Luc Moreau

Stability of multiagent systems with time-dependent communication links

We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.

Stability of continuous-time distributed consensus algorithms

We study the stability properties of linear time-varying systems in continuous time whose system matrix is Metzler with zero row sums. This class of systems arises naturally in the context of distributed decision problems, coordination and rendezvous tasks and synchronization problems. The equilibrium set contains all states with identical state components. We present sufficient conditions guaranteeing uniform exponential stability of this equilibrium set, implying that all state components converge to a common value as time grows unbounded. Furthermore it is shown that this convergence result is robust with respect to an arbitrary delay, provided that the delay affects only the off-diagonal terms in the differential equation.

A unified framework for input-to-state stability in systems with two time scales

This paper develops a unified framework for studying robustness of the input-to-state stability (ISS) property and presents new results on robustness of ISS to slowly varying parameters, to rapidly varying signals, and to generalized singular perturbations. The common feature in these problems is a time-scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory. To address various robustness problems simultaneously, the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables. The main results establish that if the boundary layer and averaged systems are ISS then the ISS bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time-scales. The generality of the framework is illustrated by making connection to various classical two time-scale problems and suggesting extensions.

Leaderless coordination via bidirectional and unidirectional time-dependent communication

We study a simple but compelling model of n interacting agents with time-dependent, bidirectional and unidirectional communication. The model finds wide application in a variety of fields including swarming, synchronization and distributed decision making. In the model, each agent updates his current state based upon the current information received from other agents according to a simple weighted average rule. Necessary and/or sufficient conditions for the convergence of the individual agents states to a common value are presented, extending recent results reported in the literature. Further, it is observed that more communication does not necessarily lead to better convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.

Practical stability and stabilization

Presents a practical stability result for dynamical systems depending on a small parameter. This result is applied to a practical stability analysis of fast time-varying systems studied in averaging theory, and of highly oscillatory systems studied by Sussmann and Liu (1999). Furthermore, the problem of practically stabilizing control affine systems with drift is discussed.

Input to state set stability for pulse width modulated control systems with disturbances

Abstract New results on set stability and input-to-state stability in pulse-width modulated (PWM) control systems with disturbances are presented. The results are based on a recent generalization of two time scale stability theory to differential equations with disturbances. In particular, averaging theory for systems with disturbances is used to establish the results. The nonsmooth nature of PWM systems is accommodated by working with upper semicontinuous set-valued maps, locally Lipschitz inflations of these maps, and locally Lipschitz parameterizations of locally Lipschitz set-valued maps.

Coordinated gradient descent: A case study of Lagrangian dynamics with projected gradient information

Abstract The paper studies gradient descent algorithms for vehicle networks. Each vehicle within the network is modeled as a double integrator in the plane. For each individual vehicle, the control input enabling coordinated gradient descent consists of a gradient descent control term and additional inter-vehicle forcing terms. When each vehicle has enough sensors to measure the full gradient at its current position, then the closed-loop system becomes Lagrangian. We focus in the present paper upon the more practical situation where each vehicle has only one sensor with which to sample the environment. We take this into account by replacing the full gradient in the closed-loop equations by its projection on the direction of motion for each individual vehicle. This gives rise to a differential equation with discontinuous right-hand side. In order to avoid the (practical and theoretical) complications that arise as a consequence of these discontinuities, we modify the inter-vehicle forcing terms and represent the velocity of each vehicle by a magnitude and an angle, resulting in a set of smooth differential equations. We demonstrate our approach with simulations

Feedback tuning of bifurcations

The present paper studies a feedback regulation problem that arises in at least two different biological applications. The feedback regulation problem under consideration may be interpreted as an adaptive control problem for tuning bifurcation parameters, and it has not been studied in the control literature. The goal of the paper is to formulate this problem and to present some preliminary results.

Stabilization by means of periodic output feedback

We consider linear time-invariant continuous-time systems x/spl dot/(t)=Ax(t)+bu(t), y(t)=cx(t) with 2-dimensional state x/spl isin//spl Rscr//sup 2/, scalar input u/spl isin//spl Rscr/, and scalar output y/spl isin//spl Rscr/. The matrices A,b and c are constant and of appropriate dimension. We discuss the problem of making the above linear system exponentially stable by means of a static time-varying output feedback u(t)=k(t)y(t). Easily verifiable necessary and sufficient conditions for this problem to be solvable are presented. Moreover, the proof of the sufficiency part is constructive; that is, it supplies the required feedback gain k(t). The paper thus solves an open problem posed by R. Brockett (1998) for the particular case of scalar input scalar output second-order systems. We assume throughout the paper that b/spl ne/(0 0)/sup T/ and c/spl ne/(0 0).

Coordinated control of networked mechanical systems with unstable dynamics

In this paper we present a coordinating control law for a network of mechanical systems with unstable dynamics. The control law is derived using the Method of Controlled Lagrangians together with potential shaping designed to couple the mechanical systems. The coupled system is Lagrangian with symmetry, and energy methods are used to prove stability and coordinated behavior. The class of mechanical systems we consider includes the planar inverted pendulum on a cart as well as the spherical inverted pendulum on a 2D cart. For these examples, the control law stabilizes each inverted pendulum and coordinates the relative motion of the carts.