# lec6 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 16.323 Lecture 6 Calculus of Variations applied to Optimal Control ˙ x = a ( x , u , t ) ˙ p = − H T x H u = • Spr 2008 16.323 6–1 Optimal Control Problems • Are now ready to tackle the optimal control problem – Start with simple terminal constraints t f J = h ( x ( t f ) ,t f ) + g ( x ( t ) , u ( t ) ,t ) dt t with the system dynamics x ˙ ( t ) = a ( x ( t ) , u ( t ) ,t ) – t , x ( t ) fixed – t f free – x ( t f ) are fixed or free by element • Note that this looks a bit different because we have u ( t ) in the inte- grand, but consider that with a simple substitution, we get x ˙ = a ( x , u ,t ) g ˜( x , x ˙ ,t ) g ˆ( x , u ,t ) → • Note that the differential equation of the dynamics acts as a constraint that we must adjoin using a Lagrange multiplier, as before: t f J a = h ( x ( t f ) ,t f )+ g ( x ( t ) , u ( t ) ,t ) + p T { a ( x ( t ) , u ( t ) ,t ) − x ˙ } dt t Find the variation: 10 t f δJ a = h x δ x f + h t f δt f + g x δ x + g u δ u + ( a − x ˙ ) T δ p ( t ) t + p T ( t ) { a x δ x + a u δ u − δ x ˙ } dt + g + p T ( a − x ˙ ) ( t f ) δt f • Clean this up by defining the Hamiltonian : (See 4– 4 ) H ( x , u , p ,t ) = g ( x ( t ) , u ( t ) ,t ) + p T ( t ) a ( x ( t ) , u ( t ) ,t ) 10 Take partials wrt each of the variables that the integrand is a function of. June 18, 2008 • • Spr 2008 16.323 6–2 Then δJ a = h x δ x f + h t f + g + p T ( a − x ˙ ) ( t f ) δt f t f + H x δ x + H u δ u + ( a − x ˙ ) T δ p ( t ) − p T ( t ) δ x ˙ dt t • To proceed, note that by integrating by parts 11 we get: t f t f − t p T ( t ) δ x ˙ dt = − t p T ( t ) dδ x T = − p T δ x t t f + t f d p ( t ) δ x dt dt t t f = − p T ( t f ) δ x ( t f ) + p ˙ T ( t ) δ x dt t t f = − p T ( t f ) ( δ x f − x ˙ ( t f ) δt f ) + p ˙ T ( t ) δ x dt t So now can rewrite the variation as: δJ a = h x δ x f + h t f + g + p T ( a − x ˙ ) ( t f ) δt f t f t f + H x δ x + H u δ u + ( a − x ˙ ) T δ p ( t ) dt − p T ( t ) δ x ˙ dt t t = h x − p T ( t f ) δ x f + h t f + g + p T ( a − x ˙ ) + p T x ˙ ( t f ) δt f t f + H x + p ˙ T δ x + H u δ u + ( a − x ˙ ) T δ p ( t ) dt t udv ≡ uv − vdu June 18, 2008 11 Spr 2008 16.323 6–3 • So necessary conditions for δJ a = 0 are that for t ∈ [ t ,t f ] ˙ x = a ( x , u , t ) ( dim n ) ˙ p = − H T x ( dim n ) H u = ( dim m ) – With the boundary condition (lost if t f is fixed) that h t f + g + p T a = h t f + H ( t f ) = 0 – Add the boundary constraints that x ( t ) = x ( dim n) – If x i ( t f ) is fixed, then x i ( t f ) = x i f ∂h – If x i ( t f ) is free, then p i ( t f ) = ( t f ) for a total ( dim n) ∂x i These necessary conditions have 2 n differential and m algebraic equa- • tions with 2 n +1...
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lec6 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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