Decentralized Adaptive Fault-Tolerant Cooperative Control for Multiple UAVs with Input Saturation and State Constraints

Abstract

This paper proposes a fault-tolerant cooperative control (FTCC) scheme for multiple UAVs in a distributed communication network against input saturation, full-state constraints, actuator faults, and unknown dynamics. Firstly, by considering physical limitations, an auxiliary control signal is designed to simplify the analysis process. Secondly, to avoid the difficulty in the back-stepping design caused by full-state constraints, virtual control signals are constructed to transform constrained variables, while the dynamic surface control is adopted to avoid the phenomenon of “differential explosion.” Thirdly, a disturbance observer (DO) is designed to estimate the unknown uncertainty caused by parameter uncertainty and actuator fault. Moreover, a recurrent wavelet fuzzy neural network (RWFNN) is used to compensate for the estimation errors of DO. Finally, it is proved that all states are uniformly ultimately bounded (UUB) by Lyapunov and invariant set theory. The effectiveness of the proposed scheme is further demonstrated by the simulation results.

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1. Introduction

In recent years, the development of unmanned aerial vehicle (UAV) technology has led to wide applications. However, single UAV provides limited capabilities, which may not be applicable to some highly complex tasks. Inspired by multiagent technology, researchers begin to investigate the application technology of multiple UAVs (multi-UAVs). Compared with a single UAV, multi-UAVs have more benefits in terms of forest fire monitoring, area detection, and disaster assistance. Unlike a single UAV, the cooperative control of multi-UAVs need the information from neighboring UAVs, which significantly increasing the control design challenge.

As the basis of multi-UAVs control, the cooperative control design is an important task. In the past few years, numerous research results of cooperative control have been reported. In [1], a cooperative control strategy for motion control of multiple unmanned vehicles was proposed, which can keep the formation during the motion. In [2], a novel distributed intermittent control framework for containment control of multiagent system was proposed, which can reduce the communication burden via a directed graph. In [3], the obstacle avoidance problem of multi-UAVs in multiple obstacle environment was studied. In [4], a robust adaptive control strategy for cooperative control of UAVs under the decentralized communication network was proposed against uncertainty. In [5], the authors investigated the cooperative transport control problem using multirotor UAVs. In [6], a system analysis method was proposed for the tracking control problem of multi-UAVs. The distributed framework was used to describe the dynamic model of UAV, and the information of nodes and networks were considered in the distributed control design. [7] studied the output feedback formation control of multi-UAVs without velocity and angular velocity sensors, which were obtained via the state observer. However, the above researches only focused on the distributed control of the first-order or second-order systems, and there exist few research on the cooperative control of fixed-wing UAVs with high-order nonlinear characteristics.

In addition, the number of components in the multi-UAV system is more than that of a single UAV. Therefore, the probability of multi-UAVs suffering from the actuator, sensor, or component faults is higher than that of a single UAV. At present, many research results show that the incidence of actuator fault in flight is the highest. Therefore, many scholars mainly focus on actuator faults [8, 9]. The probability of actuator fault occurred in the multi-UAVs will also increase due to the fact that the number of actuators is significantly increased. When an actuator fault of a UAV in the communication network occurs and is not handled in time, it will reduce the stability and threaten the safety of all UAVs [10], making the investigation on fault-tolerant cooperative control (FTCC) a necessary task. In [11], based on the design of inner-outer-loop control and back-stepping control, an FTCC strategy was designed for multi-UAVs against permanent faults. In [12], Yu et al. further studied the FTCC design method of multi-UAVs by using a similar control framework of reference generator technology. It should be emphasized that the results [11, 12] are about the FTCC scheme of multi-UAVs under the master-slave framework, which cannot be directly applied to the distributed cooperative control design. Considering the diversity of research on cooperative control of multi-UAVs in the communication network, the FTCC scheme for multi-UAVs under distributed communication network needs to be further investigated.

Moreover, the actual multi-UAV system often encounters some problems such as input saturation, inaccurate aerodynamic parameters, and external interference, which lead to system instability or performance degradation [13–16]. Recently, many results have been reported to solve the problem of input saturation. In [17], a piecewise auxiliary system was introduced to deal with the asymmetric input constraints for a class of uncertain multi-input and multi-output nonlinear systems. Then, the auxiliary system was further used to deal with the force and moment constraints on ship [18]. In [19], another auxiliary signal was constructed using the error between the desired control input and the saturation control input. In [20], to solve the disadvantages of conventional methods based on the hyperbolic tangent function, an $n$th-order auxiliary dynamic system was skillfully constructed to avoid the effect of input saturation.

It should be emphasized that although numerous studies have been reported on the above literature, few results have studied the input saturation, inaccurate aerodynamic parameters, and external interference encountered by multi-UAVs in distributed communication networks at the same time. However, such factors are inevitable in the formation flight of multi-UAVs. If these factors are not solved in time, it may lead to the instability of the networked UAV system.

Furthermore, due to physical limitations, UAV states should be constrained. However, control strategies developed recently for multi-UAVs in the distributed communication network rarely consider these constraints on the states. Based on the above discussion, this paper proposes a distributed FTCC scheme for multi-UAVs under the distributed communication network with input saturation, state constraints, actuator faults, and unknown dynamics. In this work, to avoid the difficulty in designing the control policy due to the input saturation, an auxiliary control signal is designed to transform the restricted input. To handle the full-state constraint problem, virtual control signals are defined to replace the constraints, which can simplify the back-stepping design. For the unknown nonlinear dynamics caused by actuator faults and other unknown uncertainties, disturbance observer (DO) is designed for providing the estimation, while a recurrent wavelet fuzzy neural network (RWFNN) is adopted to further compensate the estimation error. In the RWFNN, the online adaptive learning strategy of parameters is designed based on the Lyapunov theory. Compared with other existing works, the main contributions of this paper are as follows:

(1) In [21–23], actuator faults, input saturation, output constraints, and external disturbances were considered, while the state constraints were not taken into account. To obtain satisfactory control performance against such factors, the FTCC scheme is designed in this paper by simultaneously considering the state constraints, actuator faults, and external disturbances.

(2) Compared with [24–27], which assessed uncertainty dynamics by designing a DO without compensation of the DO estimation error, this work further adopts an RWFNN to offset the estimation error, in which the parameters are updated by the proposed online learning strategy.

The organization of this paper is arranged as follows. Section 2 describes the preliminaries and problem statement. The design process of the FTCC scheme and the stability analysis are given in Section 3. Section 4 shows the simulation results and analysis. Finally, the conclusion is drawn in Section 5.

2. Preliminaries and Problem Statement

2.1. System Dynamics

In this paper, the cooperative control of $N$ UAVs is investigated. The set of UAVs is denoted as $\Omega =1,2,\cdots ,N$, and the position dynamics of the $i$th UAV is described as $$\begin{array}{}\text{(1)}& \begin{array}{c}{\dot{x}}_i=V_i\cos {\gamma }_i\cos {\chi }_{i,}\hfill \\ {\dot{y}}_i=V_i\cos {\gamma }_i\sin {\chi }_i,\hfill \\ {\dot{z}}_i=V_i\sin {\gamma }_i,\hfill \end{array}\end{array}$$where $i\in \Omega$, $x_i$, $y_i$, and $z_i$ are the positions. $V_i$, ${\gamma }_i$, and ${\chi }_i$ are velocity, flight path angle, and heading angle, respectively.

The aerodynamic force equations are given by $$\begin{array}{}\text{(2)}& \begin{array}{c}{\dot{V}}_i=\frac{1}{m_i}-D_i+T_i\cos {\alpha }_i\cos {\beta }_i-g\sin {\gamma }_{i,}\hfill \\ {\dot{\chi }}_i=\frac{1}{m_i{V}_i\cos {\gamma }_i}{L}_i\sin {\mu }_i+Y_i\cos {\mu }_i+T_i\sin {\alpha }_i\sin {\mu }_i-\cos {\alpha }_i\sin {\beta }_i\cos {\mu }_i,\hfill \\ {\dot{\gamma }}_i=\frac{1}{m_i{V}_i}{L}_i\cos {\mu }_i-Y_i\sin {\mu }_i-\frac{g\cos {\gamma }_i}{V_i}+\frac{T_i}{m_i{V}_i}\cos {\alpha }_i\sin {\beta }_i\sin {\mu }_i+\sin {\alpha }_i\cos {\mu }_i,\hfill \end{array}\end{array}$$where $i\in \Omega$, $m_i$ and $g$ are the mass of the $i$th UAV and gravitational coefficient, respectively; $T_i$, $D_i$, $L_i$, $Y_i$ are the thrust, drag, lift, and lateral forces, respectively, and ${\mu }_i$, ${\alpha }_i$, and ${\beta }_i$ are the bank angle, angle of attack, and sideslip angle, respectively.

The attitude kinematic model is expressed as $$\begin{array}{}\text{(3)}& \begin{array}{c}{\dot{\alpha }}_i=q_i-\tan {\beta }_i{p}_i\cos {\alpha }_i+r_i\sin {\alpha }_i,\hfill \\ {\dot{\beta }}_i=p_i\sin {\alpha }_i-r_i\cos {\alpha }_i+{\dot{\chi }}_i\cos {\gamma }_i·\cos {\mu }_i-{\dot{\gamma }}_i\sin {\mu }_i,\hfill \\ {\dot{\mu }}_i=p_i\cos {\alpha }_i+\frac{r_i\sin {\alpha }_i}{\cos {\beta }_i}+{\dot{\gamma }}_i·\cos {\mu }_i\tanh {\beta }_i+{\dot{\chi }}_i\sin {\gamma }_i+\cos {\gamma }_i\sin {\mu }_i\tan {\beta }_i,\hfill \end{array}\end{array}$$where $i\in \Omega$. $p_i$, $q_i$, and $r_i$ are the angular rates.

The angular rate model is given as $$\begin{array}{}\text{(4)}& \begin{array}{c}{\dot{p}}_i=c_{i1}{r}_i+c_{i2}{p}_i{q}_i+c_{i3}{\mathcal{L}}_i+c_{i4}{\mathcal{N}}_i,\hfill \\ {\dot{q}}_i=c_{i5}{p}_i{r}_i-c_{i6}{p}_i^2-r_i^2+c_{i7}{\mathcal{M}}_i,\hfill \\ {\dot{r}}_i=c_{i8}{p}_i-c_{i2}{r}_i{q}_i+c_{i4}{\mathcal{L}}_i+c_{i9}{\mathcal{N}}_i,\hfill \end{array}\end{array}$$where $i\in \Omega$. ${\mathcal{L}}_i$, ${\mathcal{M}}_i$, and ${\mathcal{N}}_i$ are roll, pitch, and yaw moments, respectively.

The forces $T_i$, $D_i$, $L_i$, and $Y_i$ and the aerodynamic moments ${\mathcal{L}}_i$, ${\mathcal{M}}_i$, and ${\mathcal{N}}_i$ are expressed as $$\begin{array}{}\text{(5)}& \begin{array}{c}{T}_i=T_{i\max }{\delta }_{T_i},D_i=Q_i{s}_i{C}_{iD},\hfill \\ L_i=Q_i{s}_i{C}_{iL},Y_i=Q_i{s}_i{C}_{iY},\hfill \\ {\mathcal{L}}_i=Q_i{s}_i{b}_i{C}_{i\mathcal{L}},{\mathcal{M}}_i=Q_i{s}_i{c}_i{C}_{i\mathcal{M},}\hfill \\ {\mathcal{N}}_i=Q_i{s}_i{b}_i{C}_{i\mathcal{N}},\hfill \end{array}\end{array}$$where $Q_i=\rho V_i^2/2$ is the dynamic pressure and $s_i$, $b_i$, and $c_i$ represent the wing area, wing span, and mean aerodynamic chord, respectively. $T_{i\max }$ and ${\delta }_{T_i}$ are the maximum thrust and instantaneous thrust throttle setting, respectively. $C_{iL}$, $C_{iD}$, $C_{iY}$, $C_{i\mathcal{L}}$, $C_{i\mathcal{M}}$, and $C_{i\mathcal{N}}$ are given by $$\begin{array}{}\text{(6)}& \begin{array}{c}{C}_{iL}=C_{iL0}+C_{iL\alpha }{\alpha }_i,\hfill \\ C_{iD}=C_{iD0}+C_{iD\alpha }{\alpha }_i+C_{iD{\alpha }^2}{\alpha }_i^2,\hfill \\ C_{iY}=C_{iY0}+C_{iY\beta }{\beta }_i,\hfill \end{array}\text{(7)}& \begin{array}{c}{C}_{i\mathcal{L}}=C_{i\mathcal{L}0}+C_{i\mathcal{L}\beta }{\beta }_i+C_{i\mathcal{L}{\delta }_a}{\delta }_{ia}+C_{i\mathcal{L}{\delta }_r}{\delta }_{ir}+\frac{C_{i\mathcal{L}p}{b}_i{p}_i+C_{i\mathcal{L}r}{b}_i{r}_i}{2V_i},\hfill \\ C_{i\mathcal{M}}=C_{i\mathcal{M}0}+C_{i\mathcal{M}\alpha }{\alpha }_i+C_{i\mathcal{M}{\delta }_e}{\delta }_{ie}+\frac{C_{i\mathcal{M}q}{c}_i{q}_i}{2V_i},\hfill \\ C_{i\mathcal{N}}=C_{i\mathcal{N}0}+C_{i\mathcal{N}\beta }{\beta }_i+C_{i\mathcal{N}{\delta }_{ia}}{\delta }_{ia}+C_{i\mathcal{N}{\delta }_r}{\delta }_{ir}+\frac{C_{i\mathcal{N}p}{b}_i{p}_i}{2V_i}+\frac{C_{i\mathcal{N}r}{b}_i{r}_i}{2V_i},\hfill \end{array}\end{array}$$where ${\delta }_{ia}$, ${\delta }_{ie}$, and ${\delta }_{ir}$ are aileron, elevator, and rudder deflections, respectively. $C_{iL0}$, $C_{iL\alpha }$, $C_{iD0}$, $C_{iD\alpha }$, $C_{iD{\alpha }^2}$, $C_{iY0}$, $C_{iY\beta }$, $C_{i\mathcal{L}0}$, $C_{i\mathcal{L}\beta }$, $C_{i\mathcal{L}{\delta }_a}$, $C_{i\mathcal{L}{\delta }_r}$, $C_{i\mathcal{L}p}$, $C_{i\mathcal{L}r}$, $C_{i\mathcal{M}0}$, $C_{i\mathcal{M}\alpha }$, $C_{i\mathcal{M}{\delta }_e}$, $C_{i\mathcal{M}q}$, $C_{i\mathcal{N}0}$, $C_{i\mathcal{N}\beta }$, $C_{i\mathcal{N}{\delta }_a}$, $C_{i\mathcal{N}{\delta }_r}$, $C_{i\mathcal{N}p}$, and $C_{i\mathcal{N}r}$ are aerodynamic derivatives. The definition of the inertial terms $c_{ij}j=1,2,\cdots ,9$ in (4) can be found in [28].

2.2. Control-Oriented Model

By defining ${\mathbf{X}}_{i1}={{\mu }_i,{\alpha }_i,{\beta }_i}^T$, ${\mathbf{X}}_{i2}={p_i,q_i,r_i}^T$, and ${\mathbf{U}}_i={{\delta }_{1a},{\delta }_{1e},{\delta }_{1r}}^T$ and substituting (5), (6), and (7) into (1), (2), (3), and (4), then it follows that $$\begin{array}{}\text{(8)}& {\dot{\mathbf{X}}}_{i1}={\mathbf{F}}_{i1}+{\mathbf{G}}_{i1}{\mathbf{X}}_{i2}\text{(9)}& {\dot{\mathbf{X}}}_{i2}={\mathbf{F}}_{i2}+{\mathbf{G}}_{i2}{\mathbf{U}}_i\end{array}$$where ${\mathbf{F}}_{i1}$, ${\mathbf{G}}_{i1}$, ${\mathbf{F}}_{i2}$, and ${\mathbf{G}}_{i2}$ are given by $$\begin{array}{c}\text{(10)}& {\mathbf{F}}_{i1}=\begin{array}{ccc}0& \sin {\gamma }_i+\cos {\gamma }_i\sin {\mu }_i\tan {\beta }_i& \cos {\mu }_i\tan {\beta }_i\\ 0& -\frac{\cos {\gamma }_i\sin {\mu }_i}{\cos {\beta }_i}& -\frac{\cos {\mu }_i}{\cos {\beta }_i}\\ 0& \cos {\gamma }_i\cos {\mu }_i& -\sin {\mu }_i\end{array}\begin{array}{c}\frac{-D_i+T_i\cos {\alpha }_i\cos {\beta }_i}{m_i}-g\sin {\gamma }_i\\ \frac{1}{m_i{V}_i\cos {\gamma }_i}{L}_i\sin {\mu }_i+Y_i\cos {\mu }_i+\\ T_i\sin {\alpha }_i\sin {\mu }_i-\cos {\alpha }_i\sin {\beta }_i\cos {\mu }_i\\ -\frac{g\cos {\gamma }_i}{V_i}+\frac{1}{m_i{V}_i}{L}_i\cos {\mu }_i-Y_i\sin {\mu }_i\\ +T_i\cos {\alpha }_i\sin {\beta }_i\sin {\mu }_i+\sin {\alpha }_i\cos {\mu }_i\end{array},{\mathbf{G}}_{i1}=\begin{array}{ccc}\frac{\cos {\alpha }_i}{\cos {\beta }_i}& 0& \frac{\sin {\alpha }_i}{\cos {\beta }_i}\\ -\cos {\alpha }_i\tan {\beta }_i& 1& -\sin {\alpha }_i\tan {\beta }_i\\ \sin {\alpha }_i& 0& -\cos {\alpha }_i\end{array},\\ \begin{array}{c}{\mathbf{F}}_{i2}={F_{i21},F_{i22},F_{i23}}^{\text{T}}\\ \begin{array}{c}{F}_{i21}=c_{i1}{q}_i{r}_i+c_{i2}{p}_i{q}_i+c_{i3}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{L}0}+C_{i\mathcal{L}{\beta }_i}\\ +\frac{C_{i\mathcal{L}p}{b}_i{p}_i}{2V_i}+\frac{C_{i\mathcal{L}r}{b}_i{r}_i}{2V_i}\\ F_{i22}=c_{i5}{p}_i{r}_i-c_{i6}{p}_i^2-r_i^2+c_{i7}{\overline{q}}_i{s}_i{c}_i{C}_{i\mathcal{M}0}\\ +C_{i\mathcal{M}\alpha }{\alpha }_i+\frac{C_{i\mathcal{M}q}{c}_i{q}_i}{2V_i}\\ F_{i23}=c_{i8}{p}_i{q}_i-c_{i2}{q}_i{r}_i+c_{i4}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{L}0}+C_{i\mathcal{L}\beta }·\\ {\beta }_i+\frac{C_{i\mathcal{L}p}{b}_i{p}_i}{2V_i}+\frac{C_{i\mathcal{L}r}{b}_i{r}_i}{2V_i}+c_{i9}{\overline{q}}_i{s}_i{b}_i·\\ C_{i\mathcal{N}0}+C_{i\mathcal{N}{\beta }_i}+\frac{C_{i\mathcal{N}p}{b}_i{p}_i}{2V_i}+\frac{C_{i\mathcal{N}r}{b}_i{r}_i}{2V_i}\end{array}\end{array}\\ \begin{array}{c}{\mathbf{G}}_{i2}=\begin{array}{ccc}{a}_{11}& 0& a_{13}\\ 0& a_{22}& 0\\ a_{31}& 0& a_{33}\end{array},\hfill \\ a_{11}=c_{i3}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{L}{\delta }_a}+c_{i4}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{N}{\delta }_a},\hfill \\ a_{13}=c_{i3}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{L}{\delta }_r}+c_{i4}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{N}{\delta }_r},\hfill \\ a_{22}=c_{i7}{\overline{q}}_i{s}_i{c}_i{C}_{i\mathcal{M}{\delta }_e},\hfill \\ a_{31}=c_{i4}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{L}{\delta }_a}+c_{i9}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{N}{\delta }_a},\hfill \\ a_{33}=c_{i4}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{L}{\delta }_r}+c_{i9}{\overline{q}}_i{s}_i{b}_i{C}_{i\mathcal{N}{\delta }_r.}\hfill \end{array}\end{array}$$

It can be seen that ${\mathbf{F}}_{i2}$ and ${\mathbf{G}}_{i2}$ have many aerodynamic parameters. However, it is difficult to obtain accurate aerodynamic parameters of UAVs in practical engineering applications. To facilitate the design of the controller, ${\mathbf{F}}_{i2}$ and ${\mathbf{G}}_{i2}$ can be decomposed into known items ${\mathbf{F}}_{i20}$, ${\mathbf{G}}_{i20}$ and unknown items $\mathbf{\Delta }{\mathbf{F}}_{i2}$, $\mathbf{\Delta }{\mathbf{G}}_{i2}$, respectively.

Then, the attitude model can be described as $$\begin{array}{}\text{(11)}& {\dot{\mathbf{X}}}_{i1}={\mathbf{F}}_{i1}+{\mathbf{G}}_{i1}{\mathbf{X}}_{i2},\text{(12)}& {\dot{\mathbf{X}}}_{i2}={\mathbf{F}}_{i20}+\mathbf{\Delta }{\mathbf{F}}_{i2}+{\mathbf{G}}_{i20}+\mathbf{\Delta }{\mathbf{G}}_{i2}{\mathbf{U}}_i,\end{array}$$where $\mathbf{\Delta }{\mathbf{F}}_{i2}$, and $\mathbf{\Delta }{\mathbf{G}}_{i2}$ are unknown nonlinear functions caused by the uncertain parameters, while ${\mathbf{F}}_{i1}$, ${\mathbf{G}}_{i1}$, ${\mathbf{F}}_{i20}$, and ${\mathbf{G}}_{i20}$ are known functions.

Remark 1.

Due to the physical constraints, the sideslip angle ${\beta }_i\ne \pm \pi /2$, and $\det {\mathit{G}}_{i1}=-\sec {\beta }_i$, so ${\mathit{G}}_{i1}$ is invertible. In addition, ${\mathit{G}}_{i2}$ is related to aerodynamic parameters so that it is invertible in the flight envelope.

2.3. Actuator Fault and Input Saturation

In this paper, the actuator fault is considered, which includes gain and bias failures. Therefore, the fault model can be expressed as [29] $$\begin{array}{}\text{(13)}& {\mathbf{U}}_i={\mathit{\rho }}_{\mathit{i}}{\mathbf{U}}_{i0}+{\mathbf{U}}_{\mathit{\text{if}}},\end{array}$$where $i\in \Omega$, ${\mathbf{U}}_{i0}={u_{i01},u_{i02},u_{i03}}^T$ represents the designed control signal, and ${\mathbf{U}}_i={{\delta }_{ia},{\delta }_{ie},{\delta }_{ir}}^T$ is the actual control signal. ${\mathit{\rho }}_i=\mathrm{diag}{\rho }_{i1},{\rho }_{i2},{\rho }_{i3}$ with $0<{\rho }_{i1},{\rho }_{i2},{\rho }_{i3}\le 1$ represents the gain fault matrix, and ${\mathbf{U}}_{\mathit{\text{if}}}\in {ℝ}^3$ represents bias fault vector.

In the practical application, the output of the actuator is limited. In order to avoid the incredible phenomenon caused by actuator saturation, the designed control signal ${\mathbf{U}}_{i0}$ needs to satisfy the following constraint: $$\begin{array}{}\text{(14)}& u_{i0\tau \min }\le u_{i0\tau }\le u_{i0\tau \max },\hspace{1em}\tau =1,2,3,\end{array}$$where $\tau =1,2,3$, $u_{i0\tau \max }$ is a positive constant and $u_{i0\tau \min }$ is a negative constant, which is the maximum and minimum allowable values for the actuator, respectively.

To solve input saturation problem, an auxiliary signal ${\mathbf{v}}_i={v_{i1},v_{i2},v_{i3}}^T$ is used to get control signal ${\mathbf{U}}_{i0}{\mathbf{v}}_i$, and ${\mathbf{U}}_{i0}{v}_i={u_{i01}{v}_{i1},u_{i02}{v}_{i2},u_{i03}{v}_{i3}}^{\text{T}}$, which is expressed as $$\begin{array}{}\text{(15)}& u_{i0\tau }{v}_{i\tau }=\begin{array}{c}{u}_{i0\tau \max }\tanh \frac{v_{i\tau }}{u_{i0\tau \max }},v_{i\tau }\ge 0,\hfill \\ u_{i0\tau \min }\tanh \frac{v_{i\tau }}{u_{i0\tau \min }},v_{i\tau }<0.\hfill \end{array}\end{array}$$

By substituting (13) and (15) into (12), then the attitude model can be expressed as $$\begin{array}{}\text{(16)}& {\dot{\mathbf{X}}}_{i1}={\mathbf{F}}_{i1}+{\mathbf{G}}_{i1}{\mathbf{X}}_{i2},\text{(17)}& {\dot{\mathbf{X}}}_{i2}={\mathbf{F}}_{i20}+{\mathbf{G}}_{i20}{\mathbf{U}}_{i0}{\mathbf{v}}_i+{\mathbf{D}}_i,\end{array}$$where ${\mathbf{D}}_i=\mathbf{\Delta }{\mathbf{F}}_{i2}+{\mathbf{G}}_{i20}{\rho }_i-{\mathbf{I}}_3{\mathbf{U}}_{i0}+{\mathbf{U}}_{\mathit{\text{if}}}+\mathbf{\Delta }{\mathbf{G}}_{i2}{\mathbf{U}}_i$ is an unknown nonlinear function. Due to ${\mathbf{D}}_i$ being related to auxiliary control signal ${\mathbf{v}}_i$, designing the observer of unknown function ${\mathbf{D}}_i$ for generating the control signal ${\mathbf{v}}_i$ will cause the problem of “algebraic ring,” which is solved by introducing the following first-order filter: $$\begin{array}{}\text{(18)}& {\dot{\mathit{v}}}_i=-\mathbf{\Lambda }{\mathit{v}}_i+\mathbf{\xi },\end{array}$$where $\mathbf{\Lambda }$ is a diagnonal matrix with positive eigenvalue and $\mathbf{\xi }$ is an auxiliary control signal.

Remark 2.

As shown in (13), the fault that occurs in the actuator will diminish its ability to provide control input. For example, the range of motion of the rudder surface can reach $-25~25$ deg in the normal state, while it may deteriorate into $-20~20$ deg after the fault occurs. It seems in the fault conditions the actuator is more likely to occur saturation, i.e., cannot reach to the expected control value. In this paper, the upper and lower boundaries of the control input are fixed to the values in the normal state of the actuator, and a hyperbolic function is used to prevent actuator saturation as shown in (15). Meanwhile, using virtual control signal ${\mathbf{v}}_i$ and $\mathbf{\xi }$ to generate the expected control signal ${\mathbf{U}}_{i0}$ and using RWFNN to evaluate uncertainty item ${\mathbf{D}}_i$ which contains the actuator bias section, even though the fault could occur, the system can still keep stable.

2.4. State Constraints

The states ${\mathbf{X}}_{i1}={{\mu }_i,{\alpha }_i,{\beta }_i}^{\text{T}}$ and ${\mathbf{X}}_{i2}={p,q,r}^{\text{T}}$ generally have constraints in the practical application. In this paper, such a problem has been concerned. Due to the fact that states ${\mathbf{X}}_{i1}$ and ${\mathbf{X}}_{i2}$ have limits, inspired by works [30, 31], a transformation is used to convert the restricted states ${\mathbf{X}}_{i1}$ and ${\mathbf{X}}_{i2}$ to unrestricted states ${\mathbf{Z}}_{i1}$ and ${\mathbf{Z}}_{i2}$: $$\begin{array}{}\text{(19)}& Z_{i1\tau }=\ln \frac{X_{i1\tau }-{\underset{¯}{X}}_{i1\tau }}{{\overline{X}}_{i1\tau }-X_{i1\tau }},\text{(20)}& Z_{i2\tau }=\ln \frac{X_{i2\tau }-{\underset{¯}{X}}_{i2\tau }}{{\overline{X}}_{i2\tau }-X_{i2\tau }},\end{array}$$where $\tau =1,2,3$. $X_{i1\tau }$, $X_{i2\tau }$, $Z_{i1\tau }$ and $Z_{i2\tau }$ represent the $\tau$th element of ${\mathbf{X}}_{i1}$, ${\mathbf{X}}_{i2}$, ${\mathbf{Z}}_{i1}$, and ${\mathbf{Z}}_{i2}$, respectively. ${\overline{X}}_{i1\tau }$ and ${\overline{X}}_{i2\tau }$ represent maximum allowable range of the $\tau$th element of ${\mathbf{X}}_{i1}$ and ${\mathbf{X}}_{i2}$, respectively, while the ${\underset{¯}{X}}_{i1\tau }$ and ${\underset{¯}{X}}_{i2\tau }$ represent minimum allowable range of the $\tau$th element of ${\mathbf{X}}_{i1}$ and ${\mathbf{X}}_{i2}$, respectively.

Remark 3.

Since (19) and (20) are bijective, ${\mathbf{X}}_{i1}$ and ${\mathbf{X}}_{i2}$ will always stay in their own limits if ${\mathbf{Z}}_{i1}$ and ${\mathbf{Z}}_{i2}$ are bounded on $\forall t\ge 0$.

2.5. Basic Graph Theory

In this paper, an undirected graph $\mathcal{G}=\mathcal{V},\mathcal{C},\mathcal{A}$ is used to describe the formation flight of $N$ UAVs. The set of UAVs is described by $\mathcal{V}=v_1,v_2,\cdots ,v_N$, $\mathcal{C}\subseteq \mathcal{V}\times \mathcal{V}$ represents the communication links between UAVs, and $\mathcal{A}\in R^{N\times N}$ is the adjacency matrix. If the link between the $i$th UAV and $j$th UAV exists, $a_{ij}=a_{ji}=1$, which are the elements of $\mathcal{A}$. A path from the $i$th UAV to the $k$th UAV can be a sequence $v_i⟶v_j⟶v_k$, where $v_i,v_j,v_j,v_k\in \mathcal{C}$. If there exists a path between any two UAVs, then $\mathcal{G}$ is a connected graph. A set is defined as $N_i=j:v_i,v_j\in \mathcal{C},i\ne j$, and the degree matrix is defined as $\mathcal{D}=\mathrm{diag}{d}_i\in {ℝ}^{N\times N}$, where $d_i={\sum }_{j\in N_i}{a}_{ij}$. The Laplacian matrix $\mathcal{L}\in {ℝ}^{N\times N}$ is denoted as $$\begin{array}{}\text{(21)}& \mathcal{L}=\mathcal{D}-\mathcal{A}.\end{array}$$

Assumption 4.

The undirected graph $\mathcal{G}$ containing $N$ UAVs is connected.

Lemma 5.

Under Assumption 4, $\mathcal{L}$ and $\mathcal{L}+\mathrm{diag}\Omega$ are symmetric and positive definite [32].

2.6. Control Objective

In this paper, the control objective is to design an FTCC scheme for $N$ UAVs, such that the attitude tracking error of each UAVs can be finally uniformly bounded, while the attitude ${\mathbf{X}}_{i1}$ and ${\mathbf{X}}_{i2}$ of all UAVs are always in limits, even when a portion of UAVs is subjected to actuator saturation and actuator faults.

3. Fault-Tolerant Cooperative Controller Design and Stability Analysis

In this section, the process of designing the FTCC scheme for $N$ UAVs will be described. A main method adopted in the design is transforming the individual tracking error of each UAV to the synchronization tracking error.

3.1. Fault-Tolerant Cooperative Controller Design

Define the independent tracking error of $i$th UAV as ${\tilde{\mathbf{Z}}}_{i1}={\mathbf{Z}}_{i1}-{\mathbf{Z}}_{i1d}$, then the cooperative tracking error of $i$th UAV is defined as $$\begin{array}{}\text{(22)}& {\mathbf{E}}_{i1}={\lambda }_1{\tilde{\mathbf{Z}}}_{i1}+{\lambda }_2\sum _{j\in N_i}{a}_{ij}{\tilde{\mathbf{Z}}}_{i1}-{\tilde{\mathbf{Z}}}_{j1},\end{array}$$where ${\mathbf{E}}_{i1}={E_{i11},E_{i12},E_{i13}}^T$, ${\lambda }_1$ and ${\lambda }_2$ are positive parameters, which are used to regulate the cooperative tracking performance.

Using the Kronecker product “$\otimes$”, and define ${\mathbf{E}}_1={{\mathbf{E}}_{11}^T,{\mathbf{E}}_{21}^T,\cdots ,{\mathbf{E}}_{N1}^T}^T$, ${\tilde{\mathbf{Z}}}_1={{\tilde{\mathbf{Z}}}_{11}^T,{\tilde{\mathbf{Z}}}_{21}^T,\cdots ,{\tilde{\mathbf{Z}}}_{N1}^T}^T$, then the cooperative tracking error of all UAVs can be expressed as $$\begin{array}{}\text{(23)}& {\mathbf{E}}_1={\lambda }_1{\mathbf{I}}_N+{\lambda }_2\mathcal{L}\otimes {\mathbf{I}}_3{\tilde{\mathbf{Z}}}_1.\end{array}$$

By recalling Lemma 5, it yields ${\tilde{\mathbf{Z}}}_1={{\lambda }_1+{\lambda }_2\mathcal{L}}^{-1}\otimes {\mathbf{I}}_3{\mathbf{E}}_1\le 1/{\sigma }_{\min }{\lambda }_1+{\lambda }_2\mathcal{L}{\mathbf{E}}_1$, where ${\sigma }_{\min }·$ represents the minimum singular value of matrix “$·$.” Therefore, ${\tilde{\mathbf{Z}}}_1⟶0$ if ${\mathbf{E}}_1⟶0$.

Using (22), the synchronization error of each UAV ${\mathbf{E}}_{i1}$ can be expressed as $$\begin{array}{}\text{(24)}& {\mathbf{E}}_{i1}={\lambda }_1+{\lambda }_2\sum _{j\in N_i}^{j\ne i}{a}_{ij}{\tilde{\mathbf{Z}}}_{i1}-{\lambda }_2\sum _{j\in N_i}^{j\ne i}{a}_{ij}{\tilde{\mathbf{Z}}}_{j1}.\end{array}$$

Differentiating (24) yields $$\begin{array}{}\text{(25)}& {\dot{\mathbf{E}}}_{i1}=A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\dot{\mathbf{X}}}_{i1}-{\mathbf{g}}_{i1}{\mathbf{X}}_{i1d}{\dot{\mathbf{X}}}_{i1d}-{\lambda }_2\sum _{j\in N_i}^{j\ne i}{a}_{ij}{\dot{\stackrel{~}{\mathbf{Z}}}}_{j1}=A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\dot{\mathbf{X}}}_{i1}-{\mathbf{g}}_{i1}{\mathbf{X}}_{i1d}{\dot{\mathbf{X}}}_{i1d}-{\lambda }_2\sum _{j\in N_i}^{j\ne i}{a}_{ij}{\mathbf{g}}_{i1}{\mathbf{X}}_{j1}{\dot{\mathbf{X}}}_{j1}-{\mathbf{g}}_{i1}{\mathbf{X}}_{j1d}{\dot{\mathbf{X}}}_{j1d},\end{array}$$where $A_i={\lambda }_1+{\lambda }_2{\sum }_{j\in N_i}^{j\ne i}{a}_{ij}$, ${\mathbf{g}}_{i1}\mathbf{x}$ is ${ℝ}^3⟶{ℝ}^{3\times 3}$, $\mathbf{x}={x_1,x_2,x_3}^{\text{T}}$, which is expressed as $$\begin{array}{}\text{(26)}& \begin{array}{c}{\mathbf{g}}_{i1}\mathbf{x}=\mathrm{diag}{g}_{i1\tau }{x}_{\tau },\hspace{1em}\tau =1,2,3,\hfill \\ g_{i1\tau }{x}_{\tau }=\frac{{\overline{X}}_{i1\tau }-{\underset{¯}{X}}_{i1\tau }}{x_{\tau }-{\underset{¯}{X}}_{i1\tau }{\overline{X}}_{i1\tau }-x_{\tau }}.\hfill \end{array}\end{array}$$

Substituting (11) into (25) yields $$\begin{array}{}\text{(27)}& {\dot{\mathbf{E}}}_{i1}=A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{F}}_{i1}+{\mathbf{G}}_{i1}{\mathbf{X}}_{i2}-A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1d}{\dot{\mathbf{X}}}_{i1d}-{\lambda }_2\sum _{j\in N_i}^{j\ne i}{a}_{ij}{\dot{\stackrel{~}{\mathbf{Z}}}}_{j1}.\end{array}$$

Based on the back-stepping control architecture, (27) can be expressed as $$\begin{array}{}\text{(28)}& {\dot{\mathbf{E}}}_{i1}=A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{F}}_{i1}-A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1d}{\dot{\mathbf{X}}}_{i1d}-{\lambda }_2\sum _{j\in N_i}^{j\ne i}{a}_{ij}{\dot{\stackrel{~}{\mathbf{Z}}}}_{j1}+A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{G}}_{i1}·{\mathbf{X}}_{i2}-{\mathbf{Z}}_{i2}+{\mathbf{E}}_{i2}+{\mathbf{Z}}_{i2d},\end{array}$$where ${\mathbf{E}}_{i2}={\mathbf{Z}}_{i2}-{\mathbf{Z}}_{i2d}$ and ${\mathbf{Z}}_{i2d}$ is a virtual control signal.

By using a low-pass filter, one has $$\begin{array}{}\text{(29)}& {\dot{\mathbf{Z}}}_{i2d}=-k_{{\in }_1}{\mathbf{Z}}_{i2d}-{\overline{\mathbf{Z}}}_{i2d},\end{array}$$

where $k_{{ϵ}_1}$ is a positive constant and ${\overline{\mathbf{Z}}}_{i2d}$ is an auxiliary signal, designed as $$\begin{array}{}\text{(30)}& {\overline{\mathbf{Z}}}_{i2d}={{\mathbf{A}}_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{G}}_{i1}}^{-1}{\lambda }_2\sum _{j\in N_i}^{j\ne i}{a}_{ij}{\dot{\stackrel{~}{\mathbf{Z}}}}_{j1}-{\mathbf{A}}_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{F}}_{i1}+{\mathbf{G}}_{i1}{\mathbf{X}}_{i2}-{\mathbf{Z}}_{i2}+{\mathbf{A}}_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1d}{\dot{\mathbf{X}}}_{i1d}-{\mathbf{K}}_{i1}{\mathbf{E}}_{i1},\end{array}$$where ${\mathbf{K}}_{i1}$ is a positive diagonal matrix.

Defining the filtering error as ${\mathbf{ϵ}}_{i1}={\mathbf{Z}}_{i2d}-{\overline{\mathbf{Z}}}_{i2d}$, one can obtain $$\begin{array}{}\text{(31)}& {\dot{\mathbf{ϵ}}}_{i1}=-k_{{\in }_{i1}}{\mathbf{ϵ}}_{i1}-{\dot{\overline{\mathbf{Z}}}}_{i2d}.\end{array}$$

By substituting (30) into (28), one can obtain $$\begin{array}{}\text{(32)}& {\mathbf{E}}_{i1}^T{\dot{\mathbf{E}}}_{i1}\le -{\mathbf{E}}_{i1}^{\text{T}}{\mathbf{K}}_{i1}{\mathbf{E}}_{i1}+{\mathbf{E}}_{i1}^{\text{T}}{A}_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{G}}_{i1}·{\mathbf{E}}_{i2}+\frac{1}{2h_{i1}}{{\mathbf{E}}_{i1}^{\text{T}}{A}_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{G}}_{i1}}^2+\frac{h_{i1}}{2}{{\mathbf{ϵ}}_{i1}}^2,\end{array}$$where $h_{i1}$ is a positive constant and $·$ represents 2-norm of vector.

To estimate the unknown function ${\mathbf{D}}_i$ of each UAV, the following DO is designed for the $i$th UAV: $$\begin{array}{}\text{(33)}& {\widehat{\mathbf{D}}}_i=k_2{\mathbf{X}}_{i2}-{\widehat{\mathbf{X}}}_{i2}+k_1{k}_2\int {\mathbf{X}}_{i2}-{\widehat{\mathbf{X}}}_{i2}\text{d}t,\text{(34)}& {\dot{\widehat{\mathbf{X}}}}_{i2}={\mathbf{F}}_{i20}+{\mathbf{G}}_{i20}{\mathbf{U}}_{i0}{\mathbf{v}}_i+{\widehat{\mathbf{D}}}_i+k_1{\mathbf{X}}_{i2}-{\widehat{\mathbf{X}}}_{i2},\end{array}$$where $k_1$ and $k_2$ are positive parameters, ${\widehat{\mathbf{D}}}_i$ is the estimate of ${\mathbf{D}}_i$.

Define ${\mathbf{e}}_{X_{i2}}={\mathbf{X}}_{i2}-{\widehat{\mathbf{X}}}_{i2}$ and ${\tilde{\mathbf{D}}}_i={\mathbf{D}}_i-{\widehat{\mathbf{D}}}_i$, one can obtain $$\begin{array}{}\text{(35)}& {\dot{\mathbf{e}}}_{X_{i2}}+k_1{\mathbf{e}}_{X_{i2}}={\tilde{\mathbf{D}}}_i.\end{array}$$

Taking the derivative of (35) and using (33) give $$\begin{array}{}\text{(36)}& {\dot{\stackrel{~}{\mathbf{D}}}}_i+k_2{\tilde{\mathbf{D}}}_i=\dot{\mathbf{D}.}\end{array}$$

From (36), it can be known that the estimation error ${\tilde{\mathbf{D}}}_i$ will not converge to zero since $\dot{\mathbf{D}}\ne 0$. In order to estimate the unknown function ${\mathbf{D}}_i$ more accurate, a five-layer RWFNN is used to estimate the error ${\tilde{\mathbf{D}}}_i$ of the DO with defining ${\mathbf{\Delta }}_i={\tilde{\mathbf{D}}}_i$.

The RWFNN structure is illustrated in Figure 1, including five layers (input layer, membership layer, rule layer, composite layer, and output layer) [33]. The components of the RWFNN are introduced as follows:

[figure(s) omitted; refer to PDF]

Layer 1–Input Layer: Input layer is the first layer, where ${\mathbf{r}}_i={r_{i1},r_{i2},\cdots ,r_{i{v}_1}}^T$ is the input features of RWFNN. The output of layer 1 is expressed as $$\begin{array}{}\text{(37)}& y_{ij}^1=r_{ij},\end{array}$$where $j\in 1,2,\cdots ,v_1$, $v_1$ is the dimension of input features, and $y_{ij}^1$ represents the output of $j$th neuron of Layer 1.

Layer 2–Membership Layer: Layer 2 has $v_1$ rows and $v_2$ columns, and its output can be described as $$\begin{array}{}\text{(38)}& y_{ijk}^2=e^{-{y_{ij}^1-c_{ijk}}^2/{\sigma }_{ijk}^2},\end{array}$$where $j\in 1,2,\cdots ,v_1$, $k\in 1,2,\cdots ,v_2$. $v_2$ is a positive constant, depending on the number of neurons. $y_{ijk}^2$ denotes the neuron of layer 2 in row $j$, column $k$.

Layer 3–Rule Layer: Layer 3 has $v_2$ neurons, and the output of layer 3 is described as $$\begin{array}{}\text{(39)}& y_{ik}^3=\prod _{j=1}^{v_1}{y}_{ijk}^2,\end{array}$$where $k\in 1,2,\cdots ,v_2$, and $y_{ik}^3$ denotes the $k$th neuron of Layer 3.

Layer 4–Composite Layer: Layer 4 also has $v_2$ neurons, and the input of Layer 4 consists the output of the wavelet layer, recurrent layer, and Layer 2, where the output of wavelet layer is described as $$\begin{array}{c}\text{(40)}& {\psi }_{ik}=\sum _{j=1}^{v_1}{w}_{ijk}^F{\varphi }_{ijk}{r}_{ij},{\varphi }_{ijk}{r}_{ij}=\frac{1}{\sqrt{b_{ijk}}}1-\frac{{r_{ij}-a_{ijk}}^2}{b_{ijk}^2}{e}^{-{r_{ij}-a_{ijk}}^2/2b_{ijk}^2},\end{array}$$where $j\in 1,2,\cdots ,v_1$, $k\in 1,2,\cdots ,v_2$, ${\varphi }_{ijk}$ is the output of the $j$th neuron of Layer 1 to $k$th neuron of wavelet layer. ${\psi }_{ik}$ represents the output of $k$th neuron of wavelet. $w_{ijk}^F$, $a_{ijk}$, and $b_{ijk}$ represent the connecting weight, translation, and dilation variables, respectively.

The output of Layer 4 is expressed as $$\begin{array}{}\text{(41)}& y_{ik}^4=y_{ik}^3{\psi }_{ik}{w}_{ik}^r{y}_{ik}^{4}t-1,\end{array}$$where $y_{ik}^4$ is the output of $k$th neuron of Layer 4. $w_{ik}^r$ and $y_{ik}^{4}t-1$ represent the recurrent weight and the output at the previous time step of the $k$th neuron, respectively.

Layer 5–Output Layer: Layer 5 has $v_3$ neurons, which determines the dimension of the final output. Each neuron’s output of this layer is given by $$\begin{array}{}\text{(42)}& y_{il}^5=\sum _{k=1}^{v_2}{w}_{ikl}^5{y}_{ik}^4,\end{array}$$where $l\in 1,2,\cdots ,v_3$, $w_{ikl}^5$ is the connecting weight, and $y_{il}^5$ denotes the $l$th neuron of Layer 5.

Using (42), one can express ${\mathbf{y}}_i^5$ in the following vector form: $$\begin{array}{}\text{(43)}& {\mathbf{y}}_i^5={\mathit{\omega }}_i^5{\mathbf{y}}_i^4{\mathit{\omega }}_i^r,{\mathit{\omega }}_i^F,{\mathbf{c}}_i,{\mathbf{\sigma }}_i,\end{array}$$ where ${\mathbf{y}}_i^5=d{y}_{i1}^5,\cdots ,d{y}_{ik}^5,\cdots ,{d_{i{v}_3}^5}^{\text{T}}\in {ℝ}^{v_3\times 1}$, and ${\mathbf{y}}_i^4=d{y}_{i1}^4,\cdots ,d{y}_{ik}^4,\cdots ,{d_{i{v}_2}^4}^{\text{T}}\in {ℝ}^{v_2\times 1}$. ${\mathit{\omega }}_i^5\in {ℝ}^{v_3\times v_2}$, ${\mathit{\omega }}_i^r\in {ℝ}^{v_2\times 1}$, ${\mathit{\omega }}_i^F\in {ℝ}^{v_1{v}_2\times 1}$, ${\mathbf{c}}_i\in {ℝ}^{v_1{v}_2\times 1}$, and ${\mathbf{\sigma }}_i\in {ℝ}^{v_1{v}_2\times 1}$, which are expressed as $$\begin{array}{c}\text{(44)}& {\mathit{\omega }}_i^5={{\mathit{\omega }}_{i1}^5,{\mathit{\omega }}_{i2}^5,\cdots ,{\mathit{\omega }}_{il}^5,\cdots ,{\mathit{\omega }}_{i{v}_3}^5}^T,{\mathit{\omega }}_{il}^5={{\omega }_{i1l}^5,{\omega }_{i2l}^5,\cdots ,{\omega }_{ikl}^5,\cdots ,{\omega }_{i{v}_{2}l}^5}^T,\\ {\mathit{\omega }}_i^r={{\omega }_{i1}^r,{\omega }_{i2}^r,\cdots ,{\omega }_{ik}^r,\cdots ,{\omega }_{i{v}_2}^r}^T,\\ {\mathit{\omega }}_i^F={{\mathit{\omega }}_{i1}^F,{\mathit{\omega }}_{i2}^F,\cdots ,{\mathit{\omega }}_{ik}^F,\cdots ,{\mathit{\omega }}_{i{v}_2}^F}^T,\\ {\mathit{\omega }}_{ik}^F={\omega }_{i1k}^F,{\omega }_{i2k}^F,\cdots ,{\omega }_{ijk}^F,\cdots ,{\omega }_{i{v}_{1}k}^F,\\ {\mathbf{c}}_i={{\mathbf{c}}_{i1},{\mathbf{c}}_{i2},\cdots ,{\mathbf{c}}_{ik},\cdots ,{\mathbf{c}}_{i{v}_2}}^T,\\ {\mathbf{c}}_{ik}=c_{i1k},c_{i2k},\cdots ,c_{ijk},\cdots ,c_{i{v}_{1}k},\\ {\mathbf{\sigma }}_i={{\mathit{\sigma }}_{i1},{\mathit{\sigma }}_{i2},\cdots ,{\mathit{\sigma }}_{ik},\cdots ,{\mathit{\sigma }}_{i{v}_2}}^T,\\ {\mathbf{\sigma }}_{ik}={\sigma }_{i1k},{\sigma }_{i2k},\cdots ,{\sigma }_{ijk},\cdots ,{\sigma }_{i{v}_{1}k}.\end{array}$$

In this paper, $v_3$ is set as 3 due to the fact that the estimated variable ${\mathbf{\Delta }}_i$ is three-dimensional. Therefore, there exist optimal values ${\mathit{\omega }}_i^{5\ast }$, ${\mathit{\omega }}_i^{r\ast }$, ${\mathit{\omega }}_i^{F\ast }$, ${\mathbf{c}}_i^{\ast }$, and ${\mathit{\sigma }}_i^{\ast }$, such that $$\begin{array}{}\text{(45)}& {\mathbf{\Delta }}_i={\mathbf{y}}_i^{5\ast }+{\epsilon }_{i1}={\mathit{\omega }}_i^{5\ast }{\mathbf{y}}_i^{4\ast }{\mathit{\omega }}_i^{r\ast },{\mathit{\omega }}_i^{F\ast },{\mathbf{c}}_i^{\ast },{\mathbf{\sigma }}_i^{\ast }+{\epsilon }_{i1},\end{array}$$where ${\epsilon }_{i1}$ is the approximation error.

To design the adaptive law of weights for estimating the unknown item, it is needed to obtain the gradient of ${\mathbf{y}}_i^5$ of its variables firstly.

Differentiating both sides of ${\mathbf{y}}_i^5$, one can obtain $$\begin{array}{}\text{(46)}& d{\mathbf{y}}_i^5=d{\mathit{\omega }}_i^5·{\mathbf{y}}_i^4+{\mathit{\omega }}_i^{5}d{\mathbf{y}}_i^4,\end{array}$$

To yield $d{\mathbf{y}}_i^4$, by the same way, differentiating both sides of $y_{ik}^4$, then $$\begin{array}{}\text{(47)}& d{y}_{ik}^4=d{y}_{ik}^3{\mathbf{c}}_{ik},{\mathbf{\sigma }}_{ik}+d{\psi }_{ik}{\mathbf{\omega }}_{ik}^F+d{\omega }_{ik}^r{y}_{ik}^{4}t-1.\end{array}$$

For term $d{y}_{ik}^3$, one can obtain it by combining (37), (38), and (39), that is $$\begin{array}{}\text{(48)}& d{y}_{ik}^3=\sum _{j=1}^{v_1}d{c}_{ijk}·\frac{2y_{ij}^1-c_{ijk}}{{\sigma }_{ijk}^2}\prod _{j=1}^{v_1}{y}_{ijk}^2+\sum _{j=1}^{v_1}d{\sigma }_{ijk}·\frac{2{y_{ij}^1-c_{ijk}}^2}{{\sigma }_{ijk}^3}\prod _{j=1}^{v_1}{y}_{ijk}^2=2{\mathbf{y}}_i^{1\text{T}}-{\mathbf{c}}_{ik}\diamond {{\mathbf{\sigma }}_{ik}}^2·\prod _{j=1}^{v_1}{y}_{ijk}^2.d{\mathbf{c}}_{ik}^T+2{{\mathbf{y}}_i^{1\text{T}}-{\mathbf{c}}_{ik}}^2\diamond {{\mathbf{\sigma }}_{ik}}^3.\prod _{j=1}^{v_1}{y}_{ijk}^2·d{\mathbf{\sigma }}_{ik}^T,\end{array}$$where “$\diamond$” represents dot division between matrices, and “${\ast }^n$” represent the aligned “$\ast$” itself does $n$ times dot product. And the ${\mathbf{c}}_{ik}$ and ${\mathit{\sigma }}_{ik}$ have been defined in (44).

For the term $d{\mathbf{\psi }}_{ik}$ and $d{\omega }_{ik}^r{y}_{ik}^{4}t-1$, there exists $$\begin{array}{}\text{(49a)}& d{\mathit{\psi }}_{ik}=d\sum _{j=1}^{v_1}{\omega }_{ijk}^F{\varphi }_{ijk}{r}_{ij}=\sum _{j=1}^{v_1}{\varphi }_{ijk}{r}_{ij}·d{\omega }_{ijk}^F={\mathit{\varphi }}_{ik}·d{\mathit{\omega }}_{ik}^{F\text{T}},\text{(49b)}& d{\omega }_{ik}^r{y}_{ik}^{4}t-1={y_{ik}^4}_{t-1}·d{\omega }_{ik}^r.\end{array}$$

Then, by combining (50), (48), and (49), $d{y}_{ik}^4$ can be expressed as $$\begin{array}{}\text{(50)}& d{y}_{ik}^4={\mathbf{\Gamma }}_{ik}^{c}d{\mathbf{c}}_{ik}^T+{\mathbf{\Gamma }}_{ik}^{\sigma }d{\mathit{\sigma }}_{ik}^T+{\mathbf{\Gamma }}_{ik}^{F}d{\mathit{\omega }}_{ik}^{\mathit{\text{FT}}}+{\Gamma }_{ik}^{r}d{\omega }_{ik}^r,\end{array}$$where $$\begin{array}{}\text{(51)}& \begin{array}{c}{\mathbf{\Gamma }}_{ik}^c=2{\mathbf{y}}_i^{1T}-{\mathbf{c}}_{ik}\diamond {{\mathit{\sigma }}_{ik}}^2·\prod _{j=1}^{v_1}{y}_{ijk}^2,\hfill \\ {\mathbf{\Gamma }}_{ik}^{\sigma }=2{{\mathbf{y}}_i^{1T}-{\mathbf{c}}_{ik}}^2\diamond {{\mathit{\sigma }}_{ik}}^3·\prod _{j=1}^{v_1}{y}_{ijk}^2,\hfill \\ {\mathbf{\Gamma }}_{ik}^F={\varphi }_{ik},\hfill \\ {\Gamma }_{ik}^r={y_{ik}^4}_{t-1}.\hfill \end{array}\end{array}$$

Using (50) and (51), one can further write $d{\mathbf{y}}_i^4$ as $$\begin{array}{}\text{(52)}& d{\mathbf{y}}_i^4={\mathbf{\Gamma }}_i^{c}d{\mathbf{c}}_i+{\mathbf{\Gamma }}_i^{\sigma }d{\mathit{\sigma }}_i+{\mathbf{\Gamma }}_i^{F}d{\mathit{\omega }}_i^F+{\mathbf{\Gamma }}_i^{r}d{\mathit{\omega }}_i^r,\end{array}$$where $$\begin{array}{c}\text{(53)}& {\mathbf{\Gamma }}_i^c=\begin{array}{cccccc}{\mathbf{\Gamma }}_{i1}^c& \mathbf{0}& \cdots & \cdots & \cdots & \mathbf{0}\\ \mathbf{0}& {\mathbf{\Gamma }}_{i2}^c& \cdots & \cdots & \cdots & \mathbf{0}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ \mathbf{0}& \cdots & \cdots & {\mathbf{\Gamma }}_{ik}^c& \cdots & \mathbf{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ \mathbf{0}& \cdots & \cdots & \cdots & \cdots & {\mathbf{\Gamma }}_{i{v}_2}^c\end{array},{\mathbf{\Gamma }}_i^{\sigma }=\begin{array}{cccccc}{\mathbf{\Gamma }}_{i1}^{\sigma }& \mathbf{0}& \cdots & \cdots & \cdots & \mathbf{0}\\ \mathbf{0}& {\mathbf{\Gamma }}_{i2}^{\sigma }& \cdots & \cdots & \cdots & \mathbf{0}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ \mathbf{0}& \cdots & \cdots & {\mathbf{\Gamma }}_{ik}^{\sigma }& \cdots & \mathbf{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ \mathbf{0}& \cdots & \cdots & \cdots & \cdots & {\mathbf{\Gamma }}_{i{v}_2}^{\sigma }\end{array},\\ {\mathbf{\Gamma }}_i^F=\begin{array}{cccccc}{\mathbf{\Gamma }}_{i1}^F& \mathbf{0}& \cdots & \cdots & \cdots & \mathbf{0}\\ \mathbf{0}& {\mathbf{\Gamma }}_{i2}^F& \cdots & \cdots & \cdots & \mathbf{0}\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ \mathbf{0}& \cdots & \cdots & {\mathbf{\Gamma }}_{ik}^F& \cdots & \mathbf{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ \mathbf{0}& \cdots & \cdots & \cdots & \cdots & {\mathbf{\Gamma }}_{i{v}_2}^F\end{array},\\ {\mathbf{\Gamma }}_i^r=\begin{array}{cccccc}{\Gamma }_{i1}^r& 0& \cdots & \cdots & \cdots & 0\\ 0& {\Gamma }_{i2}^r& \cdots & \cdots & \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& \cdots & \cdots & {\Gamma }_{ik}^r& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & \cdots & \cdots & \cdots & {\Gamma }_{i{v}_2}^r\end{array}.\end{array}$$

Moreover, using the property of Kronecker product, the term $d{\mathbf{\omega }}_i^5·{\mathbf{y}}_i^4$ in (58) can be transformed to the following column vector form: $$\begin{array}{}\text{(54)}& d{\mathit{\omega }}_i^5·{\mathbf{y}}_i^4={\mathbf{I}}_n\otimes {\mathbf{y}}_i^4·d\text{vec}{\mathit{\omega }}_i^{5T},\end{array}$$where “$\otimes$” represents Kronecker product, and “$\text{vec}\ast$” represents the operation that converts the aligned “$\ast$” to a column vector, that means $$\begin{array}{}\text{(55)}& \text{vec}{\mathit{\omega }}_i^{5\text{T}}={{\omega }_{i1}^{5\text{T}},\cdots ,{\omega }_{\text{il}}^{5\text{T}},\cdots ,{\omega }_{i{v}_3}^{5T}}^T.\end{array}$$

To simplify the representation, using ${\stackrel{⟶}{\mathbf{\omega }}}_i^5$ to represent $\text{vec}{\mathbf{\omega }}_i^{5T}$. Employing (58), (52), and (54) yeilds the following total differential equation: $$\begin{array}{}\text{(56)}& d{\mathbf{y}}_i^5={\mathbf{\Gamma }}_i^{5}d{\stackrel{⟶}{\mathit{\omega }}}_i^5+{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^{c}d{\mathbf{c}}_i+{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^{\sigma }d{\mathit{\sigma }}_i+{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^{F}d{\mathit{\omega }}_i^F+{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^{r}d{\mathit{\omega }}_i^r,\end{array}$$where $$\begin{array}{}\text{(57)}& {\mathbf{\Gamma }}_i^5={\mathbf{I}}_n\otimes {\mathbf{y}}_i^4.\end{array}$$

On the other hand, the total derivative of $d{\mathbf{y}}_i^5$ can be expressed as the following total differential form: $$\begin{array}{}\text{(58)}& d{\mathbf{y}}_i^5={\frac{\partial {\mathbf{y}}_i^5}{\partial {\stackrel{⟶}{\mathit{\omega }}}_i^5}}^{T}d{\stackrel{⟶}{\mathit{\omega }}}_i^5+{\frac{\partial {\mathbf{y}}_i^5}{\partial {\mathbf{c}}_i}}^{\text{T}}d{\mathbf{c}}_i+{\frac{\partial {\mathbf{y}}_i^5}{\partial {\mathit{\sigma }}_i}}^{\text{T}}d{\mathit{\sigma }}_i+{\frac{\partial {\mathbf{y}}_i^5}{\partial {\mathit{\omega }}_i^F}}^{\text{T}}d{\mathit{\omega }}_i^F+{\frac{\partial {\mathbf{y}}_i^5}{\partial {\mathit{\omega }}_i^r}}^{\text{T}}d{\mathit{\omega }}_i^r.\end{array}$$

Hence, one can derive that $$\begin{array}{}\text{(59)}& \begin{array}{c}{\frac{\partial {\mathbf{y}}_i^5}{\partial {\stackrel{⟶}{\mathit{\omega }}}_i^5}}^T={\mathbf{\Gamma }}_i^5,{\frac{\partial {\mathbf{y}}_i^5}{\partial {\mathbf{c}}_i}}^T={\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^c\hfill \\ {\frac{\partial {\mathbf{y}}_i^5}{\partial {\mathit{\sigma }}_i}}^T={\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^{\sigma },{\frac{\partial {\mathbf{y}}_i^5}{\partial {\mathit{\omega }}_i^F}}^T={\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^F,\hfill \\ {\frac{\partial {\mathbf{y}}_i^5}{\partial {\mathit{\omega }}_i^r}}^T={\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^r.\hfill \end{array}\end{array}$$

Taking into account (58) and (59) and using the Taylor expansion, ${\mathbf{y}}_i^5$ can be expressed as $$\begin{array}{}\text{(60)}& {\mathbf{y}}_i^5={\mathbf{y}}_i^{5\ast }+{\mathbf{\Gamma }}_i^5{\stackrel{⟶}{\mathit{\omega }}}_i^5-{\stackrel{⟶}{\mathit{\omega }}}_i^{5\ast }+{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^c{\mathbf{c}}_i-{\mathbf{c}}_i^{\ast }+{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^{\sigma }{\mathit{\sigma }}_i-{\mathit{\sigma }}_i^{\ast }+{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^F{\mathit{\omega }}_i^F-{\mathit{\omega }}_i^{F\ast }+{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^r{\mathit{\omega }}_i^r-{\mathit{\omega }}_i^{r\ast }+{\epsilon }_{i2}.\end{array}$$

Using ${\widehat{\mathbf{\Delta }}}_i$ to estimate the unknown item ${\mathbf{\Delta }}_i$, and it is expressed as $$\begin{array}{}\text{(61)}& {\widehat{\mathbf{\Delta }}}_i={\mathbf{y}}_i^5+\mathrm{diag}\mathrm{sign}{\mathbf{E}}_{i2}^T{\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\widehat{\mathbf{H}}}_i\triangleq {\mathbf{y}}_i^5+{\mathbf{\Gamma }}_i^H{\widehat{\mathbf{H}}}_i,\end{array}$$where ${\mathbf{\Gamma }}_i^H=\mathrm{diag}\mathrm{sign}{\mathbf{E}}_{i2}^{\text{T}}{\mathbf{g}}_{i2}{\mathbf{X}}_{i2}$ and ${\widehat{\mathbf{H}}}_i$ is an estimated value and will be introduced later.

Defining ${\tilde{\mathbf{\Delta }}}_i={\widehat{\mathbf{\Delta }}}_i-{\mathbf{\Delta }}_i$, ${\widetilde{\stackrel{⟶}{\mathbf{\omega }}}}_i^5={\stackrel{⟶}{\mathbf{\omega }}}_i^5-{\stackrel{⟶}{\mathbf{\omega }}}_i^{5\ast }$, ${\tilde{\mathbf{c}}}_i={\mathbf{c}}_i-{\mathbf{c}}_i^{\ast }$, ${\tilde{\mathbf{\sigma }}}_i={\mathbf{\sigma }}_i-{\mathbf{\sigma }}_i^{\ast }$, ${\tilde{\mathbf{\omega }}}_i^F={\mathbf{\omega }}_i^F-{\mathbf{\omega }}_i^{F\ast }$, and ${\tilde{\mathbf{\omega }}}_i^r={\mathbf{\omega }}_i^r-{\mathbf{\omega }}_i^{r\ast }$ and combining (45), (60), and (61), it yeilds $$\begin{array}{}\text{(62)}& {\tilde{\mathbf{\Delta }}}_i={\mathbf{\Gamma }}_i^5{\widetilde{\stackrel{⟶}{\mathbf{\omega }}}}_i^5+{\mathbf{\omega }}_i^5{\mathbf{\Gamma }}_i^c{\tilde{\mathbf{c}}}_i+{\mathbf{\omega }}_i^5{\mathbf{\Gamma }}_i^{\sigma }{\tilde{\mathbf{\sigma }}}_i+{\mathbf{\omega }}_i^5{\mathbf{\Gamma }}_i^F{\tilde{\mathbf{\omega }}}_i^F+{\mathbf{\omega }}_i^5{\mathbf{\Gamma }}_i^r{\tilde{\mathbf{\omega }}}_i^r+{\epsilon }_{i2}-{\epsilon }_{i1}+{\mathbf{\Gamma }}_i^H{\widehat{\mathbf{H}}}_i.\end{array}$$

Taking the derivative of ${\mathbf{E}}_{i2}$ and using (17) and (20), one can obtain $$\begin{array}{}\text{(63)}& {\dot{\mathbf{E}}}_{i2}={\dot{\mathbf{Z}}}_{i2}-{\dot{\mathbf{Z}}}_{i2d}={\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\mathbf{F}}_{i20}+{\mathbf{G}}_{i20}{\mathbf{U}}_{i0}{\mathbf{v}}_i+{\mathbf{D}}_i-{\dot{\mathbf{Z}}}_{i2d},\end{array}$$where ${\mathbf{g}}_{i2}\mathbf{x}$ and $\mathbf{x}={x_1,x_2,x_3}^{\text{T}}$ are $$\begin{array}{}\text{(64)}& \begin{array}{c}{\mathit{g}}_{i2}\mathit{x}=\mathrm{diag}{g}_{i2\tau }{x}_{\tau },\tau =1,2,3,\hfill \\ g_{i2\tau }{x}_{\tau }=\frac{{\overline{X}}_{i2\tau }-{\underset{¯}{X}}_{i2\tau }}{x_{\tau }-{\underset{¯}{X}}_{i2\tau }{\overline{X}}_{i2\tau }-x_{\tau }}.\hfill \end{array}\end{array}$$

By using the back-stepping method and defining ${\mathbf{v}}_{id}$ as a virtual control signal, then ${\dot{\mathbf{E}}}_{i2}$ can be expressed as $$\begin{array}{}\text{(65)}& {\dot{\mathbf{E}}}_{i2}={\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\mathbf{F}}_{i20}+{\mathbf{G}}_{i20}{\mathbf{v}}_{id}+{\mathbf{G}}_{i20}{\mathbf{U}}_{i0}-{\mathbf{v}}_i+{\mathbf{G}}_{i20}{\mathbf{E}}_{i3}+{\mathbf{D}}_i-{\dot{\mathbf{Z}}}_{i2d},\end{array}$$where ${\mathbf{E}}_{i3}={\mathbf{v}}_i-{\mathbf{v}}_{id}$. In order to reduce computational burden of taking time derivative for virtual control signal ${\mathbf{v}}_{id}$, a filter is used to obtain ${\mathbf{v}}_{id}$, which is given by $$\begin{array}{}\text{(66)}& {\dot{\mathbf{v}}}_{id}=-k_{{ϵ}_2}{\mathbf{v}}_{id}-{\overline{\mathbf{v}}}_{id},\end{array}$$where $k_{{ϵ}_2}$ is a positive constant, and ${\overline{\mathbf{v}}}_{id}$ is an auxiliary signal, designed as $$\begin{array}{}\text{(67)}& {\overline{\mathbf{v}}}_{id}={{\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\mathbf{G}}_{i20}}^{-1}-{\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\mathbf{F}}_{i20}+{\mathbf{G}}_{i20}{\mathbf{U}}_{i0}{\mathbf{v}}_i-{\mathbf{v}}_i+{\widehat{\mathbf{D}}}_i+{\widehat{\mathbf{\Delta }}}_i+{\widehat{\mathbf{H}}}_i+{\dot{\mathbf{Z}}}_{i2d}-{\mathbf{K}}_{i2}{\mathbf{E}}_{i2}-A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}·{\mathbf{G}}_{i1}{\mathbf{E}}_{i1},\end{array}$$where ${\mathbf{K}}_{i2}$ is a positive diagonal matrix.

Define the filter error as ${ϵ}_{i2}={\mathbf{v}}_{id}-{\overline{\mathbf{v}}}_{id}$, then one can obtain from (66) that $$\begin{array}{}\text{(68)}& {\dot{\mathit{ϵ}}}_{i2}=-k_{{\mathit{ϵ}}_{i2}}{\mathit{ϵ}}_{i2}-{\dot{\overline{\mathit{v}}}}_{id}.\end{array}$$

Substituting (65) with (67) and (62), with considering ${\epsilon }_{i1}-{\epsilon }_{i2}\le {\mathbf{H}}_i$ in which $\ast$ represents the absolute value of the matrix “${}^{\ast }$”, while defining ${\widehat{\mathbf{H}}}_i$ as the estimation of ${\mathbf{H}}_i$ and ${\tilde{\mathbf{H}}}_i=\mathbf{H}-{\widehat{\mathbf{H}}}_i$ as the estimation error, one can obtain $$\begin{array}{}\text{(69)}& {\mathit{E}}_{i2}^{\text{T}}{\dot{\mathit{E}}}_{i2}\le -{\mathit{E}}_{i2}^{\text{T}}{\mathit{K}}_{i2}{\mathit{E}}_{i2}-{\mathit{E}}_{i2}^{\text{T}}{\mathit{A}}_i{\mathit{g}}_{i1}{\mathit{X}}_{i1}{\mathit{G}}_{i1}{\mathit{E}}_{i1}+{\mathit{E}}_{i2}^{\text{T}}{\mathit{g}}_{i2}{\mathit{X}}_{i2}{\mathit{G}}_{i20}{\mathit{E}}_{i3}+{\mathit{G}}_{i20}{\mathit{ϵ}}_{i2}-{\mathit{\Gamma }}_i^5·{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^5-{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^c{\tilde{\mathbf{c}}}_i-{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^{\sigma }{\tilde{\mathit{\sigma }}}_i-{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^F{\tilde{\mathit{\omega }}}_i^F-{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^r{\tilde{\mathit{\omega }}}_i^r+{\mathit{E}}_{i2}^T{\mathit{g}}_{i2}{\mathit{X}}_{i2}{\tilde{\mathit{H}}}_i.\end{array}$$

Taking the time derivative of ${\mathbf{E}}_{i3}$ and using (18), one can obtain $$\begin{array}{}\text{(70)}& {\dot{\mathbf{E}}}_{i3}=-\mathbf{\Lambda }{\mathbf{v}}_i+\xi -{\dot{\mathbf{v}}}_{id}.\end{array}$$

Design the auxiliary control signal $\mathbf{\xi }$ as $$\begin{array}{}\text{(71)}& \mathbf{\xi }=\begin{array}{c}\mathbf{\Lambda }{\mathit{v}}_i+{\dot{\mathit{v}}}_{id}-{\mathit{K}}_{i3}{\mathit{E}}_{i3}-{\mathit{g}}_{i2}{\mathit{X}}_{i2}{\mathit{G}}_{i20}{\mathit{E}}_{i2}-\frac{1}{{{\mathit{E}}_{i3}}^2}\frac{h_{i1}}{2}{{\mathit{ϵ}}_{i1}}^2+\frac{h_{i2}}{2}{{\mathit{ϵ}}_{i2}}^2{\mathit{E}}_{i3},\hfill \\ {{\mathit{E}}_{i3}}^2>{\mu }_{ib},\hfill \\ \mathbf{\Lambda }{\mathit{v}}_i+{\dot{\mathit{v}}}_{id}-{\mathit{K}}_{i3}{\mathit{E}}_{i3}-{\mathit{g}}_{i2}{\mathit{X}}_{i2}{\mathit{G}}_{i20}{\mathit{E}}_{i2}-\frac{1}{{\mu }_{ib}}\frac{h_{i1}}{2}{{\mathit{ϵ}}_{i1}}^2+\frac{h_{i2}}{2}{{\mathit{ϵ}}_{i2}}^2{\mathit{E}}_{i3},\hfill \\ {{\mathit{E}}_{i3}}^2\le {\mu }_{ib},\hfill \end{array}\end{array}$$ where ${\mathbf{K}}_{i3}$ is a positive diagonal matrix, and $h_{i1}$ and $h_{i2}$ are positive constants.

By combining (70) and (71), one has $$\begin{array}{}\text{(72)}& \begin{array}{c}{\mathbf{E}}_{i3}^T{\dot{\mathbf{E}}}_{i3}\le -{\mathbf{E}}_{i3}^T{\mathbf{K}}_{i3}{\mathbf{E}}_{i3}-{\mathbf{E}}_{i3}^T{\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\mathbf{G}}_{i20}{\mathbf{E}}_{i2}.\end{array}\end{array}$$

Finally, the adaptive laws of RWFNN for the $i$th UAV are developed as $$\begin{array}{}\text{(73)}& \begin{array}{c}{\dot{\stackrel{⟶}{\mathit{\omega }}}}_i^5={\eta }_{i1}^{-1}-{\gamma }_{i1}{\stackrel{⟶}{\mathit{\omega }}}_i^5+{{\mathit{E}}_{i2}^{\text{T}}{\mathit{g}}_{i2}{\mathit{X}}_{i2}{\mathbf{\Gamma }}_i^5}^T,\hfill \\ {\dot{\mathbf{c}}}_i={\eta }_{i2}^{-1}-{\gamma }_{i2}{\mathbf{c}}_i+{{\mathit{E}}_{i2}^{\text{T}}{\mathbf{g}}_{i2}{\mathit{X}}_{i2}{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^c}^T,\hfill \\ {\dot{\mathit{\sigma }}}_i={\eta }_{i3}^{-1}-{\gamma }_{i3}{\mathit{\sigma }}_i+{{\mathit{E}}_{i2}^{\text{T}}{\mathit{g}}_{i2}{\mathit{X}}_{i2}{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^{\sigma }}^T,\hfill \\ {\dot{\mathit{\omega }}}_i^F={\eta }_{i4}^{-1}-{\gamma }_{i4}{\mathit{\omega }}_i^F+{{\mathit{E}}_{i2}^{\text{T}}{\mathit{g}}_{i2}{\mathit{X}}_{i2}{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^F}^T,\hfill \\ {\dot{\mathit{\omega }}}_i^r={\eta }_{i5}^{-1}-{\gamma }_{i5}{\mathit{\omega }}_i^r+{{\mathit{E}}_{i2}^{\text{T}}{\mathit{g}}_{i2}{\mathit{X}}_{i2}{\mathit{\omega }}_i^5{\mathbf{\Gamma }}_i^r}^T,\hfill \\ {\dot{\widehat{\mathit{H}}}}_i={\eta }_{i6}^{-1}-{\gamma }_{i6}{\widehat{\mathit{H}}}_i+{{\mathit{E}}_{i2}^{\text{T}}{\mathit{g}}_{i2}{\mathit{X}}_{i2}}^T,\hfill \end{array}\end{array}$$where ${\gamma }_{i1}~{\gamma }_{i5}$ are discontinuous switching constants to prevent the weights to infinity and ${\gamma }_{i6}$ is a positive constant, where the switching constants are designed as $$\begin{array}{}\text{(74a)}& {\gamma }_{i1}=\begin{array}{cc}0\hfill & \text{if}{{\stackrel{⟶}{\mathit{\omega }}}_i^5}^2\le {\overline{\omega }}^5,\hfill \\ {\overline{\gamma }}_{i1}\hfill & \text{if}{{\stackrel{⟶}{\mathit{\omega }}}_i^5}^2>{\overline{\omega }}^5,\hfill \end{array}\text{(74b)}& {\gamma }_{i2}=\begin{array}{cc}0\hfill & \text{if}{{\mathit{c}}_i}^2\le \overline{c},\hfill \\ {\overline{\gamma }}_{i2}\hfill & \text{if}{{\mathit{c}}_i}^2>\overline{c},\hfill \end{array}\text{(74c)}& {\gamma }_{i3}=\begin{array}{cc}0\hfill & \text{if}{{\mathit{\sigma }}_i}^2\le \overline{\sigma },\hfill \\ {\overline{\gamma }}_{i3}\hfill & \text{if}{{\mathit{\sigma }}_i}^2>\overline{\sigma },\hfill \end{array}\text{(74d)}& {\gamma }_{i4}=\begin{array}{cc}0\hfill & \text{if}{{\mathit{\omega }}_i^F}^2\le {\overline{\omega }}_i^F,\hfill \\ {\overline{\gamma }}_{i4}\hfill & \text{if}{{\mathit{\omega }}_i^F}^2>{\overline{\omega }}_i^F,\hfill \end{array}\text{(74e)}& {\gamma }_{i5}=\begin{array}{cc}0\hfill & \text{if}{{\mathit{\omega }}_i^r}^2\le {\overline{\omega }}_i^r,\hfill \\ {\overline{\gamma }}_{i5}\hfill & \text{if}{{\mathit{\omega }}_i^r}^2>{\overline{\omega }}_i^r,\hfill \end{array}\end{array}$$where ${\overline{\omega }}^5$, $\overline{c}$, $\overline{\sigma }$, ${\overline{\omega }}_i^F$, and ${\overline{\omega }}_i^r$ are positive constants, which represent the boundness of weight ${\stackrel{⟶}{\mathit{\omega }}}_i^5,{\mathbf{c}}_i,{\mathit{\sigma }}_i,{\mathit{\omega }}_i^F,{\mathit{\omega }}_i^r$, respectively.

To this end, the proposed FTCC scheme is shown in Figure 2 to better illustrate the design principle and functional components in the control system.

Remark 6.

Many papers choose multiplication on input and recurrence data as an operation on the neuron of the composite layer in RWFNN. However, it is sometimes problematic. For example, when inputs from layer 3 are minuscule, the outputs of the composite neuron will also become exceedingly small under multiplication. Under the limitation of computational precision, the outputs are equal to zero. Since the outputs will loop to the next multiplication, the outputs will always be zero, which causes neuron inactivation. Therefore, this paper uses the addition operation as an alternative, and the back-propagation gradient is deduced in detail using vectorized expressions, i.e., (46)–(59).

[figure(s) omitted; refer to PDF]

3.2. Stability Analysis

Theorem 7.

Consider the $N$ UAVs (1)–(4) under the distributed communication network against the actuator faults (13), states constraints, and input saturation (15), if the control laws are chosen as (30), (67), and (71), the disturbance observers are developed as (33), and (34), and the adaptive laws are constructed as (73), (74a); then, all the states in the system are ultimately uniformly bounded and strictly confined within the limits.

Proof.

Choose a Lyapunov function as $$\begin{array}{}\text{(75)}& V=\frac{1}{2}\sum _{i=1}^N{\mathit{E}}_{i1}^{\text{T}}{\mathit{E}}_{i1}^{\text{T}}+{\mathit{E}}_{i2}^{\text{T}}{\mathit{E}}_{i2}^{\text{T}}+{\mathit{E}}_{i3}^{\text{T}}{\mathit{E}}_{i3}^{\text{T}}+{\eta }_{i1}·{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\text{T}}{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^5+{\eta }_{i2}{\tilde{\mathbf{c}}}_i^{\text{T}}{\tilde{\mathbf{c}}}_i+{\eta }_{i3}{\tilde{\mathit{\sigma }}}_i^{\text{T}}{\mathit{\sigma }}_i+{\eta }_{i4}·{\tilde{\mathit{\omega }}}_i^{F\text{T}}{\tilde{\mathit{\omega }}}_i^F+{\eta }_{i5}{\tilde{\mathit{\omega }}}_i^{\text{rT}}{\tilde{\mathit{\omega }}}_i^r+{\eta }_{i6}{\tilde{\mathit{H}}}_i^{\text{T}}{\tilde{\mathit{H}}}_i+{\mathit{ϵ}}_{i1}^{\text{T}}·{\mathit{ϵ}}_{i1}+{\mathit{ϵ}}_{i2}^{\text{T}}{\mathit{ϵ}}_{i2}.\end{array}$$

By taking the time derivative of $V$, combining (32), (69), (72), (31), and (68) and using Young inequality, one has $$\begin{array}{}\text{(76)}& \dot{V}\le \sum _{i=1}^N-{\mathbf{E}}_{i1}^{\text{T}}{\mathbf{K}}_{i1}{\mathbf{E}}_{i1}-{\mathbf{E}}_{i2}^{\text{T}}{\mathbf{K}}_{i2}{\mathbf{E}}_{i2}-{\mathbf{E}}_{i3}^{\text{T}}{\mathbf{K}}_{i3}·{\mathbf{E}}_{i3}+\sum _{i=1}^N\frac{1}{2h_{i1}}{{\mathbf{E}}_{i1}^{\text{T}}{\mathbf{A}}_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{G}}_{i1}}^2+\frac{h_{i1}}{2}{{\mathit{ϵ}}_{i1}}^2+\frac{1}{2h_{i2}}{{\mathbf{E}}_{i1}^{\text{T}}{\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\mathbf{G}}_{i20}}^2+\frac{h_{i2}}{2}{{\mathit{ϵ}}_{i2}}^2-{\gamma }_{i1}{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\text{T}}{\stackrel{⟶}{\mathit{\omega }}}_i-{\gamma }_{i2}{\tilde{\mathbf{c}}}_i^{\text{T}}{\mathbf{c}}_i-{\gamma }_{i3}·{\tilde{\mathit{\sigma }}}_i^{\text{T}}{\mathit{\sigma }}_i-{\gamma }_{i4}{\tilde{\mathit{\omega }}}_i^{F\text{T}}{\mathit{\omega }}_i^F-{\gamma }_{i5}{\tilde{\mathit{\omega }}}_i^{\text{rT}}{\mathit{\omega }}_i^r-{\gamma }_{i6}{\tilde{\mathbf{H}}}_i^{\text{T}}{\widehat{\mathbf{H}}}_i-k_{{ϵ}_{i1}}-\frac{1}{2}{{\mathit{ϵ}}_{i1}}^2-k_{{\mathit{ϵ}}_{i2}}-\frac{1}{2}{{\mathit{ϵ}}_{i2}}^2+\frac{1}{2}{{\dot{\overline{\mathbf{Z}}}}_{i2d}}^2+\frac{1}{2}{{\dot{\overline{\mathbf{v}}}}_{i2}}^2.\end{array}$$

Under the condition of (74a), the term ${\gamma }_{i1}{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\text{T}}{\stackrel{⟶}{\mathit{\omega }}}_i$ has the following property: $$\begin{array}{}\text{(77)}& -{\gamma }_{i1}{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\text{T}}{\stackrel{⟶}{\mathit{\omega }}}_i^5\le {\overline{\gamma }}_{i1}{\overline{\omega }}^5+2{{\stackrel{⟶}{\mathit{\omega }}}^{5\ast }}^2-\frac{1}{2}{{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^5}^2.\end{array}$$

The reason is as follows:

When ${\overline{\omega }}^5\ge {{\stackrel{⟶}{\mathit{\omega }}}_i^5}^2$, ${\gamma }_{i1}=0$, then $$\begin{array}{}\text{(78)}& -{\gamma }_{i1}{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\text{T}}{\stackrel{⟶}{\mathit{\omega }}}_i^5=0\le {\overline{\gamma }}_{i1}{\overline{\omega }}^5-{{\stackrel{⟶}{\mathit{\omega }}}_i^5}^2={\overline{\gamma }}_{i1}{\overline{\omega }}^5-{{\stackrel{⟶}{\mathit{\omega }}}_i^{5\ast }+{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\ast }}^2\le {\overline{\gamma }}_{i1}{\overline{\omega }}^5-{{\stackrel{⟶}{\mathit{\omega }}}_i^{5\ast }-{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\ast }}^2\le {\overline{\gamma }}_{i1}{\overline{\omega }}^5+2{{\stackrel{⟶}{\mathit{\omega }}}_i^{5\ast }}^{\text{T}}{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\ast }-{{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\ast }}^2\le {\overline{\gamma }}_{i1}{\overline{\omega }}^5+2{{\stackrel{⟶}{\mathit{\omega }}}_i^{5\ast }}^2-\frac{1}{2}{{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\ast }}^2.\end{array}$$

On the other hand, when ${\overline{\omega }}^5<{{\stackrel{⟶}{\mathit{\omega }}}_i^5}^2$, ${\gamma }_{i1}={\overline{\gamma }}_{i1}$, then, $$\begin{array}{}\text{(79)}& -{\gamma }_{i1}{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\text{T}}{\stackrel{⟶}{\mathit{\omega }}}_i^5=-{\overline{\gamma }}_{i1}{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\text{T}}{\stackrel{⟶}{\mathit{\omega }}}_i^{5\ast }+{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^5\le -\frac{1}{2}{\overline{\gamma }}_{i1}{{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^{5\text{T}}}^2-{{\stackrel{⟶}{\mathit{\omega }}}_i^{5\ast }}^2.\end{array}$$

Hence, by combining (78) and (79), one can obtain (77). By the same way, one can conclude that $$\begin{array}{}\text{(80)}& \begin{array}{c}-{\gamma }_{i2}{\tilde{\mathbf{c}}}_i^T{\mathbf{c}}_i\le {\overline{\gamma }}_{i2}\overline{c}+2{{\mathit{c}}^{\ast }}^2-\frac{1}{2}{{\tilde{\mathit{c}}}_i}^2,\hfill \\ -{\gamma }_{i3}{\tilde{\mathit{\sigma }}}_i^T{\mathit{\sigma }}_i\le {\overline{\gamma }}_{i3}\overline{\sigma }+2{{\mathit{\sigma }}^{\ast }}^2-\frac{1}{2}{{\tilde{\mathit{\sigma }}}_i}^2,\hfill \\ -{\gamma }_{i4}{\tilde{\mathit{\omega }}}_i^{\mathit{\text{FT}}}{\mathit{\omega }}_i^F\le {\overline{\gamma }}_{i4}{\overline{\omega }}^F+2{{\mathit{\omega }}^{F\ast }}^2-\frac{1}{2}{{\tilde{\mathit{\omega }}}_i^F}^2,\hfill \\ -{\gamma }_{i5}{\tilde{\mathit{\omega }}}_i^{rT}{\mathit{\omega }}_i^r\le {\overline{\gamma }}_{i5}{\overline{\omega }}^r+2{{\mathit{\omega }}^{r\ast }}^2-\frac{1}{2}{{\tilde{\mathit{\omega }}}_i^r}^2.\hfill \end{array}\end{array}$$

Furthermore, the term ${\gamma }_{i6}{\tilde{\mathbf{H}}}_i^T{\widehat{\mathbf{H}}}_i$ in (76) satisfies that $$\begin{array}{}\text{(81)}& -{\gamma }_{i6}{\tilde{\mathbf{H}}}_i^{\text{T}}{\widehat{\mathbf{H}}}_i\le -\frac{1}{2}{\gamma }_{i6}{{\tilde{\mathbf{H}}}_i}^2-{{\mathbf{H}}_i}^2.\end{array}$$

Substituting (76) with (77), (80), and (81) then yeilds $$\begin{array}{}\text{(82)}& \dot{V}\le \sum _{i=1}^N-{\mathbf{E}}_{i1}^{\text{T}}{\mathbf{K}}_{i1}-\frac{1}{2h_{i1}}{A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{G}}_{i1}}^2·{\mathbf{E}}_{i1}-{\mathbf{E}}_{i2}^{\text{T}}{\mathbf{K}}_{i2}-\frac{1}{2h_{i2}}{{\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\mathbf{G}}_{i20}}^2{\mathbf{E}}_{i2}-{\mathbf{E}}_{i3}^{\text{T}}{\mathbf{K}}_{i3}{\mathbf{E}}_{i3}+\sum _{i=1}^N-\frac{1}{2}{\overline{\gamma }}_{i1}{{\widetilde{\stackrel{⟶}{\mathit{\omega }}}}_i^5}^2-\frac{1}{2}{\overline{\gamma }}_{i2}·{{\tilde{\mathbf{c}}}_i}^2-\frac{1}{2}{\overline{\gamma }}_{i3}{{\tilde{\mathit{\sigma }}}_i}^2-\frac{1}{2}{\overline{\gamma }}_{i4}{{\tilde{\mathit{\omega }}}_i^F}^2-\frac{1}{2}{\overline{\gamma }}_{i5}·{{\tilde{\mathit{\omega }}}_i^r}^2-\frac{1}{2}{\gamma }_{i6}{{\tilde{\mathbf{H}}}_i}^2-k_{{ϵ}_{i1}}-\frac{1+h_{i1}}{2}·{{\mathit{ϵ}}_{i1}}^2-k_{{\mathit{ϵ}}_{i2}}-\frac{1+h_{i2}}{2}{{\mathit{ϵ}}_{i2}}^2+\delta ,\end{array}$$where $\delta$ is $$\begin{array}{}\text{(83)}& \delta =\sum _{i=1}^N\frac{1}{2}{{\dot{\overline{\mathbf{Z}}}}_{i2d}}^2+\frac{1}{2}{{\dot{\overline{\mathbf{v}}}}_{i2}}^2+{\overline{\gamma }}_{i1}{\overline{\omega }}^5+2{{\stackrel{⟶}{\mathit{\omega }}}^{5\ast }}^2+{\overline{\gamma }}_{i2}\overline{c}+2{{\mathbf{c}}^{\ast }}^2+{\overline{\gamma }}_{i3}\overline{\sigma }+2{{\mathit{\sigma }}^{\ast }}^2+{\overline{\gamma }}_{i4}{\overline{\omega }}^F+2{{\mathit{\omega }}^{F\ast }}^2+{\overline{\gamma }}_{i5}{\overline{\omega }}^r+2{{\mathit{\omega }}^{r\ast }}^2+\frac{1}{2}{\gamma }_{i6}{{\mathbf{H}}_i}^2.\end{array}$$

By choosing the parameters ${\mathbf{K}}_{i1}$ and ${\mathbf{K}}_{i2}$ as $$\begin{array}{}\text{(84)}& \begin{array}{c}{\mathbf{K}}_{i1}={\mathbf{K}}_{i10}+\frac{1}{2h_{i1}}{A_i{\mathbf{g}}_{i1}{\mathbf{X}}_{i1}{\mathbf{G}}_{i1}}^2\hfill \\ {\mathbf{K}}_{i2}={\mathbf{K}}_{i20}+\frac{1}{2h_{i2}}{{\mathbf{g}}_{i2}{\mathbf{X}}_{i2}{\mathbf{G}}_{i20}}^2\hfill \end{array},\end{array}$$ respectively, where ${\mathbf{K}}_{i10}$ and ${\mathbf{K}}_{i20}$ are positive diagonal matrices, (82) can be expressed as $$\begin{array}{}\text{(85)}& \dot{V}\le -cV+\delta ,\end{array}$$where $c$ is $$\begin{array}{}\text{(86)}& c=\min {\lambda }_{\min }2{\mathbf{K}}_{i10},{\lambda }_{\min }2{\mathbf{K}}_{i20},{\lambda }_{\min }2{\mathbf{K}}_{i30},{\eta }_{i1}^{-1}{\overline{\gamma }}_{i1},{\eta }_{i2}^{-1}{\overline{\gamma }}_{i2},{\eta }_{i3}^{-1}{\overline{\gamma }}_{i3},{\eta }_{i4}^{-1}{\overline{\gamma }}_{i4},{\eta }_{i5}^{-1}{\overline{\gamma }}_{i5},{\eta }_{i6}^{-1}{\overline{\gamma }}_{i6},2k_{{ϵ}_{i1}}-1-h_{i1},2k_{{ϵ}_{i2}}-1-h_{i2}.\end{array}$$

The expressions ${\dot{\overline{\mathbf{Z}}}}_{i2d}$ and ${\dot{\overline{\mathbf{v}}}}_{id}$ can be respectively obtained from (30) and (67), given by $$\begin{array}{c}\text{(87)}& {\dot{\overline{\mathbf{Z}}}}_{i2d}=\frac{\partial {\overline{\mathbf{Z}}}_{i2d}}{\partial {\mathbf{X}}_{i1}}{\dot{\mathbf{X}}}_{i1}+\frac{\partial {\overline{\mathbf{Z}}}_{i2d}}{\partial {\mathbf{E}}_{i1}}{\dot{\mathbf{E}}}_{i1}+\frac{\partial {\overline{\mathbf{Z}}}_{i2d}}{\partial {\mathbf{X}}_{i1d}}{\dot{\mathbf{X}}}_{i1d}+\frac{\partial {\overline{\mathbf{Z}}}_{i2d}}{\partial {\dot{\mathbf{X}}}_{i1d}}{\ddot{\mathbf{X}}}_{i1d}+\sum _{j\in N_i}^{j\ne i}{a}_{ij}\frac{\partial {\overline{\mathbf{Z}}}_{i2d}}{\partial {\mathbf{Z}}_{j1}}{\dot{\mathbf{Z}}}_{j1}+\frac{\partial {\overline{\mathbf{Z}}}_{i2d}}{\partial {\mathbf{X}}_{i2}}{\dot{\mathbf{X}}}_{i2}+\frac{\partial {\overline{\mathbf{Z}}}_{i2d}}{\partial {\mathbf{Z}}_{i2}}{\dot{\mathbf{Z}}}_{i2},{\dot{\overline{\mathbf{v}}}}_{id}=\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\mathbf{X}}_{i1}}{\dot{\mathbf{X}}}_{i1}+\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\mathbf{X}}_{i2}}{\dot{\mathbf{X}}}_{i2}+\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\mathbf{v}}_i}{\dot{\mathbf{v}}}_i+\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\widehat{\mathbf{D}}}_i}{\dot{\widehat{\mathbf{D}}}}_i+\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\widehat{\mathbf{\Delta }}}_i}{\dot{\widehat{\mathbf{\Delta }}}}_i+\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\widehat{\mathbf{H}}}_i}{\dot{\widehat{\mathbf{H}}}}_i+\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\dot{\mathbf{Z}}}_{i2d}}{\ddot{\mathbf{Z}}}_{i2d}+\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\mathbf{E}}_{i1}}{\dot{\mathbf{E}}}_{i1}+\frac{\partial {\overline{\mathbf{v}}}_{id}}{\partial {\mathbf{E}}_{i2}}{\dot{\mathbf{E}}}_{i2}.\end{array}$$

Define the following two vectors ${\mathbf{X}}_d$ and ${\mathbf{E}}_d$$$\begin{array}{c}\text{(88)}& \begin{array}{c}{\mathbf{X}}_d=\text{col}{\mathbf{X}}_{1d},\cdots ,{\mathbf{X}}_{id},\cdots ,{\mathbf{X}}_{Nd},\hfill \\ {\mathbf{X}}_{id}=\text{col}{\mathbf{X}}_{i1d},{\dot{\mathbf{X}}}_{i1d},{\ddot{\mathbf{X}}}_{i1d},\hfill \end{array}\begin{array}{c}{\mathbf{E}}_d=\text{col}{\mathbf{E}}_{1d},\cdots ,{\mathbf{E}}_{id},\cdots ,{\mathbf{E}}_{Nd},\hfill \\ {\mathbf{E}}_i=\text{col}{\mathbf{E}}_{i1},{\mathbf{E}}_{i2},{\mathbf{E}}_{i3}.\hfill \end{array}\end{array}$$

For any constants ${\overline{B}}_{X_d}>0$, ${\overline{B}}_E>0$, the sets $\mathcal{X}≔{\mathbf{X}}_d:{{\mathbf{X}}_d}^2\le {\overline{B}}_{X_d}$ and $\mathcal{E}≔{\mathbf{E}}_d:{{\mathbf{E}}_d}^2\le {\overline{B}}_E$ are compact ones. Then, $\mathcal{P}≔\mathcal{X}\times \mathcal{E}$ is also compact. Due to the fact that ${\dot{\overline{\mathbf{Z}}}}_{i2d}$ and ${\dot{\overline{\mathbf{v}}}}_{id}$ both are continuous function, considering the continuous property, there exists a constant ${\delta }_M>0$ when $\forall {\mathbf{X}}_d,{\mathbf{E}}_d\in \mathcal{P}$ such that $$\begin{array}{}\text{(89)}& {{\dot{\overline{\mathbf{Z}}}}_{i2d}}^2+{{\dot{\overline{\mathbf{v}}}}_{id}}^2\le {\delta }_M.\end{array}$$

Therefore, there exists a constant $\overline{\delta }>0$ satisfying $$\begin{array}{}\text{(90)}& \delta \le \overline{\delta },\forall {\mathbf{X}}_d,{\mathbf{E}}_d\in \mathcal{P}.\end{array}$$

For any constant ${\overline{B}}_E$, there exists several parameters and initial condition satisfying $$\begin{array}{}\text{(91)}& V0\le \overline{\delta }/c<2\overline{\delta }/c\le {\overline{B}}_E,\end{array}$$such that $$\begin{array}{}\text{(92)}& Vt\le \frac{\overline{\delta }}{c},\forall t\ge 0,\end{array}$$

and $\mathcal{E}$ is an invariant set.

The above result (92) can be approved by contradiction. Since the reference input ${\mathbf{X}}_d$ must be bounded, that leads to ${\mathbf{X}}_d\in \mathcal{X}$ on $\forall t\ge 0$. Assuming $\exists t_1,V{t}_1>\overline{\delta }/c$, due to $V$ is a continuous function, considering the continuous property, there exists a moment $t_0$ such that $V{t}_0=\overline{\delta }/c,V{t}_{0+}>\overline{\delta }/c$. Noticing ${{\mathbf{E}}_d}^2\le 2V$ in (75), there exists ${{\mathbf{E}}_d{t}_0}^2\le 2\overline{\delta }/c$. Considering (91) yields ${\mathbf{X}}_d{t}_0,{\mathbf{E}}_d{t}_0\in \mathcal{P}$, so $\delta t_0\le \overline{\delta }$. Moreover, by taking account of (85), one can derive $\dot{V}{t}_0\le -cV{t}_0+\delta t_0=-\overline{\delta }+\delta t_0\le 0$. Hence, that causes $V{t}_{0+}\le V{t}_0$, which leads contradiction. This completes the proof of (91). Furthermore, considering ${{\mathbf{E}}_d}^2\le 2V$ and (92), (91), one can derive $\mathcal{E}$ is an invariant set.

Therefore, all the signals in the system are uniformly bounded, and the states ${\mathbf{X}}_{i1}$, ${\mathbf{X}}_{i2}$ are always within their constraints.

4. Simulation Results and Analysis

To illustrate the effectiveness and the superiority of the proposed FTCC scheme in this paper, using MATLAB/Simulink to simulate four UAVs whose communication topology is shown in Figure 3.

[figure(s) omitted; refer to PDF]

Figure 3 shows the communication network, and the parameters of UAVs are referred to [28]. The element $a_{ij}$ of the adjacency matrix $\mathcal{A}$ is defined as 1 if the link between $i$th UAV and $j$th UAV existed; otherwise, $a_{ij}=0$. The initial attitudes of all UAVs are presented in Table 1, and the initial angular rates are defined opposite as values of attitudes. The initial values of $V_i$, ${\chi }_i$, and ${\gamma }_i$ of all UAVs are set as 30 m/s, 0.01 rad and 0.01 rad, respectively.

Table 1

Initial attitudes of all UAVs.

	${\mu }_{i}0$ (rad)	${\alpha }_{i}0$ (rad)	${\beta }_{i}0$ (rad)
UAV 1	0.01	0.01	0.01
UAV 2	-0.015	-0.015	-0.015
UAV 3	0.02	0.02	0.02
UAV 4	-0.025	-0.025	-0.025

Assuming the safe range of the attitudes ${\mu }_i,{\alpha }_i,\text{and}{\beta }_i$ of all UAVs are $-8~8$ deg, and the corresponding angular velocity $p_i,q_i,r_i$ are $-0.25~0.25$ rad/s. In addition, the maximum operation range of control surfaces ${\delta }_{ia},{\delta }_{ie},\text{and}{\delta }_{ir}$ are $-25~25$ deg.

Therefore, for any $i$th UAV, the upper bound ${\overline{\mathbf{X}}}_{i1}$ and the lower bound ${\underset{¯}{\mathbf{X}}}_{i1}$ of states ${\mathbf{X}}_{i1}$ are defined as ${8,8,8}^T$ deg and $-{8,8,8}^T$ deg, respectively. Similarly, the upper bound ${\overline{\mathbf{X}}}_{i2}$ and the lower bound ${\underset{¯}{\mathbf{X}}}_{i2}$ of states ${\mathbf{X}}_{i2}$ are ${\mathrm{0.25,0.25,0.25}}^T$ rad/s and $-{\mathrm{0.25,0.25,0.25}}^T$ rad/s, respectively. The upper bound $u_{i0\tau \max }$ and the lower bound $u_{i0\tau \min }$ are $-25$ deg and $25$ deg, respectively.

The main control parameters are chosen as ${\mathbf{K}}_{i10}=\mathrm{diag}5,5,5$, ${\mathbf{K}}_{i20}=\mathrm{diag}\mathrm{20,20,20}$, ${\mathbf{K}}_{i3}=\mathrm{diag}\mathrm{30,30,30}$, ${\eta }_{i1}={\eta }_{i2}={\eta }_{i3}={\eta }_{i4}={\eta }_{i5}=1.5$, ${\eta }_{i6}=1.8$, $k_{{ϵ}_{i1}}=k_{{ϵ}_{i2}}=100$, $h_{i1}=h_{i2}=20$, and ${\mu }_{ib}=0.001$.

To verify the effectiveness of the proposed FTCC scheme under actuator fault, the following fault signals are chosen: $$\begin{array}{c}\text{(93)}& \begin{array}{c}{\mathbf{\rho }}_1={\mathbf{\rho }}_2={\mathbf{\rho }}_3={\mathbf{\rho }}_4=1,\hfill \\ {\mathbf{U}}_{1f}={\mathbf{U}}_{2f}={\mathbf{U}}_{3f}={\mathbf{U}}_{4f}=0^{\circ }\hfill \end{array}t<8\begin{array}{c}{\mathbf{\rho }}_1=\mathrm{diag}\mathrm{0.7,0.9,0.8},\hfill \\ {\mathbf{\rho }}_2={\mathbf{\rho }}_3={\mathbf{\rho }}_4=0,\hfill \\ {\mathbf{U}}_{1f}={3^{\circ },2^{\circ },4^{\circ }}^T,\hfill \\ {\mathbf{U}}_{2f}={\mathbf{U}}_{3f}={\mathbf{U}}_{4f}=0^{\circ },\hfill \end{array}8\le t<10,\\ \begin{array}{cc}\begin{array}{c}{\mathbf{\rho }}_1={\mathbf{\rho }}_2\mathrm{diag}\mathrm{0.7,0.9,0.8},\hfill \\ {\mathbf{\rho }}_3={\mathbf{\rho }}_4=0,\hfill \\ {\mathbf{U}}_{1f}={\mathbf{U}}_{2f}={3^{\circ },2^{\circ },4^{\circ }}^{\text{T}},\hfill \\ {\mathbf{U}}_{3f}={\mathbf{U}}_{4f}={\mathbf{U}}_{4f}={\mathbf{0}}^{\circ },\hfill \end{array}\hfill & 10\le t<12,\hfill \end{array}\\ \begin{array}{c}{\mathbf{\rho }}_1={\mathbf{\rho }}_2={\mathbf{\rho }}_3=\mathrm{diag}\mathrm{0.7,0.9,0.8},\hfill \\ {\mathbf{\rho }}_4=0,\hfill \\ {\mathbf{U}}_{1f}={\mathbf{U}}_{2f}={\mathbf{U}}_{3f}={3^{\circ },2^{\circ },4^{\circ }}^T,\hfill \\ {\mathbf{U}}_{4f}=0^{\circ },\hfill \end{array}12\le t<14,\\ \begin{array}{c}{\mathbf{\rho }}_1={\mathbf{\rho }}_2={\mathbf{\rho }}_3={\mathbf{\rho }}_4=\mathrm{diag}\mathrm{0.7,0.9,0.8},\hfill \\ {\mathbf{U}}_{1f}={\mathbf{U}}_{2f}={\mathbf{U}}_{3f}={\mathbf{U}}_{4f}={3^{\circ },2^{\circ },4^{\circ }}^T.\hfill \end{array}t\ge 14.\end{array}$$

Remark 8.

The reason of choosing the main control parameter briefly describes as follows. The parameter ${\mathit{K}}_{i10}$ decides the closed-loop dynamics of ${\dot{\mathit{E}}}_{i1}$, which can be approximated as ${\dot{\mathit{E}}}_{i1}\approx -{\mathit{K}}_{i1}{\mathit{E}}_{i1}+{\mathit{E}}_{i1}^T{A}_i{g}_{i1}{\mathit{X}}_{i1}{\mathit{G}}_{i1}{\mathit{E}}_{i2}+h_{i1}/2·{{ϵ}_{i1}}^2$ according to (32) and (84), and the term ${\mathit{E}}_{i1}^T{A}_i{g}_{i1}{\mathit{X}}_{i1}{\mathit{G}}_{i1}{\mathit{E}}_{i2}+h_{i1}/2·{{ϵ}_{i1}}^2$will rapidly reduce due to the response of inner-loop dynamics ${\mathit{E}}_{i2}$and ${ϵ}_{i1}$ are converge rapidly, then, it can be further represented as ${\dot{\mathit{E}}}_{i1}\approx -{\mathit{K}}_{i10}{\mathit{E}}_{i1}$, so that the settling time of the dynamic ${\mathit{E}}_{i1}$ approximately equal to $3/{\lambda }_{\min }{\mathit{K}}_{i10}$. There exists a tradeoff – a larger value ${\mathit{K}}_{i10}$ helps to reduce the settling time while it needs larger change of the state ${\mathit{X}}_{i2}$, which may cause saturation, so ${\mathit{K}}_{i10}$ is taken as $\mathrm{diag}5,5,5$. Similarly, parameters ${\mathit{K}}_{i20}$ and ${\mathit{K}}_{i3}$ decide the convergence rate of the inner-loop control errors ${\mathit{E}}_{\text{i}2}$ and ${\mathit{E}}_{\text{i}3}$, respectively, and their values normally take several times as the ${\mathit{K}}_{\text{i}10}$ to get faster response. Moreover, for parameters ${\eta }_{\text{i}1}~{\eta }_{\text{i}6}$, their inverses serve as learning rates of neural network, which usually take as a value among $1~1000$ for fixed rate learning algorithms. In addition, parameters $k_{{ϵ}_{\text{i}1}}$ and $k_{{ϵ}_{i2}}$ are adjustment factors of dynamic surface filter, which are usually taken as a large value. Finally, for ${\mu }_{\text{ib}}$, it can adjust the astringency of the error ${\mathit{E}}_{\text{i}3}$, which is the smaller the better, but the too small value will tend to cause virtual control signal “explosion”.

The response of bank angle ${\mu }_i$, angle of attack ${\alpha }_i$, and sideslip angle ${\beta }_i$ of each UAV are shown in Figures 4–6, respectively. It can be seen that all UAVs can track their references ${\mu }_{id}$, ${\alpha }_{id}$, and ${\beta }_{id}$. Although the fault occurs to UAV 1, UAV 2, UAV 3, and UAV 4 at 8 s, 10 s, 12 s, and 14 s, respectively, all UAVs can quickly track individual references again.

[figure(s) omitted; refer to PDF]

The response of state ${\mathbf{X}}_{i2}$ is shown in Figure 7. It can be seen that the states $p_i$, $q_i$, and $r_i$ of all UAVs never exceed its upper or lower bound. Meanwhile, if the state constraints are not considered in the FTCC scheme, the states $q_1$ and $q_3$ will exceed the lower bound, which is shown in Figure 8. In addition, the control input signals ${\delta }_{ia}$, ${\delta }_{ie}$, and ${\delta }_{ir}$ are presented in Figure 9. Since the actuator constraint scheme (15) is adopted, the input signals never exceed their upper and lower bounds.

[figure(s) omitted; refer to PDF]

5. Conclusion and Future Work

This paper has explored an FTCC scheme for multi-UAVs under the distributed communication network, in which the issues including input saturation, state constraints, actuator faults, and unknown disturbances have all been taken into account.

It can be noted that the proposed FTCC scheme only considers fixed and undirected communication network. In addition, communication delay and communication interferences are not considered, and finite-time convergence technology has not been considered in the FTCC scheme, so the control performance cannot be achieved in finite time. Moreover, compared to the Euler attitude angles, the airflow attitude angles are necessary and easy to combine with the UAV’s outer loop for position control, hence in this paper, it is directly used in the attitude control. However, using the airflow attitude angles for feedback control is less reliable than the former. Furthermore, sensor fault may occur at the same time, which perhaps outweigh the risk of actuator fault, so it deserves more attention and investigation. Finally, state measurements have been directly used in the control law without considering noise filtering, so that the performance may be degraded when sensor measurements have severe noises. Taking into account the noise filtering algorithms and sensor faults simultaneously will significantly increases the difficulty of proving the closed-loop system stability, which makes the issue challenging.

Therefore, in future work, the essence of communication delays, finite-time convergence technology, the reliability of using airflow attitude angle, sensor fault, and noise filtering will be taken into account on the basis of existing research. Besides, based on the simulation results from MATLAB/Simulink, the hardware-in-the-loop verification scheme will be adopted to further verify the proposed control scheme towards more practical applications.

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (Nos. 61833013 and 62003162), Natural Science Foundation of Jiangsu Province of China (No. BK20200416), China Postdoctoral Science Foundation (Nos. 2020TQ0151 and 2020M681590), Aeronautical Science Foundation of China (No. 20200007018001), Science and Technology on Space Intelligent Control Laboratory (No. HTKJ2022KL502015), and Natural Sciences and Engineering Research Council of Canada.

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