Boundary Control for Stabilization of Large-Scale Networks through the Continuation Method

Abstract

In this work we study a continuation method which transforms spatially distributed ODE systems into PDEs that respect the spatial structure of the original ODE systems. Such PDE description can be used not only for analysis but also for a continuous control design which, being discretized back, results in a nontrivial control law for the original ODE system. In this paper we focus on the continuation for linear systems, including multidimensional inhomogeneous systems and in particular linear networks, showing that such systems can be transformed into general second-order parabolic PDEs. The method is applied to the stabilization of a chain of coupled semiconductor lasers. We obtain a PDE model of this system, design a backstepping-based boundary control to stabilize the obtained PDE and then translate the control policy back to the original laser chain, effectively stabilizing it.

Synchronization of Spin-Torque Oscillators via Continuation Method

“…For non-identical oscillators we analysed one particular class of possible equilibrium solutions, showing that its shape can be analytically reconstructed and thus opening new possibilities for more efficient modeling and future analysis of the system. Still, there are many questions that could be investigated in details regarding the system (19), its PDE approximation (21) and the synchronization condition (24). First, Corollary 3 for the general winding number k in the case of identical oscillators gives only sufficient conditions on stability and probably more rigorous statements could be made based on Theorem 2.…”

“…6a. Further, we compare them with experimental results by simulating the original ODE system (19). We initialize all oscillators in this system using an amplitude √ p i = Γ/S and a phase φ i = ik∆x for the i-th oscillator, such that the phase makes k turns along the ring.…”

“…This effect vanishes for higher values of f . This mismatch can originate from the fact that the analytic prediction was found for the PDE model (21), while the simulation was performed for the original ODE system (19).…”

“…To demonstrate how solutions to the synchronization condition (28) found by (60)-(63) approximate synchronized solutions of the original system (19) we performed numerical simulations of (19) with n = 500 oscillators being split into two types as it was described earlier in this section. Parameters of the first type of oscillators were taken the same as in Section V-C, and for the second type slight deviations in parameters were added.…”

“…We approach this problem by utilizing a recently developed continuation method, which transforms a network of coupled ODEs into a single PDE. The general continuation method was introduced by the authors in [17] and was applied in [18] to derive an urban traffic PDE model and in [19] to stabilize a Stuart-Landau laser chain. To demonstrate applicability of the method to the problem of network synchronization, in Section II we start with a toy example of analysis of the Kuramoto model synchronization on a ring.…”