Chapter_6_Dynamic_Prog.ppt

8/9/2019 Chapter_6_Dynamic_Prog.ppt

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Chapter 6

Dynamic Propgramming(315)

Chapter 6

Dynamic Propgramming(315)

- Solve control problems for Nonlinear !"S# Develope$ by %ellman

- &lternative to the "ariational approach to 'C $isc sse$ inChapter * 3 some of o r res lts from Chapter * 3 can be$erive$ by this ne+ approach, t can be se$ to solve controlproblems for nonlinear time-varying systems,

- !he 'C by this metho$ is e.presse$ as a state-variablefee$bac/ in graphical or tab lar form

Principle# f +e /no+ an optimal path then every s b-path isoptimal too


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p

0.ample# path optimi ation2 e$ges sho+ possible ights4 each has some cost2 +ant to n$ min cost ro te or path from S to N7

$ynamic programming (DP)#2 " (i) is min cost from airport i to N7 over all possible paths2 to n$ min cost from city i to N7# minimi e s m of ight cost pl s mincost to N7 from +here yo lan$ over all ights o t of city i(gives optimal ight o t of city i on +ay to N7)2 if +e can n$ " (i) for each i +e can n$ min cost path from any city

to N72 DP principle# " (i) 8 min9(c9i : " (9)) +here c9i is cost of ight fromi to 9 an$ minim m is over all possible ights o t of i


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; nitial state given# . i 8 . < $. i 8 e e,g, t+o optimal paths a i

- State variable fee$bac/# optimal choice of path $epen$s on cityat present

- or+ar$ planning $oes not +or/- 'ptimal path satisfy %ellmanEs principle

e,g, b e h i is optimal from b i e h i is optimal from e i

- Kess comp tation than e.ha stive search approach

Discrete time System


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Discrete-time Systemor nonlinear systems the state an$ co-state e> ations are har$

to solve an$ constraint f rther complicate things, Dynamicprogramming can easily be applie$ to nonlinear systems an$more constraints there are on the control an$ state variables theeasier the sol tionL

!o comp te the optimal $ecision / at c rrent time / +e ass methat the optimal costs an$ control la+ for all f t re time steps are

alrea$y /no+n, !he optimal costs @ / are given by the s m of thecosts of the c rrent $ecision an$ the optimal costs for thes bse> ent time steps#


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Solving this e> ation bac/+ar$s in time for all states yiel$s the optimal control la

; 0 l 6 1 'ptimal control


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; 0.ample 6, -1 , ptimal controlof a scalar $iscrete system


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& ro ting net+or/ is


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& o t g et o / ssho+n in ig, P6,1-1, in$ the optimalpath from . < to . 6 ifonly movementfrom left to right ispermitte$, No+ fin$the optimal pathfrom any no$e as astate-variablefee$bac/,

&ssignment


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Assignment &ircraft Go ting Net+or/ is given, in$ the optimal


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Assignment &ircraft Go ting Net or/ is given, in$ the optimalpath from city . < to . 6, N mbers in$icate f el cons mptionbet+een cities, MNote# Ge$ra+ an$ in$icate +ith arro+s optimalpaths

< 6


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0n$


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Chapter_6_Dynamic_Prog.ppt