CFU: 6
Prerequisites
Basic knowledge about closed loop control systems.
Preliminary Courses
None.
Learning Goals
The course aims at providing students with
- a set of tools for the analysis and control of networks of dynamical systems, with a special emphasis on their optimization and safety, and on their possible use for the design and management in diverse engineering applications
- a set of tools to model discrete event systems in the context of industrial automation and to design supervisory control systems. The course will focus on the finite state automata and language theory, as well as on Petri nets.
Expected Learning Outcomes
Knowledge and understanding
The students need to acquire the main methodological tools to model, analyze and control complex systems, with special emphasis on those that can be described as networks of interconnected dynamical systems. The lectures will guide the students toward a) the comprehension of the links between the topological properties of the graph and the individual dynamics of the nodes, and b) the identification of the causal mechanisms determining the spontaneous emergence of collective behaviors, such as consensus and synchronization. The analytical and numerical tools acquired by the students will be then used to understand the specificities of control design for network systems.
Applying knowledge and understanding
The students need to be capable of applying the acquired methodology to model and analyze real systems that can be described by complex network models, as for instance wireless sensor networks, population dynamics, or formation of autonomous vehicles. Furthermore, the students will need to showcase the ability apply the control techniques that they learned to design controllers for complex systems in the presence of constraints on the number of input signals and observable nodes.
Course Content - Syllabus
Part 1 Introduction and background
- Introduction
- Definition of a complex system
- Complex networks of dynamical systems
- Examples: wireless sensor networks and compartmental models
- Elements of matrix theory
- Convergent and semi-convergent matrices; eigenvalue classification
- Spectral properties of stochastic matrices
- Geršgorin disks theorem
- Perron-Frobenius theorem
- Examples
Parte 2 Graph theory
- Elements of graph theory
- Directed and undirected graphs
- Main definitions
- Paths, connectivity, and periodicity
- Condensation graphs
- Weighted graphs
- Adjacency matrix
- Linking graphs and matrices
- Properties of the adjacency matrix
- Some elementary equivalences
- Paths in the graph and powers of the adjacency matrix
- Graphs and irreducible matrices
- Graphs and primitive matrices
Parte 3 Analysis and control of networks of linear dynamical systems: consensus problem
- Discrete-time consensus problem
- Networks of discrete-time integrators
- Definition of consensus: min-max consensus, average consensus
- Condition on the graph topology for consensus in time-invariant networks
- Example: Leslie’s population model
- Continuous-time consensus problem
- Laplacian matrix of a graph: definition and properties
- Example: modeling collective dynamics in animal groups
- Network of continuous-time integrators
- Rank of the Laplacian matrix and equilibria in the network system
- Globally reachable nodes and consensus emergence
- Condition on the graph topology for consensus in time-invariant networks
- Convergence rates
- One-step convergence factor
- Asymptotic convergence factor
- Linking convergence rates and graph topology
- Consensus problems on time-varying graphs
- Examples of network systems on time-varying graphs
- Convergence over time-varying graphs connected at all times
- Convergence over time-varying graphs connected over time
Parte 4 Networks of nonlinear dynamical systems: synchronization
- Networks of nonlinear dynamical systems
- Modeling and fundamental assumptions
- Standard model of a network dynamical systems
- Example
- Synchronization
- Definition
- Example: Kuramoto oscillators
- Lyapunov-based stability analysis
- Sufficient conditions for synchronization
- Assumption on the node vector field and graph topology
Parte 5 Control of networks of nonlinear dynamical systems
- Decentralized control of network of nonlinear systems
- Centralized vs decentralized control
- Controllability of network systems
- Pinning control
- Partial control of networks
- Emerging problems and advanced network control techniques
- Elements on adaptive control of complex networks
- Control of networks with state-dependent topology
- Coevolution of graph topology and node states
- Emerging applications
Readings/Bibliography
- F. Bullo, Lectures on Network Systems, Edizione 1.3, 2019.
- M. E. J. Newman, A. L. Barabasi, and D. J. Watts, The structure and dynamics of networks, Princeton University Press, 2006.
- Additional references and lecture notes available in the tab file of the Teams class of the module
Further reading and material:
- Siljak, D. D. Decentralized control of complex systems. Courier Corporation, 2011.
- A. Barrat, M. Barthelemy, A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press, 2008.
- Uri Alon lab dataset. Available at http://www.weizmann.ac.il
- Pajek’s dataset Available at: http://vlado.fmf.uni-lj.si/pub/networks/data
Teaching Method
The teaching activities will be organized as follows: a) lectures for about 70% of the total hours, b) practical exercise in the classroom based on software tools (Matlab-simulink) for about 30% of the total hours.
Examination/Evaluation criteria
Exam type
Only oral. The oral exam focused on the discussion of a homework assigned to student by the instructor. The oral examination will also aim at assessing the knowledge of all the concepts and contents given during the lectures.
Evaluation pattern
The final mark is weighted with respect to the CFU of each module as follows:
- Module Discrete event systems and supervisory control, 6 CFU, 50%
- Module Control of complex systems and networks, 6 CFU, 50%