**3. Open-loop dynamic analysis**

In general, reactive distillation is usually associated with the occurrence of multiple steady states [3]. Occurrence of multiplicity is a consequence of the high non-linearities associated with the RD process. The cause of multiplicity is connected with the presence of multiple reactions, heat of reaction, and the crossing of non-reactive distillation boundary via reaction. Multiplicity in the form of input multiplicity or output multiplicity exists in the RD process. Input multiplicity alters

**Figure 2.** *Aspen model liquid mole fraction profile compared with industry data.*

the selection of controlled variables, whereas output multiplicity affects the choices of control structure and the operating range [4]. In open-loop analysis, a series of step changes were applied to the manipulated variable (u1 and u2) in order to check for the presence of multiplicity and to set up the operating range.

#### **4. RD-DPC control system design**

To analyze the control performance of the RD-DPC process, a two-input two-output (TITO) multivariable system with time delay is considered [5]. *Gp(s)* represent the process transfer function. Similarly, *Gc-D(s)* and *Gc-C(s)* represent the decentralized controller [6–10] and centralized controller [11–15], respectively. Controller output and process output are represented by *ui* and *yi*, respectively.

$$\mathbf{G}\_{P}(\mathfrak{s}) = \begin{bmatrix} \mathbf{g}\_{P,11}(\mathfrak{s}) & \mathbf{g}\_{P,12}(\mathfrak{s}) \\\\ \mathbf{g}\_{P,21}(\mathfrak{s}) & \mathbf{g}\_{P,22}(\mathfrak{s}) \end{bmatrix} \tag{4}$$

$$\mathbf{G}\_{\mathfrak{c}-D}(\mathfrak{s}) = \begin{bmatrix} \mathbf{g}\_{\mathfrak{c},11}(\mathfrak{s}) & \ & \ & \\ \ & \ & \mathbf{g}\_{\mathfrak{c},22}(\mathfrak{s}) \end{bmatrix} \tag{5}$$

$$\mathbf{G}\_{\mathfrak{c}-\mathbf{C}}(s) = \begin{bmatrix} \mathbf{g}\_{\mathfrak{c},\mathfrak{U}}(s) & \mathbf{g}\_{\mathfrak{c},\mathfrak{U}}(s) \\ \mathbf{g}\_{\mathfrak{c},\mathfrak{U}}(s) & \mathbf{g}\_{\mathfrak{c},\mathfrak{U}}(s) \end{bmatrix} \tag{6}$$

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

For controller settings, the SIMC tuning relations [16] are used to design the decentralized controller. Similarly, Tanttu & Lieslehto (TL) [17] tuning relations are used for calculating centralized controller settings. The controller performance is assessed by considering the setpoint tracking, settling time, and disturbance rejection tests. The controller's ability to properly move to another purity level is assessed in the grade transition test. The disturbances variables are the feed flow rate of PA to the RD column (d1) and the reboiler heat duty of the separation column (d2). These variables are typically more exposed to disturbances since they are originated from outside of the system. The controller disturbance rejection potential is evaluated by doubling the amount of disturbance to the standard reported in the industry. The controller's performance is evaluated by employing integral square error (ISE).

### **5. Results and discussion**

#### **5.1 Open-loop analysis**

In open-loop analysis, the existence of multiple steady states is observed, and the operating window for the variables is set. This section is divided into two parts. **(i) Step changes in RD reboiler heat duty:** A series of step changes were applied to the reboiler heat duty of the RD column (the manipulated variable for controlling the molar purity of DPC), to fix the operating range. A step change of 1.5% was applied to the reboiler heat duty, and the corresponding dynamic response was observed for the controlled variables y1 and y2. It was found that the column sets at other steady state and the desired molar purities are not achieved. Thus, reduced step changes were applied to u1. For a series of step changes of 1% applied to u1, the desired molar purities were not achieved. Similarly, for step changes of 0.75% to u1, the desired molar purities of DPC and MA are obtained. Thus, the manipulated variable for the RD column is operated between 886.237 kW and 899.631 kW. **(ii) Step changes in separation column condenser heat duty:** Here, step changes were applied to the condenser heat duty of the separation column (the manipulated variable for controlling the molar purity of methyl acetate (MA)), in order to fix the operating region. For step changes of 3% and 2% to the condenser heat duty of separation column, the desired molar purities are not achieved and the process sets at another steady state, indicating the presence of multiplicity. Similarly, a step change of 1% is applied to u2, **Figure 3**, and the responses in y1 and y2 are observed. It was observed that the desired molar purities are obtained, and thus the manipulated variable for separation column is operated between 293.240 kW and 299.164 kW.

#### **5.2 RD-DPC model identification**

This section describes how the model Identification for RD–DPC process is carried out [18]. When the matching process employs optimization, a model prediction is aligned with the measured values with the use of a solver. Eq. (A1) has variables *y(t)* and *u(t)* and two unknown parameters *Kp* and *τp*. These variables may be adjusted to match the data. The solver often minimizes a measure of the alignment, such as a sum of the squared errors or sum of absolute errors. The optimization solver used in excel is "generalized reduced gradient (GRG) non-linear." Here, we have two manipulated variables u1 and u2. When we give a step change in u1, we observe the response in y1 and y2, respectively, and similarly a step input to u2 gives a response in y1 and y2. So, in total we have four data sets, u1-y1, u1-y2, u2-y1, and u2-y2.

**Figure 3.** *Open-loop dynamic behavior for NL-RD-DPC process of interaction (y1) and response (y2) for a given step change (1%) in condenser heat duty (kW) of SC column (u2).*

For the obtained datasets, the variables when adjusted give us four models – g11, g21, g12, and g22, respectively. Similarly, the optimization solver "SciPy.Optimize.Minimize" function in Python, changes the unknown parameters of Eqs. (A2) and (A3) to best match the data at specified time points. The sum-of-squared errors and the obtained values of the unknown parameters for first order, first-order plus timedelay (FOPTD) and second-order plus time-delay (SOPTD) model are given in **Table 3**.

[Supporting material of process identification is given in a separate compressed file (excel, python, Aspen Plus/Dynamics and MATLAB-Simulink programs) and readers can access files from the authors home page (https://sites.google.com/site/ bcs12614/)].

From data fit and θ/τ values in **Table 3**, it can be inferred that g11 is best fit by the FO model whereas g12, g21, and g22 are best fit by the SOPTD model. The nonlinear model, under the constraint given subsequently, can be represented by the transfer function given by Eq. (7). This can also be referred to as the original plant transfer function model. For the non-linear model, the manipulated variable is varied within the given range and the corresponding molar purities are obtained in the given range.

$$\mathbf{G}p(s) = \begin{bmatrix} \frac{0.00449}{1.00226s + 1} & \frac{0.00040 \text{ } e^{-0.21647s}}{1.0116s^2 + 1.8510s + 1} \\\\ \frac{0.00205 \text{ } e^{-0.6557s}}{1.065s^2 + 1.9988s + 1} & \frac{0.00052 \text{ } e^{-0.40216s}}{4.1925s^2 + 3.07522s + 1} \end{bmatrix} \tag{7}$$


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

#### **Table 3.**

*Transfer function model parameters.*

$$
\begin{pmatrix}
886.237\ kW < u\_1 < 899.631\ kW \\
\end{pmatrix},
$$

$$
\begin{pmatrix}
0.9594 < \chi\_1 < 0.9999 \\
0.9926 < \chi\_2 < 0.9993
\end{pmatrix}
$$

To evaluate the open-loop dynamic interactions between the PVs and MVs, the relative gain array (RGA) and the Niederlinski Index (NI) are applied [19]. The RGA for the RD-DPC process is:

$$RGA(\wedge) = \begin{bmatrix} 1.5413 & -0.5413 \\ -0.5413 & 1.5413 \end{bmatrix} \tag{8}$$

The NI for the original linearized plant model is 0.6488.
