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
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
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. 6-33. Dr. Radhakant Padh, AE Dept., IISc-Bangalore 3
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
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
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
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
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
Neuro-adaptve desgn.e X = U +Δ X, X, U Dr. Radhakant Padh, AE Dept., IISc-Bangalore 9
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
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
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 0 1 0 hen e ˆ = e + Uad, X kp k Δ 1 d A b XU, Dr. Radhakant Padh, AE Dept., IISc-Bangalore 1
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
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
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
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
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
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
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
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
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 1 + 1 + kp k d k p k p = 1 1 1 1+ kp k d k p Dr. Radhakant Padh, AE Dept., IISc-Bangalore 1
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
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
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
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
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
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
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
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
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
Promsng Results: (Flght control Problem) Dr. Radhakant Padh, AE Dept., IISc-Bangalore 31
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. 6-33. 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. 97-979. 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. 1-19. Dr. Radhakant Padh, AE Dept., IISc-Bangalore 3
Dr. Radhakant Padh, AE Dept., IISc-Bangalore 33