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.
Recent years have seen an increasing amount of concerns in theoretical researches and practical applications of multiagent systems [
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 [
On the other hand, many quantized control strategies have been done for multiagent systems with linear agent dynamics. In [
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 [
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 [
(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: Section
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.
Consider a multi-UAV system which contains
The formation tracking process of multiple UAVs is illustrated in Figure
Formation tracking illustration.
(i) all the closed-loop signals are semiglobally finite-time bounded,
(ii) the
There is a common assumption for the system.
The reference trajectory
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 parameter
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
The density constant
Researchers always use sector bound property to decompose the quantized signals into linear and nonlinear parts; i.e.,
In the conventional decomposition, the nonlinear part
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.
For
For any variables
Consider a nonlinear system
Consider the system
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.
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:
Based on (
The trajectory tracking errors of the
Let
Define a Lyapunov function as
With the virtual control signals
Using (
For the
Now, we present the following theorem to summarize the analysis in this study.
Consider a multi-UAV system consisting of UAVs which are described by nonholonomic kinematic model (
(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.
To analyze the stability of the overall closed-loop system, a Lyapunov function is defined as
Since
Taking (
Similarly, the following equation is obtained:
Substituting (
According to Lemma
With the aid of Lemma
Using (
Taking the similar process of obtaining (
Substituting (
Therefore, (
Based on Lemma
Based on the former analysis, it is suggested that
Therefore, it is proved that
Unlike using
Instead of estimating the real-time values of the time-varying disturbances
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
Initial and relative positions of the UAVs.
Initial position | Relative position | |
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The kinematic and dynamic models of UAVs are described by (
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 (
The simulation results of the two situations are shown in Figures
Formation trajectories. (a) the multi-UAV system with hysteretic quantizer (
Tracking errors of the multi-UAV system with hysteretic quantizer (
Tracking errors of the multi-UAV system with logarithmic quantizer (
Quantized input in the multi-UAV system with hysteretic quantizer (
Quantized input in the multi-UAV system with logarithmic quantizer (
Parameter estimations in the multi-UAV system with hysteretic quantizer (
Parameter estimations in the multi-UAV system with logarithmic quantizer (
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.
The simulation data generated during this study have been deposited with figshare (
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors gratefully appreciate the partial support of this work by the Aeronautical Science Foundation of China under Grant 20155896025.