Said Djennoune

Controllability and Observability of Linear Discrete-Time Fractional-Order Systems

Controllability and Observability of Linear Discrete-Time Fractional-Order Systems In this paper we extend some basic results on the controllability and observability of linear discrete-time fractional-order systems. For both of these fundamental structural properties we establish some new concepts inherent to fractional-order systems and we develop new analytical methods for checking these properties. Numerical examples are presented to illustrate the theoretical results.

A NOTE ON THE CONTROLLABILITY AND THE OBSERVABILITY OF FRACTIONAL DYNAMICAL SYSTEMS

Abstract In this paper, we give some new results on the controllability and the observability of linear dynamical systems with a fractional derivative of order α , where α is a non integer number. We show that the observability and the controllability Gramians, recently introduced for a fractional order system, are solutions of fractional differential Lyapunov equations, thus generalizing the classical result for the integer case ( α = 1). Our results can be considered as a generalization of the known corresponding results in the integer order case to the fractional order one since for α = 1, the results for the integer case are recovered.

Optimal synergetic control for fractional-order systems☆

Abstract Nowadays, the control of fractional-order system is one of the most popular topics in control theory. Recent studies have demonstrated the interest of fractional calculus both for systems modeling in many areas of science and engineering and for robust controller design. Thus, several research contributions have been devoted to the extension of control theory to fractional-order systems. Synergetic control was introduced in power electronics and other industrial processes. The benefit of this control scheme has been recognized for both integer-order linear and nonlinear systems. In this paper, a fractional-order synergetic control for fractional-order systems is proposed. Both linear and nonlinear cases are considered. The macro-variable is defined by the fractional-order integral of state variables. Optimality and stability properties are analyzed. A numerical example is investigated to confirm the effectiveness of the proposed method.

New Results on the Controllability and Observability of Fractional Dynamical Systems

In this article, we describe some new results on the controllability and observability of linear dynamical systems involving a fractional derivative of order α. We show that the input control energy required to drive the state in a given direction and the energy available at the output are related to the observability and the controllability Gramians, respectively. Our results can be considered as a generalization of the known corresponding results for the integer order case to the fractional order one, since for α 1 1, the results for the integer case are recovered.

A New Approach for Stability Analysis of Linear Discrete-Time Fractional-Order Systems

In this paper, an approach using a new formalism is proposed to analyze the stability of linear discrete-time fractional-order systems. Asymptotic stability of such systems is examined. Practical asymptotic stability is introduced and illustrated by a numerical example.

Optimal low order model identification of fractional dynamic systems

This article deals with modeling, simulation, identification and model reduction of non-integer systems in time domain. The fractional order system simulation is based on a fractional integrator operating on a limited spectral range. It allows approximating the fractional system by an integer state-space representation of high dimension. An output error method is used to perform the model parameters identification including the fractional order. Non-integer order systems known to exhibit long memory behavior require high dimensional models to represent them. In this paper, an iterative non-linear programming method is used to perform the approximation reduction and numerical simulations illustrate the efficiency of the reduced model to recover the fractional order system characteristics.

Observation and sliding mode observer for nonlinear fractional-order system with unknown input

The main purpose of this paper is twofold. First, the observability and the left invertibility properties and the observable canonical form for nonlinear fractional-order systems are introduced. By using a transformation, we show that these properties can be deduced from an equivalent nonlinear integer-order system. Second, a step by step sliding mode observer for fault detection and estimation in nonlinear fractional-order systems is proposed. Starting with a chained fractional-order integrators form, a step by step first-order sliding mode observer is designed. The finite time convergence of the observer is established by using Lyapunov stability theory. A numerical example is given to illustrate the performance of the proposed approach.

Vector Fitting fractional system identification using particle swarm optimization

Abstract The optimal fractional system identification is a challenging problem as it requires the estimation of not only the numerator parameters, but also the poles transfer function model and the non-integer order reading to complex nonlinear optimization. In this paper, an algorithm using the least square method, called “Vector Fitting” (V.F.) developed by Gustavsen is extended to the fractional order system identification in frequency domain. This algorithm proceeds recursively contrarily to the well known Levy’s algorithm which uses only one iteration to calculate the model parameters. The use of an iterative method efficiently directs the model parameters evolution towards their optimal values. Indeed, during iteration the poles of the model are calculated and used as starting poles for the next one, the stability of the identified model can thus be imposed. The V.F. algorithm is then associated with the heuristic optimization method: particle swarm optimization (PSO), leading to a new fractional system identification algorithm. The algorithm works in a hierarchical way; in a higher level, PSO determines the non-integer order and in a lower stage, the V.F. algorithm identifies the other parameters.

Backstepping fault tolerant control based on second order sliding mode observer: Application to induction motors

In this paper, a fault tolerant control for induction motors based on backstepping strategy is designed. The proposed approach permits to compensate both the rotor resistance variations and the load torque disturbance. Moreover, to avoid the use of speed and flux sensors, a second order sliding mode observer is used to estimate the flux and the speed. The used observer converges in a finite time and permits to give a good estimate of flux and speed even in presence of rotor resistance variations and load torque disturbance. The stability of the closed loop system (controller + observer) is shown in two steps. First, the boundedness of the trajectories before the convergence of the observer is proved. Second, the trajectories convergence is proved after the convergence of the observer. The simulation results show the efficiency of the proposed control scheme.

Chaotic synchronisation and secure communication via sliding-mode and impulsive observers

This paper addresses the design of a secure data transmission based on the synchronisation of two chaotic systems, with the use of unknown-input observers. The approach proposed here enhances the security level against intruders thanks to an intricate encryption system. It is shown also that this approach provides more robustness with respect to channel noise. The main feature consists in separating the encryption and synchronisation operations by using two coupled continuous chaotic systems in the transmitter. Technically, the scheme is based on impulsive and sliding mode unknown-input observers. This offers the advantage of estimating the (master) states and of reconstructing the unknown inputs simultaneously. The performances of the proposed method are highlighted by simulation results.