## 4 Model Stability Analysis
Specifically, as long as a reasonable average dwell time is designed, and the ratio of the operating time of the non-stabilizable subsystems to that of the stabilizable subsystems is below a certain threshold, the switching system remains stabilizable. Let \(N_{\sigma}(k_{1},k_{2})\) denote the number of switches in time \(k_{1}\) and \(k_{2}\) for any \(k_{2}>k_{1}\geq 0\). If there exist \(T_{a}>0\) and \(N_{0}\geq 0\), such that the following conditions hold, then \(T_{a}\) is the mean residence time, and \(N_{0}\) is the jitter-bound. \[N_{\sigma}(k_{1},k_{2})\leq N_{0}+\tfrac{k_{2}-k_{1}}{T_{a}}\]  
For simplicity and without loss of generality, we choose \(N_{0}=0\), which divides the time interval into two categories, with and without isolated nodes. Where \([k_{2p},k_{2p+1})\)\((p=0,1\cdots n)\), the absence of isolated nodes, is stabilizable, and \([k_{2p+1},k_{2p+2}]\), the presence of isolated nodes, is not stabilizable. Define \(N_{l}(k_{1},k_{2})\) as the number of times a stabilizable system switches to a system with an isolated node in \([k_{1},k_{2})\); denote by \(T_{l}(k_{1},k_{2})\) the total time a subsystem with an isolated node is running in \([k_{1},k_{2}]\). \(f_{l}(k_{1},k_{2})=T_{l  
where \(\partial=\frac{1}{2}\Big{(}\alpha-\alpha^{*}\Big{)}\). \[\varepsilon_{1}\|\delta(k)\|^{2}\leq V(k)\leq\sqrt{\mu_{1}}e^{-2 \partial(k-k_{0})}V_{0}(k_{0})\leq\varepsilon_{2}\sqrt{\mu_{1}}e^{-2\partial(k- k_{0})}\|\delta(k_{0})\|^{2}\] \[\parallel\delta(k)\parallel\leq\sqrt{\frac{\varepsilon_{2}\sqrt {\mu_{1}}}{\varepsilon_{1}}}e^{-\partial(k-k_{0})}\parallel\delta(k_{0})\parallel\] The error switching system (32) is exponentially stable because \(\partial>0\). It follows from Lemma 1 that \[V(k)\leq e^{-\alpha}V(k-1)-F(k-1)\leq e^{-2\alpha}V(k-2)-F(k-1)-e^{-\alpha}F(k-2)\] \[\leq\cdots\leq e^{-\alpha(k-k_{0})}V(k-k_{0})-\sum_{s=k_{0}}^{k-1 }e^{-\alpha(k-s-1)}F(s)\] where \(F(k)=\sum_{k=0}^{\infty}z^{\mathsf{T}}(k)z(k)-\gamma^{2}\sum_{k=0}^{\infty} \omega^{\mathsf{T}}(k)\omega(k)\). The same reasoning leads to \[V(k)\leq \begin{cases}e^{-\alpha(k-k_{2p})}V(k_{2p})+\sum_{s=k_{2p}}^{k-1}e^{- \alpha(k-s-1)}F(s),k\in[k_{2p},k_{2p+1})\\ e^{\beta(k-k_{2p+1})}V_{j}(k_{2p+1})+\sum_{s=k_{2p+1}}^{k-1}e^{\beta(k-s-1)}F( s),k\in[k_{2p+1},k_{2p+2})\end{cases} \] (42) It follows from (42) that \[V(k)\leq\sqrt{\mu_{1}}\Bigg{[}e^{-2\varepsilon(k-k_{0})}V_{0}(k_{0})-\sum_{s= k_{0}}^{k-1}e^{-2\varepsilon(k-s)}F(s)\Bigg{]}\] (43) The proof process is similar to that in (40) and (41); hence, it is omitted here. Under zero initial conditions, from (43), it follows that \[0\leq-\sum_{s=k_{0}}^{k-1}e^{-2\varepsilon(k-s)}F(s)\] \[\sum_{s=0}^{\infty}z^{T}(s)z(s)\leq\gamma^{2}\sum_{s=0}^{\infty} \omega^{\mathsf{T}}(s)\omega(s)\]  
Thus, supply chain systems containing isolated nodes, provided they satisfy the conditions outlined in Theorem 3, are exponentially stable and achieve an \(H_{\infty}\) performance level of \(\gamma\). Theorems 1-3 establish the conditions for the switching topology and \(H_{\infty}\) consistency of a T-S fuzzy supply chain system based on a trust mechanism under uncertain demand. The design steps for the controller are as follows:  
* Step 1: Substitute the controller from Equation (5) into Equation (6) to derive the supply chain system represented by Equation (8). Subsequently, transform system (8) into the error system represented by Equation (10);
* Step 2: Determine the parameters of the supply chain system and the scaling gain of the consistency controller based on the fuzzy rules. This results in the supply chain error system model incorporating trust information, represented by Equation (18);
* Step 3: Utilize the Schur complement lemma and matrix transformations to obtain the control gains and necessary parameters. This is achieved by solving the linear matrix inequalities (LMIs) given by Equations (20), (21), (34) and (35), and the constraints specified in (36) and (37).