Reducing nonlinear state-space models through polynomial decoupling - PowerPoint PPT Presentation

Reducing nonlinear state-space models through polynomial decoupling Jan Decuyper, Koen Tiels, Johan Schoukens Workshop on Nonlinear System Identification Benchmarks 2019

Motivating example Black-box Reduced model Static nonlinear function Static nonlinear function 0.04 0.1 0.02 0.05 0 0 -0.05 -0.02 -0.1 -0.2 -0.2 0 -0.04 0 -0.2 0 0.2 0.2 0.2 0.015 0.01 0.04 0.005 0.02 0 0 -0.005 -0.02 -0.01 0.2 0.2 0 0 -0.015 -0.2 0 0.2 -0.2 -0.2 10 d.o.f 3 d.o.f 2/25

Outline ◮ Polynomial nonlinear state-space: a universal model ◮ Model reduction ◮ Polynomial decoupling from first-order information ◮ Reshaping the nonlinear functions ◮ Benchmark case studies ◮ Conclusions 3/25

Nonlinear state-space: a universal model Discrete-time Polynomial nonlinear state-space (PNLSS) � x ( k + 1) = Ax ( k ) + Bu ( k ) + E ζ ( x ( k ) , u ( k )) (1a) y ( k ) = Cx ( k ) + Du ( k ) + F η ( x ( k ) , u ( k )) (1b) · · · � T e.g. ζ ( x ( k ) , u ( k )) = � x 2 u 2 1 ( k ) x 1 ( k ) x 2 ( k ) x 1 ( k ) u 1 ( k ) 1 ( k ) (2) Nonlinear state-space Block-oriented NARX Volterra 4/25

Nonlinear state-space: a universal model PNLSS Toolbox v1.0 (tutorial on Thursday) x ( k + 1) = Ax ( k ) + Bu ( k ) + E ζ ( x ( k ) , u ( k )) 1. Linear subspace y ( k ) = Cx ( k ) + Du ( k ) + F η ( x ( k ) , u ( k )) identification: A , B , C , D ( E = 0 and F = 0) 2. Nonlinear optimisation Applications: ◮ Duffing oscillator (nonlinear stiffness) ◮ Van der Pol (nonlinear damping) ◮ Bouc-Wen (hysteresis) ◮ Li-Ion battery ◮ unsteady fluid dynamics ◮ . . . 5/25

Nonlinear state-space: a universal model Cons ◮ unphysical states ◮ large number of parameters ◮ little insight into the system ◮ rely on large multivariate polynomials ◮ bad extrapolation behaviour x ( k ) p ( k ) 1 1 x ( k ) n f ( x , u ) u ( k ) 1 p ( k ) n u ( k ) m 6/25

Outline ◮ Polynomial nonlinear state-space: a universal model ◮ Model reduction ◮ Polynomial decoupling from first-order information ◮ Reshaping the nonlinear functions ◮ Benchmark case studies ◮ Conclusions 7/25

Model reduction Polynomial decoupling from first-order information x ( k ) x ( k ) p ( k ) p ( k ) 1 1 g 1 ( z 1 ) 1 1 x ( k ) x ( k ) V T n n f ( x , u ) W u ( k ) u ( k ) 1 1 p ( k ) p ( k ) g r ( z r ) u ( k ) n u ( k ) n m m � � �� x V T f ( x , u ) = Wg (3) u V T g � 1 W x � z 3 � 1 � � � � 1 / 6 � � � x 2 1 / 6 − 1 / 3 1 1 x 2 x 1 z 3 f ( x , u ) = = , z = − 1 1 . (4) x 3 2 0 0 1 x 2 2 z 3 0 1 3 8/25

Model reduction Polynomial decoupling from first-order information � � �� x V T f ( x , u ) = Wg (5) u � � � ��� x g ′ v T V T J ( x , u ) = W diag (6) i i u  ∂ f ( k ) ∂ f ( k ) ∂ f ( k ) ∂ f ( k )  1 1 1 1 · · · · · · ∂ xn , ∂ x 1 ∂ u 1 ∂ um   J ( k ) = . . . . ... ... . . . .   . . . .   ∂ f ( k ) ∂ f ( k ) ∂ f ( k ) ∂ f ( k ) n n n n · · · ∂ xn , · · · ∂ x 1 ∂ u 1 ∂ um f ( x , u ) 9/25

Model reduction Polynomial decoupling from first-order information ◮ three-way tensor J V = W ◮ simultaneous diagionalisation H V = W ◮ canonical polyadic decomposition (CPD) H V = W ◮ sum of r rank-1 terms with h 1 h r r = rank J = + + . . . v 1 w 1 v r w r 10/25

Model reduction Polynomial decoupling from first-order information J = � W , V , H � (7) � � � x g i ( z i ) = h i ( z i ) dz i , z i = v i (8) u x ( k ) p ( k ) 1 g 1 ( z 1 ) 1 x ( k ) � � �� x n V T W V T f ( x , u ) = Wg u ( k ) u 1 p ( k ) g r ( z r ) u ( k ) n m 11/25

Outline ◮ Polynomial nonlinear state-space: a universal model ◮ Model reduction ◮ Polynomial decoupling from first-order information ◮ Reshaping the nonlinear functions ◮ Benchmark case studies ◮ Conclusions 12/25

Model reduction Reshaping the nonlinear functions Reducing the number of branches ◮ exploiting linear dependencies amongst branches ◮ repeated optimisation on model-level x ( k ) x ( k ) p ( k ) p ( k ) 1 1 g 1 ( z 1 ) 1 1 x ( k ) x ( k ) V T v T n n ˜ W ˜ ˜ g (˜ z ) w u ( k ) u ( k ) 1 1 p ( k ) p ( k ) g r ( z r ) u ( k ) n u ( k ) n m m ◮ Balance model accuracy to complexity 13/25

Model reduction Coupled PNLSS model � x ( k + 1) = Ax ( k ) + Bu ( k ) + E ζ ( x ( k ) , u ( k )) (9a) y ( k ) = Cx ( k ) + Du ( k ) + F η ( x ( k ) , u ( k )) (9b) Reduced decoupled PNLSS model  � � x �� V T x ( k + 1) = Ax ( k ) + Bu ( k ) + W x g x (10a)  x u  � � x �� V T y ( k ) = Cx ( k ) + Du ( k ) + W y g y , (10b)   y u 14/25

Outline ◮ Polynomial nonlinear state-space: a universal model ◮ Model reduction ◮ Polynomial decoupling from first-order information ◮ Reshaping the nonlinear functions ◮ Benchmark case studies ◮ Conclusions 15/25

Benchmark case studies The Silverbox Electrical implementation of the forced Duffing oscillator m ¨ y ( t ) + c ˙ y ( t ) + k ( y ( t )) y ( t ) = u ( t ) , (11) k ( y ( t )) = α + β y 2 ( t ) . (12) Training data: 10 realisations of random-phase multisines Validation data: ◮ 1 realisation of random-phase multisines ◮ filtered Gaussian noise with increasing amplitude (arrow) 16/25

Benchmark case studies 3-step idenfitication procedure: 1. Identify a coupled PNLSS model from the data 2. Decouple the multivariate polynomial function 3. Reduce the number of branches in the description 17/25

Benchmark case studies The Silverbox Single branch Validation error 0.01 0.06 0.04 0.008 rel. rms error 0.02 0.006 0 0.004 -0.02 0.002 -0.04 0 -0.06 0 1 2 3 4 5 -0.2 0 0.2 solid: full PNLSS model markers: reduced model blue: realisation red: arrow 18/25

Benchmark case studies The Silverbox Coupled PNLSS  x 2  1 ( k )   x 1 ( k ) x 2 ( k )   x 2 2 ( k )  � �    e 11 e 12 e 13 e 14 e 15 e 16 e 17 x 3  x ( k + 1) = Ax ( k ) + B u ( k ) +  1 ( k )  (13a) e 21 e 22 e 23 e 24 e 25 e 26 e 27  x 2  1 ( k ) x 2 ( k )   x 1 ( k ) x 2 2 ( k )   x 3  2 ( k )    y ( k ) = cx ( k ) + du ( k ) , (13b) Reduced r = 1 model  � �� w 1 θ 1 z 3 ( k ) + θ 2 z 2 ( k ) � x ( k + 1) = Ax ( k ) + B u ( k ) + (14a)  w 2    y ( k ) = cx ( k ) + du ( k ) , (14b) � �  x 1 ( k )  z ( k ) = [ v 1 v 2 ] (14c)  ,  x 2 ( k ) 19/25

Benchmark case studies The Silverbox coupled PNLSS r = 1 Linear state nonlinearity f x degrees 2, 3 degrees 2, 3 - output nonlinearity f y - - - # d.o.f 19 10 5 4 . 5 × 10 − 4 | 0 . 0084 4 . 5 × 10 − 4 | 0 . 0084 e RMSt | rel. e rms val. R 0 . 014 | 0 . 25 2 . 9 × 10 − 4 | 0 . 0054 3 . 3 × 10 − 4 | 0 . 0061 e RMSt | rel. e rms noise arrow 0 . 007 | 0 . 13 Validation arrow Validation realisation 0.3 -40 -50 0.2 -60 0.1 -70 -80 0 -90 -0.1 -100 -110 -0.2 -120 -0.3 -130 -0.4 -140 0 10 20 30 40 50 60 0 20 40 60 80 100 120 140 160 180 200 Time (s) Frequency (Hz) Black: output Blue : PNLSS error Red: 1-branch error 20/25

Benchmark case studies The Bouc-Wen system Hysteresis system with dynamic nonlinearity m ¨ y ( t ) + c ˙ y ( t ) + ky ( t ) + f H ( y ( t ) , ˙ y ( t )) = u ( t ) , (15) ˙ f H ( t ) = α ˙ y ( t ) − ( γ | ˙ y ( t ) | f H ( t ) + δ ˙ y ( t ) | f H ( t ) | ) , (16) Training data: random-phase multisine Validation data: realisation a of random-phase multisines and a swept sine 21/25

Benchmark case studies The Bouc-Wen system Validation error r = 4 1000 150 100 400 0.08 300 100 500 50 0.06 200 rel. rms error 50 0 100 0 0 0.04 0 -500 -50 -100 -50 0.02 -1000 -100 -200 -100 0 -1500 -150 -300 -10 0 10 -5 0 5 -4 0 4 -5 0 5 0 1 2 3 4 5 6 7 solid: full PNLSS model markers dotted: reduced model blue: multisine realisation red: sine sweep 22/25

Benchmark case studies The Bouc-Wen system coupled PNLSS r = 4 r = 1 Linear state nonlinearity f x degrees 2,3 degrees 2,3,4,5 degrees 2,3,4,5 - output nonlinearity f y - - - - # d.o.f 97 43 16 7 1 . 9 × 10 − 5 | 0 . 029 2 . 1 × 10 − 5 | 0 . 031 5 . 2 × 10 − 5 | 0 . 079 1 . 6 × 10 − 4 | 0 . 23 e RMSt | rel. e rms val. R 1 . 2 × 10 − 5 | 0 . 017 1 . 3 × 10 − 5 | 0 . 019 3 . 9 × 10 − 5 | 0 . 059 1 . 5 × 10 − 4 | 0 . 22 e RMSt | rel. e rms val. sweep Validation realisation Validation sine sweep -80 -90 -100 -90 -110 -100 -120 -110 -130 -140 -120 -150 -130 -160 -170 -140 -180 -150 -190 -160 -200 0 20 40 60 80 100 120 140 160 180 200 0 10 20 30 40 50 60 70 80 90 100 Frequency (Hz) Frequency (Hz) Black: output Blue : PNLSS error Red: 4-branch error 23/25

Outline ◮ Polynomial nonlinear state-space: a universal model ◮ Model reduction ◮ Polynomial decoupling from first-order information ◮ Reshaping the nonlinear functions ◮ Benchmark case studies ◮ Conclusions 24/25

Conclusions ◮ Black box PNLSS models rely on complex multivariate polynomials as generic equations ◮ hard to interpret ◮ require a large number of parameters ◮ 2 model reduction actions 1. decouple the multivariate polynomials from first order information 2. reduce the number of branches ◮ Balance model complexity to accuracy 25/25