# Adaptive Finite-Time Control for Formation Tracking of Multiple Nonholonomic Unmanned Aerial Vehicles with Quantized Input Signals

Authors: Jinglin Hu; Xiuxia Sun; Lei He
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Signal quantization can reduce communication burden in multiple unmanned aerial vehicle (multi-UAV) system, whereas it brings control challenge to formation tracking of multi-UAV system. This study presents an adaptive finite-time control scheme for formation tracking of multi-UAV system with input quantization. The UAV model contains nonholonomic kinematic model and autopilot model with uncertainties. The nonholonomic states of the UAVs are transformed by a transverse function method. For input quantization, hysteretic quantizers are used to reduce the system chattering and new decomposition is introduced to analyze the quantized signals. Besides, a novel transformation of the control signals is designed to eliminate the quantization effect. Based on the backstepping technique and finite-time Lyapunov stability theory, the adaptive finite-time controller is established for formation tracking of the multi-UAV system. Stability analysis proves that the tracking error can converge to an adjustable small neighborhood of the origin within finite time and all the signals in closed-loop system are semiglobally finite-time bounded. Simulation experiment illustrates that the system can track the reference trajectory and maintain the desired formation shape.


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## Body

## 1. Introduction

Recent years have seen an increasing amount of concerns in theoretical researches and practical applications of multiagent systems [1]. As a kind of typical multiagent systems, multi-UAV system can be used in more applications if they are equipped with formation tracking ability [2]. Formation tracking is achieved when all the UAVs in a system can track the reference trajectory and maintain a desired formation shape [3]. Formation tracking brings control challenges to multi-UAV systems, and it has been studied in many early researches [4–6]. Notably, these early efforts mainly assume that communication capacity among the UAVs is accurate and limitless. Indeed, formation communication is constrained because of limited channel bandwidth in multi-UAV systems. To reduce communication burden, control or measurement signals are always processed before being transmitted through communication networks by quantizers. The quantizer is regarded as a discontinuous map from continuous regions to a finite discrete set [7]. However, quantization introduces strong nonlinear characteristics, which may cause the systems to have worse transient performance or even instability [8, 9].It is important to take quantization into consideration in formation tracking control of multi-UAV systems. In quantized control systems, signals are always processed by various kinds of quantizers. The most commonly used quantizers are uniform quantizer and logarithmic quantizer. Uniform quantizer was used in a stabilization control scheme of linear systems to process input signals [10]. However, uniform quantizer can only lead to practical stability of control systems because of fixed quantization levels [11]. To overcome the weakness of uniform quantizer, logarithmic quantizer was introduced by increasing quantization levels near the origin. In [12], it was proved that logarithmic quantizer is the coarsest quantizer to achieve quadratic stabilization of linear systems. To avoid the system chattering caused by logarithmic quantizer, hysteretic quantizer was used in a stabilization control scheme by taking derivative information of signals into quantizer [13]. Based on backstepping technique, an adaptive control scheme is proposed for stabilization of nonlinear system with input signals being processed by hysteretic quantizer [14]. However, the nonlinearity introduced by hysteretic quantizer even makes it difficult for the existing stabilization schemes to be extended to tracking control problems, let alone to achieve trajectory tracking with guaranteed transient performance, such as finite-time convergence.On the other hand, many quantized control strategies have been done for multiagent systems with linear agent dynamics. In [15], by using uniform quantized communication, the authors proposed a gossip consensus algorithm for formation tracking of multiagent systems with linear discrete-time agent dynamics. With both uniform and logarithmic quantized information considered, an adaptive coordinated formation tracking strategy was proposed for continuous-time first-order integrator multiagent systems [16]. To improve model precision, some researchers describe UAV dynamics as nonlinear strict feedback models with uncertainties, such as unknown parameters or disturbances. With logarithmically quantized state measurements, an impulsive control strategy was proposed for formation tracking problem of multiagent systems with nonlinear dynamics [17]. Under leader-following structure, a formation tracking algorithm was proposed for second-order multiagent systems with uncertain nonlinear dynamics [18]. However, most practical applications of formation tracking control involve the agents that are nonholonomically constrained or underactuated. The multiagent systems with nonholonomic constraints cannot be controlled by simply using some control strategies like strict feedback systems. The most commonly considered nonholonomic agents are underactuated surface vessels (USVs) and wheeled mobile robots. In [19], the authors proposed an output feedback control approach for reference tracking of USVs. A cooperative quantized control scheme was proposed for a group of wheeled mobile robots with nonholonomic constraints [20]. These nonholonomic models of USVs or mobile robots cannot be applied to describe the kinematic and dynamic models of UAVs. It should be mentioned that there are few investigations about quantized control for formation tracking of multiple UAVs with nonholonomic constraints and uncertainties.The control approaches in the aforementioned papers can only guarantee that the tracking errors converge to the equilibrium with time going to infinity. On the basis of system stability, controller designs are always required to possess good transient performance. Due to some mission requirements, finite-time control schemes are investigated for formation tracking of multiple agents recently. In [21], finite-time stabilization problem was investigated for a class of high order nonholonomic systems. A homogeneous feedback control strategy was proposed for the finite-time convergence of quadrotor trajectory tracking [22]. Despite the results presented in [21, 22], the finite-time controllers are still limited to agents with completely known models or uncertainties which are bounded by known constants or continuous functions. Notably, the adaptive finite-time control schemes remain open for multiple nonholonomic UAVs with quantized input and uncertainties to the best of our knowledge.Motivated by the above discussion, we investigates the formation tracking problem of the multi-UAV systems with quantized input signals via an adaptive backstepping based finite-time control scheme. In addition, the UAVs contain nonholonomic kinematic model and autopilot model with uncertainties. These uncertainties mainly indicate unknown parameters and time-varying disturbances with unknown bounds. The main contributions are listed as follows.(i) The conventional controller for strict feedback systems cannot be applied to tracking control of the multiple nonholonomic UAVs. The UAV states with nonholonomic constraints are transformed by a transverse function method. The unknown parameters and disturbances in the models are estimated by several tuning functions in the control design.(ii) To avoid system chattering, input signals are processed by hysteretic quantizer. Many restrictions in [13, 14] are released through using new decomposition to the quantized signals. In contrast to the control signals of traditional backstepping controller, a novel transformation is introduced for the control signals to eliminate the quantization effect. In addition, a coarser quantization can be used in controller, which can reduce the communication burden more effectively.(iii) The systems are always completely known or have uncertainties with known bounds in the existing finite-time control schemes. Compared with the existing finite-time controllers, the system in our proposed control scheme has unknown parameters and time-varying disturbances, which have unknown bounds. We design the novel tuning functions to estimate the uncertainties for satisfying the system stability and the finite-time convergence of formation tracking. An adaptive finite-time control scheme is established by using finite-time Lyapunov stability theory. With the proposed finite-time controller, the trajectory tracking errors can be steered to within an adjustable small neighborhood of the origin in finite time. All the signals in the closed-loop system are semiglobally finite-time bounded.The remainder of this paper is organized as follows: Section2 introduces the control problem and some preliminaries. An adaptive finite-time control scheme is proposed for formation tracking control in Section 3. Section 4 analyzes the stability of the control scheme. A simulation experiment is given in Section 5. Finally, Section 6 makes a conclusion.


## 2. Problem Statement

In this section, the UAV model is described for later investigation firstly. Then, we introduce the time-varying formation tracking problem of multi-UAV system. Finally, a hysteretic quantizer and several preliminaries are stated.


### 2.1. Problem Description

Consider a multi-UAV system which containsN UAVs. The UAVs are assumed to be equipped with standard autopilots and maintain a certain altitude during formation tracking based on practical applications. Since the UAVs have typically nonholonomic constraints, the kinematic model of the ith UAV is described as(1)x˙iy˙iψ˙i=cos⁡ψisin⁡ψi0vi+001wi,i=1,…,N,where xi,yiT is the Cartesian position of the ith UAV, ψi is the orientation of the ith UAV, and vi and wi are the linear and angular velocities, respectively. Moreover, the autopilot model is described as(2)v˙i=1ηivqviu-vi+divt,w˙i=1ηiwqwiu-wi+diwt,where ηiv and ηiw are unknown positive parameters. div(t) and diw(t) are time-varying disturbances with unknown bounds. q(viu) and q(wiu) are quantized commanded linear and angular velocities with input signals viu and wiu to be designed. In actual UAV control practice, the autopilot inputs, viu and wiu, are computed from the information of the UAV kinematic model. From (2), it is shown that the autopilot outputs can change the linear and angular velocities in kinematic model. Thus, it achieves the control of UAV position and orientation.The formation tracking process of multiple UAVs is illustrated in Figure1. The three points represent three UAVs. With proper design of the quantized controller, the multi-UAV system can track the reference trajectory and maintain the desired formation shape. Pr(t)=(xr,yr)T and σi(t)=(σix,σiy)T represent the reference trajectory and relative position of the ith UAV in the formation shape for i=1,2,3. Pi0=(xi0,yi0)T and Pi(t)=(xi,yi)T represent its initial position and finial position. To avoid the weak point of leader failure in leader-follower structure, virtual structure is used for the formation topology structure. All the UAVs can get reference information from virtual leader. Through maintaining desired relative positions with virtual leader, the UAVs can form the desired formation shape. Therefore, the control objective is to design an adaptive controller ui=(viu,wiu)T where i=1,…,N such thatFigure 1
Formation tracking illustration.(i) all the closed-loop signals are semiglobally finite-time bounded,(ii) theN UAVs can track the reference trajectory Pr(t) and form the desired formation shape with tracking errors converging to an adjustable small neighborhood of the origin in finite time.There is a common assumption for the system.Assumption 1.
The reference trajectoryPr(t), relative position σi(t), and their first 2nd-order time derivatives are continuous and bounded.Remark 2.
It is always required for multi-UAV systems to execute various missions on the basis of time-varying formation shape. The formation shapes are changed through the adjustment of the parameterσi(t). If σi(t) is chosen to be constant, the formation shape will keep fixed.


### 2.2. Hysteretic Quantizer

In networked control systems, control signals are always quantized before being transmitted. The quantization effects may lead to worse performance or even instability. To reduce chattering, hysteretic quantizer is introduced and defined as(3)qvt=ujsign⁡v,uj1+δ<v≤uj,v˙<0,oruj<v≤uj1-δ,v˙>0,uj1+δsign⁡v,uj<v≤uj1-δ,v˙<0,oruj1-δ<v≤uj1+δ1-δ,v˙>0,0,0≤v<umin1+δ,v˙<0,orumin1+δ≤v<umin,v˙>0,qvt-,v˙=0,where uj=ρ1-jumin(j=1,2,⋯), 0<ρ<1, and δ=(1-ρ)/(1+ρ). umin represents the dead-zone for q(v(t)).Remark 3.
The density constantρ∈(0,1) is a quantization measurement. The smaller value of ρ means a coarser quantizer. The equation δ=(1-ρ)/(1+ρ) suggests that q(v(t)) has fewer quantization levels when δ is close to 1.
Researchers always use sector bound property to decompose the quantized signals into linear and nonlinear parts; i.e.,q(v)=v+τ(v), where the nonlinear part τ(v) satisfies the inequality τv⩽δv+umin [13, 14]. Different from the conventional quantization decomposition, a new decomposition method is used in this study as follows:(4)qv=ςvv+Δv,where(5)ςv=qvv,qv≠01,qv=0,Δv=0,qv≠0-v,qv=0.With the decomposition parts in (5), we can obtain that(6)1-δv⩽qv⩽1+δv,qv>01+δv⩽qv⩽1-δv,qv<0,which indicates that(7)1-δ⩽ςv⩽1+δ.In addition, we have -umin⩽Δ(v)⩽umin when q(v)=0. Thus, it is obtained that(8)Δv⩽umin.Remark 4.
In the conventional decomposition, the nonlinear partτ(v) is regarded as a disturbance, which makes it a factor that leads to the tracking error. In addition, the bound of τ(v) depends on the control input. Thus, a restriction is required for the control input in [13]. This restriction is released in [14] by using the backstepping technique. However, the proposed stabilization control approach in [14] cannot be applied to trajectory tracking problem. With the new decomposition in this study, the quantization effects can be eliminated and the additional restrictions are released.


### 2.3. Mathematical Preliminaries

Not only steady performance, but also transient performance is required to be considered in finite-time controller design. In finite-time control systems, the tracking errors of the system trajectories are required to reach the desired boundaries in finite time. Therefore, several lemmas are stated for the finite-time controller design.Lemma 5 (see [23]).
Forzi∈R(i=1,…,n) and ı∈(0,1), the inequality (z1+⋯+zn)ı⩽z1ı+⋯+znı holds.Lemma 6 (see [24]).
For any variablesd1,d2∈R, an inequality holds as follows: (9)d1h1d2h2⩽h1h1+h2sd1h1+h2+h2h1+h2s-h1/h2d2h1+h2,where h1,h2, and s can be chosen as any positive real constants.Definition 7.
Consider a nonlinear systemx˙=f(x,u), where x∈Rn and u∈Rm are system state and input, respectively. f:U→Rn is continuously differentiable on an open neighborhood U of the origin with f(0,0)=0. Suppose that, for any initial condition x(t0)=x0, there exist ϵ>0 and a settling time 0<T(x0,ϵ)<∞ such that x(t)⩽ϵ when t⩾t0+T. Then, the origin of the nonlinear system x˙=f(x,u) is semiglobal practical finite-time stability.Lemma 8 (see [25]).
Consider the systemx˙=f(x,u) in Definition 7. If there exists a continuously differentiable function V(x):U→R which is positive definite on U with V(0)=0 and the inequality V˙(x)⩽-ϕVϖ(x)+φ holds, where the real numbers ϖ∈(0,1), ϕ>0, and 0<φ<∞, the trajectory of this system is semiglobal practical finite-time stability. In addition, the finite convergence time T satisfies that T⩽V1-ϖ(x(0))/(1-ϖ)θϕ where θ∈(0,1) is a design parameter.


## 2.1. Problem Description

Consider a multi-UAV system which containsN UAVs. The UAVs are assumed to be equipped with standard autopilots and maintain a certain altitude during formation tracking based on practical applications. Since the UAVs have typically nonholonomic constraints, the kinematic model of the ith UAV is described as(1)x˙iy˙iψ˙i=cos⁡ψisin⁡ψi0vi+001wi,i=1,…,N,where xi,yiT is the Cartesian position of the ith UAV, ψi is the orientation of the ith UAV, and vi and wi are the linear and angular velocities, respectively. Moreover, the autopilot model is described as(2)v˙i=1ηivqviu-vi+divt,w˙i=1ηiwqwiu-wi+diwt,where ηiv and ηiw are unknown positive parameters. div(t) and diw(t) are time-varying disturbances with unknown bounds. q(viu) and q(wiu) are quantized commanded linear and angular velocities with input signals viu and wiu to be designed. In actual UAV control practice, the autopilot inputs, viu and wiu, are computed from the information of the UAV kinematic model. From (2), it is shown that the autopilot outputs can change the linear and angular velocities in kinematic model. Thus, it achieves the control of UAV position and orientation.The formation tracking process of multiple UAVs is illustrated in Figure1. The three points represent three UAVs. With proper design of the quantized controller, the multi-UAV system can track the reference trajectory and maintain the desired formation shape. Pr(t)=(xr,yr)T and σi(t)=(σix,σiy)T represent the reference trajectory and relative position of the ith UAV in the formation shape for i=1,2,3. Pi0=(xi0,yi0)T and Pi(t)=(xi,yi)T represent its initial position and finial position. To avoid the weak point of leader failure in leader-follower structure, virtual structure is used for the formation topology structure. All the UAVs can get reference information from virtual leader. Through maintaining desired relative positions with virtual leader, the UAVs can form the desired formation shape. Therefore, the control objective is to design an adaptive controller ui=(viu,wiu)T where i=1,…,N such thatFigure 1
Formation tracking illustration.(i) all the closed-loop signals are semiglobally finite-time bounded,(ii) theN UAVs can track the reference trajectory Pr(t) and form the desired formation shape with tracking errors converging to an adjustable small neighborhood of the origin in finite time.There is a common assumption for the system.Assumption 1.
The reference trajectoryPr(t), relative position σi(t), and their first 2nd-order time derivatives are continuous and bounded.Remark 2.
It is always required for multi-UAV systems to execute various missions on the basis of time-varying formation shape. The formation shapes are changed through the adjustment of the parameterσi(t). If σi(t) is chosen to be constant, the formation shape will keep fixed.


## 2.2. Hysteretic Quantizer

In networked control systems, control signals are always quantized before being transmitted. The quantization effects may lead to worse performance or even instability. To reduce chattering, hysteretic quantizer is introduced and defined as(3)qvt=ujsign⁡v,uj1+δ<v≤uj,v˙<0,oruj<v≤uj1-δ,v˙>0,uj1+δsign⁡v,uj<v≤uj1-δ,v˙<0,oruj1-δ<v≤uj1+δ1-δ,v˙>0,0,0≤v<umin1+δ,v˙<0,orumin1+δ≤v<umin,v˙>0,qvt-,v˙=0,where uj=ρ1-jumin(j=1,2,⋯), 0<ρ<1, and δ=(1-ρ)/(1+ρ). umin represents the dead-zone for q(v(t)).Remark 3.
The density constantρ∈(0,1) is a quantization measurement. The smaller value of ρ means a coarser quantizer. The equation δ=(1-ρ)/(1+ρ) suggests that q(v(t)) has fewer quantization levels when δ is close to 1.
Researchers always use sector bound property to decompose the quantized signals into linear and nonlinear parts; i.e.,q(v)=v+τ(v), where the nonlinear part τ(v) satisfies the inequality τv⩽δv+umin [13, 14]. Different from the conventional quantization decomposition, a new decomposition method is used in this study as follows:(4)qv=ςvv+Δv,where(5)ςv=qvv,qv≠01,qv=0,Δv=0,qv≠0-v,qv=0.With the decomposition parts in (5), we can obtain that(6)1-δv⩽qv⩽1+δv,qv>01+δv⩽qv⩽1-δv,qv<0,which indicates that(7)1-δ⩽ςv⩽1+δ.In addition, we have -umin⩽Δ(v)⩽umin when q(v)=0. Thus, it is obtained that(8)Δv⩽umin.Remark 4.
In the conventional decomposition, the nonlinear partτ(v) is regarded as a disturbance, which makes it a factor that leads to the tracking error. In addition, the bound of τ(v) depends on the control input. Thus, a restriction is required for the control input in [13]. This restriction is released in [14] by using the backstepping technique. However, the proposed stabilization control approach in [14] cannot be applied to trajectory tracking problem. With the new decomposition in this study, the quantization effects can be eliminated and the additional restrictions are released.


## 2.3. Mathematical Preliminaries

Not only steady performance, but also transient performance is required to be considered in finite-time controller design. In finite-time control systems, the tracking errors of the system trajectories are required to reach the desired boundaries in finite time. Therefore, several lemmas are stated for the finite-time controller design.Lemma 5 (see [23]).
Forzi∈R(i=1,…,n) and ı∈(0,1), the inequality (z1+⋯+zn)ı⩽z1ı+⋯+znı holds.Lemma 6 (see [24]).
For any variablesd1,d2∈R, an inequality holds as follows: (9)d1h1d2h2⩽h1h1+h2sd1h1+h2+h2h1+h2s-h1/h2d2h1+h2,where h1,h2, and s can be chosen as any positive real constants.Definition 7.
Consider a nonlinear systemx˙=f(x,u), where x∈Rn and u∈Rm are system state and input, respectively. f:U→Rn is continuously differentiable on an open neighborhood U of the origin with f(0,0)=0. Suppose that, for any initial condition x(t0)=x0, there exist ϵ>0 and a settling time 0<T(x0,ϵ)<∞ such that x(t)⩽ϵ when t⩾t0+T. Then, the origin of the nonlinear system x˙=f(x,u) is semiglobal practical finite-time stability.Lemma 8 (see [25]).
Consider the systemx˙=f(x,u) in Definition 7. If there exists a continuously differentiable function V(x):U→R which is positive definite on U with V(0)=0 and the inequality V˙(x)⩽-ϕVϖ(x)+φ holds, where the real numbers ϖ∈(0,1), ϕ>0, and 0<φ<∞, the trajectory of this system is semiglobal practical finite-time stability. In addition, the finite convergence time T satisfies that T⩽V1-ϖ(x(0))/(1-ϖ)θϕ where θ∈(0,1) is a design parameter.


## 3. Adaptive Finite-Time Controller Design

The adaptive controller design is based on finite-time Lyapunov stability theory and backstepping technique. The design process contains three steps. Firstly, a transverse function method is used to remove the nonholonomic constraints in the kinematic model of UAVs. Secondly, we design the virtual control signals of linear and angular velocities based on the transformed coordinate. Finally, the actual control signals are designed for the autopilot model.


### 3.1. Coordinate Transformation

Considering there exist difficulties in the nonholonomic constraints of UAV kinematic model, a coordinate transformation is used to the kinematic model by using transverse function method. The transformation is established as follows:(10)x¯iy¯i=xiyi+Qψ¯ihi1ζihi2ζi,ψ¯i=ψi-hi3,where (x¯i,y¯i) is the transformed position of the ith UAV, ψ¯i is the transformed orientation, and(11)Qψ¯i=cos⁡ψ¯i-sin⁡ψ¯isin⁡ψ¯icos⁡ψ¯i.Besides, hi1(ζi),hi2(ζi), and hi3(ζi) are functions of variable ζi and defined as(12)hi1ζi=ei1sinζisinhi3hi3,hi2ζi=ei1sinζi1-coshi3hi3,hi3ζi=ei2cosζi,where ei1, ei2∈(0,π/2) are positive constants. Therefore, it is obtained that(13)hi1ζi<ei1,hi2ζi<ei1,hi3ζi<ei2,which further implies that(14)x¯i-xi,y¯i-yi⩽2ei12,ψ¯i-ψi⩽ei2.Based on (12), differentiating (10) yields(15)x¯˙iy¯˙i=Riviζ˙i+∂Qψ¯i∂ψ¯ihi1ζihi2ζiwi-∂hi3ζi∂ζiζ˙i,ψ¯˙i=wi-∂hi3ζi∂ζiζ˙i,where(16)Ri=cos⁡ψisin⁡ψiQψ¯i∂hi1ζi∂ζi∂hi2ζi∂ζi.


### 3.2. Control Design of Kinematic Model

The trajectory tracking errors of theith UAV are defined as(17)zix=x¯i-σix-xr,ziy=y¯i-σiy-yr,ziψ=ψ¯i-ψr,where ψr=arctan(y˙r/x˙r).Letsiga=sign(x)xa with a>0 and x∈R. Then, we design the first virtual controller as(18)αivζ˙i=Ri-1-ki1zixziy-gi1sigqzixsigqziy+∂Qψ¯i∂ψ¯ihi1ζihi2ζiki2ziψ+gi2sigqziψ-ψ˙r+x˙r+σ˙ixy˙r+σ˙iy,αiw=-ki2ziψ-gi2sigqziψ+∂hi3ζi∂ζiζ˙i+ψ˙r,where ki1, ki2, gi1, and gi2 are positive parameters, 0<q<1.Define a Lyapunov function as(19)Vi1=12zix2+ziy2+ziψ2,whose time derivative is calculated as(20)V˙i1=-ki1zix2+ziy2-gi1zixziysigqzixsigqziy-ki2ziψ2-gi2ziψsigqziψ+Υivi-αiv+Ψiwi-αiw,where(21)Υi=zixcos⁡ψi+ziysin⁡ψi,Ψi=zixziy∂Qψ¯i∂ψ¯ihi1ζihi2ζi+ziψ.


### 3.3. Control Design of Autopilot Model

With the virtual control signalsαiv and αiw, we define the errors of linear and angular velocities as(22)ziv=vi-αiv,ziw=wi-αiw.Using (2) and (18), the time derivative of (22) can be expressed as(23)z˙iv=v˙i-α˙iv=1ηivqviu-vi+div-α˙iv,z˙iw=w˙i-α˙iw=1ηiwqwiu-wi+diw-α˙iw.For theith UAV, there exist unknown positive constants d¯iv and d¯iw such that divt⩽d¯iv and diwt⩽d¯iw when ∀t>0. Then, the quantized controller is established as follows:(24)viu=βivTvic,wiu=βiwTwic,βiv=βiv1βiv2βiv3T=1ϑ^iv1-δivuminiv1-δiv1ϑ^iv1-δivT,vic=vic1vic2vic3T=-zivκiv2ziv2κiv2+ξiv12-zivziv2+ξiv22-zivd^iv2ziv2d^iv2+ξiv32T,βiw=βiw1βiw2βiw3T=1ϑ^iw1-δiwuminiw1-δiw1ϑ^iw1-δiwT,wic=wic1wic2wic3T=-ziwκiw2ziw2κiw2+ξiw12-ziwziw2+ξiw22-ziwd^iw2ziw2d^iw2+ξiw32T,where ϑ^iv, ϑ^iw, d^iv, and d^iw are the estimations of the unknown positive parameters ϑiv=1/ηiv, ϑiw=1/ηiw, d¯iv, and d¯iw, respectively. ξa(a=iv1,iv2,iv3,iw1,iw2,iw3) are positive design parameters. uminiv and δiv are the designed parameters of the quantizer q(viu) of linear velocity. uminiw and δiw are the designed parameters of the quantizer q(wiu) of angular velocity. The parameters κiv and κiw are defined as(25)κiv=-ϑ^ivvi-α˙iv+Υi+ki3ziv+gi3sigqziv,κiw=-ϑ^iwwi-α˙iw+Ψi+ki4ziw+gi4sigqziw,where ki3, ki4, gi3, and gi4 are positive constants. Relatively, the parameter update laws are designed as(26)ϑ^˙iv=Proj-γivziv2κiv2ϑ^ivziv2κiv2+ξiv12-γivziv2d^iv2ϑ^ivziv2d^iv2+ξiv32-γivzivvi-μiϑ^iv,ϑ^˙iw=Proj-γiwziw2κiw2ϑ^iwziw2κiw2+ξiw12-γiwziw2d^iw2ϑ^iwziw2d^iw2+ξiw32-γiwziwwi-μiϑ^iw,d^˙iv=Proj-ϱivziv-μid^iv,d^˙iw=Proj-ϱiwziw-μid^iw,where μi, γiv, γiw, ϱiv, and ϱiw are positive design constants. To ensure that ϑ^iv, ϑ^iw, d^iv, and d^iw are greater than zero, the projection operator Proj{·} is utilized in the calculation of parameter update laws. It is worth mentioning that projection operator has a useful property; that is, it satisfies -(1/γ)ϑ~Proj{λ}⩽-(1/γ)ϑ~λ. It will be utilized in the following analysis.


## 3.1. Coordinate Transformation

Considering there exist difficulties in the nonholonomic constraints of UAV kinematic model, a coordinate transformation is used to the kinematic model by using transverse function method. The transformation is established as follows:(10)x¯iy¯i=xiyi+Qψ¯ihi1ζihi2ζi,ψ¯i=ψi-hi3,where (x¯i,y¯i) is the transformed position of the ith UAV, ψ¯i is the transformed orientation, and(11)Qψ¯i=cos⁡ψ¯i-sin⁡ψ¯isin⁡ψ¯icos⁡ψ¯i.Besides, hi1(ζi),hi2(ζi), and hi3(ζi) are functions of variable ζi and defined as(12)hi1ζi=ei1sinζisinhi3hi3,hi2ζi=ei1sinζi1-coshi3hi3,hi3ζi=ei2cosζi,where ei1, ei2∈(0,π/2) are positive constants. Therefore, it is obtained that(13)hi1ζi<ei1,hi2ζi<ei1,hi3ζi<ei2,which further implies that(14)x¯i-xi,y¯i-yi⩽2ei12,ψ¯i-ψi⩽ei2.Based on (12), differentiating (10) yields(15)x¯˙iy¯˙i=Riviζ˙i+∂Qψ¯i∂ψ¯ihi1ζihi2ζiwi-∂hi3ζi∂ζiζ˙i,ψ¯˙i=wi-∂hi3ζi∂ζiζ˙i,where(16)Ri=cos⁡ψisin⁡ψiQψ¯i∂hi1ζi∂ζi∂hi2ζi∂ζi.


## 3.2. Control Design of Kinematic Model

The trajectory tracking errors of theith UAV are defined as(17)zix=x¯i-σix-xr,ziy=y¯i-σiy-yr,ziψ=ψ¯i-ψr,where ψr=arctan(y˙r/x˙r).Letsiga=sign(x)xa with a>0 and x∈R. Then, we design the first virtual controller as(18)αivζ˙i=Ri-1-ki1zixziy-gi1sigqzixsigqziy+∂Qψ¯i∂ψ¯ihi1ζihi2ζiki2ziψ+gi2sigqziψ-ψ˙r+x˙r+σ˙ixy˙r+σ˙iy,αiw=-ki2ziψ-gi2sigqziψ+∂hi3ζi∂ζiζ˙i+ψ˙r,where ki1, ki2, gi1, and gi2 are positive parameters, 0<q<1.Define a Lyapunov function as(19)Vi1=12zix2+ziy2+ziψ2,whose time derivative is calculated as(20)V˙i1=-ki1zix2+ziy2-gi1zixziysigqzixsigqziy-ki2ziψ2-gi2ziψsigqziψ+Υivi-αiv+Ψiwi-αiw,where(21)Υi=zixcos⁡ψi+ziysin⁡ψi,Ψi=zixziy∂Qψ¯i∂ψ¯ihi1ζihi2ζi+ziψ.


## 3.3. Control Design of Autopilot Model

With the virtual control signalsαiv and αiw, we define the errors of linear and angular velocities as(22)ziv=vi-αiv,ziw=wi-αiw.Using (2) and (18), the time derivative of (22) can be expressed as(23)z˙iv=v˙i-α˙iv=1ηivqviu-vi+div-α˙iv,z˙iw=w˙i-α˙iw=1ηiwqwiu-wi+diw-α˙iw.For theith UAV, there exist unknown positive constants d¯iv and d¯iw such that divt⩽d¯iv and diwt⩽d¯iw when ∀t>0. Then, the quantized controller is established as follows:(24)viu=βivTvic,wiu=βiwTwic,βiv=βiv1βiv2βiv3T=1ϑ^iv1-δivuminiv1-δiv1ϑ^iv1-δivT,vic=vic1vic2vic3T=-zivκiv2ziv2κiv2+ξiv12-zivziv2+ξiv22-zivd^iv2ziv2d^iv2+ξiv32T,βiw=βiw1βiw2βiw3T=1ϑ^iw1-δiwuminiw1-δiw1ϑ^iw1-δiwT,wic=wic1wic2wic3T=-ziwκiw2ziw2κiw2+ξiw12-ziwziw2+ξiw22-ziwd^iw2ziw2d^iw2+ξiw32T,where ϑ^iv, ϑ^iw, d^iv, and d^iw are the estimations of the unknown positive parameters ϑiv=1/ηiv, ϑiw=1/ηiw, d¯iv, and d¯iw, respectively. ξa(a=iv1,iv2,iv3,iw1,iw2,iw3) are positive design parameters. uminiv and δiv are the designed parameters of the quantizer q(viu) of linear velocity. uminiw and δiw are the designed parameters of the quantizer q(wiu) of angular velocity. The parameters κiv and κiw are defined as(25)κiv=-ϑ^ivvi-α˙iv+Υi+ki3ziv+gi3sigqziv,κiw=-ϑ^iwwi-α˙iw+Ψi+ki4ziw+gi4sigqziw,where ki3, ki4, gi3, and gi4 are positive constants. Relatively, the parameter update laws are designed as(26)ϑ^˙iv=Proj-γivziv2κiv2ϑ^ivziv2κiv2+ξiv12-γivziv2d^iv2ϑ^ivziv2d^iv2+ξiv32-γivzivvi-μiϑ^iv,ϑ^˙iw=Proj-γiwziw2κiw2ϑ^iwziw2κiw2+ξiw12-γiwziw2d^iw2ϑ^iwziw2d^iw2+ξiw32-γiwziwwi-μiϑ^iw,d^˙iv=Proj-ϱivziv-μid^iv,d^˙iw=Proj-ϱiwziw-μid^iw,where μi, γiv, γiw, ϱiv, and ϱiw are positive design constants. To ensure that ϑ^iv, ϑ^iw, d^iv, and d^iw are greater than zero, the projection operator Proj{·} is utilized in the calculation of parameter update laws. It is worth mentioning that projection operator has a useful property; that is, it satisfies -(1/γ)ϑ~Proj{λ}⩽-(1/γ)ϑ~λ. It will be utilized in the following analysis.


## 4. Stability Analysis

Now, we present the following theorem to summarize the analysis in this study.Theorem 9.
Consider a multi-UAV system consisting of UAVs which are described by nonholonomic kinematic model (1), autopilot model (2), and hysteretic quantizer (3). Under the adaptive controller (24), parameter update law (26), and the condition that Assumption 1 holds, the finite-time formation tracking of multi-UAV system can be achieved such that(i) all the signals in the closed-loop system are semiglobally finite-time bounded,(ii) the formation tracking errors can converge to a small neighborhood of the origin in finite time.Proof.
To analyze the stability of the overall closed-loop system, a Lyapunov function is defined as(27)Vi=Vi1+12ziv2+12ziw2+12γivϑ~iv2+12γiwϑ~iw2+12ϱivd~iv2+12ϱiwd~iw2,where ϑ~iv=ϑiv-ϑ^iv, ϑ~iw=ϑiw-ϑ^iw, d~iv=d¯iv-d^iv, and d~iw=d¯iw-d^iw.
Sinceβiv1 and βiv3 are greater than zero according to ϑ^iv>0, it can be checked that(28)βivTvicziv=-βiv1ziv2κiv2ziv2κiv2+ξiv12-βiv2ziv2ziv2+ξiv22-βiv3ziv2d^iv2ziv2d^iv2+ξiv32⩽0.By taking similar procedure, we can prove that βiwTwicziw⩽0. Since the inequality 0⩽b1-b12/b12+b22⩽b2 holds when b2⩾0, we have (29)zivκiv-ziv2κiv2ziv2κiv2+ξiv12⩽ξiv1,ziv-ziv2ziv2+ξiv22⩽ξiv2,d^ivziv-ziv2d^iv2ziv2d^iv2+ξiv32⩽ξiv3.
Taking (4), (23)-(25), and (29) into consideration, we can obtain that(30)zivz˙iv=zivϑivqviu-vi+divt-α˙iv=zivϑivςivviuβivTvic+Δivviu-vi+divt-α˙iv⩽zivϑiv1-δivβiv1vic1+ϑiv1-δivβiv2vic2+ϑiv1-δivβiv3vic3+ϑivuminiv-ϑivvi+divt-α˙iv⩽zivκiv-ziv2κiv2ziv2κiv2+ξiv12+ϑivuminivziv-ϑivuminivziv2ziv2+ξiv22+d^ivziv-ziv2d^iv2ziv2d^iv2+ξiv32+d~ivziv-ki3ziv2-gi3zivsigqziv-ϑ~ivzivvi+ziv2κiv2ϑ^ivziv2κiv2+ξiv12+ziv2d^iv2ϑ^ivziv2d^iv2+ξiv32-zixziyΥiziv⩽ξiv1+ϑivuminivξiv2+ξiv3+d~ivziv-gi3zivsigqziv-ϑ~ivzivvi+ziv2κiv2ϑ^ivziv2κiv2+ξiv12+ziv2d^iv2ϑ^ivziv2d^iv2+ξiv32-ki3ziv2-Υiziv.
Similarly, the following equation is obtained:(31)ziwz˙iw=ziwϑiwqwiu-wi+diw-α˙iw⩽ξiw1+ϑiwuminiwξiw2+ξiv3+d~iwziw-gi4ziwsigqziw-ϑ~iwziwwi+ziw2κiw2ϑ^iwziw2κiw2+ξiw12+ziw2d^iw2ϑ^iwziw2d^iw2+ξiw32-ki4ziw2-Ψiziψ.
Substituting (20), (30), and (31) into the time derivative of Vi, we can get that(32)V˙i=V˙i1+zivz˙iv+ziwz˙iw+1γivϑ~iv-ϑ^˙iv+1γiwϑ~iw-ϑ^˙iw+1ϱivd~iv-d^˙iv+1ϱiwd~iw-d^˙iw⩽-ki1zix2+ziy2-ki2ziψ2-ki3ziv2-ki4ziw2+μiγivϑ~ivϑ^iv+μiγiwϑ~iwϑ^iw+μiϱivd~ivd^iv+μiϱiwd~iwd^iw-gi1zixsigqzix+ziysigqziy-gi2ziψsigqziψ-gi3zivsigqziv-gi4ziwsigqziw+ξiv1+ϑivuminivξiv2+ξiv3+ξiw1+ϑiwuminiwξiw2+ξiw3.
According to Lemma5, it can be checked that(33)gi1zixsigqzix+gi1ziysigqziy+gi2ziψsigqziψ+gi3zivsigqziv+gi4ziwsigqziw⩾gizix1+q+ziy1+q+ziψ1+q+ziv1+q+ziw1+q⩾gizix2+ziy2+ziψ2+ziv2+ziw21+q/2,where gi=min(gi1,gi2,gi3,gi4,).
With the aid of Lemma6, the parameters in (9) are set as d1=1, d2=μi/2γivϑ~iv2, h1=(1-q)/2, h2=(1+q)/2, s=((1-q)/2)e1+q/1-qln⁡1+q/2. Then, it can be proved that(34)μi2γivϑ~iv21+q/2⩽1-q2e1+q/1-qln⁡1+q/2+μi2γivϑ~iv2.
Using (34) and noting that ϑ~ivϑ^iv⩽(1/2)ϑiv2-(1/2)ϑ~iv2, we can obtain that(35)μiγivϑ~ivϑ^iv=-μi1+q/212γivϑ~iv21+q/2+μiγivϑ~ivϑ^iv+μi2γivϑ~iv21+q/2⩽-μi1+q/212γivϑ~iv21+q/2+μi2γivϑiv2+1-q2e1+q/1-qln⁡1+q/2.
Taking the similar process of obtaining (35), we have(36)μiγiwϑ~iwϑ^iw⩽-μi1+q/212γiwϑ~iw21+q/2+μi2γiwϑiw2+1-q2e1+q/1-qln⁡1+q/2,μiϱivd~ivd^iv⩽-μi1+q/212ϱivd~iv21+q/2+μi2ϱivd¯iv2+1-q2e1+q/1-qln⁡1+q/2,μiϱiwd~iwd^iw⩽-μi1+q/212ϱiwd~iw21+q/2+μi2ϱiwd¯iw2+1-q2e1+q/1-qln⁡1+q/2.
Substituting (33)-(36) into (32), it can be expressed as(37)V˙i⩽-gizix2+ziy2+ziψ2+ziv2+ziw21+q/2-μi1+q/212γivϑ~iv21+q/2-μi1+q/212γiwϑ~iw21+q/2-μi1+q/212ϱivd~iv21+q/2-μi1+q/212ϱiwd~iw21+q/2+μi2γivϑiv2+μi2γiwϑiw2+μi2ϱivd¯iv2+μi2ϱiwd¯iw2+21-qe1+q/1-qln⁡1+q/2+ξiv1+ϑivuminivξiv2+ξiv3+ξiw1+ϑiwuminiwξiw2+ξiw3.
Therefore, (37) can be rewritten as(38)V˙i⩽-liVi1+q/2+Θi,where li=min(gi,μi(1+q)/2), and the constant(39)Θi=μi2γivϑiv2+μi2γiwϑiw2+μi2ϱivd¯iv2+μi2ϱiwd¯iw2+21-qe1+q/1-qln⁡1+q/2+ξiv1+ϑivuminivξiv2+ξiv3+ξiw1+ϑiwuminiwξiw2+ξiw3.
Based on Lemma8 and (38)-(39), it is obtained that Vi(1+q)/2(t)⩽Θi/(1-θ)li when t⩾Ti with the settling time(40)Ti=2Vi1-q/201-qθli.
Based on the former analysis, it is suggested thatzix, ziy, ziψ, ziv, ziw, ϑ^iv, and ϑ^iw are semiglobal finite-time bounded. According to the boundedness of Pr(t) and σr(t), it is obtained that x¯i, y¯i, and ψ¯i can keep semiglobal finite-time bound. Furthermore, the virtual control signals αiv, αiw and control signals viu, wiu are bounded. Besides, according to the definition of Vi, it is obtained that zixt, ziyt, and ziψt are all smaller than (Θi/(1-θ)li)1/(1+q) when t⩾Ti. Recalling (14) and (17), we can obtain that(41)xi-δix-xr⩽xi-x¯i+x¯i-δix-xr⩽2ei1+zix,yi-δiy-yr⩽yi-y¯i+y¯i-δiy-yr⩽2ei1+ziy,ψi-ψr⩽ψi-ψ¯i+ψ¯i-ψr⩽ei2+ziψ.
Therefore, it is proved thatxi,yi, and ψi are semiglobal finite-time bounded. Based on the former analysis, it is suggested that all the closed-loop signals in the ith UAV are semiglobal finite-time bounded and the trajectory tracking errors of the ith UAV can converge to an adjustable small neighborhood of the origin in finite time. Taking the whole multi-UAV system into consideration, we define that T=max(T1,…,TN). It can be concluded that all the signals in the system are semiglobally finite-time bounded and the tracking errors are steered to within an adjustable small neighborhood of the origin when t⩾T.Remark 10.
Unlike usingκiv and κiw in (25) as control signals in conventional backstepping controllers, we make a transformation (24) for κiv and κiw. The restrictions in [13, 14] are removed through utilizing the transformed signals as control signals. In addition, with the inequalities in (29), we can eliminate the quantization effects on the system stability based on the analysis of (30) and (31). Furthermore, there are no limitation on quantizer parameters, which allows a coarser quantization to be used in the controller.Remark 11.
Instead of estimating the real-time values of the time-varying disturbancesd~iv and d~iw, we choose to estimate the upper bounds of d~iv and d~iw. For the requirement of finite-time convergence, the parameter update laws in (26) are designed to estimate the unknown parameters and upper bounds of disturbances. Due to the inequalities in (35) and (36), the control scheme can satisfy (38) and achieve the finite-time tracking.


## 5. Simulation

A simulation experiment is employed to demonstrate the effectiveness of the proposed control scheme. In the simulation experiment, we use a multi-UAV system which contains four UAVs. Our goal is to make the UAVs under the proposed control scheme achieve trajectory tracking in finite time. The reference trajectory is set as(xr,yr)=(5t,10sin⁡(0.2t)). A time-varying formation shape is designed with the initial positions and relative positions in the formation of the UAVs being listed in Table 1. The initial linear and angular velocities of the UAVs are totally set as vi=0m/s,wi=0rad/s for i=1,…,4.Table 1
Initial and relative positions of the UAVs.
Initial position Relative position UAV1 ( - 5 , - 6 ) ( 0,1 ) UAV2 ( 1,10 ) - tanh ⁡ 0.5 t - 20 - 1,4 UAV3 ( 2,1 ) - 8 tanh ⁡ 0.5 t - 20 , - 4 UAV4 ( - 2.5 , - 5 ) 3 tanh ⁡ 0.5 t - 20 - 1 , - 5The kinematic and dynamic models of UAVs are described by (1) and (2). The unknown parameters of the autopilots are set as ηiv=0.8,ηiw=1. The unknown disturbances in the autopilots are given by div(t)=0.15sin⁡(0.5t) and diw(t)=0.05cos⁡(0.75t). These values of unknown parameters and disturbances are used only for simulation and are unknown in controller design. The design parameters of hysteretic quantizers are defined as δiv=δiw=0.3, uminiv=uminiw=0.02. We choose the controller parameters as kij=gij=5(i,j=1,…,4), q=1/2, γiv=γiw=1, ϱiv=ϱiw=2, and μi=16. The initial estimates of the adaptive parameters are chosen as θ^iv(0)=3, θ^iw(0)=2, d^iv(0)=0.5, and d^iw(0)=0.2.In traditional quantized control schemes of continuous-time systems, chattering may occur in the quantized signals when the signals are processed by logarithmic quantizer. It is not desirable in the network since the transmission of signal chattering requires infinite bandwidth. To prove that the hysteretic quantizer (3) can reduce signal chattering, we also consider the same situation as above with the difference being the signal quantization. In this situation, the control input signals are processed by logarithmic quantizer instead of hysteretic quantizer. The logarithmic quantizer is defined as follows [12]:(42)Qlv=ρljul,min,ρljul,min1+δl≤v<ρljul,min1-δl,j=0,±1,±2,…,0,v=0,-Q-v,v<0,where quantization density ρl∈(0,1), δl=(1-ρl)/(1+ρl), and scaling parameter ul,min>0. Here, we set ul,min=0.2 and δl=0.3 for the logarithmic quantizer (42) used in the multi-UAV system.The simulation results of the two situations are shown in Figures2–8. From the formation trajectories in Figure 2, it is shown that the multi-UAV systems can both achieve the formation tracking fast and smoothly. However, some differences can be still observed from the tracking errors of all the UAVs given in Figures 3 and 4. As depicted in Figure 3, the UAVs with hysteretic quantizer (3) can reach the desired trajectories within the settling time T=1.57s and maintain high accuracy in steady performance. Figure 4 shows that slight oscillations and biases exist in the tracking errors of the multi-UAV system with logarithmic quantizer (42). Figure 5 shows the typical quantized input signals of the UAVs, i.e., q(v1u) of the first UAV and q(w2u) of the second UAV. These signals are processed by hysteretic quantizer (3), and signal chattering is highly reduced by observing the details of q(v1u) and q(w2u) in Figure 5. It shows that the input signals are simplified by hysteretic quantizer in favor of communication. The quantized input signals Ql(v1u) and Ql(w2u) are depicted in Figure 6 for the multi-UAV system with logarithmic quantizer (42). As the details of Ql(v1u) and Ql(w2u) show in Figure 6, signal chattering occurs during the control procedure. It is worth mentioning that these fast switchings are difficult to be transferred via multi-UAV network with a finite data rate channel in practice. In addition, the chattering does great harm to the actuators and UAVs physically. Figures 7 and 8 show the estimations of unknown parameters and disturbances of the multi-UAV systems in the two situations. It can be seen that all the signals in the closed-loop systems are semiglobal finite-time bounded from Figures 3–8. Compared with the multi-UAV system with logarithmic quantizer (42), the multi-UAV system with hysteretic quantizer (3) has better control performance and highly reduces signal chattering.Figure 2
Formation trajectories. (a) the multi-UAV system with hysteretic quantizer (3), (b) the multi-UAV system with logarithmic quantizer (42).
(a) (b)Figure 3
Tracking errors of the multi-UAV system with hysteretic quantizer (3). (a) zx, (b) zy, (c) zψ.
(a) (b) (c)Figure 4
Tracking errors of the multi-UAV system with logarithmic quantizer (42). (a) zx, (b) zy, (c) zψ.
(a) (b) (c)Figure 5
Quantized input in the multi-UAV system with hysteretic quantizer (3). (a) q(v1u) of the first UAV, (b) q(w2u) of the second UAV.
(a) (b)Figure 6
Quantized input in the multi-UAV system with logarithmic quantizer (42). (a) Ql(v1u) of the first UAV, (b) Ql(w2u) of the second UAV.
(a) (b)Figure 7
Parameter estimations in the multi-UAV system with hysteretic quantizer (3). (a) Estimations of unknown parameters, (b) estimations of unknown disturbances.
(a) (b)Figure 8
Parameter estimations in the multi-UAV system with logarithmic quantizer (42). (a) Estimations of unknown parameters, (b) estimations of unknown disturbances.
(a) (b)


## 6. Conclusion

Based on the backstepping technique, an adaptive finite-time control scheme is proposed for the time-varying formation tracking of multiple UAVs with inputs quantization. The UAVs are described by kinematic model with nonholonomic constraints and autopilot model with unknown parameters and disturbances. A transverse function method is used to transform the nonholonomic UAV states. To avoid system chattering, the input signals are processed by hysteretic quantizers. A novel decomposition method is utilized to deal with the nonlinearity caused by quantization. The adaptive finite-time controller is established through using the finite-time Lyapunov stability theory and a novel transformation of control signals. With the proposed control design, all the closed-loop signals can keep semiglobal finite-time bounded and the tracking errors can be steered to within a small neighborhood of the origin in finite time. Besides, all the UAVs form and maintain the prescribed formation shape while tracking the reference trajectory. Finally, the simulation results are provided to illustrate the effectiveness of the proposed approach.



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