b. c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 13 / 20
Solving the E-L equations Example 14.1 Launching a rocket-12 Substitute λ v = λ zv g λ zt + b into the θ E-L equation (dropping the common factor a) and we get ( λz v λ u sin θ + g λ u sin θ λ v cos θ λ z v g cos θ = 0 ) + λ v zt b cos θ λ z g cos θ = 0 λ u sin θ + (λ z t b) cos θ = 0 tan θ = λ zt b λ u c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 14 / 20
Solution Optimal control - aerospace example Example 14.1 Launching a rocket-13 Remember that λ u and λ v and b are all constants, so the equation tan θ = λ zt b λ u angle of thrust now specified ( θ = tan 1 λ ) zt b λ u but we need to determine constants c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 15 / 20
End-point conditions Example 14.1 Launching a rocket-14 Final end-points conditions T = free z(t ) = h u(t ) = u 0, orbital velocity v(t ) = free θ(t ) = free λ u = free λ v = free λ z = free c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 16 / 20
Natural boundary conditions Example 14.1 Launching a rocket-15 The free-end point boundary condition for J[q, q] = F(t, q, q)dt is In our problem [ m k=1 ( F δq k + δt F q k m k=1 )] F q k q k t=t = 0. F λ k = 0, F θ = 0, F u = λ u, F v = λ v, F ż = λ z c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 17 / 20
Natural boundary conditions Example 14.1 Launching a rocket-16 Consider δq k for each coordinate: for fixed coordinates u and z, we have δq k = 0 it is free for θ, λ u, λ v, λ z, but in each case the corresponding = 0, so we can ignore these. F q k only case where it matters is δv, which we can vary, and for which F v = λ v. Also δt is free, so we get two end-point conditions at t = T λ v (T ) = 0 H(T ) := [F uλ u vλ v żλ z ] t=t = 0 c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 18 / 20
Natural boundary conditions Example 14.1 Launching a rocket-17 Given λ v (T ) = 0, and from previous work λ v = λ zv g λ zt + b we get λ z v(t ) g = λ z T + b = λ u tan θ(t ) v(t ) = λ ug λ z tan θ(t ) c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 19 / 20
Natural boundary conditions Example 14.1 Launching a rocket-18 H(T ) = [F uλ u vλ v żλ z ] t=t = 0. Substituting F and taking into account that λ v (T ) = 0 we get aλ u cos θ(t ) + a λ zv(t ) g Combining the latter with v(t ) = λug λ z sin θ(t ) = 1 tan θ(t ) we arrive at aλ u cos θ(t ) + aλ u tan θ(t ) sin θ(t ) = 1 λ u = cos θ(t ) a c Daria Apushkinskaya (UdS) Calculus of variations lecture 14 24. Juni 2015 20 / 20