6.3 Model identification
6.3.1 Step response measurements
Step response data from the compression system can be obtained by applying a step in-put to the control valve and measuring the resulting pressure rise over a certain period of time. Before the step is applied, the compression system must be in a steady-state oper-ating point to avoid that other transients influence the measurements. Moreover, the size of the applied step must be large enough to result in a measurable response. However, the perturbation must remain small to keep the operating point of the compressor close to its initial value. Otherwise, linear approximations of the system dynamics are invalid.
For test rig A we conducted 10 different experiments at various stable operating points, rotational speeds and with different step sizes. The step signal was generated at the 0–10 V output channel of the data acquisition system and electronically converted to a 4–20 mA current. Before the step input was applied, the compressor was brought to a stable operating point. The measurement time was 6.4 s in which 8192 samples of the signals p1, p2, T1, and T2 were collected.
6.3 MODEL IDENTIFICATION 119
1 2 3 4 5 6
−900
−600
−300 0 300
t(s)
∆p(Pa)
Figure 6.9 /Filtered step response data (gray) and mean value (black); N = 9,432 rpm,
˙
mc,0= 1.13kg/s, ur,0= 0.409, ∆ur= 0.287, Lc = 0.30.
For each experiment we performed 25 measurements such that the effect of measure-ment noise on the results could be reduced through averaging. Measuremeasure-ment and flow noise were further reduced through zero-phase filtering of the data with a 5th order, low-pass Butterworth filter with a cut-off frequency between 11–17 Hz. We point out that we have chosen a very low filter cut-off frequency to remove the significant amount of 50 Hz pollution and flow noise inside the compression system. However, since the filter cut-off frequency is around the bandwidth of the used control valve, filtering is not believed to remove any important dynamic information from the raw data.
Analysis of the filtered data from all experiments showed that the smallest variance be-tween different measurements, indicating a good signal-to-noise ratio, was obtained with an initial valve opening ur,0 of 40.9% and a step size ∆ur of 28.7%. Therefore, from now on we will only present results obtained with these settings, unless stated other-wise. Results of the step response measurements for one operating condition and with the selected step input from 40.9% to 69.6% are shown in Figure 6.9. We remark that we shifted the measured signals to obtain an initial value around zero. This shifting was done by subtracting the average value of 1000 samples (≈ 0.8 s) prior to the step input from the measured data. Note that relative large differences exist between the 25 measure-ments. These differences are mainly caused by low frequent flow noise and variations of the operating conditions between subsequent measurements.
The goal of the identification method that we will discuss below is to obtain a linear model of the compression system that can be used to estimate the stability parameter B. The input of this linear model is the valve opening ur, see also Section 5.3. Hence, including the dynamics of the control valve actuator, sensors and the electronics in the data set for the identification method is not desirable. We experimentally confirmed that the dynamic response of the voltage to current converter is almost ideal up to 10 kHz.
Based on specifications of the manufacturer we considered the dynamic response of the sensors to be ideal up to at least 100 kHz.
However, a detailed analysis of the control valve dynamics revealed that the valve response to a step input is too slow to be neglected. Furthermore, we concluded that the valve has a considerable time delay of 37 samples at a sampling frequency of 1.28 kHz. In addition, we found that an additional time delay of 16 samples is present in the anti-aliasing filters of the each channel of the data-acquisition system. A detailed investigation of the valve dynamics and time delay is given byVan Helvoirt et al.(2005a).
The effect of the time delay in the valve actuator and data-acquisition system was com-pensated for by removing the first 53 samples from each step response time series. In order to remove the effect of the valve dynamics we designed a pre-filter that represents the inverse of the actual valve dynamics. The valve dynamics are given by
Hr(s) = e−128053 s 1.552·105
(s + 30.79)(s2+ 60.32s + 5042) (6.11)
We remark that this model is slightly different from the original approximation for the valve dynamics that was given byVan Helvoirt et al.(2005a), since we replaced the (non-minimum phase) zeros in the transfer function by a static gain to assure a stable inverse.
Through step response simulations we verified that this further simplification of the valve dynamics did not lead to noticeable different responses.
The pre-filter can be obtained by taking the inverse of Equation (6.11). However, to ob-tain a proper inverse we added an additional 3rd order, low-pass Butterworth filter with a cut-off frequency of 108 Hz. The cut-off frequency was chosen such that it will not compromise the accuracy of the inverse in the region of interest (< 20 Hz). The Bode2 diagram of the resulting pre-filter is shown in Figure6.10.
So far we have addressed the generation of a set of input-output data from which a linear model of the compression system can be obtained. In summary, the pre-processing of the data consists of the following steps:
• conduct step response experiments;
• calculate ensemble average of raw data;
• apply low-pass filter to reduce noise;
• remove initial steady-state offset from data;
• shift time series to remove effect of time delays;
• filter data with approximate inverse of valve dynamics.
2Hendrik Wade Bode (1905–1982) was an American scientist and pioneer of modern control theory and electronic telecommunications. In 1945 he published the classic book ’Network Analysis and Amplifier Feedback Design’ and one of his important contributions is the well-known Bode’s sensitivity integral.
6.3 MODEL IDENTIFICATION 121
10−1 100 101 102 103 104
0 20 40 60 80
10−1 100 101 102 103 104
0 90 180 270
f (Hz)
|Hr(jω)−1|(dB)∠Hr(jω)−1(◦ )
Figure 6.10 /Bode diagram of the pre-filter to compensate for non-ideal valve dynamics.
Now, we will address the method that will be used to calculate this linear model and subsequently estimate the stability parameter.