PhD Seminar Course on

Distributed Control of Multi-Agent Systems

Cagliari, Sept. 8 -- Sept. 29, 2009


Zhiyun Lin     (
Zhejiang University, China


16 hours, Sept. 8 -- Sept. 29, 2009


Lecture 1 (3 hours):    Tuesday,           3PM --- 6PM,     Sept. 8

Lecture 2 (2 hours):    Wednesday,     4PM --- 6PM,     Sept. 9

Lecture 3 (3 hours):    Tuesday,           3PM --- 6PM,     Sept. 15

Lecture 4 (2 hours):    Wednesday,     4PM --- 6PM,     Sept. 16

Lecture 5 (3 hours):    Monday,           3PM --- 6PM,     Sept. 21

Lecture 6 (3 hours):    Tuesday,           3PM --- 6PM,     Sept. 29


To be arranged


The subject of the course is the distributed control of networked multi-agent systems. Distributed control means that each agent has the same local strategy. The problem of distributed control of a network of autonomous agents is of growing interest in control and robotics because a system composed of many simple entities, each obeying the same rules of interaction, can display complex collective behaviors, which leads to a broad range of potential applications: planetary exploration, operation in hazardous environments, etc. The course is to present a coherent introduction to some fundamental problems in networked multi-agent systems and to provide a self-contained, broad exposition of the notions and mathematical tools from graph theory, nonnegative matrix theory, and nonsmooth analysis that are relevant in cooperative control problems.  

  1. Introduction to  networked multi-agent systems (2 hours)

    • Motivation

    • A bit of history

    • Research problems in multi-agent systems

  2. Connectivity in graph theory (2 hours)

    • Digraphs and undirected graphs

    • Connectedness

    • A fundamental result on connectivity

  3. Nonnegative matrices and graphs (3 hours)

    • Adjacency matrices and digraphs

    • Irreducible matrices and primitive matrices

    • Stochastic matrices and SIA matrices

    • Wolfowitz Theorem

  4. Generator matrices and graphs (2 hours)

    • Metzler matrices and M-matrices

    • Generator matrices, Laplacian, and digraphs

    • Algebraic properties of generator matrices

    • H(\alpha, m) stability

  5. Consensus and rendezvous problems (4 hours)

    • State model and interaction graph

    • Fixed topology: cyclic coupling

    • Fixed topology: arbitrary coupling

    • Dynamic topology: symmetric coupling

    • Dynamic topolgy: asymmetric coupling

    • Rendezvous via state-dependent graph

  6. Formations via rigid graph theory (3 hours)

    • Rigid-body transformations

    • Frameworks

    • Infinitesimal rigidity

    • Polygon formations


Grading homework assignments


Alessandro Giua
Dep. of Electrical and Electronic Engineering
University of Cagliari, Italy