Control-Oriented System Identification of Turbojet Dynamics

Abstract

The autonomous operation of turbojets requires reliable, accurate, and manageable dynamical models for several key processes. This article describes a practical robust method for obtaining turbojet thrust and shaft speed models from experimental data. The proposed methodology combines several data mining tools with the intention of handling typical difficulties present during experimental turbojet modeling, such as high noise levels and uncertainty in the plant dynamics. The resulting shaft speed and thrust models achieved a percentage error of 0.8561% and 3.3081%, respectively, for the whole operating range. The predictive power of the resulting models is also assessed in the frequency domain. The turbojet cut frequencies are experimentally determined and were found to match those predicted by the identified models. Finally, the proposed strategy is systematically tested with respect to popular aeroengine models, outperforming them both in the time and frequency domains. These results allow us to conclude that the proposed modeling method improves current modeling approaches in both manageability and predictive power.

1. Introduction

The propulsion systems of Unmanned Aerial Vehicles (UAVs) require as much performance and reliability (or even more) as traditional manned aircraft. This becomes clear when considering some of the challenges inherent to the operation of these vehicles, such as adverse operating conditions, multiple operating configurations, variations in the effective payload, stringent reaction-time demands, and performance variability due to component degradation [1,2]. In addition, the variables measured during flight are significantly limited [3]. This combination of operational uncertainty and limited sensor information renders the aeroengine modeling task to be particularly important yet complex.

Turbojet modeling can be useful to estimate variables that cannot be measured during flight and to obtain redundant sensor measurements. For instance, turbojet thrust, which is the ultimate purpose of aircraft propulsion systems, is typically not measured during flight. Therefore, thrust models are computed with off-board experiments and dedicated test facilities [4]. The importance of thrust models relies on their usefulness for the decision process related to aircraft speed and attitude [5]. On the other hand, the shaft speed acts as a safety constraint for turbojet operation [6], is the most commonly regulated variable in commercial set-ups [7,8], and plays an important role as a health [9] and thermal state indicator [10]. The importance of turbojet thrust and shaft speed has led to the introduction of many mathematical representations. These representations can be built either theoretically or experimentally:

• Theoretical models use physical principles to make predictions of turbojet performance [11,12,13,14]. The underlying principles of theoretical models are based on thermodynamic analyses for mass, energy, and momentum conservation [15]. Most theoretical models yield computationally intensive solutions and/or only deal with static conditions. Thus, these models are often useful for design [16,17], assessing environmental impact [18], and long-term health monitoring [19,20].
• Experimental models are data-based representations calculated with data measured during operation. Experimental model applications range from assessing aeroengine performance and emissions [21,22], component degradation, and environmental impact [23] to diagnosis [24,25]. Nonetheless, these models often require a priori knowledge of the model structure and larger data sets to compute the corresponding parameters. Complex model structures are prone to be more sensitive to parameter variations [26], especially when dealing with high levels of sensor noise and uncertainty in the system dynamics [27,28]. This is particularly true when considering that in practice, aircraft model parameters are computed from noisy flight test data [29,30,31].

In this context, hybrid combinations of theoretical and experimental models can be used to combine the advantages of each approach. The main difference that segregates fully experimental (black-box) and gray-box models is the intention of finding a consistent model structure rather than focusing solely on data fitting. Gray-box models usually possess a structure that combines well-known physical laws with data-based parameter adjustment [32], as depicted in Figure 1. The purpose of using physically-inspired structures is to have a model structure consistent with the real-world process; nonetheless, the recent development of data mining tools has enabled the discovery of consistent patterns and model structures underlying in experimental data [33,34]. Gray-box models obtained through data mining have already yielded good results in energy forecasting applications [35,36], thus presenting an interesting and novel approach to mathematical modeling.

In order to select the best-suited mathematical representation and system identification method, the application and available information play a key role. For turbojet autonomy, which is required to guarantee performance and operational safety [37,38], the preferred models are those that facilitate maintaining the engine within specific operational conditions and dealing with the system’s inherent nonlinear dynamics [39,40]. From a practical standpoint, turbojet autonomy also requires dynamic models that remain manageable enough to allow its evaluation with metrics typically used by aeronautical authorities, such as GSFC-STD-1000 [41] and MIL-HDBK-516B [42].

The previous discussion shows that current aeroengine models can still be improved to better comply with the specific requirements that involve pursuing turbojet autonomy. For instance, simpler linear approximations are commonly used for this application [7,39,40,43,44,45]. However, UAVs operate in wider regimes than manned aircraft [46,47], and the valid operating range of linear approximations is often exceeded. Neural-network-based models can be used to meet the nonlinear system requirements [48,49]; nonetheless, these models are incompatible with certification metrics, provide little insight into the process dynamic characteristics, and may involve a high computational cost for on-board implementation [50,51]. This explains why, even with the availability of highly accurate modeling techniques, computationally lean models are preferred in many recent aerospace applications [52].

Control-oriented models have been of particular interest to improve the reliability and performance of different applied energy systems, including wind turbines [53], variable-geometry turbochargers [54], battery management systems [55,56,57], fuel-cell systems [58], multienergy systems [59], and prediction of performance in internal combustion engines [60,61,62]. In particular, it is desirable for these models to comply with the following key characteristics: (i) accuracy for the whole operating range, (ii) manageability, and (iii) compatibility with typical certification metrics. Notwithstanding the widespread development of control-oriented models in other application areas, a comprehensive bibliographical review showed that the availability of turbojet control-oriented nonlinear models is limited. Thus, this paper introduces a novel system identification approach that takes these elements into consideration for the main variables that describe turbojet operation with respect to the input fuel flow.

The proposed identification method is formulated with the intention of handling the typical difficulties presented during data-based turbojet modeling, such as high noise-to-signal ratio and uncertainty in the plant dynamics. In particular, the method is based on a hybrid time–frequency domain regression with a clustering + classification technique, which allows obtaining improved low-order gray-box models even in these adverse conditions. The main objective of the proposed identification process is for typical linear control structures to be compatible with the resulting gray-box models with only very minor adaptations. This property simplifies the adoption of the model with the most widespread turbojet control techniques.

In addition, due to the lack of standardized testing procedures for turbojet dynamical model identification and validation, a comprehensive set of experimental tests is proposed for this purpose. The resulting models are compared with two other popular aeroengine modeling approaches in order to assess the performance of the proposed method with current methodologies. The results show that the proposed method is able to capture transient, steady state, and key frequency-domain properties of the engine shaft speed and thrust. Overall, the proposed models show improved predictive power while being structurally simpler.

2. Model Structure Definition

Although linearity can be attractive from an engineering point of view, assuming linearity during controller design can significantly reduce the operating range of the resulting schemes, which is required to take full advantage of the propulsion system’s capabilities. In the context of aeronautical systems, this has historically been dealt with by several techniques that extend the operating range of typical structures, such as gain scheduling [63]. Another option is the direct integration of more complex model structures into the controller design stage [64]. A salient characteristic of these approaches is that the resulting controllers are heavily dependent on the mathematical approach used to represent the plant’s nonlinear dynamics. This introduces several limitations, since the use of a particular model structure limits the viable options for control design and analysis.

Block-structured models are separable models assembled by linear and nonlinear subsystems. The most common configurations are the Hammerstein and Wiener structures, which consist in a cascade connection of a memoryless static nonlinear subsystem and a linear time-invariant subsystem; the difference being the arrangement of the subsystems, as shown in Figure 2.

Wiener-structured models are prone to be compatible with aeroengine modeling [65], since they allow to better represent static and dynamical uncertainties. In [66], the precision of both Wiener and Hammerstein representations is compared, concluding that the Wiener structures outperform Hammerstein structures in modeling a nonlinear process with less identification data requirements. Although previous studies have pursued aeroengine modeling through Wiener models, the present paper assesses the system identification problem through a more realistic approach by dealing with real-process uncertainties and noise through an experimental framework. Moreover, the proposed identification method uses a Wiener structure, specifically due to its particular properties regarding controller design and analysis, which are described as follows:

• Wiener models can retain the stability, robustness, and performance properties of the linear dynamic component for each possible operating condition [67] by exploiting the properties of the linear subsystem [68].
• Wiener models can be analyzed in terms of classical robustness margins [69].

These properties allow combining Wiener models with a wide variety of control design techniques, such as model predictive control [70], adaptive control [71], classical control [72], gain scheduling schemes [73], feed-forward strategies [74], and optimal control methods, such as $H_{\infty }$ controllers [75]. Thus, the resulting models can be seamlessly integrated into previous research based on linear representations to assess the resulting system’s stability and performance characteristics. As an example, in [38], the structure of gray-box models was exploited to design an optimal direct-thrust controller with the aid of a thrust observer, resulting in reduced fuel consumption. The previous discussion shows how this structure allows complying with the specific goal of achieving model compatibility with the most widespread control design techniques.

3. Novel Data-Based Identification Method

The proposed identification method used for the turbojet model estimation is described in the following section. This is a two-step process that segregates causal and stochastic dynamical components from the system signals and thereafter captures the nonlinearities in the process through a simple memoryless mapping function.

3.1. Linear Subsystem Identification

During this stage, the purpose of the identification algorithm is to capture the different dynamical and frequency components of the signal; however, the resulting parameters are influenced by measurement noise and other types of distortion. In experimental conditions, the measured response of the system encompasses:

• Causal components (from input variables) that represent the true dynamic behavior of the system (a consistent behavior in the response).
• Stochastic components can be considered to be contamination external to the process.

This introduces the necessity of segregating these causal and stochastic components. For instance, it has been shown that high levels of non-Gaussian noise can mistakenly be identified as additional zeros–poles, a phenomenon akin to overlearning in other data-based identification methods. The proposed method is designed to overcome the difficulties associated with dynamic subsystem identification. This method departs from a well-known approach to obtain a discrete-time-invariant model, represented by transfer function G(z), as shown next:

$$G\left(z\right)={\displaystyle \frac{v\left(z\right)}{u\left(z\right)}}={\displaystyle \frac{{\displaystyle \sum _{i=0}^{nb}}{b}_i{z}^{-i}}{{\displaystyle \sum _{j=0}^{na}}{a}_j{z}^{-j}}}$$

(1)

where v is the linear output to an arbitrary input u, and $a_i$ and $b_i$ are the coefficients of the discrete transfer function. Therefore, the number of poles and zeros of the linear subsystem is given by $na$ and $nb$, respectively. Thus, the resulting sampled time response $v\left(k\right)$ of linear Model (1), presented as a difference equation, is

$$\begin{array}{c}v\left(k\right)=-a_{1}v(k-1)-\dots -a_{na}v\left(z^{k-na}\right)\\ +b_{1}u(k-1)+\dots b_{nb}u(k-nb)+e\left(k\right)\end{array}$$

(2)

where k is the sample number. The coefficients of the polynomial function can be easily computed through parameter optimization by the minimization of modeling errors $\sigma$ of n output measurements $y_m$:

$${\sigma }^2=\sum _{k=1}^n{(y_m\left(k\right)-v\left(k\right))}^2$$

(3)

If $\exists \theta \ni min\left(\right|\left|\sigma \right|\left|\right)$, least squares algorithm can used to determine transfer Function (1) parameters:

$$\theta ={\left({\varphi }^T\varphi \right)}^{-1}{\varphi }^{T}Y$$

(4)

where $\theta$ contains the transfer Function (1) parameters $\theta ={[a_1\dots a_{na}{b}_1\dots b_{nb}]}^T$ that correspond to its poles and zeros polynomial coefficients, and $\varphi$ is the regression vector $\varphi =[-v(k-1)\dots -v(k-na\left)u\right(k)\dots u(k-1\left)u\right(k-nb\left)\right]$. This structure associates the known measured values of past states and input conditions with the unknown parameters. Although least squares algorithms can provide a good level of noise rejection allowing the identification of underlying dynamical components, their performance can be limited when dealing with nonlinear systems, time-varying systems, and high levels of non-Gaussian noise [76]. Furthermore, least squares algorithms require a priori knowledge of the zero–pole structure (i.e., $na$ and $nb$). An initial idea of the zero–pole structure can be obtained from theoretical analysis; however, complex models, such as turbojets, require one to account for possible additional unmodeled dynamics. This yields a two-pronged problem with respect to the previous formulation: (1) determining the ideal zero–pole structure and (2) discriminating valid and invalid zeros–poles (i.e., reducing overlearning). This paper introduces a practical and novel approach to these problems using data mining to analyze and classify pattern formations. The proposed method to obtain the linear subsystem consists in the following:

• Defining a set of j transfer functions, each one obtained from a small portion of the operating range using the aforementioned least squares approach. The complete set of transfer functions must cover the whole operating range. In addition, each one of these transfer functions must be estimated considering a large number of poles and zeros, generating a set of high-order identified linear models. If a large enough zero–pole structure is used, these models must then contain both valid and invalid zeros–poles. Arguably, the valid components must be consistent along all the operating ranges (with small variations due to changes in operating conditions) [77]. On the other hand, the invalid components introduced by noise, overlearning, and other stochastic elements must be of a more random nature, that is, inconsistent along the set of identified linear models. Moreover, the fundamental frequency of the identified components also reveals important information regarding their validity. For instance, turbojet thrust measurements are normally severely contaminated by high-frequency vibration noise. However, it is well-known that thrust dynamics are much slower.
• Segregating valid and invalid components of the identified models using the following considerations: (1) repeatability and (2) fundamental frequency. This can be easily achieved by superimposing the zeros–poles of identified transfer functions in a polar plane and looking for clusters. The clusters must be defined according to the number of poles–zeros within a radius determined by the experiments. The threshold defined by the circle boundaries allows determining the zero–pole values that yield a low standard deviation with respect to the circle centroid. That is, the estimates of the zero–pole locations of the system are bounded within conditions that ensure consistency and repeatability in the system time and frequency response characteristics. Inside the circumscribed area there must exist a pole–zero for each identified transfer function to ensure repeatability. Note that increasing the circle radius yields more variety in the identified system dynamics characteristics and may integrate stochastic dynamics that are not part of the system dynamics. The resulting plot allows observing dynamics that are being consistently identified in all the estimated models as clusters, while those that lack a cluster pattern correspond to the stochastic components. This pole–zero clustering analysis allows for distinguishing the causal and stochastic components present in the measurements and determining the appropriate pole–zero structure at the same time. Moreover, the frequency of the identified components is also easily assessed by the angle of the identified cluster. In the case of shaft speed and thrust dynamics, clusters near the real axis, being of the lowest frequency, are preferred (i.e., dominant in the steady state).
• Forming a transfer function with poles–zeros equal to the average of each cluster. That is, each cluster is translated into a single pole–zero. This yields a time-invariant low-order transfer function with the averaged dynamical properties found to be consistent in the complete operating range.

It is clear that the proposed method depends on the clustering formation, which must be adjusted according to the experiment, mainly the size and number of elements are relevant parameters (i.e., dispersion and aggregation conditions). In particular, the number of elements should be equal or close to the number of identified linear models (i.e., operating points) to ensure that the dynamic modes are effectively present in all cases. This aspect defines the main aggregation condition for the clustering. On the other hand, cluster size can be determined experimentally. A smaller cluster size indicates more homogeneity of the dynamic modes along the complete operating range and is desirable. A larger cluster size could indicate either a higher level of nonlinearity in the experimental dynamic response or a higher noise level. That is, the dispersion of the clusters is related to these aspects. In practice, it was observed that the method lends well to a visual adjustment of this variable since a 2D polar plot is always obtained no matter the number of operating points. For instance, if the Euclidean distance is used to define the clusters, the poles and zeros would have to remain within a distance $d_c$ circumscribed with respect to the centroid $c_c$. That is, a cluster of poles exists if and only if

$$\left|c_c-p_j\right|<d_c\forall j$$

(5)

where $p_j$ belongs to the roots of $b_0+b_1{z}^{-1}+\dots b_{nb}{z}^{-nb}$ of each $j-th$ identified transfer function. Respectively, a cluster of zeros exists if and only if

$$\left|c_c-z_j\right|<d_c\forall j$$

(6)

where $z_i$ belongs to the roots of $a_0+a_1{z}^{-1}+\dots a_{na}{z}^{-na}$ for each $j-th$ identified transfer function. The automated adjustment of cluster properties, such as k-means [78] and hierarchical clustering [79], can also be used, as long as the specific goal of clearly distinguishing between consistent and stochastic system dynamics is met.

3.2. Static Function Nonlinear Approximation

The typical operation regimes, which involve operating at steady state conditions, introduce the necessity of accurate prediction of the static characteristics. If there exists a family of equilibrium conditions parameterized by the output and each of these conditions represents a feasible operating condition, then an operator can be designed for each local time-invariant approximation. Hereby, gain scheduling provides a simple approach to nonlinear analysis by interpolating between the members of each family. Thus, the nonlinear static function output, defined as $y=f\left(v\right)$, can be obtained through any nonlinear mapping. Consider a ratio $r(y_m,v)$ among the identified linear model response $v\left(k\right)$ and the measurements $y_m\left(k\right)$ in the form of

$$r(y_m,v)={\displaystyle \frac{y_m\left(k\right)}{v\left(k\right)}}$$

(7)

This ratio represents the remaining nonlinearities that were not captured by the linear model alone, ideally with $y_m\left(k\right)=r(y_m,v)v\left(k\right)$. If $\exists f\left(v\right)\ni f\left(v\right)\approx r(y_m,v)v\left(k\right)$, it is possible to define and minimize a residual $\delta$ for a given experiment length $k=1\to n$ as

$${\delta }^2=\sum _{k=1}^n{(y_m\left(k\right)-f\left(v\left(k\right)\right))}^2$$

(8)

Then, the resulting model estimates $y\left(k\right)=f\left(v\right(k\left)\right)$ are close to the experimental data available. That is, $\left|\right|\delta \left|\right|\approx 0$ with the nonlinear static gain approximation. On the other hand, the compatibility of the resulting model with specific control design tools depends on the properties of the nonlinear output function.

With this in mind, simpler nonlinear approximations are preferred for this application. The studies presented in [15,80] analyzed several Exhaust Gas Temperature (EGT) models, providing an experimental criterion for model selection based on the application and the available input data. The results suggest that, for this particular application, static microturbojet nonlinearities can be approximated with polynomials. These structures have also presented good results at capturing the stationary properties of the shaft speed of large- and small-scale turbojets [81,82]. In addition, one such approximation was experimentally tested in [83] with the thrust of a small-scale turbojet. Following these results, in this article, polynomial approximations of order i (which is determined through exploratory data analysis) are used to model the shaft speed and thrust nonlinearities:

$$r(y_m,v)v\left(k\right)\approx f\left(v\right)={\beta }_0+{\beta }_{1}v\left(k\right)\dots {\beta }_i{\left(v\left(k\right)\right)}^i$$

(9)

with a vector of polynomial coefficients $\beta ={[{\beta }_0{\beta }_1\dots {\beta }_i]}^T$. Since different nonlinear static gain approximations are possible, it is convenient to define the optimization problem in a generalized form. For instance, gradient descent can be applied in this case to compute the coefficients of Approximation (9) with l iterations. That is,

$${\beta }_l:={\beta }_{l-1}-\gamma \sum _{k=1}^n\left[\left(f\left(v\left(k\right)\right)-y_m\left(k\right)\right)v\left(k\right)\right]$$

(10)

where $\gamma$ is the learning rate. The goal of this algorithm is to find $\beta \ni min\left|\right|\delta \left|\right|$, which implies fitting the nonlinear static gain Approximation (9) to the Ratio (7). The role of the nonlinear static gain is to narrow the gap between the linear model estimates and the real process behavior.

3.3. Identification Method Summary

Figure 3 presents a flowchart of the novel identification method. Additionally, a summary of this method is presented next.

• High-order linear dynamic subsystem set identification: A set of high-order transfer functions is obtained. Each transfer function is obtained from a small portion of the operating range. The parameters of each transfer function are determined through a squared error minimization problem.
• Cluster analysis: The parameter identification results are superimposed in a polar plot, in the form of the poles and zeros of the identified transfer functions.
• Model simplification: Pole–zero clusters are formed and identified according to the number of elements and their proximity, the dynamics being identified consistently are considered to represent the causal components of the input.
• Nonlinear approximation: A nonlinear gain is characterized through a polynomial function with the information from the different steady-state values.
• Final model determination: The time-invariant transfer function and the nonlinear static gain function are assembled in a Wiener structure.

Note that the resulting models are as good as the input data used for identification. Therefore, different techniques of data reconciliation are advised as a preliminary step to perform a consistency check on the measurements, detect errors, and validate that the data are useful.

4. Experimental Set-Up

In this article, the SR-30 gas turbine from Turbine Technologies (Chetek, WI, USA) was used as an experimental case study involving the modeling of a commercial microturbojet. The obtained information allows assessing the performance of the proposed gray-box approach in an experimental framework.

4.1. SR-30 Engine

The microturbojet comprises an inlet, a centrifugal compressor, an annular combustion chamber, a single-stage axial flow turbine, and a subsonic nozzle (Figure 4). The maximum operational rotational speed is 87,000 RPM with a maximum thrust of 178 N. The fuel flow is adjusted by a manual lever, which is connected to a valve that controls the flow. The fuel blend consists of Jet A with a specific energy of 42.80 MJ/kg. The thrust force is measured through a strain gauge cell, while a pressure transducer system is used to measure the fuel flow and a 2-pole tachometer generator measures the rotational shaft speed.

4.2. Data Acquisition System

The platform includes the PCI 4351 data acquisition system; an A/D board with 24-bit resolution for 16 analog inputs with a 60 samples/s capability and a Virtual Bench Logger data acquisition program for real-time monitoring of the measured variables on a host PC. The maximum practical sampling frequency (8.85 Hz) was selected for all the experiments.

4.3. Design of Experiments

Attempting to model with poorly informative data and/or using an inappropriate model structure are the most common errors during system identification [65]. Therefore, the identification and validation data sets must contain enough information regarding the system properties. Although there are well-known testing regimes used for static turbojet modeling, used for example to calculate performance maps [84], this is not the case for dynamic turbojet identification. Therefore, the following identification and validation tests for microturbojet modeling are proposed to help standardize microturbojet dynamic model identification and validation.

Three separate tests are specified. These tests are designed to provide appropriate data sets for model identification and extensive evaluation of the resulting models. The main objectives and practical considerations for each test are described next and presented in Figure 5.

• Identification regime. For the excitation signal, stepped inputs of varying duration and magnitude are preferred in this application since they allow extracting information of the dynamic and static characteristics of the engine [85] and are required to test the repeatability as stated by the identification method. The suggested identification procedure is presented in Figure 5a. The measured data are later divided according to each step (labeled as A, B, C, D, E, F, and H). The experiment must include the whole operating range and each step should be maintained sufficient time for the system to stabilize. The proposed experimental design weights the mid-lower shaft speed ranges higher than the top range (only one step in the upper range is used), because this is the most common operating region. The experimental design (i.e., number of steps and operating range) can be modified to weight any desired region according to particular applications.
• Time-domain validation. To validate the resulting models a validation data set must be obtained and the resulting models should perform well under these untrained test conditions. In this case, both the transient and steady characteristics of the turbojet must be excited. The excitation signals suggested for this purpose must contain a variety of steady-state conditions and dynamical behaviors, simulating a wide range of real operating conditions. In this case, a series of fast steps, ramp signals, sinusoidal inputs, and whole-rage step signals with high frequency content are suggested. This combination of signals allows testing for transient, steady state as well as slower thermodynamic and other nonmodeled phenomena. These excitation signals can be applied in separate experiments or combined in a single experiment as the one shown in Figure 5b.
• Frequency-domain validation. Finally, the frequency-domain properties of the plant are fundamental for the development of model-based controllers and estimators because they are responsible for key stability and robustness properties. The predictive power of the resulting models in the context of the frequency domain can be assessed with a chirp-like input as the one shown in Figure 5c. The frequency range of the input signal depends on the turbojet capabilities, with the upper bound being the most important. However, it is easy to find this upper bound experimentally by increasing the input frequency until the measured output variables reach a negligible amplitude. For instance, for the tested microturbojet the frequency range obtained using this procedure was 0.574 rad/s–13.78 rad/s.

In all cases, relevant environmental variables must be registered during the tests, including temperature, pressure, and relative humidity. The tests include steady flow at the engine inlet (no induced airspeed) and constant altitude (390 m asl) conditions.

5. Applied System Identification

In this section, the identification of a microturbojet considering the model structure and the method described in the previous sections is presented. In addition, two baseline models are incorporated in this section for further assessment of the model performance.

5.1. Shaft Speed Identification

The proposed high-order transfer functions comprise 19 poles and 3 zeros before the cluster analysis, with a total of 7 identified transfer functions, one for each step input of the identification regime. The identified zeros and poles of each transfer function are superimposed in Figure 6a. The clusters were identified according to the amount of elements (one for each identified model to ensure repeatability) and their proximity. The shaft speed pole clusters are located near 0 and $\pm 25$ deg/s. The high-frequency pole clusters near the Nyquist–Shannon frequency limit are not included in the reduced model to reduce the effects of sampling distortions. The rest of the poles and zeros are randomly scattered over the Z plane. Although some elements resemble clusters, the number of elements condition was not satisfied. That is, these were not identified consistently in all operating conditions and are of higher frequency, suggesting that these components are due to over-learning. Considering the previous procedure, the reduced shaft speed model yields

$$G_s\left(z\right)={\displaystyle \frac{82.70}{(z-0.936)(z^2-1.608z+0.692)}}$$

(11)

Figure 6b shows the frequency response of the resulting reduced Model (11) in comparison with the high-order models. This figure allows to evaluate the effects of the cluster simplification.

The static function approximation is obtained considering the ratio among the prediction of the reduced linear dynamic model and each steady state from Figure 6a. The shaft speed static function ($f_s$) was approximated through the following polynomial:

$$f_s=2.878\times {10}^4+4.021\times {10}^{-1}{v}_s+1.532\times {10}^{-6}{v}_s^2$$

(12)

Figure 7b presents the accuracy of the nonlinear function estimation in relation with the experimental data.

The complete shaft speed nonlinear model is arranged in the structure shown in Figure 2. This representation consists in transfer function $G_s\left(z\right)$ in series with static function $f_s$. Figure 7a presents the performance of the linear and nonlinear models in comparison with the experimental data. This figure shows that although the dynamic components may be well represented by the reduced linear subsystem; the static gain can diverge depending on the operation condition. This is especially true for the shaft speed dynamics, where the nonlinear behavior is more dominant.

5.2. Thrust Identification

Thrust modeling presents different challenges. In contrast to the measured shaft speed data, thrust measurements contain higher levels of noise, while the nonlinear behavior is less dominant. The proposed high-order transfer functions consist in 21 poles and 19 zeros before the cluster analysis. The zeros and poles of all transfer functions are presented in Figure 8a. This figure shows that the main thrust pole-cluster is located near 0 deg/s. The rest of the poles and zeros appear to be randomly scattered over the polar plane, showing that the clustering method allows extracting the main transient characteristics of the turbojet under the uncertainty and noise conditions commonly found in these devices. Again, although some elements resemble clusters, the number of elements condition was not satisfied.

Figure 8b shows that the reduced model, obtained through the cluster analysis, also captures the averaged frequency–response of the high-order models up to the frequency where they are consistent. The resulting transfer function is

$$G_t\left(z\right)={\displaystyle \frac{0.7962}{z-0.936}}$$

(13)

The nonlinear static function approximation was calculated similarly to the shaft speed presented before, resulting in the nonlinear output Function (14).

$$f_t=3.881\times {10}^1-1.004v_t+2.356\times {10}^{-2}{v}_t^2$$

(14)

The resulting approximation and the fitted data are presented in Figure 9b. The final thrust model has the structure presented in Figure 2 with a linear time-invariant model $G_t\left(z\right)$ and the memoryless nonlinear model $f_t$. Lastly, Figure 9a presents the simulation results of the complete thrust model compared with experimental data. These results show that the proposed gray-box identification approach allows accurate modeling of the thrust behavior even in the presence of high levels of measurement noise.

5.3. Baseline Models

In general, assessing the performance of a particular model requires comparing the quality of the model to some relevant baseline. With this purpose, Hammerstein–Wiener and Neural Network AutoRegressive with eXogenous input (NNARX) models were obtained using a commercial parameter-computing tool for both the shaft speed and thrust. These models were obtained using the same identification regime data set.

5.3.1. Hammerstein–Wiener Model

This methodology has been used successfully in commercial applications by computing the parameters of each stage from measured data using least-squares estimation [86]. The accessibility of this approach has led to widespread use in different engineering applications with control and estimation purposes, such as modeling borefield thermal dynamics [87] and two-stage shape memory alloys [88]. Using this approach, Models (15) and (16) were computed for the shaft speed and thrust, respectively, using the same data set from the identification regime test and the structure presented in Figure 10.

The same pole–zero configuration obtained in the previous section was maintained. However, in this case, a more classical Hammerstein–Wiener identification method (as implemented in popular commercial toolboxes) was used mainly for comparison purposes. This methodology does not include the high-order clustering, and simplification steps proposed in this article. Therefore, comparing the resulting models will highlight the importance of these steps. The shaft speed Hammerstein–Wiener model is given by

$$\begin{array}{c}{\widehat{u}}_s=0.007u^2+0.873u-0.7745\\ G_{sh}\left(z\right)={\displaystyle \frac{z(0.026+z)}{(z-0.786)(z^2-1.043z+0.725)}}\\ {\widehat{y}}_{sh}=-1.526v_{sh}^2+1.517\times {10}^3{v}_{sh}+2.990\times {10}^4\end{array}$$

(15)

Similarly, the thrust model is

$$\begin{array}{c}{\widehat{u}}_t=0.038u^2+0.404u-1.357\\ G_{th}\left(z\right)={\displaystyle \frac{1}{(z-0.773)}}\\ {\widehat{y}}_{th}=3.118\times {10}^{-1}{v}_{th}^2-4.0531v_{th}+34.823\end{array}$$

(16)

5.3.2. Neural Network AutoRegressive with Exogenous Input

Dynamic neural network models are widespread and have been used in applications such as the control of heat exchangers [89] and hydro-power plants [90]. Although few reports deal with the use of NN to capture the transient characteristics of turbojets, there are several studies where NNs have been implemented to estimate different static conditions in turbojets [5,15,91]. This shows an interesting opportunity to test the neural network capabilities in estimating the transient characteristics of turbojets. For training purposes, an NN with the structure presented in Figure 11a is used. This allows training the network with the true measured output values, achieving a more accurate representation of the turbojet.

The neural network consists of a feed-forward neural network with one hidden layer composed of 10 neurons, as suggested in [15,91]. The inputs of the NN are the fuel flow and three delayed outputs to maintain an agreement with the third-order dynamics identified during the linear model identification from the previous sections. This structure is congruent with the results presented in [49,92]. These results were obtained through a sensitivity analysis of the network structure (i.e., different number of layers, neurons, activation functions, etc.) for turboshaft dynamic modeling. The best-fitting neural network was selected after an iterative process considering different initial conditions with parameters computed through Bayesian regularization. After the network identification, the neural network is manipulated into an Output Error (OE) structure (Figure 11b), where the delayed estimation is operated in a closed loop. The resulting NNARX allows addressing the estimation problem in a practical sense, where the output is not measurable. This is a representative scenario particularly for thrust, since this variable can be measured in ground stations but is not available during flight. The resulting NNs for both shaft speed and thrust are included as a novel baseline for evaluating the developed gray-box models.

5.4. Models with the Identification Data Set

The resulting models obtained through the identification methods presented in this section are shown in Figure 12a,b for shaft speed and thrust, respectively. Regarding the shaft speed, it is clear that the double nonlinearity of the Hammerstein–Wiener model captures the lower steady-state conditions with greater accuracy than the proposed gray-box model with a single nonlinearity; while the NN clearly outperforms both in terms of steady-state conditions. In transient conditions, the gray-box model provided greater accuracy than the Hammerstein–Wiener model, while the NN provided the slowest response. The resulting models during the thrust identification performed similarly to the shaft speed identification, although the NN was more heavily influenced by the measurement noise.

6. Experimental Validation

The models are evaluated using extensive validation conditions, which are divided into time domain and frequency domain. It is important to note that this additional experimental data was not used to refine the existing models. Although the scope of the paper is to develop control-oriented models rather than evaluate particular parameters of the engine, it is important to note that the resulting measurements (i.e., thrust and shaft speed), specific fuel consumption (SFC) and overall efficiency are within the same range of values as those reported in studies using the same engine, that is within 0–87,000 RPM, 20 N–178 N and up to 1.2 lb/(lb h) for each one of these parameters [8,93,94].

6.1. Time Domain

The time-domain validation results are presented in Figure 13. For the shaft speed (Figure 13a), the Hammerstein–Wiener model tends to have large errors during transient conditions. This suggests that while the nonlinear characteristics are accurately identified, the transient information is not well captured. The NN, on the other hand; struggles to provide the same performance when structured in the OE form, which is necessary for some practical applications. This loss of accuracy when restructuring the model to its OE form has been reported in previous studies [95]. The gray-box model provides a response close to the actual measured data during all the test conditions.

The major difficulty presented during the thrust modeling was handling the effects of measurement noise. The performance of the models with the thrust validation data is similar, with the main differences being that the NN shows more consistent results and the Hammerstein–Wiener model filters high-frequency components of the signal. These validation experiments confirm that the gray-box models are able to predict the microturbojet dynamic and static behaviors with a high level of accuracy, reliability, and simplicity.

6.2. Frequency Domain

An innovative framework for the evaluation of the resulting models based on the frequency domain properties of the system is presented in this section. Figure 14 presents the results of an experimental frequency-swept test of the microturbojet. These results can be compared with the theoretical cut-off frequency of the resulting models. The results show that the gray-box model successfully captures the shaft speed and thrust dynamic characteristics. The experimental amplitude decay observed in Figure 14 corresponds to the behavior of the identified models right after the cut-off frequency. For the shaft speed, the NN response does not follow the sine wave accurately, while the Hammerstein–Wiener model overpredicts the shaft speed values and presents an out-of-phase response. For the thrust, both the NN and Hammerstein–Wiener models capture the low-frequency components of the measured response, but the higher frequencies are amplified. This indicates that the frequency-domain characteristics and the order of the system have been identified better by the gray-box models up to the test bench capabilities.

While some modeling approaches are based solely on frequency response identification [93,96], in this article, frequency-domain elements were indirectly considered during the clustering stage of the identification process. In this section, an experimental frequency response analysis is used for validation. This validation is important to ensure that the models can be used properly with frequency domain controller and estimator design techniques [97].

The results are also comparable with other existing literature reports. For instance, in [98] a microturbojet engine was modeled using a single linear third-order state-space representation with good experimental results. However, the use of a fixed linear representation yields a very limited operating range, as noted in [98]. In addition, instead of adjusting the model parameters using a set of three experimental measurements (in this case, fuel flow, shaft speed, and thrust), in [98], very precise and extensive knowledge of the machine parameters and variables is necessary. In particular, the method proposed in [98] requires the measurement, calculation, reconstruction, or previous knowledge of over 30 machine parameters, which can be difficult to achieve in many setups. In another report [10], a microturbojet is modeled as a single linear discrete transfer function calculated from averaging a set of linear models obtained at different operating points. The average model is then used with a combination of robust control design techniques to account for the model uncertainty. While the approach presented in [10] yields good results, it requires special control techniques to account for the model uncertainty due to the use of a single average model. Instead, the proposed approach achieves a low level of modeling error with a single model. In another study [99], the authors use recurrent neural networks to model the dynamic response of the fuel flow of a microturbojet with good results. Although limited to this variable (which in our case is assumed to be measured), the results shown in [99] suggest that input valve fuel flow may be easier to model than the variables studied here. Finally, an older study using the same engine was presented in [8]; in this study, the authors model the shaft speed using a linear autoregressive model and a neural network, with the neural network yielding the best fit. This is in line with our results, since a single linear model fails to capture the complete operating range, and while a neural network model can achieve a better fit, its modeling performance lags behind both the HW model and the proposed gray-box model. Finally, it is important to note that in these literature reports (and in others not discussed here due to lack of space), several features of the proposed method are lacking, mainly (1) a wide set of identification and validation tests (i.e., not being the same kind of maneuvers), (2) validation tests including frequency-domain testing, and (3) a proposal for obtaining an average linear representation based on overfitting, clustering simplification, and frequency-domain considerations instead of simple averaging. Overall, the results presented in this section, particularly the benchmark models (i.e., the HW and the NN models), are in line with the literature reports, thus validating their use for the assessment of the proposed gray-box model.

7. Predictive Power Analysis

The following section presents the performance assessment of the developed models using different accuracy and precision metrics, as well as some important remarks regarding the computed models.

7.1. Models Error Analysis

In this section, a brief analysis of the model estimation error is presented. Figure 15 presents the Box Plot of the modeling error of each model during the time-domain validation. In the case of the gray-box model, the nonlinear shaft speed modeling error is bounded within a band of $\pm 5\%$ error, while the thrust nonlinear modeling error is delimited among $\pm 12\%$ (this is highly influenced by the noise in the measurements). This benchmark allows for evaluating the propagation error of the sensors’ uncertainty to the model outputs. The NNs follow in precision, with the Hammerstein–Wiener models having the greater error spread with the validation data.

On the other hand, Figure 16 presents the statistical analysis of the modeling errors during the frequency-domain validation. This validation data set is particularly characterized by the absence of steady states and wide dynamic conditions. Thus, it is designed to reveal identification errors in the process dynamics rather than in the static gain. This analysis shows that the gray-box model retains the accuracy and precision properties presented during the time-domain validation (i.e., similar characteristics in the error Box Plot during the time-domain and frequency-domain validations). This contrasts with the NN and Hammerstein–Wiener models, which presented an increased error distribution during this test. The previous observations suggest that the estimation errors of the gray-box are not strongly related to a specific dynamic condition (static or dynamic), implying that the proposed identification method captures the turbojet dynamics more accurately.

Additionally, the accuracy and precision of the models are studied using the Root Mean Squared Percentage Error (RMSPE) and Standard Error (SE), respectively. These metrics are presented in

Table 1. Nonlinear models performance metrics.

Variable	Model	RMSPE %	SE	RMSPE %	SE
		Time Domain		Frequency Domain
Shaft speed	G-B	$1.3429$	$26.613$	$2.0327$	$11.641$
	W-H	$5.5167$	$121.06$	$8.5607$	$47.835$
	NN	$3.1928$	$52.280$	$3.8578$	$27.555$
Thrust	G-B	$4.3739$	$0.0580$	$6.4324$	$0.0309$
	W-H	$8.0221$	$0.1431$	$14.838$	$0.0552$
	NN	$7.4271$	$0.1466$	$17.136$	$0.0489$

with both validation data sets. The RMSPE is a classical error indicator included to analyze the models’ accuracy (i.e., low error percentage) even with high noise-to-signal ratios. In this case, the gray-box models provide the highest accuracy. On the other hand, the SE in combination with Figure 15 allows one to conclude that the gray-box models are also highly precise (i.e., bounded error in a small region with low variability with respect to the experiments). When comparing similar test conditions, the resulting models can match complex estimation architectures in terms of accuracy and robustness [22]. In summary, the models obtained with the proposed identification method provide greater precision and accuracy in untrained conditions when compared with other popular modeling alternatives, as well as being more robust to propagation errors while also having a simpler structure.

7.2. Error Propagation Analysis

In addition to the total error assessment, the error propagation properties of each model are studied next. For this application, the main source of uncertainty for the models is the sensor noise. The experimental sensor noise was determined by operating the engine in a steady-state regime so that the transient dynamics were minimized. Then, 300 data points of each variable were obtained and fitted to a normal distribution by optimizing the likelihood [100].

Figure 17 shows the resulting histogram of the measured data and the fitted distributions. The resulting log-likelihoods were 1359.71, −1688.49, and −679.634 for the fuel flow, shaft speed, and thrust, respectively. These log-likelihoods were compared with several other fitted distributions, including Nakami, uniform, inverse Gaussian, and extreme value. The normal distribution proved to be either the best fit or very close to the best; thus, it was decided to move forward using this distribution due to its better-known properties. The resulting standard deviations of the fitted distributions were 0.00327, 50.7, and 2.08 for fuel flow, shaft speed, and thrust, respectively. The mean values for each variable were 5.43, 73,849.6, and 87.79 for fuel flow, shaft speed, and thrust, respectively. This demonstrates that both shaft speed and fuel flow have a low level of measurement noise, while thrust has the highest level of noise. This is normal since thrust is measured using a load cell, which is well known to suffer from higher levels of noise.

Although error propagation in static models is a well-established topic, in the case of dynamic autoregressive (particularly nonlinear autoregressive models), the error propagation properties can be more involved to determine (see, for instance, [101]). On the other hand, a simple tool that allows numerically assessing the error propagation is a Monte Carlo (MC) simulation [102]. Thus, an MC simulation is presented next to assess the error propagation. The MC simulation was performed by comparing the output of each model to a noiseless constant fuel flow input from a zero initial condition with a noisy fuel flow input for a simulation time of 20 s (enough time so that all models reach the steady state). This allowed for assessing both the transient and stationary properties of each model. For each simulation, the noise added to the input fuel flow was obtained using the experimentally fitted distribution of Figure 17. A set of 100 simulations for each variable and each model were performed using this methodology. The difference between the nominal (i.e., using the noiseless fuel flow input) and the noisy outputs (i.e., using the noisy fuel flow input) was calculated and aggregated for all the simulations to evaluate the error propagation of each model.

Finally, Figure 18 shows the aggregated error histogram of each model together with the best-fit normal distribution for the thrust and the shaft speed. In the case of the shaft speed obtained with the neural-network-based model, the maximum-likelihood normal distribution does not represent the data properly; thus, a logistic and a stable distribution were also fitted, with the stable distribution resulting in the best log-likelihood. This could be the result of the nonlinear activation functions within the neural network, which tend to distort the normal distribution from the noise source.

The resulting standard deviations of the best-fit normal distributions for the shaft speed were 6.035, 14.08, and 26.5 for the gray-box, the HW, and the NN models, respectively. On the other hand, the resulting standard deviations for the thrust were 0.0136, 0.0352, and 0.1064 for the gray-box, the HW, and the NN models, respectively.

One of the main observations regarding the MC simulation is that the gray-box model yields the lowest error propagation among the compared models (as measured by the standard deviation), while the neural network model has the highest. Another important observation is that all the models have a smaller standard deviation than that of the experimentally measured signals (i.e., 50.7 and 2.08 for the shaft speed and thrust, respectively). This can be explained by observing that, although all models amplify the noise from the noisy fuel flow signal, this particular measurement has the lowest noise level of all the experimental measurements (i.e., 0.00327); thus, the error propagation from the noisy fuel flow yields a lower noise level than the experimental measurements of the output variables. It is important to note that this only quantifies the error propagation from the noisy input signal. Other sources of error and uncertainty are not included and are responsible for the total modeling error observed in the previous section (see Figure 15).

As a final note, the resulting error propagation characteristics can be considered a salient feature of the proposed gray-box model. In addition to yielding the lowest total error and the lowest input-signal error propagation, the gray-box model is suitable for thrust-control applications. In many of these applications, thrust measurements suffer from a high level of sensor noise (as noted in the previous paragraphs) and are difficult/expensive to implement. Therefore, a particularly attractive configuration is that of using a state observer for reconstructing the thrust by measuring the fuel flow as the input signal and the shaft speed as the output signal (see [38] for an example of this configuration). This application demands the lowest level of sensor noise in the fuel flow and shaft speed, and a low level of error propagation from the fuel flow to both thrust and shaft speed, which is the particular combination of characteristics that the gray-box model yields.

7.3. Remarks Regarding Microturbojet Nonlinear Behavior

Finally, important remarks regarding the cluster analysis and the static function approximation of the gray-box identification method are presented. During the cluster analysis, the model poles were located within a small region, allowing their location to vary slightly from one identification condition to another. These variations can be induced due to changes in the engine dynamics, which are dependent on the engine’s thermal state (at any given operating condition). The latter effect has been observed previously for larger turbojets [103,104]. On the other hand, this document presents, an experimental evaluation of this nonlinear behavior in microturbojets. It is important to note that the static function obtained through the results of the proposed identification process is similar in structure to those used for larger engines. The importance of this finding lies in the fact that this structural similitude has not been found using thermodynamic modeling approaches. That is, thermodynamic models for large turbojets have struggled to capture microturbojet static and dynamic behavior [105,106], but the present results demonstrate that there are indeed similarities.

8. Conclusions

A novel model for the main variables that describe the operation of turbojets is presented in this article. The resulting models are compatible with typical controller, observer, and estimator topologies. The identification methodology is formulated with the intention of providing an identification approach capable of handling the typical difficulties presented during data-based turbojet modeling, such as high noise-to-signal ratio and uncertainty in the plant dynamics.

Experimental results show that the proposed method is capable of segregating stochastic and causal components present in the measurements of a microturbojet in an experimental setup. This identification method has the following stages (1) high-order linear model identification, (2) cluster analysis, (3) dynamic model simplification, (4) nonlinear static gain approximation, and (5) final model integration using a Wiener structure. The resulting model encompasses information on the whole turbojet operation regime, allowing one to increase the operating range of linear control schemes.

The experimental validation of the models included a wide spectrum of operating regimes to test their accuracy and precision with untrained conditions. The developed models were also systematically tested along with other widespread models to identify the best-fitting aeroengine representation. These widespread models included Hammerstein–Wiener and Neural Network AutoRegressive with Exogenous Input structures. Along the experimental tests, the gray-box models performed remarkably well under several error metrics, including their error propagation characteristics, outperforming the baseline models. Moreover, the main frequency-domain properties of the turbojet were obtained experimentally and were found to closely match those predicted by the proposed gray-box models. This contrasts with the failure of the baseline models in predicting these important features. These results support the conclusion that the proposed gray-box models have successfully captured the main dynamic and nonlinear static features required for high-performance and robust turbojet control while also maintaining the lowest error propagation among the compared models.