## 3 Stability Analysis for the Slipping Phase
### Critical Period Regarding Stability  
According to the stability criterion expressed in the \(Z\)-domain [29], the control system can be stable once all the poles are inside the unit circle of the \(Z\)-plane. Using the parameters of the car listed in Table 1[6], the scheduling periods of \(T\) = 0.2 s and 0.4 s are given as examples to show the derivation process of the poles. Then, additional calculations for different scheduling periods are performed. The path of the poles changing with respect to the task scheduling period \(T\) is illustrated in Figure 4.  
\[H(z)=\frac{1.439z^{6}-1.179z^{5}-0.1924z^{4}-0.003649z^{3}-2.88\cdot 10^{-5}z^{2}+ 1.785\cdot 10^{-9}z-1.345\cdot 10^{-14}}{z^{7}-1.972z^{6}+0.9451z^{5}+0.02713z^{4}+0.0002 246z^{3}-2.398\cdot 10^{-9}z^{2}+8.489\cdot 10^{-15}z} \tag{24}\]  
The seven poles of the closed-loop transfer function \(G(z)\) in (22) are calculated as follows.  
\[\begin{array}{c}z_{1}=8.78\cdot 10^{-6},\ z_{2}=5.28\cdot 10^{-5},\ z_{3}=0.959 \\ z_{4}=-0.203-0.355\cdot i\\ z_{5}=-0.203+0.355\cdot i\\ z_{6}=-0.011-0.008\cdot i\\ z_{7}=-0.011+0.008\cdot i\\ \end{array} \tag{25}\]  
There are three real poles and four conjugate poles. As shown in Figure 5, it can be seen that the modules of the seven roots are all less than 1, i.e., they are all located inside the unit circle in the \(Z\)-plane. Thus, the closed-loop system is stable.  
Next, let \(T\) = 0.4 s. The open-loop transfer function \(H(z)\) in (21) is derived as follows.  
\[H(z)=\frac{3.1z^{6}-2.693z^{5}-0.1553z^{4}+2.854\cdot 10^{-5}z^{3}-6.524\cdot 10^{- 9}z^{2}+1.014\cdot 10^{-18}z-1.142\cdot 10^{-28}}{z^{7}-2z^{6}+1.001z^{5}-0.0003118z^{4}+5.057\cdot 10^{-8}z^{3}-1.938\cdot 10^{-18}z^{2}+7.206\cdot 10^{-29}z} \tag{26}\]  
The seven poles of the closed-loop transfer function \(G(z)\) in (22) are calculated as follows.  
\[\begin{array}{c}z_{1}=-1.933,\ z_{2}=-0.087,\ z_{3}=0.921\\ z_{4}=9.19\cdot 10^{-5}-\ 1.83\cdot 10^{-4}\cdot i\\ z_{5}=9.19\cdot 10^{-5}+1.83\cdot 10^{-4}\cdot i\\ z_{6}=7.77\cdot 10^{-11}-\ 1.07\cdot 10^{-10}\cdot i\\ z_{7}=7.77\cdot 10^{-11}+1.07\cdot 10^{-10}\cdot i\\ \end{array} \tag{27}\]  
Again, there are three real poles and four conjugate poles. However, as shown in Figure 5, the first pole \(z_{1}\) is located outside the unit circle in the \(Z\)-plane; thus, the closed-loop system is unstable.  
Additional calculations for 0.001 s \(\leq T\leq\) 0.5 s are performed every 0.001 s. The changing path of the maximum module \(|z|_{max}\) of the seven poles is plotted in Figure 4. It can be seen that \(|z|_{max}<\) 1 when \(T\) < 0.3 s, while \(|z|_{max}>\) 1 when \(T\) > 0.3 s. Therefore, the critical period, represented by \(T_{\max}\), is defined as \(T\) = 0.3 s. That is, the closed-loop system will be stable when the task scheduling period is \(T\) < 0.3 s, whereas it becomes unstable when the task scheduling period is \(T\) < 0.3 s.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbol** & **Value** & **Symbol** & **Value** & **Symbol** & **Value** \\ \hline \(J_{e}\) & 0.1 kg\(\cdot\)m\({}^{2}\) & \(J_{c1}\) & 0.5 kg\(\cdot\)m\({}^{2}\) & \(J_{c2}\) & 0.5 kg\(\cdot\)m\({}^{2}\) \\ \(J_{v}\) & 10 kg\(\cdot\)m\({}^{2}\) & \(k_{e}\) & 1000 Nm/rad & \(k_{p}\) & 10,000 Nm/rad \\ \(c_{e}\) & 10 kg\(\cdot\)m\({}^{2}\) & \(c_{p}\) & 20 kg\(\cdot\)m\({}^{2}\) & \(k_{p}\) & 5 \\ \(k_{i}\) & 1 & \(\tau\) & 10 ms & & \\ \hline \hline \end{tabular}  
* Let \(T\) = 0.2 s. Then, the open-loop transfer function \(H(z)\) in (21) is calculated as follows.
* \[\begin{array}{c}z_{1}=8.78\cdot 10^{-6},\ z_{2}=5.28\cdot 10^{-5},\ z_{3}=0.959 \\ z_{4}=-0.203-0.355\cdot i\\ z_{5}=-0.203+0.355\cdot i\\ z_{6}=-0.011-0.008\cdot i\\ z_{7}=-0.011+0.008\cdot i\\ \end{array}\] (28)  
\end{table}
Table 1: Parameters used in the stability analysis.