Eventos Anais de eventos
COBEM 2023
27th International Congress of Mechanical Engineering
A Tutorial on Linear Gap Metrics for Robust Control
Submission Author:
Jose Luiz Montandon Neto , SP
Co-Authors:
Jose Luiz Montandon Neto, André Vecchione, Thiago Boaventura
Presenter: André Vecchione
doi://10.26678/ABCM.COBEM2023.COB2023-0836
Abstract
Robust control theory aims to develop control systems that can effectively handle variations in system parameters, modeling errors, external disturbances, and other uncertainties. Linear gap metrics play a critical role in the field of robust control, as they measure the dynamic distance between two linear time-invariant (LTI) systems in the frequency and time domains. Gap metrics combined with stability margins or thresholds can guarantee the stability of a set of systems subjected to the same controller or the stability of a set of controllers subjected to the same system. Linear gap metrics are derived from functional analysis and have multiple calculation methods available in the literature; however, some of these methods can be difficult to implement. This paper presents a concise roadmap for calculating linear gap metrics, including the various steps in the process. By exploring these steps, we can better understand the different gap metrics, methods, and their implications. The main conclusions are as follows. 1) The current literature has at least 11 linear gap metrics. 2) Only two of the 11 linear gaps are extensively applied in classic control problems. 3) The latest linear gap metric, called kernel-gap metric, allows online analysis of the system's robustness through a data-driven algorithm applied to a state-space system representation, transfer functions in the $z$- domain, and Stable Kernel/Image realizations. 4) The kernel gap is a big step for the gap metric theory since the previous gaps are used for offline and frequency response analysis. 5) Linear gap metrics can be a practical and faster approach than the $H$-infinity and $\mu$-synthesis analysis. 6) The kernel gap metric may be a feasible path to approximate a distance measure between two nonlinear systems, since it calculates a gap in the time domain. The measurement of the distance between two nonlinear systems is still an open problem in control theory and functional analysis.
Keywords
Linear gap metrics, tutorial, robust control, robust stability guarantee, functional analysis
