Design example with evaluation of control input u の変更点

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[[制御工学特論]]

Mixed sensitivity problem is modified to evaluate control input u with a high pass filter F(s) as following figure.
#ref(HPF_u.png)
Moreover, WT(s) is set to improper for over estimation of modeling error in high frequency range.

#ref(nominal_fixed.m)
#ref(weight.m)
#ref(cont_fixed.m)

Execution:
 >> nominal_fixed
 >> weight
 >> cont_fixed

- Comparison the designed result with the base design

Suppose two design results are stored in folders `base' and `HPF_u' respectively. 
The following m-file load the designed and experimental results then plots some figures.  

#ref(compare_fixed.m)
#ref(realfft.m)

Execution:
 >> compare_fixed

For realfft.m see [[FFTの使い方]].

The following figure shows bode diagram of discretized controllers.
#ref(compare_Kd.png)
Lower gain characteristic is obtained in high frequency range due to evaluation of control input. 

The following figure shows gain characteristic of sensitivity function.
#ref(compare_S.png)
In the high frequency range above 200Hz, amplification of tracking error is improved, although system performance might be reduced since the gain becomes slightly larger than base design. 

The following figure shows time response of motor speed.
#ref(compare_y.png)
As expected above, base design seems to have better performance. However, during control experiment, high frequency vibration was occured for base design.

This problem is confirmed by FFT analysis. The following figure shows FFT analysis result of the above figure.
This problem is confirmed by FFT analysis. The following figure shows FFT analysis result of the above figure from 4sec to 5sec.
#ref(compare_y_fft.png)
There are large peaks at around 100 Hz and 200 Hz, and peaks of base design are larger than the design with HPF. 
These peaks might be occured by nature of discretized controllers, since the order is reversed in continuous-time system. (In the figure of sensitivity gain, base design has smaller gain characteristic in the frequency range from 100 Hz to 200 Hz.)    

This problem is also shown in control input. The following figure shows time response of control input (motor torque). 
#ref(compare_u.png)
The base design controller generates larger amplitude of control input.
This does not imply that base design controller is worse, since generally agressive controllers make large control input. 
However, FFT analysis reveals that this result for base design is not desired.
The following figure shows the FFT analysis result of control input.
#ref(compare_u_fft.png)
The control input for base design controller has remarkable components in high frequency range, which might be the cause of the high frequency vibration during control experiment.