Ball and Plate System Controller Using State Observer and Geometric Approach

Abstract

The primary objective of this paper is to present a feedback regulator using a Luenberger observer for state estimation of the ball-plate system, which is characterized by high instability and non-linearity. The novelty of this work lies in the design of an innovative control approach that explicitly considers time delay in the feedback loop—an aspect often neglected in prior studies. The adopted methodology involves modeling the system in state space while accounting for delay, and then constructing a state-feedback observer using a geometric approach. Numerical simulations were conducted to validate the proposed design. For instance, with an observer gain of L₂ = [1.58, 1.35], the controller minimizes response time along the x-axis and remains stable for delays up to 0.6364 seconds. Similarly, along the y-axis, a gain of L₅ = [0.58, 0.27] ensures robustness even with delays up to 1.4084 seconds, while effectively reducing initial overshoot. In all tested scenarios, estimation errors converged to zero, confirming the effectiveness of the observer-based controller. These findings support future work on automatic gain tuning based on performance specifications.