Adaptive estimation using multiagent network identifiers with undirected and directed graph topologies

Adaptive estimation of external fields in reproducing kernel Hilbert spaces

“…This proves that g ∈ W, and hence W is norm closed. The analysis above in Equations ( 10) and (11) has been carried out under the assumption that the number agents N = 1. But modifications for N > 1 simply replaces the operator E *…”

“…Problems of this type are studied in the family of papers. [9][10][11] These formulations are quite general and are formulated for systems where the evolution laws for the agents are defined in terms of Gelfand triples, with the constituent operators satisfying some standard boundedness and coercivity properties for common DPS. In these papers the agent states evolve in an infinite dimensional state space, in contrast to the situation here.…”

“…This argument employs a definition of PE in the weak topology on the RKHS and is the first of its kind for this class of evolution problems in an RKHS. It is possible to view this strategy as an adaptation of the weak PE analyses that have been used in the study of DPS associated with some types of partial differential equations (PDEs) as in References 9‐11. For the DPS in Equation () we exploit the uniform embedding of the RKHS space and the proof of output convergence to simplify some of the arguments of convergence in the weak topology in this case.…”

“…In addition to the discrete time formulation of adaptive estimation summarized above, there also is a body of work that studies infinite dimensional consensus estimation problems for various classes of DPSs generated by families of PDEs. Problems of this type are studied in the family of papers 9‐11 . These formulations are quite general and are formulated for systems where the evolution laws for the agents are defined in terms of Gelfand triples, with the constituent operators satisfying some standard boundedness and coercivity properties for common DPS.…”

“…In contrast to prior work, in this article we explore weak PE conditions that are stated in terms of the weak topology on for adaptive estimation of an external scalar field. The strategy based on the weak topology is qualitatively similar to that in Reference 33 for the study of certain classes of PDEs as in References 9‐11, but the details of the proofs must be modified to exploit properties of the RKHS and the RKHS embedding formulation. This characterization of convergence is made precise by defining a weak norm $$$ {\left|\cdotp \right|}_w $$$ on and defining a linear subspace that consists of functions that, in some sense, “fail to be weakly persistently excited.” It is later shown that any solution $$$ \tilde{g}(t) $$$ of the governing ideal error equation () is contained in $$$ \overline{B_r(0)} $$$, the closed ball in of radius $$$ r $$$ centered at the origin, for a suitably large $$$ r>0 $$$.…”