**1. Introduction**

A distributed electric propulsion aircraft is a new type of aircraft that converts mechanical energy into electrical energy through an engine-driven generator. It is used in conjunction with energy storage devices, such as lithium batteries, to power multiple electric propulsion devices distributed on the wings or fuselage. The DEP aircraft studied in this work is presented in Figure 1. With distributed propulsion, an aircraft's propeller slipstream can significantly increase the airflow velocity behind its propeller disks, which will improve the aircraft performance in flight [1], enhance the stability of the wing structure [2], and realize short take off. Electric propulsion can increase efficiency of the propulsion system [3] and reduce noise [4]. The fuel consumption and pollution emission of an aircraft diminish as the DEP system improves the working condition of the gas turbines and aerodynamic efficiency of the vehicle, which satisfies the green requirements for the future [5,6]. In addition, the DEP system has multiple redundancy of a power system, which is safer and labeled as a very promising propulsion type.

In the study of methods for modeling of DEP aircraft, Joseph W. Connolly et al. of the NASA Glenn Research Center developed a nonlinear dynamic model with full flight envelope controller for the propulsion system of a partially turboelectric single-aisle aircraft. Optimization strategies for efficiency of the aircraft were investigated by adjusting the power between the energy for turbofan thrust and the extracted energy used to power the tail fan [7]. Nhan T. Nguyen et al. from the NASA Ames Research Center proposed an adaptive aeroelastic shape control framework for distributed propulsion aircrafts, which

**Citation:** Li, J.; Yang, J.; Zhang, H. Research on Modeling and Fault-Tolerant Control of Distributed Electric Propulsion Aircraft. *Drones* **2022**, *6*, 78. https://doi.org/10.3390/ drones6030078

Academic Editor: Abdessattar Abdelkefi

Received: 26 February 2022 Accepted: 11 March 2022 Published: 17 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). *drones*

allows the wing-mounted distributed propulsion system to twist the wing shape in flight to improve aerodynamic efficiency through the flexibility of an elastic wing. In addition, an aero-propulsive-elastic model of a highly flexible wing distributed propulsion transport aircraft was established, and analysis of the initial simulation results showed that the scheme could solve the potential flutter problem and effectively improve the aerodynamic efficiency quantity of the lift-to-drag ratio [8]. Zhang Jing et al. from Beihang University systematically investigated the integrated flight/propulsion modeling and optimal control of distributed propulsion configuration with boundary layer ingestion and supercirculation features and proposed an integrated flight/propulsion optimal control scheme to deal with the strong coupling effects and to implement comprehensive control of redundant control surfaces as well as the distributed engines [9]. Lei Tao et al. from Northwestern Polytechnic University built a complete simulation model of the DEP aircraft power system and comparatively analyzed the pros and cons of three evaluation indexes, namely the propulsion power, the propulsion efficiency, and the range in pure electric propulsion and turboelectric propulsion architectures, based on a flight profile [10]. Da Xingya et al. from the High-Speed Aerodynamics Research Institute under China Aerodynamics Research and Development Center introduced the power-to-thrust ratio as a parameter. They analyzed the effects of the state of a boundary layer and propulsion system parameters on system performance through a numerical analysis method based on the integral equation of boundary layer and verified the reliability of the calculation method by comparing the baseline state with N3-X [11]. For future electric airliners, Shanghai Jiao Tong University and the NASA Glenn Research Center developed a design method and a propulsion electric grid simulator for a turboelectric distributed propulsion (TeDP) system, explored the influence of the motor size and the spread length and air inlet conditions on the number of thrusters, and established a simulation system of a generator driven by a gas turbine engine and a system constituting two permanent magnet motors to simulate the drive of motor propelling fans. These techniques can convert a common motor system into a unique TeDP electric grid simulation program [12,13]. P.M. Rothhaar, a research engineer from the NASA Langley Research Center, developed the full process of testing, modeling, simulation, control, and flight test of a distributed propulsion vertical takeoff and landing (VTOL) tilt-wing aircraft and established methods for self-adaptive control architectures, control distribution research and design, trajectory optimization and analysis, flight system identification, and incremental flight testing [14]. J.L. Freeman performed a dynamic flight simulation of directional control authority-oriented spreading DEP and developed a linear time-invariant state space model to simulate the six-degree-of-freedom flight dynamics of a DEP aircraft controlled by a throttle lever. The study showed that further development of this technology could reduce or eliminate the vertical tail of an aircraft [15]. *Drones* **2022**, *6*, x FOR PEER REVIEW 2 of 25

**Figure 1.** The DEP aircraft studied in this paper. **Figure 1.** The DEP aircraft studied in this paper.

In the study of methods for modeling of DEP aircraft, Joseph W. Connolly et al. of the NASA Glenn Research Center developed a nonlinear dynamic model with full flight envelope controller for the propulsion system of a partially turboelectric single-aisle aircraft. Optimization strategies for efficiency of the aircraft were investigated by adjusting the power between the energy for turbofan thrust and the extracted energy used to power the tail fan [7]. Nhan T. Nguyen et al. from the NASA Ames Research Center proposed an adaptive aeroelastic shape control framework for distributed propulsion aircrafts, which In the research of the coordinated control technology of thrust and fault-tolerant control technology for DEP aircrafts, Jonathan L. Kratz et al. from the NASA Glenn Research Center designed a flight control plan for a single-aisle turboelectric aircraft with aft boundary layer thruster, and the designed controller was validated by simulation within the flight envelope. The results also showed that the engine efficiency was greatly improved [16]. Eric Nguyen Van et al. proposed a method to calculate a motor's bandwidth and control law for an active DEP aircraft with designed longitudinal/lateral control law

allows the wing-mounted distributed propulsion system to twist the wing shape in flight to improve aerodynamic efficiency through the flexibility of an elastic wing. In addition,

that the scheme could solve the potential flutter problem and effectively improve the aerodynamic efficiency quantity of the lift-to-drag ratio [8]. Zhang Jing et al. from Beihang University systematically investigated the integrated flight/propulsion modeling and optimal control of distributed propulsion configuration with boundary layer ingestion and supercirculation features and proposed an integrated flight/propulsion optimal control scheme to deal with the strong coupling effects and to implement comprehensive control of redundant control surfaces as well as the distributed engines [9]. Lei Tao et al. from Northwestern Polytechnic University built a complete simulation model of the DEP aircraft power system and comparatively analyzed the pros and cons of three evaluation indexes, namely the propulsion power, the propulsion efficiency, and the range in pure electric propulsion and turboelectric propulsion architectures, based on a flight profile [10]. Da Xingya et al. from the High-Speed Aerodynamics Research Institute under China Aerodynamics Research and Development Center introduced the power-to-thrust ratio as a parameter. They analyzed the effects of the state of a boundary layer and propulsion system parameters on system performance through a numerical analysis method based on the integral equation of boundary layer and verified the reliability of the calculation method by comparing the baseline state with N3-X [11]. For future electric airliners, Shanghai Jiao Tong University and the NASA Glenn Research Center developed a design method and a propulsion electric grid simulator for a turboelectric distributed propulsion (TeDP) system, explored the influence of the motor size and the spread length and air inlet conditions on the number of thrusters, and established a simulation system of a generator driven by a gas turbine engine and a system constituting two permanent magnet motors to simulate the drive of motor propelling fans. These techniques can convert a common motor system into a unique TeDP electric grid simulation program [12,13]. P.M. Rothhaar, a research engineer from the NASA Langley Research Center, developed the full process of testing, modeling, simulation, control, and flight test of a distributed propulsion vertical takeoff and landing (VTOL) tilt-wing aircraft and established methods for self-adaptive control architectures, control distribution research and design, trajectory optimization and analysis, flight system identification, and incremental flight testing [14]. J.L. Freeman performed a dynamic flight simulation of directional control authority-oriented spreading DEP and developed a linear time-invariant state space model to simulate the six-degreeof-freedom flight dynamics of a DEP aircraft controlled by a throttle lever. The study and distribution modules, and the results demonstrated that the method can reduce the surface area of a vertical tail by 60% [17,18]. The NASA Glenn Research Center developed an 11 kw lightweight and efficient motor controller for X-57 DEP aircrafts. The controller includes a control processor and a three-phase power inverter weighing 1 kg and not requiring a heat sink, and its efficiency is over 97% [19]. Garrett T. Klunk et al. considered the stability and control effectiveness in the event of engine fault. An active thruster-based control system can redistribute thrust to offer dynamic directional stability when a thruster is unable to recover symmetric thrust. This capability satisfies the function of a vertical tail in an aircraft and, if permitted during certification, can completely replace the vertical tail [20]. The University of Michigan investigated the fault detection and control of DEP aircraft engines. For thruster faults in DEP aircrafts, Kalman filtering was adopted to detect motor faults, and a model predictive controller was leveraged to recover the altitude of cruising flight and redistribute thrust to a properly operating motor [21]. In recent years, the development of artificial intelligence provides a new technical way for fault-tolerant control. R. Shah from Cornell University proposed adaptive and learning methods and compared them to control DC motors actuating control surfaces of unmanned underwater vehicles. The result showed that deterministic artificial intelligence (DAI) outperformed the model-following approach in minimal peak transient value by approximately 2–70% [22]. S.M. Koo from Cornell University determined the threshold for the computational rate of actuator motor controllers for unmanned underwater vehicles necessary to accurately follow discontinuous square wave commands. The results showed that continuous DAI surpassed all modeling approaches, making it the safest and most viable solution to future commercial applications in unmanned underwater vehicles [23]. It can be seen that DAI has broad application prospects in the field of fault-tolerant control of DEP aircraft actuator in the future and should be deeply studied.

The redundant thrusters of DEP aircrafts also increase the risk of fault in the propulsion system, so it is necessary to study fault-tolerant control to ensure flight safety. At present, there is little research on coordinated thrust control, and research on fault-tolerant control of propulsion system for DEP aircrafts is also in the preliminary stage. In this context, a power system model for DEP aircrafts, including the engine module, the generator and energy storage system module, and the thruster module, is established in Section 2. A mathematical model of a six-degree-of-freedom DEP aircraft was built based on the principles of aerodynamics and flight dynamics. In Section 3, research on control methods to coordinate thrust from multiple thrusters is discussed based on the mathematical model of DEP aircrafts. The lateral and longitudinal control loops of DEP aircrafts were set up based on the principles of total energy control and total heading control, and a fault-tolerant control method was developed for the case where a thruster of a DEP aircraft has failed. In Section 4, experiments simulating flight tests and fault-tolerant control within the mission segment are outlined, and the experimental results are used to verify the effectiveness of the designed coordinated thrust control system and the fault-tolerant control method. Finally, all the major results are summarized and discussed in Section 5. In this study, the correctness and effectiveness of the designed coordinated thrust control method and the fault-tolerant control method for DEP aircrafts were theoretically verified, providing a theoretical basis for future engineering application and development of the control system for DEP aircrafts.

#### **2. Modelling of the DEP Aircraft**

Unlike traditional aircrafts, a DEP aircraft is powered by electrical energy converted from the mechanical energy of its engine, so the energy flow of its propulsion system differs from that of traditional aircraft. In this study, a mathematical model of the DEP aircraft's propulsion system was established, including its engine, generator, energy storage, thruster, and other modules. Then, a mathematical model of the DEP aircraft was built according to aerodynamics and flight dynamics to deepen understanding of the drive

mode and flight mechanism of DEP aircrafts and lay the foundation for flight control and simulation research.

#### *2.1. Mathematical Model of the DEP Aircraft's Propulsion System*

#### 2.1.1. Engine Module

In this study, two turboshaft engines were adopted to convert mechanical energy into electrical energy stored in the energy storage system. The turboshaft engines follow the ideal Brayton cycle.

Flow in the inlet was considered as an isentropic process with no total pressure loss and temperature loss, so the isentropic flow equation is as follows:

$$\frac{P\_t}{P\_s} = \left(1 + \frac{k-1}{2} M\_a^{\;2}\right)^{\frac{k}{k-1}}\tag{1}$$

$$\frac{T\_l}{T\_s} = 1 + \frac{k-1}{2} M\_a^{\;2} \tag{2}$$

where *P<sup>t</sup>* is the total pressure, *T<sup>t</sup>* is the total temperature, *P<sup>s</sup>* is the static pressure, *T<sup>s</sup>* is the static temperature, *µ* is the specific heat ratio of the ideal gas, and *M<sup>a</sup>* is the Mach number.

The pressure ratio of a compressor is as follows, where *Pt*<sup>2</sup> is the total inlet pressure of the compressor, and *Pt*<sup>3</sup> is the total outlet pressure of the compressor.

$$P\_{\text{ratio}} = \frac{P\_{t3}}{P\_{t2}} \tag{3}$$

It was assumed that the compressor is ideal and therefore provides isentropic compression. The temperature ratio can be calculated from the isentropic relations, where *Tt*<sup>2</sup> is the total inlet temperature of the compressor, and *Tt*<sup>3</sup> is the total outlet temperature of the compressor.

$$\frac{T\_{t3}}{T\_{t2}} = \left(\frac{p\_{t3}}{p\_{t2}}\right)^{\frac{k-1}{k}}\tag{4}$$

The increase in heat in the airflow within the combustor is proportional to the fuel consumption rate and the fuel heat value, as described below:

$$dm\_0 \mathbb{Q} = dm\_f H\_V \tag{5}$$

where *dm*<sup>0</sup> is the mass flow of air, *dm<sup>f</sup>* is the mass flow of fuel, *Q* is the heat exchanged with the system, and *H<sup>V</sup>* is the heat value of fuel.

With the ideal burner efficiency and constant specific heat, the equation is as follows:

$$\left(dm\_0 + dm\_f\right)\mathbb{C}\_p T\_{t4} - (dm\_0)\mathbb{C}\_p T\_{t3} = dm\_f H\_V \tag{6}$$

The maximum mass flow of fuel *dm<sup>f</sup>* max can be calculated using the highest temperature of the turbine inlet temperature *T IT* at a constant-pressure specific heat *Cp*.

$$d\_{m\_{f\text{max}}} = \frac{-(TIT\_{\text{max}}) - T\_{t3} \mathcal{C}\_{p} dm\_{0}}{\mathcal{C}\_{p} TIT\_{\text{max}} - H\_{V}} \tag{7}$$

The turbine provides enough power to drive the compressor. Therefore, there is a condition to be satisfied, namely the turbine power should be equal to the compressor power. Under ideal conditions, the equation for this condition is as follows, where *Tt*<sup>4</sup> is the total inlet temperature of the gas turbine, and *Tt*<sup>41</sup> is the total inlet temperature of the power turbine.

$$dm\_0 \mathbb{C}\_p(T\_{t3} - T\_{t2}) = \left(dm\_0 + dm\_f\right) \mathbb{C}\_p(T\_{t4} - T\_{t41})\tag{8}$$

It was assumed that the turbine is ideal and is therefore isentropically depressurized. The temperature ratio can be calculated based on the isentropic relations. The isentropic relations were then adopted to change the pressure ratio of the turbine according to the following equation, where *Pt*<sup>4</sup> is the total inlet pressure of the gas turbine, and *Pt*<sup>41</sup> is the total inlet pressure of the power turbine.

$$\frac{P\_{t41}}{P\_{t4}} = \left[1 - \frac{T\_{t2}}{T\_{t4}} \frac{1}{\left(1 + \frac{dm\_f}{dm\_0}\right)} \left\{ \left(\frac{P\_{t3}}{P\_{t2}}\right)^{\frac{k-1}{k}} - 1 \right\} \right]^{\frac{k}{k-1}} \tag{9}$$

The power turbine extends the flow to ambient pressure to obtain the maximum power. It was assumed that the turbine is ideal and therefore it is isentropically depressurized. The isentropic relations were adopted to change the temperature ratio as follows:

$$\frac{T\_{l3}}{T\_{l2}} = \left(\frac{p\_{l3}}{p\_{l2}}\right)^{\frac{k-1}{k}}\tag{10}$$

The nozzle works isentropically, and there is no loss of total pressure and temperature. The total inlet pressure of nozzle *Pt*<sup>5</sup> is equal to the total outlet pressure of nozzle *Pt*7.

$$P\_{l\5} = P\_{l\7} \tag{11}$$

Power recovery of the turboshaft engine is a function of the total enthalpy change of the turbine:

$$P\_{Recovery} = \left(dm\_0 + dm\_f\right)\mathbb{C}\_p(T\_{t4} - T\_{t41})\tag{12}$$

The specific fuel consumption *SFC* is shown below:

$$SFC = \frac{dm\_f}{P\_{\text{Recovery}}}\tag{13}$$

#### 2.1.2. Electric Power Generation and Energy Storage Module

Mechanical energy generated by the turboshaft engine is mechanically connected to a generator through the reduction gear box, and the generator then stores the generated electrical energy in the energy storage battery. Ports of the generator and the energy storage system were defined as presented in Figure 2. *Drones* **2022**, *6*, x FOR PEER REVIEW 6 of 25

**Figure 2.** Definition of ports of the electric power generation and energy storage system. **Figure 2.** Definition of ports of the electric power generation and energy storage system.

The mechanical power of the generator *Pmec* is calculated based on the following equation: The mechanical power of the generator *Pmec* is calculated based on the following equation:

> *P P lost mec* 1

> > 3

*I* is negative, the battery is in a discharged state. The state of charge *SOC* is a

*I* represents the current of a battery at Port 3 of the energy storage system.

100 *norn*

*l dq <sup>I</sup> dt*

*dSOC dq dt dt C*

where *Cnorn* is the rated capacity of a battery. The output power *Pbat* of the energy stor-

where *Rcell* is the internal resistance of a battery cell, *cell I* is the battery current, *cell S* is the number of cells in series in a battery, and *Pcell* is the number of cells in parallel in a

The thruster module of a DEP aircraft consists of 16 sets of motors connected to pro-

2 4 *F C n D P Thrust T p* 

$$P\_{\rm mec} = T\_m w\_s \tag{14}$$

(15)

are the torque and the rotational speed of the shaft at

*P P P elec mec lost* (16)

(17)

(18)

and the torque *T<sup>P</sup>* of a single pro-

(20)

<sup>2</sup> *P R I S P bat cell cell cell cell* (19)

*q* extracted from the energy storage system for use is calcu-

The electrical energy generated is as follows:

state variable, and its derivative is calculated as follows:

age system at Port 1 is calculated as follows:

pellers through a reduction gear box. The thrust *F<sup>P</sup>*

is the efficiency defined by the motor's characteristics.

and *w<sup>s</sup>*

In the equation, *T<sup>m</sup>*

where

lated as follows:

where <sup>3</sup>

When <sup>3</sup>

battery.

2.1.3. Thruster Module

peller are calculated as follows:

the generator's Port 2, respectively.

The load of the charge *<sup>l</sup>*

In the equation, *T<sup>m</sup>* and *w<sup>s</sup>* are the torque and the rotational speed of the shaft at the generator's Port 2, respectively.

The lost power *Plost* is calculated according to the following equation:

*Plost* = (1 − *η*) · |*Pmec*| (15)

where *η* is the efficiency defined by the motor's characteristics. The electrical energy generated is as follows:

*Pelec* = *Pmec* − *Plost* (16)

The load of the charge *q<sup>l</sup>* extracted from the energy storage system for use is calculated as follows:

$$\frac{dq\_l}{dt} = -I\_3 \tag{17}$$

where *I*<sup>3</sup> represents the current of a battery at Port 3 of the energy storage system. When *I*<sup>3</sup> is negative, the battery is in a discharged state. The state of charge *SOC* is a state variable, and its derivative is calculated as follows:

$$\frac{dSOC}{dt} = -\frac{dq}{dt} \cdot \frac{100}{\mathcal{C}\_{norm}}\tag{18}$$

where *Cnorn* is the rated capacity of a battery. The output power *Pbat* of the energy storage system at Port 1 is calculated as follows:

$$P\_{\rm bat} = \mathcal{R}\_{\rm cell} I\_{\rm cell}^2 \mathcal{S}\_{\rm cell} P\_{\rm cell} \tag{19}$$

where *Rcell* is the internal resistance of a battery cell, *Icell* is the battery current, *Scell* is the number of cells in series in a battery, and *Pcell* is the number of cells in parallel in a battery.

#### 2.1.3. Thruster Module

The thruster module of a DEP aircraft consists of 16 sets of motors connected to propellers through a reduction gear box. The thrust *F<sup>P</sup>* and the torque *T<sup>P</sup>* of a single propeller are calculated as follows:

$$F\_P = \mathcal{C}\_{\text{Trrust}} \rho n\_T \, ^2 D\_p ^4 \tag{20}$$

$$T\_P = \frac{\mathcal{C}\_{power} \rho m\_T \,^3 D\_p^{-5}}{\omega} \tag{21}$$

where *ρ* is the air density, *n<sup>T</sup>* is the rotational speed, *ω* is the rotational speed in the international system of units, *D<sup>p</sup>* is the propeller's diameter calculated from the propeller's radius, *CThrust* is the thrust coefficient, and *Cpower* is the power coefficient.

The thrust coefficient *CThrust* and power coefficient *Cpower* of the propeller are related to the geometric characteristics of the propeller, such as diameter, number of blades, blade area, rotating area, blade angle, theoretical pitch angle, etc. The *CThrust* and *Cpower* map of the propeller can be generated by the propeller performance map generator tool. According to the propeller shaft speed, aircraft speed, and actual pitch angle, the value of *CThrust* and *Cpower* at this time can be interpolated.

The propulsion ratio *J* of the propeller is calculated as follows:

$$J = \frac{V\_a}{n\_T D\_p} \tag{22}$$

In the equation, *V<sup>a</sup>* is the norm of the airspeed vector → *Va*. The thrust and the torque coefficients are equal to zero when the rotational speed is opposite to the rotation (counterclockwise or clockwise) direction.

The propeller's aerodynamic efficiency *λ* is defined as follows:

$$
\lambda = J \frac{\mathbf{C}\_{Thrust}}{\mathbf{C}\_{power}} \tag{23}
$$

Based on the position of the propeller relative to the body, the thrust and the moment of the propulsion system acting on the aircraft can be calculated.

#### *2.2. Mathematical Model of the DEP Aircraft*

In this study, a mathematical model of DEP aircrafts was established based on the principles of aerodynamics and aircraft dynamics. (*u*, *v*, *w*) is the linear velocity of the aircraft, (*p*, *q*,*r*) is its angular velocity, and (*φ*, *θ*, *ψ*) represents its roll angle, pitch angle, and yaw angle.
