There is considerable interest in the study of fractional‐order systems but obtaining accurate time domain responses is a difficult problem. This is because all methods reported on to date use approximations for the fractional derivative both for analytical‐based computations and more relevantly in simulation studies. This means unlike in integer systems exact simulations are not available and thus for considering non‐linear problems and comparisons with measured data no exact solution reference exists. In this study, the authors provide a major breakthrough for this situation by developing methods which allow the exact computation of the time responses of fractional‐order systems.

The choice of battery type for electric vehicles is of great importance. For electric vehicle manufacturers, this issue is the biggest engineering problem. An electric vehicle produced with the right battery type will undoubtedly be preferred by users. In this study, the charge and discharge characteristics of four different battery types with a fixed charger were investigated. The batteries were then placed on a hybrid electric vehicle model and the charge and discharge characteristics were examined. The batteries were then placed on a hybrid electric vehicle model and the charge and discharge characteristics were examined.

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.

Model reference adaptive control (MRAC) could be designed using different adaptation mechanisms. An adaptation mechanism could affect the Model Reference Adaptive System (MRAS) performance. High adaptive gain increases the transient response performance while decreasing the robustness of the MRAS even under perfect conditions (no time delay, without unmatched uncertainty and absence of disturbance) and increasing control signal oscillation in classical MRAC. Classical MRAS do not perform satisfactorily on such time-delayed adaptive signals. This study presents and compares two different adaptation mechanisms in MRAS, which are the low-frequency learning method on MRAC and MRAC with PI adjustment mechanism. We have proposed a modification to the low-frequency learning method on MRAC to get a transient response as good as MRAC with a PI adjustment mechanism on the time delay system. The best performance is obtained from MRAS with PI adaptation mechanism.

Recently, studies on delay-based controllers that consider the positive effects of time delay have been increasing in the literature. In this context, a proportional integral retarded (PIR) controller structure has been proposed. There are two important factors in the design of a delay-based controller: closed-loop stability and controller tuning. The motivation for this study is that there is no previous literature on stability region analysis on PIR controllers. To address this gap in the literature, this paper presents, for the first time, a comprehensive stability region analysis for PIR controllers. The study derives the stability boundary locus (SBL) equations for time-delay processes with PIR controllers, allowing stable regions to be defined analytically and graphically in the two-dimensional controller parameter space. In order to obtain the SBL, the stability boundary line equations are derived, and the variations in the case of different types of time-delay processes are presented with mathematical equations and proved with examples. Another novelty of this work is its systematic investigation of how the controller time delay affects the stability region when it is smaller, equal or larger than the process delay. The proposed method fills a critical gap in the design and tuning of PIR controllers by providing a reliable tool by which to visualize and understand stability properties.