## 2 The Physical Model  
We will consider the mechatronic system schematically shown in Figure 1. It consists of a rigid DC motor and a mechanical load in the form of a solid disk with the mass moment of inertia \(J_{l}\) (compare Reference [19]). We begin deriving the equations for rotor dynamics and current distribution in the electrical system by defining the relationship between the torque \(T_{ma}\) of the motor (\"a\" denotes the actual model) and the current \(i_{a}\) in the motor's armature circuit,  
\[T_{ma}(t)=K_{t}i_{a}(t). \tag{1}\]  
The parameters used in the work are listed in Table 1. The electromotive force \(\mathcal{E}\) depends on the rotational speed in a linear manner,  
\[\mathcal{E}(t)=K_{b}\dot{\theta}_{ma}(t). \tag{2}\]Adding the moments of forces acting on the rotor shaft in the mechanical system and taking into account the moment of force from the damping of the mechanical system of the motor, we obtain:  
\[\bar{\theta}_{ma}(t) =T_{ma}(t)-T_{la}(t)-b_{m}\dot{\theta}_{ma}(t), \tag{3}\] \[\bar{\theta}_{ma}(t) =\frac{1}{J}\big{(}K_{t}\dot{i}_{a}(t)-T_{la}(t)-b_{m}\dot{\theta}_ {ma}(t)\big{)}. \tag{4}\]  
Deriving the second differential equation describing the dynamics of current variations in the electrical system, we will use the second Kirchhoff's law and the mathematical description of electronic components, i.e., the resistance \(R\) and the inductance \(L\) included in it (analytical calculations of parameters of permanent magnets motors are described in Reference [20]). Based on [PERSON]'s second law, the sum of voltage drops on resistive elements in a closed circuit is equal to the sum of electromotive forces present in this circuit:  
\[L\dot{i_{a}}(t) =u_{a}(t)-R\dot{i_{a}}(t)-\mathcal{E}(t), \tag{5}\] \[\dot{i_{a}}(t) =\frac{1}{L}\big{(}u_{a}(t)-R\dot{i_{a}}(t)-K_{b}\dot{\theta}_{ma} \big{)}, \tag{6}\]  
where \(u_{a}\) is defined by Equation (13). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Notation** & **Value** & **Unit** \\ \hline rotor inertia & \(J_{m}\) & \(2.3\cdot 10^{-3}\) & \(\mathrm{kg}\cdot\mathrm{m}^{2}\) \\ load inertia & \(J_{l}\) & \(10\cdot J_{m}\) & \(\mathrm{kg}\cdot\mathrm{m}^{2}\) \\ motor damping & \(b_{m}\) & \(1.2\cdot 10^{-4}\) & \(\mathrm{N}\cdot\mathrm{m}\cdot\mathrm{s}/\mathrm{rad}\) \\ load damping & \(b_{l}\) & \(0.02\) & \(\mathrm{N}\cdot\mathrm{m}\cdot\mathrm{s}/\mathrm{rad}\) \\ torque constant & \(K_{t}\) & \(0.5\) & \(\mathrm{N}\cdot\mathrm{m}/\mathrm{A}\) \\ inductance & \(L\) & \(1.4\cdot 10^{-4}\) & \(\mathrm{H}\) \\ resistance & \(R\) & \(0.9\) & \(\Omega\) \\ back-EMF & \(K_{b}\) & \(0.5\) & \(\mathrm{V}\cdot\mathrm{s}/\mathrm{rad}\) \\ stiffness & \(k_{rl}\) & \(400\) & \(\mathrm{N}\cdot\mathrm{m}/\mathrm{rad}\) \\ damping & \(b_{rl}\) & \(0.01\) & \(\mathrm{N}\cdot\mathrm{m}\cdot\mathrm{s}/\mathrm{rad}\) \\ reference control & \(\bar{\beta}\) & \(1\) & \(\mathrm{V}/\mathrm{rad}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the investigated motor model. Figure 1: Physical model of the Direct Current (DC) permanent magnet motor driving a softly attached mechanical load. The load is connected to the rotor by a flexible connector, which has damping and elastic properties dependent on the relative angular displacement and velocity, respectively, between the rotor and the disk-shaped inertial load. Let the load moment \(T_{l}\), operating in a flexible connection of both shafts, depends on the sum of its two components, i.e., the moment \(T_{ls}\), resulting from the difference of angular positions \(\delta\theta_{a}\) and the moment \(T_{ld}\), resulting from the difference in angular velocities \(\delta\omega_{a}\), which we write in the form  
\[ \begin{split} T_{l}(t)&=T_{ls}(t)+T_{ld}(t)\\ &=k_{rl}\delta\theta_{a}(t)+b_{rl}\delta\omega_{a}(t)\\ &=k_{rl}(\theta_{ma}(t)-\theta_{la}(t))+b_{rl}(\theta_{ma}(t)- \theta_{la}(t)).\end{split}