Systems theory (nonlinear dynamics) – 5 cfu

Master of Science (“Laurea magistrale”) in Automation and Control Engineering

School of Industrial and Information Engineering – Leonardo Campus

Aims and learning outcomes

The course illustrates methods for the theoretical and numerical analysis of nonlinear dynamical systems and their application to case studies in the field of automation engineering. The focus is on the use of nonlinear systems theory, in particular bifurcation analysis, to classify the behaviours of the system and to understand the critical transitions occurring when some model or control parameters are varied. To this aim, the main bifurcations occurring in nonlinear systems are discussed in detail and illustrated first on simple examples and then, with the help of tools for numerical analysis, on case studies. A few bifurcations occurring in non-smooth systems (e.g., systems with impacts or switches) are also considered.

Syllabus

Topic 1: Asymptotic behavior of dynamical systems: equilibria, cycles, tori, chaos. Multi-stability.

Topic 2: Parameterized systems and structural stability. Bifurcations of equilibria and cycles. Robustness analysis. Hysteresis and catastrophes. Transitions from periodic to aperiodic oscillations.

Topic 3:Discontinuous (non-smooth) systems: dynamical properties and bifurcations.

Topic 4:Fundamentals of numerical bifurcation analysis: continuation methods.

Topic 5:Case studies: Application to power systems, vehicle dynamics, and spacecrafts attitude analysis.

Course organization and final assessment

The course is organized in: lectures: the theoretical notions are presented and illustrated with the help of simple examples and exercises; seminars: case studies are introduced and discussed; computer labs: standard software for numerical bifurcation analysis is used for analyzing examples and case studies.

The final assessment consists in a written exam (possibly replaced by a specific project).

Bibliography

Suggested textbooks are:

S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, 1994

K. T. Alligood, T.D. Sauer, J.A. Yorke, Chaos - An Introduction to Dynamical Systems, Springer, 1996

Lecture notes will be made available to students, as well as scientific papers on specific topics and case studies.

Course organization and final assessment

The course is organized in: lectures: the theoretical notions are presented and illustrated with the help of simple examples and exercises; seminars: case studies are introduced and discussed; computer labs: standard software for numerical bifurcation analysis is used for analyzing examples and case studies.

The final assessment consists in a written exam (possibly replaced by a specific project).