H2 Performance of Relative Sensing Networks: Analysis and Synthesis

Abstract

This work provides a general framework for the analysis and synthesis of a class of relative sensing networks (RSN) in the context of its H2 performance. In an RSN, the underlying connection topology couples each agent at their outputs. A distinction is made between RSN with homogeneous agent dynamics and RSN with heterogeneous RSN. In both cases, an expression for the system H2 norm is developed that explicitly shows the dependance of the connection topology on that property. In the homogeneous setting, the norm expression reduces to the Frobenius norm of the underlying connection topology incidence matrix, E(G), scaled by the H2 norm of the agents comprising the RSN. In the heterogeneous case, the H2 norm becomes the weighted Frobenius norm of the incidence matrix, where the weights appear on the nodes of the graph, and correspond to the H2 norm of each agent in the RSN. The H2 norm characterization is then used to synthesize RSN with certain H2 performance. Speciflcally, a semi-deflnite programming solution is presented to design a local controller for each agent when the underlying topology is flxed. A solution using Kruskal’s algorithm for flnding a minimum weight spanning tree is used to design the optimal RSN topology given flxed agent dynamics. Nomenclature (A;B;C;D) State-space realization for a linear system xi(t); x(t) State vector ui(t); u(t) Control vector wi(t); w(t) Exogenous input vector yi(t); y(t) Measured output vector for an individual agent and all agents zi(t); z(t) Controlled variable y G (t) Global RSN output Yo; Xc Observability and controllability grammian G; V; E A graph and its vertex and edge sets E(G); L(G); ¢(G); A(G) Incidence matrix, graph Laplacian, degree matrix, and adjacency matrix §hom(G); §het(G) Homogeneous and heterogeneous RSN T w7!y i ; T w7!z i Closed-loop map from wi(t) to yi(t) and zi(t) T w7!G hom ; T w7!G het Homogeneous and heterogeneous map from w(t) to RSN output A < B Equivalent to (A i B) a symmetric negative-deflnite matrix 1 Vector with all entries equal to one R n Real n-dimensional Euclidean space