M14 – L’AQUILA 21/03/2016 ‐ 24/03/2016

Tools for nonlinear control, Lyapunov function, positivity, applications

Frederic Mazenc
Laboratoire des Signaux et SystĂšmes (L2S)
CNRS-CentraleSupélec-U PSUD, France


We will present fundamental results pertaining to ordinary differential equations, discrete-time systems and nonlinear control theory. In particular, we will review the notion of Lyapunov function, the LaSalle Invariance Principle, the Jurdjevic-Quinn’s theorem and the techniques called backstepping and forwarding. We will perform construction of strict Lyapunov functions. We will study the notion of positive systems. We will study several applied problems (chemostats, PVTOL, cart-pendulum system).
The module is partially based on the research monograph: M. Malisoff, F. Mazenc, Constructions of Strict Lyapunov Functions,   Spinger-Verlag, serie : Communications and Control Engineering, 2009


1) Introduction to dynamical systems:
  Ordinary  Differential Equations, discrete-time systems, time-varying systems, basic notions (existence and uniqueness of solutions, finite escape time phenomenon). Notions of stability (local, global, basin of attraction), notion of  input-to-state stability.

2) Fundamental results. Linear systems: stability analysis, linearization. Hartman-Grobman Theorem, Two dimentional systems : Poincaré–Bendixson theorem. Dulac’s criterion, properties of ω-limit sets.

3) Lyapunov functions: Lyapunov theorem, converse Lyapunov theorem, LaSalle Invariance Principle. Weak Lyapunov functions, strict Lyapunov functions, Matrosov Theorem. Construction of strict Lyapunov functions. Determination of an estimate of a basin of attraction via a strict Lyapunov functions. Notion of ISS Lyapunov function.

4) Control design: Lyapunov design, Jurdjevic-Quinn theorem, classical backstepping, bounded backstepping, backstepping for time-varying systems, strabilization and tracking though forwarding, Sontag‘s formula.

5) Positive systems: Cooperative nonlinear systems, linear positive systems, linear Lyapunov function.  Notion of interval observer.