PNLSS 1.0 A polynomial nonlinear state-space Matlab toolbox Koen - - PowerPoint PPT Presentation

PNLSS 1.0 A polynomial nonlinear state-space Matlab toolbox

Koen Tiels Vrije Universiteit Brussel

Applications

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Wiener-Hammerstein benchmark

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Goal: Capture nonlinear dynamic system behavior

System Model + Input Output Modeled

• utput

Error signal

+ –

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Goal: Capture nonlinear dynamic system behavior

System Model + Input Output Modeled

• utput

Error signal

+ –

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Goal: Capture nonlinear dynamic system behavior

System Model + Input Output Modeled

• utput

Error signal

+ –

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Goal: Capture nonlinear dynamic system behavior

System Model + Load profile u(t) Terminal voltage ymeas(t) Modeled terminal voltage y(t) Error signal ǫ(t)

+ –

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Goal: Capture nonlinear dynamic system behavior

System PNLSS model + Load profile u(t) Terminal voltage ymeas(t) Modeled terminal voltage y(t) Error signal ǫ(t)

+ –

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Polynomial nonlinear state-space model (PNLSS)

9/28                                  x(t + 1) = A x(t) + B u(t) + E ζ(x(t), u(t)) y(t) = C x(t) + D u(t) + F η(x(t), u(t)) linear state-space model polynomials in x and u

Polynomial nonlinear state-space model (PNLSS)

10/28                                                               x(t + 1) = A x(t) + B u(t) + E ζ(x(t), u(t)) y(t) = C x(t) + D u(t) + F η(x(t), u(t)) linear state-space model polynomials in x and u

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

1

x1x2 x1u . . . x2

2u

u3 . . .            

Identification of a PNLSS model

11/28                                                               x(t + 1) = A x(t) + B u(t) + E ζ(x(t), u(t)) y(t) = C x(t) + D u(t) + F η(x(t), u(t)) linear state-space model polynomials in x and u

θ =         vec(A) vec(B) vec(C) vec(D) vec(E) vec(F)         ǫ(k, θ) = Y (k, θ) − Ymeas(k) VWLS(θ) =

NF

• k=1

ǫH(k, θ)W (k)ǫ(k, θ) ˆ θ = arg min

θ

VWLS

Problem is nonlinear in the parameters

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Nonlinear optimization Starting values?

VWLS(θ) θ

Starting value: best linear approximation (BLA)

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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))

Nonlinear optimization

x(t + 1) = A x(t) + B u(t) + E ζ(x(t), u(t)) y(t) = C x(t) + D u(t) + F η(x(t), u(t))

Collect data

load(’WH EstimationExample.mat’) % Multisine data

Time (s) 0.05 0.1 u(t)

• 4
• 2

2 4

• Freq. (Hz) #104

2 4 6 8 |U(k)| (dB)

• 80
• 40

40

• Freq. (Hz) #104

2 4 6 8 (rad) 0 :

• :

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Best linear approximation and distortion levels

N = 4096; % Number of samples P = 2; % Number of periods R = 9; % Number of realizations m = 1; % Number of inputs p = 1; % Number of outputs lines = 2:787; % Excited frequency lines . . . % Reshape data U = fft(u); % N x m x R x P Y = fft(y); % N x p x R x P U = U(lines,:,:,:); % Input spectrum at excited frequencies Y = Y(lines,:,:,:); % Output spectrum at excited frequencies % BLA, total distortion, and noise distortion [G, covGtotal, covGnoise] = fCovarFrf(U,Y);

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Best linear approximation and distortion levels

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Frequency (Hz)

5000 10000 15000

Amplitude (dB)

• 120
• 100
• 80
• 60
• 40
• 20

BLA Total distortion Noise distortion

Linear state-space model

nAll = 2:8; % Model orders maxr = 10; % Subspace dimensioning parameter maxIter = 100; % Max. nbr. of Levenberg-Marquardt iterations fs = 78125; % Sampling frequency freq = (lines-1)*fs/N; % Frequency vector forcestability = true; % Force stable model showfigs = true; % Show overview and best model for each order % Frequency domain subspace + nonlinear optimization % McKelvey et al., 1996; Pintelon, 2002 models = fLoopSubSpace(freq,G,covGtotal,nAll,maxr,. . . maxIter,forcestability,showfigs,fs); . . . % Select best model on last period of validation data [A,B,C,D] = models{6}{:}; % Select 6th-order model [A,B,C] = dbalreal(A,B,C); % Balanced realization

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6th-order linear state-space model

18/28 Frequency [kHz] 5 10 15 Amplitude [dB]

• 120
• 100
• 80
• 60
• 40
• 20

G11

BLA (parametric) BLA (nonpar) residual standard deviation

Starting value: best linear approximation (BLA)

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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))

Nonlinear optimization

x(t + 1) = A x(t) + B u(t) + E ζ(x(t), u(t)) y(t) = C x(t) + D u(t) + F η(x(t), u(t))

Transient settings

% Concatenate all data u = u(:); % N*P*R x m y = y(:); % N*P*R x p % Transient settings NTrans = N; % Add a period before each realization startReal = [1 4097 . . . 32769]; % Starting indices realizations T1 = [NTrans startReal]; % Transient setting periodic data T2 = 0; % No nonperiodic transient handling

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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))

% Settings nonlinear terms nx = [2 3]; % Nonlinear degrees in ζ ny = [2 3]; % Nonlinear degrees in η % Linear model in PNLSS form model init = fCreateNLSSmodel(A,B,C,D,nx,ny,T1,T2,0); % Set all monomials free for optimization model init.xactive = fSelectActive(’full’,n,m,n,nx); model init.yactive = fSelectActive(’full’,n,m,p,ny);

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Poor man’s stabilization method

% Simulate validation data during estimation model init.u val = uval(:); % Set bound on modeled validation output model init.max out = 1000*max(abs(yval(:)));

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Nonlinear optimization

% Settings Levenberg-Marquardt (LM) optimization nIter = 50; % Number of iterations W = []; % No (frequency) weighting lambda = 100; % Starting value LM damping factor % Nonlinear optimization [model,y mod,models] = fLMnlssWeighted(u,y,model init,. . . nIter,W,lambda); % Select the best model on the validation data . . . model = models(min i);

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Results on test data

% Change transient setting model pnlss test = model; model pnlss test.T1 = [16384 1]; % Compute rms error on multisine test data e = rms(ytest - fFilterNLSS(model pnlss test,utest)); multisine swept sine linear 0.03817 0.02319 PNLSS 0.03817 0.02258

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Pros and cons of a PNLSS model

                                                              x(t + 1) = A x(t) + B u(t) + E ζ(x(t), u(t)) y(t) = C x(t) + D u(t) + F η(x(t), u(t)) linear state-space model polynomials in x and u

• Flexibility

X

Number of parameters

• Multiple input

X

Stability multiple output

X

Interpretability

• Initial estimates

X

Extrapolation

• Software available

X

Process noise

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What to expect in PNLSS 2.0?

Object-oriented Modular components (initialization, cost, Jacobian, optimization) Beyond polynomials Structure specification/retrieval

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Success story: battery cell

27/28 Lithium Ion Polymer Battery (EIG-ePLB-C020, Li(NiCoMn)) 3.65 V, 20 A h, AC impedance (1 kHz < 3 mΩ)

Nonparametric distortion analysis

modified by courtesy of Rishi Relan

Success story: battery cell

28/28 4 states up to third-degree nonlinearities

Time domain Frequency domain

modified by courtesy of Rishi Relan