Beyza Billur İskender

Analysis of an axis-symmetric fractional diffusion-wave problem

This paper presents an axis-symmetric fractional diffusion-wave problem which is considered in polar coordinates. The dynamic characteristics of the system are described with a partial fractional differential equation in terms of the Riemann–Liouville fractional derivative. This continuum problem is reduced to a countable infinite problem by using the method of separation of variables. In this way, the closed form solution of the problem is obtained. The Grunwald–Letnikov approach is applied to take a numerical evaluation. The compatibility and effectiveness of this approach are realized by some simulation results which are obtained by a MATLAB program. It can be seen that the analytical and numerical solutions overlap.

Fractional optimal control problem of an axis-symmetric diffusion-wave propagation

This paper presents the formulation of an axis-symmetric fractional optimal control problem (FOCP). Dynamic characteristics of the system are defined in terms of the left and right Riemann‐Liouville fractional derivatives (RLFDs). The performance index of a FOCP is described with a state and a control function. Furthermore, dynamic constraints of the system are given by a fractional diffusion-wave equation. It is preferred to use the method of separation of variables for finding the analytical solution of the problem. In this way, the closed form solution of the problem is obtained by a linear combination of eigenfunctions and eigencoordinates. For numerical evaluation, the Grunwald‐Letnikov approximation is applied to the problem. Consequently, some simulation results show that analytical and numerical solutions overlap for = 1. This numerical approach is applicable and effective for such a kind of FOCP. In addition, the changing of some variables related to the problem formulation is analyzed.

Fractional Order Control of Fractional Diffusion Systems Subject to Input Hysteresis

This paper concerns the control of a time fractional diffusion system defined in the Riemann-Liouville sense. It is assumed that the system is subject to hysteresis nonlinearity at its input, where the hysteresis is mathematically modeled with the Duhem operator. To compensate the effects of hysteresis nonlinearity, a fractional order Proportional +Integral +Derivative (PID) controller is designed by minimizing integral square error. For numerical computation, the Riemann-Liouville fractional derivative is approximated by the Grunwald-Letnikov approach. A set of algebraic equations arises from this approximation, which can be solved numerically. Performance of the fractional order PID controllers are analyzed in comparison with integer order PID controllers by simulation results, and it is shown that the fractional order controllers are more advantageous than the integer ones.

Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation

This article presents an optimal control problem for a fractional heat conduction equation that describes a temperature field. The main purpose of the research was to find the boundary temperature that takes the thermal stress under control. The fractional derivative is defined in terms of the Caputo operator. The Laplace and finite Fourier sine transforms were applied to obtain the exact solution. Linear approximation is used to get the numerical results. The dependence of the solution on the order of fractional derivative and on the nondimensional time is analyzed.

Parameter Optimization of Fractional Order PI λ D μ Controller Using Response Surface Methodology

This chapter presents optimization of fractional order PI λ D μ control parameters by using response surface methodology. The optimization process is observed on a fractional order diffusion system subject to input hysteresis which is defined with Riemann–Liouville fractional derivative. The system is transferred to a fractional order state space model by using eigenfunction expansion method and then Grunwald–Letnikov approximation is applied to solve the system numerically. The necessary data for response surface analysis are read from the obtained numerical solution. Finally, second-order polynomial response surface mathematical model for the experimental design is presented and the optimum control parameters are predicted from this response surface model. The proposed optimization method is compared with the technique of minimization of integral square error by means of settling time and the results are discussed.

Tuning of fractional order PI λ D μ controller with response surface methodology

This paper presents response surface methodology for tuning of fractional order PIλDμ controller of a fractional order diffusion system subject to input hysteresis which is defined with Riemann-Liouville fractional derivative. Eigenfunction expansion method and the Grünwald-Letnikov numerical technique are used to solve the system. The necessary data for response surface analysis are read from the obtained numerical solution. Then second-order polynomial response surface mathematical model for the experimental design is presented and the optimum controller parameters are predicted from this model. The proposed tuning method is compared with the technique of minimization of integral square error by means of settling time and the results are discussed.

Time-fractional boundary optimal control of thermal stresses

In this paper, a temperature field described by a fractional heat subconduction equation with a boundary temperature control is considered. The foundation of an optimal boundary control to take the thermal stress under constraints is purposed. Problem is formulated in terms of Caputo time-fractional derivative. The solution is found by applying Laplace and finite Fourier sine transforms. In addition, linear approximation is used to get the numerical solution. Consequently, the graphics of numerical results obtained by MATLAB are illustrated.