A leader-following consensus problem of multi-agent systems in heterogeneous networks

Abstract

Within a differential–algebraic framework, this paper studies the leader-following consensus problem for networks of a class of strictly different nonlinear systems. In this case, by allowing any type of interplay between followers and assuming a directed spanning tree in the network, a dynamic consensus protocol with diffusive coupling terms is designed to achieve the leader-following consensus. In this setting, it is revealed that the full closed loop network can be interpreted as an input-to-state convergent system. Moreover, with the premise that differential–algebraic techniques allow us to completely characterize its synchronization manifold, a stability analysis for this manifold is presented. Finally, the effectiveness of the approach is shown in two numerical examples that consider a network of chaotic systems.

Introduction

Consensus and synchronization are closely related problems of multi-agent systems, as described in Wieland, Sepulchre, and Allgöwer (2011), on a complex relationship between the characteristics of the network: types of systems, coupling and topology involved. Yet another general description for these problems can be given in terms of the synchronization manifold. If it exists, the behavior of the whole network can be described in terms of this set, whose elements are the variables of interest subject to some constraints due to the characteristics of the network.

As seen in nature, common dynamical behavior is expected within groups of interacting units, i.e., coupled oscillators (Pikovsky, Rosenblum, & Kurths, 2001) or multi-agent systems (Olfati-Saber, Fax, & Murray, 2007). When this correlated motion occurs, synchronization is achieved for all nodes in the network. It is currently known that it could happen not only for groups of integrator nodes, limit cycle oscillators but in the case of chaotic systems as well (Pecora & Carroll, 1990). Thus synchronization manifold can be defined as the stable synchronous solution, that results from the differences between the dynamics of all individual units and their interactions within the network. Roughly speaking, for networks of identical units (homogeneous networks) corresponds a trivial synchronization manifold, this is given as the equality of the states; otherwise, for heterogeneous networks, it does not necessarily exist. As stated, the definition of synchronization manifold is a key element to describe the synchronous behavior of the network as a whole. Its stability determines if a stable synchronous behavior can be achieved (Kocarev & Parlitz, 1996).

In networks of homogeneous linear dynamics, it is well understood (Ren and Beard, 2008, Scardovi and Sepulchre, 2009) that dynamical controllers lead all diffusively coupled linear systems to synchronize to an open loop linear system, i.e., a trivial exponential stable synchronization manifold is the synchronous solution of the network. However, in the problems of networks of (nearly) identical units (Dörfler and Bullo, 2014, Strogatz, 2000, Zhang et al., 2017), the control theory community has tried to solve the problems of synchronization of heterogeneous networks with the appropriate design of dynamic consensus protocols (Cruz-Ancona et al., 2017, Isidori et al., 2014, Panteley and Loría, 2017, Wieland et al., 2011). Recently, a great deal of attention has been given towards the study of the latter. Some of those efforts are given in terms of stability in a practical sense (Panteley & Loría, 2017), where it has been found that the behavior of linearly diffusively coupled units can be synchronized to an emerging motion (an estimate of the synchronization manifold) that depends only on the structure of the network. Other efforts consider the existence of a common virtual dynamical behavior, that could be imposed as the case of the exosystem (Isidori et al., 2014) or as the result from the characteristics of the network (endosystem) (Wieland et al., 2011) for all units in the network, such that an appropriate dynamical controller is capable of forcing all the units to behave in the same manner as the virtual system. The synchronous solution in this sense is given in the form of a nonlinear output regulator (Isidori et al., 2014) or an internal model principle (Wieland & Allgöwer, 2009), respectively.

Hence, the problems just described are very similar to those in consensus literature, it can be seen that the case of the endosystem or emergent dynamics correspond to a generalization of classical leaderless consensus problems and the case of the exosystem to the leader-following consensus problem. It is worth mentioning that the way of studying those kinds of problems depends on the characterization of the collective behavior between interactive units, i.e., the stability of its synchronization manifold.

Lately, interesting results on the leader following consensus problem have been motivated from an output regulation perspective (Liu and Huang, 2018, Su and Huang, 2012) and the robustification of consensus protocols in order to compensate communication link failures and actuator perturbations (Cai and Huang, 2016, Das and Lewis, 2010, Liu and Huang, 2017, Shen et al., 2019, Shi and Shen, 2017, Wang et al., 2018, Wu et al., 2017, Yu and Xia, 2012). Even though, the aforementioned efforts consider the cases where the designed control is able to deal with the perturbations in networks of nearly identical dynamic agents, the complete characterization of the synchronization manifold is still an open problem. This paper addresses the latter while considering a fixed communication topology and ideal nonlinear and nonidentical dynamics of the agents of the network.

In a differential–algebraic setting, this paper studies a leader-following consensus problem of multi-agent systems for strictly different nonlinear units (allowing any type of interplay between followers, cf. Cruz-Ancona et al. (2017) for a regime with no interacting followers). Different from existing approaches (Isidori et al., 2014, Panteley and Loría, 2017), the following issues are addressed:

• The exact characterization of an asymptotically stable algebraic synchronization manifold for completely heterogeneous networks in a directed spanning tree. This is obtained from very simple differential–algebraic properties of the systems involved from a new perspective involving explicit nonlinear mappings of the states of the systems involved and controls. It is a counterpart of the estimate of the network’s collective behavior in leaderless case (Panteley & Loría, 2017), and it is not considered as the trivial case of equality of the states or outputs of the systems.

• For a large class of systems, the leader following consensus problem is solved by means of a new distributed dynamic consensus protocol with a single coupling scalar gain. Within this protocol, an implicit internal model is presented in the form of a function that contains the differential information (Martínez-Guerra & Cruz-Ancona, 2017) of the leader. Without solving any PDE, a single coupling scalar gain is suitable for the design and the interactions with the leader depend exclusively on the characteristics of the communication graph in comparison with Isidori et al. (2014).

• A detailed stability analysis for the algebraic synchronization manifold is provided for two cases where complete or partial information from the leader is provided to all followers in the network. In the worst case scenario, the distance from all trajectories of the systems to the synchronization manifold can be made arbitrarily small with a sufficiently large coupling gain.

• Towards a separation principle, it is revealed that the full closed loop can be regarded as an input-to-state convergent system. This allows to freely design the coupling gain from the properties of the directed communication graph in the network.

The rest of this paper is organized as follows: The description of the class of systems involved is contained in Section 2. The problem statement is given within Section 3. In this section a detail description of the design of a stable synchronization manifold for heterogeneous networks is considered. The main results are presented in Section 4. Two numerical examples, on a network of chaotic systems, illustrate our method in Section 5. Concluding remarks, discussions and future directions of this research are stated in Section 6. Finally, some auxiliary results and important proofs (e.g. Theorem 9 and Corollary 10) are given in Appendix A Auxiliary results, Appendix B Proof of, Appendix C Proof of.

Notation

For two matrices $A\in {\mathbb{R}}^{m\times n}$ and $B\in {\mathbb{R}}^{p\times q}$, $A\otimes B=\left[a_{ij}B\right]\in {\mathbb{R}}^{np\times mq}$ where $\otimes$ denotes the Kronecker product of matrices, Matrix $0_{\mu \times \nu }\in {\mathbb{R}}^{\mu \times \nu }$ represents a matrix with all entries equal to zero, Matrix $I_{\nu }\in {\mathbb{R}}^{\nu \times \nu }$ is the identity matrix, assume matrix $A_{\nu }=$ $\left(\begin{array}{cc}\hfill 0_{(\nu -1)\times 1}\hfill & \hfill I_{\nu -1}\hfill \\ \hfill 0\hfill & \hfill 0_{1\times (\nu -1)}\hfill \end{array}\right)$ $\in {\mathbb{R}}^{\nu \times \nu }$, and vector $B_{\nu }=$ $\left(\begin{array}{c}\hfill 0_{(\nu -1)\times 1}\hfill \\ \hfill 1\hfill \end{array}\right)$ $\in {\mathbb{R}}^{\nu }$ for a system in canonical form, Vector ${\mathbf{1}}_{\nu }={(1,\dots ,1)}^T\in {\mathbb{R}}^{\nu }$ denotes a $\nu$-dimensional column vector with all elements being $1$ and $\mathrm{diag}\left({\mathbf{1}}_{\nu }\right)=I_{\nu }$; ${\mathcal{G}}_{\nu }\triangleq ({\mathcal{V}}_{\nu },{\mathcal{E}}_{\nu },{\mathcal{A}}_{\nu })$ denotes a directed graph with a set of nodes ${\mathcal{V}}_{\nu }=\left\{1,\dots ,\nu \right\}$, a set of edges ${\mathcal{E}}_{\nu }={\mathcal{V}}_{\nu }\times {\mathcal{V}}_{\nu }$, and an adjacency matrix ${\mathcal{A}}_{\nu }=\left[a_{ij}\right]\in {\mathbb{R}}^{\nu \times \nu }$ where $a_{ij}=1$ if $(j,i)\in {\mathcal{E}}_{\nu }$ and $a_{ij}=0$ otherwise; the in-degree of node $v_i$ is represented as ${\mathrm{deg}}_{\mathrm{in}}\left(v_i\right)={\sum }_{j=1}^{\nu }{a}_{ij}$, ${\mathcal{L}}_{\nu }=\left[l_{ij}\right]\in {\mathbb{R}}^{\nu \times \nu }$ denotes a nonsymmetrical Laplacian matrix associated with ${\mathcal{G}}_{\nu }$ where $l_{ij}={\sum }_{j=1,i\ne j}^{\nu }{a}_{ij}$ if $i,j=1,\dots ,\nu$ and $l_{ij}=-a_{ij}$ otherwise; ${\Vert \cdot \Vert }_p$ represents the usual vector $p$-norm, ${\Vert \omega \left(t\right)\Vert }_{[t_0,t]}={\Vert \omega \left(t\right)\Vert }_{\infty }={sup}_{t_0\le \tau \le t}\Vert \omega \left(\tau \right)\Vert$ and ${\Vert x\Vert }_{\mathcal{M}}\triangleq {inf}_{\hat{x}\in \mathcal{M}}\left\{{\Vert x-\hat{x}\Vert }_2\right\}$ denotes the point $x$ to set $\mathcal{M}$ distance; a positive definite symmetric matrix $P$ is denoted by $P>0$ with ${\bar{\lambda }}_P>{\underline{\lambda }}_P>0$ as the maximum and minimum eigenvalues of $P$, respectively; $\nabla J$ and $[\partial ∕\partial x]f$ denote the gradient vector of a scalar valued function $J$ and the Jacobian matrix of a vector valued function $f$, respectively. A continuous function $\rho :[0,a)\to [0,\infty )$ is: $\mathcal{K}$ function if it is strictly increasing and $\rho \left(0\right)=0$, or ${\mathcal{K}}_{\infty }$ if it is a $\mathcal{K}$ function, $a=\infty$ and $\rho \left(s_1\right)\to \infty$ as $s_1\to \infty$. A continuous function $\beta :[0,a)\times [0,\infty )\to [0,\infty )$ is a $\mathcal{K}\mathcal{L}$ function if: for each fixed $s_2$, the mapping $\beta (s_1,s_2)$ belongs to class $\mathcal{K}$ with respect to $s_1$ and, for fixed each $s_1$, the mapping $\beta (s_1,s_2)$ is decreasing with respect to $s_2$ and $\beta (s_1,s_2)\to 0$ as $s_2\to \infty$.