A Hybrid Open/Closed-Loop μ Control Method for Achieving Consistent Transient Performance in Turbofan Engines

Abstract

The inconsistency in acceleration and deceleration performance between high and low altitudes is a significant challenge in aircraft engine control today. In the past, neither open-loop fuel–air ratio control nor closed-loop N-dot control could resolve this issue perfectly; the difference in acceleration and deceleration performance between high and low altitudes is even more than three times. The operational characteristics of aircraft engines vary significantly between high and low altitudes, posing challenges for transient state control in high-performance aircraft engines. To address these transient performance inconsistencies due to altitude uncertainties, a μ-synthesis adaptive tracking transition control design method with hybrid open-loop and closed-loop direct thrust control is proposed. The main innovation lies in proposing a new segmented hybrid control scheme. Under a high-power state, it employs a dual closed-loop μ-synthesis adaptive tracking framework, using fuel flow to control thrust and nozzle area to control the turbine pressure ratio. In a low-power state, a single-variable closed-loop and open-loop control architecture is applied. Simulation results show that the hybrid open/closed-loop control method can suppress the inconsistency of acceleration and deceleration performance caused by altitude uncertainties in turbofan engines, ensuring consistent robustness in acceleration and deceleration performance across different altitudes. From the ground to an altitude of 11 km, the new method has an acceleration time range of 3.44 s–3.84 s and a deceleration time range of 4.83 s–5.98 s; compared with the previous fuel–air ratio acceleration time of 4.17 s–9.12 s and deceleration time of 6.12 s–14.48 s, its high and low-altitude acceleration and deceleration consistency performance is greatly improved.

1. Introduction

Turbofan engine acceleration and deceleration control is crucial for enabling the engine to swiftly transition between states, with the requirement that performance remains consistent across high and low altitudes during flight [1,2]. However, traditional open-loop fuel–air ratio (FAR) control laws, designed using engine similarity principles [3,4,5], do not guarantee this consistency. The difference in acceleration and deceleration performance between high and low altitudes is even more than three times, severely affecting the aircraft’s maneuverability. Ensuring consistent acceleration and deceleration performance across different altitudes has become a significant challenge in engine transient state control.

To address this issue, the commonly used approach is to employ a FAR correction function that adjusts with changes in ambient pressure [6,7], aiming to reduce the performance discrepancies between high and low altitudes during transition. Although these methods alleviate the problem to some extent, performance losses still occur during in-flight acceleration and deceleration, particularly under harsh weather conditions. Some researchers have explored transition state control using optimization (variable replacement method and particle swarm optimization [8], by changing the independent variables describing engine transient state performance [9] and the fixed-states method [10]) or machine learning (Model-Free Deep Reinforcement Learning Method [11], DNN-LPV Model [12] and Neural Network Learning-Based Global Optimization [13]) methods, but these approaches have not fundamentally resolved the issue of inconsistent performance between high-altitude and low-altitude transitions. On the other hand, Morrison et al. and Howlett et al. proposed a closed-loop acceleration control method based on rotor speed acceleration rate for turboshaft engines [14,15]. While this approach improves helicopter performance across a wide range of operating conditions, it still depends on similar parameters of rotor acceleration rate within the flight envelope. Subsequently, many researchers have explored rotor speed acceleration rate closed-loop control. Wang et al. [16] proposed an N-dot transient PI control design method. Although simulation results indicate that this method outperforms the traditional FAR method in transient performance, it fails to address the inconsistency in engine acceleration and deceleration performance under different flight conditions. Huang et al. [17] proposed an N-dot control method based on constrained nonlinear optimization of PI parameters, but this approach only ensures that the controller meets basic acceleration performance requirements. Yang et al. [18] and Peng et al. [19] designed two similar N-dot-based transient control strategies for APU to ensure consistent acceleration performance. However, these methods were only validated on the ground, without addressing acceleration and deceleration consistency under varying flight conditions. Li et al. [20] proposed an active switching N-dot control structure based on tracking error to improve the acceleration performance of turbofan engines. Although this method effectively enhances acceleration performance, it does not address the issue of inconsistent acceleration and deceleration performance at high and low altitudes. Liu et al. proposed a model reference adaptive control method for aeroengines, which has not yet been validated in engineering. This method attempts to achieve consistency in engine performance under different flight conditions by designing a reasonable reference model [21]. However, these studies mainly focus on control design methods and validation against FAR plans, without addressing the issue of performance consistency in acceleration and deceleration across different altitudes. Although rotor speed acceleration rate control has shown some improvement in performance consistency, it still depends on similar engine principles, as the acceleration rate commands are derived from FAR control laws. To date, there has been no significant progress in achieving consistent acceleration and deceleration performance across varying altitudes.

Pilots expect a linear relationship between power lever angle (PLA) and engine thrust, but traditional speed control creates a nonlinear connection, increasing pilot workload [22]. Litt et al. [23] proposed a method to linearize this relationship by compensating for the non-linearities in engine speed, but it only applies to steady-state control and does not ensure linear thrust response during acceleration and deceleration. Therefore, a direct thrust control mode is considered to be applied in the transition state to ensure this linear relationship. Moreover, the engine transition state changes drastically, and the resulting uncertainty problem has a significant impact, which is difficult to be solved by the traditional control method, so the μ-synthesis control method can be used. μ control theory, developed from modern control theory, addresses robust control in uncertain systems. Introduced by Doyle in 1982, it uses structured singular values for frequency-domain loop shaping and reformulates uncertainty issues into structured singular value μ problems for controller design [24]. Subsequently, the method has been widely used in various fields, such as tooltip tracking in active magnetic bearing (AMB) spindle applications [25] and the nanosatellite rendezvous and docking problem [26]. The key advantage of μ control is the suppression of system uncertainties and dynamic decoupling of multivariable loops through sensitivity weighting functions and ${\mathrm{H}}_{\infty }$ norm constraints.

According to engine operating principles [4], the engine similarity principle applies under stable conditions but does not hold during transient states, serving only as an approximation. Thus, it is difficult to ensure the consistency of acceleration and deceleration performance of transition states at varying flight conditions by the traditional open-loop FAR transition state control method based on the similar principle. Meanwhile, to enhance engine thrust performance, direct thrust control is anticipated to be a key technology and a leading direction for future engine control development.

This paper proposes a hybrid open-loop and closed-loop thrust control method. The main fuel loop ensures closed-loop thrust control for a linear thrust–throttle relationship, while the nozzle loop switches between open-loop and closed-loop control based on engine conditions. Compared to the traditional FAR transition state control method based on the similar principle, this approach maintains consistent transition state performance and engine safety across different altitudes, as validated by simulations.

2. Hybrid Open/Closed-Loop Segmented μ Control

2.1. Hybrid Open/Closed-Loop Segmented Control Scheme for Acceleration and Deceleration

Traditional turbofan engine acceleration and deceleration control uses a combination of FAR functions varying with the relative corrected speed of the high-pressure rotor (${\overline{N}}_{2,\mathrm{cor}}$) and steady-state control, applying Min–Max logic to manage transient states throughout the flight envelope. During acceleration, the Min–Max logic selects the lower fuel flow rate between the open-loop FAR acceleration controller and the steady-state PID controller as the actual output. While decelerating, it selects the higher one. Rapid state changes causing large command-feedback errors may result in the PID output greatly exceeding or falling below the acceleration/deceleration controller output. In such cases, the transition is dominated by the open-loop FAR controller, with performance largely dependent on the FAR control schedule design. This control scheme performs well on the ground but deteriorates in effectiveness as flight altitude increases. The problem arises because the FAR control scheme relies on the engine similarity principle, which applies only under stable conditions. In transient states, this principle does not hold and becomes an approximation, leading to performance issues. Thus, after designing the FAR control plan for acceleration and deceleration, high-altitude corrections are necessary. However, the precise corrections are challenging, leading to inconsistent performance between ground and airborne conditions. For instance, acceleration time on the ground might be within 5 s, but at an altitude of 15km, it can exceed this by a significant margin. Additionally, the open-loop FAR control loses performance under flight condition disturbances or system uncertainties.

To address this issue, engine acceleration and deceleration characteristics and performance requirements were considered:

• The transition time between states should be minimized for dynamic acceleration and deceleration.
• Strong resistance to disturbances caused by changing flight conditions is essential.
• Engine thrust should respond linearly to the power lever angle.
• The aircraft’s specific maneuverability and agility requirements.

In direct thrust control, if the engine thrust response tracks the desired commands, it meets Requirements 3 and 4. Integrating adaptive control with closed-loop feedback addresses Requirements 1 and 2.

Additionally, since the nozzle area ($A_8$) of a turbofan engine with an afterburner is adjustable, the afterburner should not affect the core engine’s operating point. Hence, a dual-loop adaptive closed-loop feedback system is necessary for both the main fuel ($W_{\mathrm{f}}$) and nozzle circuits; meanwhile, the nozzle area should be enlarged in idle or near-idle states for surge margin and then gradually reduced when transitioning to higher power to minimize thrust loss. Therefore, an open-loop structure can be used for the nozzle circuit during this transition, while the main fuel circuit should maintain a thrust closed-loop structure for adaptive feedback control. The proposed segmented switching control scheme for thrust acceleration and deceleration transitions is as follows:

$$\begin{array}{l}\mathrm{if}{\overline{N}}_{2,\mathrm{cor}}\ge {\overline{N}}_{2,\mathrm{cor},\mathrm{cr}}\\ \hspace{1em}{W}_{\mathrm{f}}\to F_{\mathrm{N}}=f(PLA,H,Ma)\\ \hspace{1em}\hspace{1em}{A}_8\to {\pi }_{\mathrm{T}}=g(PLA,H,Ma)\\ \mathrm{else}\\ W_{\mathrm{f}}\to F_{\mathrm{N}}=f(PLA,H,Ma)\\ A_8=h\left({\overline{N}}_{2,\mathrm{cor}}\right)\end{array}$$

(1)

where H and Ma are flight height and Mach number, respectively, $F_{\mathrm{N}}$ means engine net thrust, and ${\pi }_{\mathrm{T}}$ represents the turbine pressure ratio. ${\overline{N}}_{2,\mathrm{cor},\mathrm{cr}}$ is the segmented switching point for the engine’s relative high-pressure rotor speed. If ${\overline{N}}_{2,\mathrm{cor}}\ge {\overline{N}}_{2,\mathrm{cor},\mathrm{cr}}$, the system uses dual-loop control for acceleration and deceleration. If ${\overline{N}}_{2,\mathrm{cor}}<{\overline{N}}_{2,\mathrm{cor},\mathrm{cr}}$, it switches to a combination of open-loop and closed-loop control.

Compared to traditional open-loop FAR control method, the proposed hybrid open/closed-loop control strategy maintains the fuel flow loop in closed-loop control throughout the transition phase. This ensures engine performance and robustness against flight condition variations, forming the foundation for consistent acceleration and deceleration performance across altitudes. Additionally, the nozzle control loop switches between open-loop and closed-loop modes based on the engine state parameter ${\overline{N}}_{2,\mathrm{cor}}$, ensuring aerodynamic stability in low-power states and minimizing thrust loss in high-power states.

2.2. Design of the Open-Loop Nozzle Control Circuit

In the proposed segmented switching control scheme for thrust acceleration and deceleration, the nozzle operates in open-loop mode when the ${\overline{N}}_{2,\mathrm{cor}}$ is below the switching point. To ensure the desired thrust varies linearly with the power lever angle, the nozzle area should adjust accordingly. This adjustment helps achieve the linear thrust response more effectively. To reflect high-altitude conditions, a linear relationship for the nozzle area based on the ${\overline{N}}_{2,\mathrm{cor},\mathrm{cr}}$ is proposed:

$$A_8=h({\overline{N}}_{2,\mathrm{cor}})=A_{8,\max }+{\displaystyle \frac{A_{8,\min }-A_{8,\max }}{{\overline{N}}_{2,\mathrm{cor},\mathrm{cr}}-{\overline{N}}_{2,\mathrm{cor},\mathrm{idle}}}}\left({\overline{N}}_{2,\mathrm{cor}}-{\overline{N}}_{2,\mathrm{cor},\mathrm{idle}}\right)$$

(2)

where $A_{8,\max }$ and $A_{8,\min }$ are the maximum and minimum nozzle area, respectively, and ${\overline{N}}_{2,\mathrm{cor},\mathrm{idle}}$ represents the relative corrected speed of the high-pressure rotor in engine idle state.

In engine idle conditions, to ensure a sufficient surge margin for the fan and compressor, the nozzle area is designed to remain at its maximum value:

$$A_8=A_{8,\max }$$

(3)

Although the nozzle circuit switches between open-loop and closed-loop control modes according to ${\overline{N}}_{2,\mathrm{cor}}$, the fuel flow rate circuit always maintains a closed-loop control mode to ensure the engine thrust response during the transition process. Above the switching point, a dual-loop multivariable closed-loop control mode of $W_{\mathrm{f}}$ and $A_8$ is used to simultaneously control the thrust and turbine pressure ratio; below that point, a single-variable closed-loop control mode of $W_{\mathrm{f}}$ for thrust and an open-loop regulation of $A_8$ according to the laws shown in Equations (2) and (3) are engaged to ensure the rapidity of the thrust response in the transition process as well as the surge margin of those compression components. In order to deal with the problem of excessive model uncertainty brought about during the transition process, the closed-loop modes are all based on the reference model μ-synthesis control method.

2.3. Modeling Description of the Turbofan Engine

The configuration of the turbofan engine in this paper is illustrated in Figure 1. The engine model input parameters are $W_{\mathrm{f}}$ and $A_8$, output parameters are $F_{\mathrm{N}}$,${\pi }_{\mathrm{T}}$, high-pressure rotor speed $N_2$, low-pressure rotor speed $N_1$, turbine inlet temperature $T_4$, and surge margins of three compressor components, SMFD, SMFC, and SMC, respectively.

The engine component characteristics and geometric parameters are given from the “ABFAN.mxl” model in GSP 11 [28] and the nonlinear dynamic model is developed using the component-based method [29]. The steady comparison between the dynamic model used in this study and the “ABFAN.mxl” model is shown in Figure 2.

However, in practical applications, $F_{\mathrm{N}}$, $T_4$, and the surge margin are unmeasurable, making closed-loop feedback control challenging. This study adopts the estimation algorithm from Reference [27] to calculate these parameters using the following formulas:

$$F_{\mathrm{N}}=\left[W_8\cdot V_8+\left(P_8-P_0\right)\cdot A_8\right]-W_2\cdot V_0$$

(4)

where $W$ represent the mass flow rate, $V$ is the velocity of working fluid, and $P$ means the total pressure of working fluid. The number of subscripts represents the corresponding engine section.

$$T_4={\displaystyle \frac{{\eta }_{\mathrm{B}}\cdot FA{R}_4\cdot Hu+C{p}_3\cdot T_3}{(FA{R}_4+1)\cdot C{p}_4}}$$

(5)

where ${\eta }_{\mathrm{B}}$ is the combustion efficiency, $Hu$ is the low level calorific value of the fuel, $Cp$ denotes the constant-pressure specific heat capacity of the working fluid, and $T$ denotes the total temperature of working fluid.

$$SM={\displaystyle \frac{{\pi }_{\mathrm{surge}}-{\pi }_{\mathrm{working}}}{{\pi }_{\mathrm{working}}}}$$

(6)

$SM$ is the surge margin, $\pi$ represents the pressure ratio.

It should be noted that these formulas cannot be used independently. Their application requires simultaneous consideration of measurable sensor parameters (see

Table 1. Required sensor signals for estimation of unmeasurable engine parameters.

Definition	Variables	Definition	Variables
Fan Inlet Total Temperature	$T_2$	Ambient Static Pressure	$P_{\mathrm{amb}}$
Fan Inlet Total Pressure	$P_2$	Fan Bypass Outlet Total Pressure	$P_{13}$
Fan Core Outlet Total Pressure	$P_{25}$	HPC Outlet Total Pressure	$P_3$
HPT Outlet Total Pressure	$P_{45}$	LPT Outlet Total Pressure	$P_5$
LP Rotor Speed	$N_1$	HP Rotor Speed	$N_2$

), engine component characteristics, inherent working fluid properties, and the aerodynamic and thermodynamic relationships between engine components. As these are not the focus of this paper, they will not be elaborated further. For details, please refer to Reference [27].

2.4. Thrust Closed-Loop Model Adaptive Tracking μ-Synthesis Transient State Control Design Method

In the transition direct thrust control scheme, when the ${\overline{N}}_{2,\mathrm{cor},\mathrm{cr}}$ is below the switching point, the main fuel loop uses single-variable adaptive closed-loop feedback. Leveraging μ-synthesis theory’s robustness to system uncertainties, this section introduces a design method for thrust closed-loop model adaptive tracking μ-synthesis control to ensure rapid and accurate thrust servo performance.

2.4.1. Augmented Controlled Plant Description

The normalized linear model of the engine is as follows:

$$G_{\mathrm{engine}}:\left\{\begin{array}{c}\Delta \dot{\overline{x}}(t)=A\Delta \overline{x}(t)+B\Delta \overline{u}(t)\\ \Delta \overline{y}(t)=C\Delta \overline{x}(t)+D\Delta \overline{u}(t)\end{array}\right.$$

(7)

The equivalent first-order transfer function of the fuel actuator with small closed-loop control is as follows:

$$G_{\mathrm{act}}\left(s\right)={\displaystyle \frac{1}{{\tau }_{W_{\mathrm{f}}}s+1}}$$

(8)

where ${\tau }_{W_{\mathrm{f}}}$ represents the equivalent inertia time constant of the fuel actuator loop.

For direct thrust control with servo performance requirements, an integrator should be included in the forward path of the control loop

$$G_{\mathrm{inte}}(s)={\displaystyle \frac{1}{s}}$$

(9)

then, the augmented controlled plant is

$$G_{\mathrm{Aug}}=G_{\mathrm{engine}}\cdot G_{\mathrm{act}}\cdot G_{\mathrm{inte}}$$

(10)

2.4.2. μ-Synthesis Direct Thrust Closed-Loop Control Structure

The proposed adaptive tracking μ-synthesis transient state control structure for the thrust closed-loop model is shown in Figure 3.

In Figure 3, $G_{\mathrm{ref}}$ represents the transfer function matrix of the reference model, $W_{\mathrm{u}}$ is the weighting function for suppressing controller output energy, $W_{\mathrm{P}}$ is the weighting function for tracking model error sensitivity, $\Delta$ is an unstructured uncertainty matrix with an ${\mathrm{H}}_{\infty }$ norm less than or equal to 1, and $W_{\mathrm{I}}$ is the uncertainty bound weighting function; $r_{\mathrm{ref}}=F_{\mathrm{N},\mathrm{cmd}}$ is the reference command, $u=W_{\mathrm{f}}$ is the controller output, $z={\left[\begin{array}{cc}{z}_p& z_u\end{array}\right]}^{\mathrm{T}}$ is the performance evaluation vector, $y=F_{\mathrm{N}}$ is the engine measurement output, $y_{\mathrm{ref}}=F_{\mathrm{N},\mathrm{ref}}$ is the output of the reference model, and $d=d_{F_{\mathrm{N}}}$ represents external disturbance inputs.

By employing the model reference adaptive μ-synthesis control structure shown in Figure 3, and through the appropriate selection of $G_{\mathrm{ref}}$ and $W_{\mathrm{P}}$, the closed-loop control system performance aligns with $G_{\mathrm{ref}}$. This indirectly ensures that critical engine safety parameters, such as turbine inlet temperature, rotational speed, and surge margin, remain within their respective limits [27].

Using $r{u}_0$ to represent the relative uncertainty in steady state, ${\omega }_{\mathrm{I}}$ is the frequency at which the relative uncertainty reaches 100%, and $r{u}_{\infty }$ is the amplitude of the uncertainty weight function in the high-frequency range. Thus, the uncertainty bound weighting function is

$$W_{\mathrm{I}}(s)={\displaystyle \frac{\frac{s}{{\omega }_{\mathrm{I}}}+r{u}_0}{\frac{s}{r{u}_{\infty }{\omega }_{\mathrm{I}}}+1}}$$

(11)

Considering model tracking servo performance requirements in the low-frequency range, $W_{\mathrm{P}}$ is designed:

$$W_{\mathrm{P}}(s)={\displaystyle \frac{\alpha }{{\epsilon }_{\mathrm{P}}}}{\displaystyle \frac{\frac{s}{{\omega }_{\mathrm{P}}}+1}{\frac{s}{{\omega }_{\mathrm{P}}{\epsilon }_{\mathrm{P}}}+1}},(0.1<\alpha \le 1,0<{\epsilon }_{\mathrm{P}}<<1)$$

(12)

where $\alpha$ and ${\epsilon }_P$ are adjustment factors for balance model tracking servo performance and low-frequency disturbances suppression, and ${\omega }_{\mathrm{P}}$ is the bandwidth of closed-loop system.

To suppress control output saturation, $W_{\mathrm{u}}$ can be designed:

$$W_{\mathrm{u}}(s)=\gamma {\displaystyle \frac{\frac{s}{{\omega }_{\mathrm{u}}}+1}{\frac{s}{\left({\omega }_{\mathrm{u}}/{\epsilon }_{\mathrm{u}}\right)}+1}}$$

(13)

where $\gamma$, ${\omega }_{\mathrm{u}}$, and ${\epsilon }_{\mathrm{u}}$ are control energy suppression adjustment factors.

Convert the μ-synthesis control structure shown in Figure 3 to the standard $\Delta$-P-K structure as shown in Figure 4.

For the standard structure shown in the Figure 4, the D-K iterative algorithm [30,31] or inspired algorithm [32] can be used to obtain the μ-synthesis controller.

3. Simulation Verification and Discussions

3.1. Single-Variable Augmented Controlled Plant and Controller Design

The normalized linear model for a small bypass ratio mixed-exhaust turbofan engine is as follows:

$$\begin{array}{l}A=\left[\begin{array}{cc}-2.21& 6.30\\ 0.23& -3.49\end{array}\right],B=\left[\begin{array}{c}1.01\\ 1.65\end{array}\right]\\ C=\left[\begin{array}{cc}0.25& 0.22\end{array}\right],D=0.12\end{array}$$

(14)

The parameters for this steady-state point are as follows:

$$\begin{array}{cccc}PLA={17}^{\circ }& dPs=0\mathrm{kPa}& dTs=0\mathrm{K}& \\ H=0\mathrm{km}& Ma=0& W_{\mathrm{f}}=0.1898\mathrm{kg}/\mathrm{s}& A_8=0.5118{\mathrm{m}}^2\\ N_1=5002\mathrm{rpm}& N_2=9989\mathrm{rpm}& F_{\mathrm{N}}=3812.39\mathrm{N}& {\pi }_{\mathrm{T}}=5.543\\ T_4=950.55\mathrm{K}& SMFD=13.97\%& SMFC=15.42\%& SMC=35.09\%\end{array}$$

(15)

The equivalent transfer function of the fuel actuator’s small closed-loop is as follows:

$$G_{\mathrm{act}}\left(s\right)={\displaystyle \frac{1}{0.16s+1}}$$

(16)

Meanwhile, considering the integration of an integral component in the forward channel, the augmented controlled plant is obtained as follows:

$$G_{\mathrm{Aug}}={\displaystyle \frac{0.70307(s^2+10.98s+43.18)}{\mathrm{s}(\mathrm{s}+1.473\left)\right(\mathrm{s}+4.22\left)\right(\mathrm{s}+6.078)}}$$

(17)

Assuming the model uncertainty is 10% in the low-frequency range, 100% in the mid-frequency range, and 1000% in the high-frequency range, the uncertainty weighting function is given as follows:

$$W_{\mathrm{I}}={\displaystyle \frac{10s+20}{s+200}}$$

(18)

The tuning parameters in the performance sensitivity function are designed as $\alpha =0.6$, ${\epsilon }_{\mathrm{p}}={10}^{-6}$, ${\omega }_{\mathrm{p}}=3\mathrm{rad}/\mathrm{s}$, then the performance sensitivity function is

$$W_{\mathrm{P}}(s)={\displaystyle \frac{1.8s+5.4}{3s+9\times {10}^{-6}}}$$

(19)

The control energy suppression adjustment factors are designed as $\gamma =0.01$, ${\omega }_{\mathrm{u}}=20\mathrm{rad}/\mathrm{s}$, and ${\epsilon }_{\mathrm{u}}=0.002$; then, the weighting function is

$$W_{\mathrm{u}}\left(s\right)={\displaystyle \frac{100s+2000}{20s+200000}}$$

(20)

$$G_{\mathrm{ref}}(s)={\displaystyle \frac{25}{\left(s^2+9s+25\right)}}$$

(21)

By applying the D-K iteration method, an 11th-order μ-synthesis controller is obtained:

$$\begin{array}{c}{G}_{\mathrm{K}}\left(s\right)={\displaystyle \frac{123.59(s+1.90\times {10}^5)(s+{10}^4)(s+200)(s+16.53)(s+6.08)}{(s+1.90\times {10}^5)(s+1059)(s+199.8)(s+17.37)(s+3.57\times {10}^{-6})}}\\ \times {\displaystyle \frac{(s+4.22)(s+1.47)(s+2.028\times {10}^{-5})(s^2+9.07s+24.88)}{(s^2+8.83s+24.09)(s^2+10.7s+42.32)(s^2+35.76s+372.1)}}\end{array}$$

(22)

The structured singular value representing the robust stability and robust performance of the closed-loop system is shown in Figure 5 below.

The figure shows that the robust stability metric ${\mu }_{\Delta }\left(\mathrm{M}\right)$ is 0.1296 and the robust performance metric ${\mu }_{\widehat{\Delta }}\left(\mathrm{M}\right)$ is 0.9542, both below 1, indicating that the closed-loop system with the designed single-variable controller achieves robust stability and robust performance under uncertainty [25,26]. The detailed design process and results of the dual-loop closed-loop controller can be found in Ref [27].

The engine’s high-pressure rotor corrected speed percentage serves as the state identifier, with 90% as the switching point. When ${\overline{N}}_{2,\mathrm{cor}}>90\%$, the controller uses a multivariable closed-loop control for thrust and turbine pressure ratio via fuel flow rate $W_{\mathrm{f}}$ and nozzle area $A_8$. When ${\overline{N}}_{2,\mathrm{cor}}<90\%$, it switches to $W_{\mathrm{f}}$ single-loop closed-loop control for thrust, with $A_8$ adjusted by an open-loop scheme.

3.2. Ground Transition Simulation

Flight conditions are set as follows:

$$H=0\mathrm{km},Ma=0$$

(23)

The power lever angle (PLA) input is illustrated in Figure 6. It starts at 15°, which denotes an idle state of engine. At 10 s, it quickly increases to 65° for an acceleration transition, maintaining PLA = 65° until 30 s, then decreases to 18° quickly for a deceleration transition.

The simulation runs from 0 s to 50 s, and PLA input is depicted in Figure 6. At ground conditions, the engine thrust command and response are shown in Figure 7. Acceleration from 10 s to 13.45 s takes 3.45 s, with a smooth transition (“Accel Switch”) at 10.94 s when the nozzle control switches from open-loop to closed-loop, causing no thrust disturbance. Deceleration from 30 s to 35.91 s takes 5.91 s, with a nearly seamless switch (“Decel Switch”) back to open-loop at 31.86 s. Both acceleration and deceleration times are measured as the response enters the ±2% target error band. The relative error between engine thrust command and response can be found in Figure 8.

Figure 9 shows the engine rotor speed response, which remains stable without overspeed during acceleration and deceleration.

Figure 10 shows the surge margin response curves for the fan bypass, fan core, and high-pressure compressor during ground acceleration and deceleration. The surge margins remain above the limits, with minimum values of 7.1% for the fan bypass, 11% for the fan core, and 9.1% for the high-pressure compressor.

3.3. Low-Altitude and High-Mach Transition Simulation

Flight conditions are set as follows:

$$H=0\mathrm{km},Ma=0.8$$

(24)

Simulation time ranges from 0 s to 50 s, and the PLA input is still illustrated in Figure 6. Under low-altitude high-Mach conditions, the engine thrust response can be seen in Figure 11. During this, acceleration (10 s~13.84 s) takes 3.84 s, and deceleration (30 s~35.82 s) takes 5.82 s. No significant thrust fluctuations occurred during control mode switching (10.61 s: “Accel Switch” and 32.86 s: “Decel Switch”). Acceleration and deceleration times were calculated based on the response value entering the target value’s ±2% error band. The relative error between engine thrust command and response can be seen in Figure 12.

Figure 13 shows the engine’s rotor speed response during acceleration and deceleration, with no overspeed observed. Figure 14 displays the turbine inlet temperature, indicating no overheating during acceleration.

Figure 15 shows the surge margin response curves for the fan bypass, fan core, and high-pressure compressor during acceleration and deceleration at H = 0 km, Ma = 0.8. The surge margins remained above the limit values throughout, with minimum values of 14.3% for the fan bypass, 9.3% for the fan core, and 21.1% for the high-pressure compressor.

3.4. High-Altitude and Low-Mach Transition Simulation

Flight conditions are set as follows:

$$H=9\mathrm{km},Ma=0.3$$

(25)

Simulation time ranges from 0 s to 50 s, and the PLA input is illustrated in Figure 6. In the H = 9 km, Ma = 0.3 conditions, the engine’s thrust command and response are depicted in Figure 16. Acceleration occurs from 10 s to 13.44 s, taking 3.44 s, with the nozzle control mode switching at 11.08 s (“Accel Switch”). Deceleration happens between 30 s and 34.81 s, lasting 4.81 s, with a mode switch at 32.38 s (“Accel Switch”). Both mode switches cause minimal impact on thrust response. Acceleration and deceleration times are based on the response reaching within ±2% of the target value. The relative error between engine thrust command and response is illustrated in Figure 17.

Figure 18 shows the rotor speed response. During acceleration, the high-pressure rotor exhibits a 0.11% overshoot, and the low-pressure rotor shows a 0.24% overshoot. Despite this, no overspeed occurs, ensuring safe engine operation.

The turbine inlet temperature response curve is shown in Figure 19, with no overheating observed during the acceleration process.

In the H = 9 km, Ma = 0.3 conditions, Figure 20 shows that surge margins for the fan bypass, fan core, and high-pressure compressor remain above the limits during acceleration and deceleration. Minimum margins are 7.5%, 6.1%, and 9.3%, respectively.

3.5. High-Altitude and High-Mach Transient State Simulation

Flight conditions are set as follows:

$$H=11\mathrm{km},Ma=1.2$$

(26)

Under high-altitude, high-Mach conditions, Figure 21 shows the engine thrust command and response curves. Acceleration occurs from 10 s to 13.74 s (3.74 s duration), with minimal thrust fluctuation during the nozzle control switch from open-loop to closed-loop at 10.73 s (“Accel Switch”). Deceleration takes place from 30 s to 35.98 s (5.98 s duration), with no thrust fluctuation during the switch from closed-loop to open-loop at 32.93 s (“Deccel Switch”). Transition times are measured when the response value falls within ±2% of the target value. The relative error between engine thrust command and response is shown in Figure 22.

Figure 23 shows the high- and low-pressure rotor speed response curves, with no over-speeding observed during the transition, and Figure 24 shows the turbine inlet temperature response, with no over-temperature throughout the transition.

The response curves for the fan bypass, fan core, and high-pressure compressor surge margin are shown in Figure 25. Throughout the acceleration and deceleration process, the surge margin stayed above the limit, preventing any surges. The minimum surge margins were 10.7% for the fan bypass, 9.6% for the fan core, and 16.7% for the high-pressure compressor.

3.6. Comparative Validation of High and Low-Altitude Acceleration and Deceleration

The engine fuel–air ratio (FAR) can be defined as

$${\left({\displaystyle \frac{W_{\mathrm{f}}}{P_3}}\right)}_{\mathrm{cor}}=f\left({\displaystyle \frac{N_2}{\sqrt{T_2/288.15}}}\right)$$

(27)

Traditional engine acceleration and deceleration control is based on an open-loop fuel–air ratio schedule. It uses the relationship between corrected speed and corrected fuel–air ratio, along with altitude and Mach number information, to determine the physical fuel flow rate for acceleration and deceleration under given flight conditions. This fuel flow is then compared with the fuel flow command from the steady-state controller, selecting the higher or lower value as the final transitional fuel flow. However, this open-loop fuel–air-ratio-based approach struggles to achieve consistent transitional performance across different altitudes. Figure 26 and Figure 27 compare the transitional thrust response of the method proposed in this study with that of a traditional open-loop fuel–air ratio Min–Max selection method at ground conditions (H = 0 km, Ma = 0) and high-altitude low-Mach conditions (H = 9 km, Ma = 0.3). Their comparison in thrust relative error can be found in Figure 28 and Figure 29, relatively.

From Figure 26, it can be seen that the traditional open-loop fuel–air ratio method achieves acceleration and deceleration performance similar to the hybrid open-loop and closed-loop μ-synthesis direct thrust control approach at ground conditions. However, as shown in Figure 28, when the flight condition changes, the acceleration and deceleration performance of the traditional open-loop fuel–air ratio method deteriorates significantly compared to ground conditions, making it difficult to ensure consistent performance across the flight envelope.

Furthermore, a comparison of acceleration and deceleration times for the traditional open-loop fuel–air ratio method and the hybrid open-loop and closed-loop μ-synthesis direct thrust control method across four typical states used in this paper is presented in

Table 2. Comparison of acceleration and deceleration times for traditional open-loop fuel–air ratio and hybrid open/closed-loop μ-synthesis thrust control under typical flight conditions.

Flight Condition	Miu Acc. Time(s)	FAR Acc. Time(s)	Miu Dec. Time(s)	FAR Dec. Time(s)	Engine Inlet Total Temperature (K)
Ground (H = 0 km, Ma = 0)	3.45	4.17	5.91	6.12	288.15
Low altitude, High Mach number (H = 0 km, Ma = 0.8)	3.84	3.90	5.82	5.86	325.03
High altitude, Low Mach number (H = 9 km, Ma = 0.3)	3.44	9.21	4.81	14.48	233.78
High altitude, High Mach number (H = 11 km, Ma = 1.2)	3.74	6.73	5.98	10.28	279.05

.

Table 2 shows that the control method proposed in this paper ensures consistent engine transition performance under different flight conditions. The acceleration times range from 3.44 s to 3.84 s, with a standard deviation of 0.2; the deceleration times range from 4.83 s to 5.98 s, with a standard deviation of 0.55. In contrast, the traditional open-loop fuel–air ratio method yields a standard deviation of 2.49 for acceleration time and 4.18 for deceleration time. These results indicate that the proposed method effectively mitigates inconsistencies in engine acceleration and deceleration performance between high and low altitudes compared to the traditional open-loop fuel–air ratio control approach.

4. Conclusions

To address the inconsistency in turbofan engine acceleration and deceleration performance at different altitudes, this paper proposes a hybrid open-loop and closed-loop direct thrust control method. This method maintains the fuel control loop in closed-loop thrust control during transient states, while the nozzle control loop switches between open-loop $A_8$ control and closed-loop turbine pressure ratio control based on the engine state parameter ${\overline{N}}_{2,\mathrm{cor}}$. At low engine power states, single-variable closed-loop μ-synthesis control is applied to the fuel control loop, demonstrating robust stability under uncertainties, while the nozzle control loop adjusts the nozzle throat area in open-loop mode based on the state parameter ${\overline{N}}_{2,\mathrm{cor}}$. In high engine power states, dual-loop multivariable μ-synthesis control is applied to the fuel and nozzle loops, ensuring robust stability against a wide range of uncertainties. Compared to traditional open-loop FAR-based transient control methods, this control approach effectively ensures consistency in acceleration and deceleration performance. Nonlinear simulations at different flight conditions yield the following conclusions:

• The hybrid open-loop and closed-loop direct thrust control method ensures consistent robustness in acceleration and deceleration performance at different altitudes. The acceleration times range from 3.44 s to 3.84 s, with the standard deviation of acceleration time being 0.20; the deceleration times range from 4.83 s to 5.98 s, with the standard deviation of deceleration time being 0.55.
• The open-loop mode with nozzle area linearly varying according to the $N_{2,\mathrm{cor}}$ is suitable for acceleration and deceleration in low-power settings. Mode switching ensures that $A_8$ open-loop control in low-power settings maintains surge margin and reduces acceleration/deceleration times. In high-power settings, closed-loop control of $A_8$ adjusts the turbine pressure ratio, assisting in thrust adjustment and ensuring engine operation stays within safe limits, avoiding overheating, over-speeding, and surge.
• The thrust closed-loop model adaptive tracking μ-synthesis acceleration and deceleration control avoids the issue of inconsistent high- and low-altitude performance found in traditional fuel–air ratio control schemes.

In the future, designing a switching control strategy could be considered to ensure a smooth transition between open-loop and closed-loop control in the nozzle circuit. Additionally, the estimated engine thrust value could be directly used as a state indicator for the transient switching control.