## Abstract

The fundamental principle of the transient state control method for turbofan engines, which is based on the acceleration ratio of high-pressure rotational speed (N-dot), involves sacrificing a portion of the safety margin to obtain satisfactory acceleration performance. However, it could induce surge in the engine's compressor. To prevent the destructive damage caused by surge to both the engine and its components, a surge-elimination control strategy for the engine based on an N-dot controller is proposed. First, the engine mathematical model, which incorporates the effects of engine volumetric dynamics, stall zone characteristics, and combustion chamber flameout characteristics, is established to simulate surge mechanism. Subsequently, the acceleration schedule of the N-dot is calculated by employing sequential quadratic programming (SQP) algorithm to solve the multiconstraint optimization problem, while designing the transition state controller of N-dot based on a high-order filter. Finally, the surge detection logic and surge-elimination strategy based on the *μ*-correction method are proposed and designed to realize active control of surge elimination. The simulation results demonstrate that the N-dot control method offers significant advantages in mitigating the steady-state errors resulting from inevitable engine degradations. The surge state is effectively suppressed by the proposed surge-elimination control method, and the surge duration is significantly shorten during the acceleration phases. Furthermore, compared to the one without any surge-elimination control, the proposed method decreases the acceleration time by 5.53%.

## 1 Introduction

As the primary power unit of an aircraft, the aero-engine exhibits strong nonlinearity and generally operates under variable and harsh conditions, rendering its control problem exceedingly complicated. Compared to steady-state control strategy, controlling the engine during transitional states is doubly complicated as it involves switching between several steady-state points. During the transition state, the engine state changes between several steady-state points. In such cases, the objective is to achieve the shortest acceleration time under the premise of never violating the engine limits including the physical limit of rotor speed, the maximum operating temperature of turbine blades, and the surge limit of the combustion chamber and compressor [1,2]. During the development stage, a significant portion of the control law design is dedicated to designing the engine transient controller [3]. Acceleration control plays a crucial role in transition state control, as its performance significantly impacts the short-range combat capability and battlefield survivability of combat aircraft [4,5]. Conventional methods for controlling acceleration in turbofan engines primarily include schedule-based fuel (open-loop fuel) control method [1,2,6–13], rotor acceleration-based (N-dot) control method [14–16], and other advanced techniques such as model prediction control (MPC) [17–19]. The open-loop fuel control method calculates the fuel flow by interpolating in a predesigned schedule based on the current state of the engine. This approach is intended for the standard state of the engine, and it becomes inadequate to accomplish adaptive adjustment when there is degradation in both engine and components, resulting in steady-state control error. Thus, the N-dot control method is proposed, which utilizes the rotor acceleration ratio as a closed-loop control variable. In comparison to open-loop fuel control methods, N-dot control exhibits superior tracking performance and adaptive control effects. The aforementioned two methods are based on the min/max framework. In addition to the primary acceleration controller, multiple limit protection controllers are also implemented to ensure that the extreme conditions such as temperature and speed overshoot or surge never occur during the acceleration process. However, it should be noted that the application of the min/max framework may introduce parameter fluctuations in the switching control process. To address this issue, Liu and Guan and Li [20,21] proposes a limit protection method based on dispatching regulation management. Additionally, the MPC method is widely employed in aero-engine acceleration control due to its exceptional ability to handle multiple constraint problems using advantageous algorithms [17,19]. Nevertheless, because of the limitations in the calculation capabilities of engine electronic controllers, implementing MPC poses significant challenges. Moreover, MPC relies heavily on precise onboard models, which further complicates its application.

The research on N-dot focuses on the design of the acceleration schedule and addressing the instability problem arising from the differential introduced in rotor acceleration calculation. The common methods include the fixed dynamic method [2,8], the power extraction method [1,7], and the dynamic solution method based on optimization algorithms such as the genetic algorithm [6,9] and the transient-state reverse method [22]. In addition, the variable replacement method [23] establishes an inverse solution model considering surge margin, temperature, and other limiting conditions. This enables the calculation of the curve relationship between rotational speed and acceleration in a manner that meets the specified requirements. In order to address the issue of instability, Yao et al. [16] eliminated the differential component by integrating the acceleration command and reduced the noise sensitivity of the N-dot controller. Furthermore, a modified approach for handling the differential part was proposed and implemented in the N-dot controller, which was validated through semiphysical testing [24]. However, these approaches overlook the problem that arises when engine degradation or inlet distortion occurs, leading to an unstable surge state.

To develop the surge-elimination control law, a thorough understanding of engine surge mechanisms is essential. The conventional engine component level model (CLM) fails to accurately capture the surge characteristics, necessitating research on surge-elimination control through engine ground tests. During these tests, adjustments are made to the angle of fan guide vanes and fuel flow in order to eliminate surges [25–30]. Nevertheless, the cost associated with conducting an engine ground test is substantial, thereby indicating the necessity of employing a numerical approach for surge simulation and elimination in order to develop an N-dot controller.

The present study focuses on investigating the acceleration control of a turbofan engine with a small bypass ratio through the N-dot method. The primary contribution lies in establishing an engine volume dynamics model capable of accurately capturing surge characteristics and proposing a surge-elimination strategy based on *μ*-correct method to prevent irreversible adverse effects on engine components. The paper is structured as follows: Section 2 analyzes the causes of the surge and develops a component-level model of a turbofan engine that accurately reflects compression component surge characteristics through volume dynamics methodology. In Sec. 3, an N-dot acceleration controller based on a high-order filter is designed. First, the optimization problem for acceleration schedule is built and solved. Then, in order to mitigate the high-frequency noise generated during rotors acceleration, a finite impulse response (FIR) filter is designed. In Sec. 4, the surge judgment criteria based on the pressure detection signal is discussed. A *μ*-correct strategy is also designed to eliminate surge states. In Sec. 5, the control effect of the N-dot controller and surge-elimination strategy is validated by numerical simulations. Section 6 concludes the paper.

## 2 Engine Mathematical Model With Surge Characteristics

In general, the surge characteristics of turbofan engines are modeled through the Moore-Greitzer third-order model [31–33], volume dynamics model [34,35], or other methods. In this paper, an engine mathematical model based on volume dynamics is established to depict the surge characteristics of turbofan engines during acceleration (hereinafter referred to as “surge models”). This model holds significant importance for the investigation of surge-elimination control strategies.

The typical schematic diagram of a small bypass ratio turbofan engine is illustrated in Fig. 1. It is primarily composed of an inlet, fan, compressor, combustion chamber, high-pressure turbine, low-pressure turbine, afterburner, and nozzle.

The numerical labels in Fig. 1 correspond to distinct engine sections, with their respective meanings detailed in Table 1.

Station | Section definition | Station | Section definition |
---|---|---|---|

1 | Bypass entry | 5 | Low-pressure turbine exit |

2 | Fan entry | 6 | Core stream mixer entry |

22 | Fan exit | 65 | Afterburner entry |

23 | Core engine entry | 75 | Exhaust nozzle entry |

25 | Compressor entry | 8 | Nozzle throat |

3 | Burner entry | 9 | Nozzle exit |

4 | High-pressure turbine entry | 13 | Bypass entry |

42 | High-pressure turbine exit | 16 | Bypass exit |

44 | Low-pressure turbine entry |

Station | Section definition | Station | Section definition |
---|---|---|---|

1 | Bypass entry | 5 | Low-pressure turbine exit |

2 | Fan entry | 6 | Core stream mixer entry |

22 | Fan exit | 65 | Afterburner entry |

23 | Core engine entry | 75 | Exhaust nozzle entry |

25 | Compressor entry | 8 | Nozzle throat |

3 | Burner entry | 9 | Nozzle exit |

4 | High-pressure turbine entry | 13 | Bypass entry |

42 | High-pressure turbine exit | 16 | Bypass exit |

44 | Low-pressure turbine entry |

Compared with the conventional engine CLM, the surge model incorporates engine volumetric dynamics, stall characteristics, and burner flameout characteristics.

When calculating the fan and compressor, the surge model must take into account the stall characteristics of both components. The flow and pressure ratio as well as efficiency in the stall area of the fan should be established, as shown in Fig. 2. In the figure, *PRF* denotes the pressure ratio of fan, *W*_{cor} is the corrected airflow, *ETAF* represents the efficiency of fan, and *N _{f,}*

_{cor}is the corrected rotational speed of fan. In case of a surge, the flow and pressure ratio of the fan will exhibit characteristics resembling those found in the stall zone. The stall zone of compressor shares similar characteristics with the fan.

When the turbofan engine enters a surge state, significant pulsations in compressor discharge pressure occur, resulting in gas flow and fuel gas ratio (FAR) pulsation that may lead to flameout. Therefore, apart from demonstrating stable combustion characteristics above idle, it is also necessary to reflect unstable combustion and burner flameout under unstable operating conditions. The modeling method of the burner is illustrated in Fig. 3, where the working area of the burner is divided into a stable flame zone and a flameout zone based on the boundaries of rich fuel and lean fuel flameout limits.

where *W*_{fb} is the main fuel flow and *Q*_{3} represents the airflow of the burner entrance.

where *P*_{3} is the stagnation pressure of the burner entrance and *T*_{3} denotes the stagnation temperature of the burner entrance.

*η*decreases toward zero in accordance with a first-order function. Upon returning to stable combustion and re-ignition, the combustion efficiency increases back to its normal value following a similar trend. However, if flow reversal occurs, the combustion efficiency drops to zero. The calculation of combustion efficiency is as follows:

_{b}where *T*_{xh} is the delay time of flameout, *T*_{dh} represents the delay time of ignition, and *η _{s}* is the efficiency before re-ignition.

The solution process for the turbofan engine model is depicted in Fig. 4, where initial values of *N _{f}*,

*N*,

_{c}*P*

_{4},

*P*

_{44},

*P*

_{16}, and

*P*

_{8}are selected for aerodynamic and thermodynamic calculations. Subsequently, air or gas flow through the fan, compressor, high-pressure turbine, low-pressure turbine, and nozzle can be interpolated from their respective characteristic maps. However, there exists a D-value between the interpolated air or gas flow and that calculated along the flow path. To ensure precision below 10-5, an iterative calculation using Runge–Kutta algorithm is employed to minimize this D-value. Ultimately equilibrium is achieved in Eqs. (9)–(12) as evidenced by vanishing differential term of pressure.

The variation in inlet distortion directly influences the surge boundary or working line of the engine compression components, resulting in the unstable operational conditions (stall or surge) for these components. The distortion of the engine's inlet flow field typically comprises a complex combination of stagnation temperature and stagnation pressure. The present study focuses on the steady-state distortion at the inlet and employs numerical simulation to simulate the dynamic process of engine compression components approaching and entering surge under the influence of inlet distortion. The simulation results of the surge state induced by stagnation temperature and pressure distortion at flight altitude *H *=* *0 km, Mach number *Ma *=* *0, and power lever angle *PLA *=* *60 deg are presented in Fig. 5.

The surge margin gradually decreases when both circumferential stagnation pressure distortion (DP/PC) and radial stagnation pressure distortion (DP/PR) occur simultaneously in the engine inlet, as illustrated in Figs. 5(a) and 5(b). As shown in Figs. 5(c)–5(f), the fan enters an unstable state at 3.55 s, accompanied by a significant fluctuation in pressure and flow at the exit. Additionally, the rotational speed (as shown in Fig. 5(g)) experiences a decrease and fluctuates within a specific frequency range. Figure 5(h) illustrates the change in fan operating conditions on the characteristic map, with mass flow as the abscissa and fan pressure ratio (*PRF*) as the ordinate axis. It demonstrates that the working point during the surge cycle adheres to surge characteristics.

## 3 Acceleration Control Based on the N-Dot Method

The engine's acceleration performance is a crucial metric that necessitates swift and seamless transitions between steady-states. Currently, there are primarily two types of acceleration control structures. One involves the open-loop control of *W*_{fb}/*P _{s}*

_{3}combining low selector, as depicted in Fig. 6(a), where

*P*

_{s}_{3}denotes compressor discharge static pressure. The lower value between the output of the speed closed-loop controller and the interpolation value of the acceleration control schedule is sent to the actuator. The other approach is the closed-loop control based on N-dot, as presented in Fig. 6(b). Its objective is to regulate the acceleration of rotational speed in accordance with given commands. As actual systems cannot measure acceleration directly, rotational speed differential serves as its proxy. In Fig. 6,

*N*

_{c}_{,}

*denotes the measured rotational speed of compressor, $N\u02d9c$ represents the acceleration ratio, and $N\u02d9c,r$ corresponds to the acceleration command. The control error between $N\u02d9c,r$ and $N\u02d9c$ is calculated as the input for the controller to determine the fuel flowrate. Subsequently, this fuel flow command is transmitted to an actuator, which can be modeled as a first-order segment with inertia described by 1/(*

_{m}*T*+ 1) in this paper. During designing, two pivotal issues must be considered. First, the acceleration schedule reflects the mapping relationship between rotational speed and acceleration and serves as the foundation of the N-dot controller. It affects acceleration performance, including acceleration time and overshoot. In addition, parameter jumps caused by differential parts can lead to control system instability such as large instantaneous fuel flow resulting in temperature or speed overshoots. This chapter focuses on handling these problems.

_{s}### 3.1 Optimization Design of Acceleration Schedule.

*N*) and the inlet stagnation temperature of the high-pressure turbine (

_{c}*T*

_{4}) are selected as the optimization objective for acceleration phase, which can be expressed as the following equation:

*ω*

_{1}and

*ω*

_{2}are the weight coefficients and can be determined according to the following equation:

During the acceleration process, the turbofan engine is subject to various constraints. As illustrated in Fig. 7, the curves depict the relationship between rotational speed and maximum acceleration calculated under different influences. It indicates that during initial acceleration, compressor surge margin SM* _{c}* restricts acceleration. As the acceleration process progresses, the engine operating point moves away from the surge boundary. At this stage, with an increase in fuel flow, the stagnation temperature at the high-pressure turbine inlet rises rapidly and limits further acceleration due to

*T*

_{4}constraints [36]. In conclusion, when designing an acceleration schedule, various limitations must be taken into account.

where *N _{f}* denotes the rotational speed of the fan, SM

*is the surge margin of the fan, and*

_{f}*u*and Δ

*u*represent the input of turbofan engine and input increment, respectively, and the main fuel flow

*W*

_{fb}is selected as input variable.

where, *g _{u}* represents the upper limits (e.g.,

*g*

_{u}_{,1}=

*N*(

_{f}*k*) −

*N*

_{f}_{,max }≤ 0).

*g*denotes the lower limits (e.g.,

_{l}*g*

_{l}_{,1 }= SM

_{f}_{,min}− SM

*(*

_{f}*k*) ≤ 0).

*X*and simplifies the constraint conditions into linear forms, resulting in a quadratic programing problem

^{k}*S*=

*X*−

*X*, the quadratic programming problem can be reformulated in terms of variable

^{k}*S*, leading to the general form of a quadratic programming problem

where *H *= $\u22072$*f*(*X ^{k}*),

*C*= $\u2207$

*f*(

*X*),

^{k}*A*= [$\u2207$

*g*

_{u}_{,1}(

*X*),…, $\u2207$

^{k}*g*

_{u}_{,}

*(*

_{m}*X*), $\u2207$

^{k}*g*

_{l}_{,1}(

*X*),…, $\u2207$

^{k}*g*

_{l}_{,}

*(*

_{n}*X*)]

^{k}*, $\u2207$ denotes the differential operator, and*

^{T}*B*= [

*g*

_{u}_{,1}(

*X*),…,

^{k}*g*

_{u}_{,}

*(*

_{m}*X*),

^{k}*g*

_{l}_{,1}(

*X*),…,

^{k}*g*

_{l}_{,}

*(*

_{n}*X*)]

^{k}*.*

^{T}Taking the optimal solution *S ^{*}* from Eq. (10) as the subsequent search direction for the original problem

*S*, a constrained one-dimensional search of the objective function is performed in this direction to derive an approximate solution

^{k}*X*

^{k}^{+ 1}, and this process is repeated iteratively until the optimal solution is calculated.

### 3.2 Acceleration Estimator Based on High-Order Filter.

As the rotor acceleration is unmeasurable, it is estimated based on the rotational speed. However, in actual engine systems, system noise and other factors can cause the calculated rotor acceleration to exhibit sudden jumps. Therefore, it is crucial to design an acceleration estimator with a filtering function that can remove high-frequency noise and prevent abrupt changes in main fuel flow. A control structure, as shown in Fig. 8, was designed to verify the filtering effect of the acceleration estimator. The approach involved recording the compressor speed of the actual engine system from idle to intermediate state using a sensor and extracting the noise signal. In numerical simulation, this noise signal was superimposed on the output speed of CLM to simulate real system speed signals.

*N*data obtained from ground testing, which was subsequently processed to extract the noise signal. The processing method is outlined in Eq. (11), where the average value of the steady-state period is considered as baseline data and utilized to calculate the noise at step

_{c}*k*. The noise of

*N*calculated is illustrated in Fig. 9(b)

_{c}*T*=

*0.025 s represents the sample time, and*

*k*denotes the simulation step

where the coefficient *h*(0), *h*(1),…, *h*(*N* − 1) denotes the unit impulse response of the system and *N* represents the order of the filter. Its structure is illustrated in Fig. 10.

*h*(

*n*) = −

*h*(

*N*− 1 −

*n*), while in the case of even symmetry,

*h*(

*n*) =

*h*(

*N*− 1 −

*n*), as shown in the following equation:

According to Eq. (14), when *h*(*n*) is even symmetric and *N* is even, *H*(−1) = −*H*(−1). It can be observed from Fig. 11 that *z *= −1 corresponds to *ω* = π, which represents the highest frequency of the digital frequency and filters out higher-frequency noise. Therefore, an FIR filter with even symmetry and even *N* should be selected.

*N*

*θ*(

*ω*) = − (

*N*− 1) ·

*ω*/2 and the group delay is

*τ*(

*ω*) = (

*N*− 1)/2. Assuming the speed sampling frequency is 0–5000 Hz and the sample period of the closed-loop control system is 25 ms,

*N*should satisfy the following equation:

*t*

_{sf}denotes the sampling frequency. According to Eq. (16), the maximum order can be determined as

*N*≤

*12. Assuming*

*N*=

*12, and the designed filter is as follows:*

The fuel control bandwidth of an aero-engine is typically ranged from 0 to 3 Hz. Within the frequency range of 0–19 rad/s, a high-order filter is designed with noticeable truncation in the higher frequencies to effectively suppress high-frequency interference and preserve the characteristics of the original first-order difference.

As shown in Fig. 12, the magnitude–frequency characteristic of the designed high-order filter within 3 Hz is consistent with that of a first-order differential, indicating that the estimation effect of the high-order filter on the rotational acceleration is equivalent to that of a first-order differential in this frequency band. However, the high-order filter exhibits a frequency cutoff characteristic when the frequency exceeds 8 Hz, which facilitates the attenuation of high-frequency noise and alleviates abrupt changes in estimated acceleration within a single sampling step over a wide range. The black line in Fig. 12 represents the system's cutoff frequency, which is determined to be 125.6 rad/s according to the sampling step.

In order to simulate the impact of different orders on the control effect, the order *N* is set to 8 and 12. The results are depicted in Fig. 13. The black curve represents the result of the first-order differential equation (DE), while the blue and red curves represent the simulation results of high-order filters with *N *=* *8 and *N *=* *12.

As depicted in Fig. 13(a), the higher-order filter yields a smoother acceleration estimate compared to the first-order differential, albeit with a nonlinear relationship between filter order and benefit. As illustrated in Fig. 13(b), the application of a high-order filter results in delayed speed signal, with delay time increasing proportionally to the order of the filter.

## 4 Surge Detection and Elimination Control

The airflow intake by the inlet will be reduced due to the influence of high-temperature exhaust gas emitted by aircraft during weapon launch or large-scale airflow separation during high-angle of attack maneuvers, resulting in inlet distortion, including circumferential and radial distortion. The example of the influence of circumferential distortion on fan characteristics shown in Fig. 14 is utilized to analyze the necessity for surge-elimination control. As the circumferential distortion angle increases, there is a significant decrease in surge margin. Similarly, inlet distortion has a comparable effect on compressors and will not be reiterated here. The acceleration schedule is designed based on the engine's rated working condition, which may induce surge in certain circumstances. Hence, surge-elimination control is crucial to ensure engine safety. Although shallow surges can dissipate during acceleration, irreversible damage to the engine may still occur. In order to preserve the engine's service life, the frequency of surge needs to be reduced without compromising the acceleration performance.

In this chapter, the surge-elimination control logic is integrated into the N-dot controller, as depicted in Fig. 15. When a surge signal is detected, two countermeasures are implemented to mitigate the surge phenomenon. As the compressor enters surging conditions, its discharge pressure experiences a decrease. By deliberately slowing down the pressure recovery process, the fuel flow and its increase rate can be effectively reduced. Furthermore, modifying the acceleration also contributes to a reduction in fuel flow.

The acceleration command from the acceleration schedule is expressed as a function of the stagnation temperature of fan inlet *T*_{2} and *N*_{c}, that is $N\u02d9cPs3=f(T2,Nc)$. The correction factor *μ* is defined to adjust the acceleration command when surge occurs, resulting in a corrected command $(N\u02d9cPs3)cor=\mu \xd7N\u02d9cPs3$. *K* refers to the PI controller, whose function has been previously introduced.

The detection of surges forms the premise of surge-elimination control. A reliable surge detection method ensures timely intervention by detecting surges at their inception. In engineering, *P _{s}*

_{3}is generally employed as a criterion for surge identification due to its ability to capture surge characteristics and ease of simultaneous measurement. The commonly used sensors comprise static pressure surge sensors, differential pressure surge sensors, as well as high-precision and high-response pressure sensors.

When a surge occurs, performance parameters such as engine mass flow and compressor discharge pressure will oscillate at a low frequency over time. This can result in flow blockage or even backflow within the compressor, preventing stable matching between it and downstream components and causing the entire compression system to become unstable. The decrease in fuel flow results in a reduction in the temperature of the main burner, thereby decreasing the thermal resistance to airflow and subsequently enhancing compressor airflow acceleration. Ultimately, it boosts the surge margin of the compressor.

*P*

_{s}_{3}decreases and the amplitude exceeds the critical value

*A*, as indicated in Eq. (19), it is determined that the surge state has been reached

The acceleration process is segmented based on *N _{c}*, with each segment having correction points to adjust

*μ*, as illustrated in Fig. 16. During the design phase, the correction factor is set at 1, and the initial value is transmitted as the acceleration command to the controller. The correction factor corresponding to the current state is obtained by means of speed interpolation. When point

*k*is identified as a surge point,

*μ*is automatically modified and stored in the schedule. At this juncture, the acceleration command transmitted to the controller undergoes reduction.

*μ*, as presented in Eq. (20), is a function of

*E*. The coefficient

*δ*is intricately linked to the system's characteristics

## 5 Simulation and Analysis

### 5.1 Simulation of Acceleration Control.

The turbofan engine is being validated under the simulation condition of flight altitude *H *=* *0 km and Mach number *Ma *=* *0. The idle state serves as the starting point for acceleration, while the intermediate state marks the termination of acceleration (*PLA *=* *15 deg → 64 deg). The acceleration schedule design should satisfy the constraints specified in Eq. (18), and the limit is presented in Table 2. All parameters except for surge margin are normalized with respect to the engine parameters at the intermediate state serving as the reference value, and other states are expressed as multiples of this value. Figure 17 illustrates a comparison of various methods. The blue dotted line denotes the results obtained from the open-loop *W*_{fb}/*P _{s}*

_{3}control method based on original acceleration control schedule (abbreviated as OL), while the black curve depicts those achieved through the open-loop

*W*

_{fb}/

*P*

_{s}_{3}control approach based on optimized acceleration control schedule (IOL). Finally, the red curve indicates the outcomes attained via optimized N-dot control methodology (abbreviated as N-dot).

Symbol | Definition | Limit | Unit |
---|---|---|---|

N_{f} | Rotational speed of fan | <1.05 | — |

N_{c} | Rotational speed of compressor | <1.05 | — |

T_{4} | Temperature of high-pressure turbine entrance | <1.01 | — |

FAR | Fuel air ratio | <1.2 | — |

SM_{f} | Surge margin of fan | >10% | % |

SM_{c} | Surge margin of compressor | >10% | % |

W_{fb} | Fuel flow | <1.02 | — |

ΔW_{fb} | Increment in fuel flow | <0.02 | — |

Symbol | Definition | Limit | Unit |
---|---|---|---|

N_{f} | Rotational speed of fan | <1.05 | — |

N_{c} | Rotational speed of compressor | <1.05 | — |

T_{4} | Temperature of high-pressure turbine entrance | <1.01 | — |

FAR | Fuel air ratio | <1.2 | — |

SM_{f} | Surge margin of fan | >10% | % |

SM_{c} | Surge margin of compressor | >10% | % |

W_{fb} | Fuel flow | <1.02 | — |

ΔW_{fb} | Increment in fuel flow | <0.02 | — |

As depicted in Figs. 17(a) and 17(b), the engine performances of IOL are significantly enhanced compared to OL. From the perspective of the main fuel flow, the slope of IOL exhibits a steeper inclination than that of OL, resulting in a shortened acceleration time as illustrated in Fig. 17(b). During the acceleration process, the engine is constrained by various factors as presented in Figs. 17(d)–17(f). Initially, IOL is limited by the surge margin of the compressor within a time range of *t* = 0.8–2.0 s (as shown in Fig. 17(d)), and thus, the acceleration line follows this constraint. As the acceleration proceeds further, stagnation temperature of high-pressure turbine becomes another limiting factor when *t* = 1.65–3.0 s (as shown in Fig. 17(e)). Eventually, at the end of acceleration phase, steady-state is reached.

As shown in Figs. 17(a)–17(c), compared to IOL, the main fuel flow of the N-dot controller exhibits a noticeable phase lag due to the inclusion of an integral link in the close loop, which can be theoretically weakened by adjusting controller gains. However, the calculating process for increasing main fuel flow takes into account various constraints that also impact the response performance of the N-dot controller. Hence, the response performance of the N-dot will hardly surpass that of IOL when employing the same acceleration schedule, and it is challenging to equalize their performances through adjusting control gains in practical applications. The acceleration time is defined as the duration from beginning to when *N _{c}* reaches the 95% Δ

*N*(without any speed overshoot), for instance, if

_{c}*N*is 68.72% of the design value at the start and 101.38% of the design value at the end of acceleration, then 95% Δ

_{c}*N*equals to 0.95 × (101.38–68.72) = 31.03, and thus the acceleration time is

_{c}*t*

_{ac }= $tNc=99.75$ − $tNc=68.72$. The acceleration times for IOL, N-dot, and OL are

*t*

_{ac1 }= 2.75 s,

*t*

_{ac2 }= 3.07 s,

*t*

_{ac3 }= 4.30 s, respectively. The trend of the N-dot along the boundary limit aligns consistently with that observed for the IOL, as depicted in Figs. 17(d)–(17f).

As presented in Fig. 17(h), the characteristic map is depicted, where *W*_{a25,cor} denotes the corrected airflow of the compressor. The design process must take into account engine distortion, manufacturing tolerance, and performance degradation; therefore, a certain margin is reserved based on surge and temperature boundaries. The purple solid line represents the actual surge boundary of the engine, while the purple dotted line serves as the limit boundary for designing acceleration schedules. Similarly, the green solid curve is replaced by its dotted counterpart as the *T*_{4} boundary. The acceleration line (black and red curve across the characteristic map) in Fig. 17(h) provides additional evidence to support previous analysis of Figs. 17(d)–17(f).

The proposed method is tested for its effectiveness in simulating the operation of engine components after degradation, by introducing fan and compressor flow degradations (*DegF *=* *0.02, *DegC *=* *0.02) as well as efficiency degradations in high- and low-pressure turbine (*DegH *=* *0.02, *DegL *=* *0.02).

When the fuel schedule is designed according to the nominal state, and considering that *N _{c}* command remains constant, the calculated main fuel flow remains consistent with the nominal state. Nevertheless, as degradation occurs, both turbine output power and rotational speed decrease at the same fuel flowrate. Consequently, the speed can hardly precisely track the command, resulting in a steady-state error in its response. As depicted in Figs. 18(a) and 18(b), the steady-state error of

*N*at the acceleration endpoint is 0.15%. However, N-dot accurately tracks the speed command and eliminates steady-state error (the blue curve represents the results of the PID controller, which serves as the control group). With respect to acceleration time, IOL and N-dot exhibit similar values of

_{c}*t*

_{ac1 }= 2.87 s and

*t*

_{ac2 }= 3.05 s, respectively, as shown in Fig. 18(b), which remains consistent with predegradation performance.

Unfortunately, the acceleration line can no longer adhere to the constraint. In Figs. 18(d)–18(f), both the IOL and N-dot acceleration lines exceed the surge, temperature, and *FAR* limit. In subsequent maintenance, the controller gains can be adjusted to restore compliance with the constraint in the N-dot structure; however, a redesign of the fuel flow schedule for IOL is necessary, which poses a burden on designers.

### 5.2 Simulation of Surge Elimination.

The surge boundary and operating point of engine compression components vary with changes in inlet distortion, leading to the engine approaching or reaching an unstable operational state, thereby resulting in rotating stall or surge occurrences. To simulate the surge phenomenon during acceleration, both circumferential stagnation pressure distortion and radial stagnation pressure distortion are simultaneously applied to the engine inlet. The degree of distortion never directly induces surging from a stable state; however, during acceleration, the compressor experiences surging due to a reduction in its surge margin, causing significant fluctuations in compressor output pressure and airflow.

As illustrated in Fig. 19, the blue line represents the control results of the N-dot controller without surge-elimination control, while the red line depicts the results with surge-elimination control. During the acceleration process, the operation point of turbofan engine gradually approaches the surge boundary due to inlet distortion, resulting in a decrease in surge margin. At t = 3 s, as shown in Fig. 19(a), the turbofan engine enters a surge state that lasts for 1.2 s. However, due to its shallow degree, the engine automatically exits this state as it continues to accelerate and moves further away from the surge boundary. If left unattended, another acceleration process would lead to reaching a surge state.

As depicted in Fig. 19(a), *P _{s}*

_{3}serves as an indicator of the engine's state. When encountering a surge for the first time, the turbofan engine under surge-elimination control exits the surge state earlier than that without surge-elimination control by adjusting the correction factor. During subsequent acceleration, due to modifications made to the correction factors during the first three cycles when surges occur, the entry point shifts backward and results in a significant reduction in surge duration; thus, after only one cycle, the engine exits from its surged state. Upon further acceleration, it is ensured that the turbofan engine remains completely free from any surging. Figure 19(b) indicates that the acceleration command of $N\u02d9c$ changes due to the oscillation of

*P*

_{s}_{3}during the surge, resulting in main fuel flow jitter. The slight decrease and subsequent recovery of main fuel flow during the third acceleration phase can be attributed to a change in the correction factor for the acceleration schedule. The rotational speed of compressor and its partially enlarged illustrations are depicted in Figs. 19(c) and 19(d). During the acceleration phases, the main fuel flow decreases due to a reduction in the correction factor, resulting in a lagging surge-elimination control acceleration curve compared to that without surge-elimination control before surging occurs. However, the rotational speed under surge-elimination control operates smoothly at all the phases, with an acceleration time approximately 0.26 s (5.53%) shorter than the one without surge-elimination control. This indicates that the surge-elimination control proposed successfully accomplishes its design objective.

## 6 Conclusion

This paper investigates the design methodology of a turbofan engine acceleration controller based on rotational acceleration and proposes a surge-elimination control strategy built upon this foundation. A high-order filter of FIR is designed to estimate rate of change and the surge detection logic and the surge control loop are designed to monitor and suppress the surge in real-time. Several conclusions can be drawn:

The high-order filter applied to the acceleration estimation is equivalent to that of a first-order differential in designed frequency band. Moreover, it offers greater advantages in attenuating high-frequency noise and mitigating sudden changes.

The introduction of an integral link results in a slightly longer acceleration time for the N-dot controller compared to the conventional acceleration control method. Nevertheless, it lies in its ability to maintain superior tracking performance in the event of engine degradation, thereby avoiding the steady-state error of rotational speed.

The N-dot controller with surge-elimination control exhibits a significant reduction in surge duration compared to the one without. The rotational speed operates more smoothly throughout all phases, and the acceleration time decreases by approximately 5.53%.

## Funding Data

The National Natural Science Foundation of China (Grant/Award No. 52202474; Funder ID: 10.13039/501100001809).

National Science and Technology Major Project (Grant/Award No. J2019-I-0020-0019).

In part by Innovation Center for Advanced Aviation Power, China (Grant No. HKCX2022-01-026-03).

Project funded by China Postdoctoral Science Foundation (Grant No. 2023M731655; Funder ID: 10.13039/501100002858).

In part by the Fundamental Research Funds for the Central Universities (Grant No. NT2023004; Funder ID: 10.13039/501100012226).

Fund of Prospective Layout of Scientific Research for NUAA, China (Grant No. 1002-ILA22037-1A22).

Jiangsu Province Key Laboratory of Aerospace Power System (Grant No. CEPE2022008).

## Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

## Nomenclature

- ETAF =
efficiency of fan

- FAR =
fuel gas ratio

*H*=flight altitude, km

*J*=_{c}moment of inertia of the high-pressure shaft, kg·m

^{2}*J*=_{f}moment of inertia of the low-pressure shaft, kg·m

^{2}*Ma*=Mach number

*N*=_{c}rotational speed of compressor, rad/s

*N*=_{f}rotational speed of fan, rad/s

*N*_{f}_{,cor}=corrected rotational speed of fan, rad/s

*P*_{ext}=power extraction, W

*P*_{3}=the stagnation pressure of the burner entrance, Pa

*Q*_{3}=the airflow of the burner entrance, kg/s

*R*=the gas constant, J/(kg·K)

*T*_{dh}=the ignition delay time, s

*T*_{xh}=the delay time of flameout, s

*T*_{3}=the stagnation temperature of the burner entrance, K

*W*=_{c}compressor power, W

*W*_{cor}=corrected airflow, kg/s

*W*=_{f}fan power, W

*W*_{fb}=main fuel flow, kg/s

*W*_{ht}=high-pressure turbine power, W

*W*_{lt}=low-pressure turbine power, W

*η*=_{b}the combustion efficiency

*η*=_{s}the efficiency before re-ignition