Finite-Time Mass Estimation Using ℋ_∞ and Sliding Mode Control for a Multicopter

Abstract

Nonlinear control theory applied to unmanned aeronautical vehicles is an engineering topic that has received higher and higher popularity during the last decade. Model-based control approaches have shown increased performance in flight control accuracy and robustness compared to model-free proposals based on parameter adaptation and estimation. However, model-based structures need more computational efforts in terms of spatial and temporal variables. To avoid these constraints, the latest drone flight controls are based on quaternion models, ensuring more advanced computational performances. To this aim, this paper deals with a flight control algorithm of a quadrotor, in which the mathematics model of the plant is defined in terms of quaternions. Additionally, when aerial vehicles are used in specific applications such as slung load transportation and agriculture fields, among others, the variation of the mass receives high importance since it could make the entire system unstable. In the same line of ideas, this paper presents a ${\mathcal{H}}_{\infty }$ strategy, combined with a Super-Twisting Sliding-Mode Control, ensuring the control objective of the mass variations identification, and trajectory tracking, to be solved. The stability analysis of the proposed control approach is also discussed, and the quality and performances of the presented control strategy are tested by simulations, in an interesting case in which mass variations and external perturbations cannot be negligible.

1. Introduction

Unmanned Aeronautical Vehicles (UAVs) are becoming increasingly ubiquitous. Their popularity is due to their prominent maneuverability in a plethora of fields, to perform a wide range of applications, such as, for instance, monitoring roads as in [1,2,3,4], where the authors make use of images and deep learning techniques for road damage detection, extraction of urban traffic information, road patrol, and monitoring the deterioration of the road surface. UAVs are also utilized in the field of remote surveillance, as explained in [5,6,7,8], where detection, classification, and localization of pedestrians in social distance monitoring are presented for civil applications in general. In the field of power lines inspection, it is also possible to employ drones, as explained in [9,10,11] for failure inspection of the photovoltaic sector and high-voltage insulators in emergency prevention.

Monitoring roads, remote surveillance, and power lines inspection are some of the plethora of applications one can find related to the use of aerial vehicles. However, a more complex scenario is presented when considering slung load transportation, since UAVs are exposed to mass variation that could generate instability of the closed-loop control loop, as explained in [12,13,14,15,16,17,18]. Fertilization and irrigation in agriculture [19,20], automated deliveries [21], or humanitarian work, where supplies are transported to remote locations [22], are some of the applications that match the study in consideration. In this case, in fact, a more accurate control strategy based on the estimation of mass variation, even in the presence of external wind perturbations, is desirable.

Several works address the estimation of mass and inertia, such as [23], where an online closed-loop identification of mass and inertia of a UAV is proposed, but the use of extra hardware, which is a set of linear accelerometers, is necessary. In [24,25], instead, a ${\mathcal{H}}_{\infty }$ control is applied to a linearized model, where the tracking problem is referred to as a simplified model for small angles. Other works use two input integrators to apply feedback linearization with a Sliding-Mode observer, as explained in [26]. Though both controller schemes show good results in terms of robustness, the description of the vehicle position may cause singularities in the determinant of the direction cosine matrix. However, the latter approach is not robust to unmatched disturbances.

In the same line of ideas, this work presents a new solution of the trajectory tracking problem for UAVs subjected to external disturbance and parametric variations. This approach considers the plant model based on unitary quaternions, to avoid singularities in the control and ensuring better computation performances. The novelty and originality of the proposed paper is as follows:

• The exact adaptation of the mass despite additive perturbations without the need for additional sensors;
• The inclusion of an estimator of state-dependent functions as part of the aircraft rotational dynamics,

The stability proof, using Lyapunov formalism to support the proposed scheme, is also given.

This paper is organized as follows. In Section 2, the background of the quaternion is briefly introduced. In Section 3, a dynamic model of a quatrotor using unit quaternions is detailed, whereas Section 4 shows the proposed control–estimator scheme based on the adaptive backstepping technique with the ${\mathcal{H}}_{\infty }$-Control and Super-Twisting algorithm, including the stability proof of the closed-loop system. Section 5 shows the results of the quaternion feedback control under external disturbances and parameter variations. Finally, some conclusions and future works are given in Section 6.

2. Background and Preliminaries

The quaternion is a mathematical tool used to represent affine transformations, rotations, and projections [27]. A quaternion is a magnitude represented by $q=(q_0,\mathbf{q})\in \mathbb{H}$, where the set of quaternions $\mathbb{H}$ is the vector space over $\mathbb{R}\times {\mathbb{R}}^3$. The quaternion contains a scalar part $q_0$, and a three-dimensional vector $\mathbf{q}={[q_i,q_j,q_k]}^T$, with orthonormal base $\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, such that ${\mathbf{i}}^2={\mathbf{j}}^2=\mathbf{i}\mathbf{j}\mathbf{k}=-1$.

In particular, a unit quaternion provides a convenient mathematical notation for representing orientations and rotations of objects in three dimensions, and it is described as $q=(q_0,\mathbf{q})$ with:

$$q_0^2+q_i^2+q_j^2+q_k^2=1,$$

where $\mathbf{q}={[q_i,q_j,q_k]}^T$. In addition, a pure quaternion is defined by $x=(x_0,\mathbf{x})\in {\mathbb{H}}_0$ with $x_0=0$ and $\mathbf{x}\in {\mathbb{R}}^3$.

The product of two quaternions, $q_a=(q_{0a},{\mathbf{q}}_a)$, $q_b=(q_{0b},{\mathbf{q}}_b)$, is defined as:

$$\begin{array}{ccc}\hfill q_a{q}_b& =& (q_{0a},{\mathbf{q}}_a)(q_{0b},{\mathbf{q}}_b)\hfill \\ \hfill & =& (q_{0a}{q}_{0b}-{\mathbf{q}}_a·{\mathbf{q}}_b,q_{0a}{\mathbf{q}}_b+q_{0b}{\mathbf{q}}_a+{\mathbf{q}}_a\times {\mathbf{q}}_b).\hfill \end{array}$$

(1)

A pure quaternion rotation of $x^{\prime },x\in {\mathbb{H}}_0$ is computed as:

$$x=q{x}^{\prime }{q}^{*}$$

(2)

where x represents the rotated vector, $x^{\prime }$ is the vector to be rotated, and $q^{*}$ is the quaternion conjugate. To obtain the rotated term, it follows that:

$$x^{\prime }=q^{-1}x{\left(q^{*}\right)}^{-1}=q^{*}xq.$$

(3)

Then, from (1) and (2), it follows that:

$$\begin{array}{ccc}\hfill x& =& (q_0,\mathbf{q})(0,{\mathbf{x}}^{\prime })(q_0,-\mathbf{q})\hfill \\ & =& (0,(q_0^2-{\left|\mathbf{q}\right|}^2){\mathbf{x}}^{\prime }+2(\mathbf{q}·{\mathbf{x}}^{\prime })\mathbf{q}+2q_0(\mathbf{q}\times {\mathbf{x}}^{\prime }))\hfill \end{array}$$

(4)

with $x^{\prime }=(0,{\mathbf{x}}^{\prime })$.

Consider now an isomorphism $\mathcal{G}$ that describes the transformation of rotation from a projection of three-dimensional space to the pure quaternion space [28]

$$\begin{array}{ccc}\hfill \mathcal{G}:{\mathbb{R}}^3& \to & {\mathbb{H}}_0\hfill \\ \hfill {\mathcal{G}}^{-1}:{\mathbb{H}}_0& \to & {\mathbb{R}}^3\hfill \end{array}$$

The advantage of the model based on the representation of quaternions is the reduction of computational burden [29], and this model allows a larger range of angle values before reaching singularities in the determinant of the direction of the cosine matrix.

3. Dynamic Model

The model of the multicopter includes the hub forces, the rolling moments, the gyroscopic forces, and the aerodynamic forces dependent on the air velocity. As shown in Figure 1, the propellers $(1,3)$ rotate clockwise and propellers $(2,4)$ rotate counterclockwise. To increase the aircraft’s altitude, the rotor speeds are increased at the same amount. To force the pitch and roll motion in order to have the system move forward, either the speed of rotor 3 is increased and that of 1 is reduced or the speed of rotor 2 is increased and the speed of 4 is reduced, respectively. Backward motions can be accomplished in a similar manner. The yaw motion can be performed by speeding up or slowing down the clockwise rotors depending on the desired angle direction.

The multicopter dynamic model can be derived using the Newton–Euler formalism in the body fixed frame $E_B$, about the multicopter subject to external forces $\Sigma F_{ext}$ and moments $\Sigma T_{ext}$ applied to the center of mass. Thus, the dynamic equations of motion are described by:

$$\begin{array}{ccc}\hfill F_{prop}-F_{aero}-F_{grav}& =& m{\dot{V}}_B+\Omega \times m{V}_B\hfill \\ \hfill T_{prop}-T_{aero}-T_{gyro}& =& I\dot{\Omega }+\Omega \times I\Omega \hfill \end{array}$$

(5)

where $V_B={[u,v,w]}^T$ is the translational velocity of the airframe, $\Omega ={[p,q,r]}^T$ is the angular velocity of the aircraft, $I=diag(I_x,I_y,I_z)$ is the inertia tensor, and m is the mass. The force $F_{prop}$ and torque $T_{prop}$ are produced by propeller system, $F_{aero}$ and $T_{aero}$ are the aerodynamic forces and moments vectors which act on the multicopter by blade flapping, $T_{gyro}$ is the gyroscopic moment effect produced by the change in orientation of the rotor-craft, and $F_{grav}=m{q}^{*}Gq$ is the gravity effect forces vector with $G={[0,0,g]}^T$ m/s². These variables are defined as:

$$\begin{array}{cc}\hfill & \begin{array}{cc}{F}_{prop}=\left[\begin{array}{c}0\\ 0\\ \sum _{i=1}^4{F}_i\end{array}\right]\hfill & F_{aero}=\left[\begin{array}{c}{A}_w\\ A_u\\ A_v\end{array}\right]\hfill \\ T_{aero}=\left[\begin{array}{c}{A}_p\\ A_q\\ A_r\end{array}\right]\hfill & T_{prop}=\left[\begin{array}{c}d(F_4-F_2)\\ d(F_3-F_1)\\ c\sum _{i=1}^4{(-1)}^i{F}_i\end{array}\right]\hfill \end{array}\\ & T_{gyro}={\sum }_{i=1}^4{J}_R(\Omega \times e_3^E){(-1)}^{i+1}{\omega }_i\end{array}$$

(6)

where d is the distance from the center of mass to the rotor shaft, c is the drag factor, $J_R$ is the rotor inertia, ${\omega }_i$ is the speed of the rotor i, $A_j={\displaystyle \frac{1}{2}}\rho C_j{W}_j^2$ is the aerodynamic function with $j=p,q,r,u,v,w$, where $\rho$ the air density, $C_j$ is the aerodynamic coefficient, and $W_j$ is the element of $W={[\Omega ,V_B]}^T-{\Omega }_{air}$, which is the velocity of the multicopter with respect to the air [30].

Using the unit quaternion, the equations from (5) and (6) are represented as:

$$\begin{array}{ccc}\hfill \ddot{X}& =& {\mathcal{G}}^{-1}\left\{{\displaystyle \frac{1}{m}}q\mathcal{G}\left\{\left[F_{prop}-F_{aero}\right]\right\}{q}^{*}\right\}-G\hfill \\ \hfill \dot{\Omega }& =& I^{-1}\left[T_{prop}-T_{aero}-T_{gyro}-\Omega \times I\Omega \right]\hfill \end{array}$$

(7)

where $X={[x,y,z]}^T$ is translational motion vector and the quaternion propagation described the differential kinematic is defined by [27]:

$$\begin{array}{ccc}\hfill \dot{q}& =& {\displaystyle \frac{1}{2}}q\mathcal{G}\{\Omega \}\hfill \\ & =& {\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}-q_i& -q_j& -q_k\\ q_0& -q_k& q_j\\ q_k& q_0& -q_i\\ -q_j& q_i& q_0\end{array}\right]\Omega .\hfill \end{array}$$

(8)

4. Control

The control objective is to track a three-dimensional reference, $(x_{7r},x_{9r},x_{11r})$, as well as the yaw angle reference, ${\psi }_r$, with known derivatives. Adaptive backstepping [31], nonlinear–${\mathcal{H}}_{\infty }$ [32], and Sliding–Mode techniques [33] are combined to ensure trajectory tracking.

The models (7) and (8) are presented in a block state space as:

$$\begin{array}{ccc}\hfill SB1& \left\{\begin{array}{cc}{\dot{x}}_7\hfill & \hfill =\\ {\dot{x}}_8\hfill & \hfill =\end{array}\right.& \begin{array}{c}{x}_8\hfill \\ (2(x_1^2+x_{13}^2)-1)u_1/m+w_8-g\hfill \end{array}\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill SB2& \left\{\begin{array}{cc}{\dot{x}}_9\hfill & \hfill =\\ {\dot{x}}_{11}\hfill & \hfill =\\ {\dot{x}}_{10}\hfill & \hfill =\\ {\dot{x}}_{12}\hfill & \hfill =\end{array}\right.& \begin{array}{c}{x}_{10}\hfill \\ x_{12}\hfill \\ 2(x_{13}{x}_3+x_1{x}_5)u_1/m+w_{10}\hfill \\ 2(-x_1{x}_3+x_{13}{x}_5)u_1/m+w_{12}\hfill \end{array}\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill SB3& \left\{\begin{array}{cc}{\dot{x}}_3\hfill & \hfill =\\ {\dot{x}}_5\hfill & \hfill =\\ {\dot{x}}_{13}\hfill & \hfill =\end{array}\right.& \begin{array}{c}1/2(x_1{x}_2-x_{13}{x}_4+x_5{x}_6)\hfill \\ 1/2(x_{13}{x}_2+x_1{x}_4-x_3{x}_6)\hfill \\ 1/2(-x_5{x}_2+x_3{x}_4+x_1{x}_6)\hfill \end{array}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill SB4& \left\{\begin{array}{cc}{\dot{x}}_2\hfill & \hfill =\\ {\dot{x}}_4\hfill & \hfill =\\ {\dot{x}}_6\hfill & \hfill =\end{array}\right.& \begin{array}{c}{a}_1{x}_4{x}_6-a_2{x}_4\omega +b_1{u}_2+w_2\hfill \\ a_3{x}_2{x}_6-a_4{x}_2\omega +b_2{u}_3+w_4\hfill \\ a_5{x}_2{x}_4+b_3{u}_4+w_6\hfill \end{array}\hfill \end{array}$$

(12)

where the variables are replaced as follows:

$$\begin{array}{ccccc}{x}_1=q_0& x_2=p& x_3=q_i& x_4=q& x_5=q_j\\ x_6=r& x_7=z& x_8=\dot{z}& x_9=x& x_{10}=\dot{x}\\ & x_{11}=y& x_{12}=\dot{y}& x_{13}=q_k,\end{array}$$

Also, ${\omega }_k(k=1,\cdots ,4)$ is the speed of the rotor i, $w_i,(i=8,10,12)$ is the aerodynamic acceleration expressed in the inertia reference frame $E^E$, and $w_j,(j=2,4,6)$ is the equivalent rotational acceleration, that is, the acceleration computed from the aerodynamic functions $A_i$ and $A_j$, respectively. Let us define the outputs as ${\left[x_7{x}_9{x}_{11}\psi \right]}^T$, where $\psi ={tan}^{-1}\left({\displaystyle \frac{2(x_1{x}_{13}+x_3{x}_5)}{1-2(x_5^2+x_{13}^2)}}\right)$, $\omega =-{\omega }_1+{\omega }_2-{\omega }_3+{\omega }_4$, and

$$\begin{array}{cccc}{a}_1={\displaystyle \frac{I_y-I_z}{I_x}}& a_2={\displaystyle \frac{J_R}{I_x}}& a_3={\displaystyle \frac{I_z-I_x}{I_y}}& a_4={\displaystyle \frac{J_R}{I_y}}\\ a_5={\displaystyle \frac{I_x-I-y}{I_z}}& b_1={\displaystyle \frac{d}{I_x}}& b_2={\displaystyle \frac{d}{I_y}}& b_3={\displaystyle \frac{1}{I_z}}\end{array}$$

Hereafter, tracking is designed for altitude ($x_7$) with subblock $SB1$ through input $u_1$, subsequently, for longitude ($x_9$) and latitude ($x_{11}$) control with subblock $SB2$ using the variables $x_3$ and $x_5$, which are manipulated along with $x_{13}$ to track $\psi$ through $u_2$, $u_3$, and $u_4$ with subblocks $SB3$ and $SB4$. Figure 2 shows the details of the proposed control.

4.1. Altitude Control

Define the tracking error as follows:

$$\begin{array}{c}\hfill z_1=x_{7r}-x_7\end{array}$$

(13)

where $x_{7r}$ is the reference signal. Then, using (9), the tracking error dynamics are obtained as:

$$\begin{array}{ccc}\hfill {\dot{z}}_1& =& {\dot{x}}_{7r}-x_8=z_2\hfill \\ \hfill {\dot{z}}_2& =& {\ddot{x}}_{7r}+g-(x_1^2+x_{13}^2-x_3^2-x_5^2)u_1/m-w_8.\hfill \end{array}$$

(14)

The control $u_1$ in (14) is formulated as:

$$u_1=-{\displaystyle \frac{\widehat{m}}{x_1^2+x_{13}^2-x_3^2-x_5^2}}\left(-g-{\ddot{x}}_{7r}-K_1{\zeta }_1+v_1\right)$$

(15)

where ${\zeta }_1={\left[z_1{z}_2\right]}^T$ is the vector error of altitude tracking and the gain $K_1=[k_{11},k_{12}]$ is chosen to achieve the nominal system be stable. In addition, $v_1$ will be designed to reject the unknown perturbation. The mass estimation $\widehat{m}$ is calculated from Reinforcement Learning, as in [34], using state estimator:

$$\begin{array}{ccc}\hfill {\dot{\widehat{x}}}_8& =& (2(x_1^2+x_{13}^2)-1)u_1/\widehat{m}-g+{\epsilon }_1\hfill \\ \hfill {\epsilon }_1& =& {\lambda \left|e\right|}^{1/2}\mathrm{sign}\left(e\right)-{\epsilon }_2\hfill \\ \hfill {\dot{\epsilon }}_2& =& -\alpha \mathrm{sign}\left(e\right)\hfill \end{array}$$

where $e=x_8-{\widehat{x}}_8$ is the state estimation error. Hence, the adaptive law for the estimation of mass is:

$$\dot{\widehat{m}}=\gamma \mathrm{sign}\left((g+{\ddot{x}}_{7r}+K_1{\zeta }_1-v_1){\epsilon }_1\right)$$

(16)

with a positive scalar $\gamma$.

Furthermore, the closed-loop subsystem (14) and (15) becomes:

$$\begin{array}{ccc}\hfill {\dot{z}}_1& =& z_2\hfill \\ \hfill {\dot{z}}_2& =& -K_1{\zeta }_1+v_1+{\displaystyle \frac{\tilde{m}}{m}}\left(-g-{\ddot{x}}_{7r}-K_1{\zeta }_1+v_1\right)+w_8,\hfill \end{array}$$

(17)

where $\tilde{m}=m-\widehat{m}$ is the mass estimation error.

The system (17) can be represented in the matrix form as:

$$\begin{array}{ccc}\hfill {\dot{\zeta }}_1& =& A_1{\zeta }_1+B_{11}{v}_1+B_{12}\left(w_8\left(t\right)+\Gamma {\displaystyle \frac{\tilde{m}}{m}}\right)\hfill \\ \hfill y_1& =& C_1{\zeta }_1\hfill \end{array}$$

(18)

where

$$\begin{array}{cccc}\hfill A_1& =\left[\begin{array}{cc}0& 1\\ -k_{11}& -k_{12}\end{array}\right],\hfill & \hfill B_{11}& =\left[\begin{array}{c}0\\ 1\end{array}\right],\hfill \\ \hfill \Gamma & =-k_{11}{z}_1-k_{12}{z}_2-g-{\ddot{x}}_{7r}+v_1,\hfill & \hfill B_{12}& =\left[\begin{array}{c}0\\ 1\end{array}\right].\hfill \end{array}$$

and the pair $\{C_1,A_1\}$ is assumed to be observable.

The robust control part $v_1$ is formulated using the ${\mathcal{H}}_{\infty }$-control technique [35,36,37], as follows:

$$v_1=-B_{11}^T{P}_1{\zeta }_1$$

where $P_1$ is a positive definite matrix solution of Riccati equation:

$$\begin{array}{c}\hfill 0=P_1(B_{11}{B}_{11}^T-{\kappa }_1^{-2}{B}_{12}{B}_{12}^T)P_1-P_1{A}_1-A_1^T{P}_1-C_1^T{C}_1-{\rho }_{1}I\end{array}$$

(19)

where ${\rho }_1$ and ${\kappa }_1$ are some positive constants.

As result, the response $y_1=C_1{\zeta }_1$, for initial state ${\zeta }_1\left(t_0\right)=0$, satisfies:

$${\int }_{t_0}^{t_1}{∥y_1\left(t\right)∥}^{2}dt<{\kappa }_1^2{\int }_{t_0}^{t_1}{∥w_8^{\prime }\left(t\right)∥}^{2}dt.$$

where $w_8^{\prime }\left(t\right)=\Gamma {\displaystyle \frac{\tilde{m}}{m}}+w_8\left(t\right)$.

Moreover, according to the work of Arellano-Muro et al. [34], the adaptation error $\tilde{m}\left(t\right)$ is bounded by:

$$∥\tilde{m}\left(t\right)∥\le {\displaystyle \frac{\gamma }{\eta (\gamma -b_2)}}\left(1+{\displaystyle \frac{d_1}{{\lambda }_{min}^{{\Gamma }^T\Gamma }}}\right){\displaystyle \frac{\sqrt{1+\eta }}{\eta }}$$

for some $t\ge t_1$ and a constant $0<\eta <1$.

To show the stability of the closed-loop subsystem (18) with (16), para $t\ge t_1$ y ${\zeta }_1\in D_1\subset {\mathbb{R}}^2$, define a Lyapunov function:

$$V_{z1}={\zeta }_1^T{P}_1{\zeta }_1.$$

Taking the time derivative of the Lyapunov function and using (18) results in:

$$\begin{array}{c}\hfill {\dot{V}}_{z1}={\zeta }_1^T\left(A_1^T{P}_1+P_1{A}_1-P_1{B}_{11}{B}_{11}^T{P}_1\right){\zeta }_1+w_8^{\prime T}{B}_{12}^T{P}_1{\zeta }_1+{\zeta }_1^T{P}_1{B}_{12}{w}_8^{\prime }.\end{array}$$

(20)

Suppose that, in the equilibrium point of the closed-loop subsystem $SB1$, the perturbation term holds under the limits:

$$∥w_8^{\prime }∥\le b_1∥{\zeta }_1∥+b_2.$$

where $b_1$ and $b_2$ are real positive constants in some admissible region ${\zeta }_1\in D_1$.

Setting ${\overline{w}}_8:=B_{12}^T{P}_1{\zeta }_1$ from (20), results in:

$$\begin{array}{ccc}\hfill {\dot{V}}_{z1}& =& {\zeta }_1^T\left(A_1^T{P}_1+P_1{A}_1-P_1{B}_{11}{B}_{11}^T{P}_1+{\kappa }^{-2}{P}_1{B}_{12}{B}_{12}^T{P}_1\right){\zeta }_1-{\kappa }_1^{-2}{\zeta }_1^T{P}_1{B}_{12}{B}_{12}^T{P}_1{\zeta }_1+\hfill \\ & & 2{\zeta }_1^T{P}_1{B}_{12}{w}_8^{\prime }\hfill \\ & =& -{\zeta }_1^T(C_1^T{C}_1+\rho I_2){\zeta }_1-{\kappa }_1^{-2}{\zeta }_1^T{P}_1{B}_{12}{B}_{12}^T{P}_1{\zeta }_1+2{\zeta }_1^T{P}_1{B}_{12}{w}_8^{\prime }\hfill \\ & \le & -{\zeta }_1^T{C}_1^T{C}_1{\zeta }_1-({\kappa }_1^{-2}-2b_1{\lambda }_{min}^{B_{12}^T{P}_1}){∥{\overline{w}}_8∥}^2+2b_2∥w_8^{\prime }∥.\hfill \end{array}$$

Choosing:

$${\kappa }_1\le \sqrt{{\displaystyle \frac{1}{2b_1{\lambda }_{min}^{B_{12}^T{P}_1}}}},$$

the norm $∥{\zeta }_1\left(t\right)∥$ is ultimately bounded in an admissible region $D_1$, that is:

$$\begin{array}{c}\hfill ∥{\zeta }_1\left(t\right)∥\le \beta ,\beta =\mu \sqrt{{\displaystyle \frac{{\lambda }_{max}^{P_1}}{{\lambda }_{min}^{P_1}}}},\\ \hfill \mu ={\displaystyle \frac{2b_2\sqrt{{\lambda }_{max}^{P_1{B}_{12}{B}_{12}^T{P}_1}}}{{\lambda }_{min}^{C_1^T{C}_1+\rho I_2}\theta }},\end{array}$$

where $0<\theta <1$ for all $∥{\zeta }_1\left(t\right)∥\ge \mu$.

As a result, the following proposition is enunciated: the closed-loop solution (18) ${\zeta }_1$ using the adaptive law (16) is asymptotically stable in an admissible region $D_1\in {\mathbb{R}}^2$ and the adaptive error $\tilde{m}$ is bounded for $t\ge t_1$.

Since ${\zeta }_1$ stays on the ball $\beta$, so does the tracking error $z_1$ and the mass estimate since the error is also bounded.

4.2. Latitudinal, Longitudinal, and Yaw Control

Considering the subsystem (10), define the tracking error vector $z_3\in {\mathbb{R}}^2$ as:

$$\begin{array}{ccc}\hfill z_3& =& \left[\begin{array}{c}{x}_{9r}-x_9\\ x_{11r}-x_{11}\end{array}\right]\hfill \end{array}$$

(21)

where $x_{9r}$ and $x_{11r}$ are the references signals. Then, the tracking error dynamics are obtained as:

$$\begin{array}{ccc}{\dot{z}}_3\hfill & =& \left[\begin{array}{c}{\dot{x}}_{9r}-x_{10}\\ {\dot{x}}_{11r}-x_{12}\end{array}\right]=z_4\hfill \\ {\dot{z}}_4\hfill & =& \left[\begin{array}{c}{\ddot{x}}_{9r}\\ {\ddot{x}}_{11r}\end{array}\right]-{\displaystyle \frac{2u_1}{m}}\left[\begin{array}{cc}{x}_{13}& x_1\\ -x_1& x_{13}\end{array}\right]\left[\begin{array}{c}{x}_3\\ x_5\end{array}\right]-\left[\begin{array}{c}{w}_{10}\\ w_{12}\end{array}\right].\hfill \end{array}$$

(22)

Considering the vector ${[x_3,x_5]}^T$ as a virtual control in (22), we choose its desired value ${[x_{3d},x_{5d}]}^T$ as:

$$\begin{array}{ccc}\hfill \left[\begin{array}{c}{x}_{3d}\\ x_{5d}\end{array}\right]& =& B_{21}^{-1}\left(\left[\begin{array}{c}-{\ddot{x}}_{9r}\\ -{\ddot{x}}_{11r}\end{array}\right]-K_{21}{\zeta }_{21}+v_{21}\right)\hfill \end{array}$$

(23)

with

$$\begin{array}{c}{B}_{21}=-{\displaystyle \frac{2u_1}{\widehat{m}}}\left[\begin{array}{cc}{x}_{13}& x_1\\ -x_1& x_{13}\end{array}\right],\\ K_{21}=\left[\begin{array}{cccc}{k}_{21,1}& 0& k_{21,3}& 0\\ 0& k_{21,2}& 0& k_{21,4}\end{array}\right],\end{array}$$

and $v_{21}$ is formulated using ${\mathcal{H}}_{\infty }$-control to attenuate the disturbance $w_{10}\left(t\right)$ and $w_{12}\left(t\right)$ of the closed-loop subsystem (21) and (23), where the vector ${\zeta }_{21}={[z_3^T,z_4^T]}^T$ is defined as:

$$\begin{array}{ccc}\hfill {\dot{\zeta }}_{21}& =& A_{21}{\zeta }_{21}+B_{21,v}{v}_{21}+B_{21,w}\left[\begin{array}{c}{w}_{10}\\ w_{12}\end{array}\right]\hfill \\ \hfill y_{21}& =& C_{21}{\zeta }_{21}\hfill \end{array}$$

(24)

where

$$\begin{array}{c}{B}_{21,v}=B_{21,w}={\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}^T,\\ A_{21}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -k_{21,1}& 0& -k_{21,3}& 0\\ 0& -k_{21,2}& 0& -k_{21,4}\end{array}\right],\\ C_{21}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\end{array}$$

so that we can establish $v_{21}=-B_{21,u}^T{P}_{21}{\zeta }_{21}$, where $P_{21}$ is a symmetric positive definite matrix solution of Riccati equation:

$$\begin{array}{c}\hfill 0=P_{21}(B_{21,v}{B}_{21,v}^T-{\kappa }_{21}^{-2}{B}_{21,w}{B}_{21,w}^T)P_{21}-P_{21}{A}_{21}-A_{21}^T{P}_{21}-C_{21}^T{C}_{21}-{\rho }_{21}I\end{array}$$

(25)

with sufficiently small constant ${\kappa }_{21}>0$ and some positive constant ${\rho }_{21}$.

Define now the following:

$$z_5=\left[\begin{array}{c}{x}_{3d}\\ x_{5d}\\ x_{13r}\end{array}\right]-\left[\begin{array}{c}{x}_3\\ x_5\\ x_{13}\end{array}\right],$$

(26)

where $x_{13r}$ is calculated as a recursive solution to the following expression:

$${\psi }_r=ta{n}^{-1}\left({\displaystyle \frac{2\left(\sqrt{1-x_{3d}^2-x_{5d}^2-x_{13r}^2}{x}_{13r}+x_{3d}{x}_{5d}\right)}{1-2(x_{5d}^2+x_{13r}^2)}}\right)$$

as a function of the given reference ${\psi }_r$ for the system output and the desired values $x_{3d}$ and $x_{5d}$.

Then, using (26), the subsystem (11) is reduced to:

$$\begin{array}{c}\hfill {\dot{z}}_5=-{\displaystyle \frac{1}{2}}\left[\begin{array}{c}{x}_1{x}_2-x_{13}{x}_4+x_5{x}_6\\ x_{13}{x}_2+x_1{x}_4+x_3{x}_6\\ -x_5{x}_2+x_3{x}_4+x_1{x}_6\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{d}{dt}}{B}_{21}^{-1}\left(\left[\begin{array}{c}-{\ddot{x}}_{9r}\\ -{\ddot{x}}_{11r}\end{array}\right]-K_{21}{\zeta }_{21}+v_{21}\right)+f_5\\ {\dot{x}}_{13r}\end{array}\right]\end{array}$$

(27)

where

$$f_5=B_{21}^{-1}\left(\left[\begin{array}{c}-x_{9r}^{\left(3\right)}\\ -x_{11r}^{\left(3\right)}\end{array}\right]-(K_{21}+B_{21,u}^T{P}_{21}){\dot{\zeta }}_{21}\right).$$

Calculating the derivative ${\displaystyle \frac{d}{dt}}{B}_{21}^{-1}$ as:

$${\displaystyle \frac{d}{dt}}{B}_{21}^{-1}={\dot{x}}_1{\displaystyle \frac{\partial }{\partial x_1}}{B}_{21}^{-1}+{\dot{x}}_{13}{\displaystyle \frac{\partial }{\partial x_{13}}}{B}_{21}^{-1}+{\dot{u}}_1{\displaystyle \frac{\partial }{\partial u_1}}{B}_{21}^{-1}$$

the system (27) becomes:

$${\dot{z}}_5={\overline{f}}_5+(B_{22}+B_{23})\left[\begin{array}{c}{x}_2\\ x_4\\ x_6\end{array}\right]$$

(28)

where the following matrix:

$$B_{22}=-{\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}{x}_1& -x_{13}& x_5\\ x_{13}& x_1& x_3\\ -x_5& x_3& x_1\end{array}\right]$$

is considered as known, while the following matrix:

$$\begin{array}{ccc}\hfill B_{23}& =& {\displaystyle \frac{1}{2}}\left[\begin{array}{cc}{\displaystyle \frac{\partial B_{21}^{-1}}{\partial x_1}}\left(h_{\infty x_q}\right)& {\displaystyle \frac{\partial B_{21}^{-1}}{\partial x_{13}}}\left(h_{\infty x_q}\right)\\ 0& 0\end{array}\right]{\left[\begin{array}{cc}-x_3& -x_5\\ -x_5& x_3\\ -x_{13}& x_1\end{array}\right]}^T\hfill \\ \hfill h_{\infty x_q}& =& \left[\begin{array}{c}-{\ddot{x}}_{9r}\\ -{\ddot{x}}_{11r}\end{array}\right]-K_{21}{\zeta }_{21}+v_{21}\hfill \end{array}$$

and the vector

$${\overline{f}}_5=\left[\begin{array}{c}-{\displaystyle \frac{1}{2}}{\dot{u}}_1{\displaystyle \frac{\partial }{\partial u_1}}{B}_{21}^{-1}\left(\left[\begin{array}{c}-{\ddot{x}}_{9r}\\ -{\ddot{x}}_{11r}\end{array}\right]-K_{21}{\zeta }_{21}+v_{21}\right)-f_5\\ {\dot{x}}_{13r}\end{array}\right]$$

are considered unknown.

The following assumption is instrumental to get the desired result.

Assumption 1.

The unknown vector ${\overline{f}}_5$ and matrix $B_{23}$ are bounded, and they can be expressed as a Kronecker series:

$$\begin{array}{ccc}\hfill {\overline{f}}_5& =& {\displaystyle \sum _{k=0}^{\infty }\frac{1}{k!}{D}_k{x}_q^{\otimes k}}\hfill \\ \hfill B_{23}& =& {\displaystyle \sum _{k=0}^{\infty }\frac{1}{k!}{C}_{k}X,}\hfill \end{array}$$

(29)

where $x_q={[x_1,x_3,x_5,x_{13}]}^T$ and $X=I_3\otimes x_q^{\otimes k}$ are defined. Moreover, $x_q^{\otimes k}=x_q^{\otimes (k-1)}\otimes x_q$ with $x_q^{\otimes 0}=1$ and the operator $\otimes$ is a Kronecker product. We have $C_k\in {\mathbb{R}}^{3\times 3^k}$ and $D_k\in {\mathbb{R}}^{3\times 3\left(4^k\right)}$ as constants.

Then, defining in (26) the vector $x_d={[x_{3d},x_{5d},x_{13r}]}^T$, and using (28) yields:

$${\dot{x}}_d={\overline{f}}_5+B_{23}{x}_{\Omega }$$

(30)

where $x_{\Omega }={[x_2,x_4,x_6]}^T$

Defining ${\widehat{x}}_d$ as an estimate of $x_d$ with the following dynamics:

$${\dot{\widehat{x}}}_d={\widehat{\overline{f}}}_5+{\widehat{B}}_{23}{x}_{\Omega }+v$$

(31)

where

$$\begin{array}{ccc}\hfill {\widehat{\overline{f}}}_5& =& {\displaystyle \sum _{k=0}^N\frac{1}{k!}{\widehat{D}}_k{x}_q^{\otimes k}}\hfill \\ \hfill {\widehat{B}}_{23}& =& {\displaystyle \sum _{k=0}^N\frac{1}{k!}{\widehat{C}}_{k}X}\hfill \end{array}$$

(32)

with N any positive constant, also:

$$v={\alpha }_1{\tilde{x}}_d+{\alpha }_2\mathrm{sign}\left({\tilde{x}}_d\right)$$

(33)

where ${\tilde{x}}_d=x_d-{\widehat{x}}_d$ is the adaptation error and ${\widehat{C}}_k$ and ${\widehat{d}}_k$ are constants to be adapted with the law:

$$\begin{array}{ccc}\hfill {\dot{\widehat{C}}}_k& =& {\displaystyle \frac{{\gamma }_2}{k!}}({\tilde{x}}_d{x}_{\Omega }^T+z_5{x}_{\Omega ,d}^T)X^T\hfill \\ \hfill {\dot{\widehat{D}}}_k& =& {\displaystyle \frac{{\gamma }_1}{k!}}({\tilde{x}}_d+z_5){\left[x_q^{\otimes k}\right]}^T,\hfill & \hfill {\gamma }_1>0,{\gamma }_2>0.\end{array}$$

(34)

Using (30) and (31) with (29) and (32), the estimation error dynamics become:

$${\dot{\tilde{x}}}_q={\tilde{\overline{f}}}_5+{\tilde{B}}_{23}{x}_{\Omega }+\eta -{\alpha }_1{\tilde{x}}_d-{\alpha }_2\mathrm{sign}\left({\tilde{x}}_d\right)$$

(35)

where ${\tilde{B}}_{23}={\sum }_{k=0}^N{\displaystyle \frac{1}{k!}}{\tilde{C}}_{k}X$ and ${\tilde{\overline{f}}}_5={\sum }_{k=0}^N{\displaystyle \frac{1}{k!}}{\tilde{D}}_k{x}_q^{\otimes k}$ are the adaptation errors, ${\tilde{C}}_k=C_k-{\widehat{C}}_k$, and ${\tilde{D}}_k=D_k-{\widehat{D}}_k$ are the parameters errors, and the term $\eta$ is considered a bounded disturbance:

$${\displaystyle \eta =\sum _{k=N+1}^{\infty }\left(\frac{1}{k!}{\widehat{C}}_{k}X\right)x_{\Omega }+\sum _{k=N+1}^{\infty }\frac{1}{k!}{\widehat{D}}_k{x}_q^{\otimes k}}$$

On the next step of the design procedure, considering the variables $x_2$, $x_4$, and $x_6$ in (28) as the virtual controls, its desired values are chosen of the form:

$$\begin{array}{c}\hfill \left[\begin{array}{c}{x}_{2d}\\ x_{4d}\\ x_{6d}\end{array}\right]=x_{\Omega ,d}={(B_{22}+{\widehat{B}}_{23})}^{-1}\left(-{\widehat{\overline{f}}}_5-K_{22}{z}_5\right),\end{array}$$

Defining the control error:

$$z_6=x_{\Omega ,d}-x_{\Omega },$$

and using the subsystem (12) yields

$${\dot{z}}_6=B_6\left({\Gamma }_6{\theta }_6-\left[\begin{array}{c}{u}_2\\ u_3\\ u_4\end{array}\right]-\left[\begin{array}{c}{w}_2/b_1\\ w_4/b_2\\ w_6/b_3\end{array}\right]\right)$$

(36)

where

$$\begin{array}{c}{B}_6=diag\{b_1,b_2,b_3\}>0,\\ {\theta }_6={[1/b_1,a_1/b_1,a_2/b_1,1/b_2,a_3/b_2,a_4/b_2,1/b_3,a_5/b_3]}^T,\\ {\Gamma }_6=\left[\begin{array}{cccccccc}{\dot{x}}_{2d}& -x_4{x}_6& x_4\omega & 0& 0& 0& 0& 0\\ 0& 0& 0& {\dot{x}}_{4d}& -x_2{x}_6& x_2\omega & 0& 0\\ 0& 0& 0& 0& 0& 0& {\dot{x}}_{6d}& -x_2{x}_4\end{array}\right].\end{array}$$

Defining ${\widehat{\theta }}_6$ as the estimate of the parameter ${\theta }_6$, the control is formulated as:

$$\begin{array}{ccc}\hfill \left[\begin{array}{c}{u}_2\\ u_3\\ u_4\end{array}\right]& =& {\Gamma }_6{\widehat{\theta }}_6+\Lambda {∥z_6∥}^{1/2}\mathrm{sign}\left(z_6\right)-\sigma \hfill \\ \hfill \dot{\sigma }& =& -M\mathrm{sign}\left(z_6\right)\hfill \end{array}$$

(37)

where $\Lambda$ is a symmetric positive definite matrix and $M>0$ is a diagonal matrix. The adaptation law is proposed of the form:

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_6={\Gamma }_6^{T}M\mathrm{sign}\left(z_6\right).\end{array}$$

(38)

Then, the closed-loop subsystem (36) and (37) becomes:

$$\begin{array}{ccc}\hfill {\dot{z}}_6& =& B_6\left({\Gamma }_6\tilde{\theta }-\Lambda {∥z_6∥}^{1/2}\mathrm{sign}\left(z_6\right)+\sigma -w^{\prime }\left(t\right)\right)\hfill \\ \hfill \dot{\sigma }& =& -M\mathrm{sign}\left(z_6\right).\hfill \end{array}$$

(39)

with

$$\begin{array}{cc}{\tilde{\theta }}_6={\theta }_6-{\widehat{\theta }}_6,& w^{\prime }\left(t\right)={[w_2/b_1,w_4/b_4,w_6/b_3]}^T\end{array}.$$

Thus, the closed-loop system (24), (28), (35) and (39) is represented as follows:

$$\begin{array}{ccc}\hfill {\dot{\zeta }}_{21}& =& A_{21,CL}{\zeta }_{21}+B_{21,w}\left[\begin{array}{c}{w}_{10}\\ w_{12}\end{array}\right]+Q_1{z}_5\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill \begin{array}{c}{\dot{z}}_5\\ {\dot{\tilde{x}}}_q\\ {\dot{z}}_6\\ \dot{\sigma }\end{array}& \begin{array}{c}=\\ =\\ =\\ =\end{array}& \begin{array}{c}-K_{22}{z}_5+{\tilde{\overline{f}}}_5+{\tilde{B}}_{23}{x}_{\Omega }+Q_2{z}_6\hfill \\ {\tilde{\overline{f}}}_5+{\tilde{B}}_{23}{x}_{\Omega }+\eta -{\alpha }_1{\tilde{x}}_d-{\alpha }_2\mathrm{sign}\left({\tilde{x}}_d\right)\hfill \\ B_6({\Gamma }_6{\tilde{\theta }}_6-\Lambda {∥z_6∥}^{1/2}\mathrm{sign}\left(z_6\right)+\sigma -w^{\prime })\hfill \\ -M\mathrm{sign}\left(z_6\right)\hfill \end{array}\hfill \end{array}$$

(41)

where $A_{21,CL}=A_{21}+B_{21,v}{B}_{21,v}^T{P}_{21}$, $Q_1={\displaystyle \frac{u_1}{\widehat{m}}}\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ x_{13}& x_1& 0\\ -x_1& x_{13}& 0\end{array}\right]$ and $Q_2=-B_{22}-B_{23}$.

The following assumption will be used to analyze stability of the system (40) and (41).

Assumption 2.

The external disturbances in the system (40) and (41) are bounded by:

$$\begin{array}{ccc}\hfill ∥{[w_{10}\left(t\right),w_{12}\left(t\right)]}^T∥& \le & b_{21}∥{\zeta }_{21}∥+b_{31},\hfill \\ \hfill ∥w^{\prime }\left(t\right)∥& \le & b_{22}{∥z_6∥}^{1/2}+b_{32},\hfill \end{array}$$

where $b_{ij}>0,j=1,2;i=2,3$.

Additionally, from Assumption 1, it follows that $∥\eta ∥$ is also bounded, and thus:

$$∥\eta ∥\le {\delta }_1∥{\tilde{x}}_d∥+{\delta }_2,{\delta }_1>0,{\delta }_2>0.$$

To demonstrate the stability of the closed-loop system (40), (41) note input-to-state stability in (40) with $z_5$ and ${[w_{10},w_{12}]}^T$ as input and asymptotic stability of system (41). Along with [38], let us consider the first Lyapunov function candidate to unforced system (40):

$$V_1={\zeta }_{21}{P}_{21}{\zeta }_{21}$$

choosing a constant ${\kappa }_{21}>0$, from (25), we obtain:

$$\begin{array}{ccc}\hfill {\dot{V}}_1& =& {\zeta }_{21}(A_{21}^T{P}_{21}+P_{21}{A}_{21}-P_{21}{B}_{21,v}{B}_{21,v}^T{P}_{21}){\zeta }_{21}\hfill \\ & =& -{\zeta }_{21}^T(C_{21}^T{C}_{21}+{\mu }_{21}I){\zeta }_{21}-{\kappa }_{21}^{-2}{\zeta }_{21}^T{P}_{21}{B}_{21,w}{B}_{21,w}^T{P}_{21}{\zeta }_{21}<0,\hfill \end{array}$$

(42)

Thus, the solution ${\zeta }_{21}\left(t\right)$ is exponentially tends to zero. Now, consider the second Lyapunov function candidate:

$$\begin{array}{ccc}\hfill V_2& =& \hfill {\displaystyle \frac{1}{2}{z}_5^T{z}_5+\frac{1}{2}{\tilde{x}}_d^T{\tilde{x}}_d+\frac{1}{2{\gamma }_1}\sum _{k=0}^{\infty }\sum _{i=1}^3\sum _{j=1}^{4^k}{\tilde{d}}_{ij,k}^2+\frac{1}{2{\gamma }_2}\sum _{k=0}^{\infty }\sum _{i=1}^3\sum _{j=1}^{3\left(4^k\right)}{\tilde{c}}_{ij,k}^2+\frac{1}{2}{\widetilde{{\theta }_6}}^T\widetilde{{\theta }_6}+\sqrt{z_6^T{M}^2{B^{-1}}^2{z}_6}+\frac{1}{2}{\sigma }_1^T{\sigma }_1}\end{array}$$

where ${\sigma }_1=\sigma -{\sigma }_{ss}$ and ${\sigma }_{ss}$ is a positive constant.

Calculating its time derivative along the trajectories of the system (41) yields:

$$\begin{array}{ccc}\hfill {\dot{V}}_2& \le & -z_5^T{K}_{22}{z}_5+z_5^T\left({\tilde{\overline{f}}}_5+{\tilde{B}}_{23}{x}_{\Omega ,d}\right)+z_5^T{Q}_2{z}_6+{\displaystyle \frac{1}{2{\gamma }_2}}\sum _{k=0}^{\infty }\sum _{i=1}^3\sum _{j=1}^{3\left(4^k\right)}{\tilde{c}}_{ij,k}{\dot{\tilde{c}}}_{ij,k}+{\tilde{x}}_d^T{\tilde{\overline{f}}}_5+{\tilde{x}}_d^T{\tilde{B}}_{23}{x}_{\Omega }\hfill \\ & & +{\displaystyle \frac{1}{2{\gamma }_1}}\sum _{k=0}^{\infty }\sum _{i=1}^3\sum _{j=1}^{4^k}{\tilde{d}}_{ij,k}{\dot{\tilde{d}}}_{ij,k}-({\alpha }_1-{\delta }_1){∥{\tilde{x}}_d∥}^2-({\alpha }_2-{\delta }_2)∥{\tilde{x}}_d∥+{\dot{\tilde{\theta }}}_6^T{\tilde{\theta }}_6+\mathrm{sign}\left(z_6^T\right)M\Gamma {\tilde{\theta }}_6\hfill \\ \hfill & & -\mathrm{sign}\left(z_6^T\right)M{w}^{\prime }-{\lambda }_{min}(M\Lambda ){∥z_6∥}^{1/2}+{\sigma }_{ss}^{T}M\mathrm{sign}\left(z_6\right).\hfill \end{array}$$

(43)

Substituting (38) with ${\dot{\widehat{\theta }}}_6=-{\dot{\tilde{\theta }}}_6$, and (34) with ${\dot{\tilde{c}}}_{lj,ki}=-{\dot{\widehat{c}}}_{lj,ki}$ and ${\dot{\tilde{d}}}_{ki}=-{\dot{\widehat{d}}}_{ki}$, the derivative (43) becomes:

$$\begin{array}{c}{\dot{V}}_2\le -z_5^T(K_{22}-{\displaystyle \frac{1}{2}}I)z_5-({\alpha }_2-{\delta }_2)∥{\tilde{x}}_d∥-({\alpha }_1-{\delta }_1){∥{\tilde{x}}_d∥}^2+{\displaystyle \frac{1}{2}}{z}_6^T{Q}_2^T{Q}_2{z}_6+({\sigma }_{ss}-b_{23})M\mathrm{sign}\left(z_6\right)- \hfill \\ \hfill ({\lambda }_{min}(M\Lambda )-{\lambda }_{max}\left(M\right)b_{22}){∥z_6∥}^{1/2}.\end{array}$$

Let us introduce a positive constant $0<{\theta }_1<1$, and assign ${\sigma }_{ss}=b_{23}$, to obtain:

$$\begin{array}{ccc}\hfill {\dot{V}}_2& \le & -(c_2-{\displaystyle \frac{1}{2}}){∥z_5∥}^2-({\alpha }_2-{\delta }_2)∥{\tilde{x}}_d∥-({\alpha }_1-{\delta }_1){∥{\tilde{x}}_d∥}^2+{\beta }_1{∥z_6∥}^2-(c_3-p_3){∥z_6∥}^{1/2}+{\theta }_1(c_3-p_3){∥z_6∥}^{1/2}-\hfill \\ \hfill & & {\theta }_1(c_3-p_3){∥z_6∥}^{1/2}\hfill \\ \hfill & \le & -(c_2-{\displaystyle \frac{1}{2}}){∥z_5∥}^2-({\alpha }_2-{\delta }_2)∥{\tilde{x}}_d∥-({\alpha }_1-{\delta }_1){∥{\tilde{x}}_d∥}^2-(c_3-p_3)(1-{\theta }_1){∥z_6∥}^{1/2}-\hfill \\ & & {∥z_6∥}^{1/2}({\theta }_1(c_3-p_3)-{\beta }_1{∥z_6∥}^{3/2})\hfill \end{array}$$

(44)

where

$$\begin{array}{c}\hfill c_2={\lambda }_{min}\left(K_{22}\right),c_3={\lambda }_{min}(M\Lambda ),p_3={\lambda }_{max}\left(M\right)b_{22},{\beta }_1={\displaystyle \frac{1}{2}}{\lambda }_{max}\left(Q_2^T{Q}_2\right)\end{array}$$

for a neighbor close to $z_6=0$, such that ${\theta }_1(c_3-p_3)-\beta {∥z_6∥}^{3/2}>0$, this is:

$$∥z_6∥<{\left({\displaystyle \frac{{\theta }_1(c_3-p_3)}{{\beta }_1}}\right)}^{2/3}.$$

Equation (44) results in:

$$\begin{array}{c}\hfill {\dot{V}}_2\le -(c_2-{\displaystyle \frac{1}{2}}){∥z_5∥}^2-({\alpha }_2-{\delta }_2)∥{\tilde{x}}_d∥-({\alpha }_1-{\delta }_1){∥{\tilde{x}}_d∥}^2-\\ \hfill (c_3-p_3)(1-{\theta }_1){∥z_6∥}^{1/2},\end{array}$$

(45)

if we choose

$${\alpha }_1>{\delta }_1, {\alpha }_2>{\delta }_2, c_2>{\displaystyle \frac{1}{2}}, c_3>p_3.$$

Thus, the Lyapunov function derivative is negative semi-definite. Through Barbalat’s lemma and LaSalle’s invariance principle, it is easy to see that the solution $∥{[z_5\left(t\right),{\tilde{x}}_d\left(t\right),z_6\left(t\right)]}^T∥$ tends asymptotically to zero, and the parameter estimates are bounded.

Finally, choosing the Lyapunov function of cascade system (40) and (41) as $V=V_1+V_2$ [39], whose derivative is represented in (42) and (45), yields:

$$\begin{array}{c}\hfill \dot{V}\le -c_1{∥{\zeta }_{21}∥}^2-(c_2-{\displaystyle \frac{1}{2}}){∥z_5∥}^2-({\alpha }_2-{\delta }_2)∥{\tilde{x}}_d∥-({\alpha }_1-{\delta }_1){∥{\tilde{x}}_d∥}^2-(c_3-p_3){∥z_6∥}^{1/2}\\ \hfill -{\kappa }_{21}^{-2}{\zeta }_{21}^T{P}_{21}{B}_{21,w}{B}_{21,w}^T{P}_{21}{\zeta }_{21}+2{\zeta }_{21}^T{P}_{21}{B}_{21,w}\left[\begin{array}{c}{w}_{10}\\ w_{12}\end{array}\right]+2{\zeta }_{21}^T{P}_{21}{Q}_1{z}_5.\end{array}$$

Replacing the bounds of the disturbances results in:

$$\begin{array}{cc}\hfill \dot{V}\le & -c_1{∥{\zeta }_{21}∥}^2-(c_2-{\displaystyle \frac{1}{2}}){∥z_5∥}^2-({\alpha }_2-{\delta }_2)∥{\tilde{x}}_d∥-({\alpha }_1-{\delta }_1){∥{\tilde{x}}_d∥}^2-(c_3-p_3){∥z_6∥}^{1/2}-\\ & {\kappa }_{21}^{-2}{\lambda }_{min}\left(P_{21}{B}_{21,w}{B}_{21,w}^T{P}_{21}\right){∥{\zeta }_2∥}^2+2{\lambda }_{max}\left(P_{21}{B}_{21}\right)∥{\zeta }_{21}∥(b_{21}∥{\zeta }_{21}∥+b_{31})+2{\zeta }_{21}^T{P}_{21}{Q}_1{z}_5\\ \hfill \le & -(c_1-p_1){∥{\zeta }_{21}∥}^2-({\kappa }_{21}^{-2}{p}_{11}-p_{12}){∥{\zeta }_{21}∥}^2+{\beta }_2∥{\zeta }_{21}∥-(c_2-p_2){∥z_5∥}^2-({\alpha }_2-{\delta }_2)∥{\tilde{x}}_d∥-({\alpha }_1-{\delta }_1){∥{\tilde{x}}_d∥}^2-\\ & (c_3-p_3){∥z_6∥}^{1/2}\end{array}$$

choosing

$$\begin{array}{ccc}\hfill c_1>p_1,& c_2>p_2,& {\kappa }_2>{\left({\displaystyle \frac{2b_{21}{\lambda }_{max}\left(P_{21}{B}_{21,w}\right)}{{\lambda }_{min}\left(P_{21}{B}_{21,w}{B}_{21,w}^T{P}_{21}\right)}}\right)}^{-1/2},\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill c_1=& {\lambda }_{min}(C_{21}^T{C}_{21}+{\mu }_{21}I),& p_1={\lambda }_{max}\left(P_{21}{P}_{21}\right),\hfill \\ \hfill {\beta }_2=& 2b_{31}{\lambda }_{max}\left(P_{21}{B}_{21,w}\right),& p_2={\lambda }_{max}(Q_1^T{Q}_1+{\displaystyle \frac{1}{2}}I).\hfill \end{array}$$

Thus, the Lyapunov function is negative semi-definite for all:

$$∥{\zeta }_{21}∥>{\displaystyle \frac{{\beta }_2}{(c_1-p_1){\theta }_2}}={\mu }_{\zeta },0<{\theta }_2<1;$$

however, a solution $∥{\zeta }_{21}\left(t\right)∥$ is ultimately bounded as follows [40]:

$$∥{\zeta }_{21}\left(t\right)∥\le b_{\zeta },b_{\zeta }={\mu }_{\zeta }\sqrt{{\displaystyle \frac{{\lambda }_{max}\left(P_{21}\right)}{{\lambda }_{min}\left(P_{21}\right)}}}.$$

Thus, the tracking error $∥z_3\left(t\right)∥$, from (21), is also ultimately bounded. However, since the solution $∥z_5\left(t\right)∥$ tends asymptotically to zero then, the output $∥\psi ∥$ tends to $∥{\psi }_r∥$ asymptotically.

5. Simulation Results

To demonstrate the effectiveness of the proposed control scheme, a hypothetical scenario is presented in which an aircraft with an initially unknown load travels through four points to unload part of it. For simulation purposes, a sequence of checkpoints i ($C{P}_i$), where $i=\{0,1,\cdots ,4\}$ are chosen as $C{P}_0=(0,0,5)$, $C{P}_1=(1,0,5)$, $C{P}_2=(1,1,5)$, $C{P}_3=(0,1,5)$, and $C{P}_4=(0,0,5)$ meters at intervals of 4 s; each checkpoint is a set of reference $(x_r,y_r,z_r)$. In $C{P}_i$ the mass $m_i$ is released, and

Table 1. Loads to be released at each checkpoint.

Checkpoint	Load
$C{P}_1$	$m_1=1.500$	$\mathrm{kg}$
$C{P}_2$	$m_2=3.460$	$\mathrm{kg}$
$C{P}_3$	$m_3=2.050$	$\mathrm{kg}$

shows the loads to be released.

The parameters used are shown in

Table 2. Parameters of quadrotor.

Nominal Parameter	Value
m	2.450	$\mathrm{kg}$
$I_{x,y}$	$21.6\times {10}^{-3}$	$\mathrm{kg}{\mathrm{m}}^2$
$I_z$	$43.2\times {10}^{-3}$	$\mathrm{kg}{\mathrm{m}}^2$
$J_R$	$3.357\times {10}^{-5}$	$\mathrm{kg}{\mathrm{m}}^2$
d	$0.45$	$\mathrm{m}$

. During the simulation, the inertias are modified with respect to the relation of the total and nominal mass.

For the altitude block $SB1$ in control law (15), $K_1=[9,6]$ is defined, and for the adaptive dynamics (16), $\gamma =2.5$ is established. The gain of the ${\mathcal{H}}_{\infty }$ part is to find resolving (19) con ${\kappa }_1=0.2818$ y ${\rho }_1=0.1$, resulting in:

$$P_1=\left[\begin{array}{cc}2.3853& 0.1143\\ 0.1143& 0.3723\end{array}\right].$$

Likewise, for the latitudinal–longitudinal block $SB2$, the gain control (23) is established as $K_{21}=\left[\begin{array}{cccc}4& 0& 3& 0\\ 0& 4& 0& 3\end{array}\right]$. The matrix gain for the part that attenuates the unmatched disturbances $v_{21}$ is to be found through (25):

$$P_{21}=\left[\begin{array}{cccc}2.2961& 1.0485& 0.2792& 0.1487\\ 1.0485& 2.2961& 0.1487& 0.2792\\ 0.2792& 0.1487& 0.7397& 0.5059\\ 0.1487& 0.2792& 0.5059& 0.7397\end{array}\right],$$

using ${\kappa }_{21}=0.1961$ and ${\rho }_{21}=0.1$.

Finally, the gains in (37), are as follows: $\Lambda =\mathrm{diag}\{4,4,2\}$ and $M=\mathrm{diag}\{8,8,3\}$.

To prove control robustness, the velocity wind in aerodynamic functions [30], is set in time dependence as:

$${\Omega }_{air}=\left[\begin{array}{c}10cos\left(0.2t\right)\\ 10cos(0.2t-0.3\pi )\\ 10cos(0.2t-0.6\pi )\\ 10cos(0.2t-0.9\pi )e^{-0.5t}\\ 10e^{-{(t-7)}^2/8}-15e^{-{(t-16)}^2/8}\\ 10e^{-{(t-7)}^2/8}-15e^{-{(t-16)}^2/8}\end{array}\right].$$

(46)

In the graphs of Figure 3, tracking using the proposed adaptive control scheme (blue line) is seen. Another one (red line) is also shown, with the control without adapting the parameters, that is, setting Equations (16) and (38) as zero at all times. Figure 3a–d are depicted the tracking of positions z, x, y, and $\psi$ orientation, respectively. The tracking in the Earth reference frame is shown in Figure 4. Notice that the chattering is not significant due to the ST action. Figure 5 shows the detail at $C{P}_4$, when all the surfaces are zero in the ST algorithm.

Estimation of the total mass using the Discontinuous Gradient algorithm is shown in Figure 6. In the first 8 s, the mass is the sum of all the loads and the nominal one, that is $m+m_1+m_2+m_3=9.460\mathrm{kg}$; then, in $C{P}_1$, the resulting mass is the sum of all of them when $m_1$ is released, that is $m+m_2+m_3=7.960\mathrm{kg}$. Afterward, at time $t=13\mathrm{s}$, the load of mass $m_2$ is released, and hence, the total mass is $m+m_3=4.500\mathrm{kg}$; finally, when the quadrotor arrives to $C{P}_3$, at time $t=18\mathrm{s}$, the mass $m_3$ is discharged, where the total mass equals m until the end of simulation. On the other hand, chattering in mass adaptation is not noticeable, thanks to the integral action in the finite-time mass estimation algorithm.

6. Conclusions

This paper presents a model-based nonlinear ${\mathcal{H}}_{\infty }$ controller for quadrotors aerial vehicles in which the plant model is given in terms of quaternions reducing the computational cost of executing model-based control laws. The study is developed for a scenario in which mass variations and external wind perturbations cannot be neglected, and a Super-Twisting Sliding-Mode control strategy combination ensures that the trajectory tracking control task is performed even in the presence of external perturbation and system mass variations. Simulation results, obtained with MATLAB R2023a/Simulink, show that the accuracy of the proposed control approach, in terms of trajectory tracking error, is higher if compared with other control strategies that do not consider parameter variations. To compare the quality and performance of the proposed approach concerning existing control methods, in future works, the use of a disturbance observer along with a finite-time parameter estimator will be explored.