## 3 Control Method
### Explicit Dead-Time Compensation-Based MPC (EDT-MPC)  
We compared the proposed and a conventional method in MPC. The explicit dead-time compensation-based MPC (EDT-MPC) was selected as the conventional method. Generally, a controlled system involves a certain dead-time. Additionally, the time lag behavior of the actuator must also be considered. To avoid the presence of high order matrices in the augmented representation, the EDT compensation was developed by [PERSON] et al. [5], who examined this aspect in terms of stability and constraint satisfaction for discrete-time linear systems. The state prediction by EDT represents the current time, including the dead-time. In this study, the state prediction by the EDT was extended to the future state prediction for comparison with the proposed method using ESN. Consider the following uncertain discrete-time linear system with dead-time:  
\[\mathbf{x}_{k+1}=\mathbf{A}\mathbf{x}_{k}+\mathbf{B}\mathbf{u}_{k-d}+\mathbf{B}_{e}\mathbf{w}_{k} \tag{23}\]  
where \(d\) represents the nominal process dead-time, and \(\mathbf{w}_{k}\) is the vector of disturbance. In this study, \(d\) is treated as an artificial dead-time in EDT-MPC for comparison. The proposed ESN-MPC uses the time margin, which is the time difference between the current and actual execution times (Figure 2). Consider the following prediction model, where the system to be controlled depends only on past control actions:  
\[\mathbf{x}_{k+1}=\mathbf{A}\mathbf{x}_{k}+\mathbf{B}\mathbf{u}_{k-d}. \tag{24}\]  
The predicted value for \(\mathbf{x}_{k+d}\), based on the information available at the \(k\) step, denoted by \(\mathbf{x}_{k+d|k}\), can be obtained using Equation (24) recursively, as follows:  
\[\mathbf{x}_{k+d|k}=\mathbf{A}^{d}\mathbf{x}_{k}+\sum_{j=1}^{d}[\mathbf{A}^{j-1} \mathbf{B}\mathbf{u}_{k-j}], \tag{25}\] \[\mathbf{\tilde{x}}_{k}:=\mathbf{x}_{k+d|k}. \tag{26}\]  
By considering the explicit compensation scheme ([PERSON] et al. [5]), given by Equations (25) and (26), the predicted behavior can be noted as  
\[\mathbf{\tilde{x}}_{k+1} = \mathbf{A}\mathbf{\tilde{x}}_{k}+\mathbf{B}\mathbf{u}_{k}+\mathbf{B}_{e}\mathbf{\tilde{w} }_{k} \tag{27}\]  
where \(\mathbf{\tilde{w}}_{k}:=\mathbf{w}_{k+d|k}\) is the effect of \(\mathbf{w}_{k}\) on the predicted state \(\mathbf{\tilde{x}}_{k}\), which is uncertain at the timing of the \(k\) step. From Equation (27), \(\mathbf{B}_{e}\mathbf{\tilde{w}}_{k}\) can be obtained as:  
\[\mathbf{B}_{e}\mathbf{\tilde{w}}_{k} = \mathbf{\tilde{x}}_{k+1}-\mathbf{A}\mathbf{\tilde{x}}_{k}-\mathbf{B}\mathbf{u}_{k}. \tag{28}\]  
By applying Equations (23), (25) and (26), Equation (28) transforms into the following equation:  
\[\mathbf{B}_{e}\mathbf{\tilde{w}}_{k} = \mathbf{A}^{d}\mathbf{x}_{k+1}+\sum_{j=1}^{d}[\mathbf{A}^{j-1}\mathbf{B}\mathbf{u}_{k -j+1}]-\mathbf{A}\Bigg{\{}\mathbf{A}^{d}\mathbf{x}_{k}+\sum_{j=1}^{d}[\mathbf{A}^{j-1}\mathbf{B} \mathbf{u}_{k-j}]\Bigg{\}}-\mathbf{B}\mathbf{u}_{k} \tag{29}\] \[= \mathbf{A}^{d}[\mathbf{x}_{k+1}-\mathbf{A}\mathbf{x}_{k}-\mathbf{B}\mathbf{u}_{k-d}]\] \[= \mathbf{A}^{d}\mathbf{B}_{e}\mathbf{w}_{k}.\]Then, the uncertain vector of \(B_{\epsilon}\bar{w}_{k}\) can be determined by \(B_{\epsilon}\mathbf{w}_{k}\). Using Equation (23) recursively and applying \(\mathbf{w}_{k-j+d|k}=\bar{\mathbf{w}}_{k-j}\), as shown in the following equation:  
\[\mathbf{x}_{k+d|k} = \mathbf{A}^{d}\mathbf{x}_{k}+\sum_{j=1}^{d}[\mathbf{A}^{j-1}\mathbf{B}\mathbf{u}_{k-j} ]+\sum_{j=1}^{d}[\mathbf{A}^{j-1}\mathbf{B}_{\epsilon}\mathbf{w}_{k-j+d|k}] \tag{30}\] \[\mathbf{x}_{k+1}=\mathbf{A}\mathbf{x}_{k}+\mathbf{B}\mathbf{u}_{k-d}. \tag{24}\]0  
By incorporating Equation (29) into Equation (30), the predicted value for \(\mathbf{x}_{k+d|k}\) can be obtained as  
\[\mathbf{x}_{k+1}=\mathbf{A}\mathbf{x}_{k}+\mathbf{B}\mathbf{u}_{k-d}. \tag{24}\]1  
This extended representation does not include any unknown information available at the \(k\) step.