Stabilization of Fractional‐Order Linear Systems with State and Input Delay

Abstract

This paper describes a variable structure control for fractional-order systems with delay in both the input and state variables. The proposed method includes a fractional-order state predictor to eliminate the input delay. The resulting state-delay system is controlled through a sliding mode approach where the controller uses a sliding surface defined by fractional order integral. Then, the proposed control law ensures that the state trajectories reach the sliding surface in finite time. Based on recent results of Lyapunov stability theory for fractional-order systems, the stability of the closed loop is studied. Finally, an illustrative example is given to show the interest of the proposed approach.

New Results on Stabilization of Fractional‐Order Nonlinear Systems via an LMI Approach

“…Moreover, many systems modelled with the help of fractional calculus display rich fractional dynamical behavior, such as viscoelastic systems [10], boundary layer effects in ducts [11], electromagnetic waves [12], fractional kinetics [13,14], and electrode-electrolyte polarization [15,16]. Many stability conditions have been proposed for linear fractional-order systems [17][18][19][20][21][22][23], fractional-order nonlinear systems [24][25][26][27][28][29][30], fractional-order neural networks [31,32], fractional-order switched linear systems [33,34], fractional-order singular systems [35,36], positive fractional-order systems [37,38] and fractional chaotic complex networks systems [39]. Many stability conditions have been proposed for linear fractional-order systems [17][18][19][20][21][22][23], fractional-order nonlinear systems [24][25][26][27][28][29][30], fractional-order neural ne...…”

Lyapunov Functional Approach to Stability Analysis of Riemann‐Liouville Fractional Neural Networks with Time‐Varying Delays

“…Because of the memory and hereditary properties of fractional‐order differential operators, the fractional‐order derivatives can be used to describe the actual phenomena more accurately than integer‐order models. Recently, some important advances on dynamical behaviors such as chaos synchronization , consensus control , stabilization problem and Hopf bifurcation for fractional‐order systems or fractional‐order practical models have been reported in the current literature, which sufficiently show the superiority and importance of fractional calculus, and effectively motivate the development of the related applied fields. For more details on fractional calculus and fractional‐order differential equations theory, one can refer to the monographs of Miller and Ross , Podlubny , Kilbas et al.…”

“…. nization [9], consensus control [10], stabilization problem [11][12][13][14] and Hopf bifurcation [15,16] for fractional-order systems or fractional-order practical models have been reported in the current literature, which sufficiently show the superiority and importance of fractional calculus, and effectively motivate the development of the related applied fields. For more details on fractional calculus and fractional-order differential equations theory, one can refer to the monographs of Miller and Ross [17], Podlubny [18], Kilbas et al [19] and Diethelm [20].As we all know, the stability is an important performance metric for any dynamic system.…”

Stabilization of Non‐Linear Fractional‐Order Uncertain Systems

“…Considering the robustness and performance requirements of the closed-loop systems, the uncertainties of the systems cannot be ignored. There have been some stability results about the fractional-order uncertain systems [29][30][31][32][33][34]. So far, very few works exist for the stability and stabilization issue of non-linear fractional-order uncertain systems.…”