## Abstract

Stall is a type of flow instability in compressors that sets the low-flow limit for compressor operation. During the past few decades, efforts to develop a reliable stall warning system have had limited success. This paper focuses on the small nonlinear disturbances prior to deep surge and introduces a new approach to identify these disturbances using nonlinear feature extraction algorithms including phase-reconstruction of time-series signals and evaluation of a parameter called approximate entropy. To the best of our knowledge, this is the first time approximate entropy has been used for stall warning, and thus, its definition and utility are presented in detail. The technique is applied to stall data sets from two different compressors: a high-speed centrifugal compressor that unexpectedly entered rotating stall during a speed transient and a multistage axial compressor with both modal- and spike-type stall inception. In both cases, nonlinear disturbances appear, in terms of spikes in approximate entropy, prior to surge. The presence of these presurge spikes indicates the potential of using the approximate entropy parameter for small disturbance detection and stall warning. The details of the nonlinear feature extraction algorithm, including guidelines for its application as well as results from applying the algorithm to rig-level data, are presented.

## Introduction

One challenge that has plagued the development of gas turbine engines, from early designs to the advanced engines of the present day, is stall and surge. Stall is a type of flow instability in compressors that sets the low-flow limit for compressor operation. As a result of the potentially damaging consequences of entering stall, extensive research has been performed on stall inception, stall detection, and stall control. A thorough review of the previous research and progresses made on these topics can be found in Refs. 1 and 2. The majority of the previous research focuses on stall/surge precursors and their inception. Early works from Benser and Moses [3], Jasen [4], Greitzer [5], Camp and Day [6] discuss the two routes to compressor rotating stall including the modal- and spike-type stall inception. More recent research focuses on the physical mechanisms that produce these different routes to rotating stall. Studies from Inoue et al. [7] and Pullan et al. [8] linked the formation of the stall cell with radial vorticity in the region between the casing and the blade suction surface. Despite the enhanced understanding of stall inception mechanisms (i.e., modal and spike-type stall inception) from the works, there has been limited progress achieved toward reliable stall warning or effective stall suppression.

The approaches for stall warning that focus on prestall flow irregularities can be categorized into two types of approaches: the correlation approach or the ensemble-average approach. The correlation approach was introduced by Dhingra et al. [9]. The method utilizes a parameter, known as the correlation measure, to gauge the repeatability of the pressure signature associated with the blade passing event. They found that there is a drop in the value of correlation measure as stall is approached, and the same trend was observed in both low- and high-speed compressors. In a later study, a stochastic model of correlation measure was also introduced by the same researcher [10], in which each drop in repeatability in the blade passing signature (correlation measure) is defined as an “event,” and a statistical parameter, “event rate,” is measured to gauge the margin of a compressor operating condition from stall. They found that the event rate increased rapidly as the compressor flow rate was reduced toward the stall point. In addition to model development, efforts have been made to implement the approach into engine active control systems. For example, Christensen et al. [11] implemented the correlation measure into a real-time stability management system. Experiments conducted on several laboratory compressor research articles, as well as a modern aircraft engine, showed that the parameter was able to signal increased proximity between the compressor operating point and its stability limit [11]. Liu et al. [12] laid out an active engine control scheme including a stall margin mode for protection against compressor instabilities, and assessed the system using a computer simulation of a turbofan engine.

On the other hand, Young et al. [13] characterized the blade passing irregularities using an ensemble-average approach in which the differences in individual blade-pass signatures are compared with an ensemble-averaged blade pass signature and characterized by the root-mean-square (rms) value of the difference. Then, the mean of the rms differences is evaluated to characterize the flow irregularities associated with the blade passing signature. Similar to the increasing “event rate” as stall is approached [10], there is an increase in the intensity of irregularity of the blade pass signature as the compressor is throttled toward stall. However, the study also showed that the increase in irregularity in the blade passing signature is highly dependent on both the tip-clearance size and eccentricity [13]. For example, a compressor with a small, uniform tip clearance would exhibit a modest increase in blade passing irregularity while a compressor with a large, uniform tip-clearance would give a sharp rise in irregularity at all circumferential locations as mass flow rate was reduced. In contrast, for a compressor with an eccentric tip clearance, the increase in irregularity with a reduction in compressor flow rate will only occur in the section of the annulus with the largest tip clearance. Thus, for compressors in aero engines, which can experience an increase in tip clearance over its service span, as well as eccentric tip clearance during a flight cycle, stall warning based on blade passing signature irregularity poses a challenge. For example, a stall warning system based on one pressure transducer at a fixed location would fail to give reliable results for compressors with eccentric tip clearances. On the other hand, the use of multiple transducers at different locations could lead to any number of false alarms. Therefore, a stall warning system based on blade passing signature irregularity would be very challenging to implement in an aero-engine due to changes in tip-clearance size and eccentricity during each flight cycle, or overall life cycle, of the compressor.

As mentioned previously, there are two routes to compressor rotating stall with different onset mechanisms. One is through the growth of small amplitude (compared to the mean velocity) disturbances with large-length scales (full annulus), which lead to the modal-type rotating stall. The other is through transient disturbances with much shorter length scales (several blade pitches), which leads to the spike-type rotating stall [8]. Despite the difference in these two routes, they share a common attribute: the nonlinear nature of rotating stall. The purpose of this work is to explore the potential of utilizing nonlinear feature extraction algorithms to detect small flow disturbances that occur just prior to stall/surge. Although nonlinear feature extraction algorithms including correlation dimension, entropy, and fractal dimension have been widely used in the research of chaos and recently in fault/failure diagnosis in engineering, there is very limited research that utilizes nonlinear feature extraction algorithms in turbomachinery. The only study in the open literature was performed by Wang et al. [14] for investigation of centrifugal compressor poststall characteristics using fractal dimension. Thus, this study aims to introduce a new direction for stall warning research.

## Methodology

The nonlinear feature extraction algorithm used in this study includes phase reconstruction of time-series data and evaluation of approximate entropy. The parameter, approximate entropy (ApEn), has nothing to do with thermodynamic entropy. Instead, it is a statistical parameter that measures the amount of regularity and unpredictability of fluctuations in time-series data. A time series with more repetitive patterns of variations renders smaller approximate entropy values and vice versa. The parameter was first introduced by Pincus [15]. Unlike other exact regularity statistics, including correlation dimension algorithms and various entropy measures, which require a vast amount of data and are discontinuous due to system noise, approximate entropy can discern changing complexity in a system with a relatively small amount of data [15]. This makes it an attractive parameter to explore considering the long-term goal of developing an in-flight stall warning system.

### Phase Space Reconstruction.

*m*-dimensional space as follows:

where $m$ is the embedding dimension, $t$ is the index lag, and $M=N\u2212(m\u22121)t$ is the number of embedded points in $m$-dimensional space. The time delay for a signal of sampling frequency, $fs$, is $td=t/fs$. An illustration for the phase construction of a time series is shown in Fig. 1.

According to Takens's theorem, the choice of time delay could almost be arbitrary for an infinite noise-free dataset [16]. However, for real data sets with the presence of noise and finite size, delay time has an important role in the reconstruction of the attractor. For example, Casdagli et al. [17] showed compressed reconstructed attractor (*redundance*) for an undersized time delay and also showed discontinued attractor dynamics (*irrelevance*) for an oversized time delay. There are several commonly used approaches for selection of time delay. One widely used method is the autocorrelation function. However, it has been pointed out that the autocorrelation function may not be appropriate for nonlinear systems, and instead, time delay should be chosen as the first local minimum of the mutual information (MI) [18].

### Approximate Entropy.

$Ckm$ represents the fraction of pairs of points whose maximum difference in their respective scalar components (also known as the sup-norm) separation with respect to $xk$ is no greater than $r$, where $r$ is the radius of similarity.

There are four parameters involved in evaluating the approximate entropy including data size, $N$, embedding dimension, $m$, time delay, $td$, and radius of similarity, $r$. In this paper, the effects of different choices for these parameters are evaluated and presented in the Considerations for Selection of *N*, *m*, *r*, and *t** _{d}* section.

## Results From Analysis of Compressor Data

This section presents a summary of results from two case studies using the nonlinear feature extraction algorithm. Analyses were performed using data sets acquired in two compressor research facilities at Purdue University including a high-speed single-stage centrifugal compressor and an intermediate-speed three-stage axial compressor facility. For the high-speed centrifugal compressor, stall was encountered unexpectedly during speed transients. The three-stage axial compressor features both modal- and spike-type stall inception depending on the rotor tip clearance levels. Therefore, these data sets provide a unique opportunity of examining the capability of the nonlinear feature extraction algorithm for different prestall signatures as well as different modes of operations (transient versus quasi-steady-state). In both experiments, the fast-response transducers were chosen for stall inception, and the ranges of the sensors were selected to cover the maximum pressure oscillations during the surge cycle.

### High-Speed Centrifugal Compressor With Rotating Stall During Speed Transients.

The nonlinear feature extraction algorithm is first applied to data acquired on a high-speed single-stage centrifugal compressor, which experienced unexpected rotating stall during speed sweeps. The layout of the compressor facility is shown in Fig. 2(a). The driveline includes a 1400 horsepower AC electric motor, a 30.46:1 ratio gearbox, an exhaust plenum with shaft housing, and a high-speed centrifugal compressor. The speed of the motor is precisely controlled using a variable frequency drive. The throttle valve is downstream of the compressor. The compressor stage has a configuration representative of aero-engine applications. The compressor has a design speed around 45,000 rpm and produces a total pressure ratio near 6.5 at the design condition. The compressor was instrumented with both steady flow and fast-response instrumentation. Total pressure and total temperature rakes were installed at the compressor inlet and exit to characterize the compressor performance. Fast-response pressure transducers were placed along the outer diameter of the flow path from impeller leading edge to downstream of the diffuser throat for detecting the location of stall inception. Details of the research facility, including instrumentation, can be found in Ref. 19. In this study, signals from a single transducer located at impeller leading edge, indicated in Fig. 2(b), is utilized for the purpose of stall warning.

Compressor speed sweeps (from subidle to full speed) were performed at four throttle positions (from choke to near surge). Each sweep starts with an acceleration and ends with a deceleration. A constant sweep rate was used for all sweeps. The throttle position for each sweep is listed in Table 1. Both compressor transient performance and unsteady pressure along the flow path were real-time monitored and continuously recorded during the speed sweeps. Figure 3 shows the compressor transient performance during the third and fourth sweeps at constant throttle settings of 34.0% and 36.2%, where a higher percentage indicates a more closed throttle and, thus, a higher compressor loading. Compressor transient performance is characterized by total pressure ratio using the area-averaged flow properties measured at the compressor inlet and exit. The difference in the compressor transient performance is associated with the heat transfer between the flow and the hardware [20]. In the experiment, it is observed that heat is extracted from the flow and warms up the compressor hardware during accelerations while heat is added to the flow (from the warm hardware) during decelerations. Even though no flow instability is observed during either of the accelerations, a flow instability (mild surge) occurs unexpectedly during the decelerations near 90% corrected speed. In both cases, the compression system recovers to stable operating conditions (lines below 90% corrected speed) once the throttle valve is opened.

Parameter | Value |
---|---|

Minimum speed (rpm) | 25,000 |

Maximum speed (rpm) | 48,000 |

Sweep rate (rpm/s) | 66.7 |

Throttle position (% close) | 21.6 (first sweep) |

29.0 (second sweep) | |

34.0 (third sweep) | |

36.2 (fourth sweep) |

Parameter | Value |
---|---|

Minimum speed (rpm) | 25,000 |

Maximum speed (rpm) | 48,000 |

Sweep rate (rpm/s) | 66.7 |

Throttle position (% close) | 21.6 (first sweep) |

29.0 (second sweep) | |

34.0 (third sweep) | |

36.2 (fourth sweep) |

During compressor sweeps, the unsteady pressure from the casing-mounted fast-response pressure transducers was continuously recorded. A sample rate of 100 kHz was used, which provides approximately 1300 data points per rotor revolution near 90% corrected speed (where the flow instability occurs). Figure 4(a) shows the instantaneous pressure traces obtained at the impeller leading edge during the fourth sweep with the 36.2% closed throttle. The abscissa is time in compressor speed, and the ordinate is the static pressure. In contrast to the stable operation during the acceleration, flow instability occurred during the deceleration. Due to the nature of mild surge, there are no significant pressure oscillations at the impeller leading edge. As a result, there are no evident differences in the raw pressure signal acquired during acceleration and deceleration shown in Fig. 4(a). However, considerable pressure oscillations were observed in the vaneless space and diffuser vane passage during the deceleration of the fourth sweep. A thorough investigation of the stall mechanism documented in Ref. 20 shows that the flow instability originated in the impeller with a small pressure perturbation. This small disturbance quickly developed into a rotating stall burst (olive). The rotating burst appears periodically and sends the compressor into mild surge (orange). Furthermore, the flow instability developed into a continuous high-frequency rotating stall (red). The rotating stall was a single lobe and moved opposite the direction of rotor rotation. Its propagation speed varied from approximately one-quarter to one-third of the impeller speed as the loading increases (from the third sweep with 34% closed throttle to the fourth sweep with 36.2% closed throttle). The compressor stays in the stalled condition for approximately 5800 revolutions (8 s) and returns to stable operation (blue) after opening the throttle.

Figure 4(b) shows the corresponding approximate entropy of the instantaneous pressure traces during the fourth sweep. The approximate entropy results shown in Figs. 4–6 were obtained with data from ten rotor revolutions, an embedding dimension $m=2$, a radius of similarity $r=0.2\sigma $ (where $\sigma $ is the standard deviation of the dataset), and time delay obtained from the average mutual information (AMI) method. Different from the time-domain raw pressure signal shown in Fig. 4(a), there are evident changes in the approximate entropy during acceleration and deceleration, as shown in Fig. 4(b). The approximate entropy of the unsteady pressure measurement during the acceleration without flow instabilities stays fairly constant. In contrast, the approximate entropy of the unsteady pressure measurement spikes during the deceleration as the flow instabilities occur. The approximate entropy during the first disturbance is more than two times larger than the approximate entropy at the stable operating conditions. During the phase of mild surge (orange color in Fig. 4(a)), spikes in approximate entropy occur more frequently. Finally, the approximate entropy remains at a similar level as that of the first disturbance during the phase of high-frequency rotating stall.

For the purpose of stall warning, it is of most interest to capture that first disturbance. Thus, results over a smaller range of speed transients focusing on the occurrence of the first few disturbances are shown in Fig. 5. The top graph shows the instantaneous pressure traces acquired at the impeller leading edge, and the bottom graph shows the corresponding approximate entropy. The spikes in approximate entropy align perfectly with the disturbances shown in the pressure traces.

In addition, the approximate entropy for the unsteady pressure acquired during the third sweep with a 34% close in throttle was also analyzed, and a spike in approximate entropy was observed as the first disturbance arrives during the deceleration, shown in Fig. 6. The spike in approximate entropy shows that approximate entropy is able to capture small, nonlinear disturbances in the compression system. Furthermore, in this study, there is a 10 s interval during the third deceleration and 8 s interval during the fourth deceleration between the occurrence of the first disturbance and the fully developed high-frequency stall, thus indicating the potential of using approximate entropy for stall warning in aero engines since this would be enough time to adjust the operation to avoid entering stall.

### Multistage Axial Compressor With Both Modal- and Spike-Types of Rotating Stall.

In addition to the high-speed centrifugal compressor, the algorithm is also applied to a multistage axial compressor with both modal- and spike-type stall inception. The compressor features an inlet guide vane and three stages that model the rear stages of a high-pressure core compressor. The design speed of the compressor is 5000 rpm, which produces an appreciable density rise at design point (of the order of 8% per stage). The compressor is powered by a 1400 HP AC motor, shown in Fig. 7(a). The facility features an open-loop air system including a settling chamber, inlet ducts with an ASME standard long-form venturi, and an exhaust collector (scroll) with embedded sliding annulus throttle. Details of the compressor research facility can be found in Ref. 21. The dataset used for the analysis presented herein was acquired in a previous test campaign conducted by Berdanier and Key [21] to better understand the effects of large rotor tip clearances on small-core compressor performance and operability. Experiments were conducted and detailed measurements were acquired at three rotor tip clearance configurations including 1.5, 3.0, and 4.0% span (accounts for 1.6%, 3.3%, and 4.4% of annulus, respectively) to study the effects of large clearances on small core compressors. The rotor tip clearance was adjusted using separate casings with different depths of recesses over the rotor, as shown in Fig. 8. Three sets of fast-response pressure transducers were flush-mounted into the outer diameter of the flow path at an axial position of 15% axial chord upstream of each rotor. At each axial location, six sensors were placed circumferentially around the compressor. Additionally, an axial array of sensors was also installed at a selected circumferential location over each rotor and distributed axially 15% axial chord upstream of the rotor leading edge to 15% axial chord downstream of the rotor trailing edge. Instrumentation details can be found in Ref. 22. In this study, the pressure signal from a single transducer instrumented upstream of rotor 1 was used for the purpose of stall warning, as indicated in Fig. 7(b).

Different from the stall experienced by the centrifugal compressors during speed transients, the stall inception measurements on this three-stage axial compressor were acquired using a quasi-steady approach. At a particular corrected speed, the stall point was mapped by closing the throttle in incremental steps to increase the loading of the compressor. After the throttle was adjusted, the compressor was allowed to reach a steady-state operation. When the compressor was sufficiently close to stall (as determined from an a priori stall test to map out the flow rate where stall occurs), the fast-response measurements were continuously recorded at a sample rate of 100 kHz and low-pass filtered at 40 kHz as the throttle was slowly closed. This allowed the capture of prestall activity, as well as stall inception. A detailed analysis of the tip clearance effects on stall inception for this machine can be found in Ref. 22. A few key findings can be summarized: (1) for this compressor, stage 1 is always the limiting stage, which was indicated by the consistent rollover in stage 1 static-to-static characteristics and also supported by the unsteady pressure measurements; (2) stall cells develop in stage 1, further extends to stages 2 and 3, and eventually cause the whole machine to go into stall; (3) the compressor showed a change in prestall signature with changes in rotor tip clearance. At design speed (100% Nc), the compressor has a strong prestall modal behavior at smaller rotor tip clearances (1.5 and 3% tip clearance) but is dominated by a spike-type stall at 4.0% tip clearance; (4) This change in stall inception mechanism is led by a combined impact associated with changes in tip leakage flow and changes in stage matching at different tip clearances. A summary of stage 1 results, static pressure characteristic and representative stall signatures at all the three tested tip clearance configurations are shown in Fig. 8. Note that an arbitrary offset was applied to the static pressure characteristics acquired at the three tip clearance configurations for better illustration of the changes in slope near stalled condition.

Figure 9 shows the instantaneous pressure traces over rotor 1 during stall inception and the corresponding approximate entropy. Figure 9(a) shows the results acquired at 1.5% tip clearance, and results from 3.0% to 4.0% tip clearances are shown in Figs. 9(b) and 9(c), respectively. In all the three cases, the approximate entropy was evaluated using the same data size (2 rotor revolutions), embedding dimension $m=4$, radius of similarity $r=0.2\sigma $, and AMI method for time delay calculation.

For all three tip clearance configurations, a significant increase in the magnitude of pressure traces and the calculated approximate entropy (as shown in red in the figure) occurred at compressor stall. In addition, prestall disturbances are detected using the approximate entropy parameter. Each prestall disturbance results in a peak in approximate entropy, shown in dark yellow in the figure. For example, the first disturbance during one stall inception campaign conducted at 1.5% tip clearance appears approximately 5 s prior to stall and results in a 75% increase in approximate entropy, as shown in Fig. 9(a). Similarly, the first prestall disturbance observed in the 4.0% tip clearance configuration with spike-type stall occurs approximately 6 s prior to stall and results in an almost 200% increase in approximate entropy, as indicated in Fig. 9(c). Despite the different onset criteria associated with the modal- and spike-type rotating stall, these peaks in approximate entropy prior to compressor stall indicates the nonlinear nature in both types of rotating stall. Additionally, these peaks in approximate entropy show the potential utility of using approximate entropy for prestall disturbance detection and stall warning.

In summary, the method using a nonlinear feature extraction algorithm is applied to stall data sets from two different types of compressors for aero-engine applications with success. In both cases, the algorithm was able to extract the small nonlinear disturbances prior to the compressor rotating stall, indicating the potential of using this method for compressor stall warning. Even though both experiments are related to compressors in aero-engines, the method can be used to a broad scope of turbomachines such as industrial compressors and turbochargers. Despite the different types of machines, there are only two routes to compressor rotating stall (modal- and spike-type), and the method has been shown success in detecting the small nonlinear disturbances prior to rotating stall in both scenarios.

Besides the promising results, there still exist questions and challenges for robust stall warning implementation. For example, questions regarding the repeatability of prestall disturbances, the optimal transducer location, criterion for threshold, etc., must be considered. While they are not addressed in this paper, they are the focus of continued effort and future work. The Considerations for Selection of *N*, *m*, *r*, and *t*_{d} section of this paper does, however, address the sensitivity of the process in choosing the parameters needed to evaluate the approximate entropy parameter.

## Considerations for Selection of $N,\u2009m,\u2009r,\u2009and\u2009td$

As discussed in the Methodology section, there are four parameters involved in evaluating the approximate entropy including: data size ($N)$, embedding dimension ($m)$, time delay ($td)$, and radius of similarity $(r)$. In this section, the influence of each parameter is presented using the dataset acquired on the high-speed centrifugal compressor. Considerations and recommended guidelines for selecting the individual parameter are provided.

### Considerations for Selection of $N$.

Figure 10 shows the distribution of $C(m,N,r,1)$ for a variety of data sizes ranging from $Nrev=2$ to $Nrev=25$ during stable operation with an embedding dimension of 2. For $Nrev\u22655$, the distribution of $\Phi m(r)$ remains nearly the same over the entire range for the selected radius of similarity with further increase in data size and, thus, indicates that a data size greater than five rotor revolutions represents the true correlation of the time series. However, for $Nrev\u22642$, $C(m,N,r,1)$ fails to represent the true correlation. It deviates slightly from the value at the same radius of similarity for $Nrev\u22655$. In addition, the value of $C(m,N,r,1)$ is nearly 1 for $r\u22654\sigma $, and this indicates that the distance between nearly every pair of two vectors in the constructed $m$-dimensional space is within four times of the standard deviation of the dataset. This observation agrees with the findings reported in Ref. 23.

The second consideration is to avoid saturation of the stranger extractor due to the large data sets. The approximate entropy of the unsteady pressure at the impeller leading edge throughout for the first disturbance during the deceleration in the fourth sweep is shown in Fig. 11 over a variety of selections for $N$. In an ideal case, each disturbance results in a single peak in approximate entropy. Thus, a selection of $Nrev=10$ or $Nrev=25$ gives a nice evolution of the disturbance. In both cases, it renders a single peak in approximate entropy and smooth ramp up and down as the disturbance arrives and leaves, respectively. However, a selection of larger data sets, i.e.,$\u2009Nrev=50$, results in saturation of the “stranger” attractor, and therefore, yields worse result (smaller peak in approximate entropy). Besides, a selection of larger data sets such as $Nrev=50$ yields a smaller update rate in approximate entropy. On the other hand, a smaller data size, i.e., $Nrev=2$, or $Nrev=5$, renders multiple peaks in approximate entropy but at a smaller magnitude during a single disturbance. Comparing to a single, dominant peak, these small multipeaks may introduce false alarms and make it difficult to implement in real engine stall warning applications.

### Considerations for Selection of $m$.

The influence of the embedding dimension was investigated, and the results are shown in Fig. 12. In this study, an embedding dimension of $m=2$ provides the best extraction of the disturbance for the high-speed centrifugal compressor dataset. This agrees with the study reported in Ref. 14, in which the same embedding dimension, $m=2$, was used to characterize compressor poststall signatures using fractal dimensions. For the case $m=3$, the calculated approximate entropy within the duration of disturbances gets substantially attenuated compared to the case for $m=2$. Furthermore, a selection of $m=4$ results in small drops in approximate entropy as the disturbances occur and, thus, fails to capture the disturbances.

### Considerations for Selection of $r$.

The radius of similarity sets the threshold for the definition of a “stranger” and, thus, care must be taken to find an optimal value for each application. In general, a small value of $r$ allows one to discern small levels of change in the system complexity, but this increase in sensitivity also comes at a price of reliability. For example, selection of a small $r$ may detect flow irregularities associated with turbulence and render a low signal-to-noise ratio. Figure 13 shows the distribution of approximate entropy over a wide range of radius of similarity $0.1\sigma \u2264r\u22644.0\sigma $. Results from stable conditions (light gray lines) and the time period when flow instabilities occur (dark yellow lines) are shown in the figure. The distributions of approximate entropy during stable conditions (light gray lines) are very repetitive over the entire range of selected $r$. In contrast, the distributions of approximate entropy during the time period when disturbances occur (dark yellow lines) are different and much less repetitive compared with those from stable conditions. This decrease in repeatability for approximate entropy distribution during unstable conditions is caused by the transient behavior (development and decay) of the disturbances.

where the average approximate entropy is calculated from approximate entropy at four different radii of similarity including $r=0.1\sigma ,\u20090.2\sigma ,\u20090.5\sigma ,\u2009and\u20091.0\sigma $. This method is motivated by the study from Kim et al. [23], in which they used the same formula for calculation of the correlation dimension. A similar set for similarity of radius, $r=0.2\sigma ,\u20090.5\sigma ,\u20091.0\sigma ,\u2009and\u20092.0\sigma $, was used and also provided good results.

To examine the effectiveness of the average approximate entropy, Fig. 14 shows the results of average and individual approximate entropy. Among the four selected radii of similarity, the choice of $r=0.2\sigma $ gives the largest peak magnitude of approximate entropy as the disturbances arrive. This agrees with the results shown in Fig. 11 that the optimal radius of similarity is $r=0.26\sigma $. Additionally, a selection of $r=0.5\sigma $ also gives good results in terms of the magnitude of the peak in approximate entropy as the disturbance arrives. However, the choices of $r=0.1\sigma $ and $r=1.0\sigma $ do not extract the disturbances well and give much smaller peaks in approximate entropy as the disturbances arrive. As discussed previously, a selection of a small radius of similarity, $r=0.1\sigma $, may allow the detection of small flow irregularities due to turbulence, which agrees with the observation of larger fluctuations in approximate entropy for stable operating conditions. In contrast, a selection of an excessive radius of similarity $r=1.0\sigma $ does render smaller and more steady approximate entropy for stable conditions. However, it also filters out the relatively small irregularities associated with the disturbances and, thus, results in a smaller peak in approximate entropy during the appearance of disturbances. At last, the averaged approximate entropy (dark yellow stars) provides a good balance between the level of approximate entropy during stable conditions and the magnitude of peaks in approximate entropy. Therefore, it is recommended to use the average approximate entropy for applications without a priori knowledge of the typical compressor stall signatures.

### Considerations for Selection of $td$.

As discussed in the Methodology section, the choice of time delay plays an important role in reconstructing the “stranger” attractor. The time delay could be obtained from either an autocorrelation (AC) function or MI method. Although the AC method requires less data and computational time, previous research in Ref. 18 has pointed out that it may not be appropriate for nonlinear systems and suggests using the first local minimum of the mutual information as the time delay. In this study, the influences of both methods on the calculation of approximate entropy were investigated, and the results are shown in Fig. 15. The results shown in the figure were obtained using the same number of data $Nrev=10$, embedding dimension $m=10$, and radius of similarity $r=0.2\sigma $. The black squares are the approximate entropy calculated using the time delay obtained from the AMI method, the olive circles are the approximate entropy based on the time delay selected as the first maxima (local maxima) from the autocorrelation function, and the blue triangles are the approximate entropy based on the time delay selected as the global maxima from the autocorrelation function. In general, the influence of the method utilized for obtaining the time delay in this study is minor. All three approaches give similar magnitudes of peaks in approximate entropy as disturbances arrive. However, the time delay obtained from the AC method with global maxima introduces a small fluctuation, at a very low frequency, to the approximate entropy at stable operating conditions. This is indicated by the slow increase in approximate entropy during stable operating conditions. In contrast, the approximate entropy obtained using time delays from the AMI method and AC method with local maxima stays fairly constant for stable compressor operating conditions.

In summary, general guidelines for the selection of parameters such as $N,\u2009m,\u2009r,\u2009and\u2009td$ are presented in this section. The AMI method is recommended to determine the value of $td$ for all types of machines and an average approximate entropy, according to Eq. (7), is recommended based on the values calculated at four different radii of similarity, including *r* = 0.1*σ*, 0.2*σ*, 0.5*σ*, and 1.0*σ*. Additionally, it is worth noting that the optimal values for date length ($N$) and embedding dimension ($m$) may differ among different types of machines. For instance, the optimal data size for the high-speed single-stage centrifugal compressor is $N=10$ rotor revolutions while its value is $N=2$ rotor revolutions for the medium-speed, three-stage axial compressor. The optimal embedding dimension for the centrifugal compressor is $m=2$, while the optimal embedding dimension for the three-stage axial compressor is $m=4$. Thus, intelligent choices of these parameters must be exercised in implementing approximate entropy for optimal extraction of flow disturbances.

## Conclusions

Stall is a type of flow instability in compressors, which sets the low flow limit for compressor operation. As a result of the damaging consequences, extensive research has been put toward understanding stall inception in hopes of being able to detect prestall signatures to allow appropriate measures to be taken to avoid stall, also known as stall control. However, there is limited progress in developing a reliable stall warning or effective stall suppression system, which motivates the work presented in this paper.

The contributions of this paper are twofold. First, it introduces a new approach to identify the small disturbances prior to stall using nonlinear feature extraction algorithms. The method is different from the well-known stall warning techniques in the time domain including the correlation measure method introduced by Dhingra et al. [9] and the ensemble-average method introduced by Young et al. [13]. The analysis of the new method is performed in phase space using the approximate entropy parameter. Approximate entropy is a measure of the amount of irregularity and unpredictability of fluctuations in time-series data and was first introduced by Pincus. In general, a time series with more repetitive patterns of fluctuations renders smaller approximate entropy and vice versa. A detailed procedure of the nonlinear feature extraction algorithm was presented. Furthermore, the method is applied to a high-speed centrifugal compressor, which experienced unexpected rotating stall during speed transients, and a multistage axial compressor, with both modal and spike types of stall. For both compressors, the signals from casing-mounted transducers were first reconstructed in the phase domain. Then, the approximate entropy for the phase-reconstructed signal was evaluated. In both cases, the appearance of nonlinear disturbances, in terms of spikes in approximate entropy, occur prior to stall. The concurrent spike in approximate entropy with the occurrence of the pressure disturbance shows that the parameter is capable of capturing small disturbances in a compression system and also indicates the potential of using the approximate entropy parameter for stall warning in aero engines. This is the first use of approximate entropy for the purpose of stall warning in compressors in the open literature.

As with other stall warning techniques, the intelligent choice of several parameters must be exercised. To implement approximate entropy, there are four parameters involved including the number of data points $(N)$, embedding dimension $(m)$, time delay ($td)$, and radius of similarity ($r)$. The influence of these four parameters on the effectiveness of approximate entropy for disturbance extraction was explored. Additionally, considerations and guidelines for selection of each parameter were also provided. The technique can be implemented within a real-time health monitoring system for compressors or gas turbine engines for cases with and without preliminary experimental results. In cases without preliminary test data, the method needs to establish a baseline condition by real-time monitoring and collection of stable operation data. Those data can be used to tune parameters such as $N,\u2009m,\u2009r,\u2009and\u2009td$. For cases with test data, determination of the optimal parameters of $N,\u2009m,\u2009r,\u2009and\u2009td$ can be conducted offline. Then, the values of these parameters can be programmed in the online monitoring system.

In summary, this paper introduces a new approach for identifying small prestall or presurge disturbances using nonlinear feature extraction algorithms. Analysis of the unsteady pressure signals acquired in two compressor research facilities shows the potential of using approximate entropy for stall warning in a broad scope of turbomachines such as gas turbine engines and industrial compressors connected to large volumes. The work presented in the paper serves as a foundation for future work on stall warning using nonlinear algorithms.