Semi-global leader-following output consensus for heterogeneous fractional-order multi-agent systems with input saturation via observer-based protocol

doi:10.1016/j.neucom.2020.03.028

Neurocomputing

Volume 402, 18 August 2020, Pages 298-306

https://doi.org/10.1016/j.neucom.2020.03.028 Get rights and content

Abstract

In this paper, semi-global leader-following output consensus problem for heterogeneous fractional-order multi-agent systems with input saturation and external disturbances is investigated. The case where the dynamics are integer-order can be viewed as a special case of fractional-order systems. First, a novel observer-based consensus algorithm is designed to solve the semi-global output consensus problem. It is worth noting that the consensus is semi-global since the set of initial states is bounded. Then, the low gain feedback technique is utilized based on the solution of a parametric algebraic Riccati equation. Moreover, sufficient conditions of output consensus for heterogeneous fractional-order multi-agent systems are derived by using algebraic graph theory, matrix theory and the stability theory of fractional-order systems. It can be proven that the output signals of followers can reach synchronization with the leader’s. Finally, numerical examples are provided to illustrate the effectiveness of our conclusions.

Introduction

Over several decades, distributed cooperative control problem of multi-agent systems (MASs) has already aroused the concern of a large number of scholars due to its wide applications [1], [2], [3], [4], [5], [6]. The consensus is a fundamental problem in cooperative control of MASs, which includes leaderless consensus and leader-following consensus (tracking consensus) [7], [8], [9], [10], [11]. For the latter, the leader generates the reference signal to be tracked by followers to achieve state or output consensus.

In practice, agents always have different dynamics due to the impacts of communication and external environment. Hence, it is necessary to research the consensus problem for heterogeneous MASs (HMASs). At present, some conclusions on output consensus for HMASs have been published in [12], [13], [14], [15], [16]. The output consensus problem for uncertain HMASs was addressed in [12], where an internal model was embed into the control law of each agent. In [13], event-triggered and self-triggered control algorithms were designed to address the output consensus problem for HMASs, and Zeno behavior can be excluded. The robust output regulation problem for nonlinear MASs was investigated based on the distributed observer in [14]. In [16], the output formation-containment problem for MASs was addressed, where each agent had different dimensions and dynamics.

In general, since there are the maximum and minimum limits in the vast majority of practical models, control input subject to the bounded saturation is extremely prevalent in practical applications. Therefore, it is important and meaningful to take into account input saturation and some results have been obtained in [17], [18], [19], [20]. In [17], semi-global consensus for MASs with input saturation was achieved, where suppose that each agent was asymptotically null controllable. The low-gain state feedback method was adopted to construct the consensus protocol, such that semi-global output consensus was reached [18]. The output consensus problem for HMASs with input saturation was dealt with based on the observer in [19]. Moreover, in [20], the consensus problem for MASs with input saturation was studied under a directed switching proximity topology.

On the other hand, the existing results in consensus problem primarily assume an integer-order dynamics. It is noted that many practical problems can be better described by fractional-order dynamics, which have been applied in the areas of science, engineering and biochemistry [21], [22]. In the field of systems and control, some works on fractional-order systems (FOSs) [23], [24], [25], [26] and fractional-order MASs (FOMASs) [27], [28], [29], [30], [31], [32], [33] have been done. The authors of [24] investigated the Mittag-Leffler Stability of nonlinear FOSs and extended the Lyapunov direct method to the case of FOSs, in which some crucial theorems were obtained for the future work. In [25], new lemmas were obtained for FOSs with 0 < α ≤ 1 to analyze the stability problem. Moreover, the authors of [27] adopted Lyapunov function method to guarantee leader-following consensus for nonlinear FOMASs with 0 < α ≤ 1. Containment control problem for FOMASs with time-varying delays was solved based on two consensus schemes, such that the sufficient and necessary condition was derived in [28]. In [29], group multiple lags leader-following consensus for nonlinear FOMASs was achieved, and parametric uncertainties were considered via an adaptive control protocol. Furthermore, algebraic conditions of leader-following consensus for nonlinear FOMASs with delays were obtained in [30], where the interaction topology is directed or undirected graph. The consensus problem for FOMASs was addressed by sampled-data control in [31]. However, most of the existing consensus results were for homogeneous FOMASs, which means agents with identical dynamics.

Motivated by aforementioned analysis, the study on heterogeneous FOMASs (HFO-MASs) subject to input saturation is of practical implications. In addition, due to various uncertainties, disturbances are also inevitable in the practical systems [34], [35]. Therefore, the semi-global output consensus problem for HFO-MASs with input saturation and external disturbances is investigated in this paper. The main contributions are summarized as follows.

(1) From the perspective of its content, it is the first time to investigate the semi-global leader-following output consensus problem for HFO-MASs with input saturation and external disturbances. Most investigators devoted themselves to investigate the consensus problem for MASs with input saturation or FMASs without input saturation. The research of this paper is meaningful and challenging with novelty.

(2) A novel distributed observer-based control scheme is designed by employing the low gain feedback technique based on the solution of a parametric algebraic Riccati equation, such that the semi-global output consensus for HFO-MASs is guaranteed. Moreover, a proper low-gain parameter can eliminate the saturation nonlinearity.

(3) Since the well-known Leibnitz rule is not applicable for FOSs, thus, fractional-order Lyapunov direct method are presented to overcome the difficulty of this paper.

The rest of this paper is organized as follows. In Section 2, preliminaries are presented. Output consensus problem for general HFO-MASs is solved via observer-based consensus protocol in Section 3. In Section 4, the semi-global leader-following output consensus for HFO-MASs with input saturation and external disturbances is reached by low gain feedback approach. Section 5 provides numerical examples, and conclusions are given in Section 6.

Section snippets

Notations and graph theory

Notations. Let $R^{m \times n}$ and $C$ be the sets of m × n real matrices and complex field, respectively. $Z^{+}$ is the set of positive integers. I_N represents the N-dimensional identity matrix. For a constant T ≥ 0 and $ς = {[ς_{1}, ς_{2}, \dots, ς_{p}]}^{T},$ ${∥ ς ∥}_{\infty} = \max | ς_{i} |$ and ${∥ ς ∥}_{\infty, T} = \sup_{t \geq T} | ς |$ . 1_N represents an N-dimensional column vector with all elements being 1. 0 represents the matrix with all elements being 0. ⊗ denotes the Kronecker product. sgn(x) is the sign function of x. For matrix Ω, $s p e c (Ω) = {λ | d e t (λ I - Ω) = 0, λ \in C}$ . For an

Problem statement

Consider a general HFO-MASs with one leader and N followers, where the dynamics of the leader and followers are described as follows, respectively, $D^{α} x_{0} = S x_{0},$ $y_{0} = D x_{0},$ $D^{α} x_{i} = A_{i} x_{i} + B_{i} u_{i} + Q_{i} x_{0},$ $y_{i} = C_{i} x_{i}, i \in \tilde{N}$

where 0 < α ≤ 1, $x_{i} \in R^{n_{i}}, u_{i} \in R^{p_{i}}, y_{i} \in R^{q}$ denote the state, input and output of agent i, respectively. $x_{0} \in R^{s}, y_{0} \in R^{q}$ represent the state and output of the leader, respectively. $A_{i} \in R^{n_{i} \times n_{i}},$ $B_{i} \in R^{n_{i} \times p_{i}},$ $C_{i} \in R^{q \times n_{i}},$ $S \in R^{s \times s},$ $D \in R^{q \times s}$ and $Q_{i} \in R^{n_{i} \times s}$ are constant matrices.

Definition 2

(Problem 1) Given HFO-MASs (1) and (2) with the

Semi-global output consensus for HFO-MASs with input saturation

In view of the ubiquity of input saturation, consensus problems for MASs with input saturation are of certain significance. Therefore, in this section, semi-global leader-following output consensus problem for HFO-MASs subject to input saturation is investigated.

Numerical examples

Consider a HFO-MASs with $α = 0.7$ consisting of one leader and four followers with the topology shown in Fig. 1.

Conclusions

In this paper, we investigated semi-global leader-following output consensus problem for HFO-MASs with input saturation and external disturbances under fixed topology. A novel distributed observer was constructed to estimate the leader’s state. Moreover, the observer-based consensus protocol was designed to achieve the semi-global output consensus by employing the low gain feedback method. Finally, examples were presented to illustrate the conclusions we proposed.

It is worth noting that

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (61627809, 61433004, 61621004), and Liaoning Revitalization Talents Program (XLYC1801005).

Zhiyun Gao received the B.S. degree in mathematics and applied mathematics from Changzhi University, Changzhi, China, in 2015, and the M.S. degree in basic mathematics from Northeastern University, Shenyang, China, in 2017, where she is currently pursuing the Ph.D. degree in control theory and control engineering. Her current research interests include descriptor multi-agent systems, fractional-order multi-agent systems, and event-triggered control.

References (40)

T. Feng et al.
Globally optimal distributed cooperative control for general linear multi-agent systems
Neurocomputing
(2016)
Z. Gao et al.
Leader-following consensus conditions for fractional-order descriptor uncertain multi-agent systems with 0 < α< 2 via output feedback control
J. Frankl. Inst.
(2020)
Z. Gao et al.
Guaranteed-performance consensus for descriptor nonlinear multi-agent systems based on distributed nonlinear consensus protocol
Neurocomputing
(2020)
T. Liu et al.
Cooperative robust output regulation for a class of nonlinear multi-agent systems subject to a nonlinear leader system
Automatica
(2019)
Y. Wang et al.
Output formation-containment of interacted heterogeneous linear systems by distributed hybrid active control
Automatica
(2018)
H. Geng et al.
Consensus of a heterogeneous multi-agent system with input saturation
Neurocomputing
(2015)
Y. Li et al.
Mittag–leffler stability of fractional order nonlinear dynamic systems
Automatica
(2009)
M.A. Duarte Mermoud et al.
Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems
Commun. Nonlinear Sci. Numer. Simul.
(2015)
M.S. Tavazoei et al.
A note on the stability of fractional order systems
Math. Comput. Simul.
(2009)
Z. Yu et al.
Leader-following consensus of fractional-order multi-agent systems under fixed topology
Neurocomputing
(2015)

H. Liu et al.

Containment control of fractional-order multi-agent systems with time-varying delays

J. Frankl. Inst.

(2019)

R. Yang et al.

Consensus analysis of fractional-order nonlinear multi-agent systems with distributed and input delays

Neurocomputing

(2019)

J. Chen et al.

Multiconsensus of fractional-order uncertain multi-agent systems

Neurocomputing

(2015)

Q. Ma et al.

Output consensus for heterogeneous multi-agent systems with linear dynamics

Appl. Math. Comput.

(2015)

Y. Yang et al.

Distributed adaptive output consensus control of a class of heterogeneous multi-agent systems under switching directed topologies

Inf. Sci. (Ny)

(2016)

Z. Liu et al.

Formation-containment control of multiple underactuated surface vessels with sampling communication via hierarchical sliding mode approach

ISA Trans.

(2019)

M. Ge et al.

Hierarchical controller-estimator for coordination of networked euler-lagrange systems

IEEE Trans. Cybern.

(2019)

Z. Gao et al.

Consensus conditions for higher-order descriptor multi-agent systems with communication time-delays

Transactions of the Institute of Measurement and Control

(2020)

R. Wang et al.

The small-signal stability analysis of the droop-controlled converter in electromagnetic timescale

IEEE Trans. Susta. Energy

(2019)

R. Wang et al.

Line impedance cooperative stability region identification method for grid-tied inverters under weak grids

IEEE Trans. Smart Grid

(2020)

Cited by (12)

Non-negative scaled edge-consensus of saturated networked systems via adaptive output-feedback control
2024, Neurocomputing
This paper investigates fully distributed scaled non-negative edge-consensus problems of networked systems with actuator saturation and unmeasurable internal states. By using the adaptive control method, output-feedback-based low-gain technique, and graph theory, a new adaptive algorithm is designed to obtain scaled edge-consensus conditions, under which the difficulties caused by the constraints on edge states and controllers are overcome. There are three interesting characteristics that any global information of networks is not used in this paper’s algorithm and result, including the number of vertexes and edges; the feasible solutions of the scaled edge-consensus conditions exist and can be easily obtained; the convergence rate of the controller is adjustable. Furthermore, the designed algorithm is expanded in the cases without actuator saturation. Finally, three examples are given to verify the theoretical results.
Output synchronization of multi-agent systems via reinforcement learning
2022, Neurocomputing
Citation Excerpt :
Researches on multi-agent systems (MAS) have received extensive attention due to the widespread applications in many fields [1–4].
In this paper, the measured input–output data sequences with pinning gain are proposed via the topology of multi-agent systems (MAS), where the requirement of reinforcement learning algorithm on internal state is avoided by pinning gain and the measured data of leader and neighbors. Besides, a data-based tracking state is given, which can be applied to MAS with different control matrices. According to the sequences and tracking state, a distributed control policy and corresponding reinforcement learning algorithm are proposed for the output synchronization. The proposed algorithm overcomes the shortcoming that previous algorithms can not be applied to MAS with different control matrices in the absence of model information and full-state vector. Finally, the effectiveness of proposed algorithm is verified by simulation examples.
Suboptimal linear quadratic tracking control for multi-agent systems
2022, Neurocomputing
Citation Excerpt :
Distributed control for multi-agent systems has received extensive attention due to its potential advantages and practical applications in recent decades, e.g., multi-vehicle systems [1], mobile sensor networks [2], formation control of spacecrafts [3] and intelligent transportation systems [4], and so on. There are two main types of multi-agent systems: leaderless systems [5,6] and leader–follower systems [7,8,31]. The leader–follower consensus is also called distributed tracking control or trajectory regulation.
This paper investigates the distributed linear quadratic tracking control problem for multi-agent leader–follower systems. Considering that only few followers can access the state information of the leader owing to the limited communication range, a novel distributed control law is designed to make followers convergent to the leader by introducing appropriate interconnections among the followers. Besides the tracking consensus of the leader–follower systems, the designed control law also enables the associated cost to be less than a given tolerance for any initial states of the leader and the followers. The implementation of the designed distributed control law can be executed by solving a single Riccati equation, requiring no global information of the communication topology. Two examples are finally provided to demonstrate the effectiveness of the proposed method.
Neural networks-based adaptive event-triggered consensus control for a class of multi-agent systems with communication faults
2022, Neurocomputing
Citation Excerpt :
In recent decades, the adaptive consensus issue of multi-agent systems (MASs) has been extensively studied by many researchers owing to its wide application in unmanned air, spacecraft formation flying, vehicle formations, etc [1–9].
This paper studies the adaptive distributed event-triggered fault-tolerant consensus problem for a class of multi-agent systems with time delays and external disturbance. The communication faults between the agent and its neighbours are considered, and a fault-tolerant control mechanism is designed to overcome the communication faults. Moreover, a distributed event-triggered mechanism is constructed based on the distributed errors, and hence the communication burden can be reduced. Combining the Lyapunov stability theorem with the Lyapunov-Krasovskii function, it verifies that the consensus tracking errors are bounded with the guaranteed security performance, and the time delays phenomenon can also be compensated. Finally, the effectiveness and utility of the proposed control method are fully validated by two simulation examples.
Time-varying output formation-containment control for homogeneous/heterogeneous descriptor fractional-order multi-agent systems
2021, Information Sciences
Citation Excerpt :
In [27], the event-triggered control strategy was constructed to deal with the exponential consensus problem of nonlinear fractional-order MASs (FOMASs). The semi-global output consensus of heterogeneous FOMASs with external disturbances and input saturation was achieved via a novel observer-based consensus protocol in [28]. The authors of [29] derived the necessary and sufficient conditions to solve synchronization and consensus problems of FOMASs by sampled-data control.
In this paper, the time-varying output formation-containment (TV-OFC) problem of homogeneous and heterogeneous descriptor fractional-order multi-agent systems (DFO-MASs) is investigated. The results obtained in previous works for nonsingular integer-order multi-agent systems can be regarded as particular cases of our results. This control aims at making leaders keep time-varying shape to move and followers move inside the convex hull spanned by leaders simultaneously. First, both the output feedback control strategy and observer-based control algorithm are designed to address the TV-OFC problem of heterogeneous DFO-MASs, where the virtual leader provides the reference trajectory for leaders to achieve the formation tracking. Then, sufficient conditions with less computation complexity are presented and the methods to determine gain matrices of the schemes are proposed for homogeneous and heterogeneous DFO-MASs, so as to achieve the TV-OFC. Finally, numerical examples are given to demonstrate the effectiveness of theoretical conclusions.
Leader-following Non-fragile Consensus Control of Fuzzy Multi-agent Fractional Order Interval Systems
2024, International Journal of Control, Automation and Systems

View all citing articles on Scopus

Huaguang Zhang received the B.S. degree and the M.S. degree in control engineering from Northeast Dianli University of China, Jilin City, China, in 1982 and 1985, respectively. He received the Ph.D. degree in thermal power engineering and automation from Southeast University, Nanjing, China, in 1991. He joined the Department of Automatic Control, Northeastern University, Shenyang, China, in 1992, as a Postdoctoral Fellow for two years. Since 1994, he has been a Professor and Head of the Institute of Electric Automation, School of Information Science and Engineering, Northeastern University, Shenyang, China. His main research interests are fuzzy control, stochastic system control, neural networks based control, nonlinear control, and their applications. He has authored and coauthored over 280 journal and conference papers, six monographs and co-invented 90 patents. Dr. Zhang is the fellow of IEEE, the E-letter Chair of IEEE CIS Society, the former Chair of the Adaptive Dynamic Programming & Reinforcement Learning Technical Committee on IEEE Computational Intelligence Society. He is an Associate Editor of AUTOMATICA, IEEE TRANSACTIONS ON NEURAL NETWORKS, IEEE TRANSACTIONS ON CYBERNETICS, and NEUROCOMPUTING, respectively. He was an Associate Editor of IEEE TRANSACTIONS ON FUZZY SYSTEMS (2008–2013). He was awarded the Outstanding Youth Science Foundation Award from the National Natural Science Foundation Committee of China in 2003. He was named the Cheung Kong Scholar by the Education Ministry of China in 2005. He is a recipient of the IEEE Transactions on Neural Networks 2012 Outstanding Paper Award.

Juan Zhang received the B.S. degree in Information and Computing Science from Shanxi Normal University, Linfen, China, in 2016, and the M.S. degree in basic mathematics from Northeastern University, Shenyang, China, in 2018, where she is currently pursuing the Ph.D. degree in control theory and control engineering. Her current research interests include containment control, consensus problem, output regulation problem, and formation control for multi-agent systems etc.

Shaoxin Sun received the B.S. degree in control technology and instrument from Hebei University, Baoding, China, in 2014 and the M.S. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2016. She is currently pursuing the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China. Her current research interests include fuzzy systems, time-delay systems, fault estimation, fault tolerant control, and stochastic/random systems.

View full text

Semi-global leader-following output consensus for heterogeneous fractional-order multi-agent systems with input saturation via observer-based protocol

Abstract

Introduction

Section snippets

Notations and graph theory

Problem statement

Semi-global output consensus for HFO-MASs with input saturation

Numerical examples

Conclusions

Declaration of Competing Interest

Acknowledgment

Neurocomputing

J. Frankl. Inst.

Neurocomputing

Automatica

Automatica

Neurocomputing

Automatica

Commun. Nonlinear Sci. Numer. Simul.

Math. Comput. Simul.

Neurocomputing

J. Frankl. Inst.

Neurocomputing

Neurocomputing

Appl. Math. Comput.

Inf. Sci. (Ny)

Formation-containment control of multiple underactuated surface vessels with sampling communication via hierarchical sliding mode approach

ISA Trans.

Hierarchical controller-estimator for coordination of networked euler-lagrange systems

IEEE Trans. Cybern.

Consensus conditions for higher-order descriptor multi-agent systems with communication time-delays

Transactions of the Institute of Measurement and Control

The small-signal stability analysis of the droop-controlled converter in electromagnetic timescale

IEEE Trans. Susta. Energy

Line impedance cooperative stability region identification method for grid-tied inverters under weak grids

IEEE Trans. Smart Grid