LMI conditions for robust consensus of uncertain nonlinear multi-agent systems

Abstract

Establishing consensus is a key probleminmulti-agent systems (MASs). This thesis proposes a novel methodology based on convex optimization in the form of linear matrix inequalities (LMIs) for establishing consensus in linear and nonlinear MAS in the presence of model uncertainties, i.e., robust consensus.

Firstly, this thesis investigates robust consensus for uncertain MAS with linear dynamics. Specifically, it is supposed that the system is described by a weighted adjacency matrix whose entries are generic polynomial functions of an uncertain vector constrained in a set described by generic polynomial inequalities. For continuous-time dynamics, necessary and sufficient conditions are proposed to ensure the robust first-order consensus and the robust second-order consensus, in both cases of positive and non-positive weighted adjacency matrices. For discrete-time dynamics, necessary and sufficient conditions are provided for robust consensus based on the existence of a Lyapunov function polynomially dependent on the uncertainty. In particular, an upper bound on the degree required for achieving necessity is provided. Furthermore, a necessary and sufficient condition is provided for robust consensus with single integrator and nonnegative weighted adjacency matrices based on the zeros of a polynomial. Lastly, it is shown how these conditions can be investigated through convex optimization by exploiting LMIs.

Secondly, local and global consensus are considered in MAS with intrinsic nonlinear dynamics with respect to bounded solutions, like equilibrium points, periodic orbits, and chaotic orbits. For local consensus, a method is proposed based on the transformation of the original system into an uncertain polytopic system and on the use of homogeneous polynomial Lyapunov functions (HPLFs). For global consensus, another method is proposed based on the search for a suitable polynomial Lyapunov function (PLF). In addition, robust local consensus in MAS is considered with time-varying parametric uncertainties constrained in a polytope. Also, by using HPLFs, a new criteria is proposed where the original system is suitably approximated by an uncertain polytopic system. Tractable conditions are hence provided in terms of LMIs. Then, the polytopic consensus margin problem is proposed and investigated via generalized eigenvalue problems (GEVPs).

Lastly, this thesis investigates robust consensus problem of polynomial nonlinear system affected by time-varying uncertainties on topology, i.e., structured uncertain parameters constrained in a bounded-rate polytope. Via partial contraction analysis, novel conditions, both for robust exponential consensus and for robust asymptotical consensus, are proposed by using parameter-dependent contraction matrices. In addition, for polynomial nonlinear system, this paper introduces a new class of contraction matrix, i.e., homogeneous parameter-dependent polynomial contraction matrix (HPD-PCM), by which tractable conditions of LMIs are provided via affine space parametrizations. Furthermore, the variant rate margin for robust asymptotical consensus is proposed and investigated via handling generalized eigenvalue problems (GEVPs).

For each section, a set of representative numerical examples are presented to demonstrate the effectiveness of the proposed results.