Polynomial nonlinear state-space modeling of the F-16 aircraft - - PowerPoint PPT Presentation

Polynomial nonlinear state-space modeling

• f the F-16 aircraft benchmark

Koen Tiels

2017 Workshop on Nonlinear System Identification Benchmarks

Introduction: F16 ground vibration test

2/20

Goal: Capture system dynamics

System Model + Inputs Outputs Modeled

• utputs

Error signals

+ –

3/20

Goal: Capture system dynamics

System Model + Inputs Outputs Modeled

• utputs

Error signal

+ –

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Goal: Capture system dynamics

System Model + Inputs Outputs Modeled

• utputs

Error signals

+ –

5/20

Set-up Nonparametric analysis Parametric modeling

6/20

Two inputs and three outputs are provided in the benchmark

Signal generator Shaker F16 Voltage Force 145 accelerations

Two inputs:

reference input: Voltage actual input: Force

Three outputs:

Acceleration at excitation location Acceleration on right wing Acceleration on payload

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Multisine excitation with random frequency grid

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41 −0.4 0.4 Time (s) Voltage (V) Voltage signal 1 60 −100 −10.5 Frequency (Hz) Amplitude (dB) Voltage spectrum

u(t) =

F

• k=1

Akcos(2πkf0t + φk) 3 amplitude levels (12.2, 49.0, and 97.1 N RMS) 3 periods per amplitude level (two in steady state) 10 input realizations per level (9 for estimation, 1 for testing) 16384 points per period

Set-up Nonparametric analysis Parametric modeling

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The FRFs and the distortion levels are estimated

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The FRFs and the distortion levels are estimated

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20 40 60 −100 −50 Frequency (Hz) FRF (dB) Output 1 20 40 60 −100 −50 Frequency (Hz) FRF (dB) Output 2 20 40 60 −100 −50 Frequency (Hz) FRF (dB) Output 3

Focus on frequency band [4.7, 11] Hz

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4.75.2 6.6 7.3 9.1 11 −71 −45 −15 Frequency (Hz) FRF (dB) Output 1 4.75.2 6.6 7.3 9.1 11 −65 −39 −10 Frequency (Hz) FRF (dB) Output 2 4.75.2 6.6 7.3 9.1 11 −65 −40 −10 Frequency (Hz) FRF (dB) Output 3

Amplitude level 2 has the largest nonlinear distortion to signal ratio

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4.7 5.2 6.6 7.3 9.1 11 −65 −40 −10 Frequency (Hz) FRF (dB) Output 3 Amplitude 2: FRF Amplitude 3: FRF Amplitude 1: FRF Amplitude 2: Total distorion Amplitude 3: Total distorion Amplitude 1: Total distorion Amplitude 1: Noise distorion Amplitude 2: Noise distorion Amplitude 3: Noise distorion

The nonlinear distortions at 7.3 Hz are about 25 dB larger than the noise

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4.7 5.2 6.6 7.3 9.1 11 −65 −40 −10 Frequency (Hz) FRF (dB) Output 3 Amplitude 2: FRF Amplitude 3: FRF Amplitude 1: FRF Amplitude 2: Total distorion Amplitude 3: Total distorion Amplitude 1: Total distorion Amplitude 1: Noise distorion Amplitude 2: Noise distorion Amplitude 3: Noise distorion

Set-up Nonparametric analysis Parametric modeling

15/20

A linear state-space model captures dynamic behavior

16/20                                  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

A polynomial nonlinear state-space model captures nonlinear dynamic behavior

17/20                                                               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 polynomial nonlinear state-space model

18/20                                                               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) vec(x(0)) vec(u(0))

             

ǫ(k, θ) = Y (k, θ) − Ymeas(k) KWLS(θ) =

NF

• k=1

ǫH(k, θ)W (k)ǫ(k, θ) ˆ θ = arg min

θ

KWLS

Nonlinear interaction only within first resonance and coupled interaction in other three resonances

19/20

4.75.2 6.6 7.3 9.1 11 45 50 53 71 Output 2

• Freq. (Hz)

Amplitude (dB) 4.75.2 6.6 7.3 9.1 11 44 49 52 71 Output 3

• Freq. (Hz)

Output Linear error PNLSS error Linear error PNLSS error Output

Conclusions and future work

Nonparametric analysis:

High-quality measurements Some room for improvement with nonlinear modeling

Parametric modeling:

Frequency weighting possible Challenging benchmark

Future work:

Non-polynomial basis functions

20/20

Two inputs and three outputs are provided in the benchmark

Signal generator Shaker F16 Voltage Force 145 accelerations

Two inputs:

reference input: Voltage actual input: Force

Three outputs:

Acceleration at excitation location Acceleration on right wing Acceleration on payload

Sampling frequency:

400 Hz (upsampled from 200 Hz)

21/20

Amplitude level 2 has the largest nonlinear distortion to signal ratio

22/20

4.7 5.2 6.6 7.3 9.1 11 −65 −40 −10 Frequency (Hz) FRF (dB) Output 3 Amplitude 2: FRF Amplitude 3: FRF Amplitude 1: FRF Amplitude 2: Total distorion Amplitude 3: Total distorion Amplitude 1: Total distorion Amplitude 1: Noise distorion Amplitude 2: Noise distorion Amplitude 3: Noise distorion

The nonlinear distortions at 7.3 Hz are about 25 dB larger than the noise

23/20

6.6 7.3 −65 −40 −10 Frequency (Hz) FRF (dB) Output 3 Amplitude 1: Noise distorion Amplitude 2: Noise distorion Amplitude 3: Noise distorion Amplitude 1: FRF Amplitude 2: FRF Amplitude 1: Total distorion Amplitude 2: Total distorion Amplitude 3: Total distorion Amplitude 3: FRF

Focus on main resonance with second-order linear and nonlinear model (lowest amplitude)

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7.3 8.6 21 47 59

• Freq. (Hz)

Amplitude (dB) Output 1 7.3 8.6 27 53 65

• Freq. (Hz)

Output 2 7.3 8.6 27 53 64

• Freq. (Hz)

Output 3 Output Linear error Output Linear error Output Linear error PNLSS error PNLSS error PNLSS error

95 parameters

Focus on main resonance with second-order linear and nonlinear model (amplitude level 2)

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7.3 8.6 38 51 66

• Freq. (Hz)

Amplitude (dB) Output 1 7.3 8.6 44 57 71

• Freq. (Hz)

Output 2 7.3 8.6 44 57 71

• Freq. (Hz)

Output 3 Output Output Output Linear error Linear error Linear error PNLSS error PNLSS error PNLSS error

95 parameters

Limit the number of parameters by allowing nonlinear interaction only within a resonance

                      

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

                      

+

=

        

A1 A2 A3 A4 A5 A6

        

x(t) + Bu(t) +

                      

f1(x1, x2) f2(x1, x2) f3(x3, x4) f4(x3, x4) f5(x5, x6) f6(x5, x6) f7(x7, x8) f8(x7, x8) f9(x9, x10) f10(x9, x10) f11(x11, x12) f12(x11, x12)

                      

y(t) = Cx(t) + Du(t) + g1(x1, x2) + g2(x3, x4) + g3(x5, x6) + g4(x7, x8) + g5(x9, x10) + g6(x11, x12)

26/20

Limit the number of parameters by allowing nonlinear interaction only within a resonance

                      

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

                      

+

=

        

A1 A2 A3 A4 A5 A6

        

x(t) + Bu(t) +

                      

f1(x1, x2) f2(x1, x2) f3(x3, x4) f4(x3, x4) f5(x5, x6) f6(x5, x6) f7(x7, x8) f8(x7, x8) f9(x9, x10) f10(x9, x10) f11(x11, x12) f12(x11, x12)

                      

y(t) = Cx(t) + Du(t) + g1(x1, x2) + g2(x3, x4) + g3(x5, x6) + g4(x7, x8) + g5(x9, x10) + g6(x11, x12)

27/20

Nonlinear interaction only within a resonance (12th order models)

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4.75.2 6.6 7.3 9.1 11 37 47 50 53 71 Output 2

• Freq. (Hz)

Amplitude (dB) 4.75.2 6.6 7.3 9.1 11 37 46 49 52 71 Output 3

• Freq. (Hz)

Output Output Linear error PNLSS error Linear error PNLSS error 350 parameters (full model would have 7826 parameters)

Nonlinear interaction only within first resonance and coupled interaction in other three resonances

29/20

4.75.2 6.6 7.3 9.1 11 45 50 53 71 Output 2

• Freq. (Hz)

Amplitude (dB) 4.75.2 6.6 7.3 9.1 11 44 49 52 71 Output 3

• Freq. (Hz)

Output Linear error PNLSS error Linear error PNLSS error Output 882 parameters (full model would have 7826 parameters)