OSMOSE

This project is a Young Researcher project awarded by the French National Research Agency for the period 2024-2028. Its goal is to develop a unifying theoretical and practical framework for state observation of hybrid and nonsmooth dynamical systems with application in nonsmooth mechanics.

Dynamical systems with nonsmooth or hybrid behaviors are common in mechanical systems due to phenomena like :

• dry friction, typically creating stick-slip oscillations that lead to mechanical ware and damage, with often poor knowledge of the friction parameters especially in drilling applications; 

• impacts, for instance in walking robots, which create discontinuities in the system state with often poor knowledge of the impact model and times;

• backlash, inducing hysteresis, a typical hybrid behavior which deteriorates performance when not properly estimated and compensated.

Those may be modeled with nonsmooth differential equations or differential inclusions, or even hybrid dynamics, combining continuous-time and discrete-time equations. For purposes of control, observation and monitoring, it is instrumental to robustly estimate their state or parameters in real time. 

A possible way is to design an observer, namely a dynamical system taking as input the sensors' measurements and producing as output an estimate converging asymptotically to the true state. This observer may be smooth, nonsmooth or hybrid, . But the nonsmoothness of dynamics in differential inclusions, or the coupling of continuous and discrete dynamics in hybrid systems, as well as the dependence of their times of discontinuity (jump times) on their (unknown) initial condition render the design and analysis of nonsmooth and hybrid observation strategies challenging.