11.2 Closed-Loop Simulations
11.2.1 Simulations based on first-order identified model
, R=0.52 0 0 0.52
The process is sampled using a sample time Ts = 4 [s]. The prediction hori-zon used in the simulations is N = 500, corresponding to a prediction of the future dynamic behavior of the process for 2000 [s]. It has been chosen to be long to approximate an infinite horizon controller, hereby reducing the effect of differences between open-loop and closed-loop solutions.
In the simulations the nominal value for the deterministic part of the disturbance isddetk = [70 70]T. It should be noted that these are the same conditions which the identified models were obtained under. This corresponds to what would be considered normal operation conditions in the case study.
In the simulations a simulation profile where the process is in a steady state is used. The initial water level (height) in Tank1 is 50.2cm and the water level in Tank2is42.7 cm.
11.2.1 Simulations based on first-order identified model
As already mentioned, it is not evident that all the tuning parameters obtained in the previous section result in satisfactory behavior of the control system. To investigate what level of sensitivity results in acceptable behavior, simulations are conducted for different values ofMS,max.
The first simulation is conducted based on the results in Table11.1. A simulation based on the tuning parameters obtained forMS,max= 1.25andMS,max= 1.30 is presented in Figure11.1.
In the simulation the reference for the water level in Tank 1 is att= 170[min]
increased to 58cm, while the reference for Tank 2 is unchanged. Furthermore, at t= 35 [min] a step change in the deterministic part of the disturbance into Tank 2 is introduced with a relative change of +40[cm3/s].
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Figure 11.1: Closed-loop simulation based on first-order identified model with tuning parameters obtained usingMS,max= 1.25andMS,max= 1.30.
The simulation in Figure11.1shows that these tunings provide good reference tracking for both outputs, despite the disturbance and the high level of process and measurement noise. It is seen that the disturbance rejection for y2 is most efficient for MS,max= 1.25, but at the same time that y1 reacts quite strongly as well.
Furthermore, it is clearly seen that the variance of the control signal u2 for MS,max = 1.30 is very high and that this set of tuning parameters is not ac-ceptable and useless in practice. The parameters obtained for MS,max = 1.25 seem to provide an acceptable performance.
Next it is investigated what performance is obtained when the sensitivity bound is reduced. To illustrate this, a test is conducted for MS,max = 1.20 and MS,max = 1.15 using the same simulation scenario as used above. The sim-ulation is presented in Figure11.2.
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Figure 11.2: Closed-loop simulation based on first-order identified model with tuning parameters obtained usingMS,max= 1.15andMS,max= 1.20.
From the simulation in Figure11.2it is seen that these sets of tuning parame-ters provide a marginally better and smooth overall reference tracking for both outputs compared to the previous simulation. However, it is seen that these tunings both react more slowly to the reference step. A nice feature related to these tunings is that both outputs are relatively insensitive to the high level of process noise.
The disturbance rejection is similar for both sets of tuning parameters, but MS,max = 1.20 provides a slightly faster rejection without causing any side effects.
A very important fact about these tunings is that the variances on the control inputs are very small. In particular it is seen that the variance of the control sig-nalu2is reduced substantially for both tunings in comparison to the case where MS,max = 1.30. This nicely illustrates the benefits of lowering the sensitivity bound.
It will now be investigated if the performance can be increased even more by further reducing the sensitivity bound. This simulation is conducted for MS,max = 1.20 and the lowest sensitivity bound MS,max = 1.15. The simula-tion for these sets of tuning parameters is presented in Figure 11.3.
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Figure 11.3: Closed-loop simulation based on first-order identified model with tuning parameters obtained usingMS,max= 1.10andMS,max= 1.20.
The simulation in Figure11.3very clearly illustrates the trade-off between dis-turbance rejection and sensitivity of the system. It is seen that the disdis-turbance introduced iny2 at t= 35 [min] is rejected very slowly forMS,max= 1.10and that this has a substantial impact in the simulation.
When compared to the tuning obtained forMS,max= 1.20 it is seen that this tuning is in a sensetoo robust and probably not that useful in practice due to its overall performance.
From this simulation it can be concluded that there is no practical benefit in selectingMS,max= 1.10as this heavily increases the disturbance rejection time and does not offer a better tracking for the system. Based on the simulations conducted so far the best trade-off between low input variance and disturbance rejections seems to be obtained for a tuning usingMS,max= 1.20orMS,max= 1.15.
The simulations conducted so far are all based on the tuning parameters in Table 11.1obtained using tuning Approach I. Since the tuning parameters obtained for other deviation scenarios are very similar, simulations will not be illustrated for the tuning parameters in Table11.2 and Table11.3. It should however be noted that similar performance and behavior are observed for such simulations.
Next simulation based on the results in Table 11.4is considered. That is the tuning parameters obtained using tuning Approach II.
As already stated, the tuning parameters obtained based on Approach II are very similar to that of Approach I. Based on simulation studies it was seen that the tuning parameters resulted in similar closed-loop performance. Therefore only one simulation is presented for this approach. The simulation is conducted forMS,max= 1.25andMS,max= 1.50and is reported in Figure11.4.
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Figure 11.4: Closed-loop simulation based on first-order identified model with tuning parameters obtained usingMS,max= 1.25andMS,max= 1.50.
The simulation in Figure 11.4 shows that the tuning for MS,max = 1.25 pro-vides good reference tracking for both outputs, despite the disturbance and the high level of process and measurement noise. For MS,max = 1.50 this is not the case. It is seen that especially y1 is heavily affected by the high level of process and measurement noise and as a result oscillates around the reference.
The disturbance rejection fory2is surprisingly very efficiently rejected for both tunings.
Furthermore, it is clearly seen that the variance of the control signal u1 for MS,max = 1.50 is very high and that this set of tuning parameters is not ac-ceptable and useless in practice. It is noted that this is the opposite situation of what was seen in Figure 11.1. The parameters obtained forMS,max = 1.25 seem to provide an overall acceptable performance.