Convex conditions on decentralized control for graph topology preservation

International audienceThe paper focuses on the preservation of a given graph topology which is usually chosen to ensure its connectivity. This is an essential ingredient allowing interconnected systems to accomplish tasks by using decentralized control strategies. We consider a networked system with discrete-time dynamics in which the subsystems are able to communicate if an algebraic relation between their states is satisfied. Each subsystem is called agent and the connected subsystems are called neighbors. The agents update their state in a decentralized manner by taking into account the neighbors' states. The characterization of the local control feedback gains ensuring topology preservation is provided. The results are based on invariance and set-theory and yield to conditions in Linear Matrix Inequality (LMI) form. The conditions for topology preservation are applied to an illustrative example concerning partial state consensus of agents with double integrator dynamics

Set theory conditions for stability of linear impulsive systems

International audience— In this paper we give tractable necessary and sufficient condition for the global exponential stability of a linear impulsive system. The reset rule considered in the paper is quasi-periodic and the stability analysis is based on a standard tool in set theory that is Minkowski functional. Firstly, we reformulate the problem in term of discrete-time parametric uncertain system with the state matrix belonging to a compact but non-convex set. Secondly, we provide a tractable algorithm for testing the stability and computing the associated polyhedral Lyapunov function when the system is stable. The main result is an algorithm whose computational effort is analogous to that of classical algorithms for contractive polytopes computation for discrete-time parametric uncertain systems with the state matrix belonging to a polytopic set

Stability analysis for systems with asynchronous sensors and actuators

International audienceThe paper provides computation-oriented necessary and sufficient conditions for the global exponential stability of linear systems with asynchronous sensors and actuators. Precisely, we focus on continuous-time linear systems whose state undergoes finite jumps referred to as impulsions. The impulsions are of two types: those related to input updates and those related to measurement updates. We assume that impulsions of each type occur periodically but the periods may be different and the clocks at the sensor and at the actuator are not synchronized. We first show that the analysis can be reduced to a finite time domain. Based on that, we provide necessary and sufficient conditions for the global exponential stability of systems belonging to the class under study. An example illustrates numerically the proposed results

Time scale modeling for consensus in sparse directed networks with time-varying topologies

The paper considers the consensus problem in large networks represented by time-varying directed graphs. A practical way of dealing with large-scale networks is to reduce their dimension by collapsing the states of nodes belonging to densely and intensively connected clusters into aggregate variables. It will be shown that under suitable conditions, the states of the agents in each cluster converge fast toward a local agreement. Local agreements correspond to aggregate variables which slowly converge to consensus. Existing results concerning the time-scale separation in large networks focus on fixed and undirected graphs. The aim of this work is to extend these results to the more general case of time-varying directed topologies. It is noteworthy that in the fixed and undirected graph case the average of the states in each cluster is time-invariant when neglecting the interactions between clusters. Therefore, they are good candidates for the aggregate variables. This is no longer possible here. Instead, we find suitable time-varying weights to compute the aggregate variables as time-invariant weighted averages of the states in each cluster. This allows to deal with the more challenging time-varying directed graph case. We end up with a singularly perturbed system which is analyzed by using the tools of two time-scales averaging which seem appropriate to this system

Constructive necessary and sufficient condition for the stability of quasi-periodic linear impulsive systems

International audienceThe paper provides a computation-oriented necessary and sufficient condition for the global exponential stability of linear impulsive systems, whose impulsions are assumed to occur quasi-periodically. Based on the set-theoretic conditions for robust stability of uncertain linear systems, the existence of polyhedral Lyapunov functions is proved to be necessary and sufficient for global exponential stability of quasi-periodic linear impulsive systems. A constructive method is developed for testing the stability of the system and for computing set-induced polyhedral Lyapunov functions. The method leads to an algorithm whose complexity is similar to the standard algorithm related to discrete-time parametric uncertain systems with the state matrix belonging to a convex polytopic set

LMI conditions for topology preservation: applications to multi-agent tasks

International audienceIn this work we present several implementation strategies answering to different classical problems in multi-agent systems. The model under consideration consists of a discrete-time dynamics multi-agent system in which two agents are able to communicate when an algebraic relation between their states is satisfied. As emphasized in the literature, the connectivity of the communication network is essential for global coordination objectives. Thus, the primary goal of our methodology is to characterize the controllers that preserve a given topology allowing the global coordination. In a second step we choose the controller appropriated to the main agreement objective by solving a convex optimization problem associated to the minimization of a well-chosen cost function. Examples concerning full or partial consensus of agents with double integrator dynamics illustrate the implementation of the proposed methodology

Coordination in networks of linear impulsive agents

International audienceThe paper focuses on consensus in heterogeneous networks containing both linear and linear impulsive dynamics. This model applies for networks that are formed by several clusters. Most agents can only update their state in a continuous way using inner-cluster agent states. On top of this, few agents also have the peculiarity to update their states in a discrete way by reseting it using states from agents outside their clusters. The motivation of this behavior is that communication constraints hamper continuous inter-clusters interactions. Under appropriate assumptions we prove that all subsystems asymptotically agree and we provide an upper-bound of the convergence speed. We illustrate the behavior with an academic example containing five agents grouped in two clusters

Consensus and influence power approximation in time-varying and directed networks subject to perturbations

International audienceThe paper focuses on the analysis of multi-agent systems interacting over directed and time-varying networks in presence of parametric uncertainty on the interaction weights. We assume that agents reach a consensus and the main goal of this work is to characterize the contribution that each agent has to the consensus value. This information is important for network intervention applications such as targeted advertising over social networks. Indeed, for an advertising campaign to be efficient, it has to take into account the influence power of each agent in the graph (i.e., the contribution of each agent to the final consensus value). In our first results we analytically describe the trajectory of the overall network and we provide lower and upper bounds on the corresponding consensus value. We show that under appropriate assumptions, the contribution of each agent to the consensus value is smooth both in time and in the variation of the uncertainty parameter. This allows approximating the contribution of each agent when small perturbations affect the influence of each agent on its neighbors. Finally, we provide a numerical example to illustrate how our theoretical results apply in the context of network intervention

Some remarks on Smith predictors A geometric point of view

International audienceIn this paper we develop a method to obtain the stability crossing curves of a Smith Predictor control scheme. More explicitly, we compute the crossing set, which consists of all frequencies corresponding to all points on the stability crossing curve, and we give their complete classification. Furthermore, the directions in which the zeros cross the imaginary axis are explicitly expressed

Passivity-based tracking control of multiconstraint complementarity Lagrangian systems

In this study one considers the tracking control problem of a class of nonsmooth fully actuated Lagrangian systems subject to frictionless unilateral constraints. A passivity-based switching controller that guarantees some stability properties of the closed-loop system is designed. A particular attention is paid to transition (impacting) and detachment phases of motion. This paper extends previous works on the topic as it considers multiconstraint n-degree-of-freedom systems