Distributed Average Tracking Research Articles

Finite-Time Distributed Average Tracking for a Class of Nonlinear Multi-Agent Systems With External Disturbances

In this article, a finite-time distributed average tracking (DAT) problem is considered for unity relative degree nonlinear multi-agent systems (MASs) subject to external disturbances. The objective of this article is to design a distributed controller such that the output of each agent converges to the desired trajectory (the average of multiple nonlinear signals) within a finite time. First, we introduce a steady-state generator, which can reproduce the desired trajectory. Based on the steady-state generator, we design a distributed observer to estimate the desired trajectory, which is robustness to initialization errors. Then, an observer-based output-feedback controller is proposed for each agent such that the output of each agent can converge to its own generated signal within a finite time. Finally, it is shown that the output of each agent can converge to the desired trajectory within a finite time using the proposed control strategy, and hence the finite-time DAT problem is solved. A numerical example is provided to demonstrate the effectiveness of the proposed algorithm.

Distributed Average Tracking With Incomplete Measurement Under a Weight-Unbalanced Digraph

During the implementation of a cooperative algorithm, information about the agents’ velocity may be unavailable due to the space constraint and availability of sensors. Thus, it gives rise to the design of distributed average tracking (DAT) algorithms without using agents’ velocity measurements. These are denoted as velocity-free DAT problems. The existing literature has addressed such problems in the presence of an undirected graph for the reference signals with bounded position, velocity, and acceleration differences. We propose a velocity-free DAT algorithm under a weight-unbalanced strongly-connected digraph that represents the most general network structure for achieving DAT. Additionally, the proposed algorithm works for a broader range of time-varying references, having bounded acceleration differences among themselves. Linear stability theory is used to establish uniform ultimate boundedness of the errors for bounded acceleration differences. Asymptotic convergence of the errors is guaranteed for converging acceleration differences. Unlike the existing works, our DAT algorithm does not need any update law for the gains. Thus, the approach is computationally efficient. Numerical simulations with the comparison with the state-of-the-art demonstrate the performance of our algorithm over a wider range of time-varying references under weight-unbalanced graph.

Delay and Packet-Drop Tolerant Multistage Distributed Average Tracking in Mean Square.

This article studies the distributed average tracking (DAT) problem pertaining to a discrete-time linear time-invariant multiagent network, which is subject to, concurrently, input delays, random packet drops, and reference noise. The problem amounts to an integrated design of delay and a packet-drop-tolerant algorithm and determining the ultimate upper bound of the tracking error between agents' states and the average of the reference signals. The investigation is driven by the goal of devising a practically more attainable average tracking algorithm, thereby extending the existing work in the literature, which largely ignored the aforementioned uncertainties. For this purpose, a blend of techniques from Kalman filtering, multistage consensus filtering, and predictive control is employed, which gives rise to a simple yet comepelling DAT algorithm that is robust to the initialization error and allows the tradeoff between communication/computation cost and stationary-state tracking error. Due to the inherent coupling among different control components, convergence analysis is significantly challenging. Nevertheless, it is revealed that the allowable values of the algorithm parameters rely upon the maximal degree of an expected network, while the convergence speed depends upon the second smallest eigenvalue of the same network's topology. The effectiveness of the theoretical results is verified by a numerical example.

Robust Distributed Average Tracking for Disturbed Second-Order Multiagent Systems

This article investigates the distributed average tracking (DAT) problem for disturbed second-order multiagent systems, where a crowd of agents is required to track the average of the multiple time-varying signals. First, a new kind of distributed average estimator is developed for each agent to estimate the average of the multiple time-varying signals in finite time. The protocol possesses the distinguished feature of robustness to initialization errors, which can recover from network alterations. Then, an observer-based finite-time tracking protocol is proposed to make each agent exactly track the average of the multiple time-varying signals in finite time in the absence of velocity measurement. By carefully analyzing the dynamic properties of the tracking error system, a suitable Lyapunov function is constructed to estimate the settling time for convergence of the tracking error system theoretically. Furthermore, an adaptive DAT protocol is proposed, which is a fully distributed protocol because it can solve the DAT problem without using any global information. Finally, two simulation examples are provided to verify the effectiveness of the methods.

Distributed Average Tracking for Linear Heterogeneous Multi-Agent Systems With External Disturbances

In this paper, we address the distributed average tracking (DAT) problem for linear multi-agent systems with heterogeneous dynamics and external disturbances. The objective is that each agent can accurately track the desired tracking signal by designing controllers without initialization. The desired tracking signal means the average of individual time-varying reference inputs (one per agent). The current problem setting is technically more challenging and more practical than existing works. Based on the feedforward method, a state feedback controller is first designed to solve the DAT problem, which can guarantee the exponential convergence of tracking errors in the case where the desired tracking signal and external disturbances are known. Then the DAT problem for linear heterogeneous multi-agent systems with unknown desired tracking signal and unknown external disturbances is also solved. A distributed observer is designed for each agent to estimate the desired tracking signal. In parallel, the output of each agent is driven to track the output of the observer. Since each agent cannot access the desired tracking signal and external disturbances, a dynamic compensator independent of the exogenous signal and a dynamic internal-model based component are introduced to overcome these challenges. Finally, a numerical example is included for illustration.

Distributed Average Tracking Problem Under Directed Networks: A Distributed Estimator-Based Design

This article considers the distributed average tracking (DAT) problem that a group of agents aims to track the average of local reference signals. The communication network is strongly connected but not required to be weight balanced. A framework of distributed estimator-based algorithm design without any initialization requirement is presented. For agents with nominal first-order (FO) dynamics, a class of distributed estimators is proposed to estimate the average of local reference signals within fixed time, with which each agent finally achieves the DAT within fixed time. For agents with second-order dynamics subject to modeling uncertainty, unmodeled dynamics, as well as external disturbances, a class of distributed estimator is proposed to estimate the average of local reference signals within fixed time, and then a local disturbance-observer-based control protocol is proposed for each agent to achieve DAT asymptotically. Different from most of the existing DAT results under general directed networks that DAT error can only be made globally ultimately bounded, the exact DAT can be achieved within fixed time or asymptotically here. Further, the idea of distributed estimator is applied to solve a class of distributed optimization problems within finite time. Finally, two simulation examples are given to show the effectiveness of the theoretical results.

Design of Distributed Event-Triggered Average Tracking Algorithms for Homogeneous and Heterogeneous Multiagent Systems

This article addresses the design problem of distributed event-triggered average tracking (DETAT) algorithms for homogeneous and heterogeneous multiagent systems. The objective of the DETAT problem is to develop a group of distributed cooperative control algorithms with event-triggered strategies for agents to track the average of multiple time-varying reference signals. First, for homogeneous linear multiagent systems, based on sampling measurements and model-relied holding techniques, a class of static-gain DETAT algorithms is proposed with a couple of local event-triggered functions for estimators and controllers, respectively. Compared with the existing distributed average tracking (DAT) algorithms, the static-gain DETAT algorithms greatly reduce the cost over communication networks and the frequency of control protocol updates. Second, to reduce the chattering phenomenon caused by nonsmooth items in static-gain algorithms and requirements of the global information of networks, smooth dynamic-gain DETAT algorithms are introduced based on boundary layer approximation methods and self-adaptive principles. Third, for heterogeneous linear multiagent systems, a new algorithm is established by using the output regulation techniques for the heterogeneous DETAT problem. The outputs of heterogeneous agents can ultimately track the average of multiple time-varying reference signals. To the best of our knowledge, it is the first time to study the DETAT problem for heterogeneous multiagent systems. Finally, some examples are presented to show the validity of theoretical results.

Fast Distributed Average Tracking in Multiagent Networks: The Case With General Linear Agent Dynamics

Within the context of distributed average tracking (DAT) for multiagent networks (MANs), the convergence rate of DAT algorithms is a key evaluation index. Practically, it is generally required to construct DAT algorithms with a fast convergence rate for engineering MANs. Herein, the fast DAT problem for MANs in the presence of multiple time-varying reference signals generated by linear nonautonomous systems is investigated for the first time. Compared with most existing results on DAT for MANs with multiple time-varying external signals, the present system model is much more generic and the DAT can be guaranteed in finite time. In particular, by using combined tools from the Luenberger canonical form theory of linear multivariable systems and nonsmooth control theory, a new type of distributed observer is developed to estimate the average reference signal based on which a local tracking controller is skillfully designed and utilized. With the designed controller, the considered DAT can be achieved within a finite time depending on the initial states of agents. When the initial state of each individual agent is unavailable, a new type of fixed-time distributed observer for estimating the average reference signals and a local fixed-time tracking controller based on implicit Lyapunov functions are further proposed to solve the DAT problem in fixed time, which is independent of the initial states of agents. In addition, numerical simulations are performed to verify the validity of the designed DAT algorithms.

Distributed Average Tracking in Weight-Unbalanced Directed Networks

This article studies a distributed average tracking (DAT) problem, in which a collection of agents work collaboratively, subject to local communication, to track the average of a set of reference signals, each of which is available to a single agent. Our primary objective is to seek a design methodology for DAT under possibly weight-unbalanced directed networks-the most general and thus most challenging case from the network topology perspective, which has few results in the literature. For this purpose, we propose a distributed algorithm based on a chain of two integrators that are coupled with a distributed estimator. It is found that the convergence depends on not only the network topology but also the deviations among the reference signal accelerations. Another primary interest of this article stems from the dynamics perspective-a point perceived as a main source of control design difficulty for multiagent systems. Indeed, we devise a nonlinear algorithm that is capable of achieving DAT under weight-unbalanced directed networks for agents subject to high-order integrator dynamics. The results show that the convergence to the vicinity of the average of the reference signals is guaranteed as long as the signals' states and control inputs are all bounded. Both algorithms are robust to initialization errors, i.e., DAT is insured even if the agents are not correctly initialized, enabling the potential applications in a wider spectrum of application domains.

Design Fixed-Time Practical Distributed Average Tracking Algorithms for Nonlinear Signals With Bounded- and Lipschitz-Type Derivatives

In this brief, the fixed-time practical distributed average tracking (DAT) problem for multiple nonlinear signals is studied, where the nonlinear signals are with bounded- and Lipschitz-type derivatives respectively. Two fixed-time practical DAT algorithms are proposed for agents with local interactions to track the average of the multiple nonlinear signals within a fixed convergence time by using time-base generator techniques. Different from existing results, the proposed DAT algorithms in this brief are applicable to DAT problems for nonlinear signals with both bounded- and Lipschitz-type derivatives, which keeps more reality. Also, the proposed fixed-time practical DAT algorithms are able to provide an explicit estimation of the upper-bounded convergence time without dependence on initial conditions. The fixed convergence time in the proposed algorithms can be adjusted more flexibly. Finally, some illustrative examples are shown to manifest the validity of the fixed-time practical DAT algorithms.

Distributed Algorithm for Higher-Order Integrators to Track Average of Unbounded Signals

In this work, we design an algorithm for a group of higher-order integrators aiming to track the average of multiple time-varying and possibly unbounded reference signals. The existing literature has studied distributed average tracking (DAT) for higher-order systems in the presence of bounded or Lipschitz-type reference signals. In such DAT algorithms, each agent requires the knowledge of global bounds on signals for bounded references and state-dependent control gains for unbounded references. Addressing these issues, we propose a DAT algorithm for a group of higher-order integrators in the presence of time-varying references that can possibly be unbounded. The highest derivative of references become equal, asymptotically. Agents use neighbors’ data obtained from the local communication framework that makes the current algorithm distributed in nature. In contrast to existing work, our DAT algorithm uses constant gains to reduce high control effort, which may be caused due to state-dependent gains. Using numerical example, the performance of the current algorithm is compared with the existing state-of-the-art. This reveals the superiority of the proposed algorithm.

Finite-Time Distributed Average Tracking for Second-Order Nonlinear Systems.

This paper studies the distributed average tracking (DAT) problem for multiple reference signals described by the second-order nonlinear dynamical systems. Leveraging the state-dependent gain design and the adaptive control approaches, a couple of DAT algorithms are developed in this paper, which are named finite-time and adaptive-gain DAT algorithms. Based on the finite-time one, the states of the physical agents in this paper can track the average of the time-varying reference signals within a finite settling time. Furthermore, the finite settling time is also estimated by considering a well-designed Lyapunov function in this paper. Compared with asymptotical DAT algorithms, the proposed finite-time algorithm not only solve finite-time DAT problems but also ensure states of physical agents to achieve an accurate average of the multiple signals. Then, an adaptive-gain DAT algorithm is designed. Based on the adaptive-gain one, the DAT problem is solved without global information. Thus, it is fully distributed. Finally, numerical simulations show the effectiveness of the theoretical results.

Distributed Average Tracking in Multi-Agent Coordination: Extensions and Experiments

This paper addresses the distributed average tracking (DAT) problem for a group of agents to track the average of multiple time-varying reference signals, each of which is available to only one agent, under local interaction with neighbors and an undirected graph. We consider three cases: 1) DAT for single-integrator dynamics with a chain of integrators in algorithm design; 2) DAT with swarm behavior for single-integrator dynamics; and 3) DAT with swarm behavior for double-integrator dynamics. First, a continuous distributed algorithm with a chain of two integrators is proposed for single-integrator dynamics, where each agent needs its own and its neighbors’ filter outputs obtained through communication besides its absolute position, local relative positions, reference signal, and reference velocity. The introduced algorithm can deal with a wide class of reference signals with steady deviations among reference velocities. Then, two swarm-based DAT algorithms for, respectively, single- and double-integrator dynamics are presented. In the first swarm-based algorithm for single-integrator dynamics, each agent needs to measure the relative positions between itself and its neighbors, while in the second swarm-based algorithm for double-integrator dynamics, relative velocity information is required too. In both cases, the information can be obtained through local sensing. If correct initialization constraints can be achieved, the center of the agents will track the average of the reference signals and the agents will maintain connectivity and avoid interagent collision. Numerical simulations are also presented to illustrate the theoretical results. Finally, the algorithms presented in this paper are experimentally implemented and validated on a multirobot platform.

Distributed Average Tracking for Lipschitz-Type of Nonlinear Dynamical Systems.

In this paper, a distributed average tracking (DAT) problem is studied for Lipschitz-type of nonlinear dynamical systems. The objective is to design DAT algorithms for locally interactive agents to track the average of multiple reference signals. Here, in both dynamics of agents and reference signals, there is a nonlinear term satisfying a Lipschitz-type condition. Three types of DAT algorithms are designed. First, based on state-dependent-gain design principles, a robust DAT algorithm is developed for solving DAT problems without requiring the same initial condition. Second, by using a gain adaption scheme, an adaptive DAT algorithm is designed to remove the requirement that global information, such as the eigenvalue of the Laplacian and the Lipschitz constant, is known to all agents. Third, to reduce chattering and make the algorithms easier to implement, a couple of continuous DAT algorithms based on time-varying or time-invariant boundary layers are designed, respectively, as a continuous approximation of the aforementioned discontinuous DAT algorithms. Finally, some simulation examples are presented to verify the proposed DAT algorithms.

Distributed average tracking with input saturation

In this paper, the distributed average tracking (DAT) problem of a multi-agent system with input saturation is considered, whose objective is to make each agent track the average of a group of time-varying reference signals by local saturated inputs. This paper proposes two algorithms based on the techniques of low-gain feedback and nonsmooth feedback. The first algorithm considers reference signals without external inputs under the low-gain feedback scheme. We show that if the system matrix of the references is negative semi-definite, then the DAT error will be ultimately upper bounded. The other algorithm considers reference signals with bounded inputs, where the idea of nonsmooth feedback is employed. Finally, two simulation examples are presented to validate the theoretical results.

A Connection Between Dynamic Region-Following Formation Control and Distributed Average Tracking.

This paper studies the inherent connection between dynamic region-following formation control (DRFFC) and distributed average tracking (DAT). We propose a fixed-gain DAT algorithm with robustness to initialization errors for linear multiagent systems, which is capable of achieving DAT with a zero tracking error for a large class of reference signals. In the case that the fixed gain cannot be chosen properly, we present an adaptive control gain design, under which each agent simply chooses its own gain and the restriction on knowing the upper bounds on the reference signals and their inputs is removed. We show that the proposed DAT algorithms can be employed to solve the DRFFC problem. This is an attempt on the applications of DAT algorithms to achieve distributed control; existing works most use DAT as distributed estimation algorithms. For single-integrator, double-integrator, higher-order linear dynamics, we derive the corresponding DRFFC algorithms from the DAT algorithm. Compared with existing DRFFC algorithms, the DAT-based DRFFC algorithms do not require the desired region to have a regular shape and is capable of generating a much richer formation behavior. Numerical examples are also included to show the validity of the derived results.

Distributed Average Tracking for Reference Signals With Bounded Accelerations

In this technical note, the distributed average tracking (DAT) problem for reference signals with bounded accelerations is considered. A sliding-mode surface is designed such that trajectories on the surface will achieve DAT. A discontinuous control algorithm is then proposed to guarantee that the surface can be reached in finite time. It is shown that if the switching network topology is connected at all time, then DAT can be solved by properly choosing a gain parameter of the algorithm.

Distributed Average Tracking of Networked Euler-Lagrange Systems

Under an extended proportional-integral (PI) control scheme, a new class of distributed average tracking (DAT) algorithms has been developed for three different kinds of references: references with steady states, references with bounded derivatives, and references with a common derivative. The class of DAT algorithms are further applied to solve the DAT problem for Euler-Lagrange (EL) systems.