## 2 Mathematical Model of the Direct Current Electro-Mechanical System  
Electric motors are a fundamental part of many power industrial tools, robotic systems, and household appliances. The direct-current motor (DC motor), is an energy conversion system where the electrical power is transformed into the mechanical. The mathematical model of the DC motor can be presented as a second or third-order system. The second-order mathematical model is used for the presented research aim, chattering suppression. The position control of the electro-mechanical system has a significant characteristic, where the chattering phenomena at the output of the controller affect the wear and heat losses in the motor and electronic components directly. The second-order mathematical model covers the mechanical dynamic, where the electrical transient response is neglected, and only the final values after the transient phenomena of the electrical subsystem are used in the presented controller design. In this case, the accuracy of the mathematical model does not play an important role, because the STA coefficients can be overestimated. With the fast BF adaptation procedure in operation mode two, the overestimated STA parameters do not have a notable effect on the closed-loop response. The second-order mathematical model of the DC motor is presented as  
\[\begin{array}{l}\dot{\varphi}=\omega,\\ J\dot{\omega}=-sgn(\omega)\Big{(}\overline{f}(\omega,\varphi)+B|\omega|\Big{)} +D(i)+d_{in},\end{array} \tag{1}\]where \(D(i)\) is  
\[D(i)=\left\{\begin{array}{ll}k_{e}(i-m_{f})&for\quad i\geq m_{f}\\ 0&for\quad-m_{f}<i<m_{f}\\ k_{e}(i+m_{f})&for\quad i\leq-m_{f}\end{array}\right.. \tag{2}\]  
The coefficients in Equations (1) and (2), where \(\omega\), \(\varphi\), \(d_{in}\), \(J\), \(B\), \(k_{e}\), \(m_{f}\), \(\overline{f}\) are the angular velocity, shaft angle, input disturbance, output disturbance, inertia, viscous friction, the current-torque constant, dead zone value of the input torque and the nonlinear friction coefficient as a function of the angular velocity and the angle, respectively. For better transparency and further STA controller design, the new state vector is \(\left[\varphi\ \omega\right]^{T}=\left[x_{1}\ x_{2}\right]^{T}\). The simplified model with a new state vector is presented as  
\[\begin{array}{ll}\dot{x}_{1}&=x_{2},\\ \dot{x}_{2}&=-J^{-1}\Big{(}B+\frac{k_{e}k_{m}}{K}\Big{)}x_{2}+\frac{k_{e}}{J \overline{k}}\widetilde{D}(u)+J^{-1}d_{in},\\ &=-fx_{2}+gu+td_{in},\end{array} \tag{3}\]  
where \(R,\ k_{e}\), \(u\) are motor winding resistance, mechanical constant and input voltage, respectively. The nonlinearity \(\widetilde{D}\) is  
\[\widetilde{D}(u)=\left\{\begin{array}{ll}(u-Rm_{f})&for\quad u\geq Rm_{f}\\ 0&for\quad-Rm_{f}<u<Rm_{f}\\ (u+Rm_{f})&for\quad u\leq-Rm_{f}\end{array}\right.. \tag{4}\]  
One of the crucial objectives of the controller design is to ensure the tracking of the different changed signals, where the tracking error is introduced as  
\[\begin{array}{ll}e_{1}=x_{d}-x_{1},\\ e_{2}=\dot{x}_{d}-x_{2}.\end{array} \tag{5}\]  
where \(x_{d}\) and \(\dot{x_{d}}\) are the reference signal and its time derivative, respectively. By applying the error variable Equation (5) to the system Equation (3), the new transformed system is  
\[\begin{array}{ll}\dot{e}_{1}&=e_{2}\\ \dot{e}_{2}&=-fe_{2}-gv-td_{in}+\ddot{x}_{d}+f\dot{x}_{d},\\ &=-fe_{2}-gv+\overline{d}.\end{array} \tag{6}\]  
For the STA controller design, the lumped term of the disturbance, uncertainty and reference derivatives \(\overline{d}\) need to be a strictly bounded function, where  
\[\begin{array}{l}\overline{d}=\ddot{x}_{d}+f\dot{x}_{d}-td_{in},\\ \sup_{t\geq 0}\overline{d}\left|\overline{d}\right|\leq\Delta_{d}<\infty. \end{array} \tag{7}\]  
The presented system in Equation (6) is used further for an adaptive STA-positioning controller design, based on the BF adaptation technique.