## 4 Main Results
### Estimation Strategy  
The proposed observer-predictor scheme for system (1) is presented in Figure 2. The following results present the convergence conditions for two cases:  
\[ \begin{array}{ll}(i)&\text{unstable delayed second order ($G_{\rm s}(s)=1$).}\\ (ii)&\text{unstable delayed higher order ($G_{\rm s}(s)$ with $p\geq 2$).}\\ \end{array}  \tag{13}\]  
A state space representation of system (1) considering \(G_{\rm s}(s)=1\) is given with  
\[ \begin{bmatrix}\dot{\omega}_{1}(t)\\ \dot{\omega}_{2}(t)\\ y(t+\tau)\end{bmatrix} = \begin{bmatrix}a&b&0\\ 0&-b_{1}&0\\ 1&0&0\end{bmatrix}\begin{bmatrix}\omega_{1}(t)\\ \omega_{2}(t)\\ y(t)\end{bmatrix} + \begin{bmatrix}0\\ 1\\ 0\end{bmatrix} u(t). \tag{13}\]  
**Theorem 1**. : _Consider system (1) with \(G_{\rm s}(s)=1\) with the state space representation (13) and the observer-predictor scheme shown in Figure 2. Then, there exist constants \(g_{1}\) and \(g_{2}\) such that \(lim_{t\to\infty}[\hat{\omega}(t)-\omega(t)]=0\) if and only if \(\tau<\frac{1}{a}-\frac{1}{b_{1}}+\sqrt{\frac{1}{a^{2}}+\frac{1}{b_{1}^{2}}}\)._  
Proof. : Consider the observer-predictor scheme of Figure 2. Its complete dynamics can be described as  
\[ \begin{bmatrix}\dot{\omega}_{1}(t)\\ \dot{\omega}_{2}(t)\\ \dot{\omega}_{1}(t)\\ \dot{\omega}_{2}(t)\end{bmatrix} = \begin{bmatrix}a&b&0&0\\ 0&-b_{1}&0&0\\ 0&0&a&b\\ 0&0&0&-b_{1}\end{bmatrix}\begin{bmatrix}\omega_{1}(t)\\ \omega_{2}(t)\\ \dot{\omega}_{1}(t)\\ \dot{\omega}_{2}(t)\end{bmatrix} + \begin{bmatrix}0&0&0&0\\ 0&0&0\\ 0&0&bg_{1}&-bg_{1}\\ 0&0&g_{2}&-g_{2}\end{bmatrix}\begin{bmatrix}0\\ 0\\ y(t)\\ \dot{g}(t)\end{bmatrix} + \begin{bmatrix}0\\ 1\\ 0\\ 1\end{bmatrix} u(t), \tag{14}\]  
Figure 2: Proposed observer–predictor scheme. Defining the prediction error as \(e_{\omega}(t)=\hat{\omega}(t)-\omega(t)\), where \(e_{\omega}(t)= \begin{bmatrix}e_{\omega 1}(t)\\ e_{\omega 2}(t)\end{bmatrix} \) and the estimate of the output error as \(e_{y}(t)=\hat{y}(t)-y(t)\), it is possible to describe the behavior of the dynamic error signal error \(\dot{e}_{\omega}(t)=\hat{\omega}(t)-\hat{\omega}(t)\),  
\[ \begin{bmatrix}\dot{e}_{\omega_{1}}(t)\\ \dot{e}_{\omega_{2}}(t)\\ e_{y}(t+\tau)\end{bmatrix} = \begin{bmatrix}a&b&-bg_{1}\\ 0&-b_{1}&-g_{2}\\ 1&0&0\end{bmatrix}\begin{bmatrix}e_{\omega 1}(t)\\ e_{\omega 2}(t)\\ e_{y}(t)\end{bmatrix} . \tag{16}\]  
Whether \(e_{\omega}(t)\to 0\) as \(t\to\infty\) in (16) depends on the stability of the dynamics depicted by (16). Notice that if the dynamic error (16) is stable and the initial conditions of the error \(e_{\omega}(0)\) are different from zero, the corresponding trajectories of the error \(e_{\omega}(t)\to 0\) as \(t\to\infty\). In this way, consider now a state-space realization of the system given by (8). It is clear that the stability condition for system (8) given in Lemma 3 (see Remark 1) is the same as the prediction error given in (16) since both systems have equivalent dynamic behaviors. Therefore, the condition for the convergence of the observer is \(\tau<\frac{1}{a}-\frac{1}{b_{1}}+\sqrt{\frac{1}{a^{2}}+\frac{1}{b_{1}^{2}}}\). Now, to extend these results about the design of the observer to a more general class of systems, consider an unstable delayed higher order system given by (1) (\(G_{\rm s}(s)\) with \(p\geq 2\)) with its state space representation given by  
\[ \begin{bmatrix}\dot{\omega}_{1}(t)\\ \dot{\omega}_{2}(t)\\ \dot{x}(t)\\ y(t+\tau)\end{bmatrix} = \begin{bmatrix}a&b&0&0\\ 0&-b_{1}&C_{1}&0\\ 0&0&A_{1}&0\\ 1&0&0&0\end{bmatrix}\begin{bmatrix}\omega_{1}(t)\\ \omega_{2}(t)\\ x(t)\\ y(t)\end{bmatrix} + \begin{bmatrix}0\\ 0\\ 1\\ 0\end{bmatrix} u(t), \tag{17}\]  
\[\text{where }A_{1} = \begin{bmatrix}-b_{2}&1&0&...&0\\ 0&-b_{3}&1&\ddots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0&0&0&b_{n-1}&1\\ 0&0&0&0&b_{n}\end{bmatrix} \text{, }B_{1} = \begin{bmatrix}0\\ 0\\ \vdots\\ 0\\ 1\end{bmatrix} \text{, }C_{1} = \begin{bmatrix}1&0&...&0&0\end{bmatrix} \text{, and }x(t)= \begin{bmatrix}x_{1}(t)\\ x_{2}(t)\\ \vdots\\ x_{n-1}(t)\\ x_{n}(t)\end{bmatrix} \text{.