Nonlinear distortion analysis before linearisation.

Nonlinear distortion analysis before linearisation.

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Linearising the dynamics of nonlinear mechanical systems is an important and open research area. In this paper, we adopt a data-driven and feedback control approach to tackle this problem. A model predictive control architecture is developed that builds upon data-driven dynamic models obtained using nonlinear system identification. The overall meth...

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... 40 dB. From Table 1, it is derived that the linear natural frequency of the system is equal to 3.56 Hz. Based on this, the desired closed-loop bandwidth for an outer-loop controller is chosen equal to 14 Hz, that is well beyond the linear resonance. In other words, the frequency band of interest for the linearising MPC ranges between 0 and 14 Hz. Fig. 6 shows the nonlinear distortion analysis of system (21) before linearisation. It is excited with 10 realisations of 5 periods of an odd random-phase multisine with a RMS amplitude of 0.12 N. For every group of 4 odd frequency lines, 1 line is randomly rejected to serve as an odd detection line. Moreover, 4000 points per period are ...
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... noise covariance is set to the mean value of the output noise covariance of the nonlinear distortion analysis, i.e., R UKF = 1.13 · 10 −14 m 2 ; the process noise covariance matrix is set to Q UKF = 0.05R UKF I 2 . Fig. 8 depicts the nonlinear distortion analysis of the linearised plant. The exact same multisine excitation signal as for Fig. 6 is considered. An excellent linearisation is achieved since the odd and even distortions observed Fig. 6 are herein almost coincident with the noise floor. The residual distortions around 3.8 and 7.6 Hz are 50 dB below the validation output. Yet, it must be noted that these residuals can be made arbitrarily close to the noise level by ...
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... analysis, i.e., R UKF = 1.13 · 10 −14 m 2 ; the process noise covariance matrix is set to Q UKF = 0.05R UKF I 2 . Fig. 8 depicts the nonlinear distortion analysis of the linearised plant. The exact same multisine excitation signal as for Fig. 6 is considered. An excellent linearisation is achieved since the odd and even distortions observed Fig. 6 are herein almost coincident with the noise floor. The residual distortions around 3.8 and 7.6 Hz are 50 dB below the validation output. Yet, it must be noted that these residuals can be made arbitrarily close to the noise level by tuning the MPC and UKF parameters more aggressively. However, this would require excessive input energy. ...
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... as in the previous performance test, in this way allowing for a valid comparison between the two. As expected, the even nonlinearities are visibly more present when compared to Fig. 8. In addition, almost no improvement regarding the suppression of the even nonlinearities is noticed when compared to the original system before linearisation in Fig. 6. Only around the resonance frequency, there is a reduction of approximately 15 dB. Nevertheless, the odd nonlinearities are all reduced to a satisfactory level, similarly to Fig. 8. Regarding the even nonlinearities, one could argue that they should in fact be suppressed, even when the quadratic nonlinearity is not modelled. After all, ...
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... the control architecture, a random-phase multisine with a RMS amplitude of 0.22 N is used, compared to 0.12 N in the previous sections. Fig. 12 presents the nonlinear distortion analysis (left) before and (right) after linearisation. In the left subfigure, nonlinear distortions are observed to be markedly greater when compared to the analysis in Fig. 6. Nevertheless, a satisfactory linearisation is achieved in the right subplot, the results being similar to Fig. 8, with the peaks around 3.8 and 7.6 Hz slightly ...
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... in the setup as is explained in Fig. 15. The value of the spring constant is set to k c = 2 · 10 9 V/m 3 . By doing so, the obtained dynamics is dominated by the nonlinearity, in the sense that nonlinear distortions under a high-level multisine excitation are less than 10 dB below the linear system dynamics in the first resonance region. Fig. 16 shows the nonlinear distortion analysis of the system with artificial nonlinearity. It is excited with 5 realisations of 4 periods of an odd random-phase multisine with a RMS amplitude of 0.015 V. For every group of 4 odd frequency lines, 1 line is randomly rejected. Moreover, 3280 points per period are considered at a sampling ...
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... dB, and the even distortions are completely suppressed, even though a cubic nonlinearity only is modelled. As expected, there is no significant performance improvement around the first flexible mode of the beam. As a final performance indicator, Fig. 20 plots the output of the plant before linearisation (in light grey), i.e., the output data in Fig. 16. It is, in this figure, clearly visible that the effect of the hardening spring is eliminated; the resonance peak is shifted back to lower frequencies and the static output value at 0 Hz increases. The colouring of the noise observed in Fig. 16 is also whitened. ...
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... Fig. 20 plots the output of the plant before linearisation (in light grey), i.e., the output data in Fig. 16. It is, in this figure, clearly visible that the effect of the hardening spring is eliminated; the resonance peak is shifted back to lower frequencies and the static output value at 0 Hz increases. The colouring of the noise observed in Fig. 16 is also whitened. ...

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