POLYNOMIAL CONSTRAINED FACTORING IN P-NARX IDENTIFICATION KIA N A - - PowerPoint PPT Presentation

KIA N A KA R A M I, D A V ID W ES TW IC K

POLYNOMIAL CONSTRAINED FACTORING IN P-NARX IDENTIFICATION

Nonlinear System Identification Benchmarks

April, 2019

NARX MODEL

T he No nline ar Auto re g re ssive e Xo g e no us I nput Mo de l

Advantages

Maybe line ar in parame te rs but no t always Bro ad applic atio n T

• o lbo x is available

Disadvantages

T

• o many parame te rs to ide ntify

Diffic ult to inte rpre t A blac k bo x mo de l

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y(t) = F(u(t),u(t −1),…,u(t − nu), y(t −1),…, y(t − ny)) + e(t)

POLYNOMIAL NARX MODEL

Po lyno mial NARX Mo de l

Advantages

Output is line ar func tio n o f parame te r

Disadvantages

Po o r e xtrapo latio n Diffic ulty in de te c ting e xtrapo latio n

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y(t) = F(u(t),u(t −1),…,u(t − nu), y(t −1),…, y(t − ny)) =

cp,q(k1,k2,…,km) y(t−ki) u(t−ki)

i= p+1 m

∏

i=1 p

∏ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥

kp+1,…,km=1 nu

∑

k1,…,kp ny

∑

p=0 m

∑

m=1 M

∑

DECOUPLED POLYNOMIAL NARX MODEL

De c o uple d po lyno mial alg o rithms fo r NARX mo de l

Advantages

SI SO no nline arity L e ss parame te rto ide ntify

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Multivariate Po lyno mials Se t o f Univariate Po lyno mials CPD

MIMO POLYNOMIAL DECOUPLING

T enso r metho ds c an be used to dec ouple MI MO polynomials Applic atio ns in Parallel Wiener

• H

ammerstein model, Polynomial state spac e models

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g1(x1) g1(x1)

V

T

gr(xr) gr(xr)

z1 zm

• x1

xm

• W

f1(z) fm(z)

• f (z)

z1 zm

• f1(z)

fm(z)

JACOBIAN BASED DECOUPLING

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f (z) = W gi(vi

Tz)

⎡ ⎣ ⎤ ⎦ J f (z) = W ′ g1(v1

Tz)

• ′

gr(vr

Tz)

⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

E valuate f(z) at sampling points Collec t Jac obian matric es Jf(z(1)), Jf(z(2)), Jf(z(3)), Jf(z(4)) …

JACOBIAN BASED DECOUPLING

T ensor

• f stac ked Jac obian matr

ic es

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MISO POLYNOMIAL NARX MODEL

• 8

y

V

T

f1(x1) fr(xr) fr f (xr) ∑

x1

z1 zm

xm

• SI

SO no nline arity

• No nline ar in so me

parame te rs

DECOUPLING THE JACOBIAN

Jac o bian:

• Single o utput, e ac h Jac o bian is a ve c to r, no t a matrix

Stac king o pe rating po ints c re ate s a matrix Matrix fac to rizatio n pro ble m T e nso r unique ne ss re sults no lo nge r apply!

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HESSIAN DECOMPOSITION

= = = V H V V V v1 v1 h1 + . . . + vr vr hr

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J f (z) = V ′′ g1(v1

Tz)

• ′′

gr(vr

Tz)

⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ V T He ssian is a te nso r T

e nso r unique ne ss re sults apply!

DECOUPLING UNSTRUCTURED HESSIAN

T he H essian tensor dec omposed using CPD. Dec omposition results in V (twic e) and H matric es. V: is direc tly related to dec oupled model H : should c ontain the values of the (twic e differentiated ) polynomials

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DECOUPLING UNSTRUCTURED HESSIAN

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IMPOSING POLYNOMIAL STRUCTURE

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T he standard CPD uses Alternating L east Square. T ensor is linear in all of the elements. When impose polyno mial c onstraint, its no t linear anymore so AL S c annot be used U se Separable L east Square instead

T he te nso r value s no nline ar in V but line ar in C T re at C as a func tio n o f V and o ptimize o ve r V

IMPOSING POLYNOMIAL STRUCTURE

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OPTIMIZATION SOLUTION

Use the V fac to r to initialize an o ptimizatio n No n-Co nve x Optimizatio n

F ac to ring He ssian pro vide s e stimate o f V C appe ars line arly (SL S o ptimizatio n)

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( ˆ V , ˆ C) = argmin

V ,C || y − ˆ

y(V ,C) ||2

BOUC-WEN BENCHMARK

Represent hysteretic effec ts Nonlinear behavior depends on an internal state

No t me asurable dire c tly

B

• uc -Wen model:

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mL y(t) + cL y(t) + kLy(t) + z(y(t), y(t)) = u(t)

• z(y(t),

y(t)) = α y(t) − β(γ | y(t) || z(t) |υ−1 z(t) +δ y(t) | z(t) |υ )

BOUC-WEN BENCHMARK

Data generation using example provided with B enc hmark

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46 46.2 46.4 46.6 46.8 47 −150 −100 −50 50 100 150 200 Identification Data Force (N) 46 46.2 46.4 46.6 46.8 47 −2 −1 1 2 x 10

−3

Displacement (m) time (sec)

BOUC-WEN BENCHMARK

Model spec ific ation

Numbe r o f past input: 4 Numbe r o f past o utput: 6 Numbe r o f Branc he s: 10 Po lyno mial de gre e : 8

Considerably smaller than previo us models L imited by the size o f that Jac obian o f the H essian

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BOUC-WEN BENCHMARK

100 model with random initialization 1 model with unstruc tured CPD 1 mode with polynomial c onstr aint

Co mpare the c o nve rge nc e during o ptimizatio n Validatio n re sults

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RESULTS

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RESULTS

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Initia l So lutio n Fit Pe rc e nt

U nstruc ture d CPD o f He ssian 97.63% Struc ture d CPD o f He ssian 97.63% Rando m (Ave rage ) 97.54%

CONCLUSION

F ac toring the hessian for initialization is worthen Computing the H essian of struc tured model is very expensive Does it worth doing?!

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