Antonis Papachristodoulou

Introducing SOSTOOLS: a general purpose sum of squares programming solver

SOSTOOLS is a MATLAB toolbox for constructing and solving sum of squares programs. It can be used in combination with semidefinite programming software, such as SeDuMi, to solve many continuous and combinatorial optimization problems, as well as various control-related problems. The paper provides an overview on sum of squares programming, describes the primary features of SOSTOOLS, and shows how SOSTOOLS is used to solve sum of squares programs. Some applications from different areas are presented to show the wide applicability of sum of squares programming in general and SOSTOOLS in particular.

On the construction of Lyapunov functions using the sum of squares decomposition

A relaxation of Lyapunovs direct method has been proposed elsewhere that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In the paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain non-polynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain non-polynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.

A tutorial on sum of squares techniques for systems analysis

This tutorial is about new system analysis techniques that were developed in the past few years based on the sum of squares decomposition. We present stability and robust stability analysis tools for different classes of systems: systems described by nonlinear ordinary differential equations or differential algebraic equations, hybrid systems with nonlinear subsystems and/or nonlinear switching surfaces, and time-delay systems described by nonlinear functional differential equations. We also discuss how different analysis questions such as model validation and safety verification can be answered for uncertain nonlinear and hybrid systems.

Effects of Delay in Multi-Agent Consensus and Oscillator Synchronization

The coordinated motion of multi-agent systems and oscillator synchronization are two important examples of networked control systems. In this technical note, we consider what effect multiple, non-commensurate (heterogeneous) communication delays can have on the consensus properties of large-scale multi-agent systems endowed with nonlinear dynamics. We show that the structure of the delayed dynamics allows functionality to be retained for arbitrary communication delays, even for switching topologies under certain connectivity conditions. The results are extended to the related problem of oscillator synchronization.

Analysis of switched and hybrid systems - beyond piecewise quadratic methods

This paper presents a method for stability analysis of switched and hybrid systems using polynomial and piecewise polynomial Lyapunov functions. Computation of such functions can be performed using convex optimization, based on the sum of squares decomposition of multivariate polynomials. The analysis yields several improvements over previous methods and opens up new possibilities, including the possibility of treating nonlinear vector fields and/or switching surfaces and parametric robustness analysis in a unified way.

Consensus in Multi-Agent Systems With Coupling Delays and Switching Topology

The robustness of consensus in single integrator multi-agent systems (MAS) to coupling delays and switching topologies is investigated. It is shown that consensus is reached for arbitrarily large constant, time-varying, or distributed delays if consensus is reached without delays. This delay robustness holds under the weakest possible connectivity assumptions on the underlying graph, i.e., as long as the graph is uniformly quasi-strongly connected and switches with a dwell-time. The proof is based on a contraction argument and allows to remove technical assumptions that were used in previous publications. Moreover, the result also applies to non-scalar single integrators, an important extension toward consensus and rendezvous in higher dimensions.

Synchronization in Oscillator Networks: Switching Topologies and Non-homogeneous Delays

We investigate the problem of synchronization in oscillator networks when the delay inherent in such systems is taken into account. We first investigate a general Kuramoto- type model with heterogeneous time delays, both with a complete network as well as a nearest neighbor interaction, for which we propose conditions for synchronization around a rotating frequency. Then, we turn our attention to the problem of synchronization when the topologies are allowed to change. We show that synchronization is possible in the presence of delay, using a common Lyapunov functional argument.

Robust Consensus Controller Design for Nonlinear Relative Degree Two Multi-Agent Systems With Communication Constraints

Robust static output-feedback controllers are designed that achieve consensus in networks of heterogeneous agents modeled as nonlinear systems of relative degree two. Both ideal communication networks and networks with communication constraints are considered, e.g., with limited communication range or heterogeneous communication delays. All design conditions that are presented are scalable to large and heterogeneous networks because the controller parameters depend only on the dynamics of the corresponding agent and its neighbors, but not on other agents in the network.

Positive Forms and Stability of Linear Time-Delay Systems

We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that stability implies that there exists a quadratic Lyapunov function on the state space, although this is in general infinite dimensional. We give an explicit parametrization of a finite-dimensional subset of the cone of Lyapunov functions. Positivity of this class of functions is enforced using sum-of-squares polynomial matrices. This allows the computation to be formulated as a semidefinite program

Delay-dependent rendezvous and flocking of large scale multi-agent systems with communication delays

We study the stability of multi-agent system (MAS) formations with delayed exchange of information between the agents. The agents are described by second order systems. They communicate via a symmetric connected communication topology with constant, heterogeneous, symmetric delays between any two neighboring agents. We consider two different tasks for the MAS: rendezvous, where all agents meet at an arbitrary point, and flocking, where all agents reach a given formation and move in a predefined direction. Therefore, we propose a decentralized control algorithm with position coupling gains kji. We prove that the MAS achieves rendezvous for any constant delay if the communication topology is connected and the coupling gain is sufficiently small. For larger gains, rendezvous and flocking are delay-dependent, i.e., they are reached for any delay smaller than a bound which depends on kji. Thereby, the controllers can be tuned in a totally decentralized fashion, i.e., only based on the communication delays to their neighbors and not considering the delays in the rest of the network. For the analysis, we use both frequency and time domain methods to prove delay-independent and delay-dependent rendezvous and flocking, respectively.