Identification of a onestage axial compressor system is addressed. In particular, we investigate the underlying dynamics of tip air injection and throttle activation to the overall compressor dynamics and the dynamics around the tip of the compressor blades. A proposed subspace system identification algorithm is used to extract three mathematical models: relating the tip air injection to the overall dynamics of the compressor and to the flow dynamics at the tip of the compressor blade and relating the movement of the throttle to the overall compressor dynamics. As the system identification relays on experimental data, concerns about the noise level and unmodeled system dynamics are addressed by experimenting with two model structures. The identification algorithm entails a heuristic optimization that allows for inspection of the results with respect to unmodeled system dynamics. The results of the proposed system identification algorithm show that the assumed model structure for the system identification algorithm takes on an important role in defining the coupling characteristics. A new measure for the flow state in the blade passage is proposed and used in characterizing the dynamics at the tip of the compressor blade, which allows for the inspection of the limits for the utilized actuation.
Operation of axial compressor systems, as found in a number of aerospace applications, faces difficulties due to instabilities at the peak of its performance curves. To mitigate these instabilities, a number of different control mechanisms have been introduced. Among them are air injection into the tip of the blade passage [
As the system identification relays on experimental data, concerns about the noise level need to be addressed as well as unmodeled system dynamics. The unmodeled system dynamics can become an important issue when the experimental data do not include all system modes during the system identification experiment or the assumed model structure does not allow for accommodating one or more particular dynamic characteristics. We propose to utilize two different model structures to investigate the unmodeled system dynamics and the effects of the proposed optimization. These structures are the autoregressive moving average with exogenous input (ARMAX) model and the autoregressive with exogenous input (ARX) model. An innovation sequence is extracted by using an estimated parameter sequence. The ARMAX model and ARX model parameters are then utilized to construct a Hankel matrix, representing the sampled impulse response of the system relating tip air injection to pressure rise coefficient. A singular value decomposition of the Hankel matrix is performed for an eigensystem realization (ERA, [
In the following, the details of the system identification scheme as well as the proposed realization and optimization are introduced.
Consider a linear, timeinvariant, discrete time system:
In order to estimate the state variable
Equation (
Perform a SVD on
inputnormal form:
balanced form:
output normal form:
Equation (
Define
The Hankel matrix in (
In this work, we utilize a onestage compressor system with a blade geometry that allows for spike inception. The proposed identification scheme is used to identify the dynamics at different flow conditions, including near stall conditions. The dynamics of the compressor is given by the changes within the blade passage area as well as the changes in the pressure rise coefficient computed by the input and output pressures. Hence, we extract three separate models, one for the dynamics within the blade passage due to air injection, one for the input/output relationship of the compressor due to throttle movements, and one that characterizes the dynamics due to air injection pressure as the input of the system and the pressure rise coefficient as the output of the system. The compressor employed for this research is a lowspeed rotor with characteristic parameters as given in Table
Test compressor design parameters.
Quantity  

Design speed (rpm)  2400 
Rotor blade number  60 
Outer diameter (mm)  500 
Mass flow rate (kg/s)  2.9 
Rotor tip chord (mm)  36.3 
Rotor tip stagger angle (deg.)  39.2 
Hubtip ratio  0.75 
Sensor location for experimental setup. The dashed breakout view is rotated 90° and depicts the six sensors for capturing the dynamics of the blade passage.
Sensors 1 and 8 are utilized to compute the flow coefficient and pressure rise. A separate sensor is mounted at the injection port to measure the injection pressure. The pressure sensors measure at a sampling rate of 20,000 Hz. There are eight air injectors, equally distributed over the circumference, injecting air at an angle of 15 degrees with a steady pressure. Sensors 2 to 7 are dedicated to the dynamics within the blade passage.
For the actuation, there are two inputs. One input is given by the movement of the throttle valve, which is used to set the operating point. The other input is accomplished via highpressure air injection using the eight injectors equally distributed around the circumference of the compressor annulus. The setup of the measurement and injectors is shown in Figure
Picture of (a) side view of the onestage axial compressor system and (b) view of the throttle using a cone movement system for controlling the flow coefficient and pressure rise.
We first present the results for inferring the compressor dynamics modeling the dynamics related to the air injection and overall output of the compressor as measured by flow coefficient and pressure rise coefficient. Considering Figure
Bode plot for ARX (black) and ARMAX (blue) realization; flow coefficient = 0.55.
The first fundamental frequency is at around 557 Hz. Once optimization is used as well as different realization forms (input (IN), output (ON), or balanced (BN) normal), the Bode plots of the two models (ARX and ARMAX) start to diverge from each other. For the ARX based model set, the frequency responses converge to a very defined Bode plot, with a fundamental frequency of 477 Hz. For the ARMAX based model set, the resulting frequency responses are reduced in their emphasis to the fundamental frequency. The ARX based model set also shows a much improved magnitude for the first fundamental frequency. Regardless of the realization (balanced, input, or output normal), all of the optimized ARX based models seem to agree with the first fundamental frequency and its amplitude. The phase plot does not show much change from unoptimized to optimized system identification results, for both model structures and all realizations.
At a flow coefficient of 0.55, the compressor operates sufficiently far away from the stall inception. The optimization of the singular values for the system identification algorithm is based on minimizing the error between the simulated model output and the measured overall system output, as given by (
ARX model in block diagram form.
The corresponding system equation is given by
ARMAX model in block diagram form.
The corresponding system equation is given by
Bode plot for ARX (black) and ARMAX (blue) realization; flow coefficient = 0.51.
Without any current theory explaining the observation of the mode at 477 Hz, our focus will be directed towards the low frequency behavior of the system. Doing so, we assume that the coupling dynamics are found closer to the slow dynamics frequency region. Utilizing air injection as a control input breaks up the flow structure at the blade level. The overall dynamics is given by the flow coefficient and pressure rise coefficient. These two measures seem to reside in the low frequency area of the data. Hence, a second filter is proposed to be used to prepare the collected data. In particular, a low pass filter with a cutoff frequency of 150 Hz is utilized. The filter employs 20 filter weights and a Kaiser window. In addition, the data is downsampled from 20 kHz to 400 Hz. The filtering and downsampling are used to ensure that the system identification and its optimization focus on frequencies below 150 Hz. The proposed optimal identification algorithm is employed to extract dynamic relationships between the input (injection) and the output (flow coefficient). For system identification, the data is also prepared by subtracting the dc offset; that is, the data has a zero mean. For the results, this implies that the output of the models expresses the change in flow coefficient, rather than the flow coefficient itself.
In the following, Bode plots of the various identified models are given, using the proposed filtering and resampling. Figure
Bode plot for ARX (black) and ARMAX (blue) realization; flow coefficient = 0.58. Data is filtered and resampled prior to system identification.
Bode plot for ARX (black) and ARMAX (blue) realization; flow coefficient = 0.55. Data is filtered and resampled prior to system identification.
Bode plot for ARX (black) and ARMAX (blue) realization; flow coefficient = 0.51. Data is filtered and resampled prior to system identification.
Simulated and measured output using optimized ARX (balanced realization) model at flow coefficient 0.51.
A test to see if the extracted models can predict the output should provide indication if these frequencies are based on residuals of the rotor frequency or if they actually model the coupling dynamics between the injection and the pressure rise coefficient. Comparing Figures
Simulated and measured output using no optimized ARX (balanced realization) model at flow coefficient 0.51.
A similar observation can be made for the ARMAX models shown in Figures
Simulated and measured output using optimized ARMAX (balanced realization) model at flow coef. 0.51.
Simulated and measured output using (no optimized) ARMAX (balanced realization) model at flow coefficient 0.51.
Injection pressure plot for flow coefficient 0.51, validation data.
The system given by the optimized ARX model (balanced realization) can be given in statespace form (see (
The ARX/ARMAX model order is set to
Cost during TS optimization.
For the simulations, a total of 45 Markov parameters were used to construct the block Hankel matrix given in (
The proposed optimization embedded in the identification algorithm and the different cost functions are first tested using simulations. A simple secondorder system with a natural frequency of 17 Hz is utilized. The simulations are carried out using different process and measurement noise levels. For each case, 20 simulations are used to statistically characterize and compare the performance of the proposed identification routines. The ETS algorithm uses 200 iterations for each simulation; the model order of the ARX and ARMAX models is
Simulation results with 1% and 5% noise std.
Error squared  Correl. coef. 
Natural freq. 
Natural freq. 


ARXnom. 1%  1.5154  0.9338 


Mean 5%  1.5159  0.9311  17.0211 



ARXnom. 1%  0.0456  0.0078  0.0126  0.0388 
Stand. dev. 5%  0.0428  0.0057  0.0993  0.3552 


ARXopt. 1% 

0.9337  16.5024  2.9272 
Mean 5% 


16.5826  2.4553 


ARXopt. 1%  0.0633  0.0073  0.4595  2.7031 
Sta. Dev. 5%  0.0544  0.0058  0.4590  2.3990 


ARXopt 
1.4837  0.9321  16.9763  0.1394 
Mean 5%  1.4828  0.9317 




ARXopt 
0.0369  0.0086  0.0909  0.3359 
Stand. dev. 5%  0.0494  0.0051  0.0561  0.1756 


ARMAX 1%  23.4750  0.9200  17.1773  1.0429 
nom. Mean 5%  36.4212  0.7272  15.1183  11.069 


ARMAX 1%  24.6217  0.0721  4.4278  12.1042 
nom. Std. 5%  66.8224  0.3197  9.6057  26.5340 


ARMAX 1% 


16.4052  3.4997 
opt. Mean 5% 


16.5794  2.4741 


ARMAX 1%  0.0710  0.0070  0.4461  2.6227 
opt. Std. 5%  0.0580  0.0054  0.3955  2.1698 


ARMAX 
1.5249 



opt. Mean 5%  1.4866  0.9317 




ARMAX 
0.0607  0.0054  0.0445  0.1716 
opt. Std. 5%  0.0619  0.0045  0.0952  0.3731 
The error squared and correlation coefficients are computed based on 1000 validation data points. In Table
Simulation results with 10% and 15% noise standard deviation.
Error sqr.  Correlat. coef. 
Natural freq. 
Nat. freq. 


ARXnom. 10%  1.4850 



Mean 15%  1.5144  0.9306  17.1098  0.6459 


ARXnom. 10%  0.0780  0.0050  0.0937  0.3668 
Stand. dev. 15%  0.1067  0.0066  0.2080  1.0023 


ARXopt. 10% 

0.9329  16.3853  3.6159 
Mean 15% 


16.3792  3.6518 


ARXopt. 10%  0.0568  0.0042  0.2628  1.5461 
Sta. Dev. 15%  0.0782  0.0044  0.5484  3.1678 


ARXopt. 
1.5583  0.9313  16.9356  0.3788 
Mean 15%  1.5069  0.9342 




ARXopt. 
0.0545  0.0063  0.0673  0.3596 
Stand. dev. 15%  0.0814  0.0067  0.0687  0.2473 


ARMAX 10%  26.223  0.8083  11.9832  29.510 
nom. Mean 15%  93.074  0.5015  24.5364  44.332 


ARMAX 10%  34.234  0.1483  8.9178  30.975 
nom. Std. 15%  108.29  0.5021  26.108  137.97 


ARMAX 10% 


16.6365  2.1382 
opt. Mean 15% 


16.4029  3.5123 


ARMAX 10%  0.0573  0.0070  0.4485  2.1277 
opt. Std. 15%  0.0910  0.0071  0.3980  2.3409 


ARMAX 
1.5181  0.9297 


opt Mean 15%  1.5093  0.9320 




ARMAX 
0.0884  0.0067  0.0898  0.3178 
opt. Std. 15%  0.0555  0.0043  0.0671  0.2085 
From the inspection of these results, we conclude that the ARX and ARMAX based model identification with optimization either with or without prior knowledge of the natural frequency works sufficiently well for use of identifying the overall system dynamics of the compressor system. Considering the overall dynamics of the compressor system between the input (throttle movements) and the corresponding pressure rise coefficient, the proposed system identification algorithm with prior information of the natural frequency is used. The identified model fits well for the three different flow coefficients of 0.51, 0.55, and 0.58, indicating that the overall dynamics does not change noticeably during change of operating point. It is expected that this may change, as the operating point gets closer to stall. For the identification, the same parameters are used as given in the simulation section. The resulting natural frequency of the extracted model is 17.1 Hz using the ARMAX base model with optimum
For the identification of the dynamics within the blade passage, sensors 2 to 7 are used to capture the resulting dynamics at the blade tip area. As the sensors measure the pressures at all times, the data does not correspond to a single blade passage (the compressor runs at 2400 rpm). A hall effect sensor is used to phase lock the data and computes the pressure distribution within the blade passage. A sample pressure distribution is given in Figure
Pressure distribution within the blade passage for a section of the compressor. The units for the pressure are in [Pa], Cad is the axial tip chord, and the rotor pitch is a normalized parameter relative to blade chord.
For the system identification of the dynamics within the blade passage, the phase locking approach will not provide sufficient information, as only approximately 2% of the data can be associated with one rotation of the blade passage. We assume that the measured data of one blade passage corresponds to the state of a set of blade passages at that time. This is not to say that all blade passages have the same flow distribution at a given time. We rather state that the condition of the flow is similar. By using this assumption, we propose to use an information based entropy measure for the state of the dynamics within the blade passage. Recent work utilizes the correlation coefficient between the pressure data of the blade passage and the pressure data corresponding to the same blade passage one revolution prior. The correlation coefficient is used to characterize the state of flow in a blade passage. The lower the correlation coefficient, the more unsteady the flow. By using an entropy measure, there is no need to utilize data from over one rotation of the compressor. This entropy can be computed directly with the data present and hence has the same sampling frequency as the input. Therefore, we can associate the state of fluid flow with one characteristic number. The Shannon entropy is commonly used in information theory and measures the amount of information contained in a message. Here, one uses the entropy of a signal and classifies its predictability by a low entropy value and its randomness or amount of disorder by a large entropy value. The entropy is computed as
By assuming equivalent flow condition at a given time for a set of blade passages, we artificially increase the spectral entropy value, due to the variation of flow energy among different blade passages at a given time. The entropy can be assessed at each sensor location and hence provides a snapshot of the flow characteristics. This is shown schematically in Figures
Entropy as description for current state of flow within a blade passage.
Spectral entropy for leading sensor. The first part of time series is without injection; second part is with injection.
Figure
Spectral entropy spatial distribution within a blade passage with and without injection.
Utilizing the optimized identification approaches for the ERA with ARX and ARMAX base models, the identification results for the dynamics of the flow at the entry of the blade passage are reduced to the fundamental natural frequency and given in Table
Identification results for dynamics in blade passage.





ARX  52.4  90.4  101.6 
ARXoptimum  54.3  90.4  116.5 
ARMAXoptimum  54.4  90.4  116.7 
The identification utilized 45 Markov parameters to construct the Hankel matrix given in (
In this work, we propose a system identification algorithm utilizing a TS based realization. The TS optimization along with the system identification algorithm is used to investigate two different model structures and their characteristics applied to an axial compressor system. In particular, three dynamical models are extracted which characterize inputoutput behavior at different levels of the compressor system. One of these models describes the relationship between air injection at the blade tip and the resulting overall compressor dynamics represented by the computed change in flow coefficient. Considering the presented results: injection at the tip of the leading edge of the compressor blade has an influence on the overall dynamics of the compressor. These types of coupling dynamics may be of interest in developing more efficient control schemes for compressor control. Such controllers can contribute to extending the stall margin improvement (SMI) and run axial compressors at higher efficiencies. Another model extracted using the proposed identification algorithm captures the dynamics between the throttle movement and the overall dynamics as measured by the change in the computed flow coefficient—using measured data points. The proposed identification scheme embeds prior knowledge of the compressor and therefore guides the identification to more accurate results. Finally, the proposed identification algorithm is used to infer a model relating the air injection—treated as an input—at the blade time to the resulting flow dynamics near the tip of the compressor blade. At this stage, such models relating the air injection to flow characterizations near the tip may help in gaining a better understanding of the effect from the air injection, its reach into the blade passage, and its potential stall dynamics at this location. The proposed identification scheme with an embedded TS optimization algorithm as presented in this work seems to be able to aid in developing any of the three models stated above.
Discrete time statespace model characterizing the dynamics between throttle movements and pressure rise coefficient changes is
The authors declare that they have no conflicts of interest.
This research was supported by a generous grant from the National Science Foundation of China on Project no. 51306178.