Distributed model reference adaptive containment control of heterogeneous multi-agent systems with unknown uncertainties and directed topologies

https://doi.org/10.1016/j.jfranklin.2020.11.005Get rights and content

Abstract

In this paper, the containment control problem of heterogeneous uncertain high-order linear Multi-Agent Systems (MASs) is addressed and solved via a novel fully-Distributed Model Reference Adaptive Control (DMRAC) approach, where each follower computes its adaptive control action on the basis of local measurements, information shared with neighbors (within the communication range) and the matching errors w.r.t. its own reference model, without requiring any previous knowledge of the global directed communication topology structure. The approach inherits the robustness of the direct model reference adaptive control (MRAC) scheme and allows all agents converging towards the convex hull spanned by leaders while fulfilling at the same time local additional performance requirements at single-agent level, such as prescribed settling time, overshoot, etc. The asymptotic stability of the whole closed-loop network is analytically derived by exploiting the Lyapunov theory and the Barbalat lemma, hence proving that each follower converges to the convex hull spanned by the leaders, as well as the boundedness of the adaptive gains. Extensive numerical analysis for heterogeneous MAS composed of stable, unstable and oscillating agent dynamics are presented to validate the theoretical framework and to confirm the effectiveness of the proposed approach.

Introduction

Distributed cooperative control of Multi-Agent Systems (MASs) has received considerable attention during the last decades due to its application in a variety of research fields, as physics, economic sciences, biology and engineering [1]. Most notably, the framework allows easily dealing with distributed cyber-physical systems overcoming some of the well-known limits of centralized control architectures with respect to the spatial distribution of devices, short-range of communication, etc. [2]. Indeed, in engineering applications, cyber-physical systems can be intuitively represented as a MAS composed of different dynamical agents that aim at reaching a prescribed common behavior, leveraging local information as well as the one shared with their neighbors (within their communication range) [3]. This gives rise to a variety of consensus problems (such as synchronization, flocking, etc.) that are commonly categorized into two main groups [4]: consensus with and consensus without a leader (the first is often also referred as leader-follower consensus or distributed tracking in the technical literature [5], [6]).

However, in some practical applications, such as earth monitoring, satellite formation, coordinate motion of mobile robots or autonomous vehicles, there is more than one leader (see [7] and references therein for an overview). The MAS literature addresses this issue as a containment problem in the case when all agents within the MAS must converge within the convex hull spanned by the leaders [8], [9] and as a formation-containment problem in the case when states/outputs must track the convex combination of multiple leaders while reaching at the same time a prescribed configuration (that can be static or time-varying) [10]. Since the formation-containment has a wider aim, its solution needs more complex control protocols with respect to containment ones, that leverage specific control actions for guaranteeing the further achievement of the desired prescribed configuration (see e.g. [10], [11] and the references therein). Furthermore, with an appropriate choice of the formation vector, it is possible to recast a formation-containment problem into a containment one, while the inverse is not always possible [11].

Focusing on the only containment problem, that is the objective of our work, the technical results often refer to MASs composed of single or double integrator dynamics (see for example [12], [13] and [14], [15], respectively). Fewer attempts have been devoted to generic linear and nonlinear dynamics. Here, the linear case has been addressed leveraging a wide range of control protocols (e.g., see [16] for an overview of the literature), while robust H approaches are mainly proposed for nonlinear homogeneous agents. For example in [7], where the problem is addressed for second-order homogeneous nonlinear Lipschitz agents or in [17], [18], where robust output regulation schemes are designed for dealing with uncertain homogeneous dynamics.

However, in many real applications, agents can not be assumed as identical and, hence, the challenging case of heterogeneous MAS has been also considered in the recent literature [19]. For example in [20], [21], where the containment problem is solved via distributed H for agents dynamics subject to both unknown external disturbances and local measurements noises, or in [22], where the output containment has been addressed via the internal model principle approach. Recently, the class of singular, or descriptor, MAS has been also investigated and both distributed state and output feedback actions are proposed (e.g., see [23] and references therein), while the presence of switching topologies has been dealt in [24]. More recently, the problem of output containment control problem for heterogeneous agents has been also solved via PID-like control strategy in [19].

Due to their ability to on-line adapt the control actions, distributed adaptive protocols have been also proposed for solving the containment control problem (e.g., see [9], [25], [26], [27] and references therein) and for designing state observers in the specific case of the output containment with external disturbances [28], [29]. The output containment problem has been also solved without leveraging state estimation via adaptive strategies, as in [30] and in [31] where the dynamics of the leaders are assumed to be perfectly known only for a subset of the followers, and in [32] where the special case of networked Euler–Lagrange systems has been considered.

Within the adaptive framework, in this paper we propose a fully-Distributed Model Reference Adaptive Control (DMRAC) able to solve the containment control for uncertain linear heterogeneous MAS with underlying directed topologies containing a united directed spanning tree [33]. It is worth to note here that only few works tailor the model-reference control architecture to the distributed context and, usually, these attempts have been proposed for the leader-tracking or synchronization control problem, instead of containment. For example, distributed model-reference leader-tracking schemes, leveraging the idea that the leader dynamics act as the reference model, have been proposed for linear heterogeneous MAS under the restrictive assumption of undirected topologies [34], [35], and then extended to nonlinear agents in [36], while a virtual model reference control approach has been recently exploited in [37] for achieving synchronization. However, the latter two aforementioned proposed protocols need for a nonlinear compensator and a distributed observer, respectively, while the adaptive control actions are updated by evaluating the errors w.r.t. the identical leaders that provide the reference dynamical behavior. The specific problem of distributed model reference adaptive containment has been, instead, very recently addressed for uncertain MAS in [38], again by considering the homogeneous leaders as the reference model to be tracked and under the restrictive assumption of undirected topologies.

Conversely, here we propose a fully-distributed MRAC approach, inspired to the classical MRAC scheme proposed by Landau [39], that solves the containment control problem for heterogeneous MAS with directed communication topology and uncertain dynamics. In the distributed control architecture we consider that each follower, defined by its own dynamics, has its own proper stable reference model (that has not to coincide with any leader dynamics) with respect to which locally adapts its control action exploiting neighbors information. In so doing, the desired prescribed performances of the closed-loop network, such as, for example, stabilization of unstable follower dynamics, damping of oscillating behaviors or selection of a desired settling time, are described in terms of the local stable reference model implemented at single agent level, that hence provides to each follower the desired response to the command signal. Namely, each agent computes its control action on the basis of local measurements, information shared with neighbors (within the communication range) and the matching errors w.r.t. its reference model, without requiring any previous knowledge of the global communication topology structure, such as the smallest real part of the nonzero eigenvalues of the Laplacian matrix [9], [40]. It follows that the approach is not restricted to the all-to-all topology (broadcast communication) and no knowledge of the entire communication graph is necessary neither for designing or deploying the control action at each single agent level. Furthermore, here, each follower reference model is not limited to be Hurwitz, but it can be also neutrally stable, as commonly required in the MAS technical literature (see e.g. [37] and references therein).

The approach combines the main features of the synchronized reference generators methodology [23], [29], for dealing with agents heterogeneity exploiting information shared via the directed communication network, with the robustness of well-known direct MRAC scheme. Indeed, any mismatch among the reference model and the follower behaviors is compensated by the adaptive action, which will also provide robustness against the presence of integral bounded and time-varying disturbances, variable working conditions, unstructured unmodeled dynamics and external noise [41]. Moreover, the regulator equations, that are the typical conditions to be fulfilled for dealing with the agents heterogeneity [42], holds only on the nominal part of the follower dynamics and, hence, uncertainties are assumed as completely unknown. Furthermore, note that the synchronized references generator is designed for each agent within the MAS, not for observing the followers states, but just for providing an estimation of the convex combination of the leaders reference behaviour [23], [27], [29], [42]. In so doing, each agent locally computes its synchronized references generator based on the information about the synchronized references generator coming from neighboring agents and about the neighboring leader/s behaviour within its communication range.

The stability of the closed-loop network is analytically proven leveraging Lyapunov theory and the Barbalat Lemma and the mathematical derivation clearly discloses that all followers converge towards the convex hull spanned by the leaders. Accordingly, also the reference matching errors converge to zero, while the dynamics of the adaptive gains remain bounded.

Summarizing, the main contributions are:

  • A novel Distributed Model Reference Adaptive Control strategy is proposed to solve the containment control problem for uncertain heterogeneous linear Multi-Agent Systems sharing information via a directed communication topology. The protocol, based on short range information from neighboring agents, needs no global information of the communication topology (e.g., the smallest positive eigenvalue of Laplacian matrix) and it is thus fully distributed [43], [44], [45].

  • Since the distributed scheme inherits the advantages of the adaptive model reference direct scheme, it robustly drives and maintains the agents dynamics within the convex hull spanned by the leaders imposing at the same time some local additional performance requirements at single-agent level (such as prescribed settling time, overshoot, and so on) depending on the specific choice made for the reference model of each follower. In so doing, the containment control problem may accomplish some additional desired closed-loop performances.

  • The control architecture easily allows the embedding of additional observers that can be needed in those cases when the leader dynamics are uncertain and/or its states are not fully measurable. Indeed, since the leaders are not acting as reference model, those eventual observers are completely external to the feedback control-loop at the agent level and so they can be easily added without any significant modification for the proposed control protocol. Moreover, differently from other approaches, such as the ones proposed in [46], [47], [48], each synchronized references generator, only leveraging neighboring information, does not require a local p-copy of the leader dynamic model and this results in a reduced control computational burden at single agent level.

Finally, the paper is structured as follows. In Section 2, the mathematical preliminaries are given; in Section 3 the problem statement and the closed-loop MAS are defined, while the adaptive control approach and its stability analysis are presented in Section 4 and 5 respectively. An exhaustive simulation analysis is reported in Section 6 in order to disclose the effectiveness of the proposed approach. Finally, conclusions are drawn in Section 7.

Section snippets

Math preliminaries

Graph theory is used to model the interaction among the agents and the leaders in a MAS. Considering N agents, a corresponding graph direct or undirected G is defined by a set of nodes V=1,2,,N and a set of edges εV×V. An edge (i,j)ε indicates that agent j has access to information of agent i and the agent i is defined as a neighbor of j, but not viceversa. If (i,j)ε implies that (j,i)ε, the graph is called undirected. A directed path from node i to node j is a sequence of nodes that

Problem statement

Consider a heterogeneous MAS composed of N follower and M leaders. The dynamics of the uncertain i-th agent are described by the following high-order linear system:x˙i(t)=Aixi(t)+Biui(t)+Eidi(t),i=1,,N,where xi(t)Rn×1 and ui(t)Rq×1 are the state and the control input of the i-th agent, respectively; di(t)Rn×1 represents the i-th agent uncertainty (due to unstructured unmodeled dynamics and external noises [49]) that is assumed to be integral bounded, i.e. di(t)L2; AiRn×n, BiRn×q has full

Distributed MRAC protocol for containment

To solve the containment control problem, as defined in Problem 1, the following distributed adaptive state-feedback protocol is proposed for the i-th follower (i=1,,N):ui(t)=Ki(t)xi(t)+Hi(t)ηi(t),where Ki(t) and Hi(t)Rq×n are gains matrices that have to be updated according to the classical MRAC approach [39], being Ki(t) the gain of the control action depending on only local measurements, while Hi(t) is the gain of the networked control action depending on information shared via the

Design of the adaptive law and stability analysis of the closed-loop network

In what follows the design of the adaptive law, as well as the conditions to solve the containment control problem, are established according to the following theorem.

Theorem 1

Consider a MAS composed of M leaders and N followers with heterogeneous linear dynamics as in Eqs. (2) and (1), respectively. Assume the adaptive protocol in Eq. (7), with synchronized references generator as in Eq. (8), such that for each follower (i=1,,N) the adaptive control matrices Ki(t) and Hi(t)=Ki(t)+Ui fulfill the

Numerical validation

In this section, the effectiveness of the proposed approach is validated considering the exemplar case of an heterogeneous MAS composed of three leaders and six uncertain followers sharing information over the directed communication network depicted in Fig. 1, whose Laplacian matrix takes the form of Eq. (3) and is defined as in Eq. (38).L=[L1L203×603×3]=[310100100120000010002101000100310001001021000010001000000000000000000000000000000].Note that, according to Lemma 1, the matrix L1

Conclusions

In this work the containment control problem for uncertain heterogeneous MASs is addressed and solved via a novel fully-Distributed Model Reference Adaptive Control, where each agent is characterized, besides its own dynamics, by a stable reference model with respect to adapt its control action exploiting the leaders and/or state information obtained via a directed communication network topology. More in detail, leveraging the classical approach based on the regulator equations to deal with the

Declaration of Competing Interest

None.

References (58)

  • A. Petrillo et al.

    Adaptive synchronization of linear multi-agent systems with time-varying multiple delays

    J. Frankl. Inst.

    (2017)
  • I. Landau

    A survey of model reference adaptive techniques—theory and applications

    Automatica

    (1974)
  • Q. Jiao et al.

    Distributed l2-gain output-feedback control of homogeneous and heterogeneous systems

    Automatica

    (2016)
  • Y. Su et al.

    Cooperative adaptive output regulation for a class of nonlinear uncertain multi-agent systems with unknown leader

    Systems & Control Letters

    (2013)
  • X. Zhang et al.

    Distributed containment control of singular heterogeneous multi-agent systems

    J. Frankl. Inst.

    (2020)
  • H. Haghshenas et al.

    Containment control of heterogeneous linear multi-agent systems

    Automatica

    (2015)
  • X. Li et al.

    Output-feedback protocols without controller interaction for consensus of homogeneous multi-agent systems: a unified robust control view

    Automatica

    (2017)
  • F.L. Lewis et al.

    Cooperative Control of Multi-agent Systems: Optimal and Adaptive Design Approaches

    (2013)
  • S.H. Strogatz

    Exploring complex networks

    Nature

    (2001)
  • A. Petrillo et al.

    A secure adaptive control for cooperative driving of autonomous connected vehicles in the presence of heterogeneous communication delays and cyberattacks

    IEEE Trans. Cybern.

    (2020)
  • Z. Li et al.

    Cooperative control of multi-agent systems: a consensus region approach

    (2014)
  • G. Fiengo et al.

    Distributed robust PID control for leader tracking in uncertain connected ground vehicles with V2V communication delay

    IEEE/ASME Trans. Mechatron.

    (2019)
  • G. Fiengo et al.

    Distributed leader-tracking for autonomous connected vehicles in presence of input time-varying delay

    Proceedings of the 26th Mediterranean Conference on Control and Automation (MED)

    (2018)
  • P. Cheridito

    Convex Analysis

    Lecture Notes (Princeton University, Princeton, NJ)

    (2013)
  • Z. Li et al.

    Designing fully distributed consensus protocols for linear multi-agent systems with directed graphs

    IEEE Trans. Autom. Control

    (2015)
  • S. Zuo et al.

    Time-varying output formation containment of general linear homogeneous and heterogeneous multiagent systems

    IEEE Trans. Control Netw. Syst.

    (2018)
  • J. Qin et al.

    Recent advances in consensus of multi-agent systems: a brief survey

    IEEE Trans. Ind. Electron.

    (2017)
  • Z. Li et al.

    Containment control of a class of heterogeneous nonlinear multi-agent systems

    Int. J. Control

    (2017)
  • C. Sun et al.

    Robust output containment control of multi-agent systems with unknown heterogeneous nonlinear uncertainties in directed networks

    Int. J. Syst. Sci.

    (2017)
  • Cited by (22)

    • Exponential bipartite tracking consensus in cooperative-antagonistic nonlinear Multi-Agent Systems with multiple communication time-varying delays

      2022, IFAC Journal of Systems and Control
      Citation Excerpt :

      During the past decades, consensus problems among a group of agents received remarkable attention due to the wide range of applications in various research fields, such as intelligent transportation systems (Caiazzo, Lui, Petrillo, & Santini, 2021a), sensor networks (Adachi, Yamashita, & Kobayashi, 2020; Zheng, Liu, & Liu, 2018), power systems (Andreotti, Caiazzo, Petrillo, & Santini, 2021; Wang, Zhou, & Duan, 2018) and so on. Generally, consensus problems are based on the concept of collaboration, i.e. agents collaborate with each other to perform a specific task by exchanging information via a wireless communication networks, which are modelled as unsigned graphs with non-negative edges weights (see for example (Lui, Petrillo, & Santini, 2021; Ong & Hou, 2021) and the references therein). However, in several real-world systems such as social network and opinion scenarios, it is reasonable to assume the co-presence of collaboration and competition among agents (Xia, Cao, & Johansson, 2015).

    View all citing articles on Scopus
    View full text