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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 .
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16.323 Lecture 9 Constrained Optimal Control Bryson and Ho Section 3.x and Kirk Section 5.3
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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 ) 0 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 ) 0 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 0 Find the variation (assume t 0 and x ( t 0 ) 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 0 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 0 p T ( t ) δ x ˙ dt = p T ( t f ) ( δ x f x ˙ ( t f ) δt f ) + t 0 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 0 +( H p x ˙ T ) δ p ( t ) + C T δ ν dt June 18, 2008
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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 ) 0 if C i = 0 active = 0 if C i < 0 inactive So that ν i C i = 0 i . So necessary conditions for δJ a = 0 are that for t [ t 0 , t f ] ˙ x = a ( x , u , t ) ˙ p = ( H a ) T x ( H a ) u = 0 With appropriate boundary conditions and ν i C i ( x , u , t ) = 0 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
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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 0 with x ˙ = x + u , x (0) = 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 0 , 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 0 ? ν = 10 + e 4 t 0 e 4 t 10 4 t ln(10) 4 ln(10) t So provided t t c = 4 ln(10) then ν 0 and the assumptions are consistent.
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