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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 sub-arcs 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 sub-arcs must be treated as corners that are at unspecified times- need to impose the equivalent of the Erdmann-Weirstrass 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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lec9 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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