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

SLIDE 1

Reducing nonlinear state-space models through polynomial decoupling

Jan Decuyper, Koen Tiels, Johan Schoukens

Workshop on Nonlinear System Identification Benchmarks 2019

SLIDE 2

Motivating example

Black-box Static nonlinear function

0.1
0.05
0.2
0.2

0.05 0.1 0.2 0.2

0.02

0.2 0.2 0.02 0.04

0.2
0.2

10 d.o.f Reduced model Static nonlinear function

0.2

0.2

0.04
0.02

0.02 0.04

0.2

0.2

0.015
0.01
0.005

0.005 0.01 0.015

3 d.o.f

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SLIDE 3

Outline

◮ Polynomial nonlinear state-space: a universal model ◮ Model reduction

◮ Polynomial decoupling from first-order information ◮ Reshaping the nonlinear functions

◮ Benchmark case studies ◮ Conclusions

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SLIDE 4

Nonlinear state-space: a universal model

Discrete-time Polynomial nonlinear state-space (PNLSS)

x(k + 1) = Ax(k) + Bu(k) + Eζ(x(k), u(k)) y(k) = Cx(k) + Du(k) + Fη(x(k), u(k)) (1a) (1b)

e.g. ζ(x(k), u(k)) = x 2

1 (k)

x1(k)x2(k) x1(k)u1(k) u2

1(k)

· · · T (2)

Nonlinear state-space NARX Block-oriented Volterra

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SLIDE 5

Nonlinear state-space: a universal model

PNLSS Toolbox v1.0 (tutorial on Thursday)

1. Linear subspace

identification: A, B, C, D (E = 0 and F = 0)

2. Nonlinear optimisation

x(k + 1) = Ax(k) + Bu(k) + Eζ(x(k), u(k)) y(k) = Cx(k) + Du(k) + Fη(x(k), u(k))

Applications:

◮ Duffing oscillator (nonlinear stiffness) ◮ Van der Pol (nonlinear damping) ◮ Bouc-Wen (hysteresis) ◮ Li-Ion battery ◮ unsteady fluid dynamics ◮ . . .

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SLIDE 6

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)

1

x(k)

n

u(k)

1

u(k)

m

f(x, u) p(k)

1

p(k)

n

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SLIDE 7

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

SLIDE 8

Model reduction

Polynomial decoupling from first-order information

x(k)

1

x(k)

n

u(k)

1

u(k)

m

f(x, u) p(k)

1

p(k)

n

x(k)

1

x(k)

n

u(k)

1

u(k)

m

VT

g1(z1) gr(zr)

W

p(k)

1

p(k)

n

f(x, u) = Wg

VT
x

u

(3)

f(x, u) =

x 2

1 x2

x 3

2

=

W

1/6

1/6 −1/3

1

g

z3

1

z3

2

z3

3

,

z =

VT

1 1 −1 1 1

x
x1

x2

.

(4) 8/25

SLIDE 9

Model reduction

Polynomial decoupling from first-order information f(x, u) = Wg

VT
x

u

(5)

J(x, u) = W diag

g′

i

vT

i

x

u

VT

(6)

f(x,u)

J(k) =

   

∂f (k) 1 ∂x1

· · ·

∂f (k) 1 ∂xn , ∂f (k) 1 ∂u1

· · ·

∂f (k) 1 ∂um

. . . ... . . . . . . ... . . .

∂f (k) n ∂x1

· · ·

∂f (k) n ∂xn , ∂f (k) n ∂u1

· · ·

∂f (k) n ∂um

   

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SLIDE 10

Model reduction

Polynomial decoupling from first-order information

◮ three-way tensor J ◮ simultaneous diagionalisation ◮ canonical polyadic

decomposition (CPD)

◮ sum of r rank-1 terms with

r = rank J

= = = = W V H W H V W V w1 v1 h1 + . . . + wr vr hr

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SLIDE 11

Model reduction

Polynomial decoupling from first-order information

J = W, V, H (7) gi(zi) =

hi(zi)dzi,

zi = vi

x

u

(8)

f(x, u) = Wg

VT
x

u

x(k)

1

x(k)

n

u(k)

1

u(k)

m

VT

g1(z1) gr(zr)

W

p(k)

1

p(k)

n

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SLIDE 12

Outline

◮ Polynomial nonlinear state-space: a universal model ◮ Model reduction

◮ Polynomial decoupling from first-order information ◮ Reshaping the nonlinear functions

◮ Benchmark case studies ◮ Conclusions

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SLIDE 13

Model reduction

Reshaping the nonlinear functions

Reducing the number of branches

◮ exploiting linear dependencies amongst branches ◮ repeated optimisation on model-level x(k)

1

x(k)

n

u(k)

1

u(k)

m

VT

g1(z1) gr (zr )

W

p(k)

1

p(k)

n

x(k)

1

x(k)

n

u(k)

1

u(k)

m

˜ vT

˜ g(˜ z)

˜ w

p(k)

1

p(k)

n

◮ Balance model accuracy to complexity

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SLIDE 14

Model reduction

Coupled PNLSS model

x(k + 1) = Ax(k) + Bu(k) + Eζ(x(k), u(k))

y(k) = Cx(k) + Du(k) + Fη(x(k), u(k)) (9a) (9b)

Reduced decoupled PNLSS model

    

x(k + 1) = Ax(k) + Bu(k) + Wxgx

VT

x

u

y(k) = Cx(k) + Du(k) + Wygy
VT

y

x

u

,

(10a) (10b)

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SLIDE 15

Outline

◮ Polynomial nonlinear state-space: a universal model ◮ Model reduction

◮ Polynomial decoupling from first-order information ◮ Reshaping the nonlinear functions

◮ Benchmark case studies ◮ Conclusions

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SLIDE 16

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 + βy2(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)

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SLIDE 17

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

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SLIDE 18

Benchmark case studies

The Silverbox Validation error

1 2 3 4 5 0.002 0.004 0.006 0.008 0.01

rel. rms error

solid: full PNLSS model markers: reduced model blue: realisation red: arrow Single branch

0.2

0.2

0.06
0.04
0.02

0.02 0.04 0.06

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SLIDE 19

Benchmark case studies

The Silverbox Coupled PNLSS

            

x(k + 1) = Ax(k) + Bu(k) +

e11

e12 e13 e14 e15 e16 e17 e21 e22 e23 e24 e25 e26 e27



   

x2

1 (k)

x1(k)x2(k) x2

2 (k)

x3

1 (k)

x2

1 (k)x2(k)

x1(k)x2

2 (k)

x3

2 (k)

    

y(k) = cx(k) + du(k), (13a) (13b)

Reduced r = 1 model

        

x(k + 1) = Ax(k) + Bu(k) +

w1

w2

θ1z3(k) + θ2z2(k)

y(k) = cx(k) + du(k), z(k) = [v1 v2]

x1(k)

x2(k)

,

(14a) (14b) (14c)

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SLIDE 20

Benchmark case studies

The Silverbox

coupled PNLSS r = 1 Linear state nonlinearity fx degrees 2, 3 degrees 2, 3

utput nonlinearity fy
# d.o.f

19 10 5 eRMSt | rel. erms val. R 4.5 × 10−4|0.0084 4.5 × 10−4|0.0084 0.014|0.25 eRMSt | rel. erms noise arrow 2.9 × 10−4|0.0054 3.3 × 10−4|0.0061 0.007|0.13

Validation arrow

10 20 30 40 50 60 Time (s)

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0.1

0.1 0.2 0.3

Validation realisation

20 40 60 80 100 120 140 160 180 200 Frequency (Hz)

140
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60
50
40

Black: output Blue : PNLSS error Red: 1-branch error 20/25

SLIDE 21

Benchmark case studies

The Bouc-Wen system Hysteresis system with dynamic nonlinearity

m¨ y(t) + c ˙ y(t) + ky(t) + fH(y(t), ˙ y(t)) = u(t), (15) ˙ fH(t) = α ˙ y(t) − (γ| ˙ y(t)|fH(t) + δ ˙ y(t)|fH(t)|) , (16) Training data: random-phase multisine Validation data: realisation a of random-phase multisines and a swept sine

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SLIDE 22

Benchmark case studies

The Bouc-Wen system Validation error

1 2 3 4 5 6 7 0.02 0.04 0.06 0.08

rel. rms error

solid: full PNLSS model markers dotted: reduced model blue: multisine realisation red: sine sweep

r = 4

10 0

10

1500
1000
500

500 1000

5 0 5
150
100
50

50 100 150

4 0

4

100
50

50 100

5 0 5
300
200
100

100 200 300 400

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SLIDE 23

Benchmark case studies

The Bouc-Wen system

coupled PNLSS r = 4 r = 1 Linear state nonlinearity fx degrees 2,3 degrees 2,3,4,5 degrees 2,3,4,5

utput nonlinearity fy
# d.o.f

97 43 16 7 eRMSt | rel. erms val. R 1.9 × 10−5|0.029 2.1 × 10−5|0.031 5.2 × 10−5|0.079 1.6 × 10−4|0.23 eRMSt | rel. erms val. sweep 1.2 × 10−5|0.017 1.3 × 10−5|0.019 3.9 × 10−5|0.059 1.5 × 10−4|0.22

Validation realisation

20 40 60 80 100 120 140 160 180 200 Frequency (Hz)

160
150
140
130
120
110
100
90
80

Validation sine sweep

10 20 30 40 50 60 70 80 90 100 Frequency (Hz)

200
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160
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120
110
100
90

Black: output Blue : PNLSS error Red: 4-branch error 23/25

SLIDE 24

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