Designing multivariable PI controller with multi-response optimization for a pilot plant binary distillation column: a robust design approach

Abstract

Purpose

The purpose of this paper is to propose and validate a robust industrial control system. The aim is to design a Multivariable Proportional Integral controller that accommodates multiple responses while considering the process's control and noise parameters. In addition, this paper intended to develop a multidisciplinary approach by combining computational science, control engineering and statistical methodologies to ensure a resilient process with the best use of available resources.

Design/methodology/approach

Taguchi's robust design methodology and multi-response optimisation approaches are adopted to meet the research aims. Two-Input-Two-Output transfer function model of the distillation column system is investigated. In designing the control system, the Steady State Gain Matrix and process factors such as time constant (t) and time delay (?) are also used. The unique methodology is implemented and validated using the pilot plant's distillation column. To determine the robustness of the proposed control system, a simulation study, statistical analysis and real-time experimentation are conducted. In addition, the outcomes are compared to different control algorithms.

Findings

Research indicates that integral control parameters (K_i) affect outputs substantially more than proportional control parameters (K_p). The results of this paper show that control and noise parameters must be considered to make the control system robust. In addition, Taguchi's approach, in conjunction with multi-response optimisation, ensures robust controller design with optimal use of resources. Eventually, this research shows that the best outcomes for all the performance indices are achieved when Kp11 = 1.6859, Kp12 = −2.061, Kp21 = 3.1846, Kp22 = −1.2176, Ki11 = 1.0628, Ki12 = −1.2989, Ki21 = 2.454 and Ki22 = −0.7676.

Originality/value

This paper provides a step-by-step strategy for designing and validating a multi-response control system that accommodates controllable and uncontrollable parameters (noise parameters). The methodology can be used in any industrial Multi-Input-Multi-Output system to ensure process robustness. In addition, this paper proposes a multidisciplinary approach to industrial controller design that academics and industry can refine and improve.

1. Introduction

Background

Studying binary distillation columns is significant from an environmental and sustainable development standpoint. It can improve energy efficiency, reduce waste generation, minimise environmental impact, promote sustainable production practices and facilitate integration into a circular economy. Research and Development (R&D) efforts can aid sustainable development goals and lessen environmental impact (Mendez-B et al., 2022).

Multivariable systems or Multi-Input-Multi-Output (MIMO) systems are characterised by having more than one actuation control signal and more than one process output (Zhu, 2001). The industry places great importance on MIMO Binary Distillation Column control system design. MIMO control design enhances tracking performance, which increases spectral efficiency and reduces destructive interference between multiple transmitters and receivers (Cheng et al., 2020). By controlling several input channels, the multivariable system controls several output variables (Chen et al., 2011). An effective closed-loop control system is needed to tackle MIMO system interactions.

Further, the interaction effects on the loops and the disturbances under operating conditions make it difficult to control the system. These interactions are mainly because of the changes in one input for several outputs. As there is a significant coupling in the MIMO system, a proper pairing of Input/Output is essential to reduce the interaction effects (Coughanowr and LeBlanc, 2009).

Additionally, process control of a MIMO system is complex. High non-linearity in the system necessitates adopting computational methods such as simulation, genetic algorithms, machine learning and neural networks to optimise control systems and provide sensor fault tolerance (Zhang et al., 2011). The sustainability benefits of MIMO control design include improved process stability, energy cost and consumption reduction and reduced emissions and waste generation (Calvo and Domingo, 2017). Therefore, designing optimum MIMO control systems is necessary for efficient process control and sustainable development.

Literature review

To overcome the challenges stated in the previous section, Kumar et al. (2012) proposed two synthesis methods based on Proportional Integral (PI) controller to decrease the interactions between control loops. Garrido et al. (2012) presented a technique for centralised control of a multivariable process with Right Half Plane zero. Further, Maghade and Patre (2012) designed the decoupler and determined its equivalent, a reduced-order controller. This work designed the controller by the phase margin requirements. Moreover, a decentralised Proportional-Integral-Derivative (PID) control method can be applied effectively to a process with interaction and numerous time delays. This method uses direct synthesis and the Maclarian series' expansion to identify the controller parameters. Using a Relative Gain Array and the closed-loop time constant of the MIMO process transfer function matrix, the structure of the controller will be determined (Lengare et al., 2012). Based on the Equivalent Transfer Function (ETF) technique, Luan et al. (2014) introduced a new control scheme for a high-dimensional multivariable process. This proposed method took the loop interaction effectively for the MIMO systems.

Nandong and Zang (2014) presented a novel method for developing a multi-loop control architecture for multivariable processes. Literature has adapted the Multi-Scale Control strategy of Single-Input-Single-Output to the MIMO system. An ETF parametrisation-based multi-loop PI controller technique was used to control the high dimensional multivariable processes. The mathematical relationship for ETF was initially derived by using the relationship between the equivalent closed-loop transfer function and the inverse open-loop transfer function (Luan et al., 2015).

Based on the Steady State Gain Matrix (SSGM), Ram and Chidambaram (2015) introduced a centralised PI control algorithm for a stable and unstable multivariable process. Initially, a static decoupler controller was designed. Further, the PI controller was designed by combining the SSGM and the resulting decoupler. A decentralised PID controller using Characteristic Ratio Assignment was designed to reduce the interaction effect in the Two-Input-Two-Output (TITO) system (Hajare and Patre, 2015). Nandong (2015) discussed a heuristic-based PID tuning approach for multivariable processes.

Hajare et al. (2017) developed a decentralised control approach for the TITO system using a decoupler to lessen the interaction effect among the process variables. The interaction effect in the multivariable system was reduced by designing a decoupler with an integral action (Garrido et al., 2016). Besta and Chidambaram (2016) created a centralised PID controller based on the synthesis method. The decentralised control synthesis method was extended to the centralised control methodology. Most chemical process industrial loops are based on PI control algorithms because of their functional simplicity and robust performance with uncertainty in process parameters (Skogestad, 1997).

The multivariable PI control system design of distillation column research attempts to develop control techniques for complex systems with multiple inputs and outputs. Several research investigations have offered different control system design methodologies. Chekari et al. (2022) proposed a generalised IMC-PID-FOF controller for multivariable distillation columns. The technique is based on the IMC paradigm, selecting a control configuration with minimal interactions. In another study, Liu et al. (2012) propose adaptive Multivariable Generalised Predictive Control for process control of an Internal Thermally Coupled Distillation Column and has proven to be effective in high-purity Internal Thermally Coupled Distillation Column processes.

In a study focused on Dividing Wall Column distillation columns (Chen and Lu, 2022), the authors proposed a data-driven multivariable controller for the Dividing Wall Column distillation column system. Wahid and Ahmad (2016) suggested a Multi-Model Predictive Control and Proportional-Integral Controller Switching method for controlling nonlinear distillation columns. The method was tested and contrasted with PI and Hybrid Controllers. The Multi-Model Predictive Control and Proportional-Integral Controller Switching strategy exhibited superior setpoint tracking and disturbance rejection compared to the other controllers. Chien et al. (1986) assessed the efficacy of a multivariable self-tuning controller applied to two simulations of distillation columns. The self-tuner provided promising approaches for handling non-linearities, time delays and control-loop interactions.

In the realm of load frequency control, Çelik et al. (2021) suggested a cascade (1+PD)-PID controller. The proposed cascade technique outperformed others regarding reduced objective function values, settling time, undershoot, overshoot of frequency and tie-line power deviations. Another study validated the design using single/multi-area multi-source power systems with or without a High Voltage Direct Current link. To achieve flexible control performance with the Automatic Voltage Regulator system, it has been recommended to use a dynamic weight coefficient technique. The system performance, such as maximum overshoot and settling time, is reduced significantly under different load conditions (Eke et al., 2021).

Research gap and motivation

However, there needs to be more literature which provides the designing aspects of Multivariable PI Controllers considering multi-responses. In addition, there needs to be more publications that make good use of statistical methods to guarantee the stability of the controller architecture. Moreover, the system's robustness is critical to ensure safety in the industrial scenario. Also, industries demand optimal use of the available resources to improve the performance and productivity of the system.

Safety, performance and minimum resource consumption are essential in chemical industries. Further, in chemical industries, a distillation column is commonly used for the separation process to purify the products. The significant challenges in the distillation column control are tracking steady state setpoint and selection of control configuration. It is estimated that chemical and refining industries universally use the distillation column for 95% of the separation processes (Ionescu et al., 2015). Distillation column performance can be enhanced, and closed-loop stability can be achieved with the help of the robust controller. The objective of control engineers in these industries is to design control systems for effective closed-loop performance with robust performance (Bhaskarwar et al., 2022).

From the above discussion, the research gap is that researchers must effectively use computational techniques, statistical analysis, control engineering (especially noise parameters) and chemical engineering to design Multivariable PI Controller. So the most crucial thing right now is integrating a MIMO system's robust design with a binary distillation column in a pilot plant. As a result, this paper aims to determine a centralised PI control technique based on the process transfer function matrix's larger delay and smaller time constant in conjunction with the SSGM. Also, the article's objective is to propose a method to improve main and interaction responses based on the tuning parameter. To be more specific, the goal is to accommodate both control parameters and noise parameters (which cannot be controlled in real-time settings) by using the maximum time constant (τ_max) of the process parameters in conjunction with the minimum time delay (θ_min) of those parameters.

In addition, the design of the control system for the servo and the regulatory reactions will be accomplished with the help of a simulation study. Besides, it is intended validation of optimal control parameters through real-time experimentation. The present article presents simulation results. To demonstrate the efficacy of the suggested control method, the simulation results are contrasted with those obtained using a proportional (K_p) and integral (K_i) controller developed by Davison, IMC, both with and without a controller. The findings are experimentally validated by applying the proposed PI control method on a pilot plant binary distillation column. To ensure the reliability of the suggested control system, the product purity is ultimately tested through High-Performance Liquid Chromatography.

This study's main challenge is applying statistical tools to control engineering. The PID control system design of a binary distillation column must also address variable interaction, nonlinear behaviour of distillation columns with abrupt separations, multivariable systems with couplings, non-minimum phase characteristics, model mismatches and external disturbances. Multivariable systems with couplings, non-minimum phase characteristics, model mismatches and many external disturbances make binary distillation column controller development difficult.

The article contributes to the body of knowledge by addressing the challenges mentioned above and demonstrating how to incorporate noise factors during control system design. In addition, it addresses multivariable systems by effectively using the statistically robust multi-response optimisation technique. Most notably, the study presents a novel multidisciplinary strategy for addressing the design of control systems in an organised way.

The article's structure is as follows. The subsequent section outlines the approach to the robust design, and the third section explains the experimental methods used. The research outcomes are presented in Section 4, while corresponding discussions are presented in Section 5. The conclusion is presented in Section 6, while Section 7 describes the study's limitations and the potential scope of future research.

2. Robust design

Engineering processes or products must be established with optimal use of available resources. More specifically, the appropriate product or procedure at the right price, timing and quality. Over and above, the product/process must be robust to withstand variations in uncontrollable factors in the real-time scenario (Bhat et al., 2020a). In a nutshell, “Robust Design” of the product is essential to ensure efficiency and effectiveness. Robust design is a structured statistical method that ensures that the process and parameters are designed to use resources best. It accommodates controllable and uncontrollable parameters to ensure insensitivity to the variations during the real-time application (Bhat et al., 2021a).

A robust design approach helps to enhance the reliability of the product or process by achieving the target value with the optimal number of experimentations (Reda et al., 2018). Using an axiomatic approach, robust design can be done in three ways: Taguchi's Robust Engineering, Robust Optimization and Robust Design. Taguchi's Robust Engineering fits very well in the robust controller design, as it accommodates both noise (uncontrollable) and control parameters. In addition, it substantially reduces the number of experiments in parameter design (Gijo and Scaria, 2012). Thus, Taguchi's Robust Engineering is adopted for the robust parameter design in this study.

Dr Genichi Taguchi defines robustness as “the state where the technology, product, or process performance is minimally sensitive to variables producing unpredictability (either in manufacturing or user's environment) and ageing at the lowest production cost” (Taguchi et al., 2000). Dr Taguchi, in the late 1950s, effectively introduced Taguchi's Design of Experiments (DOE), which used statistical techniques and Orthogonal Array (OA) to minimise the number of experiments to launch the product with the assurance of quality (Singh and Sinha, 2017). Later, to ensure the product's robustness towards the uncontrollable parameters such as environmental conditions, process variations and component malfunctioning, a new concept known as the Signal-to-Noise (S/N) ratio was introduced (Vlachogiannis and Roy, 2005). Here, uncontrollable factors are considered as the noise parameters. This methodology ensures minimised variations in the target value (Bhat et al., 2020b, 2021b).

In addition, during the design process, the technique considers noise elements that are internal, external and unit-to-unit (Taguchi, 1993). It intends to ensure insensitivity to the variation (noise factors) and not abolish it. The primary objective of robust design is to ascertain the appropriate control parameters while considering noise factors throughout the experimental process (Bhat et al., 2020a, 2021a). The experimentation uses OA, which helps to determine optimal control parameters with a minimum number of experiments. Moreover, the DOE proposed by Taguchi accommodates a simultaneous variation of control and noise factors within the experimental setup (Gijo and Scaria, 2012). This approach provides the optimal parameter setting and ensures optimal usage of time and cost associated with the entire process.

Besides, to ensure the minimum deviation of product or process performance from the desired target, this methodology uses “Taguchi's loss-function” concept. This concept uses the S/N ratio to determine the optimal parameter setting. As the name itself indicates, it uses response (signal/output) and noise factors with three options, namely, “Smaller is better”, “Larger is better” and “Nominal is best” (Gupta et al., 2020). Based on the requirements of the experimentation, any one type of the S/N ratio researcher can opt for during the experimentation. For example, errors in the controlled design need to be minimised. Thus, the analysis must consider the “lower-the-better” option (Bhat et al., 2020a, 2020b). However, always larger S/N ratio values need to be selected (as the intention is to improve the noise output) irrespective of the options initially adopted.

Moreover, Taguchi's DOE fits well with physical experimentation and synchronises with simulation modelling. These qualities of Taguchi's robust design method aid in reducing the time, cost and equipment use of the entire experimental procedure in an industrial setting (Gijo and Scaria, 2012). Moreover, the literature provides evidence of using the simulation-based Taguchi robust engineering approach for industrial applications (Bhat et al., 2020a, 2020b). Nonetheless, there is scant evidence of multi-response optimisation and Taguchi's robust design method for industrial control system design (Bhaskarwar et al., 2022). Thus, the article intended to use control parameters as an internal OA and noise parameters as the outer OA in the simulation platform with multi-response optimisation to design a control system for the pilot plant distillation column for industrial application.

3. Experimental methodology

Process description

Most energy-intensive chemical companies need closed-loop distillation column control. It is also a crucial control engineering operational unit. Petrochemical, natural gas and oil refining industries use it to separate mixtures from chemical species. The separation procedure uses the boiling point of the components. The mixture is separated by thermodynamics. Heavy volatile components rise as vapours during separation. Liquids sink to the bottom. Vapours and liquids generally reach their dew and bubble points in the column.

Experimental setup

From the open-loop distillation column unit, the closed-loop experimental apparatus is created. Numerous measuring, control and data-acquisition instruments are used for the experimental configuration of a pilot plant binary distillation column. The primary instruments are Data AcQuisition unit, a Peristaltic pump, Temperature transmitters and Solid-State Relay. The feed to the distillation column consists of a 30:70 mixture of Isopropyl Alcohol and water.

Experimentation

Step 1: Find out what metrics and process variables will be used to judge performance.

Figure 1 depicts the block diagram of the multivariable TITO feedback control system. The initial input is the reflux flow rate (L), and the initial output is the tray temperature (T5) near the top of the distillation column. The second input is the reboiler power rate (Q), and the second output is the temperature at the tray (T1) near the bottom of the distillation column.

The following is a general description of a TITO system's FOPTD transfer function model. (1) $$G_p\left(s\right)=\left[\begin{array}{cc}{g}_{p11}& g_{p12}\\ g_{p21}& g_{p22}\end{array}\right]$$where $$g_{ij}=\frac{K_{ij}}{{\tau }_{ij}s+1}{e}^{-{\theta }_{ij}s}$$ $$G_c\left(s\right)=\left[\begin{array}{cc}{g}_{c11}& g_{c12}\\ g_{c21}& g_{c22}\end{array}\right]$$ (2) $$g_{cij}\left(s\right)=\left(K_p+\frac{K_i}{s}\right)*{\left[g_{pij}(s=0)\right]}^{-1}$$

The controller equation in equation (2) is the PI (K_p and K_i) controller, that is, $g_{cij}\left(s\right)=K_p\left(1+\frac{1}{{\tau }_{i}s}\right)$ in the product with the inverse steady state gain matrix (g_pij[s = 0]). The proposed method for finding k_p and k_i is explained in equations (3)–(6). The (G_p[s = 0]) is the steady state gain matrix, the proportional gain (K_p) and the integral gain (K_i) is given by: (3) $$K_p={\delta }_1\frac{\tau }{\theta }$$ (4) $$K_i=\frac{K_p}{{\tau }_i}\hspace{0.17em}\text{and} {\tau }_i={\delta }_2\theta$$

The larger delay (i.e. θ = max θ_ij[G_p(s)]) and the smaller time constant (i.e. τ = min τ_ij[G_p(s)]) are considered here for the PI controller design. The δ₁ and the δ₂ are the tuning parameters for the proportional gain (K_p) and the integral time constant (τ_I), respectively. The selection of tuning parameters is based on several trials. In this article, the limitation has been overcome with the proposed formula for tuning parameters. The minimum and the maximum range of tuning parameters (δ₁ and δ₂) proposed are based on the process parameters such as time delay (θ) and time constant (τ) as given in equations (5) and (6), and this is the major contributor to the article. Thus, the tuning parameters' minimum and maximum ranges are selected based on the minimum time delay (θ_min) and the maximum time constant (τ_max) of the process parameters as given by: (5) $${\delta }_{1(\min )}\hspace{0.17em}=\hspace{0.17em}0.01\frac{{\tau }_{\max }}{{\theta }_{\min }}\hspace{0.17em}\text{and} {\delta }_{1(\max )}\hspace{0.17em}=\hspace{0.17em}0.13\frac{{\tau }_{\max }}{{\theta }_{\min }}$$

In the reviewed literature, the approach for determining the range of tuning parameters (δ) is called “rough tuning”. The range of δ in the article (Ram and Chidambaram, 2015) varies from one case to another. However, the proposed study has been performed on the distillation column transfer function model. A generic relation is arrived at using time delay and time constant. Equation (5) originated from the study where a minimum and maximum of δ₁ are arrived at using the ratio of a maximum of the time constant (τ) and a minimum of the time delay (θ). The constant of the multiplication for δ_1(min) = 0.01 and the multiplication for δ_1(max) = 0.13 are as given in equation (5).

Equation (6) originated from equation (5). Thus, equation (6): (6) $${\delta }_{2(\min )}=\frac{0.01}{{\delta }_{1(\min )}}\hspace{0.17em}\text{and} {\delta }_{2(\max )}=\frac{13}{{\delta }_{1(\max )}}$$

The plant's time delay and time constant were simulated for the study. The results recommended the range of parameters δ₁ and δ₂ as (0.2–5.3) and (0.02–5.3), respectively. The rough tuning parameter varies based on the plant transfer function. In the proposed method formula for the tuning parameter, the δ₁ and the δ₂ are introduced as a function of time delay (θ) and time constant (τ). Wood and Berry distillation model, Vinate and Luyben distillation model, Industrial Scale Polymerization reactor model and two other pilot plant distillation column model models are all used to verify its accuracy.

The proposed PI controller is compared with Davison's PI controller and the tuning rule based on the IMC technique (Mekki, 2014). The centralised PI controller of the Davison approach is based on the SSGM of the plant transfer function. The proportional gain (K_p) and the integral gain (K_i) values are obtained using the following formulae: (7) $$K_p={\delta }_1{\left[G_p\left(s=0\right)\right]}^{-1}$$ (8) $$K_i={\delta }_2{\left[G_p\left(s=0\right)\right]}^{-1}$$where the SSGM is given by: (9) $$G_p\left(s=0\right)=\left[\begin{array}{cc}{K}_{11}& K_{12}\\ K_{21}& K_{22}\end{array}\right]$$

The inverse SSGM (G_p[s = 0])⁻¹ is also named as a tuning matrix. The δ₁ and the δ₂ are the fine-tuning parameters of the Davison method. Generally, both tuning parameters range from 0 to 1 (Ram and Chidambaram, 2015). The PI controller structure based on the Davison approach is determined by combining equations (7)–(9). (10) $$G_c\left(s\right)=K_p+\frac{K_i}{s}{\left[G_p(s=0)\right]}^{-1}$$

The PI controller parameters of the IMC technique are given by equation (11): $$K_{p\_n}=\frac{2\tau +\theta }{2K+\left(\tau +\beta \right)}$$ (11) $$K_{i\_n}=\frac{1}{K\left(\tau +\beta \right)}$$

Here, β is the tuning parameter that should satisfy β > 0.2τ and β > 0.25θ for loops 1 and 2, respectively (Ramzi et al., 2012). The general decoupling matrix is given by: (12) $$D\left(s\right)=\left[\begin{array}{cc}{v}_1\left(s\right)& d_{12}\left(s\right)v_2\left(s\right)\\ d_{21}\left(s\right)v_1\left(s\right)& v_2\left(s\right)\end{array}\right]$$where v₁(s), v₂(s), d₁₂(s) and d₂₁(s) are given by equations (13)–(16). (13) $$v_1\left(s\right)=\{\begin{array}{cc}1& {\theta }_{21}\ge {\theta }_{22}\\ e^{({\theta }_{21}-{\theta }_{22})s}& {\theta }_{21}<{\theta }_{22}\end{array}$$ (14) $$v_2\left(s\right)=\{\begin{array}{cc}1& {\theta }_{12}\ge {\theta }_{11}\\ e^{({\theta }_{12}-{\theta }_{11})s}& {\theta }_{12}<{\theta }_{11}\end{array}$$ (15) $$d_{12}\left(s\right)=-\frac{g_{12}\left(s\right)}{g_{11}\left(s\right)}{e}^{-\left({\theta }_{12}-{\theta }_{11}\right)s}$$ (16) $$d_{21}\left(s\right)=-\frac{g_{21}\left(s\right)}{g_{22}\left(s\right)}{e}^{-\left({\theta }_{21}-{\theta }_{22}\right)s}$$

The open loop test is carried out for model identification on the distillation column (Martin et al., 2013). The step change is implemented on the controlled variables, reflux (L) and reboiler (Q). The subsequent tray temperature responses T₅ and T₁ are recorded. The empirical FOPDT model is identified as shown in equation (17): (17) $$\left[\begin{array}{c}{T}_5\\ T_1\end{array}\right]=\left[\begin{array}{cc}\frac{-0.13e^{-0.03s}}{1.14s+1}& \frac{0.18e^{-0.03s}}{0.64s+1}\\ \frac{-0.34e^{-1.22s}}{1.23s+1}& \frac{0.18e^{-0.03s}}{0.32s+1}\end{array}\right]\left[\begin{array}{c}L\\ Q\end{array}\right]$$

Here, the process gain is expressed in ^oC/%, whereas the time constant and delay are expressed in hours. The tuning parameters δ₁ and δ₂ are 1.5 and 1.3, respectively, and were designed based on equations (5) and (6). Consequently, the controller for equation (18) is constructed based on the proposed PI controller and is provided as follows: (18) $$G_c\left(s\right)=\left[\begin{array}{cc}1.8733+\frac{1.1809}{s}& -1.8733-\frac{1.1809}{s}\\ 3.5385+\frac{2.231}{s}& -1.3529-\frac{0.8529}{s}\end{array}\right]$$

The IMC designed for the distillation column model is given by: (19) $$G_c\left(s\right)=\left[\begin{array}{cc}0.4928+\frac{0.4266}{s}& 0\\ 0& 0.6142+\frac{1.834}{s}\end{array}\right]$$

The IMC with the decoupler design is given by: (20) $$G_c\left(s\right)=\left[\begin{array}{cc}0.5911+\frac{0.5118}{s}& 0\\ 0& -0.1856-\frac{0.5339}{s}\end{array}\right]$$ (21) $$D\left(s\right)=\left[\begin{array}{cc}1& \frac{0.18\left(1.14s+1\right)}{0.13\left(0.64s+1\right)}\\ \frac{0.34\left(0.32s+1\right)}{0.18\left(1.23s+1\right)}{e}^{-1.19s}& 1\end{array}\right]$$

This study examined IAE and ITAE performance indexes. Performance indices assess the control algorithm's quantitative performance. Errors (e) should be minimised to improve closed-loop performance. Because of the lengthy separation procedure, the distillation column model is the main focus of the research. To get a higher-purity product, fix the long-term mistake. A good control algorithm was designed by finding the lowest IAE and ITAE values.

Step 2: Determine process parameters' levels.

Process control is unpredictable (Khaki-Sedigh and Moaveni, 2009). Process industries have uncertainty because of input flow rate, tank height and form and other considerations. Thus, this study first examined these internal noise characteristics by calculating their values (Ranganayakulu et al., 2016). Then, noise factor levels were selected within 10% of the initial values (Gijo and Scaria, 2012). External and unit-to-unit noise measures can be examined during manufacturing (Taguchi, 1993). Thus, the proposed PI control tuning was tested with “± 10%” process parameter uncertainties (K and T). Equation (18) calculates the PI controller values, and equation (19) determines the initial plant levels (1). Three noise and control parameter levels were chosen to mimic non-linearity and make the system more robust. For experimentation, Table 1 lists noise and control settings and their levels.

Step 3: Choose the proper orthogonal array and allocate the process parameters.

As mentioned, Taguchi's robust design technique is distinguished by its use of OA to reduce the number of experiments. The choice of OA is determined by the total Degrees of Freedom (DOF) required for the given control parameters to obtain the optimal settings with minimal experimentation. As the present study has eight control parameters with three levels each, {[3–1]*8} = 16 total DOF needed for the experimentation. Thus, the nearest OA L₂₇(3¹³) was selected, which performs 27 experiments and is known as the inner OA. Later, to accommodate the noise factors and their levels, eight noise factors at three levels were taken into account. L₂₇(3¹³) was selected, which performs 27 experiments and is known as the outer OA (Bhat et al., 2020a and 2021a). Finally, an experimental layout is prepared with 27 control and noise factors experiments. The experimental layout is generated using Minitab Statistical software.

Step 4: Carry out the tests following the orthogonal array layout.

The values for IAE-e1, IAE-e2, ITAE-e1 and ITAE-e2 must be calculated using a simulation study. The MATLAB/Simulink (version @2020) platform estimates the performance parameter values according to the experimentation plan to suit the system equation and IMC-based PI controller.

Step 5: Determining optimal settings for control parameters.

As discussed earlier, as per the systems requirement, either “Larger-is-better”, “Smaller-is-better” or “Nominal-is-best” options need to be selected. As control parameters (errors) need to be minimised, “smaller-is-better” is opted, which calculates the S/N ratio using the following Equation S/N = −10 *log(Σ(Y²)/n), where “Y” is the observed value of the experiment test and “n” is the number of observations in a trail. The “Main Effects Plot” is constructed for S/N ratios to find the optimum values for each control parameter.

Step 5: Multi-response optimisation.

The Taguchi robust design's main drawback is that it only calculates the optimal parameter value for one response. Thus, regression-based multi-response optimisation is considered. This approach optimises all four performance parameters (IAE-e1, IAE-e2, ITAE-e1 and ITAE-e2) equally to minimise error. Each outcome's mean (target value) and maximum observed value were considered. Eventually, “Composite Desirability (D)” values (The weighted geometric mean of the individual desirability) were established to determine the optimal settings of parameters, which will collectively minimise all the errors.

Step 6: Computational analysis and validation of optimal parameters.

A simulation study used the optimal PI control parameters from the previous stage. After multi-response optimisation determines the appropriate control parameter levels, studies are performed to validate them. The experiment was repeated three times to account for bias and non-linearity. The average result was considered for comparison with the predicted results. The following equation computes Absolute Relative Deviation (ARD) to provide optimal parameter stability. (22) $$ARD\left(\%\right)=\frac{(Experimental Result-Simulation Result)}{Experimental Result}\times 100$$

4. Results

Robust design and analysis

The Main Effects Plot can for S/N Ratios (Figure 2) indicates that K_i11, K_i12, K_i21 and K_i22 significantly affect the outputs (steep slope). As discussed earlier, these values are calculated based on the “Smaller is better” equation. Further, Figure 2 helps determine the optimal value for each output parameter. The highest value marked in Figure 2 shall be the optimal value (as discussed in the research methodology). Thus, identified optimal values are shown in Table 2. It is apparent from Table 2 that only there is a difference in the optimal values of K_p11 and K_p12 for the control parameters.

Multi-response optimisation

As the objective is to determine common optimal values for each of the outputs, it is essential to consider multi-response optimisation, which will help to resolve the conflict in values for different outputs. Thus, a regressing-based multi-response optimisation was performed. The analysis was performed based on observed data of simulation results. As the objective is to minimise errors (Control Parameter), the upper and target values (average) were identified for each control parameter. Based on the observed data, for ITAE-e2 (Upper = 99.110; Target = 30.040), ITAE-e1 (Upper = 84.920; Target = 13.480), IAE-e2 (Upper = 7.253; Target = 1.144) and IAE-e1 (Upper = 5.907; Target = 2.632) values are specified. As all the control parameters are equally important and significantly affect the output, “Weightage” and “Importance” are given the value 1 in the Minitab Statistical Software. From the analysis, “Composite Desirability (D)” is established, which helps understand the best optimal values for the parameters considering all the outputs. Table 2 provides four potential solutions. The highest “Composite Desirability” provides the optimal value from the perspective of multi-response optimisation. Thus, solution 1 in Table 2 is selected (highest D = 0.961252).

Computational analysis

The simulation study was performed by considering the optimum level of PI parameters tabulated in Table 2 (solution 1) obtained during the analysis of multi-response optimisation. A simulation and real-time experimental study was performed, and responses were recorded. The closed-loop reaction with and without disturbance, also known as regulatory and servo response, was necessary to comprehend the system's performance. In addition, as a convergence requirement, the closed-loop response must converge to a predetermined setpoint for the controller to be effective. In the present investigation, the closed-loop servo and regulatory responses were recorded for the setpoint r = 1. The servo response of the system was obtained using simulations of the original plant and the plant with 10% uncertainty in all process parameters. The servo and the regulatory responses are shown in Figure 3. The main action and interaction responses are recorded in Figures 3(a) and (b) when the SP-y₁ = 1 and the SP-y₂ = 0 with disturbance d = 0.1 at 150 min. The main action and the interaction responses are recorded in Figures 3(c) and (d) when the SP-y₁ = 0 and the SP-y₂ = 2 with a magnitude of disturbance d = 0.1 at time t = 150 min. The closed-loop response indicates that the proposed control algorithm improves the settling time without oscillation and overshoot compared to others.

Later, the percentage of error improvement is tabulated in Table 3. It indicates that the proposed control algorithm is much superior to those described in the article. Table 3 reports performance metrics such as Integral Absolute Error (IAE) and Integral Square Time Error for servo and regulatory responses, with +30% uncertainty in all process parameters. The proposed PI control algorithm improves all servo and regulatory response performance indicators. In the final column of Table 3, the percentage of error improvement of the proposed control algorithm is compared to the Internal Model Controller with Decoupler (IMC+D) control algorithm. The table implies that the percentage improvement of errors with the proposed algorithm is better than Davison, the IMC without a Decoupler controller.

The ITAE performance is improved with the proposed control algorithm compared to various control algorithms such as PI controller by Davison and IMC with and without decoupler. The article mainly concentrates on the distillation column model, as the separation process takes a long time to achieve the products. The persistence of error for a long time is critical to achieving a better purity product. With the least value of ITAE, the effectiveness of the proposed method is presented. The proposed algorithm is experimentally validated by implementing it on the pilot plant binary distillation column, as shown in Figure 4. The setpoint of tray 5 is SP-T₅ = 78°C and tray 1 SP-T₁ = 66°C. The tray temperature of T₅ and T₁ are controlled for different setpoints, manipulated variables, reflux and reboiler rate in the proposed algorithm to achieve a better purity product.

Validating simulation results

Later, laboratory experimentation was performed using the pilot plant distillation column with the optimal parameter settings obtained from the multi-response optimisation to validate the simulation results. The ARD was calculated based on the previous section's equation discussed in step 6. The ARD was 0.1%, 0.4%, 1.6% and 1.1% for ITAE-e2, ITAE-e1, IAE-e2 and IAE-e1 respectively. As ARD is less than 2% for all the responses, it can culminate that results obtained from the multi-response optimisation are robust. Because of certain inevitable noise elements during the experiment, slight discrepancies in the results may be seen.

5. Discussion

The multivariable PI controller, along with its tuning parameters, is designed based robust design approach. The present study accommodated both control and noise parameters for the controller design. Saeki et al. (2017) highlight that accommodating noise parameters is essential in designing a PI controller because it improves control performance and extends the plant's lifespan. Designing a PID controller with a noise filter can reduce sensitivity to measurement noise, leading to better disturbance attenuation in Single-Input-Single-Output plants (Saeki et al., 2017). In practical applications, it is recommended to use filtered derivative PI controllers to mitigate the negative impacts of high-frequency measurement noise (Wu et al., 2019). This study confirms that noise parameters help in robust design by optimising process parameter levels and resource use.

Taguchi's robust design approach integrated with multi-response optimisation helped to achieve the research objective. The innovative method enabled the design of the process gain and integral gain parameters based on the minimum time delay and the maximum time constant. Further, a novel multidisciplinary approach encompasses computational science, control engineering and statistical techniques to ensure a robust process with optimal available resources. The proposed PI controller is tested and validated on the distillation column transfer function model. With and without a decoupler, the suggested control algorithm is compared to the PI controller proposed by Davison, IMC. The performance indices, such as IAE and ITAE, are tabulated for servo, regulatory and process parameters with +30% uncertainty. The study results were better than the Davison, IMC, with and without a decoupler.

Theoretical implications

The academicians can adopt this structured approach for other studies in the controller design. This is because the article provides a step-by-step approach to leverage the potential of the statistical approach in designing the control system. More specifically, it provides a basis for using Taguchi's robust design with multi-response optimisation in the MIMO system to ensure the robust design of the controller. Further, researchers can use the outcome of this study to compare and contrast with other novel controller design algorithms. Also, the proposed multivariable PI control system design of pilot plant distillation columns helps to develop control strategies for complex distillation column systems with multiple inputs and outputs. The academicians can compare and contrast the outcomes of this study with several other research studies by Chekari et al. (2022), Liu et al. (2012), Chen and Lu (2022), Wahid and Ahmad (2016), and Chien et al. (1986) as discussed in the introduction section.

Further, the outcome of this research can be used to improve the resiliency of industrial process control systems. It can be applied to developing cyber-production systems that do not require permanent maintenance by personnel (Stopakevych, 2021). A state-space model-based Model Predictive Control (MPC) strategy can give improved results for controlling methanol composition inside the distillation column compared to traditional PI control (Semenov et al., 2023). Dynamic optimisation tools can integrate design and control methods to improve the performance of mixture separation and energy consumption.

Managerial implications

The present study is significant to both industry and society. Its potential benefits include improved product quality, enhanced efficiency, better safety, technology transfer, and economic advantages. Optimal parameters obtained from the Taguchi robust design approach can save energy and the environment. The research seeks to strengthen the resilience of industrial process control systems, develop techniques for developing resilient industrial process control systems, and create a robust control system for an oil-producing distillation column. The study found that the Taguchi-based robust design of PI controllers resulted in better regulatory and servo performance. This indicates that using these controllers may improve process control and efficiency. MIMO system and parameter design based on multi-response optimisation and noise parameters reduces variation between real-life and laboratory outcomes.

Compared to PID control, which is extensively used in the gas sector, the suggested method is anticipated to provide superior control performance. It can overcome any intervariable interaction in a MIMO system (Wahid et al., 2018). Improved product quality is a significant benefit of using the proposed novel control system for the distillation columns. This type of control can enhance the control performance of the columns, resulting in tighter regulation of process variables like temperature and composition. As a result, this can lead to better product quality, which is of utmost importance in industries like chemical manufacturing, food and beverage and pharmaceuticals. Overall, this research has implications for improved process control, efficiency and resiliency in various distillation column industries.

6. Conclusions

The study provided numerous outcomes from the perspective of industrial application and academia. The research determined that the best performance indices outcomes are achieved when K_p11 = 1.6859, K_p12 = −2.061, K_p21 = 3.1846, K_p22 = −1.2176, K_i11 = 1.0628, K_i12 = −1.2989, K_i21 = 2.454 and K_i22 = −0.7676. Further, it is observed that integral control parameters (K_i) significantly affect the outputs more than the proportional control parameters (K_p). The ITAE is reduced in all the proposed PI controller's closed-loop responses compared to the others. Moreover, it is evident from the study that the proposed control algorithm outperforms the Internal Model Controller with Decoupler (IMC+D) control method in terms of the percentage of error reduction.

Besides, it is observed that statistical methods systematically allow for creating a more robust controller design. More importantly, Taguchi's robust design with multi-response optimisation improves control system design by allowing for control and noise parameters. Furthermore, accommodated control and noise parameters helped to design and validate the robust control for the distillation column with optimal use of resources. Eventually, simulation analysis helped to validate the optimal parameter settings. Thus, it is ascertained that a multidisciplinary approach by tapping the potential of computational science, control engineering and statistical methodologies helps to develop robust controllers for industrial applications with optimal use of available resources.

7. Limitations and scope for future work

The research is limited because only IAE and ITAE are considered for the study. However, it must be noted that these are the main control parameters in ensuring the control systems' robustness, and the study results confirm this fact. Nevertheless, the authors are intended to work further on this area with Integral Square Error and Integral Square Absolute Error to understand their significance. Also, the authors intended to research advanced controller algorithms and configurations on interacting conical tank systems.

Another study limitation is that experimentation was performed on the pilot plant distillation column. Thus, there is a scope to extend this study to the industrial distillation column to validate the study's outcomes. Further, the present study is confined to noise and control parameters in the controller design. However, the researchers are intended to add dynamic factors as prescribed in Taguchi's parameter design to have better controllers for industrial applications.

The first author gratefully acknowledges Vision Group on Science and Technology (VGST), Department of IT, BT and S&T, Government of Karnataka, India, for the project grant of RGS/F, vide GRD Number 973 dated 17.11.2021.

Figure 1.

A block diagram of multivariable Two-Input-Two-Output feedback control system

Figure 2.

Main effects plot for S/N ratios

Figure 3.

Servo and regulatory response comparison with the magnitude of disturbance d = 0.1 at t = 150 min

Figure 4.

Experimental validation of the proposed Proportional Integral control algorithm on a pilot plant binary distillation column

Table 1.

Noise and control parameters and levels for experimentation

Noise parameters					Control parameters
Process parameters		Lower limit	Nominal	Upper limit	Controller parameters		Range	Lower limit	Nominal	Upper limit
Process gain (K)	K11	−0.14	−0.13	−0.12	Proportional gain (KP)	Kp11	1.6859–2.061	1.6859	1.8733	2.061
	K12	0.16	0.18	0.2		Kp12	1.6859–2.061	−1.6859	−1.8733	−2.061
	K21	−0.37	−0.34	−0.31		Kp21	3.1846–3.8923	3.1846	3.5385	3.8923
	K22	0.16	0.18	0.2		Kp22	1.2176–1.4881	−1.2176	−1.3529	−1.4881
Time constant (T)	T11	1.03	1.14	1.25	Integral gain (Ki)	Ki11	1.0628–1.2989	1.0628	1.1809	1.2989
	T12	0.58	0.64	0.7		Ki12	1.0628–1.2989	−1.0628	−1.1809	−1.2989
	T21	1.11	1.23	1.35		Ki21	2.1–2.454	2.1	2.231	2.454
	T22	0.29	0.32	0.35		Ki22	0.7676–0.9382	−0.7676	−0.8529	−0.9382

Source: Author's own creation.

Table 2.

Optimal parameters and potential solutions for multi-response optimization

Optimal parameters for individual responses									Results
Solution	Kp11	Kp12	Kp21	Kp22	Ki11	Ki12	Ki21	Ki22	ITAE-e2Fit	ITAE-e1Fit	IAE-e2Fit	IAE-e1Fit	CompositeDesirability (D)
1	1.6859	−2.0610	3.1846	−1.2176	1.0628	−1.2989	2.454	−0.7676	22.8151	9.8650	1.95307	2.68399	0.961252
2	1.6859	−2.0610	3.8923	−1.2176	1.0628	−1.2989	2.454	−0.7676	25.8850	11.9664	1.96719	2.73950	0.956449
3	2.0610	−2.0610	3.1846	−1.2176	1.0628	−1.2989	2.454	−0.7676	22.9171	8.8837	2.04832	2.72623	0.953754
4	1.6859	−1.6859	3.1846	−1.2176	1.0628	−1.2989	2.454	−0.7676	25.3024	9.2379	2.03902	2.73611	0.953437

Source: Author's own creation

Table 3.

Comparison of performance indices

Closed loop response	Performance Indices	Proposed	Davison	IMC	IMC+D
Servo	IAE	11.89	22.79	37.66	15.78
	ITAE	50.00	243.17	512.81	181.15
Regulatory	IAE	12.17	23.31	38.61	16.18
	ITAE	93.78	325.87	669.40	244.45
+30% uncertainty in all the process parameters	IAE	21.73	19.55	38.90	15.86
	ITAE	137.00	155.81	581.65	140.54

Source: Author's own creation