In recent years, there has been considerable interest in the study of feedback systems containing processes whose dynamics are best described by fractional order derivatives. Various situations have been cited for describing heat flow and aspects of bioengineering, where such models are believed to be superior. In many situations these feedback systems are not linear and information on their stability and the possibility of the existence of limit cycles is required. This paper presents new results for determining limit cycles using the approximate describing function method and an exact method when the nonlinearity is a relay characteristic.

System identification is an important methodology used in control theory and constitutes the first step of control design. It is known that many real systems can be better characterized by fractional-order models. However, it is often quite complex and difficult to apply classical control theory methods analytically for fractional-order models. For this reason, integer-order models are generally considered in classical control theory. In this study, an alternative approximation method is proposed for fractional-order models. The proposed method converts a fractional-order transfer function directly into an integer-order transfer function. The proposed method is based on curve fitting that uses a quadratic system model and Equilibrium Optimizer (EO) algorithm. The curve fitting is implemented based on the unit step response signal. The EO algorithm aims to determine the optimal coefficients of integer-order transfer functions by minimizing the error between general parametric quadratic model and objective data. The objective data are unit step response of fractional-order transfer functions and obtained by using the Grünwald-Letnikov (GL) method in the Fractional-Order Modeling and Control (FOMCON) toolbox. Thus, the coefficients of an integer-order transfer function most properly can be determined. Some examples are provided based on different fractional-order transfer functions to evaluate the performance of the proposed method. The proposed method is compared with studies from the literature in terms of time and frequency responses. It is seen that the proposed method exhibits better model approximation performance and provides a lower order model.

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of $\alpha \in \{0.1,0.2,0.3,\dots ,0.9\}$ . Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for $1/(s^{\alpha }+1)$ . By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of $\alpha$ for $0<\alpha <1$ . Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.

In this study, proportional-integral (PI) controller design with a geometric approach for first order time-delayed systems is presented. This method can be expressed as an improved version of the weighted geometric center method. The method is based on calculating the center of gravity of a stability triangle selected inside the stability boundary locus (SBL). Stability boundary is determined along the ( k_p , k_i ) axes with the SBL. A stability triangle is formed with the two boundary points of the SBL intersecting the k_p axis for k_i = 0 and the point corresponding to the weighted geometric center value of the angular frequency. The center of gravity of the determined stability triangle gives the optimum PI control gains. Also, analytical solution method for the PI controller design is presented. The proposed method (stability triangle) is examined on numerical examples. The time response performances of the controller calculated with the proposed method and the controllers calculated using the weighted geometric center method were compared. In addition, the comparisons of the PI controller determined by the proposed method and the different controllers selected in the neighborhood of this point are included. Comparisons with the studies in the literature including the PI controller design of the time-delayed first-order open-loop unstable transfer function are presented. As a result, it has been seen that the proposed method determines the parameters that provide optimum system performance in the tested region.

In this paper, limit cycle frequency, pulse width and stability analysis are examined using different methods for relay feedback nonlinear control systems with integer or fractional order plant transfer functions. The describing function (DF), A loci, a time domain method formulated in state space notation and Matlab/Simulink simulations are used for the analysis. Comparisons of the results of using these methods are given in several examples. In addition, the work has been extended to fractional order systems with time delay. Programs have been developed in the Matlab environment for all the theoretical methods. In particular, Matlab programs have been written to obtain a graphical solution for the A loci method, which can precisely calculate the limit cycle frequency. The developed solution methods are shown in various examples. The major contribution is to look at finding limit cycles for relay feedback systems having plants with a fractional order transfer function (FOTF). However, en route to this goal new assessments of limit cycle stability have been done for a rational plant transfer function plus a time delay.

In this paper, an optimization method is proposed to design a fractional order system with PID controller by taking Bode’s ideal transfer function as reference model. In the study, PID controllers are preferred because of their simplicity, reliability and robustness. PID controller parameters can be obtained by optimization method. In order to obtain the desired time response, it is sufficient to set two parameters in the Bode’s ideal transfer function. The Bode's ideal transfer function was considered as the reference model and compared with the generated model. The error in the output signal is minimized by the integral performance criteria, and the PID controller parameters are optimized. Integral performance criteria are frequently used in evaluating the performance of control systems. In Simulink model, Matsuda's 4th order integer approximation model is used for fractional order control system. Finally, the success of the optimization method is seen in the given numerical example.