Course objective
The objective of this course is to provide tools for the analysis of nonlinear dynamical systems modelled by ordinary differential equations. These include engineering systems, such as robotic, automotive, and aerospace systems, but also, biological and socio-economical systems. A key aspect of these equations is that their nonlinear nature usually prevents the calculation of an explicit solution.
We rely on stability theory to study the qualitative behavior of dynamical systems by introducing specific stability notions and the tools to guarantee them in practice. Furthermore, beyond stability, these tools also enable to study the systems performance, robustness to disturbances, and interconnections. We present fundamental results to study stability of interconnections, the Small Gain Theorem and the Kalman-Yakubovic-Popov lemma.
Course content
- Input-Output Analysis: Motivation and main definitions (functional spaces, operator norms, basic operator algebra, norms for linear systems). Absolute Stability, Small Gain Theorem, loop transformations;
- Circle Criterion, Passivity, Lemma de Kalman-Yakubovic-Popov;
- Specific problems posed by nonlinearities, stability of input-free systems; Lyapunov tools to guarantee stability, LaSalle’s invariance principle;
- Limit cycles, Poincaré-Bendixson theorem, notion of input-to state stability (ISS) for systems with inputs; Lyapunov tools for ISS, cascade and feedback interconnection;
- Non-autonomous systems, Uniform Stability;
- Passivity-Based control, control Lyapunov functions.
References
- Nonlinear Systems, Hassan Khalil, Pearson (3rd ed) 2001.
- Nonlinear Systems Analysis, M. Vidyasagar, SIAM Classics in Applied Mathematics (2nd edition), 2002.
- Ordinary Differential Equations, Part II, N. Rouche and J. Mawhin (Translation from: Équations Différentielles Ordinaires Tome II: Stabilité et Solutions Périodiques. Rouche, Nicolas, and Jean Mawhin. Paris: Masson, 1973).
