# Advanced Construction of the Dynamic Matrix in Numerically Efficient Fuzzy MPC Algorithms
## 2 Efficient Fuzzy MPC Algorithms with the Advanced Construction of the Dynamic Matrix
### Optimization Problem in the Numerical and Analytical Versions of the Algorithms
#### 2.3.1 Optimization with the Classical Performance Index  
Optimization task (1)-(4), which is solved by the control algorithm in each time step, changes now to the following form with corrections \(\Delta\mathbf{\tilde{u}}\) being the decision variables:  
\[\arg\min_{\Delta\mathbf{\tilde{u}}}\Bigl{\{}J_{MPC+}=(\mathbf{\overline{y}}-\mathbf{ \tilde{y}}-\mathbf{A}_{k}\cdot\Delta\mathbf{\tilde{u}})^{T}\cdot\mathbf{\kappa}\cdot(\bm {\overline{y}}-\mathbf{\tilde{y}}-\mathbf{A}_{k}\cdot\Delta\mathbf{\tilde{u}})+\Delta\mathbf{ u}^{T}\cdot\mathbf{\Lambda}\cdot\Delta\mathbf{u}\Bigr{\}} \tag{42}\]  
subject to:  
\[\Delta\mathbf{u}_{\min}\leq\Delta\mathbf{u}\leq\Delta\mathbf{u}_{\max}\, \tag{43}\] \[\mathbf{u}_{\min}\leq\mathbf{u}\leq\mathbf{u}_{\max}\,\] (44) \[\mathbf{y}_{\min}\leq\mathbf{\tilde{y}}+\mathbf{A}_{k}\cdot\Delta\mathbf{\tilde{ u}}\leq\mathbf{y}_{\max}\, \tag{45}\]  
where \(\Delta\mathbf{u}=\stackrel{{\rightarrow}}{{\Delta\mathbf{u}}}+\Delta\mathbf{ \tilde{u}}\), \(\mathbf{u}=\stackrel{{\rightarrow}}{{\mathbf{u}}}+\mathbf{\tilde{u}}\), and:  
\[\stackrel{{\rightarrow}}{{\Delta\mathbf{u}}}=\left[\begin{array}{c} \stackrel{{\rightarrow}}{{\Delta\mathbf{u}}}\\ \vdots\\ \stackrel{{\rightarrow}}{{\Delta\mathbf{u}}}\end{array}\right]\!, \stackrel{{\rightarrow}}{{\Delta\mathbf{u}}}=\left[\begin{array}{c} \Delta u^{m}_{k+1|k-1}\\ \vdots\\ \stackrel{{\rightarrow}}{{\Delta\mathbf{u}}}\end{array}\right]\!, \stackrel{{\rightarrow}}{{\mathbf{u}}}=\left[\begin{array}{c} \stackrel{{\rightarrow}}{{\mathbf{u}}}\\ \vdots\\ \stackrel{{\rightarrow}}{{\mathbf{u}}}\end{array}\right]\!, \stackrel{{\rightarrow}}{{\mathbf{u}}}=\left[\begin{array}{c} \stackrel{{\rightarrow}}{{\mathbf{u}}}\\ \vdots\\ \stackrel{{\rightarrow}}{{\mathbf{u}}}\end{array}\right]\!, \stackrel{{\rightarrow}}{{\mathbf{u}}}=\left[\begin{array}{c} \stackrel{{\rightarrow}}{{\mathbf{u}}}\\ \vdots\\ \stackrel{{\rightarrow}}{{\mathbf{u}}}\end{array}\right]\!\!, \stackrel{{\rightarrow}}{{\mathbf{u}}}=\left[\begin{array}{c} \stackrel{{\rightarrow}}{{\mathbf{u}}}\\ \vdots\\ \stackrel{{\rightarrow}}{{\mathbf{u}}}\end{array}\right]\!\!\[I_{MPC+v2}=(\mathbf{\bar{y}}-\mathbf{\tilde{y}}-\mathbf{A}_{k}\cdot\Delta\mathbf{\bar{u}})^{T} \cdot\mathbf{\kappa}\cdot(\mathbf{\bar{y}}-\mathbf{\tilde{y}}-A_{k}\cdot\Delta\mathbf{\bar{u}})+ \Delta\mathbf{\bar{u}}^{T}\cdot\mathbf{\Lambda}\cdot\Delta\mathbf{\bar{u}}. \tag{47}\]  
The change consists of the modification of the second component of the performance index, which now depends only on the corrections of the control signals \(\Delta\mathbf{\bar{u}}\). This modification leads to algorithms that generate faster responses, but the influence of the \(\lambda\) parameter on system robustness becomes significantly limited.  
If the performance index (47) is minimized without constraints, then the following analytical solution is obtained:  
\[\Delta\mathbf{\bar{u}}=\left(\mathbf{A}_{k}^{T}\cdot\mathbf{\kappa}\cdot\mathbf{A}_{k}+\mathbf{ \Lambda}\right)^{-1}\cdot\mathbf{A}_{k}^{T}\cdot\mathbf{\kappa}\cdot(\mathbf{\bar{y}}- \mathbf{\tilde{y}}). \tag{48}\]