Control-Oriented System Identification of Turbojet Dynamics
Abstract
:1. Introduction
- 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].
2. Model Structure Definition
- Wiener models can be analyzed in terms of classical robustness margins [69].
3. Novel Data-Based Identification Method
3.1. Linear Subsystem Identification
- 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.
- 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.
3.2. Static Function Nonlinear Approximation
3.3. Identification Method Summary
- 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.
4. Experimental Set-Up
4.1. SR-30 Engine
4.2. Data Acquisition System
4.3. Design of Experiments
- 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.
5. Applied System Identification
5.1. Shaft Speed Identification
5.2. Thrust Identification
5.3. Baseline Models
5.3.1. Hammerstein–Wiener Model
5.3.2. Neural Network AutoRegressive with Exogenous Input
5.4. Models with the Identification Data Set
6. Experimental Validation
6.1. Time Domain
6.2. Frequency Domain
7. Predictive Power Analysis
7.1. Models Error Analysis
7.2. Error Propagation Analysis
7.3. Remarks Regarding Microturbojet Nonlinear Behavior
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Variable | Model | RMSPE % | SE | RMSPE % | SE |
---|---|---|---|---|---|
Time Domain | Frequency Domain | ||||
Shaft speed | G-B | ||||
W-H | |||||
NN | |||||
Thrust | G-B | ||||
W-H | |||||
NN |
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Villarreal-Valderrama, F.; Liceaga-Castro, E.; Hernandez-Alcantara, D.; Santana-Delgado, C.; Ekici, S.; Amezquita-Brooks, L. Control-Oriented System Identification of Turbojet Dynamics. Aerospace 2024, 11, 630. https://doi.org/10.3390/aerospace11080630
Villarreal-Valderrama F, Liceaga-Castro E, Hernandez-Alcantara D, Santana-Delgado C, Ekici S, Amezquita-Brooks L. Control-Oriented System Identification of Turbojet Dynamics. Aerospace. 2024; 11(8):630. https://doi.org/10.3390/aerospace11080630
Chicago/Turabian StyleVillarreal-Valderrama, Francisco, Eduardo Liceaga-Castro, Diana Hernandez-Alcantara, Carlos Santana-Delgado, Selcuk Ekici, and Luis Amezquita-Brooks. 2024. "Control-Oriented System Identification of Turbojet Dynamics" Aerospace 11, no. 8: 630. https://doi.org/10.3390/aerospace11080630