Leader-following almost output consensus for linear multi-agent systems with disturbance-affected unstable zero dynamics☆
Introduction
For over a decade, the design of distributed consensus protocols for multi-agent systems has been one of the fundamental topics in distributed control (see, for example [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]). These consensus protocols are designed to achieve either leaderless consensus, where the states (outputs) of all agents converge to the same value, or leader following consensus, where the states (outputs) of all follower agents converge to the state (output) of the leader agent.
In the leader-following output consensus scenario, most of the consensus protocols designs have implicitly assumed that the agent dynamics to be minimum phase systems and not are affected by disturbances. However, many real world systems are of nonminimum phase [17]. For example, the inverted pendulum on a cart [18], and the V/STOL aircraft [19] are both nonminimum phase systems. Such nonminimum phase property of a system may prevent certain control objectives from being achieved especially if the unstable zero dynamics is affected by disturbances. Therefore, we are motivated to consider the consensus problem of multi-agent systems whose agent dynamics is of nonminimum phase and affected by the disturbance, which, to the best of our knowledge, is an open research problem.
In this paper we design consensus protocols for a class of linear multi-agent systems with disturbance-affected unstable zero dynamics. Our approach is motivated by the results on the disturbance decoupling problem for individual systems (see, for example [20], [21], [22], [23], [24]). The zero dynamics is allowed to be heterogeneous with all the poles on the -axis. The condition on the way the disturbance affects the zero dynamics of each follower agent is identified. The leader agent’s output to be followed can be any bounded signal that does not contain the frequency components of the -axis invariant zeros of the follower agents. Novel consensus protocols are constructed of a low-and-high gain feedback structure in which the low gain feedback design technique [24] is utilized to stabilize the zero dynamics of each follower agent by allowing its output to vary within a small neighborhood of the desired output. We show that, when the communication topology among the follower agents is undirected and connected, and there is a communication link between the leader agent and at least one follower agent, these protocols achieve leader-following almost output consensus, that is, the leader-following output consensus is achieved to any pre-specified degree of accuracy while the states remain bounded in the absence of the disturbances, and when the system is operating in output consensus within the desired level of accuracy, the -gain from the disturbances to the difference between each follower agent’s output with and without the disturbances from the same initial condition is attenuated to any desired level of accuracy. The protocols can be easily generalized for agents with invariant zeros on the closed left-half plane by a nonsingular transformation (see, for example, [25]) that decompose the zero dynamics into two subsystems, one with stable poles and the other with poles on the -axis. We note that, compared to the results on individual systems, where the output converges toward zero precisely in the absence of the disturbances, the outputs of the multiple agents under our proposed protocols reach consensus to a time-varying desired signal only with a pre-specified accuracy. In order to analyze the effect of the disturbance on the output consensus of the system, we have to compare the output in the absence and in the presence of the disturbance.
Organization of the paper: Section 2 recalls preliminaries in graph theory. Section 3 formulates the leader-following almost output consensus problem for linear multi-agent systems with disturbances-affected unstable zero dynamics. Section 4 presents the consensus protocols as well as the analysis of the closed-loop system. Section 5 contains simulation results that verify the theoretical conclusions. Section 6 concludes the paper.
Let () denote the () dimensional Euclidean space. For , denotes its norm. Let denote and denote of appropriate dimensions. Let () denote the identity matrix of dimensions (appropriate dimensions). For a matrix , denotes its image, denotes its norm and denotes its eigenvalues. For a symmetric matrix , indicates that the matrix is positive definite (positive semidefinite) and denotes its largest (smallest) eigenvalue. The cardinality of a finite set is denoted as .
Section snippets
Preliminaries
Consider a leader-following multi-agent system with one leader agent, labeled as , and follower agents, labeled as .
The communication topology among the follower agents is described by an undirected graph . Each follower agent is represented by a vertex from the set of vertices . The communication link between follower agents and is represented by an edge from the set of edges .
The adjacency matrix of the follower agents is defined as
Problem statement
We consider a group of follower agents and one leader agent. The dynamics of follower agents is described by , where and are the states, is the control input, is the output and is the disturbance. Let . Assume that is controllable with all eigenvalues of on the closed
Main results
In this section, the consensus protocols are designed in three steps. Then we show that these protocols solve Problem 1.
We first recall the following lemmas.
Lemma 1 Given in the form of (2) with all eigenvalues of on the -axis, let be the unique matrix such that for some . In addition, Let Then, there exists , such that, for each , [25]
A numerical example
We consider a group of five agents, including one leader agent, labeled as , and four follower agents. The follower agent dynamics is described by
The desired output is generated by the leader agent, whose dynamics is described by with .
The underlying communication topology is described by and as
The distributed consensus
Conclusions
In this paper, we studied the leader-following almost output consensus problem of linear multi-agent systems with disturbance-affected unstable zero dynamics. Under some conditions on the agent dynamics and the way the disturbances affect the zero dynamics, we constructed low-and-high gain based consensus protocols for the follower agents. These conditions are the same as those necessary for achieving almost disturbance decoupling for individual systems and are thus mild. The protocols we
CRediT authorship contribution statement
Tingyang Meng: Conceptualization, Methodology, Simulation, Writing. Zongli Lin: Conceptualization, Methodology, Writing.
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.
References (25)
- et al.
Consensus condition for linear multi-agent systems over randomly switching topologies
Automatica
(2013) - et al.
Global consensus for discrete-time multi-agent systems with input saturation constraints
Automatica
(2014) Robust consensus tracking of a class of second-order multi-agent dynamic systems
Systems Control Lett.
(2012)- et al.
Global optimal consensus for discrete-time multi-agent systems with bounded controls
Automatica
(2018) - et al.
Nonlinear control design for slightly non-minimum phase systems: application to v/stol aircraft
Automatica
(1992) A note on almost disturbance decoupling for nonlinear minimum phase systems
Systems Control Lett.
(1996)Global almost disturbance decoupling with stability for non minimum-phase single-input single-output nonlinear systems
Systems Control Lett.
(1996)Almost disturbance decoupling with global asymptotic stability for nonlinear systems with disturbance-affected unstable zero dynamics
Systems Control Lett.
(1998)- et al.
Further results on almost disturbance decoupling with global asymptotic stability for nonlinear systems
Automatica
(1999) - et al.
Consensus problems in networks of agents with switching topology and time-delays
IEEE Trans. Automat. Control
(2004)
Consensus and cooperation in networked multi-agent systems
Proc. IEEE
Consensus seeking in multiagent systems under dynamically changing interaction topologies
IEEE Trans. Automat. Control
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Work was supported in part by the US Army Research Office under Grant W911NF-17-1-0535.