Explicit formula and graphs of few special functions are derived in the paper on the basis of various definitions of various fractional derivatives and various fractional integrals. Their applications are also reviewed in the paper.

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Applications of Fractional Calculus

This paper discusses the FPGA implementation of the fractional-order derivative as well as two fractional-order chaotic systems where one of them has controllable multi-scroll attractors. The complete hardware architecture of the Grunwald-Letnikov (GL) differ-integral is realized with different memory window sizes. As an application of the proposed circuit, a complete fractional-order FPGA implementation of Liu chaotic system is introduced with different fractional-orders. Moreover, a fractional-order controllable heart and V-shape multi-scrolls chaotic systems are verified in the case of symmetric and asymmetric cases. Different interesting attractors are realized under various parametric changes with distinct step sizes for different fractional-orders. To verify the chaotic behavior of many generating attractors, the Maximum Lyapunov Exponent (MLE) is calculated for such systems. The designs have been simulated using Xilinx ISE 14.5 and realized on Xilinx FPGA Virtex 5 XC 5 VLX 50 T . The achieved throughputs are: 4.4 Gbit/s for GL, 1.986 Gbit/s for Liu system, and 2.921 Gbit/s for V-Shape multi-scroll attractor.

FPGA implementation of two fractional order chaotic systems

This paper studies the adaptive control problem for a class of uncertain fractional nonlinear systems with actuator faults, where the total number of failures is allowed to be infinite. A compensating term in a smooth function form of a conventional control law is introduced to compensate for the actuator faults. After the introduction of fractional-order adaptation laws, an adaptive controller is designed by using a modified backstepping technique. Fractional Lyapunov stability criterion is adopted to prove the convergence of the developed proposed controller even in the presence of actuator faults. Finally, a simulation example of fractional-order Chua–Hartley’s system is given to verify the effectiveness of the proposed fault-tolerant control scheme.

Fuzzy Adaptive Fault-Tolerant Control of Fractional-Order Nonlinear Systems

This paper proposes a combined Virtual Reference Feedback Tuning–Q-learning model-free control approach, which tunes nonlinear static state feedback controllers to achieve output model reference tracking in an optimal control framework. The novel iterative Batch Fitted Q-learning strategy uses two neural networks to represent the value function (critic) and the controller (actor), and it is referred to as a mixed Virtual Reference Feedback Tuning–Batch Fitted Q-learning approach. Learning convergence of the Q-learning schemes generally depends, among other settings, on the efficient exploration of the state-action space. Handcrafting test signals for efficient exploration is difficult even for input-output stable unknown processes. Virtual Reference Feedback Tuning can ensure an initial stabilizing controller to be learned from few input-output data and it can be next used to collect substantially more input-state data in a controlled mode, in a constrained environment, by compensating the process dynamics. This data is used to learn significantly superior nonlinear state feedback neural networks controllers for model reference tracking, using the proposed Batch Fitted Q-learning iterative tuning strategy, motivating the original combination of the two techniques. The mixed Virtual Reference Feedback Tuning–Batch Fitted Q-learning approach is experimentally validated for water level control of a multi input-multi output nonlinear constrained coupled two-tank system. Discussions on the observed control behavior are offered.

Data-driven model reference control of MIMO vertical tank systems with model-free VRFT and Q-Learning.

This paper introduces all the possible topologies of the Wien bridge oscillator family. This family has 72 topologies, 24 of them contain only RC or RL pairs, and the rest contain mixed pairs. The complete mathematical analysis of all twelve possible capacitive-based topologies is proposed in the fractional-order domain. The investigated circuits can be categorized into two groups, each with a similar characteristic equation. Three integer-order approximation techniques for the Laplacian operator $$s^\alpha $$
 are employed to solve and simulate the Wien bridge system. The studied approximations are those of Matsuda, Oustaloup, and Valsa’s network. Fractional-order capacitor (FOC) emulators are built using these approximations and applied in the circuit simulation. Comparisons are made on different levels, starting with the mathematical solution of the characteristic equation, followed by PSpice simulation, which compares topologies of the Wien bridge oscillator family. Hardware implementation of the FOC emulators is presented applying passive discrete components using the Foster-I technique. Additionally, sensitivity tests of the discrete components of the FOC emulators are performed using Monte Carlo analysis. Experimental results are introduced to validate the theoretical findings.

All Possible Topologies of the Fractional-Order Wien Oscillator Family Using Different Approximation Techniques

The errors-in-variables identification problem concerns dynamic systems in which input and output signals are contaminated by an additive noise. Several estimation methods have been proposed for identifying dynamic errors-in-variables rational models. This paper presents new consistent methods for order and coefficient estimation of continuous-time systems by errors-in-variables fractional models. First, differentiation orders are assumed to be known and only differential equation coefficients are estimated. Two estimators based on Higher-Order Statistics (third-order cumulants) are developed: the fractional third-order based least squares algorithm (ftocls) and the fractional third-order based iterative least squares algorithm (ftocils). Then, they are extended, using a nonlinear optimization algorithm, to estimate both the differential equation coefficients and the commensurate order. The performances of the proposed algorithms are illustrated with a numerical example.

New consistent methods for order and coefficient estimation of continuous-time errors-in-variables fractional models

In this paper, the fractional closed-loop system identification problem is addressed. Using the indirect approach, which supposes the knowledge of the controller, both coefficients and fractional orders of the process are estimated. The optimal instrumental variable method combined with a nonlinear optimization algorithm is handled to identify the fractional closed-loop transfer function. Also, two techniques are used for model selection the Akaike's information criterion and the $$R_T^2$$RT2 criterion. The performances of the proposed scheme are illustrated by a numerical example via Monte Carlo simulation and by real electronic system identification.

On the Closed-Loop System Identification with Fractional Models

This paper presents a new model-free adaptive fractional order control approach for linear time-varying systems. An online algorithm is proposed to determine some frequency characteristics using a selective filtering and to design a fractional PID controller based on the numerical optimization of the frequency-domain criterion. When the system parameters are time-varying, the controller is updated to keep the same desired performances. The main advantage of the proposed approach is that the controller design depends only on the measured input and output signals of the process. The effectiveness of the proposed method is assessed through a numerical example.

Manel Chetoui

Papers

New consistent methods for order and coefficient estimation of continuous-time errors-in-variables fractional models

On the Closed-Loop System Identification with Fractional Models

Model-free adaptive fractional order control of stable linear time-varying systems

A bias correction method for fractional closed-loop system identification

EIV methods for system identification with fractional models