Contraction theory and applications to nonlinear control systems

While classical control theory often focuses on stability analysis and the stabilization of an equilibrium point (which, in practice, corresponds to a steady-state operating condition), contraction theory shifts the focus to motion and the relationship between trajectories. The interest in contraction stems from the following question: if we initialize a dynamic system under two different conditions, do the two resulting trajectories converge toward one another over time? In other words, will the effect of the initial conditions be decisive in determining steady-state behavior, or will the trajectories “forget” their starting points? In the latter case, the system is said to be contractive.

The value of such a property for engineering applications is particularly appealing, as it offers guarantees and a high degree of predictability regarding the behavior of the system’s solutions, even in the presence of model uncertainties and disturbances. For this reason, contraction theory finds various applications in numerous fields, such as power systems , mechanical systems , biology and bioengineering , aerospace engineering , electrical circuits