## 3 Control Method
### Solving the Predictive Control Problem
The MPC, which is based on the constrained finite-horizon optimization, for the tracking problem can be formulated as follows:  
\[min\;{{I}_{k}}=\sum\limits_{i=0}^{H_{p}}\parallel{{{\hat{y}}_{k+i}}-{{\mathbf{r}} _{{{{I}}}k+i}}}\parallel_{\mathbf{Q}}^{2}+\sum\limits_{i=0}^{H_{u}}\parallel{\Delta{ {\hat{u}}_{k+i}}}\parallel_{\mathbf{R}}^{2} \tag{21}\]  
subject to  
\[\hat{\mathbf{x}}_{k+i+1}=\mathbf{A}\hat{\mathbf{x}}_{k+i}+\mathbf{B}\hat{\mathbf{u}}_{k +i}+B_{e}\hat{\mathbf{w}}_{k+i},\;\;\hat{\mathbf{y}}_{k+i}=\mathbf{C}\hat{x}_{k+i},\;\;\hat{ \mathbf{x}}_{0}=\mathbf{x}_{0},\] \[\hat{\mathbf{y}}^{min}\leq\hat{\mathbf{y}}_{k+i}\leq\hat{\mathbf{y}}^{max},\; \;\hat{\mathbf{u}}^{min}\leq\hat{\mathbf{u}}_{k+i}\leq\hat{\mathbf{u}}^{max},\;\;\Delta{{ \hat{u}}_{k+i}}=0\;\;for\;\;i\in\{H_{u}+1,\cdots,H_{p}\}\]  
where the notations \(\parallel{\cdot}\parallel_{\mathbf{Q}}^{2}\) and \(\parallel{\cdot}\parallel_{\mathbf{R}}^{2}\) represent the weighted norms as a quadratic form, which indicates, e.g., for \(\mathbf{\mathcal{Q}}\), that \(\mathbf{x}^{\mathrm{T}}\mathbf{Qx}=\parallel\mathbf{Q}^{1/2}\mathbf{x}\parallel^{2}=\parallel \mathbf{x}\parallel_{\mathbf{Q}}^{2}\). Here, \(i\) is the step number, \(\hat{\mathbf{y}}\) is the predicted output, \(\mathbf{r_{t}}\) is the reference trajectory for the output, and \(\hat{\mathbf{w}}\) is the predicted disturbance. The diagonal matrices \(\mathbf{\mathcal{Q}}\) and \(\mathbf{\mathcal{R}}\) represent the output and control input weights, respectively. The standard predictive control problem can be solved by updating the solution each time. This modification is performed each time with optimal solution changes. This type of method is called real-time iteration [20]. Herein, we consider the target HEV powertrain as a linear time-invariant system. The predictive control problem of this system involves certain constraints. To solve the constrained optimization problem, it can be formulated as a quadratic programming (QP) problem based on the discretization method in Equations (18)\(-\)(21), using a cost function and predicted output [19]. \[\underset{\Delta{{U}_{k}}}{\min}\;\;\frac{1}{2}\Delta{{{U}_{k}^{\mathrm{T}}} \mathbf{\Phi}}\Delta{{{U}_{k}}}-\mathbf{\mathcal{G}}^{\mathrm{T}}\Delta{{{U}_{k}}} \tag{22}\]  
subject to  
\[\mathbf{\Omega}\Delta{{{U}_{k}}}\leq\mathbf{\eta}\]  
where  
\[\mathbf{\mathcal{G}}=2\mathbf{\Theta}^{\mathrm{T}}\mathbf{\mathcal{Q}}\varepsilon_{k},\;\; \mathbf{\Phi}=\mathbf{\Theta}^{\mathrm{T}}\mathbf{\mathcal{Q}}\mathbf{\Theta}+\mathbf{\mathcal{R}},\]  
\[\mathbf{\Omega}=\left[ \begin{array}{c}-\mathbf{C}_{2}\\ \mathbf{C}_{2}\\ -\mathbf{\Theta}\\ \mathbf{\Theta}\\ \end{array} \right],\;\;\mathbf{\eta}=\left[ \begin{array}{c}-\mathbf{u}^{min}+\mathbf{C}_ {1}\mathbf{u}_{k-1}\\ \mathbf{u}^{max}-\mathbf{C}_{1}\mathbf{u}_{k-1}\\ -\mathbf{y}^{min}+\mathbf{\mathbf{\pi}}\mathbf{x}_{k}+\mathbf{\gamma}\mathbf{u}_{k-1}+\mathbf{\Xi}\mathbf{W}_{mk }\\ \mathbf{y}^{max}-\mathbf{\mathbf{\pi}}\mathbf{x}_{k}-\mathbf{\gamma}\mathbf{u}_{k-1}-\mathbf{\Xi}\mathbf{W}_{mk }\\ \end{array} \right],\;\;\mathbf{C}_{1}=\left[ \begin{array}{c}\mathbf{I}\\ \mathbf{I}\\ \mathbf{I}\\ \vdots\\ \mathbf{I}\\ \end{array} \right],\;\;\mathbf{C}_{2}=\left[ \begin{array}{c}\mathbf{I}&0&0&\cdots&0 \\ \mathbf{I}&\mathbf{I}&0&\cdots&0\\ \mathbf{I}&\mathbf{I}&\mathbf{I}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \mathbf{I}&\mathbf{I}&\cdots&\mathbf{I}\\ \end{array} \right],\]  
\[\mathbf{\varepsilon}_{k}=\mathbf{\mathcal{T}}_{k}-\mathbf{\Psi}\mathbf{x}_{k}-\mathbf{\gamma}\mathbf{u}_ {k-1}-\mathbf{\Xi}\mathbf{W}_{mk},\quad\mathbf{\mathcal{T}}_{k}=\left[ \begin{array}{c} \mathbf{r}_{k+1|k}\\ \vdots\\ \mathbf{r}_{k+H_{p}|k}\\ \end{array} \right]\]  
where \(\mathbf{\varepsilon}_{k}\) is the tracking error that changes in each step owing to the system disturbance. \(\mathbf{\mathcal{T}}_{k}\) is the reference matrix. \(\mathbf{u}^{min}\) and \(\mathbf{u}^{max}\) denote the input limits, and \(\mathbf{y}^{min}\) and \(\mathbf{y}^{max}\) denote the output limits of the constraints.