Fractional order tube model reference adaptive control for a class of fractional order linear systems

Abstract

We introduce a novel fractional order adaptive control design based on the tube model reference adaptive control (TMRAC) scheme for a class of fractional order linear systems. By considering an adaptive state feedback control configuration, the main idea is to replace the classical reference model with a single predetermined trajectory by a fractional order performance tube guidance model allowing a set of admissible trajectories. Besides, an optimization problem is formulated to compute an on-line correction control signal within specified bounds in order to update the system performance while minimizing a control cost criterion. The asymptotic stability of the closed loop fractional order control system is demonstrated using an extension of the Lyapunov direct method. The dynamical performance of the fractional order tube model reference adaptive control (FOTMRAC) is compared with the standard fractional order model reference adaptive control (FOMRAC) strategy, and the simulation results show the effectiveness of the proposed control method.

Time-optimal control of linear fractional systems with variable coefficients

“…Explicit solutions to linear systems of differential equations provide a basis to perform stability analysis and to solve control problems. Analytical solutions of linear systems of fractional differential equations with constant coefficients were derived by Chikrii and Eidelman (2000), Chikrii and Matichin (2008), or Kaczorek (2008), and then applied to solving control problems by Matychyn and Onyshchenko (2015;2018b;2018a;2019), Dzieliński and Czyronis (2013), Balaska et al (2020), and Si et al (2021). Explicit solutions to linear systems of differential equations are usually expressed in terms of the state transition matrix.…”

Fractional Order Fault Tolerant Control - A Survey

“…Fractional calculus is based on the generalization of the integration and differentiation to any arbitrary order; it has been successfully utilized for the design of new control techniques. One of the earliest fractional order controllers were the CRONE controller and the P I λ D µ controllers [11], it has been demonstrated that fractional order controllers can improve system performances and stability, as well as provide robustness against disturbances and modeling uncertainties [12,13]. Since then, fractional calculus has been introduced in several control schemes and methods such as sliding mode control [14], predictive control [15,16], and particularly adaptive control [17,18,19].…”