## 4 Main Results
### Improved Stability Conditions  
Considering a partition of the time delay, it is possible to improve the stability conditions for some cases, [14]. In this way, the result presented in the previous sections can be improved as follows. Consider the system (1) with the time delay partition given by,  
\[\frac{Y(s)}{U(s)}=G(s)\mathrm{e}^{-\tau s}=\mathrm{e}^{-\tau_{1}s}G(s)\mathrm{ e}^{-\tau_{2}s}, \tag{21}\]  
The observer scheme given in Figure 2 can be modified as shown in Figure 3.  
Lemma 4: _Let us consider the system (21), \(G_{s}(s)=1\) and the observer scheme presented in Figure 3. Then, there exist gains \(g_{1  
separation for linear observers, and based on the adequate convergence of \(\hat{\omega}(t)\), Lemma 1 can be applied considering the estimated signal \(\hat{y}(t+\tau_{2})\) to the design of the control stage. That is, there exists a control \(C(s)\) of type PID that stabilizes the system with the delay \(\tau_{1}\) if and only if \(\tau_{1}<\frac{1}{a}-\frac{1}{b_{1}}+\sqrt{\frac{1}{a^{2}}+\frac{1}{b_{1}^{2}}}\). Therefore, it can be concluded that the original time delay limit \(\tau\) is \(\tau=\tau_{1}+\tau_{2}<2\bigg{(}\frac{1}{a}-\frac{1}{b_{1}}+\sqrt{\frac{1}{a^{ 2}}+\frac{1}{b_{1}^{2}}}\bigg{)}\).  
Corollary 2: _Let us consider the system (21), \(G_{S}(s)\) with \(p\geq 2\), and the observer scheme presented in Figure 3. There are gains \(g_{1}\) and \(g_{2}\) and a control \(C(s)\) of type PID such that the closed-loop system \(Y(s)/R(s)\) is stable if and only if \(\tau<\frac{2}{a}-\frac{2}{b_{1}}-\sum_{i=2}^{p}\frac{1}{b_{i}}+\sqrt{\frac{1}{ a^{2}}+\frac{1}{b_{1}^{2}}}+\sqrt{\frac{1}{a^{2}}+\sum_{i=1}^{p}\frac{1}{b_{i}^{2}}}\)._  
Proof: Theorem 1 can be applied directly to the observer of Figure 3 since the convergence of the observer is not affected by the delayed input \(u(t-\tau_{1})\). Thus, there are gains \(g_{1}\) and \(g_{2}\) such that \(lim_{t\rightarrow\infty}[\hat{\omega}(t)-\omega(t)]=0\) if and only if \(\tau_{2}<\frac{1}{a}-\frac{1}{b_{1}}+\sqrt{\frac{1}{a^{2}}+\frac{1}{b_{1}^{2}}}\). Taking into account the principle of separation for linear observers and based on the convergence of \(g(t+\tau_{2})\), Lemma 1 can be applied to the control stage by considering the estimated signal \(\hat{y}(t+\tau_{2})\). Then, there is a PID type control \(C(s)\) that stabilizes the plant \(G(s)\) with a delay \(\tau_{1}\) if and only if \(\tau_{1}<\frac{1}{a}-\sum_{i=1}^{p}\frac{1}{b_{i}}+\sqrt{\frac{1}{a^{2}}+\sum_ {i=1}^{p}\frac{1}{b_{i}^{2}}}\). Therefore, it can be concluded that the original delay time limit is \(\tau=\tau_{1}+\tau_{2}<\frac{2}{a}-\frac{2}{b_{1}}-\sum_{i=2}^{p}\frac{1}{b_{ i}}+\sqrt{\frac{1}{a^{2}}+\frac{1}{b_{1}^{2}}}+\sqrt{\frac{1}{a^{2}}+\sum_ {i=1}^{p}\frac{1}{b_{i}^{2}}}\).  
Remark 2: _Since, in Lemma 4, the size of the delays \(\tau_{1}\) and \(\tau_{2}\) is bounded in the same proportion by convergence and control conditions, it is suggested to set the partition as \(\tau_{1}=\tau/2\) and \(\tau_{2}=\tau/2\). For the case of Corollary 2, it is suggested to set the delay partition as \(\tau_{1}<\tau_{2}\) since the proportion of the bounds in convergence and control conditions is different._