MIMO MPC with SFLD Control

This section will detail the MIMO MPC with SFLD starting with defining the control problem and performing tuning on two SISO systems to then complete the tuning of the MIMO system and determine if the SFL-Plant constraints can be applied successfully; the criteria for successful constraint application is feasible input profiles.

5.2.3.1 Objective Function and Constraints

The objective function for the MIMO MPC with SFLD control problem is:

min 𝑣 𝐽 = 𝑄1∑(𝑦1,𝑖− 𝑦1,𝑖,𝑠𝑒𝑡𝑝𝑜𝑖𝑛𝑡) 2 𝑁𝑝 𝑖=1 + 𝑅₁∑ ∆𝑣_1,𝑙2 𝑁𝑐 𝑙=1 + 𝑄₂∑(𝑦_2,𝑖− 𝑦_{2,𝑖,𝑠𝑒𝑡𝑝𝑜𝑖𝑛𝑡})2 𝑁𝑝 𝑖=1 + 𝑅₂∑ ∆𝑣_2,𝑙2 𝑁𝑐 𝑙=1 Equation 5-6 Subject to: 273 ≤ 𝑢1 = 𝜑(𝒙, 𝑣) ≤ 360 −2 ≤ Δ𝑢1(𝐾𝑚𝑖𝑛−1) = 𝜑(𝒙𝒊, 𝑣𝑖) − 𝜑(𝒙𝒊+𝟏, 𝑣𝑖+1) ≤ 2 0.25 ≤ 𝑢₂ = 𝜑(𝒙, 𝑣) ≤ 2.5 Given: 𝒙 𝒙̇ = 𝑓(𝑥) + 𝑔(𝑥)𝑢

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Where 𝑢1 is the jacket temperature, 𝑢2 is seed loading, 𝑦1is the supersaturation and 𝑦2 is the

crystal mean size. The SFL-Plant constraints are also applied to the absolute value and relative change of the jacket temperature, and also to the absolute value of the seed loading.

The simulation data is from the P/W system as shown in Table 5-3 (Nagy, Chew, et al., 2008a; Nagy, Fujiwara, et al., 2008), meanwhile the initial conditions for the crystallizers are a temperature of 315 K, concentration of 0.0256 g/g initialised with seed moments corresponding to a mean-size of 20 µm and seed loading of 0.5 g/L.

Value Units 𝒌𝒃 𝑒45.8 min-1g-1 𝒃 6.2 - 𝒌𝒈 𝑒−4.1 m min-1 𝒈 1.5 - 𝝆 1000 kgm-3 𝒌𝒗 0.24 - 𝝆𝒄 1296 kgm-3 𝑽 1 L 𝑪𝒊𝒏 0.0256 g/g 𝑭𝒊𝒏/𝑭𝒐𝒖𝒕 0.07 L/min 𝑲 0.1 - 𝑼𝑨𝒄 54521 J min-1 K-1 𝑻𝒊𝒏 305 K

Table 5-3 – Crystallization Data

5.2.3.2 Tuning MIMO MPC with SFLD using SISO MPC

The tuning procedure used will first consider the two decoupled input/output systems as SISO systems to perform the tuning, and the chosen parameters will be combined for the MIMO MPC. The objective function weights for 𝑄 and 𝑅 are set to 1 for the SISO tuning.

5.2.3.2.1 SISO Supersaturation Control Tuning

The SISO supersaturation control tuning was fully detailed in chapter 4. The resulting tuning parameters values of 0.5 and 1 were used for 𝛽₀ and 𝛽₁, respectively.

5.2.3.2.2 Seed loading

The SISO case for seed loading and mean size has also been evaluated using a similar iterative approach. It was determined that the seed loading should not be less than 0.25 g/L or larger

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than 2.5 g/L, thus these constraints were defined using the SFL-Plant constraints method. The MPC objective function used for tuning is:

min 𝑣 𝐽 = ∑(𝑦𝑖− 𝑦𝑖,𝑠𝑒𝑡𝑝𝑜𝑖𝑛𝑡) 2 5 𝑖=1 + ∑ ∆𝑣_𝑙2 5 𝑙=1 Equation 5-7 Subject to: 0.25 ≤ 𝑢 = 𝜑(𝒙, 𝑣) ≤ 2.5 Given: 𝒙 𝒙̇ = 𝑓(𝑥) + 𝑔(𝑥)𝑢

The main results from this tuning were that the seed size appears to be very sensitive to the seed loading and many combinations of tuning parameters needed to be used to establish a stable and desirable control response. It was found that the value of 𝛽0 had significantly less

impact on the overall control response than 𝛽1. For the latter, a value less than 25 resulted in

significant oscillations in the seed loading throughout the simulation and values over 50 resulted in a very slow response and convergence to setpoint, longer than the 300-minute simulation used for tuning. Three tuning simulations are shown where 𝛽1 is 30, 40 and 50 in

Figure 5-3, Figure 5-4 and Figure 5-5 respectively.

Figure 5-3 - Mean size control – 𝛽1= 30

The results in Figure 5-3 show the mean size of crystals increasing from 20 µm, overshooting the set-point of 40 µm and finally converging to within 2% of the target at 82 minutes and stabilized within 1% of the target after 125 minutes. The corresponding seed loading sees

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some oscillations at the beginning, including saturation at the highest and lowest allowed seed loading as per the bounds, but this subsided after 15 minutes and was common to all three results that are presented here. The oscillation could not be avoided despite efforts to change the tuning parameters, this is one of the inherent difficulties with this form of input/output linearization; tuning parameters can be chosen to ensure a convergence but in comparison to traditional linear systems with linear MPC, the parameters cannot be chosen intuitively (Kravaris and Soroush, 1990; Kravaris, 1988). There is a transient change in the loading up to 82 minutes, beyond which the loading stabilizes with some minor oscillations. Finally, the trajectory is smooth after 125 minutes, simultaneously the output trajectory also stabilises. Though the result was considered acceptable as the mean size difference between the set-point and trajectory was 0.04 µm, other controller tuning parameters were tested to see if a value closer to 40 µm could be obtained.

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Figure 5-5 - Mean size control – 𝛽1= 50

Subsequently, the results in Figure 5-4 show the best case for set-point convergence where 𝛽₁ is 40, the mean size was within 2% at 100 minutes and the trajectory finally stabilised to within 0.25% of the set-point at 130 minutes. In addition to requiring more time to reach steady state, the seed loading trajectory has a noticeably greater magnitude in oscillation which would require the seeding mechanism to change the loading every minute. The oscillations were more pronounced as compared to the previous tuning settings but eventually the input oscillations did stabilize and disappear after 250 minutes. For a final comparison, the case where 𝛽₁ is 50 shows the mean size within 2% of the set-point at 135 minutes and stabilizing within 1.25% after 190 minutes. In conjunction to this, the seed loading sees some oscillatory behaviour again followed by a transient change to seed loading through the start-up procedure in the first 120 minutes and the seed loading is stabilized at 190 minutes. The larger value of 𝛽1 in this range resulted in longer time to reach steady state.

There was no direct correlation between the tuning value to the set-point error, because when the value of 40 was chosen the mean size at steady state was closest to the set-point, but it was further from setpoint at 30 and furthest at 50.

This decaying oscillatory response at the beginning of these simulations could be due to some plant-model mismatch which causes an instability in the control moves, but further actions could be taken to mitigate these. The first option would be to apply significantly larger input weights on the seed loading to prevent the oscillations. Another way to prevent the

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oscillations would be to apply an input rate limit, thus preventing the seed loading from changing significantly from one-time step to the next. It is possible to facilitated these constraints in the constraints handling function. Furthermore, there may be a need for more complex linear models to capture the input/output behaviour for seed loading onto crystal mean size, and a further analysis on the plant-model mismatch is likely to indicate this; alternatively it may be more suitable to select a different input instead of seed loading to control the mean size.

Though these results demonstrate the impact of the controller tuning parameters for SFL, the decision was made to use the value of 40 for 𝛽1 despite the oscillatory behaviour in the input

trajectory. The SFL-Plant constraints applied to the seed loading in all three simulations were also satisfied, so all the input trajectories were again feasible. Therefore, with the two separate SISO test cases, a starting point for MIMO MPC with SFLD is now established.