Neuro-Adaptive Design - I:

Transcription

1 Lecture 36 Neuro-Adaptve Desgn - I: A Robustfyng ool for Dynamc Inverson Desgn Dr. Radhakant Padh Asst. Professor Dept. of Aerospace Engneerng Indan Insttute of Scence - Bangalore

2 Motvaton Perfect system modelng s dffcult Sources of mperfecton: Unmodelled dynamcs (mssng algebrac terms n the model) Inaccurate knowledge of system parameters and/or change of system parameters durng operaton Inaccuracy n computatons (lke matrx nverson) Objectve: o ncrease the robustness of dynamc nverson wth respect to parameter and/or modelng naccuraces Dr. Radhakant Padh, AE Dept., IISc-Bangalore

3 Reference B. S. Km and A. J. Calse, Nonlnear Flght Control Usng Neural Networks, Journal of Gudance, Control and Dynamcs, Vol. 0, No. 1, 1997, pp Dr. Radhakant Padh, AE Dept., IISc-Bangalore 3

4 Neuro-adaptve desgn System Assumptons: Goal: () ( 3 ) X = f X X (,, δ ) n X, X R, δ R 1 X, X m are avalable for control computaton m= n X, X, δ R f and X (square system,.e. ) s nvertble X X X X c c ( : commanded sgnal ) c m Dr. Radhakant Padh, AE Dept., IISc-Bangalore 4

5 Neuro-adaptve desgn Ideal Case: Defne hen δ X X X c Desgn such that X + K X + K X = 0, where K, K > 0 pdf matrces d p p d, If K = dag k, K = dag k, k k > 0 d d p p p d x + k x + k x = d p c d p 0 x x + k x + k x = 0 Dr. Radhakant Padh, AE Dept., IISc-Bangalore 5

6 Neuro-adaptve desgn x = x + k x + k x c d p u th ( compoent of "pseudo control") Repeatng ths exercse for = 1,,..., n we get X = U ( U : Pseudo control varable) f X X = U (,, δ ) δ = f 1 ( X, X, U) wth the assumton that the nverse exts Dr. Radhakant Padh, AE Dept., IISc-Bangalore 6

7 Neuro-adaptve desgn Note: Real - tme computaton of ths nverse may be dffcult. In the case, a Neural Network can learn ths relatonshp offlne (n an approxmate sense) 1 = f ( X X U ) NN 1 X, X, U.e, ˆ δ,, = Problem: he model and the neural network tranng are NO perfect...these ntroduce errors. Dr. Radhakant Padh, AE Dept., IISc-Bangalore 7

8 Neuro-adaptve desgn Soluton : Desgn another neural network to cancel ths error! Desgn of Adaptve NN After syntheszng, the system dynamcs s (,, ˆ δ ) X = f X X ˆ δ = f ( X, X, δ ) + f ( X, X, ˆ δ) f ( X, X, δ) U U Δ ( X, X, U) Dr. Radhakant Padh, AE Dept., IISc-Bangalore 8

9 Neuro-adaptve desgn.e X = U +Δ X, X, U Dr. Radhakant Padh, AE Dept., IISc-Bangalore 9

10 Neuro-adaptve desgn Note: If Δ=, then the error dynamcs behaves lke the deal case, and hence, the objectve wll be met. rck : Modfy the Pseudo control as where 0 ˆ and : U ad U U = U ˆ I U U = X + K X + K X I c d p s the P seudo control for the "deal case". Adaptve control to cancel the unwarted effect. ad Note: Pseudo-control modfcaton s the reason why technque s applcable to dynamc nverson only. Dr. Radhakant Padh, AE Dept., IISc-Bangalore 10

11 Neuro-adaptve desgn Wth ths, the closed loop error dynamcs becomes X + K X + K X = X X + K X + K X.e, d p c d p ( X ) (,, ) c KdX KpX U X X U = + + +Δ = U Queston Can we desgn I ( U ) I Uad Δ X X U ( X X U) ˆ,, X + K ˆ dx + KpX = Uad Δ X, X, U : Uˆ Uˆ Δ,, ad ad (adaptvely ) such that Dr. Radhakant Padh, AE Dept., IISc-Bangalore 11

12 Neuro-adaptve desgn If ths happens, then th Defne channel : X, X 0 (approxmately) x + k x + k x = Uˆ Δ X, X, U e d p ad x x hen e ˆ = e + Uad, X kp k Δ 1 d A b XU, Dr. Radhakant Padh, AE Dept., IISc-Bangalore 1

13 Neuro-adaptve desgn Goal for Uˆ ad Uˆ he "deal purpose" of s to capture the functon Δ "pefectly" so that e 0 asymptotcally However, ths s dffcult to acheve. Hence, am for "practcal stablty "only.e, e ad remans bounded, where the bound can be made arbtrarty small. Dr. Radhakant Padh, AE Dept., IISc-Bangalore 13

14 Neuro-adaptve desgn Let Δˆ,, be a NN realzaton of,, ( X XU) Δ X XU when a fnte number ( N) of bass functons β X, X, U, j = 1,,..., N are used, wth the j correspondng weghts of the network beng () Wˆ t, j = 1,,..., N j.e, ˆ ˆ ( X, X, U) W () t β ( X, X, U) Δ = j j Dr. Radhakant Padh, AE Dept., IISc-Bangalore 14

15 Neuro-adaptve desgn β 1 β ˆ Δ (,, ) ˆ, ˆ,..., ˆ X X U = W W 1 W N... β N ( ) () β ( ).e. Uˆ = Δ ˆ X, X, U = Wˆ t. X, X, U Ideal case: ad { X X U} ( ) () β ( ) When Wˆ s optmzed over some compact doman n,,, * * * let the result be Wˆ. In that case Δ ˆ X, X, U = Wˆ t. X, X, U Δ Δ ˆ ˆ < ε ε * and where s the deal error of functon. Dr. Radhakant Padh, AE Dept., IISc-Bangalore 15

16 Neuro-adaptve desgn Note: ε depends on the followngs: () N : Sze of the network () Choce of bass functons β ( j = 1,..., N) () Accuracy of the frst neural network th (whch determnes Δ ) j Dr. Radhakant Padh, AE Dept., IISc-Bangalore 16

17 Neuro-adaptve desgn Estmaton Error: () = ˆ () * () * β,, β (,, ) ˆ ˆ ˆ ˆ U Δ = W t X XU W X XU ad * () ˆ β ( ) = Wˆ t W X, X, U = W β,, ( X X U) W t W t Wˆ where, : Error n weght at tme Note: Wˆ s constant. Hence = = W * () t Wˆ () t Wˆ ˆ W() t = 0 * t Dr. Radhakant Padh, AE Dept., IISc-Bangalore 17

18 Neuro-adaptve desgn Notaton : β ( X, X, U) = β (,, ) t e W Δ ˆ =Δˆ ( X, X, U) (,, ) t e W * * Δ ˆ ˆ =Δ ( X, X, U) t, e, W Used n mplementaton X x Note: e, e X x used n proof Dr. Radhakant Padh, AE Dept., IISc-Bangalore 18

19 Neuro-adaptve desgn th Error dynamcs n channel : Note : ( ˆ ) e = Ae + b U Δ ad ( ˆ ˆ * * ˆ ) = Ae + b U Δ +Δ Δ ad = Ae + bw β t e W + b Δ Δ ( *,, ˆ ) A 0 1 = s a Hurwtz matrx. kp k d Characterstc equaton: s + k s+ k = 0, wth k, k > 0 d p d p Dr. Radhakant Padh, AE Dept., IISc-Bangalore 19

20 Neuro-adaptve desgn Stablty Analyss: Defne a Lyapunov Functon canddate: ( ) where, V e, W V = n = 1 (, ) V e W 1 1 e Pe + W W, e > E P γ = 1 E + W W, e E E : defnes "dead zone" P γ By defnton, e e Pe P s a pdf matrx satsfyng P PA + A P = Q, Q > 0 (pdf) Note : By the selecton, V s contnuous at the boundary of dead zone. Dr. Radhakant Padh, AE Dept., IISc-Bangalore 0

21 Neuro-adaptve desgn Note: (1) Exstence of such a pdf matrx P s guaranteed by the fact A s Hurwtz. P () If Q = I, then the soluton for s gven by kd k 1 p kp k d k p k p = kp k d k p Dr. Radhakant Padh, AE Dept., IISc-Bangalore 1

22 Neuro-adaptve desgn 1 1 V = ( e Pe + e Pe ) + W W γ ˆ * ( ˆ * β,, ) Ae + bw β t, e, W b Pe + Δ Δ 1 = + WW ˆ e P Ae bw t e W b γ Δ Δ 1 ( ˆ * ) 1 = e A P + PA e + e Pb W β + Δ Δ + W Wˆ γ 1 ˆ W e Qe + e Pb ε + W e Pb β + γ Weght update rule: Wˆ = γ e Pbβ Dr. Radhakant Padh, AE Dept., IISc-Bangalore

23 Neuro-adaptve desgn 1 e Qe + ε e P Pb P P 1 e λmn ( Q ) + ε e P P b λmax However, e Pe P e e Pe.e, e λ max ( P ) e Pe e = λ max ( s dened snce s pdf ) ( P) λ ( P) e max P Dr. Radhakant Padh, AE Dept., IISc-Bangalore 3

24 Neuro-adaptve desgn 1 e V Q e P Pe P P λmn + ε λmax λmax ( P ) 1 e λmn ( Q ) P = + ε e Pe λmax ( P ) e P ( Q) ( P ) ( Q) ( P ) ( P) max 1 e λ V e P P mn + ε λ P max λmax 1 e λ V 0, when + < 0 ( P) P mn ε λmax λmax λ Dr. Radhakant Padh, AE Dept., IISc-Bangalore 4

25 Neuro-adaptve desgn.e, ε λ max P < 1 e P λ λ max mn P Q ().e, e P > ε λ λ mn max Note: 1 hs defnes E, However t contans ε, Q whch s unkown. However, t may requre teratve smulaton study. P 3 Dr. Radhakant Padh, AE Dept., IISc-Bangalore 5

26 Neuro-adaptve desgn If Q = I ( P) E= ε λmax where, P s gven before. ( 3 ) Selectng Q = I also leads to the least bounds. Insde the dead zone: V = W Wˆ 3 Select Wˆ = 0 hen V = 0 Note: By the selecton, V s contnuous at the boundary of ths deadzone. Dr. Radhakant Padh, AE Dept., IISc-Bangalore 6

27 Neuro-adaptve Desgn: Implementaton of Controller (1) Desgn Parameter selecton: Wˆ 0 = 0 Q P= E = I [formula gven] = ε λ P max 3 ( ε s unknown ry wth teraton) Dr. Radhakant Padh, AE Dept., IISc-Bangalore 7

28 Neuro-adaptve Desgn: Implementaton of Controller () Weght update Rule: ˆ W = γ e Pbβ, f e > E 0 x 0 where, e =, b=, x 1 β,, : Bass functon selecton P = ( X X U) γ : Learng rate otherwse ( U: pseudo control ) Dr. Radhakant Padh, AE Dept., IISc-Bangalore 8

29 Neuro-adaptve Desgn: Implementaton of Controller (3) Control Computaton: Adaptve Control: Pseudo Control: Actual Control: Uˆ = Wˆ U = U ˆ I Uad = X + K X + K X Uˆ ˆ δ ad = = fˆ 1 β c d p ad ( X, X, U) NN,, 1 ( X X U) Dr. Radhakant Padh, AE Dept., IISc-Bangalore 9

30 Neuro-adaptve desgn Nce Results: () Adapton happens n fnte tme () () () As t, the error e t les nsde the deadzone and W a constant value. t approaches Dr. Radhakant Padh, AE Dept., IISc-Bangalore 30

31 Promsng Results: (Flght control Problem) Dr. Radhakant Padh, AE Dept., IISc-Bangalore 31

32 References B. S. Km and A. J. Calse, Nonlnear Flght Control Usng Neural Networks, Journal of Gudance, Control and Dynamcs, Vol. 0, No. 1, 1997, pp J. Letner, A. J. Calse and J. V. R. Prasad, Analyss of Adaptve Neural Networks for Helcopter Flght Controls, Journal of Gudance, Control, and Dynamcs, Vol. 0, No. 5, 1997, pp M. Sharma and A. J. Calse, Neural-Network Augmentaton of Exstng Lnear Controllers, J. of Gudance, Control and Dynamcs, Vol. 8, No. 1, 005, pp Dr. Radhakant Padh, AE Dept., IISc-Bangalore 3

33 Dr. Radhakant Padh, AE Dept., IISc-Bangalore 33