Adapted Speed Control of Two-Stroke Engine with Propeller for Small UAVs Based on Scavenging Measurement and Modeling

Abstract

The speed of the engine–propeller directly determines the power output for Unmanned Aerial Vehicles (UAV) with internal combustion engines. However, variable air pressure can impact the engine’s air exchange and combustion processes, causing minor changes that affect the engine speed and result in variations in propeller thrust. A single-loop control strategy was proposed incorporating a feed-forward air-intake model with throttle feedback for small UAVs equipped with a two-stroke scavenging internal combustion engine and propeller. The feed-forward model was built with a simplified model of the airpath based on the scavenging measurement, which combined the tracer gas method and CFD simulation by a two-zone combustion chamber model. The feed-forward control strategy was built by a simplified crankcase–scavenging–cylinder model with CFD results under different air pressures, demonstrating a 1% error compared with CFD simulation. An iterative method of feed-forwarding was suggested for computing efficiency. A feedback controller was constructed using fuzzy PID for minimal instrumentation in engine control for small aircraft. Finally, the single-loop control strategy was validated through simulation and experimentation. The results indicate an 89% reduction in average speed error under varying air pressure and an 83.7% decrease in average speed overshoot in continuous step speed target experiments.

1. Introduction

Small aircraft, particularly Unmanned Aerial Vehicles (UAV), demonstrate extensive applicability in both civil and military domains [1]. As one kind of power source, internal combustion engines (ICEs), especially two-stroke engines, have advantages such as prolonged endurance, high power density, and heightened reliability so that they find widespread use in small aircraft, facilitating extended hover or flight scenarios with sustained efficiency [2].

In comparison with electric UAVs, despite the advantages of UAVs with internal combustion engines mentioned earlier, the stable operation of ICEs is contingent upon a stable environment at high altitudes. Variations in air pressure and temperature impact the air exchange and combustion processes of ICEs, resulting in engine speed fluctuations and cyclic variations [3] due to minor climate changes. Simultaneously, the propeller, connected to the engine, experiences alterations in speed and thrust, mirroring the engine speed fluctuations. These fluctuations contribute to dynamic instability in the aircraft, leading to variations in flight level and airspeed instability [4]. Consequently, achieving torque balance between the engine and the propeller under a specific transmission ratio becomes imperative for the speed control of the engine–propeller system in the face of changing climatic conditions.

Nowadays, open-loop and closed-loop control methods are mainly used for range extenders or small propeller aircraft powered by ICE. Open-loop methods, which mainly use PID and improved PID methods, such as fuzzy PID and adaptive PID methods, to regulate speed deviation, have a straightforward structure and are easy to realize [5]. Huang et al. present a fuzzy adaptive PID for an unmanned helicopter engine so that the engine can start safely and maintain a constant engine speed in various power states [6]. However, the inherent nonlinearity of the engine–propeller system speed is prone to state deviation and cannot meet the control requirements in the full range of working conditions [7]. Closed-loop control strategies mainly include speed feed-forward and feedback methods. The feed-forward method calculates the engine control parameter through the current system demand speed and feed-forward parameter tables, models, or observers [8,9]. Zeng et al. used a speed feed-forward parameter table combined with the PID feedback method to reduce the speed fluctuation of the engine under rated operating conditions [10]. The control accuracy of the nonlinear systems depends on the accuracy of the models. Hu et al. presented a bivariate coordinated sliding mode constant speed control method with a mathematically modeled engine [11]. Simultaneously, by balancing the engine torque of the system, a dual closed-loop control strategy of system speed and engine torque (throttle) is adopted to improve the accuracy of speed control [12]. Chen et al. applied a Generalized Predictive Control (GPC) method as a feed-forward model to enhance the speed tracking and robustness of speed control [13]. Therefore, the improvement of feed-forward models or methods could enhance the accuracy and robustness of speed control methods [13].

Despite the dual closed-loop control strategy of speed and torque with models or observers that could effectively achieve accurate engine speed control, the weight limitations on small UAVs restrict their ability to carry numerous sensors. One parameter of the engine speed or torque is usually chosen and monitored as the control variable, with engine speed often taking precedence due to its crucial role in propeller performance. The torque is usually derived from the propeller profile and engine speed, so the dual closed-loop strategies do not necessarily need to be applied in small UAVs. Regarding the control of the internal combustion engine’s speed, the core is the control of air intake. Under closed-loop equivalent ratio conditions, the intake mass governs the total mass, energy, and compression pressure at the start of combustion, which will influence the flame speed and heat release process. Moreover, variations in atmospheric pressure resulting from changes in flight level or wind force will also impact intake mass. Therefore, the modeling of the airpath and its corresponding engine model are pivotal aspects to ensure the accuracy of engine speed [14], especially for the aviation piston engine discussed in this paper.

Due to the weight constraints of aircraft, the 2-stroke engine with scavenging and exhaust ports is widely used in small aircraft. Compared with 4-stroke engines, the two-stroke engine shows its advantages. The two-stroke engines always work twice as the four-stroke engines so that the engine load and displacement can be reduced with higher energy density [15]. At the same time, the engine could work smoothly under two-stroke operating mode with lower vibration and friction losses [16]. The intake port without a valve could reduce the weight of the engine [17]. The intake and exhaust processes are usually conducted simultaneously [18]. Therefore, a portion of the fresh mixture of fuel and air entering through the scavenging port will inevitably be discharged through the exhaust port, and the short-circuit phenomena will lead to increased fuel consumption. Moreover, the accuracy of controlling the equivalent ratio of the fresh mixture and the mass of fuel injection is also affected by short-circuit phenomena [19]. Stuecke et al. measured short-circuit phenomena by a visualization method with colors [20]. At the same time, the trapped mass is influenced by the air pressure at the intake/exhaust port. Therefore, the establishment of a feed-forward model is essential to predict the scavenging process accurately, thereby enabling precise control of the engine’s speed.

Currently, engine control parameters under various operating conditions are consolidated into Engine Control Units (ECUs) as data tables. Parameters, such as injection duration and spark timing, corresponding to the engine’s speed, intake pressure, and throttle angle, are written into the controllers and are controlled by referring to the table [21]. At the same time, the lambda from the exhaust pipe could be used as the input for the feedback model to control the fuel mass [22,23]. However, this approach falls short in addressing the transient response of inlet pressure, particularly variations induced by changes in flight altitude and wind speed. The relationship between intake mass and engine operating parameters is not considered in the parameter tables. In addition, while the average value model calculated according to the average intake pressure is utilized in control strategies [14], some detailed characteristics, especially regarding scavenging parameters changing with pressure, need to be reflected through more intricate models, including but not limited to the finite element model and transient model [24,25,26]. Manish et al. used a CFD model to predict and reduce short-circuit losses [27]. Mattarelli et al. analyzed the influence of scavenging port parameters on the scavenging process [28]. Kroner et al. analyzed the whole calculation area with a combustion chamber and crankcase [29]. Although the finite element method can simulate the variations of trapped mass during the scavenging process in detail, the computational demands limit its application to real-time engine control.

The feasibility of the methods relies on the capability to measure the scavenging parameters of two-stroke engines. There are two primary methods for testing the scavenging performance of two-stroke engines. One method is direct measurement, involving the use of tracer gases or liquids in ambient or underwater. Sampling techniques are utilized to calculate the trapped mass by measuring changes in the concentration of tracer gases or liquids. The other method is indirect measurement, utilizing methods like airflow test benches [30], optical testing [31], or flow rate measurements [32]. These methods calculate instantaneous flow rates at the intake and exhaust ports to indirectly determine the trapped mass in the cylinder. However, it should be noted that these methods have certain limitations. When using the tracer gas method, it is necessary to consider the influence of the tracer gas on the combustion process within the cylinder [33,34,35]. If the tracer gas affects the combustion process, deviations in scavenging parameter measurements may occur due to incomplete combustion or unburned gases. Additionally, fuel might not burn completely in the actual combustion process. Therefore, restrictions on combustion or exhaust temperatures or alterations during tracer gas measurement may result in differences between actual scavenging performance and experimental scavenging parameters. Airflow test benches are typically used for flow testing under cold engine state and cannot measure intake and exhaust airflow velocities under actual engine operating conditions. Inaccurate scavenging measurements can affect the accuracy of simulation modeling, leading to imprecise parameters in the scavenging process. For two-stroke aircraft engines used in aviation, the combustion process is particularly sensitive to scavenging performance, given that these engines operate in high-altitude, low-pressure environments.

In this research, a single closed-loop control strategy was proposed based on air intake estimation for small aircraft with internal combustion engines, especially for the autonomous control of engine and propeller systems in UAVs. A hybrid measurement method for scavenging parameters of two-stroke engines was developed, which combined the tracer gas method and numerical simulations. The feed-forward controller was constructed based on a simplified model of the airpath derived from the measurement results, while the single feedback controller was established using engine speed as the primary control parameter, which was for the minimal instrumentation requirements for engine control in small aircraft. Finally, the proposed control strategy was validated through simulation and experimental testing, leading to the achievement of single closed-loop control of engine–propeller speed in small aircraft. In contrast with previous research, our study presents a more accurate single-loop strategy of engine speed control solution with fewer sensors by increasing the accuracy of the intake model of two-stroke engines.

2. Research Platform

2.1. Experiment Testbench

A two-stroke piston engine served as the research object in this study. The engine and its test platform are shown in Figure 1. The engine employed a scavenging strategy combined with cross-flow and loop flow. The fresh mixture was introduced into the crankcase via the throttle and the reed valve with the fuel supplied by the carburetor and entered the cylinder through the symmetrical main scavenging port and auxiliary scavenging port to facilitate the scavenging process. The main parameters of the engine are shown in

Table 1. Engine Parameters.

Subject	Parameter
Working Form	Port-Scavenging Two-Stroke
Number of Cylinders	2
Bore × Stroke/mm	36 × 28
Displacement/mL	56
Maximum Speed/rpm	8500
Compression Ratio	9.64
Efficient Compression Ratio	6.88
EVO/EVC/°CA	105/255
IVO/IVC/°CA	122/241
Fuel Supply Form	Carburetor
Ignition Form	CDI Ignition

.

To construct a precise intake feed-forward model, a variety of methods was adopted to measure the air exchange boundary of the engine. Intake pressure was measured by a pressure sensor, and the high-frequency pressure sensors were applied to measure the pressure in the crankcase (Kistler 4054) and cylinder (Kistler 6053cc) with a sampling interval of 0.36 °CA. The tracer gas method was used to measure the scavenging parameters of the engine. Compared with alternative measurement techniques, such as PIV and steady test of the air loop, the tracer gas method can reflect the real process of air exchange under the combustion state in the cylinder.

The connection between the ECU and the computer was built using the CAN protocol for engine control. The steering engine controlled the throttle, while the ECU managed the fuel injection and ignition parameters. The engine control architecture is shown in Figure 2.

2.2. CFD Simulation Platform

To comprehend the specific air exchange process of the two-stroke scavenging engine, a three-dimensional numerical simulation model was established in this paper by Converge 2.3.10 software. The model included various components, including the combustion chamber, scavenging port, exhaust port, crankcase, and other structures, which can fully simulate the air exchange and combustion process. The geometric model of the engine is shown in Figure 3. The engine numerical simulation model adopted a single cylinder with a scaled crankcase to ensure that the intake volume of a single cylinder is consistent with the actual value. Among the models, the RNG K-ε model was used for the flow simulation and the G-Equation model was used to simulate the basic flame propagation process in the cylinder. The turbulence model was chosen for the RNG K-ε model, which can be used to evaluate large-scale flow during the scavenging process [36] and has widely been adopted in the field of combustion simulation of ICEs [37]. The crankcase reed valve is limited to open when the intake pressure is larger than the pressure in the crankcase, i.e., 0.015 bar, which was obtained through experimental testing. The basic grid of the model was 4 mm, and a 1 mm density of the grid was adopted at the crankcase and the cylinder area. Speed-adaptive densification was applied in the whole domain with a 0.5 mm size, and temperature-adaptive densification was enabled in the cylinder with the same size during the combustion process (starting before the spark ignition). The grid near the spark plug was embedded during spark ignition and the minimum grid size was 0.125 mm.

The independence of grid concentration between the grid sizes of 2 mm, 1 mm without adaptive densification, 1 mm with adaptive densification, and 0.5 mm fixed embedded of combustion chamber area is shown in Figure 3c,d. The two main parameters, in-cylinder pressure at Exhaust Port Closed (EPC) and maximum in-cylinder pressure, were chosen to verify the independence of the grid concentration. The in-cylinder pressure at EPC showed the scavenging status at the end of the scavenging process, which the turbulence model mainly influences, and the maximum in-cylinder pressure showed the independence of grid concentration for both the turbulence model and combustion model. The grids of 1 mm showed the independence of the error of pressure at EPC, and the maximum pressure was less than 5%. Both parameters converged as the number of grids increased. As a result, grids of 1 mm with adaptive densification (grid number 23,425 at top dead center (TDC)) were chosen as the basic grids for the detailed simulation of the combustion process. The mesh view at TDC is shown in Figure 3e.

2.3. 1D Verification Platform

In this research, a GT-Power model of the two-stroke two-cylinder engine was constructed. To minimize the calculation process, the scavenging process was simplified into two extreme conditions: the complete mixing condition and the complete scavenging condition. In the complete mixing condition, the cylinder and exhaust pipe contain exhaust gas, whereas in the complete scavenging condition, they contain the fresh mixture. However, the scavenging process of the two-stroke engine is located between the two conditions, subject to certain changes in the external environment and combustion chamber structure. To capture the influence of high-altitude environmental pressure changes on engine scavenging efficiency accurately, the in-cylinder scavenging curves under different environmental pressures, temperatures, and speeds are obtained by three-dimensional simulation. The combustion model employed was the SI-Turb model, which could calculate the flame propagation by laminar flame speed and the intensity of turbulence. The model was validated under the atmospheric pressure condition.

The propeller was modeled according to the lobe element theory, and according to the lobe element momentum theory, the UAV propeller blade was divided into infinitely thin blade units along the radial direction. The lift and drag forces of each blade unit were analyzed. Finally, the total lift and drag forces of the UAV propeller were obtained by summing these units. The lift and drag forces on the propeller are shown in Equations (1) and (2):

$$L_{pro}=C_{Tp}\rho n^2{D}_{pro}^4$$

(1)

$$M_{pro}=C_{dp}\rho n^2{D}_{pro}^5$$

(2)

L_pro is the thrust of the propeller, M_pro is the torque of the propeller, C_Tp is the lift coefficient, and C_dp is the drag coefficient. Dpro is the diameter of the propeller, ρ is the density of air, and n is the engine speed.

3. Measurement of Scavenging and Construction of Feed-Forward Model

3.1. Hybrid Method of Scavenging Measurement

As mentioned, the tracer gas method was used to measure the scavenging parameters. Due to cost considerations, methane was selected as the tracer gas in this study. There are several reasons for this choice, including methane being a hydrocarbon fuel like gasoline with similarities. A small amount of blended methane will not significantly affect the combustion process of gasoline, and its combustion products are the same as gasoline, so it will not significantly affect the concentration of exhaust components. Methane is an easily available industrial gas, and its use and testing costs are low compared with other tracer gases.

Although methane has the above advantages as the tracer gas, it is also produced in small amounts during the combustion of gasoline as an intermediate product of combustion when the tracer gas is not introduced. To eliminate this partial effect, the following hypothesis is proposed in this study:

(1) Testing with and without tracer gas: The methane in emissions was tested both with and without methane in the air intake, and the methane concentration without tracer gas was considered to be the methane concentration produced by the uncompleted combustion of gasoline.

(2) Fixed methane concentration: The methane concentration is calculated based on the inlet gas flow rate, and the methane is introduced at a fixed volume fraction (flow rate) to ensure a constant methane volume fraction/equivalent ratio.

In this research, the volume fraction of methane was fixed according to the equivalent ratio and different speeds and throttle openings. Figure 4 shows the comparison of in-cylinder heat release rate with/without tracer gas at 6500 rpm and 50% throttle opening. This operating condition was selected for that it is the economy operating point for the engine–propeller system. The results in Figure 4 show that the in-cylinder pressure was not significantly reflected by the methane, and there was no significant impact on the engine performance.

Figure 5 shows the trapping efficiency and scavenging efficiency in the cylinder at different speeds and throttle openings. The trapping efficiency is defined as the ratio of the fresh mixture remaining in the cylinder to the fresh mixture flowing through the intake port, which is used to describe the short-circuiting degree of the two-stroke engine; the scavenging efficiency is defined as the ratio of the mass of fresh mixture remaining in the cylinder to the total mass at the end of the gas exchange, which is used to characterize the gas exchange performance of the engine. The results show that the in-cylinder trapping efficiency decreases with increasing throttle opening at the same speed, considering that it might be due to the higher intake flow rate that increases the short circuit and reduces the trapping efficiency. Although the increase in intake flow rate with the increase in throttle opening could add more fresh mixture into the cylinder, the high sweeping flow rate leads to the decrease in engine scavenging efficiency, and finally, the quality of fresh mixture in the cylinder will be reduced, which eventually leads to the decrease in engine scavenging efficiency. In the full operating range, the scavenging efficiency of the engine measured by the tracer gas method was between 30 and 45%.

To simulate the air exchange process under conditions of reduced air pressure, 3D numerical simulation was carried out with replication of experimental results. The variation in in-cylinder pressure and crankcase pressure during the air change process is shown in Figure 6 and Figure 7, in which the simulation condition was also selected as 6500 rpm@50% throttle opening condition. The results show that the in-cylinder pressure was greater than the crankcase/exhaust pressure at the beginning of the scavenging stage, so a small amount of exhaust gas flowed through the scavenging pipe. With the compression process of the crankcase, the mixture in the crankcase was forced into the cylinder and finally reached the balance between the pressure in the crankcase, the pressure in the cylinder, and the pressure in the exhaust pipe when the piston was near the bottom dead center. After that, the crankcase pressure and cylinder pressure were balanced until the end of scavenging. In this process, the crankcase pressure exceeded the intake pressure so that the reed valve kept closed and the air intake did not have a significant impact on the intake mass in the crankcase. After the reed valve was opened, as the scavenging was approaching the end of the scavenging process, the crankcase and cylinder were in a state of connection. However, due to the throttling effect, the crankcase pressure and post-throttle pressure will reach equilibrium again after the reed valve is opened and until the scavenging ports are closed. Therefore, in the scavenging process in the cylinder, the reed valve opening and closing moment did not have a significant effect on the scavenging process in the cylinder, as shown in Figure 3. The reed valve’s status was determined by the calculated pressure difference between the throttle and the crankcase. The pressure in the crankcase at 6500 rpm and 50% throttle opening from experiment and simulation is shown in Figure 7.

The in-cylinder scavenging efficiency obtained from CFD Simulation was 0.85, which was different from the experimental value measured by the tracer gas method. Since the crankcase pressure and in-cylinder pressure were close to the experimental value, it was considered that the combustion process influenced the calculation of the scavenging parameter. To further verify the cause of the deviation, the in-cylinder combustion process of the two-stroke engine was simulated using 3D numerical simulation, and it could be seen that the exothermic rate was closely matched to the experimental exothermic rate in the first half of the combustion process, but there was an evident afterburning phenomenon in the second half of the combustion process. The comparison of the experimental and 3D numerical simulations of the exothermic rate is shown in Figure 8.

Figure 9 shows the flame propagation process in the cylinder under the simulated heat release rate curve, in which the flame propagation front was expressed by a 1000 K isothermal plane. As shown in the figure, after the spark ignition, the flame was influenced by the in-cylinder flow, propagated rapidly in the clockwise direction in the longitudinal section, and gradually filled the hemispherical combustion chamber, forming the main combustion phase of the exothermic rate curve, while the combustion chamber edge exotherm generated the subsequent after-combustion phase. In this process, the main combustion phase in the cylinder exerted 34% of the total heat value in the cylinder, and the afterburning area exerted 66% of the total heat.

As can be seen from Figure 8 and Figure 9, the afterburning stage in the experiment was suppressed due to the possible pressure wave transmission in the cylinder, or the flameout in the narrow area at the edge of the combustion chamber. To simulate this phenomenon, a two-zone combustion chamber model was established, in which the G-equation was used in the central region of the cylinder to simulate the flame propagation process under turbulent conditions, while the combustion model was closed in the edge region to simulate the flame out phenomenon in the edge region of the combustion chamber. The average temperature and total molecular weight of the two zones were used to calculate the average pressure in the cylinder. The results of the exothermic rate and cylinder pressure using the two-zone combustion chamber model were compared with the experiment results, as shown in Figure 10. The test and simulation values fit well, and the model can be considered to meet the calculation requirements of the scavenging process.

The simulation results show that the in-cylinder heat release accounted for 38% of the total calorific value, which was close to the scavenging efficiency obtained by the experiment. Therefore, it can be assumed that the proportion of in-cylinder gas to the total intake volume obtained by the tracer gas is the proportion of fully burned in-cylinder gas, while the mixture in the cylinder but not burned was regarded as short-circuit gas, thus leading to a significant error in the scavenging parameters measured by the tracer gas method. This result was also verified at 8500 rpm, which was the maximum engine speed of the test engine. Based on the validation results, the model was simulated under different inlet pressures, and the relationship between the inlet pressure and the scavenging curve was obtained, as shown in Figure 11. From the results of the 3D numerical simulation, it can be seen that the exhaust residual ratio was reduced with the dropping of environment pressure under a high cylinder residual ratio for all three engine speeds. At the same time, with the increase in engine speed, the scavenging curve decreased earlier with the cylinder residual ratio reduced, especially for the high-speed condition. The scavenging curve was closer to the perfect mixing curve at the beginning of the scavenging process. With the residual ratio reduced, the scavenging curve under high-speed operating conditions showed a rapid drop when the cylinder residual ratio was less than 0.3, which was earlier than the other operating conditions. The above results were also used for the establishment of the object model and the verification of the control strategy.

3.2. Construction of Feed-Forward Model

During the intake process of the engine crankcase, with the piston going up, the expansion of the crankcase led to pressure drops. The difference between the internal and external pressure of the reed valve causes the reed valve to actuate. After the reed valve is opened, the fresh air will first enter the inlet channel through the throttle from the atmospheric environment and then enter the crankcase through the reed valve with fuel. During the time the piston is going down from the top dead center, the volume of the crankcase gradually decreases, causing the pressure to increase. When the crankcase pressure is greater than the outer pressure, the reed valve will be closed, and the crankcase becomes a closed system. With the continuous downward of the piston, the pressure of the crankcase increases gradually until the scavenging port is opened. The mixture in the crankcase enters the cylinder under the action of pressure difference and begins the scavenging process.

In gas exchange, the gas enters the crankcase from the external environment through the throttle and the reed valve. To simplify the model, the throttle is regarded as an orifice in this paper, and the intake process is isentropic. When intake begins, the flow through the throttle is shown in the following formula:

$$q_{mi}=C_{fi}·A_{th\_eff}·{\rho }_0$$

(3)

where A_{th_eff} is the effective flow area of the throttle, C_fi is the gas velocity at the throttle, and ρ₀ is the gas density at the throttle. The expression of the effective flow area at the throttle is shown in Formula (4).

$$A_{th\_eff}=C_{dth}·A_{th}$$

(4)

where C_dth is the flow coefficient of throttle, and A_th is the Total throttle area. The expression of gas density at the throttle is shown in Formula (5).

$${\rho }_0={\displaystyle \frac{P_0}{R·T_0}}$$

(5)

where P₀ is the ambient pressure, T₀ is the ambient temperature, and R is the gas constant. The expression of gas velocity at the throttle is shown in Formula (6).

$$C_{fi}=\left\{\begin{array}{cc}\sqrt{2{\displaystyle \frac{k}{k-1}}R{T}_0\left[1-{\left({\displaystyle \frac{P_i}{P_0}}\right)}^{{\displaystyle \frac{k-1}{k}}}\right]}& {\displaystyle \frac{P_i}{P_0}}>{\left({\displaystyle \frac{2}{k+1}}\right)}^{{\displaystyle \frac{k}{k-1}}}\\ 2{\displaystyle \frac{k}{k-1}}R{T}_0& {\displaystyle \frac{P_i}{P_0}}<{\left({\displaystyle \frac{2}{k+1}}\right)}^{{\displaystyle \frac{k}{k-1}}}\end{array}\right.$$

(6)

When ${\displaystyle \frac{P_i}{P_0}}>{\left({\displaystyle \frac{2}{k+1}}\right)}^{{\displaystyle \frac{k}{k-1}}}$, air flow is subcritical. When ${\displaystyle \frac{P_i}{P_0}}<{\left({\displaystyle \frac{2}{k+1}}\right)}^{{\displaystyle \frac{k}{k-1}}}$, air flow is supercritical, where k is the adiabatic coefficient.

The reed valve is regarded as a 0–1 valve; that is, it can be opened only when a specific pressure difference is satisfied. After opening, it is considered as a throttling orifice. The flow process is consistent with that at the throttle valve. The formula for gas mass change in the crankcase can be obtained from the ideal gas state equation, as shown in the following formula:

$$m_c={\displaystyle \frac{P_c{V}_c}{R{T}_c}}$$

(7)

$${\dot{m}}_c=q_{mc}$$

(8)

where P_c is the crankcase pressure, V_c is the volume of the crankcase, T_c is the temperature of the crankcase, and q_mc is the gas flow at the reed valve. The mass change in the inlet channel can be described by the ideal gas state equation, as shown in the following formula:

$$m_i={\displaystyle \frac{P_i{V}_i}{R{T}_i}}$$

(9)

$${\dot{m}}_i=q_{mi}-q_{mc}$$

(10)

where V_i is the inlet channel volume, and T_i is the intake temperature.

The model is discretized, and the calculation step is 1 CA. The calculation process is shown in Figure 12. When the reed valve is opened, the initial pressure of the inlet channel and crankcase is obtained, and the model begins to calculate until the crankcase pressure is greater than the intake pressure. It is considered that the reed valve is closed, and the crankcase intake ends at that time. Finally, the fresh mixed gas mass in the cylinder can be obtained by multiplying the intake mass of the crankcase and the sweep coefficient obtained by three-dimensional simulation. During the intake process of the crankcase, the volume gradually increases, and the pressure decreases as the piston goes up. Due to the pressure difference, the gas enters the crankcase from the reed valve. The discrete mass expression is shown in the following formula:

$$m_{c CA+1}=m_{c CA}+{\displaystyle \frac{q_{mc CA}}{6N}}$$

(11)

$$q_{mc CA}={\displaystyle \frac{P_{i CA}}{R{T}_{i CA}}}·A_{valve}·\sqrt{2{\displaystyle \frac{k}{k-1}}R{T}_{i CA}\left[1-{\left({\displaystyle \frac{P_{c CA}}{P_{i CA} }}\right)}^{{\displaystyle \frac{k-1}{k}}}\right]}$$

(12)

$$P_{c CA}={\displaystyle \frac{m_{c CA}R{T}_{c CA}}{V_{c CA}}}$$

(13)

The discrete mass expression in the inlet channel is shown in the following formula:

$$m_{i CA+1}=m_{i CA}+{\displaystyle \frac{q_{mi CA}-q_{mc CA}}{6N}}$$

(14)

$$q_{mi CA}={\displaystyle \frac{P_{0 }}{R{T}_0}}·A_{th\_eff}·\sqrt{2{\displaystyle \frac{k}{k-1}}R{T}_0\left[1-{\left({\displaystyle \frac{P_{i CA}}{P_{0 } }}\right)}^{{\displaystyle \frac{k-1}{k}}}\right]}$$

(15)

$$P_{i CA}={\displaystyle \frac{m_{i CA}R{T}_{i CA}}{V_{i }}}$$

(16)

As shown in Figure 13, through the simulation, the intake mass of the model at different openings is verified under the environmental pressure and temperature of 1.013 bar-298 K, 0.9 bar-291 K, 0.7 bar-278 K, and 0.5 bar-263 K, respectively, and the error is within 1%.

Due to the need to calculate the feed-forward throttle opening that can meet the target intake mass, the model needs to be solved reversely. However, the intake prediction model is nonlinear, and the reverse solution is complex and requires high computational power. Therefore, this paper adopts the iterative method to make the predicted intake mass calculated by the model gradually converge to the target intake mass. At this time, the throttle opening of the input model is used as the output of the feed-forward throttle opening. The calculation process is shown in Figure 14. Firstly, the initial value of the throttle opening is given to the intake prediction model. According to this initial value, the intake mass under this opening is calculated by the intake prediction. After comparing it with the target intake mass, the next time, the calculation value of the throttle opening is corrected by multiplying it with the correction coefficient and then calculating again. Each engine cycle is calculated three times. The convergence speed of the air intake prediction model was verified. It can be seen from Figure 15 that the air intake prediction model can also converge rapidly when the target mass changes to meet the calculation requirements.

3.3. Construction of Feedback Model

In this research, the fuzzy PID controller was chosen as the feedback model to keep achieving closed-loop control of the engine’s speed and enhance the stability of the UAV’s speed, as shown in Figure 16. The strategy utilizes the rotational speed deviation as the input and controls the throttle opening as the output to ensure real-time stability of the engine’s rotational speed. Precise feed-forward control is employed to compensate for external disturbances caused by environmental changes and correct intake losses resulting from these changes.

4. Validation of Algorithm

4.1. Simulation Validation

Figure 17 shows the optimization of the control effect of modeling feed-forward combined with fuzzy PID feed-back control compared with MAP feed-forward and PID feed-back. To present the speed control stability of the UAV during the climbing and descending process, the pressure variation was simulated as shown in Figure 17. The change in the pressure during the change in the UAV’s flight altitude was 0.8–0.75–0.7–0.75–0.73–0.8–0.75 bar. The pressure change is expanded compared with the actual value of pressure change when the altitude layer changes or when the wind/wind direction changes to verify the control effect of the controller.

Though the UAV was climbing with fixed thrust, the change in flight level led to a change in air pressure and density, so the speed of the propeller should also be changed. Figure 18 demonstrates the propeller speed during climbing and descending under fixed vertical speed. Compared with fixed MAP with PID, the control strategy mentioned in this research showed more obvious advantages. The overshoot was smaller, and the propeller speed was smoother during the pressure change process. The maximum speed error was 6.53 rpm, and the average speed error was 1.68 rpm for the control strategy with MAP and PID. The improved strategy with modeling and fuzzy PID showed a 2.53 rpm maximum speed error and a 0.18 average speed error, which was reduced by 89%.

Considering the sudden acceleration or deceleration process with thrust change, the continuous step condition was set to verify the effectiveness of the control strategy. As shown in Figure 18, compared with the comparison strategy, the strategy in this research showed a more minor speed overshoot and faster response of propeller speed during the change in target propeller speed. The control strategy developed in this paper reduces the average overshoot by 79% and reduces the maximum overshoot (during the step process of 6500–6000) by 78.9%. The detailed improvement is shown in

Table 2. Speed overshoot Comparison.

Speed Step Range	Overshoot by MAP + PID	Overshoot by Model + Fuzzy PI	Improvement Rate
6000–6500	33.7	5.9	82.5% (Maximum improvement)
6500–7000	14.8	3.4	77%
7000–6500	0.1	0.6	−83.3%
6500–6000	53.7	11.3	78.9% (Maximum Overshoot)
Average	25.6	5.3	79%

.

4.2. Experimental Validation

The experimental validation of the algorithm developed in this paper was carried out by continuous step changes in the propeller speed target from 4134 rpm to 4941 rpm to 4134 rpm to 4941 rpm to 4134 rpm, based on the engine performance. Changes in the target speed are implemented using CAN communication. The continuous step changes were used to simulate the acceleration and deceleration conditions with the change in lift demand of UAVs. As shown in Figure 19, the model’s feed-forward + fuzzy PI feedback control algorithm and MAP feed-forward + PID feedback control algorithm are used to demonstrate the speed-tracking performance of the UAV engine when the target speed undergoes continuous step changes. Since the control effectiveness in a steady state has already been demonstrated, this section mainly focuses on the overshoot during speed step changes.

Figure 20 demonstrates the algorithm developed in this paper showed the maximum overshoot is 1.23%, with an average overshoot of 0.76% under continuous step changes in the target speed. Compared with the MAP PID algorithm, which has a maximum overshoot of 5.85% and an average overshoot of 4.69%, the algorithm developed in this paper reduces the maximum overshoot by 78.9% and the average overshoot by 83.7%. From the results mentioned above, it is evident that the developed algorithm in this paper has a significant advantage. This is attributed to two key factors. Firstly, model-based feed-forward control calculates more precise feed-forward control values. Secondly, the fuzzy PI controller established in this study can adaptively adjust the proportional and integral coefficients based on the speed error and its first derivative. This ensures that smaller integral coefficients are applied when there is a large speed difference, preventing significant overshoot caused by the accumulation of speed errors on the feed-forward control input.

5. Conclusions

A single closed-loop control strategy based on air intake estimation for small aircraft with an internal combustion engine was proposed, especially for the autonomous control of small UAV’s engine and propeller system. The feed-forward controller was built with a simplified model of the airpath based on the measurement result, and the single feedback controller was built based on the engine speed for the minimum instrument for engine control in small aircraft. At last, the control strategy was verified by simulation and experiment, and the single closed-loop control of engine and propeller speed in small aircraft was achieved. The detailed conclusions are shown below:

• The CFD simulation was employed to address the impact of incompletely burned gas in the cylinder, and the two-zone combustion chamber model could reduce the error of the tracer gas method under actual engine operating conditions.
• The feed-forward control strategy was built using a simplified crankcase–scavenging–cylinder model with a CFD result, which showed a 1% error with CFD simulation. The iterative feed-forward method was proposed for computing efficiency.
• The fuzzy PID was used as a feedback and model-based feed-forward strategy. The strategy utilizes the rotational speed deviation as the input and controls the throttle opening as the output to ensure real-time stability of the engine’s rotational speed.
• Simulation and experiments were carried out to verify the algorithm developed in this research. The average speed error was reduced by 89% under varying air pressure, and the average overshoot of speed was also reduced by 83.7% in the continuous step speed target experiment.