## 1 Introduction  
Control systems encounter unpredictable disturbances in real-world control applications. In order to maintain optimal control performance, real control systems should be designed to be robust enough against environmental disturbances, which are mainly unpredictable in character. From a practical point of view, performance degradations of classical control systems are largely caused by their dependence on predetermined system models. In the controller design phase, classical PID controllers are tuned to produce optimal control of a predetermined nominal plant model, which is indeed an idealized representation of a real-world system. The mathematical models of real systems can maintain their validity for predefined operating conditions and ranges, and only observable dynamics of the systems in these range and conditions can be represented by system models. Hence, they omit some inaccessible system dynamics and unknown disturbances, which are not rigorously investigated in the modeling stage. To enhance the representative capability of mathematical modeling, it is well-known that non-linear modeling [1], fractional-order modeling [2], and stochastic modeling [3] methods are often used. However, nowadays, optimal PID tuning methods still rely on integer order dynamics and linear system design approaches for the sake of simplicity and practical effectiveness. Therefore, the coverage potential of these modeling types to represent real-world conditions is quite limited, while mathematical models can represent the behavior of real systems in some presumed idealized conditions. In general, optimal controller design in classical control has been addressed using integer order plant models and linear dynamics. Consequently, the designed controller law becomes mathematically optimal only for those ideal and linearized models operating in numerical simulation environments. However, when they are implemented in real-world conditions, practitioners commonly encounter the fact that the calculated controller coefficients are not practically optimal for real processes and fine-tuning efforts are needed for these controllers to achieve acceptable control performance. Firstly, non-ideal realization of controller function inherently deteriorates control optimality. Secondly, a number of factors such as unmodelled dynamics, noise, environmental disturbance, ageing of system components, and parametric perturbations may have a significant influence on control performance. Therefore, it is obvious that model-based optimal controller design methods should adhere to robust control performance requirements in the controller design stage, including considerations of disturbance rejection performance and insensitivity to model parameter variation. The current study focuses on enhancement of the disturbance rejection performance of classical control loops. The disturbance model is assumed to be additive input disturbance that has unknown characteristics. The disturbance rejection capacity of a classical negative feedback PID control loop is inherently bound [4]. A feasible solution to increase disturbance rejection performance beyond the inherent bounds of a PID control loop is by using a multi-loop control structure [5]. The findings of the current study suggest that a multi-loop model reference (ML-MR) adaptive control structure could be an effective solution to improve the disturbance rejection performance of classical PID control loops. Due to a performance tradeoff between disturbance rejection control and set-point control performances [4], increasing the disturbance rejection control performance can severely deteriorate the set-point performance of classical PID control loops. A primary motivation of this study is the improvement of disturbance rejection control performance without effecting the set-point performance of existing closed loop PID control systems. ML-MR control structures are commonly composed of two hieratical loops.