Article - Control Transactions of the Institute of Measurement and Control 1–16 Ó The Author(s) 2024 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/01423312241254878 journals.sagepub.com/home/tim Disturbance-decoupled control synthesis for conformable fractional-order linear systems: A geometric approach Hasan Abbasi Nozari 1 , Seyed Jalil Sadati Rostami 1 and Paolo Castaldi 2 Abstract This paper investigates the theoretical requirements for conformable fractional-order linear systems to satisfy the fundamental principles of invariant subspaces, which are the foundation of geometric control theory, for the first time. The problem of exact disturbance decoupling in conformable frac- tional linear systems is tackled using newly developed geometric tools. Moreover, a set of necessary and sufficient conditions has been established to address this problem, which, at the same time, stabilizes the compensated system for any initial condition in the presence of unknown disturbances. We also show that employing the conformable fractional model as a synthesis model for geometric geometric decoupling control provides a better understanding of the structural system-theoretic characteristics of fractional systems, while upholding the fundamental properties of integer systems. To illustrate the applicability and effectiveness of the theoretical findings, two numerical simulations, including an application to an active suspension sys- tem, were carried out and compared with those of the Caputo derivative. Keywords Disturbance decoupling, invariant subspace, conformable derivative, conformable fractional-order system, stability Introduction The theory of fractional calculus has attracted significant interest in applied mathematics, particularly in the fields of systems and control theory. Numerous papers and textbooks have explored this paradigm (see, for example, Balci et al., 2019; Buscarino et al., 2020; Ibrahim et al., 2021; Kumar et al., 2019; Sheng et al., 2017; Tavakoli-Kakhki et al., 2010; Wang et al., 2012 and the references therein). An important aspect of fractional differential equations is the fractional dif- ferential order, which allows for a greater degree of freedom in accurately describing a system (Podlubny, 1999). Among the various definitions of fractional derivatives, the most cele- brated are the Caputo derivative and the Riemann–Liouville (RL) derivative, which are nonlocal fractional operators defined by the fractional integral (Podlubny, 1999). However, these definitions have certain limitations. For instance, the RL derivative does not satisfy the property D a (1)= 0. Neither the Caputo derivative nor the RL derivative adheres to the beneficial properties of the chain rule, as expressed by D a (f 8 g)(t)= D a (f )(g(t))D a (g)(t), the product rule, as expressed by D a (fg)= fD a (g)+ gD a (f ), semigroup, and so on. For further details, please see Naifar et al. (2022) and Khalil et al. (2014) and the references therein. In addition, the fractional systems defined by these nonlocal operators are inherently infinite-dimensional systems, and their future beha- viors depend not only on their current status but also on their past history. This gives rise to the initialization problem of fractional systems (Sabatier et al., 2010), leading to the con- cept of pseudo-state instead of state in control systems (Trigeassou et al., 2012), where time invariance cannot be recovered and the pseudo-state cannot always be measured due to cost and technological limitations (e.g. accurate frac- tional observers are needed for this aim (Sabatier et al., 2014, 2012)). The aforementioned setbacks hinder the natural extension of fundamental concepts from the control theory of integer systems to fractional systems modeled by nonlocal operators. Recently, a simple and well-behaved fractional derivative called the conformable derivative was proposed in the work by Khalil et al. (2014), and its physical interpretation is described in the work by Khalil et al. (2019). The definition was later extended to higher orders, and significant properties such as the fractional Laplace transform, fractional exponen- tial function, and fractional chain rule were subsequently 1 Faculty of Electrical and Computer Engineering, Babol Noshirvani University of Technology, Iran 2 Department of Electrical, Electronic, and Information Engineering, University of Bologna, Italy Corresponding author: Seyed Jalil Sadati Rostami, Faculty of Electrical and Computer Engineering, Babol Noshirvani University of Technology, Shariyatee Avenue, P.O.box. 484, Babol 8514143131, Iran. Email: j.sadati@nit.ac.ir Downloaded from https://iranpaper.ir https://www.tarjomano.com https://www.tarjomano.com