## 4 Example
### Experiments  
For the considered control plant a few MPC algorithms were designed--an NMPC one based on the nonlinear model and nonlinear optimization, an LMPC one based on a linear model in the form of a step response (of DMC type) and the proposed FMPC algorithms using a fuzzy model, different methods of prediction improvements, and different forms of the performance index (FMPC1--with classical performance index, and FMPC2--with the modified performance index). The fuzzy model used in the FMPC algorithms is composed of three step responses obtained near the following operating points:  
R1 \(C_{B0}=0.91\text{ mol}/\text{L}\), \(C_{A0}=2.18\text{ mol}/\text{L}\), \(F=20\)\(\text{L}/\text{h}\);  
R2 \(C_{B0}=1.12\text{ mol}/\text{L}\), \(C_{A0}=3\text{ mol}/\text{L}\), \(F=34.3\)\(\text{L}/\text{h}\);  
R3 \(C_{B0}=1.22\text{ mol}/\text{L}\), \(C_{A0}=3.66\text{ mol}/\text{L}\), \(F=50\)\(\text{L}/\text{h}\). The model used to obtain the dynamic matrix is thus composed of three rules; the membership functions, same as in Reference [34], chosen after analysis of the steady-state characteristic of the control plant, are shown in Figure 2.  
The simulation experiments were done using Matlab. The sampling time equal to \(T_{\text{s}}=3.6\) s was assumed; the values of tuning parameters were as follows: prediction horizon \(p=70\), control horizon \(s=35\), and at the beginning weighting coefficient \(\lambda=0.001\). During the experiments operation of control systems with NMPC, LMPC and FMPC algorithms was compared. The LMPC algorithm was designed using the step response obtained near the R2 point. The FMPC algorithms use two models: nonlinear one in the form of state equations, to generate a free response and the fuzzy one with local models in the form of step responses, to obtain the dynamic matrix. Responses of the control system to changes in the setpoint and to the change of the disturbance by \(10\%\) in the 6 th minute of the experiment are shown in Figure 3. If the setpoint was changed to \(\overline{C}_{B1}=1\text{ mol}/\text{L}\), responses obtained in the control system with the FMPC1 algorithm (blue lines in Figure 3) are very close to those obtained with NMPC algorithm with nonlinear optimization (red lines in Figure 3). In both cases there is almost no overshoot and both algorithms work much better than their LMPC counterpart (magenta lines in Figure 3). In the latter case there is a significant overshoot. Figure 2: Membership functions of the fuzzy model used to obtain the dynamic matrix. Moreover, settling time is much longer than in the case of NMPC and FMPC algorithms. The FMPC1 algorithm is slightly slower than the NMPC one, whereas the FMPC2 is practically as fast as NMPC. Figure 3: Responses of the control system to changes in the setpoint to \(\overline{C}_{B1}=1\bmod/L\) and \(\overline{C}_{B2}=1.25\bmod/L\) and to the change of the disturbance by 10% in the 6 th minute of the experiment; \(\lambda=0.001\); Nonlinear Model Predictive Control (NMPC)—red lines, linear MPC (LMPC)—green lines, fuzzy MPS (FMPC)1—blue lines, FMPC2—magenta lines. In the case when setpoint changed to \(\overline{C}_{B2}=1.25\) mol/L, the responses obtained with the FMPC algorithms are even faster than the one generated with NMPC algorithm. In the case of all the algorithms the setpoint is reached without overshoot, but LMPC works much slower than the algorithms based on nonlinear models. Disturbance responses obtained near \(C_{B}=1\) mol/L with FMPC and NMPC algorithms are almost the same. There is no overshoot which is present in the response of the LMPC algorithm. The algorithms based on nonlinear models slightly faster compensate the change in disturbance. In the case of operation near \(C_{B}=1.25\) mol/L, the NMPC is the fastest in compensation of the disturbance change. The FMPC1 is slightly slower and the LMPC is much slower than its counterparts. It can be noticed that the FMPC algorithms offer control performance close to the one which can be obtained with the NMPC algorithm and with nonlinear optimization.