The simplicity, transparency, reliability, high efficiency and robust nature of PID controllers are some of the reasons for their high popularity and acceptance for control in process industries around the world today. Tuning of PID control parameters has been a field of active research and still is. The primary objectives of PID control parameters are to achieve minimal overshoot in steady state response and lesser settling time. With exception of two popular conventional tuning strategies (Ziegler Nichols closed loop oscillation and Cohen-Coon's process reaction curve) several other methods have been employed for tuning. This work accords a thorough review of state-of-the-art and classical strategies for PID controller parameters tuning using metaheuristic algorithms. Methods appraised are categorized into classical and metaheuristic optimization methods for PID parameters tuning purposes. Details of some metaheuristic algorithms, methods of application, equations and implementation flowcharts/algorithms are presented. Some open problems for future research are also presented. The major goal of this work is to proffer a comprehensive reference source for researchers and scholars working on PID controllers.

This study proposes a new fractional mathematical model to study the impact of vaccination on COVID-19 outbreaks by categorizing infected people into non-vaccinated, first dose-vaccinated, and second dose-vaccinated groups and exploring the transmission dynamics of the disease outbreaks. We present a non-linear integer order mathematical model of COVID-19 dynamics and modify it by introducing Caputo fractional derivative operator. We start by proving the good state of the model and then calculating its reproduction number. The Caputo fractional-order model is discretized by applying a reliable numerical technique. The model is proven to be stable. The classical model is fitted to the corresponding cumulative number of daily reported cases during the vaccination regime in India between 01 August 2021 and 21 July 2022. We explore the sensitivities of the reproduction number with respect to the model parameters. It is shown that the effective transmission rate and the recovery rate of unvaccinated infected individuals are the most sensitive parameters that drive the transmission dynamics of the pandemic in the population. Numerical simulations are used to demonstrate the applicability of the proposed fractional mathematical model via the memory index at different values of $0.7,0.8,0.9$ and 1. We discuss the epidemiological significance of the findings and provide perspectives on future health policy tendencies. For instance, efforts targeting a decrease in the transmission rate and an increase in the recovery rate of non-vaccinated infected individuals are required to ensure virus-free population. This can be achieved if the population strictly adhere to precautionary measures, and prompt and adequate treatment is provided for non-vaccinated infectious individuals. Also, given the ongoing community spread of COVID-19 in India and almost the pandemic-affected countries worldwide, the need to scale up the effort of mass vaccination policy cannot be overemphasized in order to reduce the number of unvaccinated infections with a view to halting the transmission dynamics of the disease in the population.