Optimal Control Formulation using Calculus of Variations

Transcription

1 Lecure 5 Opimal Conrol Formulaion using Calculus o Variaions Dr. Radhakan Padhi Ass. Proessor Dep. o Aerospace Engineering Indian Insiue o Science - Bangalore

2 opics Opimal Conrol Formulaion Objecive & Selecion o Perormance Index Necessary Condiions o Opimaliy and wo- Poin Boundary Value Problem (PBVP) Formulaion Boundary/ransversaliy Condiions Numerical Examples Dr. Radhakan Padhi, AE Dep., IISc-Bangalore

3 Opimal Conrol Formulaion: Objecive & Selecion o Perormance Index Dr. Radhakan Padhi Ass. Proessor Dep. o Aerospace Engineering Indian Insiue o Science - Bangalore

4 Objecive o ind an "admissible" ime hisory o conrol variable U,, which: 1) Causes he sysem giverned by X =, X, U o ollow an admissible rajecory ) Opimizes (minimizes/maximizes) (, ) (,, ) J = ϕ X + L X U d a "meanigul" perormance index 3) Forces he sysem o saisy "proper boundary condiions". [ our ocus: X = X (given), : ixed ] Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 4

5 Meaningul Perormance Index 1) Minimize he operaional ime ) Minimize he conrol eor ( ) 1 [ ϕ, 1] J = = d = L = 1 1 J =,, U RU d R > ϕ = L = U RU 3) Minimize deviaion o sae rom a ixed value C wih minimum conrol eor 1 J = ( X C) Q( X C) + U RU d, Q, R > Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 5

6 Meaningul Perormance Index 4) Minimize deviaion o sae rom origin wih minimum conrol eor 1 J = X QX + U RU d, Q, R > 5) Minimize he conrol eor, while he inal sae reaches close o a consan C 1 1 ( J = X ) C S X C + U RU d, S, R > X Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 6

7 Opimal Conrol Using Calculus o Variaions: Hamilonian Formulaion Dr. Radhakan Padhi Ass. Proessor Dep. o Aerospace Engineering Indian Insiue o Science - Bangalore

8 Opimal Conrol Problem Perormance Index (o minimize / maximize): Pah Consrain: J, X L, X, U d X = ϕ + =, X, U Boundary Condiions: X = X :Speciied :Fixed, X :Free Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 8

9 Necessary Condiions o Opimaliy Augmened PI Hamilonian = + + J ϕ L λ X d ( + λ ) H L Firs Variaion δj = δϕ+ δ H λ X d ( ) = δϕ + δ H λ X d ( ) Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 9

10 Necessary Condiions o Opimaliy Firs Variaion Individual erms δϕ (, ) X δ X ( δj = δϕ+ δh δλ X λ δx ) d ϕ = X δ H H H H(, X, U, λ ) ( δ X) ( U) X δ U δλ = + + λ Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 1

11 Necessary Condiions o Opimaliy ( λ δ ) d( X) X δ d = λ d d, δ X = λ δx, δ X d ( ) X ( ) X d Dr. Radhakan Padhi, AE Dep., IISc-Bangalore δx d = λ δx λ δx δx λ d = λ δ δ λ dλ 11

12 Necessary Condiions o Opimaliy Firs Variaion ϕ J = X X X δ δ δ λ H H H + δx + δu + δλ d X U λ + δx λ d δλ X d Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 1

13 Necessary Condiions o Opimaliy Firs Variaion ϕ δj = δx λ X H H + ( δx ) + λ d + ( δu ) d X U H + ( δλ ) X d λ = Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 13

14 Necessary Condiions o Opimaliy: Summary Sae Equaion Cosae Equaion X H = = λ H λ = X (,, ) X U Opimal Conrol Equaion H U = Boundary Condiion λ ϕ = X ( ) X = X :Fixed Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 14

15 Necessary Condiions o Opimaliy: Some Commens Sae and Cosae equaions are dynamic equaions Opimal conrol equaion is a saionary equaion Boundary condiions are spli: i leads o wo-poin- Boundary-Value Problem (PBVP) Sae equaion develops orward whereas Cosae equaion develops backwards radiionally, PBVPs demand compuaionallyinensive ieraive numerical procedures hese ieraive numerical procedures lead o openloop conrol soluions Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 15

16 An Useul heorem heorem: I he Hamilonian H is no an explici uncion o ime, hen H is consan along he opimal pah. Proo: dh H H H H = + X + U + λ d X U λ H H H H = + X + λ + U = X and λ X = X λ X U λ dh H = ( on opimal pah) d = i H is no an explici uncion o. Hence, he resul! Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 16

17 General Boundary/ransversaliy Condiion General condiion: ( X ) wih, ixed Special Cases: Φ Φ λ δx + + H δ = X 1) : ixed, X : ree Φ X λ δx = λ = Φ (, X ) X ) : ree, X : ixed Φ ( ) + H δ = H = Φ Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 17

18 Example 1: A oy Problem Dr. Radhakan Padhi Ass. Proessor Dep. o Aerospace Engineering Indian Insiue o Science - Bangalore

19 Example Problem: Soluion: Cosae Eq. x 1 x = x x + u J = x + x + u d 1 5 =, =, x = x = 1 ( /) λ1 λ( ) H = u + x + x + u ( H x1 ) ( H / x ) λ / 1 = = λ λ + λ 1 Opimal conrol Eq. u+ λ = u = λ Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 19

20 Example Boundary Condiions Deine Soluion x1 λ1 x1 5 =, x = λ x [ ] Z x x λ λ Z = AZ Z 1 1 = e C A A = 1 1 Dr. Radhakan Padhi, AE Dep., IISc-Bangalore

21 Example Use he boundary condiion a c = 1 c Use he boundary condiion a 1 = = x x A = e = x1 5 c 3 1 c 3 x c c4 Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 1

22 Example Four equaions and our unknowns: x x = 1 1 c c4 x x = = c c Dr. Radhakan Padhi, AE Dep., IISc-Bangalore

23 Example Soluion or Sae and Cosae () () () x1 x A = e λ 1.7 λ.4 Soluion or Opimal Conrol where A = 1 1 u = λ Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 3

24 Example : Double Inegraor Problem (Relevance: Saellie Aiude Conrol Problem) x = u Dr. Radhakan Padhi Ass. Proessor Dep. o Aerospace Engineering Indian Insiue o Science - Bangalore

25 Double Inegraor Problem u = x x = x 1 x1 = Consider a double inegraor problem as shown in he above igure. y [ ] Find such u ha he sysem iniial values X = 1 are driven o he origin by minimizing 1 J = + u d Noe : (1) : unspeciied () Conrol variable u is unconsrained Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 5

26 Double Inegraor Problem Soluion : Sysem dynamics x 1 1 x1 U AX BU x = + = + x 1 C A Boundary Condiion 1 = = ( ), X ( ) X x1 y = [ 1 ] CX (no required) = x X B Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 6

27 Double Inegraor Problem Conrollabiliy Check : Conrollabiliy Marix 1 1 M = [ B AB] = 1 = 1 1 M = 1 Hence, he sysem is conrollable. Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 7

28 Necessary Condiions o Opimaliy 1 (1) Sae Eq: H = u + λ AX + Bu X = AX + Bu H () Opimal Conrol Eq: = u u+ B λ = (3) Cosae Eq: u B H λ = = A λ X = λ = λ Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 8

29 Opimal Conrol Soluion λ 1 1 λ = A λ = = λ 1 λ λ 1 λ = λ = c λ = λ = c λ = c+ c 1 u = λ = c c 1 Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 9

30 Opimal Sae Soluion However, x 1 x x = = x u c c 1 Hence x = c1 c + c3 3 x = xd= c c + c+ c Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 3

31 Opimal Sae Soluion Using he B.C. a = : x1 c4 1 = x = c 3 () () x c1 3 c = x c1 c Using he B.C a = : c1 3 c x1 ( ) = = x ( ) c 1 c Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 31

32 ransversaliy Condiions ( : ree) ϕ = H u = + λ AX + Bu u = + [ λ λ ] 1 ( c 1 c) x u = + λ (B.C.) 1 = ( c 1 c) 4 = c cc + c 1 1 ( ) x ( ) ( c c ) 1 1 Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 3

33 ransversaliy Condiions ( : ree) In summary, we have o solve or c, c and rom: c 3c + 6= 3 1 c c = 1 ( ) c cc c = A his poin, one can solve c, c rom irs wo equaions in erms o and subiue hem in he hird equaion. hen he resuling nonlinear equaion in can be solved (preerably in closed orm). However, one mus discard unrealisic soluions (e.g. is unrealisic). Noe : One may use numerical ehniques (like Newon-Raphson echnique) Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 33

34 ransversaliy Condiions ( : ree) c 1.5 Finally, c = c /4 Hence, he opimal soluion is given by: u = c c = ( 3.95) [ ] [ ] and i will ake = = 3.91 ime unis o reach X =, 4 saring rom X = 1 Noe: (1) I is an open-loop conrol law () he applicaion o conrol has o be erminaed a Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 34

35 Reerences on Opimal Conrol Design. F. Elber, Esimaion and Conrol Sysems, Von Nosard Reinhold, A. E. Bryson and Y-C Ho, Applied Opimal Conrol, aylor and Francis, R. F. Sengel, Opimal Conrol and Esimaion, Dover Publicaions, D. S. Naidu, Opimal Conrol Sysems, CRC Press,. A. P. Sage and C. C. Whie III, Opimum Sysems Conrol (nd Ed.), Prenice Hall, D. E. Kirk, Opimal Conrol heory: An Inroducion, Prenice Hall, 197. Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 35

36 Dr. Radhakan Padhi, AE Dep., IISc-Bangalore 36