## 1 Introduction
There is a need for further research efforts to establish design methodologies and reveal some practical benefits and drawbacks. Therefore, this study aims to investigate the benefits and drawbacks of ML-MR PID-MIT control [25; 28] and addresses design and performance improvement problems in ML-MR structures. To this end, this paper revisits recent multi-loop model reference adaptive PID control techniques, and analytical tuning and performance issues are considered. In analytical optimal tuning of PID controllers, we adopt the integral time absolute error (ITAE) design rule that was proposed by [PERSON] et al. for control of the first order plus dead time dynamic systems. This design rule is based on dimensional analysis, which has been widely utilized to solve complex high-dimensional problems by reducing the number of variables to an essential set [29]. The main reason to use the Tavakoli-Tavakoli ITAE design rule for optimal tuning of PID control loops is that the method limits overshoot with faster establishment of set-point control in time delay systems [29]. The organization of this paper is given as follows. Section 2 details the mathematical background of the multi-loop model reference PID-MIT (ML-MR PID-MIT) control, and presents analytical equations for theoretical background and analytical tuning. After discussing some inherent drawbacks of ML-MR PID-MIT structures (e.g., nonlinearity and amplitude dependence of disturbance rejection performance), several ML-MR control structures are investigated to resolve these drawbacks in the following sections. In these sections, two types of hierarchical ML-MR control structures are considered. (i) Reference modeling of closed loop control loops: These types of MRC structures use a reference model to describe the desired response of closed loop control systems. Section 3 presents a multi-loop model reference PID-MIT control with controller gain modification (ML-MR PID-MIT-CGM) that allows dynamic disturbance rejection via adaptively adjusting Reference to Disturbance Ratio (RDR) performance. To deal with the drawbacks of nonlinearity from the MIT rule, a multi-loop model reference PID integral (ML-MR PID-I) control, which performs a linear adaptation rule, is introduced in this section. Section 4 introduces a multi-loop model reference PID-IM control structure (ML-MR PID-IM) that is proposed as an internal model linear adaptation rule to implement an internal model control (IMC) approach. (ii) Reference modeling of plants: These types of MRC structures use a reference model to describe desired responses of a process or a plant function. Section 5 presents the multi-loop model reference MIT-PID control, which employs the model reference control rule only for plant function adaptation (ML-MR MIT-PID-PFA) in contrast to ML-MR MIT-PID in Section 2. The main advantages of this type of structure come from the fact that all adaptation efforts aim for approximation of plant or process models compared to the reference model. This property is advantageous for time delay systems. Section 6 presents a multi-loop model reference PID-PID control with plant function adaptation (ML-MR PID-PID-PFA). This structure employs a linear adaptation rule for plant function and it can improve control performance for time delay systems. Illustrative design examples cover control the problems of an automatic voltage regulator model, the liquid level of reboilers, large time delay experimental process models, and Twin-Rotor Multi-input multi-output System (TRMS) experimental setup.