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

Polynomial nonlinear state-space modeling of 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 Error signals Inputs Outputs + System + – Modeled outputs Model 3/20

Goal: Capture system dynamics Error signal Inputs Outputs + System + – Modeled outputs Model 4/20

Goal: Capture system dynamics Error signals Inputs Outputs + System + – Modeled outputs Model 5/20

Set-up Nonparametric analysis Parametric modeling 6/20

Two inputs and three outputs are provided in the benchmark Voltage Signal Force 145 accelerations Shaker F16 generator Two inputs: reference input: Voltage actual input: Force Three outputs: Acceleration at excitation location Acceleration on right wing Acceleration on payload 7/20

Multisine excitation with random frequency grid Voltage signal 0.4 F Voltage (V) u ( t ) = � A k cos (2 π kf 0 t + φ k ) 0 k =1 3 amplitude levels −0.4 0 41 (12.2, 49.0, and 97.1 N RMS) Time (s) Voltage spectrum 3 periods per amplitude level −10.5 Amplitude (dB) (two in steady state) −100 10 input realizations per level (9 for estimation, 1 for testing) 1 60 Frequency (Hz) 16384 points per period 8/20

Set-up Nonparametric analysis Parametric modeling 9/20

The FRFs and the distortion levels are estimated 10/20

The FRFs and the distortion levels are estimated Output 1 0 FRF (dB) −50 −100 0 20 40 60 Frequency (Hz) Output 2 Output 3 0 0 FRF (dB) FRF (dB) −50 −50 −100 −100 0 20 40 60 0 20 40 60 Frequency (Hz) Frequency (Hz) 11/20

Focus on frequency band [4.7, 11] Hz Output 1 −15 FRF (dB) −45 −71 4.75.2 6.6 7.3 9.1 11 Frequency (Hz) Output 2 Output 3 −10 −10 FRF (dB) −39 FRF (dB) −40 −65 −65 4.75.2 6.6 7.3 9.1 11 4.75.2 6.6 7.3 9.1 11 Frequency (Hz) Frequency (Hz) 12/20

Amplitude level 2 has the largest nonlinear distortion to signal ratio Output 3 Amplitude 1: FRF −10 Amplitude 2: FRF Amplitude 3: FRF −40 Amplitude 1: Total distorion Amplitude 2: Total distorion FRF (dB) Amplitude 3: Total distorion −65 Amplitude 1: Noise distorion Amplitude 2: Noise distorion Amplitude 3: Noise distorion 4.7 5.2 6.6 7.3 9.1 11 Frequency (Hz) 13/20

The nonlinear distortions at 7.3 Hz are about 25 dB larger than the noise Output 3 Amplitude 1: FRF −10 Amplitude 2: FRF Amplitude 3: FRF −40 Amplitude 1: Total distorion Amplitude 2: Total distorion FRF (dB) Amplitude 3: Total distorion −65 Amplitude 1: Noise distorion Amplitude 2: Noise distorion Amplitude 3: Noise distorion 4.7 5.2 6.6 7.3 9.1 11 Frequency (Hz) 14/20

Set-up Nonparametric analysis Parametric modeling 15/20

A linear state-space model captures dynamic behavior = x ( t + 1) x ( t ) + u ( t ) + ζ ( x ( t ) , u ( t )) A B E = y ( t ) x ( t ) + u ( t ) + η ( x ( t ) , u ( t )) C D F                                  linear state-space model polynomials in x and u 16/20

A polynomial nonlinear state-space model captures nonlinear dynamic behavior = x ( t + 1) x ( t ) + u ( t ) + ζ ( x ( t ) , u ( t )) A B E = y ( t ) x ( t ) + u ( t ) + η ( x ( t ) , u ( t )) C D F                                                               linear state-space model polynomials in x and u x 2   1 x 1 x 2     x 1 u     .  .  with e.g. ζ ( x , u ) = .     x 2  2 u    u 3      .  . . 17/20

Identification of a polynomial nonlinear state-space model = x ( t + 1) x ( t ) + u ( t ) + ζ ( x ( t ) , u ( t )) A B E = y ( t ) x ( t ) + u ( t ) + η ( x ( t ) , u ( t )) C D F                                                               linear state-space model polynomials in x and u   vec( A ) ǫ ( k , θ ) = Y ( k , θ ) − Y meas ( k ) vec( B )   N F   vec( C )   � ǫ H ( k , θ ) W ( k ) ǫ ( k , θ ) K WLS ( θ ) =     vec( D )   k =1 θ =   vec( E ) ˆ   θ = arg min K WLS   vec( F )   θ   vec( x (0))     vec( u (0)) 18/20

Nonlinear interaction only within first resonance and coupled interaction in other three resonances Output 2 Output 3 71 71 Output Output Amplitude (dB) 53 52 50 49 45 44 Linear error Linear error PNLSS PNLSS error error 4.75.2 6.6 7.3 9.1 11 4.75.2 6.6 7.3 9.1 11 Freq. (Hz) Freq. (Hz) 19/20

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 Voltage Signal Force 145 accelerations Shaker F16 generator 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 Output 3 Amplitude 1: FRF −10 Amplitude 2: FRF Amplitude 3: FRF −40 Amplitude 1: Total distorion Amplitude 2: Total distorion FRF (dB) Amplitude 3: Total distorion −65 Amplitude 1: Noise distorion Amplitude 2: Noise distorion Amplitude 3: Noise distorion 4.7 5.2 6.6 7.3 9.1 11 Frequency (Hz) 22/20

The nonlinear distortions at 7.3 Hz are about 25 dB larger than the noise Output 3 Amplitude 1: FRF −10 Amplitude 2: FRF Amplitude 3: FRF Amplitude 1: Total distorion −40 Amplitude 2: Total distorion Amplitude 3: Total distorion FRF (dB) −65 Amplitude 1: Noise distorion Amplitude 2: Noise distorion Amplitude 3: Noise distorion 6.6 7.3 Frequency (Hz) 23/20

Focus on main resonance with second-order linear and nonlinear model (lowest amplitude) Output 2 Output 1 Output 3 65 64 59 Output Output Output 53 53 Amplitude (dB) 47 Linear Linear Linear 27 27 error error error 21 PNLSS error PNLSS error PNLSS error 7.3 8.6 7.3 8.6 7.3 8.6 Freq. (Hz) Freq. (Hz) Freq. (Hz) 24/20 95 parameters

Focus on main resonance with second-order linear and nonlinear model (amplitude level 2) Output 1 Output 2 Output 3 71 71 66 Output Output Output Amplitude (dB) 57 57 51 Linear Linear Linear 44 44 error error error 38 PNLSS error PNLSS error PNLSS error 7.3 8.6 7.3 8.6 7.3 8.6 Freq. (Hz) Freq. (Hz) Freq. (Hz) 25/20 95 parameters

Limit the number of parameters by allowing nonlinear interaction only within a resonance +     x 1 f 1 ( x 1 , x 2 ) f 2 ( x 1 , x 2 ) x 2         x 3 f 3 ( x 3 , x 4 )           0 0 0 0 0  f 4 ( x 3 , x 4 )  x 4  A 1          x 5 0 A 2 0 0 0 0 f 5 ( x 5 , x 6 )             0 0 0 0 0 f 6 ( x 5 , x 6 ) x 6 A 3       = x ( t ) + Bu ( t ) +       x 7 0 0 0 A 4 0 0 f 7 ( x 7 , x 8 )                   x 8 0 0 0 0 A 5 0 f 8 ( x 7 , x 8 )           0 0 0 0 0 f 9 ( x 9 , x 10 ) x 9 A 6         x 10 f 10 ( x 9 , x 10 )         f 11 ( x 11 , x 12 )  x 11        x 12 f 12 ( x 11 , x 12 ) y ( t ) = Cx ( t ) + Du ( t ) + g 1 ( x 1 , x 2 ) + g 2 ( x 3 , x 4 ) + g 3 ( x 5 , x 6 ) + g 4 ( x 7 , x 8 ) + g 5 ( x 9 , x 10 ) + g 6 ( x 11 , x 12 ) 26/20