Design specifications and parameters.

## Abstract

Reactive distillation (RD), a process-intensified technique, involves the integration of reaction and separation in a single unit. High non-linearities associated with the reactive distillation process constrict the control degrees of freedom and set the key challenge in the design of a robust control system. In this chapter, reactive distillation diphenyl carbonate (RD-DPC) design is optimized, and a decentralized as well as centralized feedback control configuration is applied to carry out the control studies. To execute the control scheme, a dynamic model of RD-DPC process is developed using Aspen Dynamic and interfaced with MATLAB Simulink for online control implementation. A comparative multi-loop feedback controller control performance study is done for different transfer function models obtained by using analytical- and optimization-based process identification techniques. The controller parameters obtained from the simple internal model control (SIMC) tuning relations for decentralized controller and Tanttu & Lieslehto (TL) tuning relations for centralized controller are applied to (i) the linear transfer function model and (ii) non-linear plant model. Set-point tracking, load rejection studies and robust stability analysis are carried out to compare the performance of different models and to investigate the controller performance of the non-linear model.

### Keywords

- reactive distillation
- diphenyl carbonate (DPC)
- decentralized controller
- centralized controller
- robustness
- non-linear model

## 1. Introduction

Polycarbonates, containing carbonate groups in their chemical structures, are an important group of thermoplastic polymers. Diphenyl carbonate (DPC), an acyclic carbonate ester, is a monomer in the production of polycarbonate polymers. The production of DPC is carried out by the transesterification reaction between dimethyl carbonate (DMC) and phenyl acetate (PA). The reactive distillation process, involving the integration of reaction and separation in one place, is usually associated with high non-linearities. The interaction of reaction and separation, responsible for the occurrence of multiple steady states, sets a challenge in designing a robust controller. Furthermore, the high non-linearity and dynamic interactions cannot be effectively controlled by single-input single-output (SISO) controller and hence urges for multi-input multi-output (MIMO) controller.

In this work, RD–DPC process model is simulated using Aspen Dynamic V11. The transfer function model and controller development are performed using MATLAB 2019b Simulink Control system and custom proportional-integral-derivative (PID) coding. An online control environment is created by interfacing Aspen Dynamic with MATLAB Simulink via AM System block and similarly linking the centralized controller to MATLAB Simulink via S-function block.

This chapter reflects the designing of RD-DPC two-column indirect sequence and a control system for maintaining the molar purity of DPC and methyl acetate (MA) greater than 99%. The chapter also shows a comparative study between control performance of decentralized and centralized feedback controllers.

## 2. RD-DPC multivariable process

DPC is produced by reacting phenyl acetate (PA) and dimethyl carbonate (DMC) in a reactive distillation column. The involved reactions and the corresponding reaction rates are mentioned subsequently (Eqs. (1)–(3)). The reaction kinetic constants for the forward and backward reactions are taken from the work done by Cheng et al. [1]. There is a rectification and a reaction zone in the RD column, as shown in Figure 1. Column design specifications and additional parameters are reported in Table 1. Although high-purity DPC is obtained at the bottoms of the RD column, the purity of methyl acetate (MA) obtained at the distillate of the RD column is low. To obtain MA at the desired purity, we have to use another separation column, thus reactive distillation plus non-reactive distillation.

Sr. No | Parameters | RD | SC |
---|---|---|---|

1 | Total number of stages | 65 | 25 |

2 | Number of reactive stages | 4–65 | — |

3 | Feed stage | 4, 59 | 11 |

4 | Reflux ratio | 1.25 | 2.48 |

5 | Operating pressure (kPa) | 125 | 101 |

6 | Stage pressure drop (kPa) | 0.625 | 0.63 |

7 | Tray holdup | 0.097 | — |

8 | Column diameter (m) | 1.022 | 0.642 |

9 | Height of column (m) | 32.5 | 12.5 |

10 | Condenser duty (kW) | −839.704 | −296.229 |

11 | Reboiler duty (kW) | 892.996 | 297.697 |

Aspen Plus/Dynamics is used to design and simulate the RD-DPC indirect sequence. The steady-state simulation results are shown in Table 2. In terms of the model validation, the required data are taken from the original case study [1, 2]. This case study is close to a real RD-DPC process in terms of sizing, as reported by the researchers; thus, the validation data can be regarded as industrial data as well. Here, the liquid mole fraction profile is chosen for validation. Figure 2 depicts the performance of simulating the RD-DPC liquid mole fraction profile and validation results. Aspen model managed to present accurate results by validating the data having an R^{2} value of ≈0.9.

Description | PA | DMC | TDMC | BTMS1 | DIST1 | BTMS2 | DIST2 |
---|---|---|---|---|---|---|---|

T (C) | 204.54 | 103.5 | 95.5 | 330.1 | 84.7 | 94.2 | 57.1 |

P (kPa) | 127 | 152 | 163 | 165.32 | 125 | 116.12 | 101 |

Mole Flow (kmol/hr) | 10 | 5.06 | 35.26 | 5.02 | 40.24 | 30.2 | 10.04 |

Mole fraction | |||||||

MA | 0 | 0 | 0.00618 | 0 | 0.26 | 0.0072 | 0.994 |

PA | 1 | 0 | 0 | 0.0054 | 0 | 0 | 0 |

DMC | 0 | 1 | 0.994 | 0 | 0.74 | 0.993 | 0.0067 |

DPC | 0 | 0 | 0 | 0.992 | 0 | 0 | 0 |

MPC | 0 | 0 | 0 | 0.0028 | 0 | 0 | 0 |

## 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 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 (u_{1} and u_{2}) 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]. * G* represent the process transfer function. Similarly,

_{p(s)}

*and*G

_{c-D(s)}

*represent the decentralized controller [6, 7, 8, 9, 10] and centralized controller [11, 12, 13, 14, 15], respectively. Controller output and process output are represented by*G

_{c-C(s)}

*and*u

_{i}

*, respectively.*y

_{i}

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 (d_{1}) and the reboiler heat duty of the separation column (d_{2}). 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 y

_{1}and y

_{2}. 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 u

_{1}. For a series of step changes of ±1% applied to u

_{1}, the desired molar purities were not achieved. Similarly, for step changes of ±0.75% to u

_{1}, 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.

**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 u**(ii) Step changes in separation column condenser heat duty:

_{2}, Figure 3, and the responses in y

_{1}and y

_{2}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

*and two unknown parameters*u(t)

*and*K

_{p}

*. 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 u*τ

_{p}

_{1}and u

_{2}. When we give a step change in u

_{1}, we observe the response in y

_{1}and y

_{2}, respectively, and similarly a step input to u

_{2}gives a response in y

_{1}and y

_{2}. So, in total we have four data sets, u

_{1}-y

_{1}, u

_{1}-y

_{2}, u

_{2}-y

_{1}, and u

_{2}-y

_{2}. For the obtained datasets, the variables when adjusted give us four models – g

_{11}, g

_{21}, g

_{12}, and g

_{22}, 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 time-delay (FOPTD) and second-order plus time-delay (SOPTD) model are given in Table 3.

Model | First order | FOPTD | SOPTD | |
---|---|---|---|---|

(GRG – Excel) | (Opt – Python) | (Opt – Python) | ||

g_{11} | K_{p} | 0.00449 | 0.0045 | 0.0045 |

τ_{p} | 1.00226 | 1.43951 | 0.67663 | |

θ | 0.33062 | 0.21039 | ||

— | 1.10372 | |||

SSE | 0.01054 | 0.005402 | 0.005268 | |

g_{12} | K_{p} | 0.0004 | 0.0004 | 0.0004 |

τ_{p} | 1.12152 | 1.00226 | 1.00579 | |

θ (s) | 0.01 | 0.21647 | ||

— | 0.92018 | |||

SSE | 0.00127 | 0.000296 | 0.000288 | |

g_{21} | K_{p} | 0.00203 | 0.00205 | 0.00205 |

τ_{p} | 1.00058 | 2.26224 | 1.03203 | |

θ | 0.69118 | 0.65557 | ||

— | 0.96838 | |||

SSE | 0.004382 | 0.004048 | 0.00407 | |

g_{22} | K_{p} | 0.00053 | 0.00053 | 0.00052 |

τ_{p} | 2.83704 | 2.67442 | 2.04758 | |

θ | 0.52164 | 0.40216 | ||

— | 0.75094 | |||

SSE | ≈ 0 | ≈ 0 | ≈ 0 | |

θ/τ | — | |||

RGA | ||||

NI | 0.6588 | 0.6562 | 0.6562 |

[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 g_{11} is best fit by the FO model whereas g_{12}, g_{21}, and g_{22} are best fit by the SOPTD model. The non-linear 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.

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:

The NI for the original linearized plant model is 0.6488.

### 5.3 Closed-loop analysis

#### 5.3.1 Controller settings

The decentralized controller settings are calculated by using IMC tuning relations [20, 21], and the centralized controller settings are calculated using TL tuning relations. The controller transfer functions are given in Table 4.

Decentralized controller | G_{c-D} | |
---|---|---|

FO | PI | |

FOPTD | PI | |

PID | ||

SOPTD | PID |

Centralized controller | G_{c-C} | |
---|---|---|

FO | PI | |

FOPTD | PI | |

PID | ||

SOPTD | PID |

#### 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. 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 y_{r1} and y_{r2}. y_{r1} is the setpoint to the controlled variable y_{1} and similarly y_{r2} is the setpoint to y_{2}. Similarly, the disturbances are set to the input variable (u_{1} and u_{2}) 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., u_{i,0} + Δu_{i}) 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 y_{r1}, whereas y_{r2} is changed between 0.995 and 0.999. Particularly for the present case, the setpoints were set at 0.975 for y_{r1} and 0.996 for y_{r2}. Similarly, disturbances for the real model were considered as the feed flow rate of PA to the RD column (d_{1}) and reboiler heat duty of the separation column (d_{2}). The disturbances are in the range of 10 * kmol/hr* < d

_{1}< 10.0185

*, and 296.697*kmol/hr

*< d*kW

_{2}< 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.

** 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, 23, 24, 25, 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,

where

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^{−2} 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

*and above).*rad/s

## 6. Conclusion

This chapter discusses the presence of multiple steady states indicating non-linear reactive distillation process. The presence of multiple steady states urged to fix the operating range for the manipulated variable. For achieving the purity of DPC (y_{1}), more than 99% of the reboiler heat duty of the RD column (u_{1}) must be varied between 886.237 kW and 899.631 kW. Similarly, for purity of MA (y_{2}), more than 99% of the condenser heat duty of the separation column (u_{2}) must be constrained between −293.240 kW and − 299.164 kW. This smaller operating range shows that the process is highly sensitive. Furthermore, if we start up the plant avoiding the given operating ranges, we end up at another steady state and hence undesirable product purities. The controller settings, derived from IMC and TL tuning relations, when applied to linear model as well as to non-linear model show proper setpoint tracking and load rejections. From the quantitative performance measures, setpoint tracking, and load rejection tests, the SOPTD–PID controller gives the best performance amongst the FO-PI, FOPTD-PI, FOPTD-PID, and SOPTD-PID controllers. Also, the centralized controller gives better performance, as compared to decentralized controller. Thus, the centralized controller regulates away the interactions more effectively than the decentralized controller. Even for λij > 1, the centralized controller shows better performances. However, the SOPTD model-based controller gives the best performance for any magnitude of setpoint value change and low value of load change. If the load value is high, FO model-based controller gave the best performance, as indicated by robust stability analysis. In other words, for lower frequency range (10^{−2} to 1 rad/s), the centralized FO-PI controller gives better robust stability as compared to decentralized FO-PI controller, and SOPTD-PID decentralized controller shows better robustness for higher frequency range (1 rad/s and above). We also conclude that the setpoint changes are tracked effectively for higher order models, whereas the load changes may or may not be regulated by higher order models. Thus, a proper trade-off has to be done between performance and robustness when selecting the control configuration and the model-based controller.

## References

- 1.
Cheng K, Wang SJ, Wong DSH. Steady-state design of thermally coupled reactive distillation process for the synthesis of diphenyl carbonate. Computers and Chemical Engineering. 2013; 52 (June):262-271 - 2.
Haubrock J. The process of dimethyl carbonate to diphenyl carbonate: Thermodynamics, reaction kinetics and conceptional process design [thesis]. Enschede: University of Twente; 2007 - 3.
Hung S-B, Tang Y-T, Chen Y-W, Lai I-K, Hung W-J, Huang H-P, et al. Dynamics and control of reactive distillation configurations for acetic acid esterification. IFAC Proceedings Volumes. 2006; 39 (2):403-408. DOI: 10.3182/20060402-4-BR-2902.00403 - 4.
Pavan Kumar MV, Kaistha N. Steady-state multiplicity and its implications on the control of an ideal reactive distillation column. Industrial & Engineering Chemistry Research. 2008; 47 (8):2778-2787 - 5.
Nandong J, Samyudia Y, Tadé MO. Control structure analysis and design for nonlinear multivariable systems. IFAC Proceedings Volumes. 2007; 40 (5):251-256 - 6.
Lengare MJ, Chile RH, Waghmare LM. Design of decentralized controllers for MIMO processes. Computers & Electrical Engineering [Internet]. 2012; 38 (1):140-147. DOI: 10.1016/j.compeleceng.2011.11.027 - 7.
Jaibhavani KS, Hannuja B. Modeling and development of decentralized PI controller for TITO system. In: Proceedings - TIMA 2017; 9th International Conference on Trends in Industrial Measurement and Automation; 6-8 January 2017; Chennai. New Jersey: IEEE; 2017. pp. 45-48 - 8.
Tavakoli S, Griffin I, Fleming PJ. Tuning of decentralised PI (PID) controllers for TITO processes. Control Engineering Practice. 2006; 14 (9):1069-1080 - 9.
Ravi VR, Thyagarajan T. A decentralized PID controller for interacting non linear systems. In: 2011 International Conference on Emerging Trends in Electrical and Computer Technology; 23-24 March 2011; Nagercoil, India. IEEE; 2011. pp. 297-302 - 10.
Besta CS, Chidambaram M. Decentralized PID controllers by synthesis method for multivariable unstable systems. IFAC-PapersOnLine [Internet]. 2016; 49 (1):504-509. DOI: 10.1016/j.ifacol.2016.03.104 - 11.
Shen Y, Sun Y, Xu W. Centralized PI/PID controller design for multivariable processes. Industrial & Engineering Chemistry Research. 2014; 53 (25):10439-10447 - 12.
Park BE, Sung SW, Lee IB. Design of centralized PID controllers for TITO processes. In: 2017 6th International Symposium on Advanced Control of Industrial Process AdCONIP; 28-31 May 2017; Taipei. IEEE; 2017. pp. 523–528 - 13.
Swetha M, Kiranmayi R, Swathi N. Design of centralized PI control system for two variable processes based on root locus technique. International Journal of Innovative Technology and Exploring Engineering. 2019; 9 (1):4851-4855 - 14.
Besta CS, Chidambaram M. Design of centralized PI controllers by synthesis method for TITO systems. Indian Chemical Engineer. 2017; 59 (4):259-279 - 15.
Chen Q, Luan X, Liu F. Analytical design of centralized PI controller for high dimensional multivariable systems [Internet]. IFAC Proceedings Volumes. 2013; 46 (32):643-648. DOI: 10.3182/20131218-3-IN-2045.00030 - 16.
Skogestad S. Simple analytical rules for model reduction and PID controller tuning. Modeling, Identification and Control. 2004; 25 (2):85-120 - 17.
Besta CS, Chidambaram M. Tuning of multivariable PI controllers by BLT method for TITO systems. Chemical Engineering Communications [Internet]. 2016; 203 (4):527-538. DOI: 10.1080/00986445.2015.1039121 - 18.
Besta CS, Chidambaram M. Modelling of interactive multivariable systems for control. In: Regupathi I, Shetty V, Thanabalan M, editors. Recent Advances in Chemical Engineering. Singapore: Springer; 2009. pp. 285-291 - 19.
Chidambaram M, Saxena N. Relay Tuning of PID Controllers [Internet]. Singapore: Springer; 2018. DOI: 10.1007/978-981-10-7727-2 - 20.
Besta CS. Control of unstable multivariable systems by IMC method. In: 2017 Trends in Industrial Measurement and Automation (TIMA); 6-8 January 2017; Chennai. IEEE; 2017. pp. 0–5 - 21.
Besta CS, Chidambaram M. Improved decentralized controllers for stable systems by IMC method. Indian Chemical Engineer [Internet]. 2018; 60 (4):418-437. DOI: 10.1080/00194506.2017.1280422 - 22.
Dileep D, Michiels W, Hetel L, Richard JP. Design of robust structurally constrained controllers for MIMO plants with time-delays. In: 2018 European Control Conference ECC; 12-15 June 2018; Limassol, Cyprus. IEEE; 2018. pp. 1566-1571 - 23.
Panyam Vuppu GKR, Makam Venkata S, Kodati S. Robust design of PID controller using IMC technique for integrating process based on maximum sensitivity. Journal of Control, Automation and Electrical Systems. 2015; 26 (5):466-475 - 24.
Zhu ZX, Jutan A. Consistency principles for stability in decentralized control systems. Chemical Engineering Communications. 1995; 132 (1):107-123 - 25.
Dittmar R, Gill S, Singh H, Darby M. Robust optimization-based multi-loop PID controller tuning: A new tool and its industrial application. Control Engineering Practice [Internet]. 2012; 20 (4):355-370. DOI: 10.1016/j.conengprac.2011.10.011 - 26.
Taiwo O, Adeyemo S, Bamimore A, King R. Centralized robust multivariable controller design using optimization. IFAC Proceedings Volumes. 2014; 19 :5746-5751