The CSTRs in which the van de Vusse reaction takes place are a popular benchmark due to their nonlinearity and difficult dynamics, willingly used to test newly developed control algorithms; see for example, References [

46,

47,

48,

49,

50,

51]. The process model of the reactor is composed of two composition balance equations; see for example, Reference [

52]:

where

${C}_{\mathrm{A}}$,

${C}_{\mathrm{B}}$ are the concentrations of components A and B, respectively,

F is the inlet (and also outlet) flow rate,

V is the volume in which the reaction takes place (assumed constant and

$V=1\mathrm{L}$),

${C}_{\mathrm{Af}}$ is the concentration of component A in the inlet flow stream (if not declared otherwise, it is assumed that

${C}_{\mathrm{Af}}=10\mathrm{mol}/\mathrm{L}$). The values of parameters are:

${k}_{1}=501/\mathrm{h}$,

${k}_{2}=1001/\mathrm{h}$,

${k}_{3}=10\mathrm{L}/(\mathrm{h}\xb7\mathrm{mol})$. The output variable is the concentration

${C}_{\mathrm{B}}$ of substance B, the manipulated variable is the inlet flow rate

F,

${C}_{\mathrm{Af}}$ is the disturbance variable. The control plant has the inverse response, thus it is a control plant difficult to control using a standard algorithm. Therefore, it is natural to use an MPC algorithm in this case.

#### 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:

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 ${T}_{\mathrm{s}}=3.6$ 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 6th minute of the experiment are shown in

Figure 3. If the setpoint was changed to

${\overline{C}}_{B1}=1$ mol/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. 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.

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. In one case (setpoint change to ${\overline{C}}_{B2}=1.25$ mol/L) they are even faster. It should be, however, emphasized that the FMPC algorithms use reliable quadratic programming routine to generate control action.

Duration of action of the researched algorithms in the experiments depicted in

Figure 3 is compared in

Table 1. For each algorithm and for each setpoint five measurements of duration were done (time needed to simulate the control plant was subtracted) and averaged. Then these average durations were summed in the last row of the table. Both FDMC algorithms work much (around 20 times) faster than the NMPC algorithm.

There was also done an experiment with

$\lambda =0.0001$ (10 times smaller than in the first experiment;

Figure 4). All the algorithms work now faster; the control action in all cases is more aggressive, but the relations between FMPC, NMPC, and LMPC algorithms remain unchanged; they are the same as in the previous experiment, for

$\lambda =0.001$. However, as the

$\lambda $ parameter is closer to zero, the differences between FMPC1 and FMPC2 algorithms became much smaller comparing to the previous experiment. The FMPC1 algorithm is practically as fast as the FMPC2 one. Such a phenomenon is reasonable, because for

$\lambda =0$ both performance indexes (

32) and (

37) are the same. Thus, the closer value of the

$\lambda $ coefficient to 0 is, the closer responses generated with FMPC1 and FMPC2 are.

There were also done experiments with shortening the control horizon

s (

Figure 5 and

Figure 6). Note, that the smaller the control horizon is the less number of decision variables in the optimization problem. Therefore, it is simpler to solve and can be solved faster. In the case of both algorithms FMPC1 and FMPC2, with different values of

$\lambda $ for many values of the control horizon the responses generated with the algorithms do not change too much. Thus, the control horizon can be shorted significantly. In the considered examples for FMPC1 algorithm the responses generated for

$s=35$ and for

$s=10$ are almost the same; in the case of control system with FMPC2 algorithm—the responses generated for

$s=35$ and for

$s=5$ are almost the same. Further decreasing of the control horizon brings changes in the responses.

In the case of the control system with FMDC1 algorithm, for $s=5$ the disturbance compensation near ${C}_{B}=1.25$ mol/L becomes faster, but unfortunately achieving the setpoint ${\overline{C}}_{B1}=1$ mol/L lasts longer. Further decrease of the control horizon to $s=1$ causes that the disturbance compensation near ${C}_{B}=1.25$ mol/L becomes even faster as well as reaching the setpoint ${\overline{C}}_{B2}=1.25$ mol/L, however at the cost of slower response to the setpoint change to ${\overline{C}}_{B1}=1$ mol/L and worsened disturbance compensation near ${C}_{B}=1$ mol/L.

In the control system with FMDC2 algorithm, for $s=2$ and $s=1$ the disturbance compensation near ${C}_{B}=1.25$ mol/L becomes slightly faster, but unfortunately achieving both the setpoints ${\overline{C}}_{B1}=1$ mol/L and ${\overline{C}}_{B2}=1.25$ mol/L lasts longer. Moreover disturbance compensation near ${C}_{B}=1$ mol/L is now slower. Thus, in this case, as in the case of FMPC1 algorithm, decreasing the control horizon should be done also carefully and there is also a threshold which should not be crossed in order to avoid control performance deterioration.

To sum up, during designing of an algorithm it is advisable to check if the control horizon can be decreased (and optimization problem solved by the algorithm simplified). However, it should be done carefully, because for too short control horizon control performance can got worse.

The mechanisms of disturbance measurement utilization were used in the next experiments. Both FMPC algorithms generated similar responses. The first method of disturbance measurement utilization (red lines in

Figure 7 and

Figure 8) is based on the same approach as in the standard DMC algorithm, the second one employs the nonlinear process model (magenta lines in

Figure 7 and

Figure 8). Disturbance compensation near

${C}_{B}=1.25$ mol/L is almost the same in the case of both methods; differences in responses are really small. It is because the model of disturbance influence on the control plant for the first method was obtained near this output value.

Application of both methods of disturbance measurement utilization near

${C}_{B}=1$ mol/L causes reduction of the maximal control error by around 50% comparing to the case when disturbance measurement was not exploited (blue lines in

Figure 7 and

Figure 8). Both methods offer very good disturbance compensation. However, in the case of the one based on the nonlinear model (magenta lines in

Figure 7 and

Figure 8) the output value achieves the setpoint slightly faster than the one based on the linear model (red lines in

Figure 7 and

Figure 8) while changes of the control signal are smaller.