## 2 Model Predictive Control Methods
### Optimization Problem and Establishment of Performance INDEX Function
\right]=\widetilde{D}\]We substitute (10) into (7) and get the performance index function in finite time:  
\[J=\min\sum\limits_{k=0}^{\infty}\big{[}e(k+1)^{T}e(k+1)+\Delta U(k )^{T}\Delta U(k)\big{]} \tag{11}\] \[=\min\sum\limits_{k=0}^{\infty}\Big{\{}\big{[}E(k)+\widetilde{A} \Delta X(k)+\widetilde{D}\Delta W(k)\big{]}^{T}[E(k)+\widetilde{A}\Delta X(k)+ \widetilde{D}\Delta W(k)\big{]}\] \[+2[E(k)+\widetilde{A}\Delta X(k)+\widetilde{D}\Delta W(k)]^{T} \widetilde{B}\Delta U(k)+\Delta U(k)^{T}\widetilde{B}^{T}\widetilde{B}\Delta U (k)\Big{\}}\] \[J=\min\sum\limits_{k=0}^{N}\big{[}e(k+1)^{T}e(k+1)+\Delta U(k)^{T }\Delta U(k)\big{]}\] (12) \[=\min\sum\limits_{k=0}^{N}\big{\{}2[E(k)+\widetilde{A}\Delta X(k) +\widetilde{D}\Delta W(k)]^{T}\widetilde{B}\Delta U(k)+\Delta U(k)^{T} \widetilde{B}^{T}\widetilde{B}\Delta U(k)\big{\}}\]  
The model predictive control is based on the dynamic model of the process, most of which are linear empirical models obtained through system identification. The characteristics of the model predictive control is to optimize the current time block every time, and then optimize the next time block. Therefore, the problem in (12) can be transformed into a finite time, and the optimization problem can be further transformed into an intuitive quadratic objective function to solve the minimum value of the performance index function in a fixed time period, as shown in Formula (13):  
\[\text{min}\quad\left[E(k)+\widetilde{A}\Delta X(k)+\widetilde{D}\Delta W(k) \right]^{T}\widetilde{B}\Delta U(k)+\Delta U(k)^{T}\widetilde{B}^{T}\widetilde{ B}\Delta U(k) \tag{13}\]  
According to the constrained conditions of (13),  
\[s.t.\|\Delta u(k)\|\leq a \tag{14}\] \[\|y(k)\|\leq b\] \[\|y(k)-r(k)\|\leq c\Rightarrow\|e(k)\|\leq c,\ a>0,\ b>0,\ c>0\]  
However, it is difficult to find the optimal solution, because it only sets a basic fixed value for the upper limit \(\|e(k)\|\leq c\) and there is no real-time update constraint. Therefore, it is necessary to find a real-time constraint for \(\|e(k)\|\)--that is,  
\[m\|e(k)\|\leq\|e(k+1)\|\leq n\|e(k)\|,\ 0<m<n<1 \tag{15}\]  
Establishment of Upper and Lower Bounds of Error Constraint Function Based on Gradient Descent Method  
In the process of solving the quadratic objective, how to set reasonable constraints is very important for solving the predictive optimal tracking control. The main problem that may arise in constrained optimization is that propositions may not be feasible. In this case, the standard QP solver has to stop the calculation, and they can only output one information, such as if the proposition is not feasible, or may give some diagnostic information in some cases. It is obviously unacceptable to apply the suspended calculation results to the device. Therefore, when QP is used for predictive control, it is very important to try to avoid an infeasible problem or to provide an alternative method for calculating the control effect. These methods include: avoiding hard constraints on output; effectively managing the constraints defined at every moment; effectively managing the prediction time domain at each time; and using a non-standard solution algorithm. In fact, the set of active constraints changes in the process of optimization. Optimization constraints are very important for solving predictive optimal tracking control.