Abdelfatah Charef

Optimal Approximation, Simulation and Analog Realization of the Fundamental Fractional Order Transfer Function

Optimal Approximation, Simulation and Analog Realization of the Fundamental Fractional Order Transfer Function This paper provides an optimal approximation of the fundamental linear fractional order transfer function using a distribution of the relaxation time function. Simple methods, useful in systems and control theories, which can be used to approximate the irrational transfer function of a class of fractional systems for a given frequency band by a rational function are presented. The optimal parameters of the approximated model are obtained by minimizing simultaneously the gain and the phase error between the irrational transfer function and its rational approximation. A simple analog circuit which can serve as a fundamental analog fractional system is obtained. Illustrative examples are presented to show the quality and usefulness of the approximation method.

Robust Fractional Adaptive Control Based on the Strictly Positive Realness Condition

Robust Fractional Adaptive Control Based on the Strictly Positive Realness Condition This paper presents a new approach to robust adaptive control, using fractional order systems as parallel feedforward in the adaptation loop. The problem is that adaptive control systems may diverge when confronted with finite sensor and actuator dynamics, or with parasitic disturbances. One of the classical robust adaptive control solutions to these problems makes use of parallel feedforward and simplified adaptive controllers based on the concept of positive realness. The proposed control scheme is based on the Almost Strictly Positive Realness (ASPR) property of the plant. We show that this condition implies also robust stability in the case of fractional order controllers. An application to Model Reference Adaptive Control (MRAC) with a fractional order adaptation rule is provided with an implementable algorithm. A simulation example of a SISO robust adaptive control system illustrates the advantages of the proposed method in the presence of disturbances and noise.

Linear fractional order system identification using adjustable fractional order differentiator

In previous decades, it has been observed that many physical systems are well characterised by fractional order models. Hence, their identification is attracting more and more interest of the scientific community. However, they pose a more difficult identification problem, which requires not only the estimation of model coefficients but also the determination of fractional orders with the tedious calculation of fractional order derivatives. This study focuses on an identification scheme, in the time domain, of dynamic systems described by linear fractional order differential equations. The proposed identification method is based on the recursive least squares algorithm applied to an ARX structure derived from the linear fractional order differential equation using adjustable fractional order differentiators. The basic ideas and the derived formulations of the identification scheme are presented. Illustrative examples are presented to validate the proposed linear fractional order system identification approach.

Process step response based fractional PIλDμ controller parameters tuning for desired closed loop response

In this paper, a tuning method of the fractional PIλDμ controllers for classical feedback control systems is proposed. The PIλDμ controller design strategy is drawn up such that the closed loop system is equivalent to a desired fractional order model whose transfer function is Bodes ideal function , a widely used function in the fractional order control domain because of its iso-damping property which is an important robustness feature. In this tuning technique, the values of the five parameters of the fractional PIλDμ controller are derived analytically using only the step response of a stable process without requirement of its model. The derived formulations of the tuning technique are presented. Illustrative examples are given to test the effectiveness and the usefulness of the proposed PIλDμ controller tuning approach.

Analog Circuit Implementation of Fractional Order Damped Sine and Cosine Functions

This paper introduces for the first time analog circuit implementations of two fundamental linear fractional order systems whose impulse responses called fractional order damped sine and cosine functions are the inverse Laplace transform of their irrational transfer functions. These analog circuit implementations are derived through rational function approximations of their irrational transfer functions.

Stability Analysis of Fractional Adaptive High-Gain Controllers for a Class of Linear Systems General case

In this paper we demonstrate that a fractional adaptive controller based on the high gain output feedback stabilizes the class of linear, time-invariant, minimum phase, SISO systems of relative degree one. We generalize the stability theorem of integer controllers to fractional order controllers whose derivative order is a real number between one and two, and introduce a new tuning parameter for the performance behaviour of the controlled plant. A simulation example is given to illustrate the effectiveness of the proposed algorithm

Proportional integral-fractional filter controller design for first order lag plus time delay system

ABSTRACT In this work, a design approach of proportional integral-fractional filter (PI-FF) controller for first order plus time delay system (FOPTD) is proposed in order to enhance the feedback control system performance characteristics. The controller design method is drawn up such that the transfer function of the overall closed-loop system is equivalent to the transfer function of the general fractional Bagley–Torvik reference model whose behaviour ranges from relaxation to oscillation for different values of the fractional order derivative and the damping ratio-like parameter. The tuning parameters of the PI-FF controller are derived analytically from the FOPTD process model and the general fractional Bagley–Torvik reference model parameters. Illustrative examples were presented to test the effectiveness and the usefulness of the proposed PI-FF controller on the feedback control system performance characteristics enhancement.

Rational Function Approximation of a Fundamental Fractional Order Transfer Function

This paper introduces a rational function approximation of the fractional order transfer function \( H(s) = \frac{{(\tau_{0} s)^{\alpha } }}{{[1 + (\tau_{0} s)^{2\alpha } ]}},\quad for\,\,0\, < \,\alpha \, \le \,0.5 \). This fractional order transfer function is one of the fundamental functions of the linear fractional system of commensurate order corresponding to pure complex conjugate poles or eigenvalues, in sα. Hence, the proposed approximation will be used in the solution of the linear fractional systems of commensurate order. Illustrative examples are given to show the exactitude and the efficiency of the approximation method.

Indirect fractional adaptive pole placement control

The main objective of this work is to study design methods of polynomial control laws by pole placement that presents actually smart solutions to many industrial applications. Even if these regulators are widely used, most of their applications concern problems where the reference signal does not vary in time. In this paper, we propose an indirect adaptive controller by fractional order pole placement, and show by mean of simulations of its application on a DC motor, the improvement in systems behavior when compared to the classical control scheme with integer order poles.