Passivity-based control of underactuated mechanical systems with Coulomb friction: Application to earthquake prevention

TL;DR

The paper addresses controlling underactuated mechanical systems with Coulomb friction under nonautonomous, discontinuous dynamics. It extends the classical passivity framework via an invariance-like principle to enable stable negative-feedback interconnections and then designs a three-stage passivity-based controller: restore passivity, incorporate integral action for constant-reference regulation, and account for actuator dynamics. The approach yields stabilization to a zero-velocity domain and robust tracking while using fewer inputs than degrees of freedom, and is demonstrated in a concrete earthquake-prevention scenario where slow, diffusion-driven actuation dissipates stored seismic energy aseismically. The results highlight a practical method for energy dissipation in friction-dominated systems and point to extensions to PDE control and observer design for broader applicability.

Abstract

Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. A characteristic example is nonautonomous and discontinuous systems due to presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists in designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). At last, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid's diffusion.

Figures (10)

• Figure 1: Negative feedback connection.
• Figure 2: (a)-(b): Schematic representation of a component of ${F}_r^{or}(\delta,u,v,0,t)$, showing how it is passive with respect to the input $[\delta^T,v^T]^T$. (c)-(d): Loss of passivity due to the addition of the loading term $F_s^*$ resulting in the new term $F_r(x_1,x_2,x_3,0,t)$.
• Figure 3: Steps of the passivity-based design.
• Figure 4: Closed loop system.
• Figure 5: Control design: Step 1.