## 3 Controller Design  
Consider the following input performance index:  
\[J(u(k))=|y_{d}(k+1)-y(k+1)|^{2}+\lambda_{\alpha(k)}|u(k)-u(k-1)|^{2} \tag{11}\]  
where \(\lambda_{\alpha(k)}>0\) represents the weight factor, and \(y_{d}(k+1)\) correlates with the targeted or intended system output.  
Substituting Equation (10) into the performance index (11), set the derivative of (11) with respect to \(u(k)\) be zero:  
\[u(k)=u(k-1)+\frac{\rho_{\alpha(k)}\phi_{\alpha(k)}(k)}{\lambda_{\alpha(k)}+ \left|\phi_{\alpha(k)}(k)\right|^{2}}(y_{d}(k+1)-y(k)) \tag{12}\]  
in this context, the symbol \(\rho_{\alpha(k)}\) shows the factor of step size, \(\rho_{\alpha(k)}\in(0,1]\).  
Next, to estimate the parameter \(\phi_{\alpha(k)}(k)\), we design the following performance index:  
\[J\Big{(}\phi_{\alpha(k)}(k)\Big{)}=\Big{|}\Delta y(k)-\phi_{\alpha(k)}(k) \Delta u(k-1)\Big{|}^{2}+\mu_{\alpha(k)}\Big{|}\phi_{\alpha(k)}(k)-\hat{\phi}_ {\alpha(k)}(k-1)\Big{|}^{2} \tag{13}\]  
where \(\hat{\phi}_{\alpha(k)}(k)\) represents the estimate of \(\phi_{\alpha(k)}(k)\), and \(\mu_{\alpha(k)}>0\) is a weighting coefficient.  
Minimizing the performance index (13), we obtain the following PPD estimation algorithm:  
\[\hat{\phi}_{\alpha(k)}(k)=\hat{\phi}_{\alpha(k)}(k-1)+\frac{\eta_{\alpha(k) }\Delta u(k-1)}{\mu_{\alpha(k)}+\Delta u^{2}(k-1)}\times\Big{(}\Delta y(k)- \hat{\phi}_{\alpha(k)}(k-1)\Delta u(k-1)\Big{)} \tag{14}\]  
where \(\eta_{\alpha(k)}(k)\) denote step size factor.  
To broaden the applicability of the PPD estimation algorithm (14), we incorporate the following reset algorithm:  
\[\hat{\phi}_{\alpha(k)}(k)=\hat{\phi}_{\alpha(k)}(1)\text{ if }\Big{|}\hat{\phi}_{ \alpha(k)}(k)\Big{|}\leq\varepsilon\text{ or }\quad\text{ sign}\Big{(}\hat{\phi}_{\alpha(k)}(k)\Big{)}\
eq\text{ sign}\Big{(}\hat{\phi}_{\alpha(k)}(1)\Big{)} \tag{15}\]  
where \(\varepsilon>0\) is a small constant.  
\[\hat{\phi}_{\alpha(k)}(k)=\hat{\phi}_{\alpha(k)}(k-1)+\frac{\eta_{\alpha(k) }\Delta u(k-1)}{\mu_{\alpha(k)}+\Delta u^{2}(k-1)}\times\Big{(}\Delta y(k)- \hat{\phi}_{\alpha(k)}(k-1)\Delta u(k-1)\Big{)} \tag{16}\]  
\[\hat{\phi}_{\alpha(k)}(k)=\hat{\phi}_{\alpha(k)}(1)\text{, if }\Big{|}\hat{\phi}_{ \alpha(k)}(k)\Big{|}\leq\varepsilon\text{, or }\text{ sign}\Big{(}\hat{\phi}_{\alpha(k)}(k)\Big{)}\
eq\text{ sign}\Big{(}\hat{\phi}_{\alpha(k)}(1)\Big{)} \tag{17}\]  
\[u(k)=u(k-1)+\frac{\rho_{\alpha(k)}\hat{\phi}_{\alpha(k)}(k)}{\lambda_{\alpha(k )}+\Big{|}\hat{\phi}_{\alpha(k)}(k)\Big{|}^{2}}(y_{d}(k+1)-y(k)) \tag{18}\]  
In this section, we design an event-triggered, data-driven LFC strategy to conserve valuable bandwidth resources. The decision to transmit the most recent sampled data to the corresponding SSMFAC controller will be based on the following triggering conditions:  
\[k_{r+1}=k_{r}+\min_{r_{k_{r}}\in N^{+}}\Big{\{}r_{k_{r}}\mid e(k_{r})^{T} \Omega e(k_{r})\geqslant\delta\Big{\}} \tag{19}\]where \(e(k_{r})=y(k_{r})-y(k)\) represents the triggering error, \(k_{r}\) is an integer denoting the triggering instant, and \(\hat{\delta}\) is the triggering threshold parameter.  
Combining algorithms (16)-(18), we obtain the following event-triggered SSMFAC algorithm:  
\[\hat{\phi}_{a(k)}(k)\Bigg{\{} =\hat{\phi}_{a(k)}(k_{r}-1)+\frac{\eta_{a(k)}\Delta u(k-1)}{\mu_{ a(k)}+\Delta u^{2}(k-1)}\times\Big{(}\Delta y(k)-\hat{\phi}_{\alpha(k)}(k-1) \Delta u(k-1)\Big{)}\quad k=k_{r} \tag{20}\] \[\hat{\phi}_{a(k)}(k_{r}-1)\quad k\in(k_{r-1},k_{r})\] (21) \[u(k) =\left\{\begin{array}{ll}u(k_{r}-1)+\frac{\rho_{a(k)}\hat{\phi} _{a(k)}(k)}{\lambda_{x(k)}+\left|\hat{\phi}_{a(k)}(k)\right|^{2}}\times(y_{d} (k+1)-y(k_{r}))\quad k=k_{r}\\ u(k_{r}-1)\quad k\in(k_{r-1},k_{r})\end{array}\right. \tag{22}\]  
The schematic diagram of SSMFAC is shown in Figure 1.