Multi-agent adaptive estimation with consensus in reproducing kernel Hilbert spaces

Abstract

This paper presents a framework for online adaptive estimation of unknown or uncertain systems of nonlinear ordinary differential equation (ODEs) that characterize a multiagent sensor network. This paper extends recent results in [2], [36] and here the nonlinear ODEs are embedded in the real, vector-valued reproducing kernel Hilbert space (RKHS) \mathbbH:=H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> with H a real, scalar RKHS. Each agent casts its local representation of the unknown function f as a member of the RKHS H. The result defines a distributed parameter system that governs the state estimates and estimates of the unknown function. The convergence of state estimates is proven along similar lines to that encountered in conventional adaptive estimation for systems of unknown nonlinear ODEs. The analysis of the parameter estimates, which is studied by an evolution in Euclidean space in conventional methods, now concerns the convergence of error functions in the RKHS. We show that the convergence of the function estimates to the unknown function in the RKHS is guaranteed provided a newly introduced persistency of excitation (PE) condition holds. This PE condition is defined on functions defined over a subset Ω that contains the trajectory of the true dynamic system. It can be viewed as an extension of the notion of partial persistence of excitation to the RKHS embedding framework.