Aerodynamic modeling and performance analysis of model predictive controller for fixed wing vertical takeoff and landing unmanned aerial vehicle

This research focuses mainly on the aerodynamic modelling and performance analysis of a model predictive controller for a hybrid fixed-wing vertical takeoff and landing of unmanned aerial vehicle. The aerodynamics, which comprises several aerodynamic characteristics including the lift, drag, and thrust coefficients, is modelled using Newton’s second law of motion. The force and moment equations were obtained, and they were then converted into matrix and state equation form, together with the transformation matrices. The essential equations with six degrees of freedom (6 DoF) and 12 state matrices including the control input and manipulating variables were obtained. The proposed model predictive controller (MPC) was designed using the optimal model predictive controller design parameters, such as a sampling time of 0.1, a prediction horizon of 15, a control horizon of 3 and additional controller settings. With a settling period of 3 s, an overshoot of 0.5865, and a steady state inaccuracy of 0.00785, the MATLAB simulation demonstrates that the system variables, including roll, pitch, and yaw, are stabilized. This MPC control is more effective in anticipating and optimizing the UAV than other control strategies. Eventually, the controller’s simulation on MATLAB Simulink demonstrates the controller’s ability to stabilize and control the system in a real-time application. Autonomous vertical takeoff and landing operations depend heavily on the mathematical model and architecture of the flight controller. Among the uses are inspection, monitoring, and rescue.


Introduction
Since the Wright brothers built their first manned aircraft in 1903, thanks to Sperry's autopilot, aircraft technology has advanced steadily until the invention of Unmanned Aerial Vehicles (UAVs) in the twenty-first century [1][2][3].They are becoming more common in both military and civilian uses [2,[4][5][6].They have recently been widely deployed for military missions such as monitoring, surveillance, full participation with weaponry, and airborne data collecting [7,8].UAVs are in considerable demand commercially due to their low production and operating costs, configuration flexibility depending on customer requirements, and lack of risk to pilots in demanding missions [1,2,[7][8][9].
Fixed-wing vertical take-off and landing (VTOL) unmanned aerial vehicles (UAVs) have garnered significant interest in recent years due to their unique capabilities [10].These aircraft can combine the efficient long-range flight characteristics of traditional fixed-wing UAVs with the vertical takeoff and landing capabilities of helicopters, making them ideal for a variety of applications, including surveillance, reconnaissance, and delivery [8,11].One of the key challenges in designing fixed-wing VTOL UAVs is the development of an effective aerodynamic model that captures the complex flight dynamics of these aircraft [12].This model is essential for designing and implementing control algorithms that can ensure safe and stable flight under various operating conditions [13].Additionally, performance analysis is crucial for evaluating the effectiveness of the control system and identifying areas for improvement [11].In addition, UAVs are commonly used in a variety of industrial segments, including agro-based control, forest and coastal surveillance, victim lookup, broadcasting, and photography [5,6].
Recent research has explored applying advanced control techniques to enhance UAV capabilities.One study [14] successfully implemented model predictive control (MPC) on a gimbal system, demonstrating superior tracking performance against external disturbances.This highlights the potential of MPC for optimizing gimbal functionality, particularly in real-time applications.Another groundbreaking study [15] employed a NARX neural network to achieve precise path tracking for hexa-rotor UAVs with load transport systems.This innovative approach paves the way for intelligent gimbal control advancements.Furthermore, research on robust UAV control models has established a strong foundation for optimization, highlighting the need for further exploration of metaheuristic algorithms across diverse path shapes to improve UAV performance in path following tasks.
In this paper, the authors present a comprehensive aerodynamic modeling and performance analysis framework for a fixed-wing VTOL UAV equipped with a model predictive controller (MPC) taking into consideration the turbulence effect as input parameters [16][17][18].The task begins by developing a high-fidelity nonlinear aerodynamic model of the aircraft that takes into account the unique challenges of VTOL flight, such as the transition between hover and forward flight.Then design an MPC-based control system that utilizes the aerodynamic model to predict the future behavior of the aircraft and generate optimal control inputs.Finally, perform a comprehensive performance analysis of the closed-loop system, evaluating its stability, tracking performance, and energy efficiency.The results of this study demonstrate the effectiveness of the proposed approach in achieving the desired flight characteristics of the fixed-wing VTOL UAV.The MPC-based control system is able to maintain stable and accurate flight under various operating conditions, while the performance analysis provides valuable insights into the strengths and weaknesses of the system.This work paves the way for the development of more efficient and reliable fixed-wing VTOL UAVs for a wide range of applications.
Therefore, this study delves into aerodynamic model design complexities like lift, thrust, and drag factors, meticulously examining translational and rotational disturbances for each modeling stage.It tackles the tightly interwoven dynamics of aircraft geometry and flight parameters by building nonlinear models and leveraging model predictive control with optimized settings, ensuring robust performance across diverse operational scenarios.
The burgeoning field of fixed-wing vertical takeoff and landing (F-VTOL) unmanned aerial vehicles (UAVs) presents a potent technological solution for aerial missions demanding exceptional maneuverability and access to confined areas [19].These versatile platforms seamlessly transition between hovering and forward flight, offering distinct advantages over traditional rotary-wing UAVs in terms of range, speed, and payload capacity.However, effectively controlling the complex aerodynamic dynamics of F-VTOL UAVs throughout the entire flight envelope remains a significant challenge.To address this, model predictive control (MPC) has emerged as a promising control strategy due to its ability to handle nonlinear dynamics, incorporate constraints, and optimize future system behavior [20].
The mechanical orientation of fixed-wing vertical takeoff and landing unmanned aerial vehicles (UAVs) can be classified into three main groups: tail-sitter, tilt-rotor [21], and tilt-wing, as identified by more various authors [22][23][24][25].Tail-sitter UAVs are VTOL UAVs that adopt a vertical position during takeoff and switch to a fixed-wing configuration during normal flight (see figure 1(C)).This design offers mechanical simplicity by reducing the number of components compared to conventional fixed-wing UAVs.Consequently, weight is conserved, complexity is minimized, and susceptibility to failures is reduced.However, the transition process from vertical to forward flight is time-consuming, and the large radius of curvature during this phase further elongates the transition time [1].
However, despite their versatility, tail-sitter UAVs also have certain limitations.One notable drawback is the relatively long transition time during the switch between vertical and forward flight modes.This transition period not only consumes time but also poses challenges in maintaining stability and control [28].The large radius of curvature observed during the transition further contributes to the increased time required, hindering the overall operational efficiency of tail-sitter UAVs.Moreover, the mechanical simplicity that contributes to their reduced weight and complexity also limits their payload capacity and adaptability to varying mission requirements.
The tilt-rotor aircraft is similar to a glider but has a rotor on top of its fixed wing [21].It is a hybrid of a glider and a quadrotor or tri-rotor UAVs [29].An extra servomotor is used to tilt the rotor-carrying shaft.This configuration's mechanical design is slightly more sophisticated than that of tail-sitter UAVs.The tilt-rotor configuration is more controlled and less vulnerable to disturbances than the tail-sitter arrangement and less mechanically difficult than the tilt-wing configuration [25].In pursuit of enhanced maneuverability and extended range, combining a quadcopter frame with a tailless aircraft structure emerges as a promising design choice for tilt-rotor UAVs [30,31].This hybrid configuration inherits the vertical takeoff and landing capabilities of multi-rotors while leveraging the efficient forward flight of fixed-wing aircraft.Crucially, it mitigates the limitations of tail-sitter UAVs, which tend to exhibit high turning radii during transition due to their unique orientation [31,32].Tilt-rotor UAVs, including our proposed design, typically utilize four motors -two tilting rotors for vertical lift and forward propulsion, and two fixed rotors for additional stabilization and control (see figure 1(B)).Alternative configurations also exist, employing three motors with one dedicated lifting fan or even two motors with variable-pitch propellers.Regardless of the specific motor arrangement, extensive research highlights the advantages of fixed-wing tilting rotor UAVs: improved maneuverability due to the combined capabilities of rotors and wings, enhanced controllability through precise motor adjustments, and reduced susceptibility to wind gusts and other external disturbances.
Among the diverse landscape of VTOL UAVs, tilt-wing configurations (show in figure 1(A)) offer a tantalizing blend of agility and efficiency [28].Unlike tail-sitters or tilt-rotors, tilt-wing UAVs maintain a fixed fuselage throughout their flight envelope, with wings rotating to either a vertical or horizontal position [13].This approach leverages the advantages of both helicopters and fixed-wing aircraft, enabling vertical takeoffs and landings alongside high-speed, fuel-efficient forward flight.
However, the current state of tilt-wing technology reveals both promise and shortcomings.On the positive side, advancements in actuation systems and control algorithms have led to improved transition smoothness and flight stability [33].Research in aerodynamic modeling, particularly computational fluid dynamics (CFD), has provided deeper insights into the complex airflows around tilting wings, leading to optimized designs with reduced drag and improved lift characteristics [34].Additionally, tilt-wing UAVs demonstrate inherent advantages over other VTOL configurations in terms of payload capacity and range, making them attractive candidates for long-range surveillance or cargo delivery missions [26].
Despite these advancements, challenges remain.Compared to their simpler counterparts like multi-rotors, tilt-wing UAVs suffer from increased mechanical complexity, translating to higher weight and potential failure points [33].The transition phases, while smoother than those of tail-sitters, still require careful piloting or advanced control strategies to maintain stability and avoid stalling.Additionally, the trade-off between vertical and horizontal performance remains a balancing act, with wing designs optimized for one mode potentially impacting the efficiency of the other.
The author of [23,27,35] employed an Adaptive Sliding Mode Controller to achieve robust tracking control of a Quadrotor UAV.They started by proposing nonlinear and coupled equations, with the Newton-Euler equations serving as the foundation for modeling.The primary focus of this study was to compare the obtained results with the current performance during the hovering mode.However, the paper's performance indexes, including settling time, rising time, and steady-state errors, fell short of expectations.To address this limitation, we present an approach in this paper by developing a Model Predictive Controller (MPC) and modeling it using the parameters utilized in the referenced study.By employing the MPC controller, we aim to improve the control performance during the hovering mode and achieve better results.
Furthermore, the author implemented a gain programmed altitude controller for Transition Control of a Tilt-Rotor VTOL UAV.Their work began with CAD modeling, followed by mathematical modeling.The report also included estimates of various aerodynamic impacts during the transition phase of flight.To evaluate the effectiveness of our approach, we will employ the model predictive controller in the transition phase and incorporate the parameters from the aforementioned study into the MPC controller.By comparing the results obtained from our study with those of the referenced work, we can assess the performance during the transition phase and gain insights into the effectiveness of our proposed approach.
In the study conducted by another author, a Proportional Derivative (PD) controller based on the Extended State Observer (ESO) was applied for the Modeling and Design of an Aircraft-Mode Controller for a Fixed-Wing VTOL UAV.The authors developed a comprehensive six-degree-of-freedom state model that incorporated aerodynamic effects, force equations, and moment equations.Computational fluid dynamics analysis was also performed using this model.Building upon the previously mentioned PD controller with ESO, the present article recommends its implementation due to its successful application in tail-sitter VTOL UAVs, albeit with different configurations.However, this research specifically focuses on tilt-rotor VTOL UAVs and aims to enhance their longitudinal stability.Given the Model Predictive Controller's (MPC) demonstrated ability to offer superior control manipulation compared to classical control methods, the authors opted to utilize this approach and expects improved outcomes through simulation.Consequently, the investigation anticipates leveraging the controller's accurate information and higher performance index to enhance the overall study's findings.
This paper study is intended to improve the aerodynamic equations of vertical takeoff and landing unmanned aerial vehicles after a thorough review of previous literature.This will be accomplished by including additional aerodynamic parameters such as lift factor, thrust factor, drag coefficients, and UAV orientation.It is also used to boost the controller's performance parameters.The following performance measures will be improved: settling time, overshoot, steady state error, and rising time.As a result, the current paper will contribute in improving the above those performance measures at three different flight stages.

Problem statement and research gap
Fixed-wing Vertical Take-Off and Landing (F-VTOL) Unmanned aerial vehicles, or UAVs, have combined the agility and vertical flexibility of helicopters with the effective, long-range flight of fixed-wing aircraft to completely transform aerial robotics.Precision farming, urban logistics, and search and rescue operations in disaster areas are just a few of the exciting applications that can be unlocked by this special combination.But achieving precise and strong control during the delicate transition periods between hover and forward flight is a crucial obstacle that must be overcome in order to fully utilize these adaptable platforms.
The fundamental difficulty with VTOL UAV aerodynamics is its inherent complexity.In contrast to traditional fixed-wing aircraft, these aircraft have to deal with dynamic flow regimes and complex control surface interactions during transitioning.For control algorithms to be designed in a way that guarantees safe and stable flying throughout the flight envelope, these dynamics must be accurately captured in a mathematical model.The difficulty is further increased by the requirement for optimal performance, as VTOL UAVs frequently operate under strict energy limits, necessitating the effective use of their limited battery capacity.To optimize flying time and operational range, the control system must thus ensure not only stability and maneuverability but also low energy consumption.
The primary objective of this study is to tackle these interrelated challenges in order to develop a comprehensive framework for: • Developing a high-fidelity aerodynamic model that can faithfully represent the intricate flight dynamics of a fixed-wing drone operating in vertical flight (VTOL) throughout forward flight, hover, and transition.
• Designing a robust Model Predictive Controller (MPC) that leverages the aerodynamic model to predict future system behavior and generate optimal control inputs for precise trajectory tracking and disturbance rejection.
• Conducting a thorough performance analysis of the closed-loop system, evaluating its stability, tracking accuracy, energy efficiency, and robustness to environmental disturbances.
The aim of this work is to bridge the theoretical and practical potential of fixed-wing VTOL unmanned aerial vehicles.By providing a data-driven approach to modelling, control, and performance monitoring, it seeks to pave the way for the development of more efficient, dependable, and flexible VTOL systems that can fulfil the full potential of this revolutionary technology.

Aerodynamic modeling of VTOL UAV
The first step in aerodynamic modeling is defining the frame of reference, followed by obtaining translation and rotation equations.Before continue with the analysis, assumptions must be made to ensure that the kinematics and dynamics models of the UAVs remain effective.When subjected to external pressures, the rigid body of a fixed-wing Quadrotor does not bend or oscillate, and the fuselage of a fixed-wing UAV is entirely symmetrical about the central axis plane, providing the Earth has no rotation and zero curvature.

Frame of reference
The Earth and body reference frames must be agreed upon before undertaking comprehensive aerodynamic modeling of the aircraft.The purpose of this section is to go over the common coordinate blocks that are used to establish a UAV model's mobility.Each reference frame has an origin and three perpendicular axes that connect to create a right-hand system.

The earth's frame of reference
The origin of this reference frame will be the center of the Earth, as depicted in figure 2, and the axes that emerge from it are as follows: • Z E axis has a South-North direction.
• X E and Y E lie on the equatorial (half-earth) plane.
The aforementioned reference frame is not mounted and spins with an angular speed of Ω E .The Earth's configuration is not inertial since it revolves both along its rotation axis and around the Sun.In this scenario, the Earth is assumed to be flat and non-rotating.This is a reasonable assumption in UAV applications.

The standard frame of reference (North-East-Down)
The figure 3 North-East-Down frame of reference is also established, and its center coincides with the UAV's mass center.This frame's axis includes: • Z V axis direction is along the acceleration due to the gravity vector; • X V and Y V axes lie on UAV's fuselage and right wing respectively.

Body reference frame
It is critical to derive the equations that reflect the UAV's dynamic behavior, which begins with a clear identification of the inertial frame and body-fixed frame, as illustrated in figure 6.The basic frame is the inertial reference frame, whose origin and three components, xI, yI, and zI, respectively, point north, east, and down.The body-fixed frame's origin (OB) is located at the UAV's center of gravity, two axes are located with the head (xB) and right-wing tip (yB), and the third axis is orthogonal to the first two (zB).
Figure 4 is the basic frame of reference representation for translation and rotation equations of motion.As a result, the x-axis represents the nose of the aircraft, the y-axis represents the right-hand side wing, and the z-axis represents the aircraft's downward gravity acceleration side.The x, y, and z axes of the rigid body (inertial) reference frame are thus given as a right-handed Cartesian coordinate system with North, East, and Downward directions, abbreviated as the NED reference frame in this study.The VTOL UAV is a rigid body with six degrees of freedom, and the aircraft's state of motion is defined by twelve waypoints.
• Inertial reference frame FE (xN, yE, zD) with origin on the surface of the Earth, • Body reference frame FB (xB, yB, zB), • Inertial reference frame FV (xV, yV, zV) with origin in the center of the UAV's mass, • Wind reference frame Fw (xw, yw, zw)

Mechanical components of the aircraft (the basic actuators involved)
The basic actuators in this VTOL UAV are servomotors for tilting the rotors and brushless DC motors for hovering and forward flight or aircraft mode flight.Brushless DC motors are extensively utilized in UAVs for a multitude of reasons, including improved torque to weight ratio, lower maintenance requirements, no brush and commutator erosion, and elimination of ionizing radiation from the commutator.The torque produced by the actuator was calculated using equation (1), which was derived from Newton's second law of rotational motion.In aircraft dynamics, the torque generated by rotors can be calculated using the center distance r and the lift force utilized by the rotor using equation (2).

Torque momento of inertia I rotational acceleration
Torque r thrust force f the distance from the rotor rotating axis to the body center 2 The proposed UAV is a hybrid a fixed-wing UAV and a quadrotor.The UAV has two rotors in front and two motors at the back of the fixed wing, with each tilting equally via powered servo.The combination of rear rotors and front rotors with tilting servomotors that gradually tilt towards the flight direction during the transition, providing the UAV with forward speed, are the transition mechanisms deployed to convert from vertical to horizontal flight and vice versa.Their connection maintains a steady altitude of flight.
The UAV platform's flight mode is categorized into three flight modes, as illustrated in figure 5.In the initial mode, VTOL (Vertical Takeoff and Landing), the flight intended in this phase is for vertical ascent, descent, and hovering.The flying platform operates in multirotor mode, with all thrust provided by the four rotors.Once reaching the specified minimum altitude, the UAV orientation quickly switches from aircraft mode to transitional mode to save power.In the second phase, transition mode, the vertical rotors are simultaneously slanted vertically, increasing the aircraft's horizontal speed.To obtain a certain angle of attack, the angular position is adjusted by altering the thrust of specific rotors.When the horizontal speed increases, the fixed wings create lift.Until the angle of attack reaches a specific value, the difference in thrust between the left and right rotors determines the roll angle.The yaw angle is controlled by the counteractive moments created by the pairs of rotors M1, M3, and M2, M4.The pitch angle is controlled by the difference in thrust between the front and back rotors.In the third mode, Forward Flight (Aircraft), the UAV has enough translational travel speed to generate lift force.The platform operates like a normal aircraft in these conditions, with yaw, pitch, and roll motions provided by aerodynamic control surfaces such as the rudder, elevator, and ailerons.Four rotors (M1, M2, M3, M4) that are 90 degrees slanted relative to their initial position create the requisite thrust force.

Aerodynamic equations
The following assumptions are used in the aerodynamic modeling: • The lift forces exerting on each airfoil do not generate momentums in the y axis.• The UAV is a rigid body, which means the flexibility of the aircraft's wings or fuselage will be neglected.
• The aerodynamic center (AC) and the center of gravity (CG) are coincident.
With the rotor thrust motors M1, M2, M3, and M4, and the force created by each motor as f1, f2, f3, f4, and the tilting servos having an equal amount of thrust force f5.Therefore, we can get the following equations for the needed VTOL UAV.The positions of each motor are equal distances from the center, and the x-and y-axes are the symmetric centers of the body.

The Rolling motions
The rolling torque can be calculated as the product of the distance from the center to the force-generating axis and the force at the rotor.As a result, the rolling torque can be calculated by subtracting the smaller side from the bigger side, as indicated in equation (3).
Where: x t is the torque generated along the rolling direction.
r is the center distance f1 f2, f3, f4 are the thrust forces of each motor on the UAV The equation provided, denoted as equation (1), and enables the calculation of the second derivative of the roll angle.This second derivative corresponds to the angular acceleration experienced in the direction of rolling.As a result, the final derivative result is given in equation (4).
Where: f is the roll angle I x is the moment of inertia about the x-x axis.
The general second derivative of the rolling angle represents the angular acceleration, which, when integrated using the model's integrator, yields the angular position.In other words, integrating this angular acceleration leads to the angular position of the rolling object.

The pitching motion
The pitch equation of motion represents the required torque in that direction yE and applies Newton's second law of motion to obtain the desired pitch angle equation.As a result, the torque necessary is computed using equation (5), and the pitch angle is depicted in equation (6).
Where: y t is the torque generated along the pitching direction.
Where: q is the pitch angle I y is the moment of inertia about the y-y axis Where: y is the yaw angle.I z is the moment of inertia about the z-z axis

The transformation matrices
To grasp and perfect motion equations and transformation behavior, it is better to state the velocities of each before beginning to calculate the transformation matrices.The relationship between those velocities is stated in equation (9).
Let the velocities of the body, wind, and earth be denoted as follows.V U, V, W : The UAV velocity with respect to Earth, V V V V , , : The UAV velocity with respect to air, V U V W , , : The wind velocity with respect to Earth, To completely grasp the UAV movement equations, it is necessary to first generate the transformation matrices that connect the various coordinate systems, which are outlined in equation (10).
• The transformation matrix T BE from Earth's coordinate system to the body's coordinate system.
• Equation (11) depicts the D BW transformation matrix from the wind coordinate system to the body coordinate system.
q q f q f q 3.3.5.Hovering mode flight (The vertical motion) The vertical motion of this quad-tilting rotor UAV is determined by the overall thrust force and the weight of the aircraft along the z-axis, as given in equation (13).

F ma 11 ( ) =
Where: -F is force, m mass of the body, and 'a' is acceleration.
The thrust force required to hover the UAV can also be calculated using equation (14).
Thus, the vertical position can be governed as:

The horizontal motion
As demonstrated in figure 6, horizontal motion in the x and y axes is produced by decreasing the RPM of the rotor whose direction is intended to travel and then increasing the RPM of the rotor on the opposite side of either the x or y axis needed to move.It is preferable in large aircraft systems to use an aileron, elevator, and rudder to regulate the aircraft yaw angle and vertical position after the transition phase of the flight, which is forward flying mode.The utilization of independent rotors to modify the horizontal position, which may be related to the yaw angle, is preferable in this small VTOL UAV.As a result, the capacity of those four rotors to move independently will be proportional to the aircraft's weight.By balancing the forces on each side of the motion along the x-axis, equation ( 16) is obtained from Newton's Second Law of Motion.Equation ( 17) is calculated similarly for motion along the y-axis using figure 7 motion analysis.

Different forces exerted by aircraft
In the study of aircraft, four basic forces had the most impact on the aircraft's efficiency.These forces are thrust force FT, gravity force FG, lift force FL, and drag force FD.The trust force, lift force, and drag force are calculated using equations ( 18)-( 20), respectively.
Where: τ T is the induced moment due to the difference of thrust between T 3,4 and T 1,2 , τ M is the airfoil's pitching moment, τ G is the gyroscopic moment.τ T is obtained through equation (22).
Where: r is the distance from the CG to the rotors.The airfoil's pitching moment τ M is obtained from the airfoil's C m slope and the lift contribution of the elevator, as it is expressed in equation (23).
The gyroscopic moment τ G arises from the combination of the airframe's angular speed Ω b = (p, q, r) T and the rotors angular speed Ω i .
The essential governing equations in the hovering flight mode are already covered in the above aerodynamic equations section, which can be expressed using equation (24).

Model predictive controller design
Model Predictive Control (MPC) is a technique that combines the control of a dynamic system with solving an optimization problem.At each sampling time, an inline finite horizon optimal control problem is formulated, starting with the current state as the initial state, and solved to determine the control sequence.In the control sequence, only the first element is utilized by the plant.The prediction horizon is then shifted to the next time step, and the problem is solved again using the new system state as the initial point.Hence, the optimization problem's solution is influenced by the current state of the system.This approach is commonly known as 'Receding Control Horizon' [38].
Model predictive control (MPC) has revolutionized the control of fixed-wing VTOL unmanned aerial vehicles (UAVs) by offering optimal and robust trajectories across diverse flight phases [39].For hovering, MPC leverages fast and accurate inner-loop controllers alongside a receding horizon strategy to precisely maintain position and attitude, as demonstrated by [9].During transition from vertical to horizontal flight, MPC seamlessly blends the control actions for both modes, ensuring smooth and efficient maneuvers as in [40].In forward flight, MPC tackles challenges like wind disturbances and actuator constraints by optimizing future control inputs based on predicted system behavior, as showcased in [41].The efficacy of MPC for VTOL UAVs extends to various types, including linear quadratic (LQ) MPC for its simplicity and fast computation [42], nonlinear MPC for handling complex aerodynamics [12], and multi-objective MPC for optimizing conflicting goals like tracking accuracy and energy consumption [41].These advancements in MPC have paved the way for more agile, efficient, and autonomous VTOL UAV operations, opening doors to diverse applications in areas like surveillance, inspection, and logistics.
MPC is an optimal method that leverages predicted system control to minimize a cost function while considering the constraints imposed by the system dynamics within a finite receding horizon.During each time step, the MPC controller acquires the current state of the plant and estimates the next state.By solving a constrained optimization problem based on an internal plant model and the current system state, the controller identifies a sequence of control actions that minimizes the cost over the horizon.However, only the first computed control action is executed by the controller, while the subsequent actions are bypassed.This process is repeated at each subsequent time step.For better clarity, a simplified representation of the Model Predictive Control workflow is depicted in figure 8.

Mathematical model of the plant
All aspects related to the aerodynamic equations, forces, and moments have been discussed in the previous section (section 3).Therefore, the remaining task is to present the state parameters as outlined below.For the initial hovering flight, the system encompasses 6 degrees of freedom (DOF) and a total of 12 states.It is critical to transform the equations to matrix form and use the following letters to denote: x 8 q = (Pitch angle) x 9 y = (Roll angle) x 11  q = (Pitch rate) Hence, the state space model could be deduced to equation (25).The full matrix form of this equation is expressed in equation 26.

Model predictive controller parameters
Unlike other controllers, the model predictive controller is a Simulink built-in tool that manages any given system, whether linear or nonlinear, by forecasting the system's future based on the MPC assigned by the designer.The model predictive controller's fundamental workflow begins by specifying the sort of plant cellular model used by the MPC controller to forecast plant activity.This plant cellular model is a critical step in predicting the MPC controller's prediction horizon.The next critical step is to define the signal types for the MPC design process by categorizing them as input and output types.The third phase, which is the backbone of the MPC controller, is the creation of the MPC plant, which is carried out after obtaining the plant model [33,43].Therefore, after several tunes of the system model on the MPC controller are Sampling Time of 0.1, Prediction horizon of 15, and control horizon of 3 weightings and constraints of

Results and discussion
MATLAB is used to simulate the developed mathematical model on the MPC controller.The preview capability of this Model predictive controller is a crucial part.The MPC controller preview window displays findings for both the input and output scenarios individually.Finally, the outcomes of the MPC controller suggested in this research will be compared to the results of previous works done so far by different scholars.This will be performed by simulating the suggested MPC controller using parameters developed by other scholars.
5.1.The input stabilization 5.1.1.System response Stabilization of the control input is essential in a control system.This model predictive controller's best characteristic is its capacity to stabilize the control input in the shortest possible time.As illustrated in figure 9, the input is stabilized within 2-3 s of the system's start.The actual input stabilization time is 2.8 s.
In figure 9, the inputs f1, f2, f3, and f4 are stabilized in order under tuned constraints, prediction horizon, control horizon, and weights.Consequently, the input scenario is stabilized in a maximum of 3 s.This means that once the system has stabilized, the given input will not be affected.When compared to other control systems, the ability to preview is quite limited unless viewed through a separate scope.However, each input includes a previewing page where you may change the weights and limits.

Validation with previous work
To see the effectiveness of the controller in stabilizing the input, system parameters used by linear quadratic regulator [44] and sliding mode controller [45] are used for comparison.Figure 10 depicts the input response comparative performance of previous works so that due to its self-optimization capability, the MPC controller designed has stabilized the system's input with in short period of time.
The MPC controller required the shortest input stabilization time, as seen by the simulation results in figure 10 taking this into consideration, the implementation of self-optimization in the MPC controller improved the settling time of the sliding mode and linear quadratic controllers at the input side.
Based on the data presented in table 1, it is evident that the model-predictive controller exhibits superior performance metrics.The rising time, settling time, and maximum overshoot were measured at 0.4054, 3.1523, and 1.5456 respectively when the suggested MPC controller was tested.These tests were conducted using the parameters provided by earlier scholars, who validated the controller's effectiveness through the utilization of a sliding-mode controller and linear quadratic regulator.In direct comparison to the previously described controllers, the MPC controller recommended for this study demonstrated notable enhancements in terms of controller performance measures.

The outputs responses
The primary objective of this study is to obtain a diverse range of output responses.As indicated in the mathematical formulation section, the system under investigation comprises six degrees of freedom, resulting in six distinct outputs.These outputs consist of three positions (X, Y, and Z) and three rotations (Roll, Pitch, and Yaw).To initiate the analysis, the author first mapped the outcomes of their controller onto the designated plant model.Subsequently, they presented the data obtained through their controller, alongside the parameters used by other researchers for validation purposes.The resulting figure 11 provides a visual representation of the responses elicited by the planned MPC controller on the designed model plant.The preliminary outcomes of the fine-tuned MPC controller demonstrate the successful stabilization of each of the six output states, in accordance with the specified weights and limitations.
Given the importance of altitude control and longitudinal stability, greater emphasis was placed on the z-axis and yaw angle.Consequently, both variables were effectively stabilized within a maximum time frame of 2.8 s, as per the established baseline.To further explore the performance, a comparative analysis was conducted for the angular and Z-position tracks, with the corresponding results being plotted in a separate section.

Linear responses
The preview of the response (figure 11) for each output highlights the exceptional quality of the controller in handling the complexities of the nonlinear system.This comprehensive response encapsulates all six outputs within a single window, revealing their remarkably similar performances.Subsequently, the current study aims to compare one linear position (Z-position) and two angular positions (roll and yaw) with findings from earlier studies in order to validate the results.Each simulation outcome is carefully mapped to showcase the intended outcomes of the simulation.
The X-position tracker illustrated in figure 12 clearly demonstrates that the location was accurately tracked to the desired reference of 1 m within a mere 1.2 s, with an insignificant overshoot of 0.045.Furthermore, the rising time of the position tracker is notably small, measuring at 0.5056, which stands in favorable comparison to previous works.
In the previous state, previous researchers [46] took approximately thirty seconds to stabilize the fast response.However, our MPC controller demonstrated a significantly improved performance by stabilizing the system within a mere two seconds.This achievement can be attributed to several factors, including the careful consideration of constraints and weights, as well as the selection of appropriate sampling time, prediction, and control horizon values.The optimization of these parameters played a crucial role in enhancing the efficiency and responsiveness of our MPC controller, resulting in a remarkable reduction in the stabilization time compared to the previous approaches.Similar to the X-position tracker, the Y-position tracker also demonstrated a remarkable stabilization time of two seconds, as depicted in figure 13.In contrast, previous researchers [44] required up to thirty seconds to achieve the same level of settling.This notable improvement can be attributed to our diligent parameter selection and the inherent self-optimization capability of the MPC controller.The Y-position response exhibited a settling time of 1.2 s, a minimal overshoot of 0.0503 s, and a rise time of 0.8054 s.These results further confirm that the MPC controller significantly enhances the system response in comparison to prior works, thanks to its superior performance and efficient optimization of control parameters.
The z-position, which represents the altitude tracking control of the UAV, holds crucial significance.As depicted in figure 14, the MPC controller efficiently tracked the desired vertical position within a brief timeframe of only 2 s.This impressive stabilization capability extends to all flight phases, encompassing hovering, transition, and forward flight.In comparison, previous research required a minimum of thirty seconds to achieve stabilization, making the MPC controller a superior choice for future position prediction and  control.Its ability to rapidly stabilize the z-position contributes to the overall effectiveness and reliability of the controller, ensuring precise altitude control throughout various flight scenarios.
Moreover, the translational response is exhibited through three distinct positions.The results clearly demonstrate that the controller, equipped with enhanced MPC parameters, effectively optimizes the system for step response in various operational modes such as hovering, transition, and forward flight.The system achieves a remarkable settling period of 3 s, with a minimal overshoot of 0.03 and a steady-state error as low as 0.009, successfully stabilizing the positions.These findings highlight the controller's exceptional ability to maintain precise and stable control over the UAV's translational movements, ensuring accurate positioning and improved overall performance.Consequently, in figure 15, an altitude tracking correlation with past efforts is demonstrated.
The proposed MPC controller is compared to the PD-based ESO controller [35] in figure 15.The simulation is carried out with the same parameters as the previous authors' PD-based ESO, as stated in the literature.
Table 2 presents a comprehensive comparison of the proposed MPC performance with other controllers for altitude tracking.The predictive nature of the model, coupled with the integration of a self-optimizer within the MPC controller, significantly enhances its overall performance.Notably, the MPC controller, starting from the plant model, achieves a remarkable output of three seconds, whereas the alternative controller requires as long as 18 s.This substantial reduction in response time clearly demonstrates the superiority of the proposed MPC controller in terms of altitude tracking control performance, surpassing the earlier controllers put forth by other researchers.The implementation of predictive techniques and self-optimization capabilities equips the MPC controller with the ability to deliver highly efficient and precise altitude tracking, setting a new benchmark in the field.

Angular responses
After careful setup, main expected responses next to the linear (X, Y, and Z) are the angular responses.Since the MPC has and optimizer incorporated in it, it performs well than other in stabilization of the system under disturbance.
The rotation of the roll angle around the x-axis is effectively stabilized within a mere 2.8 s, as visually depicted in figure 16.The system response in this particular direction exhibits an overshoot of 0.5654 and settles within 2.8 s.It is worth noting that previous controllers required a longer time to achieve stabilization in this aspect.However, the presented MPC controller outperforms them by significantly reducing the stabilization time.This notable improvement can be attributed to the superior parameterization of prediction and control horizons, as well as other crucial controller parameters.By fine-tuning these parameters, the MPC controller demonstrates enhanced efficiency and responsiveness, resulting in faster and more precise stabilization of the roll angle rotation around the x-axis.Likewise, the response of the pitch angle, as depicted in figure 17, illustrates the successful stabilization achieved by the MPC controller.The response exhibits a minimal overshoot of 0.14 s and settles within a time frame of 3.58 s.Notably, our proposed MPC controller outperforms previous works in terms of the pitch angle response.This improvement can be attributed to the unique configuration of the rotors attached to the front and back of the wing, which directly influence the pitch angle.In the case of VTOL UAV operation, there is no requirement for the complete aircraft to pitch.Leveraging this characteristic, the MPC controller effectively stabilizes the pitch angle, ensuring smoother and more precise control.The self-optimization capability of the MPC controller plays a significant role in achieving rapid stabilization and minimizing overshoot, resulting in highly efficient and accurate operations.
The yaw response demonstrates remarkable speed, allowing for the stabilization of the z-axis movement within approximately 2.8 s, except during takeoff.Figure 18 provides a zoomed-in depiction of this response, emphasizing the minute deviations observed.The yaw response exhibits an exceptionally low overshoot of 0.00145 and settles within the same 2.8-s timeframe.This plot is specifically designed to showcase the minimal deviations achieved by the controller.The yaw response plays a crucial role in maintaining the overall stability of the unmanned aircraft.A comprehensive comparison of the yaw and roll responses is currently under examination and will be presented in the subsequent paragraph, providing further insights into their respective performances.Figure 19 presents the consolidated output of the roll, pitch, and yaw responses, providing a comprehensive view of the effective tracking of angular positions.The tracking control is executed to optimize the plant using the MPC controller equipped with optimal parameters.By considering the final angular output in figure 19, the yaw angle has a low overshoot of 0.00145.However, settling times and an average steady-state error of 0.00785 are shown by the pitching and rolling angles.
These results clearly indicate the superior performance of the controller, surpassing the achievements of earlier researchers [43] in handling similar UAV systems.The controller's ability to effectively manage the UAV system is evident from the precise and stable tracking of angular positions, further confirming the efficacy of the proposed MPC controller with optimized parameters.
Figure 20 illustrates the comparison of angular positions, specifically focusing on the roll angle and yaw angle.This comparative analysis aims to validate the simulation findings of the MPC controller and the results obtained from the PD-based ESO (Extended State Observer) [35].By simulating the MPC controller using scenarios derived from previous research [35,47] in the same domain but employing a different controller, the effectiveness and performance of the controller can be assessed.The roll and yaw angles were specifically selected to evaluate the controller's longitudinal stability and cruise stability, respectively.This comprehensive comparison provides valuable insights into the controller's ability to maintain stability and control in different flight scenarios, further supporting the validity and superiority of the MPC controller in comparison to other approaches in the field.Table 3 presents a comprehensive comparison of angular positions for the proposed MPC controller, along with settling time and overshoot values, in comparison to other controllers.Notably, the MPC controller exhibits an overshoot of 0.5865, a settling time of 2.5405, and a steady-state inaccuracy of 0.00785 for the roll angle response.In contrast, the PD-based ESO [35] approach yields higher values for the same system parameters.This highlights the improved functionality of the MPC controller, particularly due to its effective control over the longitudinal stability of the UAV through the yaw angle.Moreover, in terms of yaw stabilization capability, the proposed MPC controller outperforms other approaches.As demonstrated in table 3, it achieves an impressively low overshoot of 0.00145, a settling time of 2.5056, and a rising time of 0.05045, while the PDbased ESO exhibits an extended response period.
The superior performance of the proposed MPC controller can be attributed to its predictive and selfoptimizing capabilities.These features significantly enhance controller performance, surpassing conventional control systems.The maximum settling time, which represents the control input settling response, is reduced to 3.45 s, while the maximum overshoot for the roll angle response is 0.5865.The overall steady-state error is remarkably low at 0.00785.These findings further emphasize the efficacy of the proposed MPC controller in achieving precise and stable control over the angular positions, setting a new benchmark in controller performance.

Conclusion
This study presents a comprehensive framework for modeling the aerodynamics of fixed-wing vertical takeoff and landing unmanned aerial vehicles (UAVs) through computer simulation.The aerodynamic model is developed based on Newton's Second Law of Motion and includes a state matrix consisting of state input and controlled variables.In order to appropriately handle the system, a proposed Model Predictive Controller (MPC) is implemented, comprising an optimizer and a model prediction component.
The MPC controller is optimized with specific parameters, including a prediction horizon of 15, a control horizon of 3, and a sampling time of 0.1.The simulation results for the proposed MPC controller demonstrate its capability to effectively regulate each manipulated variable and control input.Furthermore, the controller successfully tracks various outputs, such as linear positions along the x, y, and z axes, as well as angular positions encompassing roll, pitch, and yaw.The maximum settling times achieved for these outputs are within 3 s, with a maximum overshoot of 0.5865 and a steady-state error of 0.00785.
Comparing the performance of the proposed MPC controller with earlier research efforts, it becomes evident that the system's overall performance has been significantly improved.The MPC controller showcases enhanced control and stabilization capabilities, surpassing the achievements of previous studies.Finally, the implementation of the controller in a real-time application is demonstrated through simulation on MATLAB Simulink, confirming its ability to stabilize and control the system effectively in practical scenarios.

3. 3 . 3 .
The yaw motions Equations(7) and (8) are used to determine torque and angular acceleration in the yaw direction, respectively.t is the torque generated along the yaw direction.c is the experimental factor used to convert the force to moment.

•
The matrix T w , illustrated in equation(12), describes a relationship between the Euler angles and the angular velocities of each of the UAV's P, Q, and R.

*
Once the UAV reaches the vertical required point and energizes the z-position signal it automatically goes to the combined transition and forward flight controller.

x x 1 =
(Position along the x-axis) y x 2 = (Position along the y-axis) z x 3 = (Position along the z-axis) x x 4  = (Velocity along the x-axis) y x 5  = (Velocity along the y-axis) z x 6  = (Velocity along the z-axis) x 7 f = (Roll angle)
of the form Y

Figure 9 .
Figure 9.The result of the input stabilization of the MPC.

Figure 11 .
Figure 11.The output preview of the designed controller.

Figure 12 .
Figure 12.The response to the X-axis position.

Figure 19 .
Figure 19.The responses of roll pitch and yaw angles in a single window.

Table 1 .
Comparison of performance controllers for input data.

Table 2 .
Performance comparison of controllers for altitude tracking.
Figure 16.The roll angle response.

Table 3 .
Comparison of angular results using different controllers.