## 4 Model Stability Analysis
** Obviously, \(\delta(k)\) satisfies  
\[\delta(k)=\delta(k-\tau)+\sum_{\theta=\tau}^{k-1}y(\theta) \tag{22}\]  
Consequently,  
\[y(\mathbb{k})=\left(\widehat{A}+\widehat{B}+\overline{B}-\mathbb{I}\right) \delta(k)+B_{\omega}\omega(k)-\widehat{B}\delta(0)-\widehat{B}\sum_{\theta- \tau_{1}}^{\mathbb{k}-1}y(\theta)-\overline{B}\sum_{\theta-\tau_{1}}^{ \mathbb{k}-1}y(\theta) \tag{23}\]  
Consider that when \(\omega(k)=0\), for the Lyapunov function (15) along the trajectory of the system (10), for the forward difference, we have  
\[V(k+1)-e^{-\alpha}V(k) =V_{1}(k+1)-e^{-\alpha}V_{1}(k)+V_{2}(k+1)-e^{-\alpha}V_{2}(k)+V_{ 3}(k+1)-e^{-\alpha}V_{3}(k)\] \[\text{where}\] \[V_{1}(k+1)-e^{-\alpha}V_{1}(k)=2\delta^{T}(k)Py(k)+y^{T}(k)Py(k)+(1-e^{- \alpha})\delta^{T}(k)P\delta(k) \tag{24}\]It follows from (23)  
\[ \begin{split} 2\delta^{T}(k)Py(k)&=2\delta^{T}(k)P \Big{\{}\Big{(}\widehat{A}+\widehat{B}+\overline{B}-\mathrm{I}\Big{)}\delta(k) +B_{\varphi}\omega(k)\\ -\widehat{B}\delta(0)&-\sum_{\theta=k-\tau_{1}}^{k -1}\widehat{B}y(\theta)-\sum_{\theta=k-\tau_{2}}^{k-1}\overline{B}y(\theta) \Bigg{\}}\end{split}  \tag{25}\]  
Combining this with Lemma 1 yields  
\[ \begin{split}&-2\sum_{\theta=k-\tau_{1}}^{k-1}\delta^{T}(k)P \hat{B}y(\theta)\leqslant\sum_{\theta=k-\tau_{1}}^{k-1}\left[\begin{array}{ c}\delta(k)\\ y(\theta)\end{array}\right]^{T}\times\left[\begin{array}{c}\widehat{ \mathrm{X}}\widehat{\mathrm{Y}}-P\hat{B}\\ *\hat{Z}\end{array}\right]\left[\begin{array}{c}\delta(k)\\ y(\theta)\end{array}\right]\\ &=\bar{\tau}_{1}\delta^{T}(k)\hat{\mathrm{X}}\delta(k)+2\delta^{T}(k)(\widehat{ \mathrm{Y}}-P\hat{B})\times(\delta(k)-\delta(k-\bar{\tau}_{1}))+\sum_{\theta=k -\tau_{1}}^{k-1}y^{T}(\theta)\hat{Z}y(\theta)\end{split}  \tag{26}\]  
where \(\widehat{\mathrm{X}}\widehat{\mathrm{Y}}\) are symmetric matrices of appropriate dimension and  
\[ \begin{split}\begin{bmatrix}\widehat{X}&\widehat{\mathrm{Y}}\\ *&\hat{Z}\end{bmatrix}\geq 0\end{split}  \tag{27}\]  
The same reasoning can be used to obtain  
\[ \begin{split}&-2\sum_{\theta=k-\tau_{2}}^{k-1}\delta^{T}(k)P \bar{B}y(\theta)\leqslant\sum_{\theta=k-\tau_{2}}^{k-1}\left[\begin{array}{ c}\delta(k)\\ y(\theta)\end{array}\right]^{T}\times\left[\begin{array}{c}\overline{ \mathrm{X}}\overline{\mathrm{Y}}-P\bar{B}\\ *\hat{Z}\end{array}\right]\left[\begin{array}{c}\delta(k)\\ y(\theta)\end{array}\right]\\ &=\bar{\tau}_{2}\delta^{T}(k)\overline{\mathrm{X}}\delta(k)+2\delta^{T}(k)( \overline{\mathrm{Y}}-P\bar{B})\times(\delta(k)-\delta(k-\bar{\tau}_{2}))+ \sum_{\theta=k-\tau_{2}}^{k-1}y^{T}(\theta)\hat{Z}y(\theta)\end{split}  \tag{28}\]  
where \(\overline{\mathrm{X}},\overline{\mathrm{Y}}\) are symmetric matrices of appropriate dimension and  
\[ \begin{bmatrix}\overline{\mathrm{X}}&\overline{\mathrm{Y}}\\ *&\overline{Z}\end{bmatrix} \geq 0\]  
Consequently,  
\[V_{1}(k+1)-V_{1}(k)\leq\xi^{T}(k)\psi_{1}\bar{\varsigma}(k)+y^{T}(\theta)Py( \theta)+\sum_{\theta=k-\tau_{1}}^{k-1}y^{T}(\theta)\hat{Z}y(\theta)+\sum_{ \theta=k-\tau_{2}}^{k-1}y^{T}(\theta)\hat{Z}y(\theta) \tag{29}\]  
where  
\[ \begin{split}&\xi^{T}(k)=\left[\delta^{T}(k),\delta^{T}(k-\tau_{ 1}),\delta^{T}(k-\tau_{2}),\omega^{T}(k),\delta(0)\right]\\ &\Phi_{1}=\left[P(\widehat{A}_{r}-I)+(\widehat{A}_{r}-I)^{T}P+\overline{ \tau}_{1}\widehat{\mathrm{X}}+\hat{\mathrm{Y}}+\hat{\mathrm{Y}}^{T}\right]+ \overline{\tau}_{2}\overline{\mathrm{X}}+\overline{\mathrm{Y}}+\overline{ \mathrm{Y}}^{T}+(1-e^{-a})P\right]\\ &\psi_{1}=\left[\begin{array}{cccc}\Phi_{1}&\widehat{\mathrm{Y}}-P \hat{B}&\overline{\mathrm{Y}}-P\bar{B}&PB_{\omega}&-P\hat{B}\\ *&0&0&0&0\\ *&*&0&0&0\\ *&*&*&0&0\\ \end{array}\right]\end{split} \]  
Continuing the calculations, we have  
\[