## 3 Simulation Result and Analysis  
In order to better achieve tracking control under different set values, based on different decision-making requirements in practice, the control rate of the model predictive control is solved using a different set of ranges of constraint function boundaries to achieve the optimal tracking control. The basic principle of the model predictive control is to predict the future dynamic behavior of the system under a certain control action based on the process model, and on this basis, according to the given constraints and performance requirements, solve the optimal control action and implement the current control in a rolling manner. Then, in each rolling process, real-time information is detected to correct the prediction of the future dynamic behavior of the system. In order to verify the effectiveness of the proposed method, a simulation verification was conducted on the discrete state space model of multivariate control of greenhouse environmental factors proposed in reference [28]. The model structure is shown in Formula (3). Based on the model predictive control rolling optimization, combined with the gradient descent theory and the gradual reduction of error tracking quadratic norm, the corresponding learning rate that meets the conditions is derived from the above analysis, and the real-time constraint function is obtained, as shown in Formula (41). We use the predictive control model rolling optimization solution method to solve Formula (13). The optimal control rate is obtained by optimizing Formula (13), and the parameters of Formula (3) are selected as follows:  
Forecast length and control length are set to N = 4, with the initial condition  
\[\Delta x(1)=[-0.2;-4]\]  
\[e(1)=[-2;18]\]  
\[A= \begin{bmatrix}0.98412&0.00073760\\ 0.014945&0.00011545\end{bmatrix} \]  
\[B= \begin{bmatrix}0.0087496&-0.021678&-0.0033593&-0.014482\\ 0.016515&0.080241&-0.0023759&0.0062881\end{bmatrix} \]  
\[D= \begin{bmatrix}0.014954&0.00011545&0.0007062&-0.0018781\\ 0.040419&-0.0013509&-0.00031852&-0.011573\end{bmatrix} \]  
(1) The simulation prediction and tracking control results after optimizing constraints and comparison with other situations. In order to verify the effectiveness of this method, simulations were conducted under different output error constraint conditions, and four different scenarios were compared through simulation. The four situations are:  
\[ \begin{split}&\text{(a1)}\ m\|e(k)\|\leq\|e(k+1)\|\leq n\|e(k)\|; \\ &\text{(a2)}\ m\|e(k)\|\leq\|e(k+1)\|<\|e(k)\|;\\ &\text{(a3)}\ \|e(k+1)\|<\|e(k)\|;\\ &\text{(a4)}\ p\|e(k)\|\leq\|e(k+1)\|\leq q\|e(k)\|,\ 0<p<q<m<1\end{split} \]  
(a1) is the constraint condition for obtaining suitable optimized parameters under the method proposed in this article; (a2) is a constraint condition that only optimizes some parameters to obtain a lower bound; (a3) is a constraint condition that satisfies the basic conditions; (a4) is a constraint condition that is not within the appropriate parameter range after optimization. Based on the known parameters and their initial values, we obtain \(0<\alpha<0.221\) from Formula (38). According to the control input constraints, \(\|\Delta u(k)\|<a\), the required range here is \(a=10\). According to Formulas (14), (15), and (35)-(40), we determine the upper and lower bounds of \(\|e(k+N)\|\). We select \(m=0.78\) and \(n=0.968\) as appropriate parameters through simulation debugging. We complete the constraints on the upper and lower bounds corresponding to \(\|e(k+N)\|\). In all of the simulation diagrams below, the horizontal axis represents the discrete sampling time, while the vertical axis represents the variation values of various variables themselves. The simulation results are shown in Figure 1a-d.  
Figure 1: \(Cont.\)  
Only the lower bound of \(\|e(k+N)\|\) is determined, but the specific upper bound value is not determined. The constraint range of \(\|e(k+N)\|\) is obtained, that is, \(0.78\|e(k+N-1)\|\leq\|e(k+N)\|<\|e(k+N-1)\|\).