Analysis and Optimal Control of φ-Hilfer Fractional Semilinear Equations Involving Nonlocal Impulsive Conditions

The goal of this paper is to consider a new class of φ-Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the fixed point theorem, the existence and uniqueness of mild solutions are obtained for the proposed fractional system. Symmetrically, we discuss the existence of optimal controls for the φ-Hilfer fractional control system. Our main results are well supported by an illustrative example.

Gas Turbine Aerodynamics Improvement Via a Design of Intelligent Fractional Control

Gas turbines are complex processes characterized by the instability and uncertainty of various sources. The range of useful operating in an axial compressor which is part of a turbine gas is limited by aerodynamic instabilities that are surge and rotating stall. This paper presents two intelligent fractional order sliding mode controllers. At first, a robust sliding fractional surface form is proposed to deal with hazardous phenomena which limit compression systems performance, and speed transitions, which can lead to temporary stall development, pressure drop at the output, degrade the effective operation of compressors and consequently gas turbines. Second, to reduce the chattering/fluctuation in control, a fuzzy logic and finite time criterion are used as switching control at the reaching phase in the sliding mode control. Additionally, the controller gains are obtained by offline multi-objective Particle swarm optimization (MOPSO) search. Finally, the surge and rotating stall of a Variable Speed Axial Compressor (VSAC) in a gas turbine are investigated under the system nonlinearities and also in presence of an external disturbance and perturbations. The simulation results signify the performance of the two MOPSO-based fractional sliding mode controllers.

Harvest Management Problem with a Fractional Logistic Equation

In this article, we study a fractional control problem that models the maximization of the profit obtained by exploiting a certain resource whose dynamics are governed by the fractional logistic equation. Due to the singularity of this problem, we develop different resolution techniques, both for the classical case and for the fractional case. We perform several numerical simulations to make a comparison between both cases.

An Optimal Tilt Integral Derivative Applied to the Regulation of DC Link Voltage in a Stand-Alone Hybrid Energy System

This paper presents an application of fractional control scheme named Tilt Integral Derivative (TID) to control a stand-alone hybrid energy system composed of a solar photovoltaic (PV) system and a battery bank (BB). A three-level NPC inverter is inserted in order to increase the efficiency of the energy injected into the AC load. Variation in solar radiation or AC load may cause power imbalance, which leads to variation in DC link voltage. As a solution, a buck-boost converter is connected between the DC link and the battery bank to ensures the transfer of energy in both directions. The parameters of TID controller were tuned using a powerful optimization technique known a Genetic Algorithm (GA) by minimizing the Mean Square Error (MSE) used as a performance index. The effectiveness of the proposed TID controller is demonstrated through a comparison with a conventional Proportional-Integral-Derivative (PID) controller, whose parameters are computed by the pidtool function of the Matlab/Simulink tool where the DC link voltage behavior is previously modeled by a capacitor transfer function. The obtained results show that the proposed TID controller provides a stable DC bus with low chattering, regardless of the rapid irradiation and load changes, when compared to a conventional PID controller.

On asymptotically stabilizing control of nonlinear fractional control systems using an optimization scheme

In this paper, stabilization of the nonlinear fractional order systems with unknown control coefficients is considered where the dynamic control system depends on the Caputo fractional derivative. Related to the nonlinear fractional control (NFC) system, an infinite-horizon optimal control (OC) problem is first proposed. It is shown that the obtained OC problem can be an asymptotically stabilizing control for the NFC system. Using the help of an approximation, the Caputo derivative is replaced with the integer order derivative. The achieved infinite-horizon OC problem is then converted into an equivalent finite-horizon one. According to the Pontryagin minimum principle for OC problems and by constructing an error function, an unconstrained minimization problem is defined. In the optimization problem, trial solutions are used for state, costate and control functions where these trial solutions are constructed by using a two-layered perceptron neural network. A learning algorithm with convergence properties is also provided. Two numerical results are introduced to explain the main results.

L 1 Adaptive Fractional Control Optimized by Genetic Algorithms with Application to Polyarticulated Robotic Systems

Recently, an adaptive control approach has been proposed. This approach, named L 1 adaptive control, involves the insertion of a low-pass filter at the input of the Model Reference Adaptive Control (MRAC). This controller has been designed to overcome several limitations of classical adaptive controllers such as (i) the initialization of estimated parameters, (ii) the stability problems with high adaptation gains, and (iii) the appropriate parameter excitation. In this paper, a new design of the filter is presented, used for L 1 adaptive control, for which the desired performances are guaranteed (appropriate values of the control during start-up, a high filtering of noises, a reduced time lag, and a reduced energy consumption). Parameters of the new proposed filter have been optimised by genetic algorithms. The proposed L 1 adaptive fractional control is applied to a polyarticulated robotic system. Simulation results show the efficiency of the proposed control approach with respect to the classical L 1 adaptive control in the nominal case and in the presence of a multiplicative noise.

Duality theory of fractional resolvents and applications to backward fractional control systems

We study the duality theory for fractional resolvents, extending and improving some corresponding theorems on semigroups. As applications, we develop the variational technique to analyze the finite-approximate controllability of a backward fractional control system with a right-sided Riemann-Liouville fractional derivative. Moreover, validity of our theoretical findings is given by a fractional diffusion model.

The Frequency and Real-Time Properties of the Microcontroller Implementation of Fractional-Order PID Controller

The paper presents time, frequency, and real-time properties of a fractional-order PID controller (FOPID) implemented at a STM 32 platform. The implementation uses CFE approximation and discrete version of a Grünwald–Letnikov operator (FOBD). For these implementations, experimental step responses and Bode frequency responses were measured. Real-time properties of the approximations are also examined and analyzed. Results of tests show that the use of CFE approximation allows to better keep the soft real-time requirements with an accuracy level a bit worse than when using the FOBD. The presented results can be employed in construction-embedded fractional control systems implemented at platforms with limited resources.