Pierre-Alexandre Bliman

Average consensus problems in networks of agents with delayed communications

The present paper is devoted to the study of average consensus problems for undirected networks of dynamic agents having communication delays. The accent is put here on the study of the time-delays influence: both constant and time-varying delays are considered, as well as uniform and non uniform repartitions of the delays in the network. The main results provide sufficient conditions (also necessary in most cases) for existence of average consensus under bounded, but otherwise unknown, communication delays. Simulations are provided that show adequation with these results.

Backstepping design for time-delay nonlinear systems

The backstepping approach is adapted to the problem of globally uniformly asymptotically stabilizing nonlinear systems in feedback form with a delay arbitrarily large in the input. The strategy of design relies on the construction of a Lyapunov-Krasovskii functional. Continuously differentiable control laws are constructed.

A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters

In this paper, robust stability for linear systems with several uncertain (complex and/or real) scalar parameters is studied. A countable family of conditions sufficient for robust stability is given, in terms of solvability of some simple linear matrix inequalities (LMIs). These conditions are of increasing precision, and it is shown conversely that robust stability implies solvability of these LMIs from a certain rank and beyond. This result constitutes an extension of the characterization by solvability of Lyapunov inequality of the asymptotic stability for usual linear systems. It is based on the search of parameter-dependent quadratic Lyapunov functions, polynomial of increasing degree in the parameters.

An existence result for polynomial solutions of parameter-dependent LMIs

We show in this paper that any system of linear matrix inequalities depending continuously upon scalar parameters and solvable for any value of the latter in a fixed compact set, admits a branch of solutions polynomial with respect to the parameters. This result is useful for studying, e.g. parametric robustness or gain-scheduling issues.

Lyapunov equation for the stability of linear delay systems of retarded and neutral type

In this note, the delay-independent stability of delay systems is studied. It is shown that the strong delay-independent stability is equivalent to the feasibility of certain linear matrix inequality (LMI), that is to the existence of a quadratic Lyapunov-Krasovskii functional, independent of the (nonnegative) value of the delay. This constitutes the analogue of some well-known properties of finite-dimensional systems. This result is then applied to study delay-independent stability of systems with polytopic uncertainties.

Robust LMIs with parameters in multi-simplex: Existence of solutions and applications

This paper presents new results concerning the existence of solutions for robust (parameter-dependent) LMIs with parameters lying in a Cartesian product of simplexes, called multi-simplex. These results allow to derive convergent procedures based on LMI relaxations to check the positivity of polynomial matrices with parameters in multi-simplexes. As an application, the robust stability analysis of uncertain linear systems is investigated. As an immediate advantage of this flexible representation, polynomially parameter-dependent Lyapunov functions can be constructed to handle simultaneously time-invariant, arbitrarily time-varying and bounded time-varying parameters in an appropriate way. Numerical experiments illustrate the advantages of the method.

Convergence Speed of Unsteady Distributed Consensus: Decay Estimate Along the Settling Spanning-Trees

Results for estimating the convergence rate of nonstationary distributed consensus algorithms are provided, on the basis of qualitative (mainly topological) as well as basic quantitative information (lower-bounds on the matrix entries). The results appear to be tight in a number of instances and are illustrated through simple as well as more sophisticated examples. The main idea is to follow propagation of information along certain spanning-trees which arise in the communication graph.

Stabilization of LPV systems

We study in this paper the static state-feedback stabilization of linear finite dimensional systems depending polynomially upon a finite set of real, bounded, parameters. These parameters are a priori unknown, but available in real-time for control. We state two main results. First, we show that stabilizability of the class of systems obtained for frozen values of the parameters may be expressed equivalently by some LMI conditions, linked to certain class of parameter-dependent Lyapunov functions. Second, we show that existence of such a Lyapunov function for the LPV systems subject to bounded rate of variation of the parameters with respect to time, may be in the same manner expressed equivalently by some LMI conditions. In both cases, the method provides explicitly parameter-dependent stabilizing gain.

Extension of Popov absolute stability criterion to non-autonomous systems with delays

This paper extends in a simple way the classical absolute stability Popov criterion to multivariable systems with delays and with time-varying memoryless non-linearities subject to sector conditions. The proposed sufficient conditions are expressed in the frequency domain, a form well-suited for robustness issues, and lead to simple graphical interpretations for scalar systems. Apart from the usual conditions, the results assume basically a generalized sector condition on the derivative of the non-linearities with respect to time. Results for local and global stability are given, the latter concerning in particular the linear time-varying ones. For rational transfers, the frequency conditions are equivalent to some easy-tocheck linear matrix inequalities:this leads to a tractable method of numerical resolution by rational approximation of the transfer. As an illustration, a numerical example is provided.

Extension of a result by Moreau on stability of leaderless multi-agent systems.

The paper presents a result which relates connectedness of the interaction graphs in a multi-agent systems with the capability for global convergence to a common equilibrium of the system. In particular we extend a previously known result by Moreau by including the possibility of arbitrary bounded time-delays in the communication channels and relaxing the convexity of the allowed regions for the state transition map of each agent.