Design of Strongly Stabilizing Controllers for Underactuated Systems: An Example of the Cart-Pendulum System

This paper addresses the strongly stabilizing controller design for underactuated system, the cart-pendulum system and obtains third-order strongly stabilizing controllers. First, by linearizing the system model, we demonstrate that the plant satisfies the parity interlacing property (PIP) under specific output feedback conditions, ensuring the existence of such controllers. Second, we parameterize third-order stabilizing controller for the case of two symmetric zeros on the real axis and express all feasible regions for each parameter of the controller by inequalities, which extends a class of controllers obtained through interpolation with only stable poles on the real axis to controllers with stable poles on the complex plane. Further, we prove that a second-order stabilizing controller does not exist under this circumstance. Additionally, we also discuss the third-order stabilizing controllers based on the coprime factorization of strictly proper rational functions and interpolation method. Numerical simulations confirm that the closed-loop system under strongly stabilizing controller remains stable at the upright equilibrium point (UEP) and demonstrates better robustness than observer-based state feedback controller under friction disturbance.