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Lecture

Networked Control Systems: Graph Theory and Stochastic Matrices

In course

ME-427: Networked control systems

This course offers an introduction to control systems using communication networks for interfacing sensors, actuators, controllers, and processes. Challenges due to network non-idealities and opportunities offered by communication will be analyzed. ...

Description

This lecture covers the basics of networked control systems, focusing on graph theory and stochastic matrices. Topics include adjacency matrices, graph exploration, powers of adjacency matrices, averaging in wireless sensor networks, collective models, spectral properties of non-negative matrices, and the Perron-Frobenious theorem. The instructor discusses the dynamics captured by matrices with special properties, the analysis of consensus algorithms, and the localization of eigenvalues using Gershgorin disks. The lecture also addresses the spectral properties of row-stochastic matrices, the convergence of powers of stochastic matrices, and the properties of adjacency matrices through associated digraphs.

Instructor

Giancarlo Ferrari Trecate

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (32)

Connectivity (graph theory)

In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v.

Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.

Glossary of graph theory

This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.

Graph (discrete mathematics)

In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges.

Eigenvalues and eigenvectors

In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.

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Networked Control Systems: Properties and Connectivity

Explores properties of matrices, irreducibility, and graph connectivity in networked control systems.

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Covers graphical models for probabilistic distributions using graphs, nodes, and edges.

Spectral Graph Theory: Introduction

Introduces Spectral Graph Theory, exploring eigenvalues and eigenvectors' role in graph properties.

Networked Control Systems: Opportunities

Explores coordination in networked control systems, graph theory, and consensus algorithms.

Matrices and Networks

Explores the application of matrices and eigendecompositions in networks.