Global Stability Analysis Using the Eigenfunctions of the Koopman Operator

Abstract

We propose a novel operator-theoretic framework to study global stability of nonlinear systems.Based on the spectral properties of the so-called Koopman operator, our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems. The main results establish the (necessary and sufficient) relationship between the existence of specific eigenfunctions of the Koopman operator and the global stability property of hyperbolic fixed points and limit cycles. These results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space.A. Mauroy is with the

Stability Analysis of a Multilink Flexible Manipulator: Nonlinear Observers

“…In the past decades, most researchers adopted several algebra or geometry approaches [10][11][12] to deal with different kinds of actuated systems of FMs. For instance, by changing cable lengths to represent cable flexibility, various adaptations of the lumpedmass method for cable-driven systems were found in [13][14].…”

“…More recently, it is worth noting that for FMs, the system uncertainties, nonlinearities, and disturbances are unavoidable in modeling of dynamic system [9,[16][17][18], so the robust observer design problems are not easy to derive. In order to preserve the observer action under system uncertainties as well as nonlinearities, various methods to design of robust state observers (estimation or filtering), such as algebraic, geometric, generalized inverse, and variable structure observer (VSO) techniques have been used to the observer design in [8,12,19,20]. In many practice applications, the design of robust nonlinear observers is close to the actual dynamic behavior of nonlinear systems with uncertainty disturbance rather than linear observers.…”

Comparison of Coupled Nonlinear Oscillator Models for the Transient Response of Power Generating Stations Connected to Low Inertia Systems

“…This has usually been achieved by finding relevant physical characteristics using simulations on various classical multimachine model testbeds, e.g., simulating the dynamic response [3]- [6], analysing the eigenvalue sensitivity [4], investigating the effects on the rate of change of rotor speed [7], and investigating inter-area power-flow oscillations (using a five-machine reduced model to represent The Western Electric Coordinating council transmission grid) [5]. Another method is Koopman mode decomposition [8], [9] which is relevant to the current paper as the nonlinear dynamic response of the system is represented as a sum of eigenfunctions in both cases, although the methods for obtaining them differ significantly.…”

Active Damping of Power Oscillations Following Frequency Changes in Low Inertia Power Systems

“…For example, in [4] controllers for doubly fed induction generators for wind farms were designed so that instabilities resulting from a disturbance on the wind farm could be prevented. Yet another method is Koopman mode decomposition [7], [8] which is relevant to the current paper as the nonlinear dynamic response of the system is represented as a sum of eigenfunctions in both cases, although the methods for obtaining them differ significantly (as discussed below).…”

“…as well as the stabilizing signal(7) and the output of the AVR(8), where the coefficients are for 2 () Vt…”