Stabilizing and tracking unknown steady States of dynamical systems

Area of Science:

• Control Theory
• Dynamical Systems
• Nonlinear Dynamics

Background:

• Stabilizing and tracking unknown steady states in dynamical systems is crucial for many applications.
• Existing control methods often face limitations with complex system dynamics and unknown parameters.
• Understanding the inherent topological constraints of dynamical systems is essential for effective control design.

Purpose of the Study:

• To propose an adaptive dynamic state feedback controller for stabilizing and tracking unknown steady states.
• To theoretically investigate and prove the limitations imposed by system eigenvalues on stabilization.
• To demonstrate the controller's efficacy in stabilizing challenging system states, such as saddle points.

Main Methods:

• Development of an adaptive dynamic state feedback control algorithm.
• Mathematical proof establishing a topological limitation based on the number of positive real eigenvalues.
• Numerical simulations and experimental validation using an electrochemical Nickel dissolution system.

Main Results:

• A controller capable of stabilizing and tracking unknown steady states is presented.
• A fundamental limitation is proven: stabilization is impossible if the combined system and controller have an odd number of positive real eigenvalues.
• For 2D systems, stabilization of saddle points requires an unstable degree of freedom in the feedback loop.

Conclusions:

• The proposed adaptive controller effectively stabilizes and tracks unknown steady states.
• Topological constraints related to system eigenvalues significantly impact stabilization possibilities.
• The controller's ability to handle unstable foci and saddle points is validated experimentally and numerically.