Detecting Equilibrium State of Dynamical Systems Using Sliding-Window Reduced-Order Dynamic Mode Decomposition

Indranil Nayak, Mrinal Kumar, Fernando Teixeira

Abstract: We propose a sliding-window dynamic mode decomposition (DMD) approach for identification of the equilibrium state ie. the steady state or limit cycle behavior of a dynamical system. Detecting whether a system is in transient state, steady state or evolving on a periodic orbit, bears significant importance in data-driven control, especially for reduced-order methods such as DMD. The capability of DMD to model and extrapolate steady-state or limit-cycle behavior of a dynamical system are influenced by the data harvesting region. We observe the position of DMD eigenvalues on the complex plane as the system moves from transient to equilibrium state and use that to detect such transition. We test our algorithm for Van der Pol oscillator, Lorenz-96 attractor and a simple plasma beam example. The results show clear indication regarding end of transience and onset of equilibrium state.