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

Nonlinear System Identification Benchmarks POLYNOMIAL CONSTRAINED FACTORING IN P-NARX IDENTIFICATION KIA N A KA R A M I, D A V ID W ES TW IC K 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 y ( t ) = F ( u ( t ), u ( t − 1), … , u ( t − n u ), y ( t − 1), … , y ( t − n y )) + e ( t ) Advantages � Maybe line ar in parame te rs but no t always � Bro ad applic atio n � T o o lbo x is available Disadvantages � T o o many parame te rs to ide ntify � Diffic ult to inte rpre t � A blac k bo x mo de l 2

POLYNOMIAL NARX MODEL � Po lyno mial NARX Mo de l y ( t ) = F ( u ( t ), u ( t − 1), … , u ( t − n u ), y ( t − 1), … , y ( t − n y )) = ⎡ ⎤ p m n y n u y ( t − k i ) u ( t − k i ) M m ∏ ∏ c p , q ( k 1 , k 2 , … , k m ) ∑ ∑ ∑ ∑ ⎢ ⎥ ⎣ ⎦ i = 1 i = p + 1 m = 1 p = 0 k p + 1, … , k m = 1 k 1 , … , k p 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 3

DECOUPLED POLYNOMIAL NARX MODEL � De c o uple d po lyno mial alg o rithms fo r NARX mo de l Multivariate Se t o f Univariate CPD Po lyno mials Po lyno mials Advantages � SI SO no nline arity � L e ss parame te rto ide ntify 4

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 x 1 f 1 ( z ) z 1 f 1 ( z ) z 1 g 1 ( x 1 ) g 1 ( x 1 ) � � � f ( z ) � � x m z m z m W T V f m ( z ) g r ( x r ) g r ( x r ) f m ( z ) 5

JACOBIAN BASED DECOUPLING � E � Collec t Jac obian matric es valuate f(z) at sampling points J f (z (1) ), J f (z (2) ), J f (z (3) ), J f (z (4) ) … ⎡ ⎤ f ( z ) = W g i ( v i T z ) ⎣ ⎦ ⎤ ⎡ ′ T z ) g 1 ( v 1 0 ⎢ ⎥ ⎢ ⎥ J f ( z ) = W � ⎢ ⎥ ′ T z ) 0 g r ( v r ⎢ ⎥ ⎣ ⎦ 6

JACOBIAN BASED DECOUPLING � T ensor of stac ked Jac obian matr ic es 7

MISO POLYNOMIAL NARX MODEL � SI SO no nline arity x 1 z 1 f 1 ( x 1 ) y � � No nline ar in so me � ∑ T V x m z m parame te rs f r ( x r ) f ( x r ) f r � � � � � � � � � � � �� � � � � �� � � � � � � � � � � ��� ��� ��� � �� � ��� � � � � � � � ���� � ��� � � � � 8

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! 9

HESSIAN DECOMPOSITION � He ssian is a te nso r V V = � T e nso r unique ne ss re sults H V V = apply! h 1 h r = + + . . . v 1 ⎡ ⎤ v r v 1 ′′ v r T z ) g 1 ( v 1 0 ⎢ ⎥ ⎢ ⎥ J f ( z ) = V � V T ⎢ ⎥ ′′ T z ) 0 g r ( v r ⎢ ⎥ ⎣ ⎦ 10

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 11

DECOUPLING UNSTRUCTURED HESSIAN 12

IMPOSING POLYNOMIAL STRUCTURE � 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 13

IMPOSING POLYNOMIAL STRUCTURE 14

OPTIMIZATION SOLUTION � Use the V fac to r to initialize an o ptimizatio n ( ˆ V , ˆ C ) = argmin V , C || y − ˆ y ( V , C ) || 2 � 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) 15

BOUC-WEN BENCHMARK � Represent hysteretic effec ts � Nonlinear behavior depends on an internal state � No t me asurable dire c tly � B ouc -Wen model: y ( t ) + c L � y ( t ) + k L y ( t ) + z ( y ( t ), � y ( t )) = u ( t ) m L �� y ( t ) | z ( t ) | υ ) y ( t ) || z ( t ) | υ − 1 z ( t ) + δ � y ( t )) = α � y ( t ) − β ( γ | � z ( y ( t ), � � 16

BOUC-WEN BENCHMARK � Data generation using Identification Data 200 example provided with 150 100 B enc hmark Force (N) 50 0 − 50 − 100 − 150 46 46.2 46.4 46.6 46.8 47 − 3 2 x 10 1 Displacement (m) 0 − 1 − 2 46 46.2 46.4 46.6 46.8 47 time (sec) 17

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 18

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 19

RESULTS 20

RESULTS 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% 21

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?! 22

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