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Considering the fact that in the real world, a certain agent may have some sort of advantage to act before others, a novel hierarchical optimal synchronization problem for linear systems, composed of one major agent and multiple minor agents, is formulated and studied in this article from a Stackelberg–Nash game perspective. The major agent herein makes its decision prior to others, and then, all the minor agents determine their actions simultaneously. To seek the optimal controllers, the Hamilton–Jacobi–Bellman (HJB) equations in coupled forms are established, whose solutions are further proven to be stable and constitute the Stackelberg–Nash equilibrium. Due to the introduction of the asymmetric roles for agents, the established HJB equations are more strongly coupled and more difficult to solve than that given in most existing works. Therefore, we propose a new reinforcement learning (RL) algorithm, i.e., a two-level value iteration (VI) algorithm, which does not rely on complete system matrices. Furthermore, the proposed algorithm is shown to be convergent, and the converged values are exactly the optimal ones. To implement this VI algorithm, neural networks (NNs) are employed to approximate the value functions, and the gradient descent method is used to update the weights of NNs. Finally, an illustrative example is provided to verify the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this article, we aim to investigate the synchronization problem of dynamical systems, which can be of generic linear or Lipschitz nonlinear type, communicating over directed switching network topologies. A mild connectivity assumption on the switching topologies is imposed, which allows them to be directed and jointly connected. We propose a novel analysis framework from both algebraic and geometric perspectives to justify the attractiveness of the synchronization manifold. Specifically, it is proven that the complementary space of the synchronization manifold can be spanned by certain subspaces constructed from the network structure. This allows to project the states of the dynamical systems onto these subspaces and transform the synchronization problem under consideration equivalently into a convergence one of the projected states in each subspace. Then, assuming the joint connectivity condition on the communication topologies, we are able to work out a unified convergence analysis for both types of dynamical systems. More specifically, for partial-state coupled generic linear systems, it is proven that synchronization can be reached if an extra condition, which is easy to verify in several cases, on the system dynamics is satisfied. For Lipschitz nonlinear systems with positive-definite inner coupling matrix, synchronization is realized if the coupling strength is strong enough to stabilize the evolution of the projected states in each subspace under certain conditions. The above claims generalize the existing results concerning both types of dynamical systems to so far the most general framework. Some illustrative examples are provided to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper addresses the problems of pinning synchronization control and regulation and its circuit realization of a class of perturbed coupled Chua’s circuit networks. Robust constrained control strategies for pinned circuits and adaptive regulation schemes for couplings are developed to eliminate the impact of bounded time-varying perturbations and nonlinearities. Bounded synchronization results of the closed-loop Chua’s circuit networks are obtained based on Lyapunov stability theory. Furthermore, the control strategies and adaptive regulation schemes are implemented physically by circuits to construct the closed-loop Chua’s circuit networks reality. Comparison results based on MULTISIM and MATLAB softwares are given to verify the effectiveness of the proposed control and regulation circuits. [ABSTRACT FROM AUTHOR]

In network systems, a group of nodes may evolve into several subgroups and coordinate with each other in the same subgroup, i.e., reach cluster synchronization, to cope with the unanticipated situations. To this end, the leader-following practical cluster synchronization problem of networks of generic linear systems is studied in this paper. An event-based control algorithm that can largely reduce the amount of communication is first proposed over directed communication topologies. In the proposed algorithm, each node decides itself when to transmit its current state to its neighbors and how to update its controller according to the estimations of the states of it and its neighbors. Then, the Lyapunov method is utilized to perform the convergence analysis. It shows that the practical cluster synchronization can be ensured by choosing appropriate parameters no matter what kind of estimation for the state is applied. Furthermore, the Zeno behavior is also excluded for each node under some mild assumptions. Besides, three kinds of common estimations for the states including zero-order hold model, first-order approximate model, and high-order model-based estimations are, respectively, analyzed from the perspective of the exclusion of Zeno behavior. Finally, the validity of the proposed algorithm is demonstrated, the effects of the concerned parameters are simply presented, and the effects of the three estimations are also compared through several simulations. [ABSTRACT FROM AUTHOR]

In this paper, we aim to investigate the optimal synchronization problem for a group of generic linear systems with input saturation. To seek the optimal controller, Hamilton–Jacobi–Bellman (HJB) equations involving nonquadratic input energy terms in coupled forms are established. The solutions to these coupled HJB equations are further proven to be optimal and the induced controllers constitute interactive Nash equilibrium. Due to the difficulty to analytically solve HJB equations, especially in coupled forms, and the possible lack of model information of the systems, we apply the data-based off-policy reinforcement learning algorithm to learn the optimal control policies. A byproduct of this off-policy algorithm is shown that it is insensitive to probing noise that is exerted to the system to maintain persistence of excitation condition. In order to implement this off-policy algorithm, we employ actor and critic neural networks to approximate the controllers and the cost functions. Furthermore, the estimated control policies obtained by this presented implementation are proven to converge to the optimal ones under certain conditions. Finally, an illustrative example is provided to verify the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we aim to investigate the output group synchronization problem for a network of heterogeneous linear systems such that the outputs of the nodes, each representing a linear system, synchronize with each other in every cluster. Two different setups in terms of availability of the states of the reference generators are taken into consideration. A unified approach inspired by the internal model principle is proposed. First, the coupled reference generators are constructed so as to produce synchronized reference trajectories. Second, feedback controllers are designed to steer the output of each node to its corresponding reference trajectory. The two setups differ from each other in that the coupling configuration for reference generators and design of feedback controllers are rather different. Specifically, when the states of the reference generators are not available, the small gain theorem is exploited to drive the outputs of the nodes to the synchronized ones of virtual reference generators. Finally, simulations are carried out to validate the theoretical findings. [ABSTRACT FROM PUBLISHER]

The cluster synchronization problem is investigated using intermittent pinning control for the interacting clusters of nonidentical nodes that may represent either general linear systems or nonlinear oscillators. These nodes communicate over general network topology, and the nodes from different clusters are governed by different self-dynamics. A unified convergence analysis is provided to analyze the synchronization via intermittent pinning controllers. It is observed that the nodes in different clusters synchronize to the given patterns if a directed spanning tree exists in the underlying topology of every extended cluster (which consists of the original cluster of nodes as well as their pinning node) and one algebraic condition holds. Structural conditions are then derived to guarantee such an algebraic condition. That is: 1) if the intracluster couplings are with sufficiently strong strength and the pinning controller is with sufficiently long execution time in every period, then the algebraic condition for general linear systems is warranted and 2) if every cluster is with the sufficiently strong intracluster coupling strength, then the pinning controller for nonlinear oscillators can have its execution time to be arbitrarily short. The lower bounds are explicitly derived both for these coupling strengths and the execution time of the pinning controller in every period. In addition, in regard to the above-mentioned structural conditions for nonlinear systems, an adaptive law is further introduced to adapt the intracluster coupling strength, such that the cluster synchronization for nonlinear systems is achieved. [ABSTRACT FROM AUTHOR]

Most pinning synchronization schemes for complex networks with or without time delays reported in the literature require continuous state measurements and control updates, which are not suitable for implementation on digital platforms. In this paper, we first establish some pinning synchronization criteria for complex networks modeled by directed graphs with both internal and coupling delays based on an event-triggered mechanism. We also propose the design method for control gains under some mild assumptions. The piecewise constant nature of control law can help reduce the number of updates significantly, thereby saving energy resources. Moreover, by obtaining an explicit lower bound on the inter-event time interval, we prove that Zeno behavior does not exist in the evolution of each node of the complex network. At last, simulations are provided which verify the effectiveness of the proposed controllers. [ABSTRACT FROM AUTHOR]

A unified approach to the analysis of synchronization for complex dynamical networks, i.e., networks of partial-state coupled linear systems and networks of full-state coupled nonlinear oscillators, is introduced. It is shown that the developed analysis can be used to describe the difference between the state of each node and the weighted sum of the states of those nodes playing the role of leaders in the networks, thus making it feasible to consider the error dynamics for the whole network system. Different from the other various methods given in the existing literature, the analysis employed in this paper is demonstrated successfully in not only providing the consistent convergence analysis with much simpler form, but also explicitly specifying the convergence rate. [ABSTRACT FROM AUTHOR]

• First, a general setup that the output consensu of heterogeneous systems is considered for both without model-switching delay and with model switching delay under Markov switching topologies. • Second, under mild connectivity assumption , the coupled system with or without model-switching delay can achieve output synchronization by using the Lyapunov method. • Third, the criterion in theorem is significant difficulty, we propose some certain techniques to circumvent the difficulty. • Fourth, we also prove that an exponentially stable stochastic system, in mean square, with nonvanishing perturbation can achieve ultimately bounded in a small bound. In this paper, output consensus for a series of heterogeneous systems under semi-markov switching topology is investigated. The scenario under consideration is that none of the graph is assumed to be connected and the mode-switching delay is existed. The main technical challenge is model and controller dismatch because of time delay. Assuming every individual system is stabilizable and detectable and mild connectivity assumption on communication topology, the coupled system without model-switching delay can achieve output synchronization by using the Lyapunov method. For the coupled systems with model-switching delay, a bounded output synchronization conclusion can be given. The existence of controller design is also investigated. By the way, we prove an exponentially stable stochastic system, in mean square, with non-vanishing perturbation can achieve uniform boundedness and ultimate boundedness. Finally, two examples are presented to verify our main theories. [ABSTRACT FROM AUTHOR]