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y(k − 1)

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ˆ y(k)

u(k − 1) → ˆ y(k) y(k − 1) → ˆ y(k)

ˆ y(k) = f (u(k − 1), y(k − 1)) ˆ y(k) = f (u(k − 1), ˆ y(k − 1))

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ˆ y(k) = b0u(k) + b1u(k − 1) + b2u(k − 2) + . . . + bmu(k − m) J =

N

X

k=1

(y(k) − ˆ y(k))2 + λ θT P −1θ → min

θ

θ = [b0 b1 b2 · · · bm]T θT P −1θ = θT R θ = θT F T F θ = ||F θ||2

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F pure =      a0 a1 · · · an−1 1 · · · a0 a1 · · · an−1 1 · · · . . . ... ... ... ... . . . · · · a0 a1 · · · an−1 1     

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λ a0, a1, . . . , an−1

10 20 30 40 50

• 0.4
• 0.2

0.2 0.4 10 20 30 40 50 0.02 0.04 0.06 0.08 0.1 k k g(k) g(k) a0 = −0.9 g(k) − 0.9g(k − 1) = 0 g(k) − 1.66g(k − 1) + 0.81g(k − 2) = 0 a1 = −1.66; a0 = 0.81 Pole = 0.9 Poles = 0.9 e ±iπ/8

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k k k k k k g(k) g(k) g(k) g(k) g(k) g(k)

λ = 0 λ = 0.1 λ = 1 λ = 10 λ = 100 λ = 1000

prior process model

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model

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!1 1 2 3 4 5 !5000 !4000 !3000 !2000 u2 u1 u3 u4 Number of Inputs AICc N = 625 !1 1 2 3 4 5 !10000 !8000 !6000 !4000 u2 u1 u3 u4 Number of Inputs AICc N = 1296 0.5 1 0.5 1 ui y f(u1) f(u2) f(u3) u4 F-=$2%@EM* 6"5+$,%"2$ 7$//$2

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• 5$%$?"4,"-2:*%812-$12:*%#$-.%.2#*2328"*-4@%A:1.$

0.5 1 0.5 1 ui y f(u1) f(u2) f(u3) u4 B:A$1%=)C8 ?"4,$.%"1$ <$--$1 !1 1 2 3 4 5 !5000 !4000 !3000 u2z u1z u3z u4z Number of Inputs AICc N = 625 !1 1 2 3 4 5 !10000 !8000 !6000 u2z u1z u3z u4z Number of Inputs AICc N = 1296

!"#$%&'

!1 0 1 2 3 4 5 6 7 8 9 !5000 !4000 !3000 !2000 u1zu2z u3x u2xu1x u4zu4x u3z Number of Inputs AICc N = 625 !1 0 1 2 3 4 5 6 7 8 9 !10000 !8000 !6000 !4000 u2z u1z u3x u1x u2x u4x u3z u4z Number of Inputs AICc N = 1296

!"#$#%&'()*+,*-'($./,&01%2$3%0,+,*%4$/"5"3%1%6*-'(

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ES!Results BE!Results

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1 2 3 4 5 6 7 1 2 3 4 5 6 7 Number of Inputs Input Number N09:;3+%*$2$/+9>:%6$*32+* !1 0 1 2 3 4 5 6 7 8 !700 !650 !600 !550 !500 Number of Inputs AICc

ES!Results BE!Results

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50 100 150 200 250 50 100 150 200

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10 20 30 40 50 60 70 80 90 100110120130140 0.06 0.08 0.1

Number of net inputs Error on validation data (NRMSE)

10 20 30 40 50 60 70 80 90 100110120130140 0.4 0.45 0.5 0.55 0.6

Number of net inputs Error on validation data (NRMSE) 20 25 30 35 40 Discrete time k y1 and ˆ y1 50 100 150 200 250 5 10 error (y1 − ˆ y1) process (y1) model output (ˆ y1) error (y1 − ˆ y1) 30 40 50 60 Discrete time k y2 and ˆ y2 50 100 150 200 −10 10 20 error (y2 − ˆ y2) process (y2) model output (ˆ y2) error (y2 − ˆ y2)

()*%+$,-$*"+.*$ ()*%/.,)0)+1 23%)4-.+56%0&+2)? )4%+/$%z7)4-.+%5-"8$ 9:%)4-.+56%0&+2)? )4%+/$%z7)4-.+%5-"8$

x and z space seperated

x and z space seperated

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!"#$%&'(!)*&+)"

10 20 30 40 50 60 70 80 90 100 110 120 130 140

simulationError

0.5 1 1.5 2

forward selection backward elimination

10 20 30 40 50 60 70 80 90 100 110 120 130 140

Input

u1x(k-1) u2x(k-1) u3x(k-1) u4x(k-1) u5x(k-1) u6x(k-1) y1x(k-1) u1z(k-1) u2z(k-1) u3z(k-1) u4z(k-1) u5z(k-1) u6z(k-1) y1z(k-1)

forward selection backward elimination

! X Z ,'"'-&#)*

' ($)$*+,-.%-.%+/$%0,12)"+$3%-2+42+%-5% 6"),3"+,-.%3"+"7 ' 8"*9:";3%0$)$*+,-.%#,6$0%0,#.,5,*".+)<% =$++$;%;$02)+07 ' !"#$%&'()*+,-.,-**/,*$0,-$*(11%2%-&)$1/,$0$ .//3$4/3-56 ' >))-:$3%3$)"<0?%@9ABC%D%@9ABEC7 ' F-;:";3%0$)$*+,-.?%G;,$0%+-%1-3$)%+/$% 4;-*$00%:,+/%12)+,4)$%.-.),.$";%,.5)2$.*$07 ' 8"*9:";3%0$)$*+,-.?%H-.),.$";,+,$0%-.)<%.$$3% +-%=$%*-.0,3$;$3%,.%2B@9ABC%".3%2I@9AICJ

K%".3%L%04"*$%0$4";"+$3 K%".3%L%04"*$%0$4";"+$3

!"#$%&'

!"##$%&'()*+,%-#./$#01-%(2#3+45#/$67#8*)9:;1<

=#6*+1*2#>#8)'22+)?#6*1@%22###################3+45 =#$%)(+,)4+1?2#%)@5A#1*-%*#! B#CD

"( )*+,--$."/$0%1%!" 2*0$3$*0$*/4%*,*50$+-$"62*# 7( )*+,--$."/$0%1%!" 2*0$3$*0$*/4%$839%0$+-$"62*# +( +,--$."/$0%1%!" 0$3$*0$*/4%$839%0$+-$"62*# $%&"#./$#E-+)&"#F%*?%(G $%&"#./$#EHI#F%*?%(G J?*%&'()*+,%- ./$

5 10 15 20

• 2
• 1

1 2

k bk k bk c)

5 10 15 20

• 2
• 1

1 2 5 10 15 20

• 3
• 2
• 1

1 2 3

a) b) k bk

e− 1

2 θTP θ

θ = [b1 b2 · · · bm]T