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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 10 Singular Arcs Bryson Chapter 8 Kirk Section 5.6 Spr 2008 16.323 101 Singular Problems There are occasions when the PMP u ( t ) = arg min H ( x , u , p ,t ) u ( t ) U fails to define u ( t ) can an extremal control still exist? Typically occurs when the Hamiltonian is linear in the control, and the coecient of the control term equals zero . Example: on page 910 we wrote the control law: u m b < p 2 ( t ) u ( t ) = b < p 2 ( t ) < b u m p 2 ( t ) < b but we do not know what happens if p 2 = b for an interval of time. Called a singular arc . Bottom line is that the straightforward solution approach does not work, and we need to investigate the PMP conditions in more detail. Key point : depending on the system and the cost, singular arcs might exist, and we must determine their existence to fully characterize the set of possible control solutions. Note: control on the singular arc is determined by the requirements that the coecient of the linear control terms in H u remain zero on the singular arc and so must the time derivatives of H u . Necessary condition for scalar u can be stated as d 2 k ( 1) k u dt 2 k H u k = 0 , 1 , 2 ... June 18, 2008 Spr 2008 Singular Arc Example 1 16.323 102 With x = u , x (0) = 1 and u ( t ) 4 , consider objective 2 min ( x ( t ) t 2 ) 2 dt First form standard Hamiltonian H = ( x ( t ) t 2 ) 2 + p ( t ) u ( t ) which gives H u = p ( t ) and p ( t ) = H x = 2( x t 2 ) , with p (2) = (10.15) Note that if p ( t ) > , then PMP indicates that we should take the minimum possible value of u ( t ) = 0 . Similarly, if p ( t ) < , we should take u ( t ) = 4 . Question: can we get that H u for some interval of time? Note: H u implies p ( t ) , which means p ( t ) , and thus p ( t ) x ( t ) = t 2 , u ( t ) = x = 2 t Thus we get the control law that p ( t ) > u ( t ) = 2 t when p ( t ) = 0 4 p ( t ) < June 18, 2008 Spr 2008 16.323 103 Can show by contradiction that optimal solution has x ( t ) t 2 for...
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 Spring '08
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