Nonlinear Feedback Linearization Control and Region of Attraction Analysis for a Fixed-Wing UAV

Abstract

This paper presents the design of a nonlinear Multi-Input Multi-Output (MIMO) Feedback Linearization Controller (FLC) for the longitudinal dynamics of a fixed-wing UAV. The proposed approach employs dynamic extension to achieve Feedback Linearization in a fourth-order longitudinal model, offering a more compact alternative to existing high-order formulations. The controller ensures the accurate tracking of predefined airspeed and flight path angle references, that is, the control of the magnitude and direction of the velocity vector using engine thrust and pitch moment as control inputs. Additionally, this study determines the region of attraction in which the controller design remains well-defined. This analysis provides critical insights for selecting feasible airspeed and flight path angle references, helping to prevent conditions that can lead to instability or undesirable behaviors, such as the need for negative thrust. Numerical simulations validate the effectiveness of the proposed method in handling the aircraft’s nonlinear dynamics and maintaining stable flight performance.

1. Introduction

Recent advancements in chip miniaturization, battery efficiency, sensor development, control algorithms, and communication systems have significantly expanded the use of Mini Aerial Vehicles (MAVs) across a growing range of applications, including agriculture [1,2], forestry monitoring [3,4], and meteorology [5], as well as other relevant fields. MAVs can generally be classified into two main groups: rotary-wing and fixed-wing vehicles [6]. Fixed-wing vehicles offer several advantages over their rotary-wing counterparts, such as extended range, higher payload capacity, increased flight speeds, improved energy efficiency, and extended autonomy.

Among these, Unmanned Aerial Vehicles (UAVs) have received significant attention due to their numerous advantages over piloted aircraft. The absence of a pilot results in a lighter and more cost-effective platform, making UAVs well-suited for a wide range of missions. However, achieving full autonomy requires the development of an automatic flight control system, a task that remains challenging due to the nonlinear nature of the system dynamics.

In the context of controller design, most automatic flight controllers are based on linearized aircraft models specific to certain flight conditions. Various linear control techniques can then be employed, often producing excellent results, as demonstrated by Beard and McLain [7], Etkin and Reid [8], and He et al. [9]. However, when the vehicle’s state deviates from the operating point used during the design process, the model becomes invalid, potentially leading to degraded controller performance. To overcome these limitations, nonlinear control techniques provide a more effective alternative.

The use of nonlinear control techniques is essential for developing appropriate controllers across the entire range of flight conditions. In the context of aircraft control, the most common approach consists in dividing the aircraft dynamics into lateral and longitudinal subsystems, each requiring different control strategies. Since longitudinal dynamics govern critical variables such as airspeed and flight path angle, their stabilization is typically prioritized before addressing lateral control. This is often achieved using two distinct controllers: one for airspeed and another for flight path angle. For instance, Harkegard and Torkel Glad [10] designed a flight path angle controller using Backstepping and applied Nonlinear Dynamic Inversion control (NDI) for airspeed. Similarly, Tran [11] derived a nonlinear control law for flight path angle and compared its performance with that of a Feedback Linearization Control technique. Likewise, Gavilan et al. [12] divided a longitudinal controller design into airspeed and flight path angle control, implementing an adaptive Backstepping approach that simultaneously estimates aerodynamic parameters. Although these controllers demonstrated excellent performance, their design processes relied on several assumptions that could potentially introduce inaccuracies.

When an accurate model is available and the relative degree condition is satisfied, Feedback Linearization Control can cancel strong nonlinearities, transforming the system into an equivalent linear one, where a wide range of linear control strategies can be applied [13,14]. Compared with the Backstepping method, FLC often requires fewer control gains and produces an explicit control law that clearly reveals how each input affects each output. In contrast, Backstepping introduces intermediate virtual controls, which can make the physical interpretation less direct. Moreover, the feasibility of FLC can be verified by inspecting the control matrix, whereas in Backstepping, feasibility typically becomes evident only after completing the recursive design process. Compared with Model Predictive Control (MPC) [15,16,17], FLC is less computationally demanding, as MPC requires solving an optimization problem at every sampling instant. Consequently, FLC is generally easier to implement on embedded systems with limited processing power.

Nevertheless, this method has specific requirements, one of which is that the system must have a well-defined relative degree [18]. If a system lacks a well-defined relative degree, the dynamic extension technique can be used to address this limitation. The dynamic extension technique extends the system model by adding additional state variables, thus transforming the original system into one with a well-defined relative degree that is more suitable for Feedback Linearization. A first attempt to apply dynamic extension to an aircraft model is presented in [18]. In that work, a ninth-order system is considered, with throttle, aileron, elevator, and rudder deflections as inputs, and airspeed, roll, pitch, and yaw angles as outputs. Other studies have focused solely on longitudinal dynamics using a fifth-order model, where altitude, airspeed, flight path angle, angle of attack, and pitch rate serve as state variables [19,20]. For this fifth-order model, when airspeed and altitude are treated as outputs and engine thrust along with elevator deflection as control inputs, the system similarly required a dynamic extension [21,22].

In the context of longitudinal dynamics, a well-known fourth-order dynamic model considers airspeed, flight path angle, pitch angle, and pitch rate as state variables. Compared to the fifth-order model, this reduced-order representation provides a more compact description of the system dynamics and is widely used in the literature [23,24]. Additionally, while previous studies employing the fifth-order model have used airspeed and altitude as outputs [19,20], the present work instead selects airspeed and flight path angle, aligning with standard formulations in aircraft control. Moreover, these variables are commonly utilized in outer-loop controllers for path following [25].

Achieving linearization in this model proves challenging due to the system’s unstable zero dynamics [26]. To address this, Portella and Goel [27] incorporated pitch angle as an additional output alongside airspeed and flight path angle, effectively eliminating the zero dynamics. However, in the present work, rather than introducing an extra output, dynamic extension is performed to provide the system with a well-defined relative degree, enabling Feedback Linearization while preserving the original two-input two-output structure.

As mentioned above, Feedback Linearization Control has been proposed in the literature for aircraft systems. While this paper also develops a nonlinear MIMO Feedback Linearization Controller for the simplified fourth-order longitudinal model, the main contribution lies in estimating the domain of attraction where the controller remains valid, an issue that has not been addressed in the literature. This region is particularly relevant for guiding the selection of appropriate airspeed and flight path angle references. The results demonstrate that not all combinations of reference values for airspeed and flight path angle are feasible: Some cause the control matrix to become singular, while others lead to undesirable behaviors, such as the need for negative thrust.

The remainder of the paper is organized as follows. Section 2 introduces the longitudinal aircraft model. Section 3 details the controller design process. Section 4 evaluates the controller’s performance through numerical simulations. Section 5 presents a selection criterion for reference values of airspeed and flight path angle to prevent conditions that can lead to instability. Section 6 outlines the requirements for the physical implementation of the controller. Finally, Section 7 provides the concluding remarks.

2. Longitudinal Aircraft Dynamics

When a symmetric aircraft is in level flight, climbing, or descending in the x–z plane, the longitudinal and lateral-directional motions are uncoupled to first order. Under this condition, the lateral-directional variables have a minimal influence on the longitudinal dynamics and can, therefore, be neglected. Consequently, the vehicle dynamics simplify to six equations that capture the longitudinal behavior of the aircraft.

$$\begin{array}{cc}\hfill \dot{x}& =Vcos\gamma \hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \dot{z}& =-Vsin\gamma \hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \dot{V}& ={\displaystyle \frac{1}{m}}\left(-D+Tcos\alpha -mgsin\gamma \right)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \dot{\gamma }& ={\displaystyle \frac{1}{mV}}\left(L+Tsin\alpha -mgcos\gamma \right)\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \dot{\theta }& =q\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \dot{q}& ={\displaystyle \frac{1}{J_y}}{\tau }_m\hfill \end{array}$$

(6)

where x and z denote the vehicle’s inertial position in the longitudinal plane; V represents the airspeed, $\gamma$ the flight path angle, $\theta$ the pitch angle, and q the pitch rate.

The relationship $\theta =\gamma +\alpha$ is used to incorporate the pitch angle $\theta$ as a state variable. Accordingly, the angle of attack is defined as

$$\alpha =\theta -\gamma$$

(7)

Additionally, m represents the mass, g denotes the gravitational acceleration constant, and $J_y$ denotes the moment of inertia about the $j^b$ axis.

The variable T represents the thrust; D and L denote the drag and lift forces, respectively; and ${\tau }_m$ denotes the pitch moment.

The flat-Earth assumption neglects the variation of gravitational acceleration with distance from the center of the Earth, and it disregards Coriolis and centrifugal effects. Position and velocity relative to the Earth are described using a tangent-plane Cartesian coordinate system. This approximation remains valid for horizontal displacements of up to a few hundred miles from the tangent-plane origin [23]; beyond this range, the Earth’s curvature begins to influence the vehicle dynamics. In such cases, the equations of motion become considerably more complex due to the Earth’s shape, its rotation, and the variation of gravity with position. Consequently, the Cartesian reference surface of the flat-Earth model must be replaced by a spherical or ellipsoidal surface.

Here, small variations in altitude $h=-z$ and range x are assumed, and their influence on the system dynamics is disregarded in accordance with the flat-Earth model. Under this assumption, the aircraft dynamics reduce to a set of four equations that accurately capture the dominant longitudinal dynamics [23,24]:

$$\begin{array}{cc}\hfill \dot{V}& ={\displaystyle \frac{1}{m}}\left(-D+Tcos\alpha -mgsin\gamma \right)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \dot{\gamma }& ={\displaystyle \frac{1}{mV}}\left(L+Tsin\alpha -mgcos\gamma \right)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \dot{\theta }& =q\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \dot{q}& ={\displaystyle \frac{1}{J_y}}{\tau }_m\hfill \end{array}$$

(11)

These equations describe the longitudinal dynamics of the aircraft, mainly governed by the external forces shown in Figure 1.

The drag D and lift L aerodynamic forces are generated as the aircraft moves through the air and primarily depend on the airspeed V, air density $\rho$, wing surface area S, and their corresponding aerodynamic coefficients:

$$\begin{array}{cc}\hfill D& ={\displaystyle \frac{1}{2}}\rho V^{2}S{C}_D\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill L& ={\displaystyle \frac{1}{2}}\rho V^{2}S{C}_L\hfill \end{array}$$

(13)

The aerodynamic coefficients are, in turn, functions of the angle of attack $\alpha$. Typical flight conditions are assumed, under which the flow over the wing remains attached; therefore, linear models are used [7]:

$$\begin{array}{cc}\hfill C_D& =C_{D_0}+C_{D_{\alpha }}\alpha \hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill C_L& =C_{L_0}+C_{L_{\alpha }}\alpha \hfill \end{array}$$

(15)

where the parameters $C_{D_0}$, $C_{L_0}$, $C_{D_{\alpha }}$, and $C_{L_{\alpha }}$ are related to the aerodynamic properties of the vehicle.

In the longitudinal plane, aircraft control is achieved through the thrust T and the aerodynamic moment ${\tau }_m$, which is modeled as

$${\tau }_m={\displaystyle \frac{1}{2}}\rho V^{2}Sc{C}_m$$

(16)

where c represents the mean aerodynamic chord, and $C_m$ is given by

$$C_m=C_{m_0}+C_{m_{\alpha }}\alpha +C_{m_q}q+C_{m_{{\delta }_e}}{\delta }_e$$

(17)

This coefficient depends on the angle of attack $\alpha$, pitch rate q, and the aerodynamic properties of the vehicle. However, its primary contribution comes from the elevator deflection ${\delta }_e$, which acts as the physical actuator in a real aircraft.

Substituting the corresponding expressions for D and L, the longitudinal model (8)–(11) can be rewritten in its expanded form:

$$\begin{array}{cc}\hfill \dot{V}& ={\displaystyle \frac{1}{m}}\left[-{\displaystyle \frac{1}{2}}\rho V^{2}S\left(C_{D_0}+C_{D_{\alpha }}\alpha \right)+Tcos\alpha -mgsin\gamma \right]\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \dot{\gamma }& ={\displaystyle \frac{1}{mV}}\left[{\displaystyle \frac{1}{2}}\rho V^{2}S\left(C_{L_0}+C_{L_{\alpha }}\alpha \right)+Tsin\alpha -mgcos\gamma \right]\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \dot{\theta }& =q\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \dot{q}& ={\displaystyle \frac{1}{J_y}}{\tau }_m\hfill \end{array}$$

(21)

For compactness, the angle of attack $\alpha$, defined in terms of the state variables $\theta$ and $\gamma$ as in (7), will be used throughout the remainder of the text. For design purposes, ${\tau }_m$ will be considered as the control input, although ${\delta }_e$ can later be computed using (16) and (17).

3. Controller Design

To begin the control design, the model (18)–(21) is rearranged following the standard state-space representation of a nonlinear system:

$$\begin{array}{cc}\hfill \dot{x}& =f\left(x\right)+g\left(x\right)u\hfill \\ \hfill y& =h\left(x\right)\hfill \end{array}$$

(22)

The state vector $x\in {\mathbb{R}}^4$ is defined as

$$x={[V,\hspace{1em}\gamma ,\hspace{1em}\theta ,\hspace{1em}q]}^{\top }$$

(23)

In this case, the goal is to regulate the magnitude and direction of the velocity vector:

$$y=h\left(x\right)=\left[\begin{array}{c}{h}_1\left(x\right)\\ h_2\left(x\right)\end{array}\right]=\left[\begin{array}{c}V\\ \gamma \end{array}\right]$$

(24)

Considering the control inputs:

$$u=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]=\left[\begin{array}{c}T\\ {\tau }_m\end{array}\right]$$

(25)

In this way, the vector $f\left(x\right)\in {\mathbb{R}}^4$ and the matrix $g\left(x\right)\in {\mathbb{R}}^{4\times 2}$ are expressed as follows:

$$f\left(x\right)=\left[\begin{array}{c}{\displaystyle \frac{1}{m}}\left[-{\displaystyle \frac{1}{2}}\rho V^{2}S\left(C_{D_0}+C_{D_{\alpha }}\alpha \right)-mgsin\gamma \right]\\ \\ {\displaystyle \frac{1}{mV}}\left[{\displaystyle \frac{1}{2}}\rho V^{2}S\left(C_{L_0}+C_{L_{\alpha }}\alpha \right)-mgcos\gamma \right]\\ \\ q\\ \\ 0\end{array}\right],g\left(x\right)=\left[\begin{array}{cc}{\displaystyle \frac{1}{m}}cos\alpha & 0\\ & \\ {\displaystyle \frac{1}{mV}}sin\alpha & 0\\ & \\ 0& 0\\ & \\ 0& {\displaystyle \frac{1}{J_y}}\end{array}\right]$$

(26)

3.1. Dynamic Extension

At this stage, it is worth mentioning that a key requirement of Feedback Linearization Control is that the system must have a well-defined relative degree. According to [18], a Multi-Input Multi-Output system of the form (22) has a well-defined vector relative degree $\left\{r_1,\dots ,r_m\right\}$ if the following conditions are satisfied:

1.

The matrix $L_{g_j}{L}_f^k{h}_i\left(x\right)\in {\mathbb{R}}^{m\times m}$ is a zero matrix for all $1\le j\le m$, for all $k<r_i-1$, for all $1\le i\le m$, and for all x in a neighborhood of $x^{\circ }$.

2.

The matrix $A\left(x\right)\in {\mathbb{R}}^{m\times m}$ is nonsingular at $x=x^{\circ }$, where

$$\begin{array}{cc}\hfill A\left(x\right)& =\left[\begin{array}{ccc}{L}_{g_1}{L}_f^{r_1-1}{h}_1\left(x\right)& \dots & L_{g_m}{L}_f^{r_1-1}{h}_1\left(x\right)\\ ⋮& \ddots & ⋮\\ L_{g_1}{L}_f^{r_m-1}{h}_m\left(x\right)& \dots & L_{g_m}{L}_f^{r_m-1}{h}_m\left(x\right)\end{array}\right]\hfill \end{array}$$

(27)

By evaluating the corresponding Lie derivatives, it becomes evident that the system (18)–(21) lacks a vector relative degree, since the matrix $L_{g}h\left(x\right)$ is singular, as shown in (28); consequently, the system cannot be directly linearized via state feedback.

$$L_{g}h\left(x\right)=\left[\begin{array}{cc}{\displaystyle \frac{1}{m}}cos\alpha & 0\\ & \\ {\displaystyle \frac{1}{mV}}sin\alpha & 0\end{array}\right]$$

(28)

The absence of a vector relative degree can be linked to the fact that the lowest derivatives of $y_1$ and $y_2$, namely, $\dot{V}$ and $\dot{\gamma }$, are both affected by T and neither are affected by ${\tau }_m$. To obtain a vector relative degree, it is necessary to render $\dot{V}$ and $\dot{\gamma }$ independent of T and somehow delay the appearance of T in higher derivatives of V and $\gamma$, with the expectation that ${\tau }_m$ will also appear when this occurs.

To achieve this, a dynamic extension is performed in which the thrust is defined as the output of an auxiliary dynamical system. For this, the following change of variable is introduced:

$${\xi }_1=T$$

(29)

Here, a double integrator is used to drive the output ${\xi }_1$ via the new control input $v_1$:

$$\begin{array}{cc}\hfill {\dot{\xi }}_1& ={\xi }_2\hfill \\ \hfill {\dot{\xi }}_2& =v_1\hfill \end{array}$$

(30)

For consistency in notation, let $v_2={\tau }_m$. The extended model incorporating the auxiliary variables is then given by

$$\begin{array}{cc}\hfill \dot{V}& ={\displaystyle \frac{1}{m}}\left[-{\displaystyle \frac{1}{2}}\rho V^{2}S\left(C_{D_0}+C_{D_{\alpha }}\alpha \right)+{\xi }_{1}cos\alpha -mgsin\gamma \right]\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \dot{\gamma }& ={\displaystyle \frac{1}{mV}}\left[{\displaystyle \frac{1}{2}}\rho V^{2}S\left(C_{L_0}+C_{L_{\alpha }}\alpha \right)+{\xi }_{1}sin\alpha -mgcos\gamma \right]\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill \dot{\theta }& =q\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \dot{q}& ={\displaystyle \frac{1}{J_y}}{v}_2\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\dot{\xi }}_1& ={\xi }_2\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\dot{\xi }}_2& =v_1\hfill \end{array}$$

(36)

This new model can be rearranged in the form of (22), with the state vector $x\in {\mathbb{R}}^6$ now containing the auxiliary variables.

$$x={\left[\begin{array}{cccccc}V,& \gamma ,& \theta ,& q,& {\xi }_1,& {\xi }_2\end{array}\right]}^{\top }$$

(37)

The new input vector $v\in {\mathbb{R}}^2$ is defined as

$$v=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]=\left[\begin{array}{c}\ddot{T}\\ {\tau }_m\end{array}\right]$$

(38)

While the output vector $y\in {\mathbb{R}}^2$ remains unchanged:

$$y=h\left(x\right)=\left[\begin{array}{c}{h}_1\left(x\right)\\ h_2\left(x\right)\end{array}\right]=\left[\begin{array}{c}V\\ \gamma \end{array}\right]$$

(39)

The vector $f\left(x\right)\in {\mathbb{R}}^6$ and the matrix $g\left(x\right)\in {\mathbb{R}}^{6\times 2}$ are given by

$$f\left(x\right)=\left[\begin{array}{c}{\displaystyle \frac{1}{m}}\left[-{\displaystyle \frac{1}{2}}\rho V^{2}S\left(C_{D_0}+C_{D_{\alpha }}\alpha \right)+{\xi }_{1}cos\alpha -mgsin\gamma \right]\\ \\ {\displaystyle \frac{1}{mV}}\left[{\displaystyle \frac{1}{2}}\rho V^{2}S\left(C_{L_0}+C_{L_{\alpha }}\alpha \right)+{\xi }_{1}sin\alpha -mgcos\gamma \right]\\ \\ q\\ \\ 0\\ \\ {\xi }_2\\ \\ 0\end{array}\right],g\left(x\right)=\left[\begin{array}{ccc}0& & 0\\ & & \\ 0& & 0\\ & & \\ 0& & 0\\ & & \\ 0& & {\displaystyle \frac{1}{J_y}}\\ & & \\ 0& & 0\\ & & \\ 1& & 0\end{array}\right]$$

(40)

The corresponding Lie derivatives for this system are computed as

$$\begin{array}{cc}\hfill L_{g}h\left(x\right)& =0_{2\times 2}\hfill \\ \hfill L_g{L}_{f}h\left(x\right)& =0_{2\times 2}\hfill \\ \hfill L_g{L}_f^{2}h\left(x\right)& =\left[\begin{array}{cc}{\displaystyle \frac{1}{m}}cos\alpha & -{\displaystyle \frac{1}{m{J}_y}}\left({\displaystyle \frac{1}{2}}\rho V^{2}S{C}_{D_{\alpha }}+{\xi }_{1}sin\alpha \right)\\ & \\ {\displaystyle \frac{1}{mV}}sin\alpha & {\displaystyle \frac{1}{mV{J}_y}}\left({\displaystyle \frac{1}{2}}\rho V^{2}S{C}_{L_{\alpha }}+{\xi }_{1}cos\alpha \right)\end{array}\right]\hfill \end{array}$$

(41)

Since the matrix $L_g{L}_f^{2}h\left(x\right)\in {\mathbb{R}}^{2\times 2}$ has full rank, it follows that the extended model (31)–(36) has a well-defined vector relative degree, represented by $\{r_1,r_2\}=\{3,3\}$. Given that the extended system has dimension $n=6$, the condition $r_1+r_2=n$ is satisfied, and thus, the model can be linearized through state feedback [18,28].

3.2. Feedback Linearization Control

Once the relative degree requirement is satisfied, a control law based on Feedback Linearization is designed. To proceed, the input–output map of the system (31)–(36) is defined:

$$\left[\begin{array}{c}{y}_1^{\left(3\right)}\\ \\ y_2^{\left(3\right)}\end{array}\right]=\left[\begin{array}{c}{L}_f^3{h}_1\left(x\right)\\ \\ L_f^3{h}_2\left(x\right)\end{array}\right]+\left[\begin{array}{cc}{L}_{g_1}{L}_f^2{h}_1\left(x\right)& L_{g_2}{L}_f^2{h}_1\left(x\right)\\ & \\ L_{g_1}{L}_f^2{h}_2\left(x\right)& L_{g_2}{L}_f^2{h}_2\left(x\right)\end{array}\right]\left[\begin{array}{c}{v}_1\\ \\ v_2\end{array}\right]$$

(42)

Equivalently, in compact form,

$$y^{\left(3\right)}=L_f^{3}h\left(x\right)+L_g{L}_f^{2}h\left(x\right)v$$

(43)

The vector $L_f^{3}h\left(x\right)\in {\mathbb{R}}^2$ is obtained by calculating the corresponding Lie derivatives:

$$\begin{array}{cc}\hfill L_f^3{h}_1\left(x\right)& =-{\displaystyle \frac{1}{m}}\left[\rho S{C}_D({\dot{V}}^2+V\ddot{V})+2\left(\rho VS{C}_{D_{\alpha }}\dot{V}+{\xi }_{2}sin\alpha \right)(q-\dot{\gamma })+{\xi }_1{(q-\dot{\gamma })}^{2}cos\alpha \right.\hfill \\ \hfill & \left.-\left({\displaystyle \frac{1}{2}}\rho V^{2}S{C}_{D_{\alpha }}+{\xi }_{1}sin\alpha \right)\ddot{\gamma }-mg({\dot{\gamma }}^{2}sin\gamma -\ddot{\gamma }cos\gamma )\right]\hfill \\ \hfill L_f^3{h}_2\left(x\right)& ={\displaystyle \frac{1}{mV}}\left[\rho S{C}_L({\dot{V}}^2+V\ddot{V})+2\left(\rho VS{C}_{L_{\alpha }}\dot{V}+{\xi }_{2}cos\alpha \right)(q-\dot{\gamma })-{\xi }_1{(q-\dot{\gamma })}^{2}sin\alpha \right.\hfill \\ \hfill & \left.-\left({\displaystyle \frac{1}{2}}\rho V^{2}S{C}_{L_{\alpha }}+{\xi }_{1}cos\alpha \right)\ddot{\gamma }+mg({\dot{\gamma }}^{2}cos\gamma +\ddot{\gamma }sin\gamma )-m(\ddot{V}\dot{\gamma }+2\dot{V}\ddot{\gamma })\right]\hfill \end{array}$$

(44)

The entries of the matrix $L_g{L}_f^{2}h\left(x\right)\in {\mathbb{R}}^{2\times 2}$ were computed previously in (41):

$$\begin{array}{cc}\hfill L_{g_1}{L}_f^2{h}_1\left(x\right)& ={\displaystyle \frac{1}{m}}cos\alpha \hfill \\ \hfill L_{g_2}{L}_f^2{h}_1\left(x\right)& =-{\displaystyle \frac{1}{m{J}_y}}\left({\displaystyle \frac{1}{2}}\rho V^{2}S{C}_{D_{\alpha }}+{\xi }_{1}sin\alpha \right)\hfill \\ \hfill L_{g_1}{L}_f^2{h}_2\left(x\right)& ={\displaystyle \frac{1}{mV}}sin\alpha \hfill \\ \hfill L_{g_2}{L}_f^2{h}_2\left(x\right)& ={\displaystyle \frac{1}{mV{J}_y}}\left({\displaystyle \frac{1}{2}}\rho V^{2}S{C}_{L_{\alpha }}+{\xi }_{1}cos\alpha \right)\hfill \end{array}$$

(45)

The Equation (43) can be transformed into a pair of third-order chains of integrators by applying the following state-feedback control law:

$$v={\left[L_g{L}_f^{2}h\left(x\right)\right]}^{-1}[\nu -L_f^{3}h\left(x\right)]$$

(46)

where $\nu \in {\mathbb{R}}^2$ can be interpreted as an outer-loop control input that can be designed using linear control techniques, such as pole placement. As a result, Equation (43) reduces to

$$\begin{array}{cc}\hfill V^{\left(3\right)}& ={\nu }_1\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\gamma }^{\left(3\right)}& ={\nu }_2\hfill \end{array}$$

(48)

which represent the linearized dynamics of airspeed and flight path angle.

The validity of the controller (46) depends entirely on the invertibility of the matrix $L_g{L}_f^{2}h\left(x\right)$, whose determinant must be nonzero:

$$det|L_g{L}_f^{2}h\left(x\right)|={\displaystyle \frac{1}{m^{2}V{J}_y}}\left[{\displaystyle \frac{1}{2}}\rho V^{2}S(C_{L_{\alpha }}cos\alpha +C_{D_{\alpha }}sin\alpha )+{\xi }_1\right]$$

(49)

3.3. State-Space Representation of the Linearized Subsystems

Considering (47) and (48), the state vectors are defined as follows:

$$\begin{array}{cc}\hfill {\zeta }_1& ={\left[\begin{array}{ccc}V& \dot{V}& \ddot{V}\end{array}\right]}^{\top }\hfill \\ \hfill {\zeta }_2& ={\left[\begin{array}{ccc}\gamma & \dot{\gamma }& \ddot{\gamma }\end{array}\right]}^{\top }\hfill \end{array}$$

(50)

The linearized subsystems in the state-space representation are then given by

$${\dot{\zeta }}_i=A_i{\zeta }_i+b_i{\nu }_i$$

(51)

where

$$A_i=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right],b_i=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],i=1,2.$$

(52)

The desired state vectors are defined similarly:

$$\begin{array}{cc}\hfill {\zeta }_1^d& ={\left[\begin{array}{ccc}{V}_d& {\dot{V}}_d& {\ddot{V}}_d\end{array}\right]}^{\top }\hfill \\ \hfill {\zeta }_2^d& ={\left[\begin{array}{ccc}{\gamma }_d& {\dot{\gamma }}_d& {\ddot{\gamma }}_d\end{array}\right]}^{\top }\hfill \end{array}$$

(53)

Using (50) and (53), the tracking error is computed:

$$e_i={\zeta }_i-{\zeta }_i^d$$

(54)

The error dynamics is obtained by taking the time derivative:

$${\dot{e}}_i={\dot{\zeta }}_i-{\dot{\zeta }}_i^d$$

(55)

The desired trajectory dynamics for both systems is assumed to take the following form:

$${\dot{\zeta }}_i^d=A_i{\zeta }_i^d$$

(56)

Thus, substituting Equations (51) and (56) into (55) gives

$$\begin{array}{cc}\hfill {\dot{e}}_i& =A_i{\zeta }_i+b_i{\nu }_i-A_i{\zeta }_i^d\hfill \\ \hfill & =A_i({\zeta }_i-{\zeta }_i^d)+b_i{\nu }_i\hfill \\ \hfill & =A_i{e}_i+b_i{\nu }_i\hfill \end{array}$$

(57)

To stabilize the error dynamics, the following control law is proposed:

$${\nu }_i=-k_i^{\top }{e}_i$$

(58)

Therefore, the closed-loop error dynamics is obtained:

$${\dot{e}}_i=(A_i-b_i{k}_i^{\top })e_i$$

(59)

To ensure asymptotic stability, the gains $k_i^{\top }$ are selected so that the matrices $(A_i-b_i{k}_i^{\top })$ are Hurwitz. The parameters of the outer-loop controller (58) consist of the two vectors $k_i^{\top }$, together with the desired values of airspeed and flight path angle. The gains $k_i^{\top }$ determine the closed-loop behavior of the tracking errors $e_i$. Fast convergence of $e_i$ to zero requires large control inputs, and likewise, large desired values of airspeed or flight path angle also demand large control inputs. Consequently, bounds on the control inputs impose limits on the allowable desired values of airspeed and flight path angle. Furthermore, since the thrust control input is strictly positive, a negative desired flight path angle is feasible only with an increase in airspeed; otherwise, the airplane will dive out of control.

The complete control framework for stabilizing the longitudinal dynamics (18)–(21) is illustrated in Figure 2. The errors $e_1$ and $e_2$ are obtained by subtracting the desired values ${\zeta }_1^d$ and ${\zeta }_2^d$ from the current states of the linearized subsystems ${\zeta }_1$ and ${\zeta }_2$. These errors are used by the outer-loop controller (58) to compute ${\nu }_1$ and ${\nu }_2$. The resulting signals are then fed to the inner-loop controller (46), and, together with the system state x and the auxiliary variables ${\xi }_1$ and ${\xi }_2$, are used to compute $v_1$ and $v_2$, acting as the control inputs for the extended model. The virtual control input $v_1$ must be integrated twice to obtain T, which, together with ${\tau }_m$, forms the input vector u for the real system. The state vector x is provided to the inner-loop controller and is also used in the coordinate transformation to compute ${\zeta }_1$ and ${\zeta }_2$, thus completing the feedback loop.

4. Simulation Results

In this section, the performance of the proposed control strategy is evaluated through numerical simulations. This analysis enables the assessment of the system’s behavior under different scenarios, including variations in the initial conditions and reference values. It is assumed that the full system state is available.

The high-order derivatives required for computing the control law (46) are obtained by analytically differentiating the output vector (39). The vehicle parameters, listed in

Table 1. Physical and aerodynamic parameters of the Aerosonde UAV, sourced from [7].

Parameter	Description	Value	Units
m	Vehicle mass	13.5	kg
g	Gravitational constant	9.81	m/s2
$J_y$	Inertia along $j^b$ axis	1.135	kg· m2
$\rho$	Air density	1.2682	kg/m3
S	Wing area	0.55	m2
c	Mean chord	0.18994	m
$C_{D_0}$	Zero-lift drag coefficient	0.03	dimensionless coefficient
$C_{D_{\alpha }}$	Drag coefficient due $\alpha$	0.3	dimensionless coefficient
$C_{L_0}$	Zero $\alpha$ lift coefficient	0.28	dimensionless coefficient
$C_{L_{\alpha }}$	Lift coefficient due to $\alpha$	3.45	dimensionless coefficient
$C_{m_0}$	Zero $\alpha$ moment coefficient	−0.02338	dimensionless coefficient
$C_{m_{\alpha }}$	Moment due to $\alpha$	−0.38	dimensionless coefficient
$C_{m_q}$	Moment due to q	−3.6	dimensionless coefficient
$C_{m_{{\delta }_e}}$	Moment due to ${\delta }_e$	−0.5	dimensionless coefficient
${\alpha }_{stall}$	Stall angle	24.07	∘
$T_{max}$	Maximum thrust	150	N

, correspond to those of the Aerosonde UAV. The simulation procedure follows the block diagram shown in Figure 2.

4.1. Performance in Reference Tracking

For the airspeed subsystem, the outer-loop gain vector $k_1^{\top }=[9,22.5,9.5]$ was selected by trial and error so that the closed-loop poles of the linearized system are located at ${\lambda }_1=[-0.5,-6,-3]$, ensuring that the actuators do not saturate while maintaining an acceptable airspeed response.

In most applications, it is desirable to maintain a constant airspeed; therefore, the desired value is set to 175 m/s. Starting from 160 m/s, the control algorithm induces a step change of 15 m/s. As shown in the top plot of Figure 3, the system reaches the reference value in approximately 10 s, without overshoot or steady-state error.

Similarly, for the flight path angle subsystem, the outer-loop gain vector $k_2^{\top }=[13,31,10.5]$ was selected by trial and error so that the closed-loop poles of the linearized system are located at ${\lambda }_2=[-5\pm 1i,-0.5]$.

For this case, a sinusoidal reference signal with an amplitude of ${20}^{\circ }$ was adopted. As shown in the bottom plot of Figure 3, the proposed controller achieves accurate tracking of the desired value with a settling time of approximately 5 s.

In Figure 4, the control input signals required to drive the outputs to their reference values are shown. As can be seen, thrust demand increases during the climb phase to compensate for the weight component when the flight path angle is positive. Conversely, during the descent phase, thrust is reduced as gravity assists the maneuver. As shown, thrust remains within its physical limits.

For the pitch moment, an initial overshoot is observed, which can be attributed to the increased energy demand of the controller as it drives the vehicle from its initial state to the reference values. After the transient, the pitch moment settles to zero.

4.2. Robustness Under Realistic Conditions

To assess the robustness of the proposed control strategy, numerical simulations were performed considering realistic conditions, including density changes, sensor noise, and external wind.

In the design process, all parameters were assumed constant and independent of environmental effects. In practice, however, this assumption does not always hold. For instance, when air is modeled as an ideal gas, its density depends on pressure and temperature. At higher altitudes, both pressure and temperature decrease, leading to a rapid reduction in air density. To evaluate the controller’s performance under these conditions, the atmospheric model in [29] is used to simulate density variations with altitude. The corresponding system response is shown in Figure 5.

The commanded references cause the vehicle to perform altitude variations, describing a bell-shaped trajectory in the inertial reference frame. This, in turn, induces variations in air density. As can be seen in Figure 5, when the aircraft reaches its highest point, the air density drops to a minimum. This unavoidably degrades the controller’s performance, since the proposed controller relies entirely on the model for exact cancellation of the nonlinear terms, and parameter variations were not considered during the design process. As a result, steady-state errors arise, which could be mitigated by adopting a robust version of the proposed controller.

To emulate sensor measurements, Gaussian noise with zero mean and variance $0.01$ was added to the state signals. As shown in Figure 6, the system is still able to track the reference values in the presence of noise. In practice, however, sensor data are typically filtered before being used by the controller to improve measurement accuracy and performance.

In a more realistic scenario, the controller is evaluated under atmospheric disturbances, such as wind. To emulate external wind, the Dryden turbulence model is employed for altitudes ranging from low to medium, with light turbulence conditions [7]. As shown in Figure 7, the system demonstrates satisfactory performance even under these adverse conditions.

5. Analysis of the Region of Attraction

The proposed control algorithm is based on Feedback Linearization and has been previously addressed in references such as [18,21]. However, several issues arise regarding the application of this method, such as actuator saturation due to the introduction of additional integrators. These studies have not ensured that the vehicle thrust remains within the acceptable range, $0<T<T_{max}$, for given reference values. Furthermore, no analysis has been provided on the conditions under which the determinant in (49) remains sufficiently far from zero, as required by the control law in (46).

To address these issues, a method is proposed for selecting appropriate reference values of V and $\gamma$ so that the thrust remains within its physical limits. First, it is assumed that the desired reference values, $V_d$ and ${\gamma }_d$, are constant. The analysis is carried out in the permanent regime, i.e., when $V=V_d$ and $\gamma ={\gamma }_d$. It should be understood that if, initially, V differs from $V_d$ and/or $\gamma$ differs from ${\gamma }_d$, a transient will occur. Nevertheless, due to the closed-loop stability, V and $\gamma$ will eventually converge to their desired values. Consequently, if the initial errors are small, the properties of the permanent regime also hold during the transient.

Taking the above into account, and noting that ${\xi }_1$ and T can be used interchangeably, Equations (31) and (32) take the following form:

$$\begin{array}{cc}\hfill -{\displaystyle \frac{1}{2}}\rho V^{2}S(C_{D_0}+C_{D_{\alpha }}\alpha )+Tcos\alpha -mgsin\gamma & =0\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\rho V^{2}S(C_{L_0}+C_{L_{\alpha }}\alpha )+Tsin\alpha -mgcos\gamma & =0\hfill \end{array}$$

(61)

This means that the aircraft has reached the desired values of V and $\gamma$, and is, therefore, in steady-state flight with constant airspeed and flight path angle, either climbing ($\gamma >0$), descending ($\gamma <0$), or cruising ($\gamma =0$) [29]. In this steady-state regime, both the thrust T and the pitch angle $\theta$ remain constant.

For the purpose of estimating the region of attraction, a numerical simulation is performed. In Equations (60) and (61), V and $\gamma$ are treated as known variables, while T and $\alpha$ are treated as unknowns. Under these conditions, the expressions form a system of nonlinear equations, whose solutions cannot be obtained analytically but can be determined numerically.

Accordingly, the simulation is conducted over the ranges $\gamma \in {[-90,90]}^{\circ }$ and $V\in [15,300]$ m/s. Solving the system of Equations (60) and (61) yields two solutions, corresponding to T and $\alpha$, as shown in Figure 8 and Figure 9.

Figure 8a presents a two-dimensional representation of the thrust T, with its magnitude indicated by the color scale. As observed, low airspeed combined with small flight path angles requires relatively low thrust, whereas high airspeed combined with large flight path angles demands significantly higher thrust.

Also in Figure 8b, two contour curves are drawn representing the minimum and maximum thrust values that the aircraft motor can deliver. Since a vehicle with a conventional configuration is considered, only positive thrust ($T>0$) can be produced. It is assumed that the maximum thrust is limited to $T<150$ N. Both curves enclose a region for T, which allows setting reference values for V and $\gamma$ without exceeding the thrust constraints.

In the same plot, it can be observed that at low speeds, the aircraft can perform steep climbs, as there are no constraints for positive values of $\gamma$. However, under the same conditions, aggressive descents are not permissible. As airspeed increases, smaller values of $\gamma$ are allowable, enabling faster descents.

It is advisable to select values of V and $\gamma$ that are physically feasible. As an example, a range for $\gamma$ is established; specifically, $\gamma \in {[-20,20]}^{\circ }$ is considered an adequate range. This, in turn, defines a suitable range for V, resulting in $V\in [150,200]$ m/s. The corresponding region is shown in Figure 8b.

Regarding the solution for $\alpha$ shown in Figure 9, it can be observed that smaller values of V require larger angles of attack, eventually exceeding limits that may lead to adverse conditions such as stall. Therefore, in addition to selecting values within the region defined in Figure 8b, it is also important to consider values of V that result in acceptable angles of attack, as indicated in Figure 9b.

Based on the computed values of T and $\alpha$ shown in Figure 8 and Figure 9, the determinant of the control matrix (49) can be evaluated as a function of V, T, and $\alpha$, where ${\xi }_1$ is equal to T.

This evaluation provides insights into the validity of the proposed controller in (46) for given reference values of V and $\gamma$. The results are presented in Figure 10.

As shown in Figure 10b, when smaller values of V are considered, the determinant approaches zero, which may lead to singularities in the control matrix. A possible physical interpretation of this behavior is that at low speeds, the aircraft is unable to produce enough lift, potentially resulting in a loss of control. This idea is supported by the behavior of the angle of attack in Figure 9b, where a higher demand for $\alpha$ is observed at lower speeds in order to maintain lift. Furthermore, Figure 10b also shows that $\gamma$ does not have a significant influence on the validity of the controller, since for a given airspeed, the value of the determinant remains almost the same across the entire range of $\gamma$. This is confirmed by the fact that Expression (49) does not explicitly depend on the flight path angle.

In the previous analysis, it was assumed that the desired reference values are constant, that is, ${\dot{V}}_d=0$ and ${\dot{\gamma }}_d=0$. Nevertheless, there are exceptions. For example, in Figure 3, the reference value for the flight path angle is slowly time-varying. Likewise, Figure 3 shows that both V and $\gamma$ exhibit variations during the transient state. Therefore, it is recommended to maintain smooth transitions of the state variables by appropriately tuning the control gains. The main objective is to ensure that the rates of change of the state variables remain sufficiently small so that the initial assumptions remain valid.

5.1. Lyapunov Stability Analysis

With an appropriate choice of the vector gains $k_i^{\top }$ in (58), the matrices $(A_i-b_i{k}_i^{\top })$ become Hurwitz, ensuring that $P_i\in {\mathbb{R}}^{3\times 3}$ are the unique solutions of the Lyapunov equation [28]. Thus, to test the stability of both the airspeed and flight path angle control systems, a quadratic Lyapunov function is proposed:

$$V_i\left(e_i\right)=e_i^{\top }{P}_i{e}_i>0$$

(62)

By differentiating $V_i$ along the trajectories of (59), the following is obtained:

$${\dot{V}}_i\left(e_i\right)=-e_i^{\top }{Q}_i{e}_i<0$$

(63)

where $Q_i\in {\mathbb{R}}^{3\times 3}$ are real symmetric positive definite matrices.

The proposed Lyapunov function (62) has the following property:

$${\lambda }_{\min }\left(P_i\right){\parallel e_i\parallel }^2\le V_i\left(e_i\right)\le {\lambda }_{\max }\left(P_i\right){\parallel e_i\parallel }^2$$

(64)

This property is useful for establishing the following inequalities:

$$\begin{array}{cc}\hfill {\lambda }_{\min }\left(P_1\right){(V-V_d)}^2& \le {\lambda }_{\min }\left(P_1\right){\parallel e_1\parallel }^2\hfill \\ \hfill & \le V_1\left(e_1\right)\hfill \\ \hfill & \le V_1\left(e_1\left(0\right)\right)\hfill \end{array}$$

(65)

and

$$\begin{array}{cc}\hfill {\lambda }_{\min }\left(P_2\right){(\gamma -{\gamma }_d)}^2& \le {\lambda }_{\min }\left(P_2\right){\parallel e_2\parallel }^2\hfill \\ \hfill & \le V_2\left(e_2\right)\hfill \\ \hfill & \le V_2\left(e_2\left(0\right)\right)\hfill \end{array}$$

(66)

The choice of $e_1\left(0\right)$ and $e_2\left(0\right)$ in expressions (65) and (66), respectively, should be guided by Figure 8b. The objective is to stabilize the system around a desired point in the plane; for example, $(V_d,{\gamma }_d)=(175,0)$ was selected. The initial error states must satisfy the following conditions:

$$\begin{array}{cc}\hfill |V-V_d|& <{\displaystyle \frac{\epsilon }{2}}\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill |\gamma -{\gamma }_d|& <{\displaystyle \frac{\ell }{2}}\hfill \end{array}$$

(68)

This ensures that the constraints established for T are not violated. Under these conditions, a region of attraction is defined with $\epsilon =50$ m/s and $\ell =40$ degrees, within which the vehicle operates safely. Furthermore, after establishing that $V_1$ and $V_2$ are decreasing, as shown in (63), the error vectors $e_1$ and $e_2$ asymptotically converge to zero. Consequently, the values of V and $\gamma$ converge to their reference values.

5.2. Controller Performance in Different Flight Scenarios

To better evaluate the controller’s performance, numerical simulations were conducted for climb, cruise, and descent phases. Figure 11 shows the results.

The aircraft begins in level flight at $V=100$ m/s. After a time interval, a step change of ${\gamma }_d={20}^{\circ }$ in the flight path angle is commanded, inducing a climb, followed by the vehicle’s return to level flight. As shown in Figure 8b, performing a steep descent requires higher airspeed. For this purpose, the airspeed is gradually increased to $V=175$ m/s. Upon reaching this speed, a descent maneuver is executed by commanding ${\gamma }_d=-{20}^{\circ }$, followed by a return to level flight. Finally, the airspeed is gradually reduced to $V=100$ m/s, restoring the aircraft to its initial state.

As can be seen, the controller demonstrates good performance across the three phases throughout the entire maneuver. The thrust remains within its limits in each phase for the given reference values. However, thrust peaks occur during changes in airspeed. To address this, a thrust saturation constraint is applied within its physical limits. The results, shown in Figure 12, indicate that this constraint does not significantly degrade system performance. The only noticeable difference is a less smooth vehicle deceleration.

5.3. Stall Prevention

By assuming that the aircraft is in steady-state cruise level flight, meaning it has reached desired values for V and $\gamma$, and considering small values of the angle of attack, the following balance of forces is obtained from (18) and (19):

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\rho V^{2}S{C}_D-T& =0\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\rho V^{2}S{C}_L-mg& =0\hfill \end{array}$$

(70)

These equations describe an equilibrium condition in which thrust compensates drag and lift compensates weight.

The stall angle ${\alpha }_{stall}$ is defined as the angle of attack on the $\alpha$ versus $C_L$ curve at which the maximum lift coefficient $C_{L_{max}}$ occurs [7]. Thus, from (15), $C_{L_{max}}$ can be defined as

$$C_{L_{max}}=C_{L_0}+C_{L_{\alpha }}{\alpha }_{stall}$$

(71)

which can be rearranged as

$${\alpha }_{stall}={\displaystyle \frac{C_{L_{max}}-C_{L_0}}{C_{L_{\alpha }}}}$$

(72)

In order to prevent stall, the following constraint is established:

$$\alpha \le {\alpha }_{stall}$$

(73)

Consequently, the stall constraint sets an upper bound on the aircraft angle of attack. Using (70), this constraint can also be expressed in terms of airspeed:

$$V_{stall}=\sqrt{{\displaystyle \frac{2mg}{\rho S{C}_{L_{max}}}}}$$

(74)

In this representation, the stall constraint imposes a minimum allowable airspeed:

$$V_{stall}\le V$$

(75)

This behavior is illustrated in Figure 9b. As can be seen, decreasing airspeed causes the angle of attack to increase, eventually exceeding ${\alpha }_{stall}$. For the platform considered, the stall angle is approximately ${\alpha }_{stall}\approx 24.{07}^{\circ }$, corresponding to $C_{L_{max}}\approx 1.73$ and $V_{stall}\approx 14.82$ m/s. Therefore, to prevent stall, the reference values of V must be selected such that $V>14.82$ m/s.

6. Implementation Requirements

The proposed controller (46) requires precise knowledge of the system’s dynamics to effectively cancel the nonlinear terms. To this end, it is essential to have access to the vehicle’s state and parameters. In this case, as shown in Equations (18)–(21), measurement of the state vector (23) is required. Airspeed V is typically measured using a Pitot tube. To our best knowledge, no sensor exists for directly measuring the flight path angle $\gamma$; however, it can be determined indirectly using the relation $\gamma =\theta -\alpha$ from (7). The angle of attack $\alpha$ can be obtained using airflow vanes as in [30]; the pitch angle $\theta$ together with the pitch rate q can be measured with an Inertial Measurement Unit (IMU). In addition to the vehicle’s state, the proposed control algorithm requires knowledge of the auxiliary variables ${\xi }_1$ and ${\xi }_2$, as illustrated in the block diagram in Figure 2. These variables represent the thrust and the thrust rate, respectively, and can be obtained by numerically integrating the control input $v_1$.

The vehicle’s physical parameters, such as mass, wing area, mean aerodynamic chord, and moment of inertia, can be determined without major difficulty. Aerodynamic data for various airfoils, used to obtain $C_D$, $C_L$, and $C_m$ as functions of the angle of attack, can be found in [31]. The availability of these measurements will enable the practical implementation of the proposed controller in an embedded system. However, before deploying it on a physical vehicle, it is advisable to conduct Software-In-the-Loop (SIL) or Hardware-In-the-Loop (HIL) tests to minimize the risk of accidents. When implementing the control algorithm on hardware, it is advisable to apply a first-order low-pass filter with a cutoff frequency of around 5 Hz to attenuate high-frequency sensor noise without significantly affecting the system’s dynamic response.

7. Conclusions

The proposed nonlinear Feedback Linearization Controller successfully maintained aircraft stability during cruise flight, enabling control of both the magnitude and direction of the velocity vector. Notably, the controller achieved simultaneous stabilization of airspeed and flight path angle using engine thrust and pitch moment as control inputs. The adoption of a Multi-Input Multi-Output control strategy proved to be effective, yielding satisfactory results and demonstrating its ability to handle the nonlinear longitudinal dynamics of the aircraft.

The key contribution of this work is the estimation of the region of attraction in which the proposed controller remains valid. It was found that the stability of the longitudinal dynamics of an aircraft depends on both the airspeed and the flight path angle. The analysis showed that as the airspeed decreases, the matrix used in computing the control input tends to become singular. This study is useful for selecting appropriate reference values for airspeed and flight path angle to avoid singularities and undesirable behaviors.

Future Work

The proposed approach can be improved by incorporating aerodynamic models that account for behaviors beyond the stall angle, considering altitude variations, which directly affect air density, and accounting for external wind disturbances. Implementing FLC on hardware requires precise knowledge of the vehicle’s full state to accurately cancel the nonlinear terms numerically. This, in turn, requires suitable sensors capable of measuring aerodynamic variables, such as the angle of attack, particularly for small-scale aircraft. Therefore, validating our method on a real vehicle remains part of our future work.