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In this paper, we will study coupled dynamical systems and investigate dimension properties of the subspace spanned by solutions of each individual system. Relevant problems on collinear dynamical systems and their variations are discussed recently by Montenbruck et al. in [1] , while in this paper we aim to provide a unified analysis to derive the dimensional-invariance principles for networked coupled systems, and to generalize the invariance principles for networked systems with more general forms of coupling terms. To be specific, we consider two types of coupled systems, one with scalar couplings and the other with matrix couplings. Via the rank-preserving flow theory , we show that any scalar-coupled dynamical system (with constant, time-varying, or state-dependent couplings) possesses the dimensional-invariance principles, in that the dimension of the subspace spanned by the individual systems’ solutions remains invariant. For coupled dynamical systems with matrix coefficients/couplings, necessary and sufficient conditions (for constant, time-varying, and state-dependent couplings) are given to characterize dimensional-invariance principles. The proofs via a rank-preserving matrix flow theory in this paper simplify the analysis in [1] , and we also extend the invariance principles to the cases of time-varying couplings and state-dependent couplings. Furthermore, subspace-preserving property and signature-preserving flows are also developed for coupled networked systems with particular coupling terms. These invariance principles provide insightful characterizations to analyze transient behaviors and solution evolutions for a large family of coupled systems, such as multiagent consensus dynamics, distributed coordination systems, formation control systems, etc.

The recently proposed DeGroot-Friedkin model describes the dynamical evolution of individual social power in a social network that holds opinion discussions on a sequence of different issues. This paper revisits that model, and uses nonlinear contraction analysis, among other tools, to establish several novel results. First, we show that for a social network with constant topology, each individual's social power converges to its equilibrium value exponentially fast, whereas previous results only concluded asymptotic convergence. Second, when the network topology is dynamic (i.e., the relative interaction matrix may change between any two successive issues), we show that each individual exponentially forgets its initial social power. Specifically, individual social power is dependent only on the dynamic network topology, and initial (or perceived) social power is forgotten as a result of sequential opinion discussion. Last, we provide an explicit upper bound on an individual's social power as the number of issues discussed tends to infinity; this bound depends only on the network topology. Simulations are provided to illustrate our results., Comment: Extended version of submitted journal paper. Includes additional simulation details

In this note, a switching-type delay-adaptive predictor feedback controller is proposed to globally stabilize linear systems with unknown input delay. The idea of our method is to take advantage of the robustness of feedback laws to delay estimate error that is sufficiently small. We first prove that, when the delay estimate is sufficiently close to the true delay, there is a Lyapunov–Krasovskii functional that is strictly decreasing, and asymptotic regulation is achievable. Then, with the help of the Lyapunov–Krasovskii functional as a supervisor, the switching logic plays the role of adjusting the estimated delay online to make it sufficiently close to the true delay. The result of this note shows that, global adaptive stabilization of linear systems with unknown input delay is achievable, with a logic-based switching control law.

The distributed averaging problem is a consensus problem whose objective is to devise a protocol which will enable all the members of a group of autonomous agents to compute the average of the initial values of their individual consensus variables in a distributed manner. Periodic gossiping is a deterministic method for solving the distributed averaging problem by stipulating that each pair of agents which are allowed to gossip, do so repeatedly in accordance with a prespecified periodic schedule. Agent pairs which are allowed to gossip correspond to edges on a given connected, undirected graph. In general, the rate at which the agents' consensus variables converge to the desired average value depends on the order in which the gossips occur over a period. The main contributions of this paper are first to characterize the classes of periodic gossip sequences which have the same convergence rate and second to prove that if the graph of allowable gossips is a tree with each edge restricted to gossiping once per period, the convergence rate is quite surprisingly, fixed and invariant over all possible periodic gossip sequences the graph allows. To arrive at these results, a new and unusual graph theoretic concept, namely the transfer function of a node of an undirected graph, is used. Among all the trees with the same number of edges, optimal tree structures, which yield the fastest convergence rate, can then be sought.

In this paper, the problem of frequency-limited ${\mmb H}_{\infty} $ model reduction for positive linear time-invariant systems is investigated. Specifically, our goal is to find a stable positive reduced-order model for a given positive system such that the ${\mmb H}_{\infty} $ norm of the error system is bounded over a frequency interval of interest. A new condition in terms of matrix inequality is developed for characterizing the frequency-limited ${\mmb H}_{\infty} $ performance. Then an equivalent parametrization of a positive reduced-order model is derived, based on which, an iterative algorithm is constructed for optimizing the reduced-order model. The algorithm utilizes coarse reduced-order models resulting from (generalized) balanced truncation as the initial value. Both continuous- and discrete-time systems are considered in the same framework. Numerical examples clearly show the effectiveness and advantages of the proposed model reduction method.

The problem of localization and circumnavigation of a slowly moving target with unknown speed has been considered. The agent only knows its own position with respect to its initial frame, and the bearing angle to the target in that frame. We propose an estimator to localize the target and a control law that forces the agent to move on a circular trajectory around the target such that both the estimator and the control system are exponentially stable. We consider two different cases where the agent's speed is constant and variable. The performance of the proposed algorithm is verified through simulations.

This note aims to develop the nonnegative matrix theory, in particular the product properties of infinite row-stochastic matrices, which is widely used for multiple integrator agents, to deal with the convergence analysis of general discrete-time linear multi-agent systems (MASs). With the proposed approach, it is finally shown both theoretically and by simulation that the consensus for all the agents can be reached exponentially fast under relaxed conditions, i.e. the individual uncoupled system is allowed to be strictly unstable (in the discrete-time sense) and it is only required that the joint of the communication topologies has a spanning tree frequently enough. Moreover, a least convergence rate as well as an upper bound for the strictly unstable mode, which are independent of the switching mode of the system, are specified as well.

This paper solves an n -agent formation shape control problem in the plane. The objective is to design decentralized control laws so that the agents cooperatively restore a prescribed formation shape in the presence of small perturbations from the prescribed shape. We consider two classes of directed, cyclic information architectures associated with so-called minimally persistent formations: leader-remote-follower and coleader. In our framework the formation shape is maintained by controlling certain interagent distances. Only one agent is responsible for maintaining each distance. We propose a decentralized control law where each agent executes its control using only the relative position measurements of agents to which it must maintain its distance. The resulting nonlinear closed-loop system has a manifold of equilibria, which implies that the linearized system is nonhyperbolic. We apply center manifold theory to show local exponential stability of the desired formation shape. The result circumvents the non-compactness of the equilibrium manifold. Choosing stabilizing gains is possible if a certain submatrix of the rigidity matrix has all leading principal minors nonzero, and we show that this condition holds for all minimally persistent leader-remote-follower and coleader formations with generic agent positions. Simulations are provided.