## 2 Preliminaries
### State Feedback Control System  
As a preliminary treatment for fuzzy control, the state feedback control was analyzed [35]. The structure of the position state feedback control system is presented in Figure 5.  
Figure 4: Block diagram of digital control system with limitations.  
Figure 3: Block diagram of digital control system.  
The state equations of systems are:  
\[\begin{split}\frac{d\omega}{dt}&=-\frac{k_{f}}{f}\omega+ \frac{1}{f}M\\ \frac{d\theta}{dt}&=\omega\\ \frac{d\omega_{\text{e}i}}{dt}&=\theta^{*}-\theta \end{split} \tag{10}\]  
where the system matrices are:  
\[A=\left[\begin{array}{ccc}-\frac{k_{f}}{f}&0&0\\ 1&0&0\\ 0&-1&0\end{array}\right],\;\;b=\left[\begin{array}{c}\frac{1}{f}\\ 0\\ 0\end{array}\right],\;\;b_{r}=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],\;\;c=\left[\begin{array}{cc}0&1&0\end{array}\right] \tag{11}\]  
and the command low is:  
\[M=K_{M}\big{(}K_{\Theta p}e_{0i}-K_{\omega}\omega-K_{\Theta}\theta\big{)} \tag{12}\]  
It can be seen that a third state variable \(e_{0i}\) was introduced at the output of an integrator block on the path of the position error. This leads to zero steady-state error:  
\[\lim_{t\rightarrow\infty}e_{\Theta}(t)=0 \tag{13}\]  
The design of the state feedback command low is presented below. First, the controllability matrix is calculated:  
\[M_{c}=\left[\begin{array}{cc}b&Ab&A^{2}b\end{array}\right] \tag{14}\]  
The state feedback vector \(f\) is determined with relation  
\[f=-m_{c3}.\mu(A), \tag{15}\]  
where \(m_{c3}\) is the third line of controllability matrix.  
The poles of the closed loop control system are imposed such that the control efficiency criteria are to be ensured: overshoot \(\sigma_{1}\%\) = 0 and a small response time. The characteristic polynomial \(\mu(s)\) is calculated, and  
\[f=K_{M}\big{[}\begin{array}{cc}K_{\omega}&K_{\Theta}&K_{\Theta p}\end{array} \right] \tag{16}\]  
The values obtained for the feedback coefficients are, with approximation: \(K_{M}\) = 800, \(K_{\Theta}\) = 15, \(K_{\Theta p}\) = 10 and \(K_{\omega}\) = 5.  
This design assures the pole-zero map presented in Appendix A. The chosen poles assure an asymptotic aperiodic behavior of position at step input.  
The step signal responses of the variables of state feedback control system are shown in Appendix A.  
It can be seen that the torque and speed are limited. The response time is shorter than in the case of the error feedback control system. So, the object rotates rapidly with a  
Figure 5: Block diagram of state feedback control system.  
limitation in command power. It is observed that the position response time has zero error in the steady state and asymptotic, aperiodic behavior.  
The linear control systems, namely error feedback control, digital control with limitation, linear state feedback control, and design with classical methods, are analyzed as a base of comparison. Their transient responses at the step input signal are presented, analyzed, and compared. The control systems analyzed have good control performance criteria: small response time compared to the high moment of inertia, and zero overshoot at an asymptotic aperiodic behavior. The differences between the efficiency criteria of the linear control structures analyzed are very small.