## 2 Preliminaries
### Digital Control System  
Due to the fact that the implementation of the regulators is done digitally, an example of numerical implementation of the conventional position control system is presented below. The speed controller is approximated with the trapezoid method:  
\[s=\frac{2}{h}\frac{z-1}{z+1}, \tag{7}\]  
and its function in \(z\) results:  
\[H_{R\omega}(z)=K_{R\omega}\frac{\left(1+\frac{h}{2T_{R\omega}}\right)z-\left( 1-\frac{h}{2T_{R\omega}}\right)}{z-1} \tag{8}\]  
The sampling period \(h\) is chosen to be much shorter than the shortest time constant \(h=0.01\) s << \(T_{T\omega}=0.1\) s.  
The numerical algorithm for implementing the speed controller is:  
\[M_{k+1}^{*}=M_{k}^{*}+a_{1}e_{\omega k+1}+a_{2}e_{\omega k} \tag{9}\]  
where \(M_{k}^{*}\) is the torque reference at moment \(t_{k}=k\cdot h\), \(e_{\omega k}\) is the speed error at moment \(t_{k}\), and \(a_{1}=1877\) and \(a_{2}=-1873\) are two coefficients.  
The speed controller may be implemented digitally, using a simple microcontroller, with two multiplication and two summation operations and memorizing reference torque and error values at times \(t_{k}\) and \(t_{k+1}\).  
The block diagram of the numerical speed control system is shown in Figure 3.  
Figure 1: Block diagram of the linear control system.  
Figure 2: Block diagram of the linear control system.  
The characteristics of position and speed, in response to the step signal, for the discrete time system, are identical to those of the system in continuous time. This fact demonstrates that the dimensioning of the speed controller in discrete time, by approximation with the trapezoid method, starting from the values of the coefficients of the controller in continuous time, was correct. This control system assures the same values for the efficiency criteria as the control system with error feedback.  
Closer to reality modeling of the control system includes speed, torque, and position limitations, as seen in Figure 4.  
In the control structure from Figure 4, an anti-wind-up circuit has been introduced, in parallel with the speed controller, which also ensures the limitation of the prescribed torque. A prescribed speed limit and position limit have also been introduced.  
The step responses of the variables of the digital position control system with limitations are shown in Appendix A. It is observed that the position response time is short, as in the case of the linear control system, with zero error in the steady state and asymptotic, aperiodic behavior. The sampling period and the controller coefficients were well chosen.  
Analyzing these characteristics, the following observations arise: The speed reference is limited at 0.3 rad/s, which leads to a decrease in control torque at the start. A lower torque results in a longer response time. It is observed that the position response time is still short compared to the high moment of inertia of an object in motion. Regarding the speed characteristic, it is observed that the speed is not limited at the start, but it is limited in a permanent functioning regime. So, the object rotates rapidly with a limitation in command power. Simulation of the linear control system was made considering infinite command power. In reality, the command has finite power, which leads to a longer response time. It is observed that the position response time is short, as in the case of the linear control system, with zero error in the steady state and asymptotic, aperiodic behavior. The sampling period and the controller coefficients were well chosen.