## 4 Main Results  
Before providing the estimation strategy, a result that establishes the stability conditions for the injection scheme shown in Figure 1 is presented. This result will be used later in the proof of the convergence of the proposed observer.  
Figure 1: Proposed injection scheme.  
**Lemma 3**.: _Consider the system given by Equation (1) with an output injection described by Figure 1. Then, there exist gains \(g_{1}\) and \(g_{2}\) such that the closed-loop system,_  
\[\frac{Y(s)}{U_{\rm I}(s)}=\frac{be^{-\tau s}}{\left((s-a)(s+b_{1})+g_{1}(s+b_{1 })be^{-\tau s}+g_{2}be^{-\tau s}\right)\prod_{i=2}^{p}(s+b_{i})}, \tag{6}\]  
_is stable if and only if \(\tau<\frac{1}{a}-\frac{1}{b_{1}}+\sqrt{\frac{1}{a^{2}}+\frac{1}{b_{1}^{2}}}\)._  
Proof.: The proof of this lemma is presented in Appendix A.  
For future reference, consider a state space representation of the system (1) with \(G_{S}(s)=1\), given as  
\[\begin{bmatrix}\dot{\phi}_{1}(t)\\ \dot{\phi}_{2}(t)\\ y(t+\tau)\end{bmatrix}=\begin{bmatrix}a&b&0\\ 0&-b_{1}&0\\ 1&0&0\end{bmatrix}\begin{bmatrix}\phi_{1}(t)\\ \phi_{2}(t)\\ y(t)\end{bmatrix}+\begin{bmatrix}0\\ 1\\ 0\end{bmatrix}u(t). \tag{7}\]  
Therefore, a closed-loop state space representation (related to the closed-loop transfer function (6)) considering \(G_{S}(s)=1\) of the output injection shown in Figure 1 can be written as  
\[\begin{bmatrix}\dot{\phi}_{1}(t)\\ \dot{\phi}_{2}(t)\\ y(t+\tau)\end{bmatrix}=\begin{bmatrix}a&b&-bg_{1}\\ 0&-b_{1}&-g_{2}\\ 1&0&0\end{bmatrix}\begin{bmatrix}\phi_{1}(t)\\ \phi_{2}(t)\\ y(t)\end{bmatrix}+\begin{bmatrix}0\\ 1\\ 0\end{bmatrix}u_{1}(t). \tag{8}\]  
On the other hand, consider a state space representation of the system (1) with \(G_{S}(s)\) considering \(p\geq 2\) given with  
\[\begin{bmatrix}\dot{\phi}_{1}(t)\\ \dot{\phi}_{2}(t)\\ \dot{\alpha}(t)\\ y(t+\tau)\end{bmatrix}=\begin{bmatrix}a&b&0&0\\ 0&-b_{1}&C_{1}&0\\ 0&0&A_{1}&0\\ 1&0&0&0\end{bmatrix}\begin{bmatrix}\phi_{1}(t)\\ \phi_{2}(t)\\ \alpha(t)\\ y(t)\end{bmatrix}+\begin{bmatrix}0\\ 0\\ 1\\ 0\end{bmatrix}u(t). \tag{9}\]  
Thus, by using (9), a closed-loop state space representation of the closed-loop transfer function (6) or equivalently the output injection shown in Figure 1 can be written as  
\[\begin{bmatrix}\dot{\phi}_{1}(t)\\ \dot{\phi}_{2}(t)\\ \dot{\alpha}(t)\\ y(t+\tau)\end{bmatrix}=\begin{bmatrix}a&b&0&-bg_{1}\\ 0&-b_{1}&C_{1}&-bg_{2}\\ 0&0&A_{1}&0\\ 1&0&0&0\end{bmatrix}\begin{bmatrix}\phi_{1}(t)\\ \phi_{2}(t)\\ \alpha(t)\\ y(t)\end{bmatrix}+\begin{bmatrix}0\\ 0\\ 1\\ 0\end{bmatrix}u_{1}(t). \tag{10}\]  
**Remark 1**.: _Notice that the closed-loop state space representations (8) and (10) have the same stability condition established in Lemma (3) due to the fact that (8) is a particular case of (10) and by the assumption \(b_{i}>0\), for \(i=1,2,\cdots,p\) previously made._  
The parameters of the injection scheme depicted in Lemma 3 are presented in the following result.  
**Corollary 1**.: _Let us consider the system given by Equation (1) with an output injection as in Figure 1. The parameters \(g_{1}\) and \(g_{2}\) such that the closed-loop system \(Y(s)/U_{\rm I}(s)\) (Equation (6)) is stable can be obtained as follows:_  
\[\frac{b_{1}(a\tau-1)+a}{b}<g_{1}<\frac{b_{1}(a\tau-1)+a}{b}+\epsilon_{1}, \tag{11}\]and,_  
\[\frac{b_{1}(a-bg_{1})}{b}<g_{2}<\frac{b_{1}(a-bg_{1})}{b}+\epsilon_{2}, \tag{12}\]  
_for some \(\epsilon_{1},\epsilon_{2}>0\)._  
Proof.: The proof of this corollary is presented in Appendix B.