Predefined-Time Platoon Control of Unmanned Aerial Vehicle with Range-Limited Communication

Abstract

In this paper, the predefined-time platoon control for multiple uncertain unmanned aerial vehicles (UAVs) under range-limited communication and external disturbance constraints is considered. A novel control scheme, which can guarantee communication connectivity, collision avoidance, and the predefined convergence time simultaneously, is proposed. To achieve disturbance robustness, an observer-based distributed control law is firstly proposed with a time-varying gain. Then, a radial basis function neural network (RBFNN) with an adaptive tuning law is applied to approximate uncertainties of the system. Under the time and error transformation techniques, uniformly ultimate boundedness (UUB) stability of the closed-loop system is guaranteed within predefined convergence time. Compared with the existing results, the proposed method allows the system to have UUB within any predefined time without depending on the initial conditions or system parameters. Finally, simulation results are presented to verify the derived theorem.

1. Introduction

In the past decades, unmanned aerial vehicles (UAVs) have been widely used in various fields [1,2]. The networked multiple UAVs greatly expand the capabilities of a single system, such as scalability, robustness, and execution ability [3,4]. To reduce the wind resistance of UAVs in flight, platoon control, which ensures that the UAVs maintain desired spaces between adjacent subsystems, is an effective way to solve this problem [5,6,7,8,9,10]. Platoon control can be achieved by using relative measurements. The existing works mainly focus on vehicular platoon control in one dimension for the single-integral system, double-integral system, or third-integral system [11,12,13]. Although the application of vehicle platoon control has been partially solved, the results in actual UAVs under external disturbances and system uncertainties lack of consideration [14,15,16,17,18].

1.1. Related Works

Recently, researchers have proposed several methods to deal with external disturbance issues in platoon control [19,20,21,22]. For example, a sliding mode based robust controller was proposed for vehicular systems with external disturbances [20], where the control scheme is independent of the object parameters and insensitive to parameter changes and inevitable high frequency chattering due to the high frequency switch of the sliding mode. By representing the uncertainties of the system and the unknown dynamics and as a lumped nonlinearity, a fuzzy approximation based approach was proposed for the platoon tracking control of marine vehicles. In Reference [22], a hybrid control framework that ensures practical finite time stability and asymptotical convergence was proposed by using neural network approximation. It is worth noting that these approaches cannot guarantee the transient and steady state performances such as communication connectivity and collision avoidance.

Due to the requirements of system performance, transient and steady state performance methods have been introduced [11,17,23], in which the constrained control problem of error dynamic is transformed into an unconstrained control problem. For example, an adaptive backstepping controller was proposed for the trajectory tracking problem of an uncertain UAV with guaranteed transient and steady state performance in [24]. In Reference [25], a transient and steady-state-constraint-based approach was proposed for a group of nonholonomic mobile robots to achieve leader–follower formation with range-limited communication. In Reference [26], collision avoidance was guaranteed for the platoon formation control of multiple unmanned surface vehicles. However, these methods can only guarantee the transient and steady-state performance of the closed-loop system but cannot guarantee the convergence time. The convergence time is a crucial indicator for evaluating the system performance, especially for UAV systems with severe time-of-flight constraints due to limited energy. The finite time control method is one of the effective methods to solve this issue [27,28,29,30,31,32,33]. In References [31,32,33], the authors considered the finite time trajectory tracking problem for nonlinear systems with uncertainties. However, the convergence time of the above-mentioned methods is highly relying on the system parameters and the system’s initial conditions. In addition, it is more difficult to steer the system to converge within an user predefined time.

1.2. Contributions

In this paper, we consider the predefined-time platoon control for uncertain UAVs subject to collision, range-limited communication, and external disturbances. By using the prescribed performance approach, the collision avoidance and connectivity preservation problems are transformed into the limitations on specific position error constraints. Then, a platoon control law with a dynamic gain is designed for the UAVs. To guarantee the stability of the closed-loop UAV system within the predefined time, the time transformation technique and UUB stability are applied. The contributions can be summarized as follows:

• This paper focuses on the platoon control of UAV systems modeled by nonlinear dynamics with uncertainties and external disturbances. The RBFNN and the disturbance observer approaches are used to estimate the unknown bounded external disturbance and system uncertainties. Compared to [34,35,36], which is considered the platoon of linear systems, the control design in this paper is more complex and challenging.
• Compared to [20], connectivity preservation and collision avoidance between UAVs can be guaranteed simultaneously by the proposed control scheme. The state errors are bounded, with the bounds decay exponentially by using the prescribed performance control approach.
• Compared to the finite time convergent results in [37,38], the proposed control scheme can guarantee the platoon of UAVs converge into a small neighborhood around zero at a predefined time without relying on the initial conditions and system parameters.

Section 2 formulates the problem of interest. Section 3 gives the control design for UAVs platoon system, which is the main results. Then, the simulations are given in Section 4. Finally, Section 5 concludes this paper, and the Appendix gives some proof details.

2. Problem Formulation

2.1. UAV Model

In this paper, the rigid-body UAV with six degrees of freedom (DOF) is considered [39]. The kinematics model of the ith UAV is given as follows:

$$\begin{array}{c}\hfill {\dot{\eta }}_i=R_i{V}_i,\end{array}$$

(1)

where ${\eta }_i={[x_i,y_i,z_i,{\varphi }_i,{\theta }_i,{\psi }_i]}^T$ is the state of the UAV, with $(x_i,y_i,z_i)$ being the position in a global coordinate and $({\varphi }_i,{\theta }_i,{\psi }_i)$ being the pose. $V_i={[u_i,v_i,w_i,p_i,q_i,r_i]}^T$ is the linear and angular velocity vector in the body-fixed reference frame. $R_i=diag\{R_{i1},R_{i2}\}$ is the rotation matrix between the body-fixed reference frame and the earth-fixed reference frame, where

$$\begin{array}{cc}\hfill R_{i1}& =\left[\begin{array}{ccc}cos{\theta }_{i}cos{\psi }_i& sin{\varphi }_{i}sin{\theta }_{i}cos{\psi }_i-cos{\varphi }_{i}sin{\psi }_i& cos{\varphi }_{i}sin{\theta }_{i}cos{\psi }_i+sin{\varphi }_{i}sin{\psi }_i\\ cos{\theta }_{i}sin{\psi }_i& sin{\varphi }_{i}sin{\theta }_{i}sin{\psi }_i-cos{\varphi }_{i}cos{\psi }_i& cos{\varphi }_{i}sin{\theta }_{i}sin{\psi }_i+sin{\varphi }_{i}cos{\psi }_i\\ -sin{\theta }_i& cos{\theta }_{i}sin{\varphi }_i& cos{\theta }_{i}cos{\varphi }_i\end{array}\right],\hfill \\ \hfill R_{i2}& =\left[\begin{array}{ccc}1& sin{\varphi }_{i}sin{\theta }_i/cos{\theta }_i& cos{\varphi }_{i}sin{\theta }_i/cos{\theta }_i\\ 0& cos{\varphi }_i& -sin{\varphi }_i\\ 0& sin{\varphi }_i/cos{\theta }_i& cos{\varphi }_i/cos{\theta }_i\end{array}\right],\hfill \end{array}$$

The dynamics model of the ith UAV is given as follows:

$$\begin{array}{c}\hfill {\overline{M}}_i{\dot{V}}_i=-{\overline{C}}_i\left(V_i\right)V_i-{\overline{Z}}_i\left(V_i\right)-{\overline{G}}_i\left(V_i\right)+F_i-\Delta f_i,\end{array}$$

(2)

where ${\overline{M}}_i$, ${\overline{C}}_i\left(V_i\right)$, ${\overline{G}}_i\left(V_i\right)$, and ${\overline{Z}}_i\left(V_i\right)$ denote the inertia matrix, the coriolis matrix, gravitational matrix, and the modeling uncertainties of the ith UAV, respectively. $F_i$ is the exerted torque of the UAV in the ith body-fixed reference frame. $\Delta f_i$ is the external disturbance.

According to (1) and (2), the UAV system model can be rewritten as

$$\begin{array}{c}\hfill M_i{\ddot{\eta }}_i=-C_i\left({\dot{\eta }}_i\right){\dot{\eta }}_i-Z_i\left({\dot{\eta }}_i\right)-G_{g_i}\left({\eta }_i\right)-{\tau }_{{\varpi }_i}+{\tau }_i,\end{array}$$

(3)

where $M_i={\left(R_i^{-1}\right)}^T{\overline{M}}_i{R}_i^{-1},C_i\left({\dot{\eta }}_i\right)={\left(R_i^{-1}\right)}^T({\overline{C}}_i\left(V_i\right)-{\overline{M}}_i{R}_i^{-1}{\dot{R}}_i)R_i^{-1},Z_i\left({\dot{\eta }}_i\right)={\left(R^{-1}\right)}^T\overline{Z}\left(V\right),$ and $G_{g_i}\left(\eta \right)={\left(R_i^{-1}\right)}^T{\overline{G}}_i\left(V_i\right)$. ${\tau }_i={\left(R_i^{-1}\right)}^T{F}_i$ define the control input. ${\tau }_{{\varpi }_i}={\left(R_i^{-1}\right)}^T\Delta f_i$ is the external disturbance. For the external disturbances, the following assumption is given.

Assumption 1.

The unknown external disturbance is time-varying and bounded; its first derivative is also unknown and bounded.

Assumption 1 is widely used to model unknown external disturbances [19,20,21]. Under Assumption 1, the first derivative of the external disturbance is constrained by an unknown constant. Some important properties of the nonlinear dynamic model (3) are given:

Property 1.

$M_i$ represents inertia, which is positive and symmetric. In addition, the function has lower and upper bounds, respectively, with the positive scalars $q_{i1}$ and $q_{i2}$, i.e.,

$$\begin{array}{c}\hfill 0<q_{i1}I\le M_i\left({\eta }_i\right)\le q_{i2}I<\infty ,\end{array}$$

(4)

where I is the identity matrix with the corresponding dimension. And ${\dot{M}}_i-2C_i\left({\dot{\eta }}_i\right)$ is skew-symmetric, which means $s^T({\dot{M}}_i-2C_i{\left(\dot{\eta }\right)}_i)s=0,\forall s\in {\mathbb{R}}^n$.

Remark 1.

Note that for the UAV model, the system outputs η, $\dot{\eta }$, and $\ddot{\eta }$ can be considered as the generalized position, velocity, and acceleration, which can be measured by the omnidirectional distance sensor, gyroscope, laser speed sensor, altitude sensor, camera, GPS, acceleration transducer and other onboard sensors. Consequently, this information can be used to design the control input ${\tau }_i$.

2.2. Technical Lemmas

In this subsection, some technical lemmas are introduced.

Lemma 1

([40]). $A\in {\mathbb{R}}^{n\times n}$ is a square matrix. When $a_{ii}>0,a_{ij}\le 0$$(i\ne j)$, A is a L matrix. If all of the eigenvalues of the matrix have positive real parts, it will be an M matrix.

Lemma 2

([40]). $NP+P^{T}N$ is a positive matrix if $P\in {\mathbb{R}}^{n\times n}$ is an M matrix, and $N={\left(diag\left(P^{-1}\times 1_n\right)\right)}^{-1}$ is a diagonal positive matrix.

Lemma 3

([41]). $\Phi \left(t\right)$ is a solution of the dynamic system

$$\dot{x}\left(t\right)=f(t,x\left(t\right)),x\left(0\right)=x_0.$$

Let $t=\beta \left(\tau \right)$ be the time transformation; $\beta \left(\tau \right)$ is a strictly increasing and contiguously differentiable function of s, and $\Psi \left(\tau \right)=\Phi \left(t\right)$. Then, $\dot{\Psi }\left(\tau \right)=\dot{\beta }\left(\tau \right)f(\beta \left(\tau \right),\Psi \left(\tau \right))$, and $\Psi \left({\beta }^{-1}\left(0\right)\right)=x_0$.

Lemma 4

([42]). If $f\left(t\right)$ is a differentiable function, it has a finite limit as $t\to \infty$, and it is uniformly continuous: $\dot{f}\left(t\right)\to 0$ as $t\to \infty$.

Lemma 5

([42]). $\acute{Q}$ is a positive matrix and a solution of the system is defined as $\check{s}$, and the maximal eigenvalues and the minimal eigenvalues of $\acute{Q}$ are ${\lambda }_2\left(\acute{Q}\right)$ and ${\lambda }_3\left(\acute{Q}\right)$, respectively. The system will be UUB stable in square with ultimate bound $\frac{{\alpha }_2}{{\lambda }_2\left(\acute{Q}\right){\alpha }_1}$ when $V\left(\check{s}\right)={\check{s}}^T\acute{Q}\check{s}$ is a continuous function, and for constants ${\alpha }_1>0$, ${\alpha }_2>0$, we have

$$\pounds \left(\dot{V}\left(\check{s}\right)\right)\le -{\alpha }_1\pounds \left(V\left(\check{s}\right)\right)+{\alpha }_2,t\ge 0.$$

2.3. Problem Statements

Consider the n UAVs in a platoon; the collision and range-limited communication problem is considered. The safety state of each UAV is defined as ${\varkappa }_s={[{\varkappa }_a,{\varkappa }_g]}^T={[{\varkappa }_{ax},{\varkappa }_{ay},{\varkappa }_{az},{\varkappa }_{gp},{\varkappa }_{gr},{\varkappa }_{gy}]}^T$, with ${\varkappa }_a$ being the avoiding collision distance between each UAV. The communication distance is defined as ${\varkappa }_c={[{\varkappa }_{cx},{\varkappa }_{cy},{\varkappa }_{cz}]}^T$. The position of each UAV is defined as ${\widehat{\eta }}_i={[x_i,y_i,z_i]}^T$. The position error of the UAVs is defined as ${\widehat{\eta }}_i-{\widehat{\eta }}_{i-1}$. The desired states of each UAV is ${\varkappa }_{i,i-1}={[{\varkappa }_{u_{i,i-1}},{\varkappa }_{v_{i,i-1}},{\varkappa }_{w_{i,i-1}},{\varkappa }_{p_{i,i-1}},{\varkappa }_{q_{i,i-1}},{\varkappa }_{r_{i,i-1}}]}^T$. The desired position between each UAV is ${\varkappa }_{i,i-1}^{*}={[{\varkappa }_{u_{i,i-1}},{\varkappa }_{v_{i,i-1}},{\varkappa }_{w_{i,i-1}}]}^T$, ${\varkappa }_a\ll {\varkappa }_{i,i-1}^{*}$. Obviously, the distance of each UAV should satisfy ${\varkappa }_a<{\widehat{\eta }}_i\left(t\right)-{\widehat{\eta }}_{i-1}\left(t\right)<{\varkappa }_c$. The desired reference trajectory is given as ${\eta }_0={[x_0,y_0,z_0,{\varphi }_0,{\theta }_0,{\psi }_0]}^T$. For the initial conditions, the following assumption is given.

Assumption 2.

The initial states of the platoon cannot violate the collision and connectivity constraints, which means the distances between each UAVs should satisfy the following:

$${\varkappa }_a<∥{\widehat{\eta }}_{i-1}\left(0\right)-{\widehat{\eta }}_i\left(0\right)∥<{\varkappa }_c,$$

for all n UAVs in the platoon system.

Remark 2.

Assumption 2 requires the UAV’s initial position errors to be limited within the predefined range. The controller is designed by using the predetermined performance technique to ensure that the UAVs are always in the predefined communication range with avoiding collisions, which means that the state errors are bounded with a decaying exponential time.

When the system’s state error ${\delta }_i\left(t\right)$ evolves strictly within the predefined ranges, the prescribed performance is achieved. To avoid collision and maintain connectivity, the UAVs are maintained at a safe distance from one another and stay within the communication range, which satisfies Assumption 2. The state error ${\delta }_i\left(t\right)$ satisfies the following conditions:

$$\begin{array}{c}\hfill -{\lambda }_1{\rho }_i\left(t\right)<{\delta }_i\left(t\right)<{\lambda }_1{\rho }_i\left(t\right),\end{array}$$

(5)

where $0<{\lambda }_1<1$. Define a positive function ${\rho }_i\left(t\right)$ as

$$\begin{array}{c}\hfill {\rho }_i\left(t\right)=({\rho }_i\left(0\right)-{\rho }_i(\infty ))e^{(-lt)}+{\rho }_{i(\infty )},\end{array}$$

(6)

where ${\rho }_i\left(0\right)\in {\mathbb{R}}^{+}$, ${\rho }_i(\infty )\in {\mathbb{R}}^{+}$, $l\in {\mathbb{R}}^{+}$, and $li{m}_{t\to \infty }{\rho }_i\left(t\right)={\rho }_i(\infty )>0$. ${\rho }_i(\infty )$ represents the allowed maximum error. The decreasing rate represents the minimal convergence rate. It should be noted that ${\rho }_i(\infty )\ll {\lambda }_1$, which ensures that the practical can converge to zero.

Remark 3.

To ensure avoiding collisions within communication, $-{\lambda }_1{\rho }_i\left(0\right)$ should satisfy that $-{\lambda }_1{\rho }_i\left(0\right)>-{\varkappa }_{i,i-1}\left(t\right)+{\varkappa }_s$. Thus, we have ${\delta }_i\left(t\right)>-{\lambda }_1{\rho }_i\left(t\right)>-{\lambda }_1{\rho }_i\left(0\right)>-{\varkappa }_{i,i-1}\left(t\right)+{\varkappa }_s$, and then collision avoidance among UAVs is guaranteed.

The generalized error of the ith UAV is

$$\begin{array}{c}\hfill {\delta }_i\left(t\right)={\eta }_i\left(t\right)-{\eta }_{i-1}\left(t\right)-{\varkappa }_{i,i-1}.\end{array}$$

(7)

It can be rewritten as a matrix vector form:

$$\begin{array}{c}\hfill \delta \left(t\right)=\left[\begin{array}{c}{\delta }_1\left(t\right)\\ {\delta }_2\left(t\right)\\ ⋮\\ {\delta }_n\left(t\right)\end{array}\right]=\Xi \left(\left[\begin{array}{c}{\eta }_1\left(t\right)-{\eta }_0\\ {\eta }_2\left(t\right)-{\eta }_0\\ ⋮\\ {\eta }_n\left(t\right)-{\eta }_0\end{array}\right]-\left[\begin{array}{c}{\varkappa }_{1,0}\\ {\varkappa }_{2,0}\\ ⋮\\ {\varkappa }_{n,0}\end{array}\right]\right)=\Xi {\delta }_0\left(t\right),\end{array}$$

(8)

where ${\varkappa }_{i,0}={\sum }_{j=1}^i{\varkappa }_{j,j-1}$, and $\Xi \in {\mathbb{R}}^{n\times n}=\left[I_{i,j}\right]$, with $I_{i,i}=1$, $I_{i-1,i}=-1$, $I_{i,j}=0$, $i,j\in \{1,2,\dots ,n\}$, and ${{\delta }_0}_i\left(t\right)={\eta }_i\left(t\right)-{\eta }_0-{\varkappa }_{i,0}$.

The error transformation incorporates the state error ${\delta }_i\left(t\right)$ with regard to the appropriate performance bounds imposed by ${\rho }_i\left(t\right)$ to ensure that the state errors are restricted by the prescribed performance function. Then, the state error can be written as

$$\begin{array}{c}\hfill \sigma (\delta ,t)=\left[\begin{array}{c}{\sigma }_1({\delta }_1,t)\\ {\sigma }_2({\delta }_2,t)\\ ⋮\\ {\sigma }_n({\delta }_n,t)\end{array}\right]=\left[\begin{array}{c}\frac{{\delta }_1\left(t\right)}{{\rho }_1\left(t\right)}\\ \frac{{\delta }_2\left(t\right)}{{\rho }_2\left(t\right)}\\ ⋮\\ \frac{{\delta }_n\left(t\right)}{{\rho }_n\left(t\right)}\end{array}\right]\triangleq {\rho }^{-1}\left(t\right)\Xi {\delta }_0\left(t\right),\end{array}$$

(9)

where $\rho \left(t\right)=diag\left([{\rho }_1\left(t\right),{\rho }_2\left(t\right),\dots ,{\rho }_n\left(t\right)]\right)$. A candidate transformation function is given as

$${\zeta }_i\left(\frac{{\sigma }_i\left(t\right)}{{\lambda }_1}\right)=ln\left(\frac{H+\frac{{\sigma }_i\left(t\right)}{{\lambda }_1}}{H(1-\frac{{\sigma }_i\left(t\right)}{{\lambda }_1})}\right),$$

(10)

where $1<H<2$. The auxiliary results of the system error are given in Appendix A. Then, the control objective is as follows.

Problem 1.

Design a controller for a platoon of UAVs that allows for the maintenance of a desirable inter-UAV state while meeting the following constraints:

• The purpose of the control law is to guarantee that the UAVs can achieve the desired formation under collision avoidance and connectivity preservation. It means that each UAV should arrive at the desired position, and the state errors are bounded by a strictly decreasing exponential time function.
• The system converges within the predefined time, which is independent on the system parameters and the system’s initial values.

3. Controller Design for UAVs Platoon System without Considering Time Constraints

This section presents a control law under prescribed performance constraints to obtain the platoon control objective above. The designed controller is based on lumped state errors, RBFNN approximations, disturbance observers, and the Lyapunov function. In theory, the RBFNN can simultaneously process system uncertainties and external disturbances online. However, when using a neural network to deal with the unknown external disturbances, the control input of the neural network is difficult, if not impossible, to design. In addition, the control performance based on neural network disturbance observer is better than the method only using the neural network. Based on these two motivations, the RBFNN-based disturbance observer is designed.

Based on the dynamic surface control technique to avoid the “explosion of complexity” problem, a new lumped state error variables is defined as

$$\begin{array}{c}\hfill {\epsilon }_i={\dot{\eta }}_i+{\mu }_i{\zeta }_i,\end{array}$$

(11)

where ${\epsilon }_i={[{\epsilon }_{x_i},{\epsilon }_{y_i},{\epsilon }_{z_i},{\epsilon }_{{\varphi }_i},{\epsilon }_{{\theta }_i},{\epsilon }_{{\psi }_i}]}^T$, ${\mu }_i=diag\left\{[{\mu }_{x_i},{\mu }_{y_i},{\mu }_{z_i},{\mu }_{{\varphi }_i},{\mu }_{{\theta }_i},{\mu }_{{\psi }_i}]\right\}$ is a positive definite diagonal matrix.

Differentiating (11) and substituting (3) into it, one can obtain

$$\begin{array}{c}\hfill M_i{\dot{\epsilon }}_i={\tau }_i-C_i\left(\dot{\eta }\right){\dot{\eta }}_i-Z_i\left(\dot{\eta }\right)-G\left(\eta \right)-{\tau }_{{\varpi }_i}+M_i{\mu }_i{L}_i.\end{array}$$

(12)

To solve Problem 1, the RBFNN is firstly introduced to approximate the system uncertainties. Any smooth nonlinear function may be approximated with arbitrary precision using the neural network technique. With a simple and implementable approach, the RBFNN can estimate the nonlinear uncertainties. The RBFNN can approximate a continuous function $f\left(X\right):{\mathbb{R}}^n\to {\mathbb{R}}^m$, which can be expressed as

$$\begin{array}{c}\hfill f\left(X\right)=W^{T}S\left(X\right)+ϵ\left(X\right),\end{array}$$

(13)

where $X\in {\Omega }_X\subset {\mathbb{R}}^n$ is the input vector; $W\in {\mathbb{R}}^{i\times m}$ is the weight matrix; $i>1$ is the number of neurons in this paper; and $S\left(X\right)={[S_1\left(X\right),S_2\left(X\right),\dots ,S_i\left(X\right)]}^T$ is the Gaussian RBF function, which can be described as follows

$$\begin{array}{c}\hfill S_i\left(X\right)=e^{\left(\frac{-{∥X-C_i∥}^2}{2b_i^2}\right)},i=1,2,\dots ,n,\end{array}$$

(14)

where the center of $i_{th}$ neuron is $C_i$, and the width of it is $b_i$. From the property of the neural network method, any continuous function $f\left(X\right)$ has

$$\begin{array}{c}\hfill f\left(X\right)=W^{*T}S\left(X\right)+ϵ\left(X\right),\end{array}$$

(15)

where $W^{*}$ is the optimal constant weight vector, and $ϵ\left(X\right)$ is the approximation error with $\left|ϵ\left(X\right)\right|\le {ϵ}^{*}$, where ${ϵ}^{*}>0$. $S\left(X\right)$ is the RBF vector, and $∥S\left(X\right)∥\le S^{*}$, with $S^{*}>0.$ More details about the RBF neural network can be found in [43] and reference therein.

Denote

$$\begin{array}{c}\hfill f_i\left(X_i\right)=Z_i\left(\dot{\eta }\right)=W_i^{*}{S}_i\left(X_i\right)+{ϵ}_i\left(X_i\right),\end{array}$$

(16)

where $X_i={\dot{\eta }}_i\in {\Omega }_{X_i}\subset {\mathbb{R}}^n$. In this paper, the RBFNN method is used to estimate the uncertainties of the UAV systems. It has the function of associative memory and can fully approximate the complex nonlinear relation, which means the neural network method can approximate any nonlinear function arbitrarily so that the nonlinear problem can be handled in a linear way. Moreover, the RBFNN method has strong parallel distribution processing ability, as well as strong distribution storage and learning ability.

Substituting (16) into (12), we obtain

$$\begin{array}{c}\hfill {\dot{\epsilon }}_i=M_i^{-1}({\tau }_i-C_i\left(\dot{\eta }\right){\dot{\eta }}_i-W_i^{*}{S}_i\left(X_i\right)-{ϵ}_i\left(X_i\right)-G\left(\eta \right)-{\tau }_{{\varpi }_i}+M_i{\mu }_i{L}_i).\end{array}$$

(17)

The lumped disturbance is designed as ${\tau }_{w_i}={\tau }_{{\varpi }_i}+{ϵ}_i\left(X_i\right)$; then, (17) can be rewritten as

$$\begin{array}{c}\hfill {\dot{\epsilon }}_i=M_i^{-1}({\tau }_i-C_i\left(\dot{\eta }\right){\dot{\eta }}_i-W_i^{*}{S}_i\left(X_i\right)-G\left(\eta \right)-{\tau }_{w_i}+M_i{\mu }_i{L}_i).\end{array}$$

(18)

The disturbance observer to estimate the unknown external disturbance ${\tau }_{w_i}$ in (18) is designed as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\widehat{\tau }}_{w_i}=-{\kappa }_i-K_{\kappa i}{\epsilon }_i\hfill \\ {\dot{\kappa }}_i={\epsilon }_i-K_{\kappa i}{M}_i^{-1}({\tau }_i-C_i\left(\dot{\eta }\right){\dot{\eta }}_i-{\widehat{W}}_i{S}_i\left(X_i\right)-G\left(\eta \right)-{\widehat{\tau }}_{w_i}+M_i{\mu }_i{L}_i)\hfill \end{array}\right.,\end{array}$$

(19)

where $K_{\kappa i}$ is a positive definite diagonal matrix. ${\widehat{W}}_i$ is the estimate of the RBFNN weight $W_i$, and ${\widehat{\tau }}_{w_i}$ is the estimate of the unknown disturbance ${\tau }_{w_i}$.

The estimate errors of the disturbance and the uncertainties are

$$\begin{array}{cc}\hfill {\tilde{W}}_i=& W_i^{*}-{\widehat{W}}_i,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\tilde{\tau }}_{w_i}=& {\tau }_{w_i}-{\widehat{\tau }}_{w_i}.\hfill \end{array}$$

(21)

Substituting (18)–(20) into the differential of (21) derives

$$\begin{array}{cc}\hfill {\dot{\tilde{\tau }}}_{w_i}=& {\dot{\tau }}_{w_i}+{\epsilon }_i-K_{\kappa i}{M}_i^{-1}(-C_i\left(\dot{\eta }\right){\dot{\eta }}_i-{\widehat{W}}_i{S}_i\left(X_i\right)\hfill \\ \hfill & -G\left(\eta \right)-{\widehat{\tau }}_{w_i}+{\tau }_i+M_i{\mu }_i{L}_i)+K_{\kappa i}{M}_i^{-1}({\tau }_i\hfill \\ \hfill & -C_i\left(\dot{\eta }\right){\dot{\eta }}_i-W_i^{*}{S}_i\left(X_i\right)-G\left(\eta \right)-{\tau }_{w_i}+M_i{\mu }_i{L}_i)\hfill \\ \hfill =& {\dot{\tau }}_{w_i}+{\epsilon }_i+K_{\kappa i}{M}_i^{-1}(-{\tilde{W}}_i{S}_i\left(X_i\right)-{\tilde{\tau }}_{w_i}).\hfill \end{array}$$

(22)

The proposed controller with prescribed performance is designed as follows:

$$\begin{array}{cc}\hfill {\tau }_i=& -K_{\epsilon i}{\epsilon }_i-M_i{\mu }_i{L}_i+C_i\left(\dot{\eta }\right){\dot{\eta }}_i+{\widehat{W}}_i{S}_i\left(X_i\right)+{\widehat{\tau }}_{w_i}+G\left(\eta \right),\hfill \end{array}$$

(23)

where $K_{\epsilon i}$ is positive definite diagonal matrices. The adaptive tuning law for ${\widehat{W}}_i$ is given as follows:

$$\begin{array}{c}\hfill {\dot{\widehat{W}}}_i=-{\epsilon }_i^T{\Lambda }_i{S}_i\left(X_i\right)-M_i{\Lambda }_i{\dot{\epsilon }}_i,\end{array}$$

(24)

where ${\Lambda }_i$ is positive definite diagonal matrix.

Theorem 1.

The closed-loop system is UUB stable, and the mean square and state errors tend to an extremely small neighborhood around zero under Assumptions 1 and 2 using the UAVs model (3) in a platoon under the control input (23) with the RBFNN (16), disturbance observer (19) and adaptive control law (24).

The proof of Theorem 1 is given in Appendix B.

The control scheme can be seen in Figure 1. The control input consists of three parts: the estimated uncertainties, the external disturbance observer, and the lumped state error convergence. The designed controller only requires the derivative of its own state, and the state of the system has physical significance; thus, the derivative of the state also holds physical meaning, and its effectiveness lies in reducing the influence of the corresponding part in the system dynamics. $\eta ={[x,y,z,\varphi ,\theta ,\psi ]}^T$ represents the position and the angle of the UAV, so the derivative of the $\eta$ represents the velocity of each axis. Currently, sensor technology can realize the corresponding derivatives, which means the measurement of velocity. On the other hand, measurement noise does exist in the measurement of the state derivative. However, the $H_{\infty }$ control method can be used to address this issue, as cited in [44]. Nonetheless, this is beyond the scope of this paper. For mathematical simplicity, this is not considered in the present analysis.

4. Controller Design for UAVs Platoon System with Considered Time Constraint

In the previous section, although the platoon control of multiple UAVs can be guaranteed, the cost of time is not taken into account in the application process. As is known to all, the flight time of the UAV is limited, and the UAV needs to complete all the tasks within the limited flight time. The finite time control approach is one of the methods that considers the cost of time. However, the convergence time of the finite time control depends on its initial conditions and system parameters. Therefore, in this section, we focus on how to achieve the platoon control of multiple UAVs within a predefined time.

The proposed controller with prescribed performance and time constraint is designed as follows:

$$\begin{array}{c}\hfill {\tau }_{ti}=-{\Gamma }_{\tau i}{K}_{\epsilon ti}{\epsilon }_{ti}-M_i{\mu }_i{L}_i+C_i\left(\dot{\eta }\right){\dot{\eta }}_i+{\widehat{W}}_i{S}_i\left(X_i\right)+{\widehat{\tau }}_{w_i}+G\left(\eta \right),\end{array}$$

(25)

where

$${\Gamma }_{\tau ti}=\left\{\begin{array}{c}\frac{k_{{\tau }_i}}{T-t},t<T\\ k_{{\tau }_i},others\end{array}\right.,$$

(26)

and $K_{\epsilon ti}$ define positive definite diagonal matrices.

Remark 4.

The controller consists of two parts: one of which is used to ensure predefined time convergence; the other of which is used to compensate for the nonlinear uncertainties and disturbances. In addition, the controller (25) degenerates to (23) for $t\ge T$; thus, the proof of this part focuses on the case of $t<T$.

Remark 5.

Recall the predefined-time platoon control in [36], which considered the platoon control problem for single- and double-interrogator linear vehicle systems without disturbances and uncertainties. The main differences are as follows: (1) A predefined-time controller is designed for the nonlinear UAV system with external disturbance and system uncertainties, which will affect the stability of the system and bring additional challenges for control design. (2) A disturbance observer is proposed to handle the external disturbances, and the model uncertainties of the system are estimated by the RBFNN method. By using the adaptive control method, a lumped error is applied to avoid the “explosion of complexity” problem.

The adaptive tuning law for ${\widehat{W}}_{ti}$ is given as follows:

$${\dot{\widehat{W}}}_i=-{\epsilon }_{ti}^T{\Lambda }_{ti}{S}_i\left(X_i\right)-M_i{\Lambda }_{ti}{\dot{\epsilon }}_{ti},$$

(27)

where ${\Lambda }_i$ is positive definite diagonal matrix. Then the main results for the predefined-time platoon control of nonlinear UAV systems are presented as follows.

Theorem 2.

The closed-loop system is UUB stable, and the mean square and state errors are converged to an arbitrary small zero neighborhood within a predefined time T under Assumptions 1 and 2 by applying the UAVs model (3) in a platoon under the proposed control input (25) with the adaptive RBFNN and the proposed disturbance observer (19).

Theorem 3.

Under Assumption 1 and by using the proposed control law (25) with the selected parameters, if $k_{{\tau }_i}{K}_{\epsilon ti}{M}_a^{-1}+T{e}^{-s}C\left(\dot{\eta }\right)M_a^{-1}+T{e}^{-s}{\mu }_{ti}\varsigma \Xi -T{e}^{-s}C\left(\dot{\eta }\right){\mu }_{ti}\varsigma \Xi M_a^{-1}-I_{n\times n}$ is positive, the local control signals of each vehicle are bounded.

The details for the proof of Theorems 2 and 3 are given in Appendix C and Appendix D, respectively.

Calculating the boundary of the control input in the real nonlinear system is crucial. Set ${\Theta }_i\left(0\right)=0$ and $y\left(t\right)={\Theta }_i\left(s\right)$, and rewrite (A28) with Fourier transform

$$\begin{array}{c}\hfill Y\left(j\omega \right)={\int }_0^{\infty }y\left(t\right)e^{-j\omega t}dt=G\left(j\omega \right)U\left(j\omega \right),\end{array}$$

(28)

where the Fourier transforms of the control input and output are $Y\left(j\omega \right)$ and $U\left(j\omega \right)$, respectively. $G\left(j\omega \right)$ is the frequency response transfer function.

Based on Parseval’s theorem [45], we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {∥y∥}_{L_2}^2=& {\int }_0^{\infty }{y}^T\left(t\right)y\left(t\right)dt\hfill \\ \hfill =& \frac{1}{2\pi }{\int }_{-\infty }^{+\infty }{Y}^{*}\left(j\omega \right)Y\left(j\omega \right)d\omega \hfill \\ \hfill =& \frac{1}{2\pi }{\int }_{-\infty }^{+\infty }{U}^{*}\left(j\omega \right)G^T(-j\omega )G\left(j\omega \right)U\left(j\omega \right)d\omega \hfill \\ \hfill \le & {\left(\underset{\omega \in \mathbb{R}}{sup}{∥G\left(j\omega \right)∥}_2\right)}^2\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }{U}^{*}\left(j\omega \right)U\left(j\omega \right)d\omega \hfill \\ \hfill =& {\left(\underset{\omega \in \mathbb{R}}{sup}{∥G\left(j\omega \right)∥}_2\right)}^2{∥\vartheta \left(s\right)∥}_{L_2}^2\hfill \\ \hfill =& \sqrt{{\lambda }_{max}\left(G^T(-j\omega )G\left(j\omega \right)\right)}{∥\vartheta \left(s\right)∥}_{L_2}^2\hfill \\ \hfill \le & (k_{{\tau }_i}{K}_{\epsilon ti}{\lambda }_{max}\left(M_a^{-1}\right)+T{e}^{-s}{\lambda }_{max}\left(C\left(\dot{\eta }\right)M_a^{-1}\right)\hfill \\ \hfill +& T{e}^{-s}{\lambda }_{max}({\mu }_{ti}\varsigma \Xi )-T{e}^{-s}{\lambda }_{max}(C\left(\dot{\eta }\right){\mu }_{ti}\varsigma \Xi M_a^{-1})-I_{n\times n}){∥\vartheta \left(s\right)∥}_{L_2}^2.\hfill \end{array}\end{array}$$

(29)

Remark 6.

The proof of the control input’s boundedness is required to ensure the control input’s acceptability in real-world applications. The control input is inversely proportional to the predefined time, so there will be a minimum convergence time in the actual system. The needed input may surpass the maximum input value of the system when the convergence time is defined as a very short amount, so the system cannot be stable within the predefined time. It is critical to demonstrate that the control input is bounded when the system can be stable within a predefined convergence time.

5. Simulation Results

The simulation results were implemented to demonstrate the effectiveness of the control laws (23) and (25), respectively. The simulation setup and parameters are given in

Table 1. The simulation setup and parameters.

Name	Parameters	Setup
UAV number	N	5
UAV mass	$M_i$	5kg
Desired reference	${\eta }_0$	${[0.2\pi ,-0.3\pi ,0.1\pi ,0.6,0.3,0.5]}^T$
Initial states	${\eta }_i\left(0\right)$	${[0,0,0,0,0,0]}^T$
Safety distance	${\varkappa }_a$	0.05m
Communication distance	${\varkappa }_c$	10m
Control gain	$k_{{\tau }_i}$	12
Performance parameters	${\varrho }_{\infty }$ 0.05
	l	0.2

. In general, the disturbance with $i=1,2,\dots ,5$ is given as ${\tau }_{{\varpi }_i}={[sin\left(t\right)-cos\left(0.2t\right)+cos\left(5t\right),cos\left(1.2t\right)+sin\left(t\right),sin\left(0.9t\right)+cos\left(0.5t\right),cos\left(0.3t\right)+sin\left(t\right),sin\left(3t\right)+cos\left(6t\right),cos\left(5t\right)+sin\left(0.9t\right)]}^T$. The uncertainties of the model and the external disturbance are both unknown in the control inputs (23) and (25). The objective under the controllers is to form the platoon with limited communication while avoiding collision and maintaining connectivity between two UAVs. The desired distance between each UAVs is designed as

$${\varkappa }_{i,i-1}=\left\{\begin{array}{c}{[0,0,0,0,0,0]}^T,i=1\hfill \\ {[2,0,0,0,0,0]}^T,\mathrm{others}\hfill \end{array}\right..$$

(30)

Results and Analysis

In this part, some simulation results for the proposed control schemes are presented. Some existing control methods are also given as comparisons.

1. Simulations for the UAVs platoon without prescribed performance and time constraint

The results of applying the control law in [20] without prescribed performance or a time constraint as a comparison are shown in Figure 2. To see the estimation of the RBFNN weight succinctly, only the weight of the fifth UAV is given in Figure 2d. From Figure 2, it can be seen that, although the platoon of UAVs could be guaranteed by the controller in [20], both the position errors and the angle errors were uncontrolled, that is to say, under the controller in [20], the position errors and the angle errors could not satisfy the expected performance index. It also shows that there was a certain overshoot in the transient response. In addition, although the controller in [20] could deal with the system uncertainty to some extent, the final result shows that the controller could not suppress the external disturbances.

2. Simulations for the UAVs platoon with prescribed performance but without time constraint

The simulation results are shown in this section to evaluate how effective the controller (23) for the UAV system (3) is at avoiding collisions and maintaining connection. Figure 3 shows that the state errors could satisfy the transient and steady state performance constraints, and the system converged within a small neighborhood around zero. The control inputs were bounded under the controller (23), which can be seen in Figure 3c. The estimation of the RBFNN weight of the fifth UAV is given in Figure 3d.

Compared with the results in Figure 2, it can be seen that the overshoot of the system will not exceed the expected limits and could also effectively suppress impacts of the system uncertainties and the external disturbances.

3. Simulations for the UAVs platoon with time constraint but without prescribed performance

To clarify the convergence time constraint, the simulation results are presented under the controller with time constraint but without prescribed performance in Figure 4. The state errors did not evolve within the predefined regions that were bounded by exponentially decaying time functions. But the system could converge at the designed predefined time $T=20$ s. Figure 4c shows the bounded control inputs. As stated in Theorem 3, the boundness of the control inputs can be guaranteed even in the presence of the convergence time constraint. The estimation of the RBFNN weight of the fifth UAV is given in Figure 3d.

4. Simulations for the UAVs platoon with time constraints and prescribed performance

In this part, the simulation results for the platoon of UAVs with a convergence time constraint and prescribed performance under the proposed control law (25) are presented. The convergence time was predefined as $T=20$ s. As shown in Figure 5, the performance of the closed-loop system satisfied the constraints, and the system could converge in a predefined time. The control inputs are shown in Figure 5c, which are also bounded. The estimation of the RBFNN weight of the fifth UAV is given in Figure 5d.

Compared with the other three groups of simulation results, the state errors were bounded and evolved in the predefined regions. The system was finally converged within a small zero neighborhood within the defined predefined time $T=20$ s, which further demonstrates the validity and practicability of the proposed controller.

$$\begin{array}{cc}\hfill \frac{1}{2}{\widehat{\epsilon }}_{ti}^T{M}_i{\widehat{\epsilon }}_{ti}& +\frac{1}{2}tr\left({\tilde{W}}_i^T{\Lambda }_{ti}^{-1}{\tilde{W}}_i\right)+\frac{1}{2}{\tilde{\tau }}_{w_i}^T{\tilde{\tau }}_{w_i}\le V_t\left(0\right)exp(-{\chi }_{ti}t)+\frac{{\beta }_{ti}}{{\chi }_{ti}},\hfill \end{array}$$

(31)

which means that when $t\to \infty$, one obtains $∥{\widehat{\epsilon }}_{ti}∥\le \sqrt{2{\beta }_{ti}/{\chi }_{ti}},∥{\tilde{W}}_i∥\le \sqrt{2{\beta }_{ti}/{\chi }_{ti}}$, and $∥{\tilde{\tau }}_{w_i}∥\le \sqrt{2{\beta }_{ti}/{\chi }_{ti}}$.

6. Conclusions

This paper investigated platoon control for multiple UAVs under time constraints and limited communication constraints. The realistic nonlinear system model was applied to describe the dynamics of the UAV. An adaptive disturbance observer created by embedding an RBFNN approximation for the system uncertainties was proposed to estimate the external bounded disturbance. Then, a versatile controller was proposed with a switching dynamic gain. By using the time transformation approach, rigorous proofs were proposed for the closed-loop system in the sense of uniformly ultimate boundedness (UUB). Under the proposed control approach, the platoon of UAVs could be guaranteed within a predefined time independent of initial conditions and system parameters. In order to better describe the UAV system, a higher fidelity model, which includes nondifferentiable external disturbances, measurement noises and unmodeled dynamics, will be considered in future work. In addition, how to achieve predefined-time stabilization of such a complex nonlinear UAV system is also an interesting and meaningful topic.