## 7 Discussions and Conclusions
* Stability analysis of linear ML-MR control structures is relatively straightforward and is performed by obtaining the transfer function of the whole structure and checking for pole placements for left half-plane (LHP) stability. The satiability of a nonlinear ML-MR structure, such as ML-MR PID-MIT control, along with the stability and convergence conditions, are derived on the basis of the bounded input and bounded output stability state of the model error (see Appendix A). * Results in this study are based on simulation models. The reference models perfectly match the controlled systems. It is very important to note that model mismatches due to limitations in mathematical modeling and parametric model perturbations in real systems may have severe effects on the control performance and stability of the presented ML-MR adaptive control structures in real-world applications. The sensitivity of these structures to model mismatching should be carefully investigated in future works. The fragility of the system stability due to model imprecision should be carefully considered when designing real control systems. Moreover, the resolution of sensors and I/O ports of control cards are important in practical applications. * This study is a theoretical demonstration of six different ML-MR adaptive control structures. Future works should address experimental performance evaluations and practical realization issues of ML-MR adaptive control structures. \begin{table} \begin{tabular}{c c c c c c} \hline \hline
**ML-MR Control** & \multirow{2}{*}{**Character**} & \multirow{2}{*}{**Dependence to**} & **Negative Effects** & \multirow{2}{*}{**Applicability for Time**} & **Indications in** \\
**Structures** & & **Set-Point** & & **on Set-Point** & **Daly Systems** & **Simulation** \\  & & **Level** & **Performance** & & **Results** \\ \hline ML-MR PID-MIT & Nonlinear & Dependent & None & None & Figures 3 and 4 \\ ML-MR & Nonlinear & Dependent & None & None & Figure 8 \\ PID-MIT-CGM & Linear & Independent & None & None & Figure 10 \\ ML-MR PID-IM & Linear & Independent & None & None & Figure 14 \\ ML-MR & Nonlinear & Dependent & None & Applicable to some degree & Figure 17 \\ MIT-PID-PFA & & & & & \\ \hline \hline \end{tabular}  Some notable remarks: * ML-MR adaptive control structures are applicable in all classical control loops in order to improve the disturbance rejection performance without modifying the parameters of the existing control system or degrading the set-point performance. This is an important advantage of hierarchical ML-MR structures in control engineering * Integration approaches of the model reference control loop into classical control loops are carried out in two steps: outer loop integration uses closed loop control systems as reference models, such as ML-MR PID-MIT, ML-MR PID-I, and ML-MR PID-IM controls; and inner loop integration uses models of plant or process functions as reference models, such as ML-MR MIT-PID-PFA and ML-MR PID-PID-PFA. Outer loop integrations cover the whole closed control loop to deal with disturbances, and several works have indicated that these type integrations are preferable solutions for improving fault tolerance and disturbance rejection in the whole control loop [25; 26; 27; 28]. Inner loop integration only encloses the plant function, and hence it can be employed to deal with external disturbances or parametric perturbations that influence the controlled system. * ML-MR PID-MIT-CGM and ML-MR PID-I structures can adaptively increase the RDR performance of the closed control loops without affecting set-point performance. The ML-MR PID-IM control structure benefits from having an internal model controller design, however it should be noted that its performance is very sensitive to the accuracy of plant and controller models. * Large-time delay systems that set the point and disturbance rejection performance of ML-MR PID-PID-PFA control structures can surpass the disturbance rejection performance of a single-loop optimal FOPID control system.