## 2 System Modeling  
The 2 MM system represents an actuating motor coupled to a mechanical system with two lumped inertias connected by a low-stiffness shaft and couplings [7; 18; 37; 38; 39; 40]. The block diagram in Figure 1 shows the 2 MM system with its equivalent current control loop. In this diagram, \(i_{ref}\) is the reference current (\(A\)), \(K_{t}\) is the torque constant (N\(\cdot\)m/A), \(J_{1}\) is the equivalent motor-side inertia (kg\(\cdot\)m\({}^{2}\)), and \(J_{2}\) is the equivalent load-side inertia (kg\(\cdot\)m\({}^{2}\)). The motor-side and load-side angular velocities are represented by \(\omega_{1}\) and \(\omega_{2}\), respectively (rad/s). The inertias are coupled through a low-stiffness shaft \(K_{s}\), which is subjected to torsional torque \(\tau_{s}\) (N\(\cdot\)m). Here, the Coulomb friction on both sides is taken into account, while external forces are neglected, as the system is considered to move freely. The following describes the mechanical dynamics of the 2 MM system. The mathematical model of the 2 MM system is described by the following dynamic equations:  
\[J_{1}\frac{d\omega_{1}}{dt}= \tau_{m}-B_{1}\omega_{1}-\tau_{s}-C_{1}sign(\omega_{1}) \tag{1}\] \[J_{2}\frac{d\omega_{2}}{dt}= \tau_{s}-\tau_{l}-B_{2}\omega_{2}-C_{2}sign(\omega_{2})\] (2) \[\tau_{s}= K(\theta_{1}-\theta_{2}) \tag{3}\]  
To aid in the identification of system parameters using the proposed method, these equations are normalized with respect to the rated speed and torque. Normalizing parameters is crucial in optimization methods, ensuring faster convergence, preventing numerical instability, and enabling the consistent comparison of system performance. It eliminates scale differences, improves interpretability, robustness, and result consistency. Additionally, normalization facilitates uniform distribution in the search space, reducing the likelihood of encountering local optima and enhancing optimization efficiency and robustness [41; 42]. The normalized parameters for the motor and load side inertias are calculated as \(J_{1n}=J_{1}(\omega_{rated}/\tau_{rated})\) and \(J_{2n}=J_{2}(\omega_{rated}/\tau_{rated})\), respectively. The normalized shaft stiffness is calculated as \(K_{n}=K_{n}/\omega_{rated}\). The normalized Coulomb and viscous friction coefficients for both the motor and load sides are calculated as \(C_{1n}=C_{1}/\tau_{rated}\), \(C_{2n}=C_{2}/\tau_{rated}\), \(B_{1n}=B_{1}(\omega_{rated}/\tau_{rated})\), and \(B_{2n}=B_{2}(\omega_{rated}/\tau_{rated})\). The motor and load speeds are normalized using \(\omega_{1n}=\omega_{1}/\omega_{rated}\) and \(\omega_{2n}=\omega_{2}/\omega_{rated}\). The normalized equations are denoted with a subscript '\(n\)' to indicate that they are normalized with respect to the rated speed and torque of the actuator. This includes variables and parameters, such as the normalized angular velocities (\(\Omega_{1}\), \(\Omega_{2}\)), normalized angular positions (\(\Theta_{1}\), \(\Theta_{2}\)), normalized inertias (\(J_{1n}\), \(J_{2n}\)), and normalized viscous damping and Coulomb friction coefficients (\(B_{1n}\), \(B_{2n}\), \(C_{1n}\), \(C_{2n}\)). Additionally, the normalized motor and load torques are represented by \(\tau_{mn}\) and \(\tau_{ln}\), respectively, and the normalized shaft torsional torque is represented by \(\tau_{sn}\). To analyze the system behavior, the transfer functions of the 2 MM system can be obtained by linearizing the system equations and assuming that the Coulomb friction is small and can be neglected.