Brief PaperCooperative iterative learning for uncertain nonlinear agents in leaderless switching networks☆
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.
- (1)
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.
- (2)
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.
- (3)
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 , , and represent the sets of real numbers, -dimensional real vectors, and ()-dimensional real matrices, respectively. We employ , , , and , and let for , as well as . For a matrix (respectively, a vector) , is the maximum row sum matrix norm (respectively, norm). A matrix is called a nonnegative matrix and denoted by if , and is called a stochastic matrix if and . For two matrices and , is the Kronecker product of and . For a sequence , denote if and (the -dimensional identity matrix) if . If a scalar function is continuous and strictly increasing, and fulfills and , then it is said to belong to the class .
Section snippets
Problem description
Consider a network with agents denoted by . Let the agents evolve over time and iteration with the following dynamics: where , , and are the state, input, and disturbance, respectively; and are iteration-invariant vectors that represent the steady quantities of and , respectively; and , , , and
Main results
In this section, we utilize a 2-D dynamics analysis approach to simultaneously investigate the evolution of the state in the time direction and those of the input and the relative formation error in the iteration direction.
The evolution of the state in the time axis is exactly shown in (1), and to explore the evolution of the relative formation error in the iteration direction, (1) can be applied to obtain By
Technical proofs
Proof of Lemma 4 The result (1) can be verified based on the definitions of the union and the composition of digraphs, and the result (2) can be proved by following the similar way as that of Cao et al. (2008, Proposition 3). Thus, the proof details are omitted here for simplicity. □
Proof of Lemma 5 We can validate two facts under the condition (13) as: holds for all and ; is a subgraph of .
With the condition (A2) and the fact (ii), we can leverage the result (1) of Lemma 4 to ,
Conclusions
In this paper, we have investigated the cooperative iterative learning problem for leaderless networks subject to topologies switching in both iteration and time directions, where the agents’ locally Lipschitz nonlinear dynamics have also been involved. By utilizing the properties of products of stochastic matrices and the digraphs induced by them, we have proposed a Lyapunov-like analysis approach to derive the ISC results for the discrete parameterized system. These motivate a powerful
Jingyao Zhang received the B.S. degree in information and computing science and the M.S. degree in mathematics from Beihang University (BUAA), Beijing, China, in 2015 and 2018, respectively. He is currently pursuing the Ph.D. degree with School of Automation Science and Electrical Engineering at Beihang University (BUAA).
His current research interests include iterative learning control and nonlinear control. He was a co-recipient of the “Best Paper Award” from the IEEE 7th Data Driven Control
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Jingyao Zhang received the B.S. degree in information and computing science and the M.S. degree in mathematics from Beihang University (BUAA), Beijing, China, in 2015 and 2018, respectively. He is currently pursuing the Ph.D. degree with School of Automation Science and Electrical Engineering at Beihang University (BUAA).
His current research interests include iterative learning control and nonlinear control. He was a co-recipient of the “Best Paper Award” from the IEEE 7th Data Driven Control and Learning Systems Conference in 2018.
Deyuan Meng received the B.S. degree in mathematics and applied mathematics from Ocean University of China (OUC), Qingdao, China, in June 2005, and the Ph.D. degree in control theory and control engineering from Beihang University (BUAA), Beijing, China, in July 2010. From November 2012 to November 2013, he was a Visiting Scholar with the Department of Electrical Engineering and Computer Science, Colorado School of Mines, Golden, CO, USA. He is currently a Full Professor with the Seventh Research Division and the School of Automation Science and Electrical Engineering, Beihang University (BUAA).
His current research interests include iterative learning control, data-driven control, and multi-agent systems.
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This work was supported by the National Natural Science Foundation of China under Grants 61922007 and 61873013. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Bert Tanner under the direction of Editor Christos G. Cassandras.