Cooperative iterative learning for uncertain nonlinear agents in leaderless switching networks

Abstract

This paper is aimed at cooperative iterative learning tasks for leaderless networks with nonlinear agents, and the effects arising from switching topologies, locally Lipschitz nonlinearities, initial state shifts, and external disturbances of agents are addressed. By proposing a learning-based distributed algorithm, desired relative formation behaviors of leaderless networks can be realized, where all agents’ trajectories can be ensured to be uniformly bounded. A Lyapunov-like analysis approach is introduced to ensure learning convergence with an exponential rate by leveraging the properties of products of stochastic matrices, which can also be employed to develop input-to-state consensus results of discrete parameterized systems.

Introduction

Cooperative control of networks has attracted considerable attention owing to its potential applications in, e.g., biological systems (Olfati-Saber, 2006), multiple-vehicle systems (Fax & Murray, 2004), and sensor networks (Schenato & Fiorentin, 2011). Typically, a distributed protocol is designed for each agent by using the relative information between its nearest neighbors and itself such that the relative state deviations between agents can asymptotically converge to specified values, which implies the relative formation behaviors of networks (Olfati-Saber, Fax, & Murray, 2007). In particular, if all specified values equal zeros, then the relative formation behavior collapses to the consensus behavior, which indicates that the states of agents can asymptotically achieve agreement. For detailed explanations of cooperative control, see, e.g., Oh, Park, and Ahn (2015), Ren, Beard, and Atkins (2005) and the references therein.

Motivated by the practical applications of satellite networks (Ahn, Moore, & Chen, 2010) and marching bands (Meng and Moore, 2016, Meng and Moore, 2017), in which networks operate in a repetitive manner and the objective is to maintain a high-precision relative formation behavior between agents within a finite time interval, cooperative iterative learning—a combined study of cooperative control and iterative learning control (ILC) for networks is explored (Ahn et al., 2010, Bu et al., 2018, Bu et al., 2019, Hui et al., 2020, Jin, 2016, Li et al., 2018, Meng and Moore, 2016, Meng and Moore, 2017, Shen and Xu, 2018, Yang et al., 2016). Typically, under a learning-based distributed algorithm, the input signal of each agent is corrected based on the relative formation deviations between its nearest neighbors and itself from the previous repetitions. In such an approach, the relative formation behavior can be gradually achieved over the finite time interval with increasing iterations.

Most of the above-mentioned cooperative iterative learning problems fall into the leader–follower framework (Bu et al., 2018, Bu et al., 2019, Hui et al., 2020, Jin, 2016, Li et al., 2018, Shen and Xu, 2018, Yang et al., 2016). Benefiting from the specified reference trajectory of the leader agent, the analysis of the relative formation behaviors can be transformed into an ILC stability problem, where we directly analyze the errors between the reference trajectory and the states of agents. For this class of problems, various mature results and analysis techniques can be leveraged, such as the contraction mapping-based analysis approach in, e.g., Bu et al., 2018, Bu et al., 2019, Hui et al., 2020 and Yang et al. (2016) and the composite energy function-based Lyapunov-like approach in, e.g., Jin, 2016, Li et al., 2018 and Shen and Xu (2018). However, in the leaderless networks, the trajectories to which all agents will converge are unknown in advance. As a result, the behavior analysis problem of agents cannot be transformed into an ILC stability problem, and these two methods cannot be applied, which makes the problem more challenging. For some class of leaderless networks, e.g., sensor networks (Schenato & Fiorentin, 2011), those results investigated in the leader–follower framework may not be effective. Thus, it is of great significance to study the cooperative iterative learning for leaderless networks (Meng and Moore, 2016, Meng and Moore, 2017).

Another critical problem of cooperative iterative learning is about the network topologies. For a network with a fixed and quasi-strongly connected network topology, some remarkable results of cooperative iterative learning have been given in Bu et al., 2018, Jin, 2016, Li et al., 2018 and Shen and Xu (2018). When the switching topologies are taken into account, the strict model repetitiveness viewed as one of the basic assumptions of ILC is no longer satisfied. In some recent studies of ILC, this assumption can be relaxed (see the ILC results of, e.g., Chien et al., 2018, Yu and Li, 2017 and Altın, Willems, Oomen, and Barton (2017) accommodating the nonrepetitive plant models). However, the topology may switch from one to another totally different one. Thus, the essential model nonrepetitiveness from the switching topologies cannot be modeled as those used in, e.g., Chien et al., 2018, Yu and Li, 2017 and Altın et al. (2017). To investigate the cooperative iterative learning for networks with switching topologies (Yang et al., 2016), some severe requirements are imposed such that each of the possible topologies is strongly connected. To release the requirement, some cooperative iterative learning results of, e.g., Bu et al., 2019, Meng and Moore, 2016, Meng and Moore, 2017 and Hui et al. (2020) can admit a joint quasi-strong connectivity condition, which has been verified to be the necessary condition for ensuring the relative formation behaviors of networks with switching topologies.

From the perspective of the dynamics of agents, the existing results of the cooperative iterative learning mainly concentrate on linear agents and a special class of nonlinear agents whose dynamics fulfill a globally Lipschitz condition (Bu et al., 2018, Bu et al., 2019, Hui et al., 2020, Meng and Moore, 2016, Yang et al., 2016). This is because these results are developed based on ILC, in which the globally Lipschitz nonlinearity is seen as one of the basic assumptions (Ahn, Chen, & Moore, 2007). Naturally, they may no longer be feasible if agents’ non-globally Lipschitz nonlinearities are considered, as shown in Meng and Moore (2020). Toward this end, some efforts have been devoted in, e.g., Meng and Moore (2017) such that agents subject to a class of non-globally Lipschitz nonlinearities can be accommodated. However, it has not been disclosed whether and how the locally Lipschitz nonlinearities can be dealt with in the framework of cooperative iterative learning. In addition, robustness is also regarded as one of the most significant problems in the cooperative iterative learning because the uncertainties that arise from initial state shifts and external disturbances are generally unavoidable for agents.

In this paper, we contribute to the cooperative iterative learning issue for leaderless networks by simultaneously handling switching topologies, locally Lipschitz nonlinear dynamics, initial shifts, and external disturbances. By comparisons with the existing results of, e.g., Bu et al., 2018, Bu et al., 2019, Hui et al., 2020, Jin, 2016, Li et al., 2018, Meng and Moore, 2016, Meng and Moore, 2017, Yang et al., 2016 and Shen and Xu (2018), the main contributions are summarized in the following three aspects.

• A novel learning-based design of distributed algorithms is presented, in which the consistency between topology and relative formation error information is investigated. By contrast with, e.g., Meng and Moore (2016) and Meng and Moore (2017), the relative formation errors among neighboring agents that are determined by the current network topology are employed for the input updating of each agent.

• A new Lyapunov-like approach incorporating properties of stochastic matrices is leveraged to show input-to-state consensus results of discrete parameterized systems, and is rarely reported in ILC, of which convergence analysis is generally executed with contraction mapping methods (e.g., Bu et al., 2018, Bu et al., 2019, Hui et al., 2020, Yang et al., 2016). The construction of the Lyapunov function expresses a distinct difference from those of the composite energy functions in, e.g., Jin, 2016, Li et al., 2018 and Shen and Xu (2018), and thus the employed approach enriches the analysis tools of ILC for network systems.

• The relative formation behaviors for leaderless networks can be accomplished with an exponential rate, of which the convergence analysis cannot be transformed into an ILC stability problem as done in, e.g., Bu et al., 2018, Bu et al., 2019, Hui et al., 2020, Jin, 2016, Li et al., 2018, Yang et al., 2016 and Shen and Xu (2018). Despite this good result, the switching topologies and the locally Lipschitz nonlinear dynamics of agents that violate basic assumptions of ILC—the strict model repetitiveness and the globally Lipschitz nonlinearity, respectively, are well treated. These properties make our cooperative iterative learning results greatly extend those of, e.g., Bu et al., 2018, Bu et al., 2019, Hui et al., 2020, Jin, 2016, Li et al., 2018, Meng and Moore, 2016, Meng and Moore, 2017, Yang et al., 2016 and Shen and Xu (2018).

We organize this paper as follows. The problem description is given in Section 2. Main results including the input-to-state consensus results of the discrete parameterized system and the exponential convergence of the relative formation error among agents are given in Section 3 and the corresponding technical proofs are provided in Section 4. The conclusions are made in Section 5.

Notations

$\mathbb{R}$, ${\mathbb{R}}^n$, and ${\mathbb{R}}^{m\times n}$ represent the sets of real numbers, $n$-dimensional real vectors, and ($m\times n$)-dimensional real matrices, respectively. We employ $[0,\infty )=\{x\in \mathbb{R}:x\ge 0\}$, ${\mathbb{Z}}_{+}=\{0,1,2,\dots \}$, ${\mathbb{Z}}_T=\{0,1,2,\dots ,T\}$, and ${\mathcal{F}}_n=\{1,2,\dots ,n\}$, and let $t^{+}=t+1$ for $t\in {\mathbb{Z}}_{+}$, as well as ${\mathbf{1}}_n={[1,1,\dots ,1]}^{\mathrm{T}}\in {\mathbb{R}}^n$. For a matrix (respectively, a vector) $A$, ${\Vert A\Vert }_{\infty }$ is the maximum row sum matrix norm (respectively, $l_{\infty }$ norm). A matrix $A=\left[a_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is called a nonnegative matrix and denoted by $A\ge 0$ if $a_{ij}\ge 0$, $\forall i,j\in {\mathcal{F}}_n$ and $A$ is called a stochastic matrix if $A\ge 0$ and $A{\mathbf{1}}_n={\mathbf{1}}_n$. For two matrices $A$ and $B$, $A\otimes B$ is the Kronecker product of $A$ and $B$. For a sequence $\{A_i\in {\mathbb{R}}^{n\times n},i\in {\mathbb{Z}}_{+}\}$, denote ${\prod }_{i=h}^j{A}_i=A_j{A}_{j-1}\cdots A_h$ if $j\ge h$ and ${\prod }_{i=h}^j{A}_i=I_n$ (the $(n\times n)$-dimensional identity matrix) if $j<h$. If a scalar function $\lambda :[0,\infty )\to [0,\infty )$ is continuous and strictly increasing, and fulfills $\lambda \left(0\right)=0$ and ${lim}_{x\to \infty }\lambda \left(x\right)=\infty$, then it is said to belong to the class ${\mathcal{K}}_{\infty }$.