*5.3.2 Simulations on linear and real non-linear model*

The simulations on the linear model are carried out in MATLAB SIMULINK, first by employing the decentralized controller and then the centralized controller.


**Table 4.** *Controller transfer functions.* *Centralized and Decentralized Control System for Reactive Distillation Diphenyl Carbonate… DOI: http://dx.doi.org/10.5772/intechopen.101981*

Here, setpoint tracking (servo problem) and load rejection (regulator problem) simulations to the linearized plant transfer function model are carried out. The setpoint tracking is done by giving setpoint changes in yr1 and yr2. yr1 is the setpoint to the controlled variable y1 and similarly yr2 is the setpoint to y2. Similarly, the disturbances are set to the input variable (u1 and u2) of the process.

The simulations on the non-linear model are done by replacing the linearized plant transfer function model with the original non-linear model. Here, the same controller settings of the linear model are applied along with base value (i.e., ui,0 + Δui) to the non-linear model in order to check the controller performance. The setpoint tracking is carried out by changing the setpoint in the range of 0.921 to 0.996 for yr1, whereas yr2 is changed between 0.995 and 0.999. Particularly for the present case, the setpoints were set at 0.975 for yr1 and 0.996 for yr2. Similarly, disturbances for the real model were considered as the feed flow rate of PA to the RD column (d1) and reboiler heat duty of the separation column (d2). The disturbances are in the range of 10 *kmol/hr* < d1 < 10.0185 *kmol/hr*, and 296.697 *kW* < d2 < 297.697 *kW*.

**Linear model: Figures 4** and **5** show the response for all the transfer function models to setpoint changes and load changes, respectively, indicating the SOPTD model-based controller giving the best load rejection, less settling time, and reduced interactions. Similarly, **Figure 6** shows the comparative performance of the decentralized and the centralized SOPTD-PID controller for setpoint change, indicating centralized controller giving best performance.

#### **Figure 4.**

*Centralized controller – setpoint tracking and interactions for change in yr1 and yr2. (a) and (d) represent the responses in y1 and y2, respectively, for the setpoint change in yr1 and yr2. (b) and (c) represent the corresponding interactions.*

**Figure 5.**

*Decentralized controller – load rejections and interactions for change in d1 and d2. (a) and (d) represent the responses in y1 and y2, respectively, for the load change in d1 and d2. (b) and (c) represent the corresponding interactions.*

**Non-linear model:** Similarly, for non-linear model, it can be observed from **Figures 7**–**9** that SOPTD-PID centralized controller gives better load rejections and reduced interactions, as compared to other model-based controllers.

It is clear that both centralized and decentralized SOPTD-PID controllers show faster settling time, reduced interactions, and lower oscillations. From **Figures 6** and **9**, it is clear that the centralized controller gives faster settling, reduced interactions, and lower oscillations, as compared to the decentralized controller.

#### *5.3.3 Robust stability analysis*

The presence of model uncertainties necessitates the stability robustness of the multi-loop control system [22–26]. The dynamic perturbations existing in the system can be lumped into one single perturbation block Δ. To evaluate the robustness of the control system, inverse maximum singular value method is considered [17]. First, for a process multiplicative input uncertainty,*G s*ð Þ½ � *I* þ Δ*I*ð Þ*s* , the closed-loop system is stable if:

$$\left\|\left|\Delta\_I(j\omega)\right\|\right\| < \frac{1}{\sigma} \left\{ \left[I + \mathcal{G}\_D(j\omega)\mathcal{G}(j\omega)\right]^{-1} \mathcal{G}\_C(j\omega)\mathcal{G}(j\omega) \right\} \tag{9}$$

where *σ* is the maximum singular value of the closed-loop system. Similarly, for process multiplicative output uncertainty, ½ � *I* þ Δ*O*ð Þ*s G s*ð Þ, the closed-loop system is stable if:

*Centralized and Decentralized Control System for Reactive Distillation Diphenyl Carbonate… DOI: http://dx.doi.org/10.5772/intechopen.101981*

**Figure 6.**

*SOPTD-PID controller – setpoint tracking and interactions for a given step change in yr1 and yr2. (a) and (d) represent the responses in y1 and y2, respectively, for the setpoint change in yr1 and yr2. (b) and (c) represent the corresponding interactions.*

$$\left\|\left|\Delta\_O(j\alpha)\right\|\right\| < \frac{1}{\sigma} \left\{ \left[I + \mathbf{G}(j\alpha)\mathbf{G}\_C(j\alpha)\right]^{-1} \mathbf{G}(j\alpha)\mathbf{G}\_C(j\alpha) \right\} \tag{10}$$

The closed-loop system stability bounds are indicated by the frequency plots for the right-hand side part of Eqs. (9) and (10). The controller stability can be easily compared by comparing the area under the curve (more the area, more is the stability).

**Figures 10** and **11** show the stability bounds for decentralized and centralized RD-DPC control, respectively. In these figures, the region above the curve indicates the instability region and that below the curve indicates the stable region. From **Figures 10** and **11**, it is clear that the FO-PI controller has more area under the curve, as compared to other controllers. Thus, the FO-PI controller gives robust control as compared to others, but this contradicts the above conclusions of SOPTD-PID controller performance being the best model. This can be explained as follows: For any magnitude of change to setpoint and lower magnitudes for disturbances, the SOPTD model-based controller gives the best performance. However, if the magnitude of disturbances is high, the first-order model-based controller gives the best performance. This can be easily interpreted from **Figures 12** and **13**. **Figure 12** shows that for lower frequency range (10�<sup>2</sup> to 1 *rad/s*), the centralized FO-PI controller gives better robust stability as compared to the decentralized FO-PI controller. Similarly, from **Figure 13**, SOPTD-PID decentralized controller shows better robustness for higher frequency range (1 *rad/s* and above).

#### **Figure 7.**

*Decentralized controller – setpoint tracking and interactions for a given step change in yr1 and yr2. (a) and (d) represent the responses in y1 and y2, respectively, for the setpoint change in yr1 and yr2. (b) and (c) represent the corresponding interactions.*

#### **Figure 8.**

*Centralized controller – load rejections and interactions for a given step change in d1 and d2. (a) and (d) represent the responses in y1 and y2, respectively, for the load change in d1 and d2. (b) and (c) represent the corresponding interactions.*

*Centralized and Decentralized Control System for Reactive Distillation Diphenyl Carbonate… DOI: http://dx.doi.org/10.5772/intechopen.101981*

#### **Figure 9.**

*SOPTD-PID controller – load rejections and interactions for a given step change in d1 and d2. (a) and (d) represent the responses in y1 and y2, respectively, for the load change in d1 and d2. (b) and (c) represent the corresponding interactions.*

**Figure 10.** *Decentralized controller—Robustness—(a) input multiplicative and (b) output multiplicative uncertainties.*

**Figure 11.** *Centralized controller—Robustness—(a) input multiplicative and (b) output multiplicative uncertainties.*

**Figure 12.** *FO-PI controller—Robustness—(a) input multiplicative and (b) output multiplicative uncertainties.*

*Centralized and Decentralized Control System for Reactive Distillation Diphenyl Carbonate… DOI: http://dx.doi.org/10.5772/intechopen.101981*

**Figure 13.** *SOPTD-PID controller—Robustness—(a) input multiplicative and (b) output multiplicative uncertainties.*
