PNLSS 1.0 demo Koen Tiels Uppsala University Jan Decuyper Vrije - - PowerPoint PPT Presentation

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PNLSS 1.0 demo Koen Tiels Uppsala University Jan Decuyper Vrije - - PowerPoint PPT Presentation

Mar 04, 2024 194 likes •344 views

PNLSS 1.0 demo Koen Tiels Uppsala University Jan Decuyper Vrije Universiteit Brussel Toolbox available at http://homepages.vub.ac.be/~jschouk Goal: Capture nonlinear dynamic system behavior Error signal Input Output + System +

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SLIDE 1

PNLSS 1.0 demo

Koen Tiels Uppsala University Jan Decuyper Vrije Universiteit Brussel

Toolbox available at http://homepages.vub.ac.be/~jschouk

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SLIDE 2

Goal: Capture nonlinear dynamic system behavior

System Model + Input Output Modeled

  • utput

Error signal

+ –

2/13

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SLIDE 3

Goal: Capture nonlinear dynamic system behavior

System Model + Input Output Modeled

  • utput

Error signal

+ –

3/13

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SLIDE 4

Goal: Capture nonlinear dynamic system behavior

System Model + Input Output Modeled

  • utput

Error signal

+ –

4/13

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SLIDE 5

Linear model in PNLSS form

x(t + 1) = ABLA x(t) + BBLA u(t) + E = 0 ζ(x(t), u(t)) y(t) = CBLA x(t) + DBLA u(t) + F = 0 η(x(t), u(t))

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with e.g. ζ(x, u) =             x2

1

x1x2 x1u . . . x2

2u

u3 . . .            

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SLIDE 6

Linear model in PNLSS form

x(t + 1) = ABLA x(t) + BBLA u(t) + E = 0 ζ(x(t), u(t)) y(t) = CBLA x(t) + DBLA u(t) + F = 0 η(x(t), u(t))

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with e.g. ζ(x, u) =             x2

1

x1x2 x1u . . . x2

2u

u3 . . .             % Set which monomials are free for optimization model init.xactive = fSelectActive(’statesonly’,n,m,n,nx);

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SLIDE 7

Linear model in PNLSS form

x(t + 1) = ABLA x(t) + BBLA u(t) + E = 0 ζ(x(t), u(t)) y(t) = CBLA x(t) + DBLA u(t) + F = 0 η(x(t), u(t))

7/13

with e.g. ζ(x, u) =             x2

1

x1x2 x1u . . . x2

2u

u3 . . .             % Set which monomials are free for optimization model init.xactive = fSelectActive(’statesonly’,n,m,n,nx); model init.yactive = fSelectActive(’empty’,n,m,p,ny);

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SLIDE 8

Selection of best model on validation set

5 10 15 20 25 30 35 40 45 Successful iteration number

  • 92
  • 90
  • 88
  • 86
  • 84
  • 82
  • 80
  • 78
  • 76

Validation error [dB] Selection of the best model on a separate data set

8/13

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SLIDE 9

Estimation initial conditions x0, u0

500 1000 1500 2000 2500 Sample number

  • 4
  • 2

2 4 6 Output 10-4 Output Error with x0 = 0 Error with estimated x 0

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model.x0active = [1:n]; % Select initial states to estimate model.u0active = [1:m]; % Select initial inputs to estimate % Levenberg-Marquardt optimization [model, y mod, models] = fLMnlssWeighted x0u0(u,y,model init,nIter,W,lambda);

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SLIDE 10

Applications

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Mechanical

combine harvester semi-active damper quarter car setup robot arm hydro-static drive-line

Electronics

crystal detector

Electrochemical

Li-ion battery

Medical

interaction insulin-glucose for diabetic patients

Fluid dynamics

vortex-induced vibrations (VIV)

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SLIDE 11

VIV application

Modelling unsteady fluid dynamics: relate displacement to lift force

Cl = sin(2πfStt)

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coupled PNLSS state nonlinearity ζ degrees 0,3,5,7

  • utput nonlinearity η

degrees 0,3,5,7 # d.o.f 1396 erms validation realisation 0.06

11/13

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SLIDE 12

VIV application – result

PNLSS model

5 10 15 20 25 30 35 40 Time (s)

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

CFD model

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tic y sim = fFilterNLSS(model,u); toc

Computation time is 0.43 seconds.

simulation time of 40 s 20 x @2.40 GHz

Computation time is 45h 29min.

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SLIDE 13

Future features

◮ Decoupling ◮ Sine basis functions ◮ Outputs in the regressors

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