## 1 Introduction
These are the reference model loop, which describes the desired response of a system, and the control loop, which deals with the stability of control systems. The control loop is the primary loop that maintains the stability of the system. The secondary loop is the reference model loop, also called the adaptation loop, which handles the preservation of the desirable control performance of the system, which is described by the reference model response. When a disturbance acts on the control loop, the adaptation loop contributes to the control efforts for the reduction of the negative effects of disturbances on system outputs, thus restoring control performance by means of reference responses that are generated by the reference model. In literature, numerous ML-MR control structures have been proposed and their improvements in terms of control performance have been demonstrated. In former works, direct model reference adaptive control (MRAC) structures have been addressed [5; 6; 7; 8; 9; 10]. Direct MRAC structures perform online self-tuning of controller parameters according to a predefined reference model [5; 6; 7; 8]. Updating controller parameters in control action, known as online controller tuning, may lead to short-term instability and control performance degradations. Therefore, the stability and robust control performance of MRAC structures in real conditions remain questionable and direct MRAC control methodologies have not been accepted or adopted by practitioners [8]. The industrial applications of the MRAC method have remained limited in comparison to the widespread adoption of classical PID control loops. PID control loops have been used as industrial standard control systems because of their well-established design scheme. The simple structure and reliability of PID control law, which facilitates realization in real systems, are prominent in industrial control applications. [PERSON] has addressed performance and stability issues of direct MRAC methods and presented a stable direct MRAC methodology according to passivity conditions [8]. However, to avoid online controller parameter updates in control action,one branch of ML-MR control research studies aims to combine a classical PID control system with reference model control in a hierarchical multi-loop manner in order to benefit from the stability and set-point performance of the classical control loop and the robust control performance improvements of the model reference control (MRC) system. The topic of multi-loop control involves a wide variety of control systems that employ multi-loop control structures, including cascaded control loops [11; 12], multi-input multi-output control systems [13], use of additional control loops for performance improvements [14; 15], direct model reference control based on output matching [16; 17], direct model reference adaptive control with online controller tuning [5; 6; 7; 8; 9; 10], model reference adaptive control based on Massachusetts Institute of Technology (MIT) rule [18; 19; 20; 21; 22; 23], backstepping-based adaptive PID control [24], and hierarchical ML-MR adaptive control [25; 26; 27; 28]. The current study investigates several configurations of hierarchical ML-MR adaptive control structures. Hierarchical ML-MR adaptive PID control has been investigated, and certain benefits of this structure related to fault tolerance [25] and disturbance rejection [26] have been discussed. This structure can be easily applied to the existing closed loop PID control systems, thus transforming the PID control systems into model reference adaptive PID control systems. This presents a significant potential to improve the robust control performance of industrial PID control systems without largely modifying the existing PID loops. Although a growing research trend has been initiated for the use of ML-MR adaptive PID structures to improve the control performance of the closed loop PID control family [25; 26; 27], this control structure is still in the early stages of development.