Event-Triggered Finite-Time Attitude Cooperative Control for Multiple Unmanned Aerial Vehicles

Abstract

The finite-time attitude cooperative control problem for a group of multiple unmanned aerial vehicle systems with external disturbances and uncertain parameters is discussed in this paper. The dynamics of the systems is described by quaternion avoiding the singularity. Based on the attitude error and angular velocity error, a novel nonsingular terminal sliding mode surface is proposed for the controller with event-triggered scheme. The lumped disturbances are estimated by neural networks with adaptive law. The communication frequency is decreased by the proposed distributed event-triggered based sliding mode controller. Lyapunov theory is utilized to analyze the stability of the systems, and the Zeno behavior is avoided by rigorous proof. Finally, simulation examples are presented to illustrate the efficiency of the proposed control algorithm.

1. Introduction

Attitude cooperative control of multiple unmanned aerial vehicle systems (MUAVs) is significantly important in the formation flying missions. Compared to a single unmanned aerial vehicle (UAV), MUAVs can accomplish more complex and dangerous missions by collaboration, such as search and rescue, forest fire fighting, emergency rescue, low-attitude reconnaissance, and combat military missions [1–3]. Attitude cooperative control problem has been of growing interests in last several years due to its engineering and theoretical implications. Many scholars proposed different attitude control scheme to improve the accuracy and stability of the MUAVs. Variable structure control combined with decentralized communication scheme was proposed for spacecraft formation flying [4], based on the development of consensus theory, leader-follower was employed in the multiple aircrafts [5, 6]. As the amount of the MUAVs increases, the communication burden among each UAV will increase and may cause the network communication jam, and it would seriously affect the stability of the systems due to the band width is limited. It is significant to consider the network communication strategy of MUAVs when designing the attitude cooperative controller.

Even-triggered scheme is employed in the multiagent systems for considering the limited band width and energy consumption, instead of continuous control input update, the controller updates the input depending on the event-triggered function, which is relevant to the measurement error, and when the estimation error comes up to the given threshold value the update of the controller will be updated [7]. An event-triggered-based controller was proposed in first-order multiagent systems (MAS) by introducing the event-triggered mechanism, and the triggered condition was designed associated with the states of agents [8]. Distributed rendezvous problem was investigated for second-order multiagent systems with combinational measurement by event-triggered mechanism [9]. Based on the measurement error, the event triggered function was built for linear MAS, and all the states of the agents reach to consensus [10, 11]. Event-triggered scheme was employed in many system dynamical model which can be described as second-order dynamics [12]. A distributed sliding mode controller based on event triggered finite time mechanism was designed for formation of multirobot systems [13]. The event triggered was widely used in attitude control of spacecrafts to save the communication resources [14–16]. However, the mentioned works with attitude control did not consider the uncertain parameters. Based on event-triggered strategy, time-varying formation problem was investigated for MUAVs under switching topology [17], attitude formation on SO (8) [18], and dynamical consensus formation problem which was limited by time-varying disturbances [19]. However, external disturbances and inertial matrix uncertainty cannot avoid in practical environment. The event-based formation control for MUAVs only guarantees asymptotically convergence in the aforementioned works [17–19]. Finite-time control is a useful tool which has high accuracy and robustness property, enabling the control systems to approach the stable region in finite time. Finite-time control has been extensively utilized in Euler-Lagrange systems [20–22]. Distributed attitude tracking problem of spacecrafts was proposed considering disturbances and uncertain parameters in finite time [20]. Adaptive control was introduced into the finite-time controller extended the mentioned work [20] for attitude tracking problem of spacecrafts [21]. Feedback control was employed in formation control for finite time convergence of nonholonomic wheeled mobile robots [22]. Distributed finite-time control (FTC) problem was studied for multiple quadrotor formation with the information of leader not available to all the followers [23]. However, no external disturbances were considered. The disturbance was estimated by the observer, and FTC was investigated for a single quadrotor [24, 25]. However, attitude cooperative problem was not considered. There is less work associated with the finite-time attitude cooperative or formation control with event-triggered mechanism for MUAVs. Most recently, FTC based on event-triggered was investigated for quadrotor flying control [26, 27]. However, attitude tracking problem was not considered, and the controller designed was limited to the specific UAV. So, attitude cooperative control with FTC theory and event-triggered mechanism is more interesting.

Motivated by the aforementioned works and analysis, finite-time attitude cooperative control problem of MUAVs with event-triggered mechanism is investigated, and the network communication resources are reduced. The contribution of this paper is illustrated in the following aspects: (1) external disturbances and uncertain parameters are considered in the attitude dynamics, and the attitude cooperative problem is described by the quaternion avoiding the singularity. The attitude tracking consensus errors are measured by employing a positive error function, a novel integral sliding mode surface is proposed, the FTC is designed for the closed loop systems, and neural network is utilized to estimate the lumped uncertainties. (2) The communication frequency of the controller among the followers is reduced due to the event-triggered strategy which is employed in the controller, so the proposed novel event-triggered function saves the communication burden and energy of each UAV. The deduced lower bound between triggering intervals guarantee no Zeno behavior occurs. (3) Fast terminal sliding mode control is utilized in control law which guarantees that the attitude achieves the desired value in finite time.

The rest of this paper is organized as follows. In Section 2, preliminaries of graph theory, quaternion-based attitude dynamics of MUAVs, and some useful lemmas are given, and Section 3 gives main results. The performance of the controller is proved by numerical simulation examples in Section 4. Finally, conclusion is given in Section 5.

2. Preliminaries and Problem Formulation

2.1. Notations

The following convenience notations are adopted throughout the paper: Rⁿ denotes n × 1 real column vector, I_n = [1, ...1]^T denotes n × 1 column vector with each element being, and I_n denotes a n × n dimensional identity matrix. ⊗ stands for Kronecker product. ‖.‖ stands for the induced matrix 2-norm or the Euclidean vector norm. In addition, for a given vector x ∈ R_n, x_i denotes the ith element of the vector x, sig^r(x) = sgn(x)|x|^r.

Graph theory is utilized to describe the communication flow among the MUAVs. Let G = (ν, ξ) denotes the graph, in which, ν = {ν₁, ν₂, ⋯, ν_n} is a nonempty set containing a group number of nodes which denotes the UAV, and ξ⊆ν × ν is called edge which is a set of nodes. If there any two nodes could communicate with each other, the graph G is called connected graph. A = [a_ij]_N×N ∈ R^n×n is weighted adjacency matrix representing the communication between each node, in which, a_ii = 0, otherwise, a_ij = a_ji ≥ 0. D = diag{d₁, d₂, ⋯, n} denotes the degree matrix of associated with weighted graph, the elements of the degree matrix are d_i = ∑_j=1ⁿa_ij. The Laplacian matrix of the weighted graph is denoted by L = D − A, and L is symmetric matrix.

Throughout this paper, leader-follower MUAVs are considered which contains one leader and n followers. The followers are marked as i(i = 1, ⋯, n), and the leader is marked as 0. Let $\overline{G}$ denotes the topology graph associated with MUAVs containing one leader and n followers. A diagonal matrix B = diag{b₁, ⋯, b_n} is utilized to denote the communication between the follower UAV and the leader UAV. If b_i > 0 means that the ith follower can obtain the communication flow of the leader, otherwise, b_i = 0.

Assumption 1 . —

Consider the MUAVs consisting of N followers and one leader, the topology of the MUAVs is described by $\overline{G}$, and $\overline{G}$ is directed connected graph.

Based on Assumption 1 and graph theory, we define a matrix C = L + B.

Lemma 1 (see [28]). —

If Assumption 1 holds, then, matrix C is invertible.

Lemma 2 (see [29]). —

Consider a system modeled as $\dot{z}=f\left(z\right),f\left(0\right)=0,x\in {ℝ}^n$, a continuous function V(z) ∈ ℂ¹ which is defined on a neighbourhood of the origin. If the function V(z) satisfies that it is positive definite and $\dot{V}\left(z\right)\le -{\nabla }_1{V}^{\partial }\left(z\right)$, where ∂ ∈ (0, 1), ∇₁ is positive parameters. The systems converge to the origin in finite-time, the converge time T which depends on the initial state of z(0):

$$\begin{array}{c}T\left(z\left(0\right)\right)\le \frac{V^{1-\partial }\left(z\left(0\right)\right)}{{\nabla }_1\left(1-\partial \right)}.\end{array}$$ (1)

Lemma 3 (see [30]). —

If there is a real number x_i ∈ ℝ, i = 1, ⋯, n, α ∈ (0, 1], then

$$\begin{array}{c}{\left(\sum _{i=1}^n\hspace{0.17em}\mid \hspace{0.17em}{x}_i\hspace{0.17em}\mid \hspace{0.17em}\right)}^{\alpha }\le \sum _{i=1}^n{\left|x_i\right|}^{\alpha }\le n^{1-\alpha }{\left(\sum _{i=1}^n\hspace{0.17em}\mid \hspace{0.17em}{x}_i\hspace{0.17em}\mid \hspace{0.17em}\right)}^{\alpha }.\end{array}$$ (2)

For x ∈ ℝⁿ, |α| ∈ (0, 1), then

$$\begin{array}{c}‖x^{\alpha }‖\le n^{1-\alpha }{‖x‖}^{\alpha }.\end{array}$$ (3)

2.2. Attitude Dynamics Model of MUAVs

Throughout this paper, the attitude dynamics of UAV is described by quaternion which could avoid singular problem and analyze conveniently [31].

$$\begin{array}{c}{J}_i{\dot{\omega }}_i=u_i-S\left({\omega }_i\right)J_i{\omega }_i+{\vartheta }_i,\\ {\dot{Q}}_i=\frac{1}{2}\varphi \left(Q_i\right){\omega }_i,\end{array}$$ (4)

where Q_i = [q_i q_i0]^T represents the attitude of the ith UAV, q_i ∈ ℝ³, q_i0 ∈ ℝ, Q_i ∈ ℝ⁴, Q_i ∈ ℝ⁴, |Q_i| = 1, and ω_i ∈ ℝ³ is the angular velocity. J_i denotes the inertia matrix of the ith UAV and is positive definite; u_i denotes the control torque of the ith UAV; ϑ_i denotes the external disturbances. ϕ(Q_i) is given by

$$\begin{array}{c}\varphi \left(Q_i\right)=\left(\begin{array}{c}{q}_{i0}{I}_3+S\left(q_i\right)\\ -{q_i}^{{\rm T}}\end{array}\right).\end{array}$$ (5)

For a vector given as n = [n₁, n₂, n₃]^T ∈ R^3×1, S(n) is defined as

$$\begin{array}{c}S\left(n\right)=\left(\begin{array}{ccc}0& ‐n_3& n_2\\ n_3& 0& ‐n_1\\ ‐n_2& n_1& 0\end{array}\right).\end{array}$$ (6)

Let R(Q_i) denote the rotation matrix which is given as R(Q_i) = (2q_i0² − 1)I₃ + 2q_iq_i^T − 2q_i0Π(q_i), Q_i = [q_i q_i0]^T for the attitude control of UAV, the rotation matrix denotes the inertial frame of the ith UAV into the body frame. The multiplication between two unit quarternions is given by

$$\begin{array}{c}{Q}_1\odot Q_2=\left(\frac{q_{10}{q}_2+q_{20}{q}_1+S\left(q_1\right)q_2}{q_{10}{q}_{20}-q_1^T{q}_2}\right),\end{array}$$ (7)

where Q₁ = [q₁ q₁₀]^T and Q₂ = [q₂ q₂₀]^T.

In the leader-following MUAVs, the followers adjust itself attitude to be consistent with the attitude of the leader. The attitude tracking errors and angular velocity errors of the ith UAV are given as follows

$$\begin{array}{c}{\tilde{Q}}_i=Q_i^{-1}\odot Q_d,\end{array}$$ (8)

$$\begin{array}{c}{\tilde{\omega }}_i={\omega }_i-R\left({\tilde{Q}}_i\right){\omega }_d,\end{array}$$ (9)

where Q_d≜[q_d, η_d]^T denotes the desired attitude which is given Q_d≜[q_d, η_d]^T, and the desired angular velocity is denoted by ω_d. Based on the definition of the tracking errors, ${\tilde{\omega }}_i$ and ${\tilde{Q}}_i$ represent the attitude velocity tracking error and angular tracking errors, respectively.

The attitude tracking error systems can be obtained,

$$\begin{array}{c}{J}_{\text{i}}{\dot{\tilde{\omega }}}_{\text{i}}=-{{\omega }_{\text{i}}}^{\times }·J_{\text{i}}{\omega }_{\text{i}}+u_{\text{i}}+T_{\text{i}}·\left[\Pi \left({\tilde{\omega }}_{\text{i}}\right)·R\left(Q_{\text{id}}\right)·{\omega }_d-R\left(Q_{\text{id}}\right)·{\dot{\omega }}_d\right],\end{array}$$ (10)

$$\begin{array}{c}{\dot{\tilde{Q}}}_i=\frac{1}{2}\varphi \left({\tilde{Q}}_{\text{i}}\right)·{\tilde{\omega }}_{\text{i}}.\end{array}$$ (11)

Assumption 2 . —

Three positive constants d_m, σ₁, σ₂ exist and are satisfying |d_i| ≤ d_m, |ω_i| ≤ σ₁, and $\left|{\dot{\omega }}_i\right|\le {\sigma }_2$, respectively.

Assumption 3 . —

The inertia matrix ${\overline{J}}_i$ is known and nonsingular. ΔJ denotes the uncertainties and is bounded.

Lemma 4 (see [32]). —

Considering the system formulated as eqs. (10) and (11), for sliding mode surface ${\kappa }_i={\tilde{\omega }}_i+r_1{q}_i+r_2{q}_i^c$, where 0 < c < 1, r₁ > 0, r₂ > 0, for i = 1, ⋯, n. If the sliding mode surface reaches zero, then, ${\tilde{\omega }}_i=0,q_{0,i}=1$ and q_i = 0 can be reached in finite time, respectively.

3. Main Results

3.1. Event-Triggered Finite-Time Control Design

In this section, the control objective is to design a finite-time control law such that the angular velocity errors ${\tilde{\omega }}_i$ and the error quaternions ${\tilde{Q}}_i$ of the closed-loop system (10) and (11) can converge to small regions in finite time, respectively.

First, the sliding mode surface ${\overline{s}}_i$ is defined as

$$\begin{array}{c}{\overline{s}}_i={\overline{J}}_i\left[{\tilde{\omega }}_i+k_1{\tilde{q}}_i+k_2{{\rm T}}_i\left({\tilde{q}}_i\right)\right],\end{array}$$ (12)

with ${{\rm T}}_i\left({\tilde{q}}_i\right)\left({\tilde{q}}_i\right)={\left[{{\rm T}}_{i1}\left({\tilde{q}}_{i1}\right),{{\rm T}}_{i2}\left({\tilde{q}}_{i2}\right),{{\rm T}}_{i3}\left({\tilde{q}}_{i3}\right)\right]}^T\in R^{3\times 1}$,

$$\begin{array}{c}{{\rm T}}_{ij}\left({\tilde{q}}_{ij}\right)=\left\{\begin{array}{c}{\text{sig}}^{\ell }\left({\tilde{q}}_{ij}\right),\text{if\hspace{0.17em}}{s}_{ij}^{\ast }=0\text{\hspace{0.17em}or\hspace{0.17em}}{s}_{ij}^{\ast }\ne 0,\left|{\tilde{q}}_{ij}\right|>\Upsilon ,\hfill \\ {\varpi }_1{\tilde{q}}_{ij}+{\varpi }_2\text{si}{\text{g}}^2\left({\tilde{q}}_{ij}\right),\text{if\hspace{0.17em}}{s}_{ij}^{\ast }\ne 0,\left|{\tilde{q}}_{ij}\right|\le \Upsilon \hfill \end{array}\right.\end{array}$$ (13)

where i = 1, ⋯, n, j = 1, 2, 3, s_i^∗ = [s_i1^∗, s_i2^∗, s_i3^∗]^T, and $s_i^{\ast }={\tilde{w}}_i+k_1{\tilde{q}}_i+k_2{\text{sig}}^r\left({\tilde{q}}_i\right)$, where k₁ and k₂ are positive constants. Define ${\text{sig}}^{\iota }\left({\tilde{q}}_i\right)={\left[{\text{sig}}^{\iota }\left({\tilde{q}}_{i1}\right),{\text{sig}}^{\iota }\left({\tilde{q}}_{i2}\right),{\text{sig}}^{\iota }\left({\tilde{q}}_{i3}\right)\right]}^T$, ι ∈ (0, 1), ϖ₁ = (2 − r)ϒ^r−1, ϖ₂ = (r − 1)ϒ^r−2, ϒ is a small positive constant.

To develop the control law, the following equations are derived from (10) and (11):

$$\begin{array}{c}{\overline{J}}_i\left({\dot{\tilde{w}}}_i+k_1{\dot{\tilde{q}}}_i+k_2{\dot{{\rm T}}}_i\left({\tilde{q}}_i\right)\right)=z_i+{\delta }_i+u_i,\end{array}$$ (14)

where

$$\begin{array}{c}{z}_i=-S\left(w_i\right){\overline{J}}_i{w}_i+{\overline{J}}_i\left(S\left({\tilde{w}}_i\right)R\left({\tilde{Q}}_i\right)w_d-R\left({\tilde{Q}}_i\right){\dot{w}}_d\right)+k_1{\overline{J}}_i{\dot{\tilde{q}}}_i+k_2{\overline{J}}_i{\dot{\alpha }}_i\left({\tilde{q}}_i\right),\end{array}$$ (15)

$$\begin{array}{c}{\dot{\alpha }}_i\left({\tilde{q}}_i\right)=\left\{\begin{array}{c}r\mathrm{diag}\left({\left|{\tilde{q}}_{ij}\right|}^{r-1}\right){\tilde{q}}_i,\hspace{1em}\text{if}\hspace{1em}{s}_{ij}^{\ast }=0\hspace{1em}\text{or}\hspace{1em}{s}_{ij}^{\ast }\ne 0,\hspace{1em}\left|{\tilde{q}}_{ij}\right|>\varphi ,\hfill \\ l_1{\dot{\tilde{q}}}_i+2l_2{\tilde{q}}_i\mathrm{sgn}\left({\tilde{q}}_i\right){\dot{\tilde{q}}}_i,\hspace{1em}\text{if}\hspace{1em}{s}_{ij}^{\ast }\ne 0,\hspace{1em}\left|{\tilde{q}}_{ij}\right|\le \varphi ,\hfill \end{array}\right.\end{array}$$ (16)

$$\begin{array}{c}{\delta }_i={\vartheta }_i-{\tilde{J}}_i{\dot{\tilde{w}}}_i-S\left(w_i\right){\tilde{J}}_i{w}_i+{\tilde{J}}_i\left[S\left({\tilde{\omega }}_{\text{i}}\right)R\left({\tilde{Q}}_i\right){\omega }_d-R\left({\tilde{Q}}_i\right){\dot{\omega }}_d\right].\end{array}$$ (17)

δ _i is the lumped disturbances containing model uncertainty and external disturbances.

By (12) and (14), we can obtain

$$\begin{array}{c}{\dot{\overline{s}}}_i=z_i+{\delta }_i+u_i.\end{array}$$ (18)

Based on the sliding mode surface ${\overline{s}}_i={\overline{J}}_i\left[{\tilde{\omega }}_i+k_1{\tilde{q}}_i+k_2{{\rm T}}_i\left({\tilde{q}}_i\right)\right]$, a novel integral sliding mode surface is proposed which is given as follows

$$\begin{array}{c}{s}_i={\overline{s}}_i-{\int }_0^t{x}_i^{\eta }dt,\end{array}$$ (19)

where $x_i=-{\sum }_{j\in N_i}{a}_{ij}\left({\overline{s}}_i-{\overline{s}}_j\right)+b_i{\overline{s}}_i$, and η ∈ (0.5, 1) is strictly the ratio of positive odd numbers. The derivative of (19) is

$$\begin{array}{c}{\dot{s}}_i={\dot{\overline{s}}}_i-x_i^{\eta }.\end{array}$$ (20)

An event-triggered finite-time sliding mode consensus controller is designed as follows

$$\begin{array}{c}{u}_i\left(t\right)=x_i^{\eta }\left(t_k^i\right)-k_3\mathrm{sign}\left(s_i\left(t_k^i\right)\right)-k_4{s}_i\left(t_k^i\right)-z_i\left(t_k^i\right)-{\widehat{\delta }}_i\left(t_k^i\right),\end{array}$$ (21)

where k₃ and k₄ are positive constants, respectively. For t ∈ [t_kⁱ, t_k+1ⁱ), t_kⁱ is the latest event-triggered time for the ith UAV, and the UAV only updates the control protocol at its own event-triggered time.

An adaptive radial basis function neural networks (RBFNNs) scheme is proposed for the unknown disturbance δ_i, as RBFNNs can estimate the unknown continuous functions δ_i and ensure tracking error ultimately converges to an adequately small compact. Illustrated in Figure 1, the adaptive RBFNNs can be written as

$$\begin{array}{c}{\widehat{\delta }}_i={\widehat{W}}_i^T{H}_i\left(X_{\mathit{\text{in}}}\right),\end{array}$$ (22)

where ${\widehat{\delta }}_i\in {ℝ}^3$ is the RBFNNs output vector, $X_{\mathit{\text{in}}}={\left[{{\tilde{q}}_i}^T,{{\tilde{w}}_i}^T\right]}^T\in {ℝ}^6$ is the input vector of the RBFNNs, ${\widehat{W}}_i\in {ℝ}^{J\times 3}$ is the weight vector, J > 1 is the nodes number of middle hidden layer, H_i = [h_i1, ⋯,h_iJ]^T ∈ ℝ^J is the basis function vector, and h_iJ is being the commonly used Gaussian functions, which is simplified as

$$\begin{array}{c}{h}_{iJ}\left(X_{\mathit{\text{in}}}\right)=\exp \left(-\frac{{‖X_{\mathit{\text{in}}}-c_{iJ}‖}^2}{{\sigma }_{iJ}}\right),\end{array}$$ (23)

where c_iJ is the center of the receptive field, and σ_iJ is the width of the Gaussian function. There exists an optimal vector W_i^∗ such that

$$\begin{array}{c}{\delta }_i=W_i^{\ast T}{H}_i\left(X_{\mathit{\text{in}}}\right)+{\epsilon }_i,\end{array}$$ (24)

where ε_mⁱ denotes the maximum value of the RBFNNs estimation error ‖ε_i‖.

Figure 1.

Adaptive RBFNN control.

Let ${\tilde{W}}_i=W_i^{\ast }-{\widehat{W}}_i$ denotes the vector of weight errors, and the adaptive weight update law are designed as

$$\begin{array}{c}{\dot{\widehat{W}}}_i={AH}_i\left(X_{\mathit{\text{in}}}\right)s_i^T,\end{array}$$ (25)

where A is a positive-definite symmetric matrix of gains.

The measurement error of the event-triggered mechanism is defined as

$$\begin{array}{c}{e}_i\left(t\right)=x_i^{\eta }\left(t_k^i\right)-x_i^{\eta }\left(t\right)-k_3\mathrm{sign}\left(s_i\left(t_k^i\right)\right)+k_3\mathrm{sign}\left(s_i\left(t\right)\right)-k_4{s}_i\left(t_k^i\right)+k_4{s}_i\left(t\right)-z_i\left(t_k^i\right)+z_i\left(t\right)-{\widehat{\delta }}_i\left(t_k^i\right)+{\widehat{\delta }}_i\left(t\right).\end{array}$$ (26)

3.2. Stability Analysis

In the following, the stability of the attitude cooperative under event-triggered adaptive RBFNNs control law is analyzed in detail.

Theorem 1 . —

On the basis of Assumptions 1, and considering that the system (10) and (11) under the action of the controller (21) and the adaptive weight update law (25), the following event-triggered function is given as follows

$$\begin{array}{c}Y\left(t\right)=‖e_i‖-k_3-k_4‖s_i‖+{\rho }_i,\end{array}$$ (27)

where ε_mⁱ < ρ_i < k₃, and when the event-triggered function Y(t) > 0, that is, ‖e_i‖ > k₃ + k₄‖s_i‖ − ρ_i, the event is triggered. The ith UAV performs information interaction and update the control protocol at its own event-triggered time. And the system can achieve finite-time consensus under this action.

Proof —

First, selecting the Lyapunov function as follows

$$\begin{array}{c}{V}_1=\frac{1}{2}{s}_i^T{s}_i+\frac{1}{2}\text{tr}\left({\tilde{W}}_i^T{A}^{-1}{\tilde{W}}_i\right).\end{array}$$ (28)

Taking the derivative of (28), substituting (21) and (26) into the derivative, we can obtain

$$\begin{array}{c}{\dot{V}}_1=s_i^T\left(e_i-k_3\mathrm{sign}\left(s_i\right)-k_4{s}_i+{\delta }_i-{\widehat{\delta }}_i\right)+\text{tr}\left(-{\tilde{W}}_i^T{A}^{-1}{\dot{\widehat{W}}}_i\right)\le ‖s_i‖‖e_i‖-k_3‖s_i‖-k_4{‖s_i‖}^2+s_i^T\left({\tilde{W}}_i^T{H}_i+{\epsilon }_i\right)+\text{tr}\left(-{\tilde{W}}_i^T{A}^{-1}{\dot{\widehat{W}}}_i\right)\le ‖s_i‖\left(‖e_i‖-k_3-k_4‖s_i‖+{\epsilon }_m^i\right)+\text{tr}\left({\tilde{W}}_i^T{H}_i{s}_i^T\right)+\text{tr}\left(-{\tilde{W}}_i^T{A}^{-1}{\dot{\widehat{W}}}_i\right).\end{array}$$ (29)

By the adaptive weight update law (25), it is obtained that

$$\begin{array}{c}{\dot{V}}_1\le ‖s_i‖\left(‖e_i‖-k_3-k_4‖s_i‖+{\epsilon }_m^i\right).\end{array}$$ (30)

When the event-triggered function Y(t) ≤ 0, ‖e_i‖ ≤ k₃ + k₄‖s_i‖ − ρ_i,

$$\begin{array}{c}{\dot{V}}_1\le -‖s_i‖\left({\rho }_i-{\epsilon }_m^i\right)<0.\end{array}$$ (31)

According to the Lyapunov stability theory, it can be seen that under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), the sliding mode surface s_i can realize s_i = 0 and ${\dot{s}}_i=0$. By lemma 2, the reaching time is given as follows

$$\begin{array}{c}{t}_r=\frac{\sqrt{2}{V}_1^{1/2}\left(0\right)}{{\rho }_i-{\epsilon }_m^i}.\end{array}$$ (32)

Then, select the Lyapunov function as

$$\begin{array}{c}{V}_2=\frac{1}{2}{\overline{S}}^{{\rm T}}{\left[\left(L+B\right)\otimes I_3\right]}^{{\rm T}}\left[\left(L+B\right)\otimes I_3\right]\overline{S},\end{array}$$ (33)

where $\overline{S}={\left[{{\overline{s}}_1}^T,\cdots ,{{\overline{s}}_n}^T\right]}^T$.

Define

$$\begin{array}{c}{p}_i=\sum _{j\in n_i}{a}_{ij}\left({\overline{s}}_i-{\overline{s}}_j\right)+b_i{\overline{s}}_i,\\ P={\left[p_1^{{\rm T}},\cdots ,p_n^{{\rm T}}\right]}^{{\rm T}}=\left(L+B\right)\otimes I_3·\overline{S}.\end{array}$$ (34)

Then, we can get

$$\begin{array}{c}{V}_2=\frac{1}{2}{P}^{{\rm T}}P.\end{array}$$ (35)

Let S = [s₁^T, ⋯,s_n^T]^T, when ${\dot{s}}_i=0$, we know ${\dot{\overline{s}}}_i=x_i^{\eta }$, then$\dot{\overline{S}}=-{\left[\left(L+B\right)\otimes I_3·\overline{S}\right]}^{\eta }$.

Under Assumption 1 and Lemma 2, it can be obtained that λ_min(L + B) > 0. And taking the derivative of (35), we can obtain

$$\begin{array}{c}{\dot{V}}_2=-P^{{\rm T}}\left[\left(L+B\right)\otimes I_3\right]{\left[\left(L+B\right)\otimes I_3·\overline{S}\right]}^{\eta }=-P^{{\rm T}}\left[\left(L+B\right)\otimes I_3\right]P^{\eta }\le -{\lambda }_{\min }\left(L+B\right)P^{{\rm T}}{P}^{\eta },\end{array}$$ (36)

Since the positive odd ratio parameter η ∈ (0.5, 1), and combined with Lemma 3, we can find

$$\begin{array}{c}{\dot{V}}_2\le -{\lambda }_{\min }\left(L+B\right)\left(\sum _{i=1}^{3n}{\left|p_i\right|}^{1+\eta }\right)\le -{\lambda }_{\min }\left(L+B\right){\left({‖P‖}^2\right)}^{1+\eta /2}\le -2^{1+\eta /2}{\lambda }_{\min }\left(L+B\right)V_2^{1+\eta /2}.\end{array}$$ (37)

According to Lemma 1, under the action of the controller (21), the state of the system can reach and remain on the sliding mode surface $\overline{S}=0$ within finite time. The settling time is t_f.

$$\begin{array}{c}{t}_f=\frac{V_2^{1-\eta /2}}{2^{1+\eta /2}{\lambda }_{\min }\left(L+B\right)\left(1-\eta /2\right)}.\end{array}$$ (38)

When $\overline{S}=0$, then ${\overline{s}}_i={\overline{J}}_i\left[{\tilde{w}}_i+k_1{\tilde{q}}_i+k_2{\alpha }_i\left({\tilde{q}}_i\right)\right]=0$, so ${\tilde{w}}_i+k_1{\tilde{q}}_i+k_2{\alpha }_i\left({\tilde{q}}_i\right)=0$. According to Lemma 4, we know ${\tilde{w}}_i⟶0$, ${\tilde{q}}_i⟶0$ in finite time will be satisfied.

Next, we need to analyze whether the system has a minimum event-triggered time interval strictly greater than zero, which means that there is no Zeno behavior. When the event-triggered function (27) satisfies Y(t) > 0, the event is triggered. Combining (26), we can see that between any two adjacent event-triggered moments, ‖e_i‖ increases from zero to k₃ + k₄‖s_i‖ − ρ_i, therefore, when the growth rate is the fastest, the event-triggered time interval is the smallest. In this case, when the minimum time interval is a value greater than zero, it can be guaranteed that there is no Zeno behavior.☐

Theorem 2 . —

Based on Assumption 1, the system (10) and (11) under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), the system does not have Zeno behavior under any initial conditions.

Proof —

Let ${\beta }_i\left(t\right)=k_3\mathrm{sign}\left(s_i\left(t\right)\right)+k_4{s}_i\left(t\right)+z_i\left(t\right)+{\widehat{\delta }}_i\left(t\right)$, since the system (10) and (11) can achieve consensus under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), $‖{\dot{\beta }}_i\left(t\right)‖$ have upper bounds, which are taken as ${‖{\dot{\beta }}_i\left(t\right)‖}_{\max }$. And combining (26) to derive ‖e_i‖ as follows

$$\begin{array}{c}\frac{d‖e_i‖}{dt}\le ‖\frac{{de}_i}{dt}‖\le ‖\frac{d\left[-x_i^{\eta }\left(t\right)+{\beta }_i\left(t\right)\right]}{dt}‖\le ‖\frac{d}{dt}{x}_i^{\eta }\left(t\right)‖+‖{\dot{\beta }}_i\left(t\right)‖\le \eta ‖x_i^{\eta -1}‖‖{\dot{x}}_i‖+{‖{\dot{\beta }}_i\left(t\right)‖}_{\max }.\end{array}$$ (39)

Let X = [x₁^T, ⋯,x_n^T]^T, we can obtain $\dot{X}=-\left(L+B\right)\otimes I_3·\dot{\overline{S}}$. When $\dot{S}=0$, $\dot{\overline{S}}=X^{\eta }$ will be satisfied, so $\dot{X}=-\left(L+B\right)\otimes I_3·X^{\eta }$. And according to Lemma 3, we know

$$\begin{array}{c}‖x_i^{\eta -1}‖\le ‖X^{\eta -1}‖\le {\left(3n\right)}^{2-\eta }{‖X‖}^{\eta -1},\\ ‖{\dot{x}}_i‖\le ‖\dot{X}‖\le ‖\left(L+B\right)\otimes I_3·X^{\eta }‖\le ‖L+B‖‖X^{\eta }‖\le {\left(3n\right)}^{1-\eta }‖L+B‖{‖X‖}^{\eta }.\end{array}$$ (40)

By (38) and (39), we can obtain

$$\begin{array}{c}\frac{d‖e_i‖}{dt}\le \eta {\left(3n\right)}^{3-2\eta }‖L+B‖{‖X‖}^{2\eta -1}+{‖{\dot{\beta }}_i\left(t\right)‖}_{\max }.\end{array}$$ (41)

Due to X = −P and $‖P‖=\sqrt{2}{V}_2^{1/2}\left(t\right)\le \sqrt{2}{V}_2^{1/2}\left(0\right)$, we have

$$\begin{array}{c}\frac{d‖e_i‖}{dt}\le 2^{2\eta -1/2}\eta {\left(3n\right)}^{3-2\eta }‖L+B‖V_2^{2\eta -1/2}\left(0\right)+{‖{\dot{\beta }}_i\left(t\right)‖}_{\max }.\end{array}$$ (42)

For any t ∈ [t_kⁱ, t_k+1ⁱ), t_kⁱ is the latest event-triggered time for the ith UAV, the time interval T_mⁱ = t_k+1ⁱ − t_kⁱ, and ‖e_i(t_kⁱ)‖ = 0 at the event-triggered moment, and let ${\lambda }_i=2^{2\eta -1/2}\eta {\left(3n\right)}^{3-2\eta }‖L+B‖V_2^{2\eta -1/2}\left(0\right)+{‖{\dot{\beta }}_i\left(t\right)‖}_{\max }$, we can obtain

$$\begin{array}{c}‖e_i\left(t\right)‖-‖e_i\left(t_k^i\right)‖=‖e_i\left(t\right)‖\le \left(t-t_k^i\right){\lambda }_i\le T_m^i{\lambda }_i.\end{array}$$ (43)

When the event-triggered function (27) satisfies Y(t) > 0, the event is triggered, we have

$$\begin{array}{c}‖e_i\left(t\right)‖>k_3+k_4‖s_i‖-{\rho }_i>k_3-{\rho }_i.\end{array}$$ (44)

Combining (41) and (42), we can know

$$\begin{array}{c}{T}_m^i>\frac{k_3-{\rho }_i}{{\lambda }_i}.\end{array}$$ (45)

It can be concluded from (43) that the event-triggered time interval is strictly greater than zero, so there is no Zeno behavior.☐

4. Example Simulation

Considering the system composed of four UAVs includes three follower UAVs and one leader UAV, and the leader node is marked as 0. The directed communication topology is shown in Figure 2. Hence, we have

$$\begin{array}{c}L=\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ -1& 0& 1\end{array}\right]B=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right].\end{array}$$ (46)

Figure 2.

Communication topology.

The actual inertia matrices are assumed to be

$$\begin{array}{c}{J}_1=\left[\begin{array}{ccc}15& 1& 1\\ 2& 16& 0.5\\ 0& 0.5& 14\end{array}\right]J_2=\left[\begin{array}{ccc}13& 0.5& 0\\ 1& 15& 0.5\\ 0& 1.5& 14\end{array}\right]J_3=\left[\begin{array}{ccc}14& 1& 2\\ 0& 13& 0\\ 2& 1& 15\end{array}\right].\end{array}$$ (47)

With the existence of model uncertainties and external disturbances, the nominal inertia matrices of the UAV are given by ${\overline{J}}_1={\overline{J}}_2={\overline{J}}_3=\mathrm{diag}\left({\left[202020\right]}^T\right)$. Take the disturbances as

$$\begin{array}{c}{d}_1={\left[0.1\sin \left(t\right),0.2\cos \left(0.5t\right),0.15\cos \left(0.7t\right)\right]}^T,\\ d_2={\left[0.1\cos \left(t\right),0.2\sin \left(0.5t\right),0.15\sin \left(0.7t\right)\right]}^T,\\ d_3={\left[0.1\cos \left(t\right),0.2\cos \left(0.5t\right),0.15\sin \left(0.7t\right)\right]}^T.\end{array}$$ (48)

The initial quaternions of the follower UAVs are selected as Q₁(0) = [1, 0, 0, 0]^T, Q₂(0) = [0, 1, 0, 0]^T, and Q₃(0) = [0, 0, 1, 0]^T, the initial angular velocities of the follower UAVs are selected as w₁(0) = [0, 0, 1]^T, w₂(0) = [0, 0, 1]^T, and w₃(0) = [0, 0, 1]^T. The initial quaternion of the leader UAV is selected as Q₀(0) = [0, 0, 0, 1]^T, and the angular velocity of the leader UAV is given asw₀(t) = [0.1cos(0.2t),−0.1sin(0.2t),−0.1cos(0.2t)]^T.

The controller parameters are chosen with k₁ = 1, k₂ = 0.001, k₃ = 0.01, k₄ = 1, r = 0.6, ϕ = 0.1, η = 0.1, and ρ_i = 0.009. The adaptive RBFNN controller parameters are adjusted as J = 7, c_i = [−1.5, −1, −0.5, 0, 0.5, 1, 1.5], σ_i = 5, and A = diag([0.5 0.5 0.5]^T).

Figures 3 and 4, respectively, show the attitude tracking Q_i and the attitude tracking errors ${\tilde{Q}}_i$ of the ith UAV,i = 1, 2, 3.

Figure 3.

Attitude tracking.

Figure 4.

Attitude tracking errors.

Figures 5 and 6, respectively, show the angular velocity tracking w_i and the attitude tracking errors ${\tilde{w}}_i$ of the ith UAV. From Figures 3–6, it can be seen that the attitude and the angular velocity of all follower UAVs can accurately track the leader UAV over time under the action of the controller (17) and the event-triggered function (23).

Figure 5.

Angular velocity tracking.

Figure 6.

Angular velocity errors.

Figure 7 shows the evolution process of the measurement error norm of the system, where threshold ‖e_i‖_max = k₃ + k₄‖s_i‖ − ρ_i. When the value of ‖e_i‖ increases from zero to ‖e_i‖_max, the event is triggered.

Figure 7.

Variation trend of measurement error norm ‖e_i‖ and threshold‖e_i‖_max.

Figures 8–10 show the event-triggered time of the UAV i, i = 1, 2, 3, and the denser part marked by the rectangular box is enlarged. At the event-triggered time of the UAV i, the UAV i interacts with information and updates the controller. Figure 11 shows the control torque u_i of the UAV i.

Figure 8.

Event-triggered time for UAV1.

Figure 9.

Event-triggered time for UAV2.

Figure 10.

Event-triggered time for UAV3.

Figure 11.

Control torques.

From Figures 7–11, it can be seen that the superior performance of the proposed event-triggered control strategy in reducing the energy dissipation of the system and the update frequency of the controller.

The approximation error ‖ε_i‖ of the RBFNNs to unknown lumped disturbance δ_i is shown in Figure 12. It can be seen that the RBFNNs can approach δ_i at a faster speed under the action of the adaptive weight update law (23).

Figure 12.

The RBFNN estimation error ‖ε_i‖.

5. Conclusions

In this paper, a distributed finite time event-triggered control strategy with RBFNNs is proposed for attitude cooperative control of MUAVs. Under the leader-following framework, the tracking errors of attitude converge to zero in finite time, the communication resources is saved and the Zeno behavior is excluded by utilizing the event-triggered scheme. Finally, theory and numerical simulation proof is given for the proposed control law. In the future, we will consider actuator saturation problem by fault-tolerant technology and self-triggered scheme to be used in finite-time control.

Data Availability

We have no data to share for this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.