Advances in Sliding Mode Control of Earthquakes via Boundary Tracking of Wave and Diffusion PDEs

Abstract

Two recent results on earthquake control are summarized. A simplified model of an earthquake phenomenon is addressed by means of a cascade system of a 1D wave equation, representing the fault slip and wave propagation, and a 1D diffusion equation, representing the actuator dynamics as a diffusion process. In order to avoid a fast slip (earthquakelike behaviour), the control is designed to follow a slow reference. The control strategies are presented separately for both Partial Differential Equations (PDEs). For the wave PDE, a homogeneous boundary tracking control is developed to achieve exponential Input to State Stability (eISS) of the error closedloop dynamics. For the diffusion PDE, Proportional Integral (PI) and a discontinuous integral term are coupled to exponentially stabilize the error origin despite model uncertainties and perturbations. Simulations are additionally conducted to support the robustness and stability properties of the proposed control algorithms, by separately, obtaining critical remarks that will lead to the design of the single control for the cascade system in a future stage.

Robust Tracking for the Diffusion Equation using Sliding-Mode Boundary Control

“…where Q c (t) is the original control input designed as (13) and W ∈ ℜ (mc+mr )×mc is the null space of W . Note that if we replace (22) in (21), the demand over the controlled injection points will be strictly fulfilled at any time t. Furthermore, if we replace (22) in (19), the link between the new input matrix, Bc , and the original input matrix, B c , defined in ( 11) is stated as Bc W = B c . Control (22) will ensure the linear combination of the new controlled fluxes Qc (t) to be equal to a predetermined flux D(t), which we called demand, according to (21), while keeping the original output tracking result of the previous section.…”

“…More recently, significant progress has been made in controlling the earthquake instability of specific, well-defined, mature seismic faults [17]- [22]. These studies have employed various control algorithms to stabilize the complex and uncertain nature of the underlying underactuated physical system.…”

Control discontinuo por frontera de una red de vehículos autónomos basados en una ecuación de onda

“…Las ecuaciones en derivadas parciales (Krstic y Smyshlyaev, 2008) han demostrado su eficacia para modelar y entender de manera precisa sistemas complejos en diversos cam-pos científicos y tecnológicos. Por ejemplo, reactores químicos (Christofides, 2000), entender y predecir movimientos telúricos (Gutierrez-Oribio et al, 2022), analizar las oscilaciones complejas de sistemas biológicos (Lienard, 1928;Li et al, 2022) y sistemas con retardo (Zhu y Fridman, 2022).…”