## 2 Preliminaries and Problem Description
### PPD Prediction Algorithm  
It is important to note that the controller design incorporates not only the current information \(\phi_{i}(\ell)\) but also the information \(\phi_{i}(\ell+m)\) at future time steps, where \(m=1,2,\ldots,N-1\). However, since Equation (18) does not provide a means to calculate future estimates \(\phi_{i}(\ell+m)\), so there is no direct method to compute the values of \(\phi_{i}(\ell+1),\ldots,\phi_{i}(\ell+N)\) in \(\hat{\mathbf{A}}_{ui}(\ell)\) using the system's input and output data up to time \(\ell\). Therefore, these values must be predicted based on the previously estimated value sequence \(\hat{\phi}_{i}(1),\hat{\phi}_{i}(2),\ldots,\hat{\phi}_{i}(\ell)\). Assume that the estimated values \(\hat{\phi}_{i}(1),\hat{\phi}_{i}(2),\ldots,\hat{\phi}_{i}(\ell)\) have been obtained using Algorithm (11). Based on this estimation sequence, an autoregressive (AR) model can be constructed for prediction [28]. \[\hat{\phi}_{i}(\ell+1)=\theta_{1}(\ell)\hat{\phi}_{i}(\ell)+\theta_{2}(\ell) \hat{\phi}_{i}(\ell-1)+\cdots+\theta_{np}(\ell)\hat{\phi}_{i}(\ell-n_{p}+1) \tag{17}\]  
where, \(n_{p}\) denotes the appropriate order, typically chosen within the range of 2 to 7 [28], and \(\theta_{a}\), for \(a=1,\ldots,n_{p}\), represents the coefficients. The prediction algorithm is as follows:  
\[\hat{\phi}_{i}(\ell+m)=\theta_{1}(\ell)\hat{\phi}_{i}(\ell+m-1)+\theta_{2}(\ell) \hat{\phi}_{i}(\ell+m-2)+\cdots+\theta_{n_{p}}(\ell)\hat{\phi}_{i}\big{(}\ell+m- n_{p}\big{)} \tag{18}\]  
where \(m=1,2,\cdots,N_{u}-1\). Define \(\boldsymbol{\theta}_{i}(\ell)=\Big{[}\theta_{1}(\ell),\cdots,\theta_{n_{p}}( \ell)\Big{]}^{\text{T}}\), which can be expressed as follows:  
\[\boldsymbol{\theta}_{i}(\ell)=\boldsymbol{\theta}_{i}(\ell-1)+\frac{ \boldsymbol{\varphi}_{i}(\ell-1)}{\delta+\|\hat{\boldsymbol{\varphi}}_{i}( \ell-1)\|^{2}}\Big{[}\hat{\phi}_{i}(\ell)-\hat{\boldsymbol{\varphi}}_{i}^{ \text{T}}(\ell-1)\boldsymbol{\theta}_{i}(\ell-1)\Big{]} \tag{19}\]  
where \(\hat{\boldsymbol{\varphi}}_{i}(\ell-1)=\big{[}\hat{\phi}_{i}(\ell-1),\cdots, \hat{\phi}_{i}\big{(}\ell-n_{p}\big{)}\big{]}^{\text{T}},\delta\in(0,1]\). The proposed DMFAPC scheme is developed by combining the controller algorithms (12) and (13), the PPD estimation algorithm (15), the PPD reset mechanism (16), and the PPD prediction algorithms (18) and (19). \[\hat{\phi}_{i}(\ell)=\hat{\phi}_{i}(\ell-1)+\frac{\eta\Delta F_{i}(\ell-1)}{ \mu+\Delta F_{i}(\ell-1)^{2}}\big{[}\Delta v_{i}(\ell)-\hat{\phi}_{i}(\ell-1) \Delta F_{i}(\ell-1)\big{]} \tag{20}\]  
\[\hat{\phi}_{i}(\ell)=\hat{\phi}_{i}(1),\quad\big{|}\hat{\phi}_{i}(\ell)\big{|} \leqslant\epsilon,|\Delta F_{i}(\ell-1)|\leqslant\epsilon,\text{sign}\big{(} \hat{\phi}_{i}(\ell)\big{)}\
eq\text{sign}\big{(}\hat{\phi}_{i}(1)\big{)} \tag{21}\]  
\[\boldsymbol{\theta}_{i}(\ell)=\boldsymbol{\theta}_{i}(\ell-1)+\frac{\hat{ \boldsymbol{\varphi}}_{i}(\ell-1)}{\delta+\|\hat{\boldsymbol{\varphi}}_{i}( \ell-1)\|^{2}}\Big{[}\hat{\phi}_{i}(\ell)-\hat{\boldsymbol{\varphi}}_{i}^{ \text{T}}(\ell-1)\boldsymbol{\theta}_{i}(\ell-1)\Big{]} \tag{22}\]  
\[\hat{\phi}_{i}(\ell+m)=\theta_{1}(\ell)\hat{\phi}_{i}(\ell+m-1)+\theta_{2}( \ell)\hat{\phi}_{i}(\ell+m-2)+\cdots+\theta_{n_{p}}(\ell)\hat{\phi}_{c}\big{(} \ell+m-n_{p}\big{)} \tag{23}\]  
\[F_{i}(\ell)=F_{i}(\ell-1)+\mathbf{g}^{\text{T}}\Delta\mathbf{F}_{N_{ui}}(\ell) \tag{24}\]  
\[\Delta\mathbf{F}_{N_{ui}}(\ell)=\Big{[}\hat{\mathbf{A}}_{ui}^{\text{T}}(\ell) \hat{\mathbf{A}}_{ui}(\ell)+\lambda\mathbf{I}\Big{]}^{-1}\hat{\mathbf{A}}_{ui }^{\text{T}}(\ell)\mathbf{E}(\ell)\xi_{i}(\ell) \tag{25}\]  
where the parameter \(\eta\) represents the step size, while parameters \(\mu\) and \(\lambda\) serve as weighting factors. Algorithm 1 shows the pseudo-code of the DMFAPC algorithm proposed in this paper, and its structure is shown in Figure 1.  
\[\hat{\phi}_{i}(\ell+m)=\theta_{1}(\ell)\hat{\phi}_{i}(\ell+m-1)+\theta_{2}(\ell) \hat{\phi}_{i}(\ell+m-2)+\cdots+\theta_{n_{p}}(\ell)\hat{\phi}_{i}\big{(}\ell+m- n_{p}\big{)} \tag{18}\]0 System block diagram.