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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 61 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 ( 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 J a = h x x f + h t f t f + g x x + g u u + ( a x ) T p ( t ) t + p T ( t ) { a x x + a u u x } dt + g + p T ( a x ) ( t f ) t 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 62 Then J a = h x x f + h t f + g + p T ( a x ) ( t f ) t f t f + H x x + H u u + ( a x ) T p ( t ) p T ( t ) x dt t To proceed, note that by integrating by parts 11 we get: t f t f t p T ( t ) x dt = t p T ( t ) d x T = p T x t t f + t f d p ( t ) x dt dt t t f = p T ( t f ) x ( t f ) + p T ( t ) x dt t t f = p T ( t f ) ( x f x ( t f ) t f ) + p T ( t ) x dt t So now can rewrite the variation as: J a = h x x f + h t f + g + p T ( a x ) ( t f ) t f t f t f + H x x + H u u + ( a x ) T p ( t ) dt p T ( t ) x dt t t = h x p T ( t f ) x f + h t f + g + p T ( a x ) + p T x ( t f ) t f t f + H x + p T x + H u u + ( a x ) T p ( t ) dt t udv uv vdu June 18, 2008 11 Spr 2008 16.323 63 So necessary conditions for J 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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