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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 9 Constrained Optimal Control Bryson and Ho – Section 3.x and Kirk – Section 5.3 • Spr 2008 16.323 9–1 Constrained Optimal Control • First consider cases with constrained control inputs so that u ( t ) ∈ U where U is some bounded set. – Example: inequality constraints of the form C ( x , u ,t ) ≤ – Much of what we had on 6– 3 remains the same, but algebraic con dition that H u = 0 must be replaced – Note that C ( x ,t ) ≤ is a much harder case • Augment constraint to cost (along with differential equation con straints) t f J a = h ( x ( t f ) ,t f ) + H − p T x ˙ + ν T C dt t • Find the variation (assume t and x ( t ) fixed): t f δJ a = h x δ x f + h t f δt f + H x δ x + H u δ u + ( H p − x ˙ T ) δ p ( t ) t − p T ( t ) δ x ˙ + C T δ ν + ν T { C x δ x + C u δ u } dt + H − p T x ˙ + ν T C ( t f ) δt f Now IBP t f t f − t p T ( t ) δ x ˙ dt = − p T ( t f ) ( δ x f − x ˙ ( t f ) δt f ) + t p ˙ T ( t ) δ x dt then combine and drop terminal conditions for simplicity: t f δJ a = H x + p ˙ T + ν T C x δ x + H u + ν T C u δ u t +( H p − x ˙ T ) δ p ( t ) + C T δ ν dt June 18, 2008 Spr 2008 16.323 9–2 Clean up by defining augmented Hamiltonian • H a ( x , u , p ,t ) = g + p T ( t ) a + ν T ( t ) C where (see 2– 12 ) ν i ( t ) ≥ if C i = active = if C i < inactive – So that ν i C i = 0 ∀ i . • So necessary conditions for δJ a = 0 are that for t ∈ [ t ,t f ] ˙ x = a ( x , u , t ) ˙ p = − ( H a ) T x ( H a ) u = – With appropriate boundary conditions and ν i C i ( x , u , t ) = Complexity here is that typically will have subarcs to the solution • where the inequality constraints are active (so C i ( x , u ,t ) = 0 ) and then not (so ν i = 0 ). – Transitions between the subarcs must be treated as corners that are at unspecified times need to impose the equivalent of the ErdmannWeirstrass corner conditions for the control problem, as in Lecture 8. June 18, 2008 Spr 2008 Constrained Example 16.323 9–3 • Design the control inputs that minimize the cost functional 4 min J = − x (4) + u 2 ( t ) dt u with x ˙ = x + u , x (0) = , and u ( t ) ≤ 5 . • Form augmented Hamiltonian: H = u 2 + p ( x + u ) + ν ( u − 5) • Note that, independent of whether the constraint is active or not, we have that p ˙ = − H x = − p p ( t ) = ce − t and from transversality BC, know that p (4) = ∂h/∂x = − 1 , so have that c = − e 4 and thus p ( t ) = − e 4 − t • Now let us assume that the control constraint is initially active for some period of time, then ν ≥ , u = 5 , and H u = 2 u + p + ν = 0 so we have that ν = − 10 − p = − 10 + e 4 − t – Question: for what values of t will ν ≥ ?...
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This note was uploaded on 11/07/2011 for the course AERO 16.323 taught by Professor Jonathanhow during the Spring '08 term at MIT.
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