International Journal of Aerospace Engineering

Volume 2019, Article ID 6015270, 9 pages

https://doi.org/10.1155/2019/6015270

## Study on Identification Method for Parameter Uncertainty Model of Aero Engine

^{1}School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, China^{2}Key Laboratory for Civil Aircraft Airworthiness Certification Technology, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to Shuai Liu; moc.361@iauhsuilcuac

Received 9 September 2019; Revised 12 November 2019; Accepted 22 November 2019; Published 2 December 2019

Academic Editor: Zhiguang Song

Copyright © 2019 Jie Bai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The linear model of an aero engine is effective in a small range of the neighborhood of equilibrium points. According to this problem, the identification method for the parameter uncertain linear model of the aero engine was proposed. The identification problem is solved by calculating nonlinear programming. Considering the parameter uncertainty of the model is the critical point of this research during the optimization process. A parameter uncertain model of an aero engine can be obtained, which has large use range. This method is used for DGEN380 aero engine. The two parameters, VDD and VE, are defined for describing error range. Compared with experimental data, the uncertain model of DGEN 380 can simulate the real state of DGEN380 within 1% error range when . Compared with another conventional method of identification (recursive least squares), the parameter uncertain model, established by the method of this research, has a broad application area through parameter uncertainty of the model.

#### 1. Introduction

The mathematical model describes the relationship among aero engine state variables and input variables through mathematical logic and mathematical language which is often used for the engineering design of engine control system and engine fault diagnosis widely [1]. The nonlinear mathematical model is hardly used for the engineering design of aero engine control system and fault diagnosis on account of the complicated structure, complex form, and highly nonlinear behavior which can describe the variation of each parameter in a full envelope range of aero engine [2]. There is an approximately linear correlation between every parameter of aero engine when the state of aero engine approaches the steady state point. The nonlinear mathematical model is transformed into a linear model by linearizing at the steady state point. The linear model is usually used for controller design. The function of this controller is steady control. The transient control process between two steady states is realized by interpolating the gain parameters, namely, gain scheduling which is a conventional method. Only in this way, can the control system of aero engine exert the control function in the full envelope [3].

The aero engine linear model has characteristics of having a simple form, fast calculation speed, and others [4], which are usually used in control system design and fault diagnosis [1]. However, the difference between the first order differential of engine state variables and the linear model slope increases as the distance between the engine state point and the steady state point increases, which narrows the application range of linear model and influences the application range of the controller and observer near the steady state point. If the application scope of the aero engine linear model is more than this limited range, the accuracy of the aero engine linear model is less than 90%. The small application range is usually less than 10% of the neighborhood of equilibrium point in engineering.

The control problem and diagnosis problem of small-range fluctuation near the steady state point are the current research hotspots [5–7]. It not only needs to improve the control algorithm, the design methods of the observer, and the estimation method of model uncertainty [8, 9] but also needs to improve the model to expand the application range of the linear model. The purpose of developing the application range of the linear model is to make the model more accurate within a wide range of state parameters. In the controller design process, the influence of uncertainty has been considered in many types of research. In these researches, the upper bound of weight, the influence scope of parameter uncertainty, is added into the control algorithm. The analysis of influencing parameter uncertainty on the application range of the linear model is nearly blank in the aero engine modeling, and a few people studied it.

We focus on the identification model of the aero engine, which has both a broad application range and a simple form. The identification model of an aero engine can be obtained easily. The identification method on the combination of the least squares with a nonlinear filtering method is developed by Michael and Farrar, which is used in the model identification of F100 aero engine during the early stage of modeling [10]. The multivariable instrumental variable/approximate maximum likelihood method of recursive time-series analysis, proposed by Merrill, is used to identify the multivariable (four inputs-three outputs) dynamics of the Pratt & Whitney aero engine [11]. Torres et al. [12] attempted to identify the dynamic of the gas turbine engine offline, mainly at steady states with stochastic signals. Arkov et al. [13] focused on real-time identification for transient operations and concluded that an engine system could be averaged to a time-invariant first- or second-order transfer function by the extended recursive least squares [13]. The tracking speed and accuracy for the recursive least squares could be improved with a different design of forgetting factors. The effect of using a forgetting factor was to shift the estimating average toward the most recent data, such as that in the work of Paleologu et al. [14]. Li et al. [15] have investigated classic and modified recursive least squares algorithms for online dynamic identification of gas turbine engines. It seemed that the recursive least squares algorithm is well known for tracking dynamic systems, which is an effective conventional method of aero engine model identification. However, considering the parameter uncertainty of the identification model by recursive least squares is difficult. An identification method for the aero engine is proposed by this paper which can evaluate the parameter uncertainty of the aero engine model. Additionally, this method deals with the identification of the model by solving an optimization problem. The aero engine model may have an extensive application range, considering the parameter uncertainty of the model.

In this paper, the identification method for the aero engine parameter uncertain model is proposed. This method can identify a linear model involving model parameter uncertainty by solving the optimization problem, which is detailed in Section 2. The DGEN380 aero engine [16] is regarded as an object. The parameter uncertain model of DGEN380 is identified by DGEN380 experimental data in Sections 3.1 and 3.2. The analysis of the identification of the DGEN380 model is given in Section 3.3. Meanwhile, the parameters, VDD and VE, are defined for the error analysis of the parameter uncertain model, which is stated in Section 3.4. An example of comparing a typical least squares algorithm and the identification method for aero engine parameter uncertain model is given in Section 3.5. Section 4 is the conclusion of this paper.

#### 2. The Identification Method for the Parameter Uncertain Model of Aero Engine

The aero engine model can be formulated as follows [17]:
where represents the aero engine state vector which concludes shaft speed, temperature, and pressure; represents the aero engine input vector, including throttle angle corresponding to the fuel flow, the nozzle area, VSV (variable stator vanes), and VBV (variable bleed valve); represents the aero engine output vector. is the flight altitude; Ma is flight Mach. and serve as the nonlinear functions of aero engine state variables which are vector functions of real values. The Taylor expansion is used to linearize equation (1) at a particular steady state point, by which the linear model of aero engine is obtained. This model can be represented by as follows:
where represents the state deviation vector of dimension, represents the deviation vector of dimension, and represents the output deviation vector of one dimension. These deviation variables can be described as follows:
where ^{sta} represents the aero engine steady state vector, ^{sta} represents the aero engine steady input vector, and ^{sta} represents the aero engine steady output vector. And the steady variables are used to normalize the deviation variable in this research.

A steady state point of the aero engine is selected. The input pulse signal is set, which is the orthogonal vector. The matrixes and in equation (2) are estimated according to output response.

The static model matrix is discussed first (). The state of the aero engine is steady. The state deviation vector and the input deviation vector are formulated as follows:

The columns of matrix could be confirmed by the state response vector of a given input signal .

When , the state of the aero engine is dynamic. There is an assumption that matrix has eigenvalues, , and corresponding eigenvectors, _{1}, _{2}, …, . Matrix has repeated eigenvalue, corresponding to linearly independent eigenvectors. Matrix is diagonalized similarly, and equation (5) can be obtained.
where , .

The estimate of Matrix means solving the programming problem ( and , , …, ) where represents the discrete values of open loop state response of the nonlinear system; represents equation (2) corresponding to distinct values of state response of the linear system, accordingly, ; represents the input vector of the nonlinear system, which is orthogonal. , corresponding to ; and represents the sampling time. In equation (6), represents the weight matrix, , which is chosen to make the differences between nonlinear and linear model time responses more similar for all state variables.

The coefficient matrixes, and , of the linear model are computed using equation (6). Furthermore, the influence of model parameter uncertainty is considered. Matrix represents matrix containing parameter uncertainty. Matrix represents diagonal matrix including uncertainty of eigenvalues. If Matrix may be undiagonalized, Matrix represents Jordan matrix including uncertainty of eigenvalues.

Equation (5) can be formulated as follows: where

means a set of subscript indexes of real characteristic roots and means a set of subscript indexes of complex characteristic roots. For each of these complex characteristic roots, there is a complex conjugate one.

The suboptimal estimation of the range of the real parts and the imaginary parts of the eigenvalues of matrix means that the programming problem is solved again. where , represents the discrete values of uncertain system state response in equation (2). The calculation of the objective function in equation (9) needs multiple nonlinear programming problems. where

The weight matrix is used to reduce the difference between the state response vector of the nonlinear system and the state response vector of the linear system containing uncertainty.

The solution of equation (6) is an initial condition to solve programming problems. The optimal solution of matrix and the eigenvalues of matrix are used as the initial condition to estimate their uncertain range. The parameter uncertain model can be gotten by the calculation of programming problems in equations (9) and (10). Equations (9) and (10) are solvable, which means optimization problem has a nonempty solution set. This problem is proved in Ref. [18].

#### 3. The Analysis of Identification of DGEN380 Model

This section is about the application of the identification method of the parameter uncertain model, which is used to identify the DGEN380 aero engine model. The start-up process and shutdown process are ignored. The maximum continuous power point is selected as a steady state in the model identification process. The flight altitude is 3048 m, Ma is 0.338, and the throttle angle is 74% at this state point.

The engine state vector includes the rotation speed of the high-pressure rotor, the rotation speed of the low-pressure rotor, the exit pressure of the high-pressure compressor, and the exit pressure and temperature of the low-pressure turbine. The engine input variable is fuel flow. As shown in equation (12), and can be known.

The state variable represents the speed of the low-pressure shaft, and the state variable is the speed of the high-pressure shaft. The state variable represents the import pressure of the combustor. The state variable represents the export temperature of the low-pressure turbine, and the state variable represents the export temperature of the low-pressure turbine. The input variable is the fuel flow of DGEN380.

A parameter uncertainty model will be obtained. Moreover, a comparison is implemented by experimental data, the parameter uncertainty model, and another DGEN380 linear model. Then an error analysis is exerted by parameters VDD and VE.

##### 3.1. DGEN380 Engine Experimental Device

The experimental device contains a test bench and a control desktop. The test bench and control desktop are shown in Figures 1 and 2, respectively. The test bench is composed of an DGEN380 engine, measuring transducers, sensor baronesses, and integrators (a blue pillar in Figure 1). The main functions of the test bench are engine operating, parameter measurement, and signal transmission. The control desktop is composed of a power lever, a FADEC controller of DGEN380, and a video screen. The primary functions of the control desktop are to control the aero engine state by the power lever, to show engine state and parameter values, and to monitor the bench state.