Disturbance observer-based backstepping formation control of multiple quadrotors with asymmetric output error constraints

https://doi.org/10.1016/j.amc.2021.126693Get rights and content

Highlights

  • The position subsystem and attitude subsystem controller based on backstepping control and disturbance observer are designed.

  • An asymmetric BLF is used to realize the asymmetric constraint on the output error.

  • The filter is used to estimate the time derivative of the virtual control input to reduce computation complexity.

Abstract

This paper presents a distributed formation tracking control strategy which acts on multiple quadrotor unmanned aerial vehicles (QUAVs) formation control under external disturbance and asymmetric output error constraints. An asymmetric barrier Lyapunov function (BLF) is applied to ensure the constraint of output error. Based on graph theory and backstepping control method, a distributed formation controller is designed to achieve the formation and maintenance of formations, where external disturbance is handled by disturbance observer (DO). In the framework of Lyapunov theory, the bounded stability of the closed-loop system is proved, and the output error is remained within the constraint range. The superiority and effectiveness of the designed control strategy is verified by the compared simulation.

Introduction

Quadrotor unmanned aerial vehicle (QUAV) behaves vertical flight, hovering, small diameter and other characteristics. It can perform forest fire monitoring, traffic supervision, atmospheric detection, cargo transportation and so on. In view of nonlinear and strong coupling characteristics of a QUAV model, PID control, adaptive control, sliding mode control (SMC) and other control methods have been applied to the QUAV control [1], [2], [3], [4], [5], [6].

Although a QUAV is capable of many advantages, its performance is always limited by its weight, which may reduce the success rate when performing missions. The multiple QUAVs formation technology is brought out to solve the problem. Multiple unmanned aerial vehicles (UAVs) cooperative formation control has attracted widespread attention due to its extensive application in military and civilian applications such as coordinated reconnaissance and coordinated rescue, see [7], [8], [9], [10] and the references therein. Aiming at the formation control problem of multiple UAVs under external disturbance, model uncertainty and parameter uncertainty, neural network-based control strategy and adaptive fully distributed SMC strategy were respectively proposed in Wang et al. [11], [12]. In order to compensate for external disturbances of a QUAV, a discontinuous integral term was incorporated into the finite time attitude control law [13]. To solve the multiple QUAVs formation control problem under nonlinearity, parameter uncertainty and external disturbances, a robust control method by combining linear quadratic regulation (LQR) and the robust compensation technique was presented in Liu et al. [14]. For the formation control problem of multiple UAVs in the presence of external disturbance, a distributed formation control strategy using the SMC and the backstepping method were respectively designed in Wang and Xu [15], Zhang et al. [16]. In [17], the distributed finite-time formation tracking control problem of multiple UAVs with collision avoidance was investigated, and a distributed model predictive control (MPC) scheme based on virtual target guidance was proposed. In [18], a multi-variable adaptive consensus control system for UAVs was proposed. The UAV formation keeping control problem based on multi-agent system consensus was investigated in Zhen et al. [19]. The above research have achieved good performance. However, output errors constraints problem wasn’t considered during the control system design procedure.

In recent years, BLF and prescribed performance control (PPC) methods have been used to improve the transient performance of the system. In [20], the performance constraint problem was converted into the unconstrained problems by using error conversion method, and the conversion error was stabilized to ensure that the original tracking error is always kept in the prescribed boundary. In [21], PPC was applied for the attitude formation control of multiple UAVs without performing the PPC on the position tracking control. The synchronization tracking error with performance function was defined, and the finite time tracking control strategy was proposed using neural network and finite time differentiator in Yu and Zhang [22]. What should be pointed out that the differentiation of the transfer function is involved in the error conversion, complexity and implementation difficulty in controller design are increased. BLF was introduced to solve this problem. In [23], BLF was used to solve the state constraint problem of multi-agent systems. In [24], [25], BLF was employed to solve full-state constraint problem of continuous and stochastic nonlinear multi-agent systems. In [26], the multiple UAVs formation control under model uncertainty and external disturbance was studied. Based on BLF, a distributed prescribed performance formation control scheme was designed by combining adaptive control, neural network and DO. However, most of the existing work using BLF to solve symmetric constraint.

Based on above mentioned, a backstepping tracking control strategy is proposed for multiple QUAVs formation control with output error constraints and external disturbance in this paper. Firstly, the dynamic model of the QUAVs is analyzed and simplified into two second-order subsystems (position subsystem and attitude subsystem) with external disturbance. Secondly, for the attitude and position subsystems, the controller is designed by using the backstepping method, where the DO is designed to estimate the external disturbance, the asymmetric BLF is employed to realize the asymmetric constraint on the output error, ensuring that the output error always obeys the constraint range. Compared with PPC method used in Liu et al. [20], Bechlioulis and Rovithakis [21], Yu and Zhang [22] which needs to compute the time derivative of the transformation function, an asymmetric BLF is applied to solve the output error constraint problem in this paper. It can ensure the output error is kept in the constrained boundary without introducing a performance function. Therefore, the time derivative of the transformation function is avoided and the controller design process is simplified. Thirdly, as we know that in the framework of backstepping control method, the control design procedure requires the time derivative of the virtual control input. In this paper, an asymmetric BLF is employed to solve the output error constraint problem and design the controller based on backstepping method. The computation complexity caused by calculating the time derivative of the virtual control input may increase induced by asymmetric BLF. Moreover, the time derivative of the virtual control input requires calculating the second order time derivative of the attitude angle reference signal, which is obtained by the inverse solution of the position subsystem control input. The filter is used to estimate the time derivative of the virtual control input and reduce computation complexity. Finally, the effectiveness of designed control strategy is verified by compared simulation. Compared with the existing work, the control strategy adopted in this paper can effectively achieve asymmetrical constraints on the tracking errors of multiple QUAVs formations, and improve the accuracy of the formation.

The other parts are organized as follows. In Section 2, the ith QUAV model and problem description are described. Controller design and stability analysis are provided in Section 3. Then, simulation is shown in Section 4. At last, conclusions is given in Section 5.

Section snippets

Notations

Assume that the communication topology of multiple QUAVs is represented by a directed graph G(υ,ε), which contains a set υ={υ1,υ2,,υN} consisting of a set of nodes and a set of edges ευ×υ. In the directed graph G(υ,ε), the relationship of each node can be represented by the adjacency matrix A=[aij]RN×N. If the edge between node i and node j is connected, then aij>0, otherwise aij=0. Here, the Laplacian matrix L=[lij]RN×N of the adjacency matrix A is introduced, and define lij=aij,lii=j=1Na

Formation controller design with output error constraints

In this section, the distributed formation controller is designed for the position subsystem and attitude subsystem, respectively. Fig. 1 shows the control structure of the ith QUAV. Firstly, for the position subsystem and attitude subsystem, the BLF is introduced to ensure the asymmetrical constraints of the output error, DO is used to deal with the external disturbance. In the controller design process, the time derivative of the virtual control input is required, which increases the amount

Simulation and analysis

In this section, to illustrate the effectiveness of the proposed control scheme, a formation system consisting of a virtual leader QUAV and three followers is simulated. We consider the formation simulation verification with the communication topology relationship shown in Fig. 2, where 0 represents the virtual leader and 1–3 represents the followers. In order to better reflect the effectiveness of the designed control strategy, the following two cases are simulated and compared.

  • Case 1: output

Conclusion

Based on BLF and backstepping control method, a distributed formation tracking control strategy is presented, which acts on the formation control of multiple QUAVs under external disturbance and output error constraints. The dynamic model of the QUAV is firstly divided into a position subsystem and an attitude subsystem. To proceed, the DO is used to estimate external disturbance, and the asymmetric BLF is applied to constraint the output error, and the controller is designed based on graph

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Project supported by Natural Science Foundation of Hebei Province no. F2020203105 and the National Science Foundation of China nos. 61503323, 62073234.

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