Reduced order in domain control of distributed parameter port-Hamiltonian systems via energy shaping

TL;DR

This paper introduces a novel energy shaping control method for distributed parameter port-Hamiltonian systems, using an early lumping approach to achieve stabilization and improved dynamic performance, demonstrated through simulations on a vibrating string.

Contribution

It extends the Control by Interconnection method with an early lumping energy shaping approach, addressing both fully-actuated and under-actuated control scenarios for distributed systems.

Findings

The proposed method stabilizes the closed-loop system.

Simulation shows improved performance with early lumping.

Under-actuated control is feasible with piecewise constant controllers.

Abstract

An in-domain finite dimensional controller for a class of distributed parameter systems on a one-dimensional spatial domain formulated under the port-Hamiltonian framework is presented. Based on [25] where positive feedback and a late lumping approach are used, we extend the Control by Interconnection method and propose a new energy shaping methodology with an early lumping approach on the distributed spatial domain of the system. Our two main control objectives are to stabilize the closed loop system, as well as to improve the closed loop dynamic performances. With the early lumping approach, we investigate two cases of the controller design, the ideal case where each distributed controller acts independently on the spatial domain (fully-actuated), and the more realistic case where the controller is piecewise constant over certain interval (under-actuated). We then analyze the asymptotic stability of the closed loop system when the infinite dimensional plant system is connected with the finite dimensional controller. Furthermore we provide simulation results comparing the performance of the fully-actuated case and the under-actuated case with an example of an elastic vibrating string.