Observer-based robust controller design for fractional-order systems with stochastic perturbations: A diffusive representation approach

We investigate the fractional-order systems that are perturbed by stochastic input to achieve stabilization via sliding mode control (SMC) approach. It is assumed that the system states are unknown and there is uncertainty and time-delay in the system. We utilize the diffusive representation of the stochastic fractional-order dynamics to transform the system into an integer-order system perturbed by Brownian motion. Provided that some linear matrix inequalities (LMIs) are feasible, it is proven that the estimation error system is stochastically stabilized and the overall closed-loop system is stable in probability. A numerical simulation shows the effectiveness of the results.

Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions

Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.

A diffusive representation approach toward H∞ sliding mode control design for fractional-order Markovian jump systems

This article proposes a new [Formula: see text] sliding mode control strategy for stabilizing controller design for fractional-order Markovian jump systems. The suggested approach is based on the diffusive representation of fractional-order Markovian jump systems which transforms the fractional-order system into an integer-order one. Using a new Lyapunov–Krasovskii functional, the problem of [Formula: see text] sliding mode control of uncertain fractional-order Markovian jump systems with exogenous noise is investigated. We propose a sliding surface and prove its reachability. Moreover, the linear matrix inequality conditions for stochastic stability of the resultant sliding motion with a given [Formula: see text] disturbance attenuation level are derived. Eventually, the theoretical results are verified through a simulation example.

New Design Method for Fractional Order Proportional Integral (FO-PI) Controller of 3x3 Multivariable Systems

This paper presents a new design method of Fractional Order Proportional Integral Controller (FO-PI) for 3x3 multivariable system (three-input-three-output). The Optimal parameters of the FO-PI controllers are tuning by minimizing performance index criterion as objective function. The irrational transfer function of the fractional operator is performed by means of diffusive representation and allows to formulate the optimization problem as a function of fractional order. The simulation results show that the performance of the response obtained by diffusive approach -based FO-PI are better than whose obtained by the classical controllers.

Acceleration of the Measurement Time of Thermopiles Using Sigma-Delta Control

This work presents a double sliding mode control designed for accelerating the measurement of heat fluxes using thermopiles. The slow transient response generated in the thermopile, when it is placed in contact with the surface to be measured, is due to the changes in the temperature distributions that this operation triggers. It is shown that under some conditions the proposed controls keep the temperature distribution of the whole system constant and that changes in the heat flux at the thermopile are almost instantaneously compensated by the controls. One-dimensional simulations and experimental results using a commercial thermopile, showing the goodness of the proposed approach, are presented. A first rigorous analysis of the control using the Sliding Mode Control and Diffusive Representation theories is also made.