Time domain modeling and control of complex non-linear chemical processes using relay feedback test

Abstract

This study analyzes with the technology of auto tuning using relay feedback test for non-square MIMO system through process modeling, identification, input-output pairing and control strategies. However, the control configuration selection based on conventional steady-state Relative Gain Array (RGA) matrix sometimes degrades the loop performance and it needs attention. This study also deals with the real challenges in time domain modeling and appropriate pairing of loops for non-linear chemical processes. The choice of input-output pairing for square system has been extended to non-linear chemical processes. This ensures the system’s stability by selecting an appropriate manipulated-controlled variable pairing in non-linear chemical processes and the same is also tested for benchmark of non-linear chemical processes. In this study, two types of non-square systems are considered: one is systems with excess input than output variables and the other is systems with excess output than input variables. Some benchmark non-linear chemical processes, such as processes with mild nonlinearity, moderate to high nonlinearity and highly nonlinearity, are also taken for this study. Three standard benchmark processes are used in analysis of nonlinear chemical processes, namely: (1) Continuous Stirred Tank Reactor (CSTR) process; (2) hydrogen-ion- concentration (pH) process and (3) distillation column and this procedure is illustrated via simulation of 3-by-2 and 2-by-3 non-linear chemical processes.

Keywords

Input-output pairing non-square MIMO systems over defined system under defined system interaction non-linear chemical processes relay feedback test

Introduction

In a process having only one input variable used for controlling one output variable, the design of controller is simple. In most chemical and industrial processes the controller design is complex, as the system has multiple inputs and multiple outputs, making it multivariable in nature. Thus, with a multivariable process having equal number of input variables and output variables, such systems are termed as square systems because their transfer function matrices are square. Many benchmark square systems are given in the literature (Luyben, 1986). However, some of the multivariable systems have a number of input variables that are not equal to number of output variables. Such multivariable systems are termed as non-square systems. There are two types of non-square systems: systems that have fewer inputs than output variables will be referred to as under defined systems, while the systems with more input than output variables will be referred to as over defined systems. The under defined systems have a deficiency in input variables and over defined systems have an excess of input variables. Thus, non-square multivariable systems with n output variables and m input variables whose transfer function matrix will therefore be of dimension n x m. The system is under defined if m < n, and the system is over defined if m > n.

In the case of non-square systems, the most obvious problem with that after input-output pairing is there will be always a residual of unpaired input or output variables, depending on which of these are in excess. The innovation in this study, the nontrivial issue in non-square systems is addressed. Apart from these nontrivial issues, the other issue is which variable should be left redundant and which ones should take an active part in the control scheme.

The main issue in under defined Multi Input Multi Output (MIMO) system is that not all the outputs can be controlled since we do not have enough input variables. In the case of over defined MIMO system, we have an excess of input variables, and therefore we can actually achieve arbitrary control of the fewer output variables in more than one way. The limits of applicability of both under defined and over defined MIMO systems are very difficult to control.

Some of the multivariable processes are non-linear chemical processes exhibiting non-square MIMO systems. For example, mixing tank process (Arkun and Reeves, 1989) 2-by-3 system, shell control problem (Vlachos et al., 1999) 7-by-5 system, and so forth, are typical non-square multivariable process in industry. Three benchmark non-linear chemical processes are identified as non-square MIMO processes, namely, a mild non-linear Continuous Stirred Tank Reactor (CSTR) process (Khalid and Dawood, 2015) having two manipulated variables and three controlled variables; a Hydrogen-ion-concentration (pH) neutralization process (Henson and Seborg, 1994) having moderate to high non-linearity and highly non-linear distillation column (Marlin, 1986) having three manipulated variables and two controlled variables.

Astrom and Hagglund (1984) first proposed the relay feedback test for process control practitioners as a closed-loop tool for system identification and control. As the non-linear chemical processes are multivariable in nature, Shen and Yu (1994) proposed sequential method of identification for automatic control of multivariable systems. To synthesize a control system, singular value decomposition (SVD) framework for non-square linear systems was proposed by Lau et al. (1985). For non-square linear systems, block relative gain array, relative gain array are derived by Arkun and Reeves (1989) and Chang and Yu (1990). He et al. (2007) proposed equivalent transfer function method to non-square MIMO systems for Proportional Integral/Proportional Integral Derivative (PI/PID) decoupled controller design. The method has been applied to example with right half plane zero as given in the literature by Sarma and Chidambaram (2005). Further for two input two output system modelling and parameter estimation was proposed by Sujatha and Panda (2012). Sujatha and Panda (2013b) derived the novel method of interaction measure to select the proper input-output pairs. Mnasser and Bouani (2015) proposed the robust predictive controller for a class of multi-input–multi-output non-linear systems. Modeling and control of non-square systems using relay-feedback method were discussed by Kalpana et al. (2015). Saidi et al. (2019) studied the comparsion of internal model controller for non-square systems. In many of the above works, a dummy variable has been added to make the system as squared one for making the design to follow conventional way.

As the non-linear chemical processes are sensitive to plant parametric uncertainty, it is desired to design the robust feedback control system to utilize the technique that addresses all the plant variations up front; incorporates the information on desired output tolerance and mantaining the reasonable loop gain. Using the quantitative feedback theory, a compensator and prefilter may be designed to achieve a specified robust design which will be presented in future research work.

In general, all the literatures, the non-square systems are treated as square systems for control purpose either by discarding some inputs or by adding new outputs. Similarly, by eliminating some inputs can cause performance degradation and by appending new outputs can incur unnecessary cost. This focuses the motivation of research in these non-square systems. The contribution of this study is to bring the clear picture on how to model and control both the types of non-square MIMO systems.

The entire paper is organized as follows: Section 1 deals with the introduction. Section 2 discusses non-square MIMO systems and its types. Case studies on non-linear systems are discussed in Section 3. Section 4 highlights the proposed methodology and it includes algorithm and flow chart. Modeling, validation and parameter estimation of non-linear chemical processes are explained in Section 5. Control configuration selection based on conventional and proposed method for both the types of non-square MIMO systems are presented in Section 6. Closed loop studies and stability analysis are given in Sections 7 and 8, respectively. Conclusions are given in Section 9.

Non-square MIMO systems and its types

Block diagram of under defined non-square systems

Consider an open-loop under defined non-square MMO systems with n-1 inputs and n outputs. The system considered in this work has 2-inputs and 3-outputs. The general formulation of 3-by-2 MIMO system is shown in Figure 1. Figure 2 represents the block diagram of open-loop 3-by-2 nonsquare MIMO system. Here, m is the input variables, y is the output variables. The process transfer function matrix, G is expressed by equation (1).

The input and output variables are

\begin{matrix} m = (\begin{matrix} m_{1} \\ m_{2} \end{matrix}) y = (\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \end{matrix}) \\ G (s) = (\begin{matrix} \begin{matrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix} \\ \begin{matrix} G_{31} & G_{32} \end{matrix} \end{matrix}) \end{matrix}

(1)

The transfer function of Orgunnaike and Ray (OR) (1979) distillation column given by equation 2 represents the multiproduct plant distillation column for the separation of binary mixture of ethanol-water. The top compositions are controlled by manipulating the flow rates, pressure of the reflux and the reboiler stream. In this study, feed flow rate is considered as the disturbance. The input variables of the process are overhead reflux flow rate, side stream draw off rate and reboiler stream pressure. The output variables are overhead mole fraction of ethanol, side stream mole fraction of ethanol and the 19^th tray temperature.

G (s) = [\begin{matrix} \frac{0.66 e^{- 2.6 s}}{6.7 s + 1} & \frac{- 0.61 e^{- 3.6 s}}{8.64 s + 1} & \frac{- 0.0049 e^{- s}}{9.06 s + 1} \\ \frac{1.11 e^{- 6.5 s}}{3.25 s + 1} & \frac{- 2.36 e^{- 3 s}}{5 s + 1} & \frac{- 0.01 e^{- 1.2 s}}{7.09 s + 1} \\ \frac{- 34.68 e^{- 9.2 s}}{8.15 s + 1} & \frac{46.2 e^{- 9.4 s}}{10.9 s + 1} & \frac{0.87 (11.61 s + 1) e^{- s}}{(3.89 s + 1) (18.8 s + 1)} \end{matrix}]

(2)

Consider the situation in which the binary distillation column used in this case has its sidestream draw-off rate set at a fixed amount that cannot be changed, and yet all three of its output variables – the overhead mole fraction of ethanol, the sidestream mole fraction of ethanol, and the Tray #19 temperature – are still being monitored, and are to be controlled. The benchmark underdefined non-square system considered here is Orgunnaike and Ray distillation column, given in equation (3)

(\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \end{matrix}) = (\begin{matrix} \begin{matrix} \frac{0.66 e^{- 2.6 s}}{6.7 s + 1} & \frac{- 0.0049 e^{- s}}{9.06 s + 1} \\ \frac{1.11 e^{- 6.5 s}}{3.25 s + 1} & \frac{- 0.012 e^{- 1.2 s}}{7.09 s + 1} \end{matrix} \\ \begin{matrix} \frac{- 33.68 e^{- 9.2 s}}{8.15 s + 1} & \frac{0.87 (11.61 s + 1) e^{- s}}{(3.89 s + 1) (18.8 s + 1)} \end{matrix} \end{matrix}) (\begin{matrix} m_{1} \\ m_{2} \end{matrix})

(3)

where the input variables m₁ and m₂ are, respectively, the overhead reflux rate and the reboiler steam pressure. It is impossible to control all three output variables with only two input variables; however, the side stream mole fraction of ethanol is deemed the least important of the output variables, leaving two output variables to be controlled with the two input variables.

Upon deciding to leave the control of the side stream composition out of the control scheme, we may now rewrite the model as

(\begin{matrix} y_{1} \\ y_{3} \end{matrix}) = (\begin{matrix} \frac{0.66 e^{- 2.6 s}}{6.7 s + 1} & \frac{- 0.0049 e^{- s}}{9.06 s + 1} \\ \frac{- 33.68 e^{- 9.2 s}}{8.15 s + 1} & \frac{0.87 (11.61 s + 1) e^{- s}}{(3.89 s + 1) (18.8 s + 1)} \end{matrix}) (\begin{matrix} m_{1} \\ m_{2} \end{matrix})

(4)

along with the additional relation

y_{2} = \frac{1.11 e^{- 6.5 s}}{3.25 s + 1} m_{1} + \frac{- 0.012 e^{- 1.2 s}}{7.09 s + 1} m_{2}

From equation (4), the 3-by-2 underdefined non-square MIMO systems are treated as 2-by-2 square MIMO systems, as shown in Figure 3.

Using Skogestad’s half rule (2003), elements in the transfer function matrix are approximated to First Order Plus Dead Time (FOPDT) model. Hence, the modified transfer function matrix of equation (4) is represented as

(\begin{matrix} y_{1} \\ y_{3} \end{matrix}) = (\begin{matrix} \frac{0.66 e^{- 2.6 s}}{6.7 s + 1} & \frac{- 0.0049 e^{- s}}{9.06 s + 1} \\ \frac{- 33.68 e^{- 9.2 s}}{8.15 s + 1} & \frac{0.86 e^{- 0.88 s}}{(6.6 s + 1)} \end{matrix}) (\begin{matrix} m_{1} \\ m_{2} \end{matrix})

(5)

Block diagram of over defined non-square systems

Similarly, an over defined non-square MMO systems has n inputs and n-1 outputs. The system considered in this work has 3-inputs and 2-outputs. Figure 4 and Figure 5 represents the general block diagram and open loop transfer function of 2-by-3 MIMO system. Here, m is the input variables, y is the output variables and G is process transfer function matrix, expressed by equation (6)

m = (\begin{matrix} m_{1} \\ m_{2} \\ m_{3} \end{matrix}) y = (\begin{matrix} y_{1} \\ y_{2} \end{matrix})

G = (\begin{matrix} \begin{matrix} G_{11} & G_{12} & G_{13} \end{matrix} \\ \begin{matrix} G_{21} & G_{22} & G_{23} \end{matrix} \end{matrix})

(6)

From the block diagram it is obvious that the non square is difficult to control as it is has an unequal number of inputs and output.

For the systems with 3 inputs and 2 outputs, that is, 2-by-3 MIMO system can be divided into three subsystems with all the possible combinations of 2-by-2 MIMO systems, whose block diagrams are represented in Figures 6, 7 and 8, respectively, with different combinations of inputs.

Through pulse testing, the following transfer function model was obtained

(\begin{matrix} y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix} \frac{0.5 e^{- 0.2 s}}{3 s + 1} \\ \frac{0.004 e^{- 0.5 s}}{1.5 s + 1} \end{matrix} \begin{matrix} \frac{0.07 e^{- 0.3 s}}{2.5 s + 1} \\ \frac{- 0.003 e^{- 0.2 s}}{s + 1} \end{matrix} \begin{matrix} \frac{0.04 e^{- 0.03 s}}{2.8 s + 1} \\ \frac{- 0.001 e^{- 0.4 s}}{1.6 s + 1} \end{matrix}) (\begin{matrix} m_{1} \\ m_{2} \\ m_{3} \end{matrix})

(7)

This is a 2-by-3 MIMO system, and as such, we have a situation in which only two of the three candidate input variables will be used for control, while the third input variable will have to be set at a fixed value, and will therefore be redundant.

To determine which variable should be active and which should be redundant, we first obtain all the possible 2-by-2 MIMO subsystems; there are three such subsystems.

Subsystem 1 (Utilizing $m_{1}$ and $m_{2}$ for control)

(\begin{matrix} y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix} \frac{0.5 e^{- 0.2 s}}{3 s + 1} \\ \frac{0.004 e^{- 0.5 s}}{1.5 s + 1} \end{matrix} \begin{matrix} \frac{0.07 e^{- 0.3 s}}{2.5 s + 1} \\ \frac{- 0.003 e^{- 0.2 s}}{s + 1} \end{matrix}) (\begin{matrix} m_{1} \\ m_{2} \end{matrix})

(8)

Subsystem 2 (Utilizing $m_{1}$ and $m_{3}$ for control)

(\begin{matrix} y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix} \frac{0.5 e^{- 0.2 s}}{3 s + 1} \\ \frac{0.004 e^{- 0.5 s}}{1.5 s + 1} \end{matrix} \begin{matrix} \frac{0.04 e^{- 0.03 s}}{2.8 s + 1} \\ \frac{- 0.001 e^{- 0.4 s}}{1.6 s + 1} \end{matrix}) (\begin{matrix} m_{1} \\ m_{3} \end{matrix})

(9)

Subsystem 3 (Utilizing $m_{2}$ and $m_{3}$ for control)

(\begin{matrix} y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix} \frac{0.07 e^{- 0.3 s}}{2.5 s + 1} \\ \frac{- 0.003 e^{- 0.2 s}}{s + 1} \end{matrix} \begin{matrix} \frac{0.04 e^{- 0.03 s}}{2.8 s + 1} \\ \frac{- 0.001 e^{- 0.4 s}}{1.6 s + 1} \end{matrix}) (\begin{matrix} m_{2} \\ m_{3} \end{matrix})

(10)

Case studies

In this research, work focuses on three benchmark non-linear chemical processes with various types of nonlinearity, starting from mild to highly non-linear systems.

The following case studies are done in this study, namely: (1) CSTR process–mildly nonlinear system; (2) pH neutralization process–moderate to high nonlinear system and (3) distillation column–highly non-linear system, and these problems are identified and the results on parametric identification and control are obtained using relay feedback test.

CSTR process

In the present case, a laboratory CSTR reactor system shown in Figure 9 is considered in which the reactants are sodium hydroxide (NaOH) and ethyl acetate (EtAc). The products due to alkaline hydrolysis of ester are sodium acetate and ethyl alcohol. This CSTR process has inlet flow rate (F) and cooling flow in jacket (F_j) as the manipulated variables. The controlled variables are concentration (C_A), reactor absolute temperature (T) and cooling water temperature (T_j).

No.	Nomenclature of device (Figure 5)
1	Tank A
2	Tank B
3	Peristaltic pump of Tank A
4	Peristaltic pump of Tank B
5	Electrical motor
6	Reactor
7	Two blades
8	Outlet flow
9	Heater
10	The digital regulator of the mixer
11	Power switch
12	Temperature controller system
13	Flow rate controller system
14	Computer

The CSTR process has two manipulated and three controlled variables and is categorized as 3-by-2 under defined non-square MIMO systems.

This process can be treated as square MIMO systems with ethyl acetate and hot water as manipulated variables and reaction temperature and ethyl acetate concentration as controlled variables. The process transfer function matrix of CSTR process is presented in equation (11)

(\begin{matrix} C_{A} \\ T \end{matrix}) = (\begin{matrix} \frac{0.022 e^{- 0.33 s}}{15 s + 1} & \frac{5 e^{- 0.33 s}}{21 s + 1} \\ \frac{0.0056 e^{- 6 s}}{21 s + 1} & \frac{5.9 e^{- 0.33 s}}{21 s + 1} \end{matrix}) (\begin{matrix} F \\ F_{j} \end{matrix})

(11)

pH process

Most of the physical pH processes exhibit nonlinear behaviour to some degree. The nonlinearities are of either mild or high type. The schematic of pH neutralization process is shown in Figure 10.

Referring to Bagher et al. (2011), the transfer function matrix of pH neutralization process is given in equation (12)

(\begin{matrix} h_{1} \\ p H_{1} \\ h_{2} \\ p H_{2} \end{matrix}) = (\begin{matrix} \frac{0.43 e^{- 0.8 s}}{4.32 s + 1} & \frac{0.43 e^{- 0.1 s}}{3.10 s + 1} & \frac{0.23 e^{- 1 s}}{5.24 s + 1} & \frac{0.22 e^{- 0.5 s}}{4.42 s + 1} \\ \frac{- 0.33 e^{- 1 s}}{2.56 s + 1} & \frac{0.32 e^{- 0.5 s}}{2.58 s + 1} & \frac{- 0.2 e^{- 1.8 s}}{2.82 s + 1} & \frac{0.2 e^{- 0.8 s}}{3.3 s + 1} \\ \frac{0.22 e^{- 1.1 s}}{5.52 s + 1} & \frac{0.23 e^{- 0.3 s}}{4.49 s + 1} & \frac{0.42 e^{- 0.4 s}}{3.32 s + 1} & \frac{0.41 e^{- 0.1 s}}{2.07 s + 1} \\ \frac{- 0.22 e^{- 1.5 s}}{3.24 s + 1} & \frac{0.22 e^{- 1.2 s}}{2.65 s + 1} & \frac{- 0.32 e^{- 0.8 s}}{2.36 s + 1} & \frac{0.32 e^{- 0.4 s}}{2.03 s + 1} \end{matrix}) (\begin{matrix} Q_{1} \\ Q_{3} \\ Q_{4} \\ Q_{6} \end{matrix})

(12)

The pH neutralization process has three input variables such as flow-rates (Q) of acid, base and buffer; and two output variables as pH and level (h) of the tank. Hence, the process is defined as 2-by-3 over defined non-square MIMO system. The process is treated as square MIMO systems where pH₂ and h₂ are to be controlled variables using Q₄ and Q₆ as the manipulated variables with Q₁ and Q₃ held constant. The resulting process transfer function matrix is represented in the following equation (13)

(\begin{matrix} p H_{2} \\ h_{2} \end{matrix}) = (\begin{matrix} \frac{- 0.32 e^{- 0.8 s}}{2.36 s + 1} & \frac{0.32 e^{- 0.4 s}}{2.03 s + 1} \\ \frac{0.42 e^{- 0.4 s}}{3.32 s + 1} & \frac{0.41 e^{- 0.1 s}}{2.07 s + 1} \end{matrix}) (\begin{matrix} Q_{4} \\ Q_{6} \end{matrix})

(13)

Distillation column

The design of binary distillation process shown in Figure 11 was suggested by Weischedel and McAvoy (1981) is a highly nonlinear chemical process.

The parameters (Table 1) of the distillation tower are listed below.

Table 1.

Operating variables of the binary distillation column.

Relative volatility	2.4
Number of trays	17
Feed tray	9
Analyzer dead time	2min
Feed light key	X_F=0.50
Distillate light key	X_D=0.98 mole fraction
Bottom light key	X_B=0.02 mole fraction
Feed flow	F_F=10Kg mole/min
Reflux flow	F_R=8.53Kg mole/min
Distillate flow	F_D = 5 kg mole/min
Reboiler flow	F_V = 13.53 kg mole/min
Tray holdup	H = 1.0 kg mole
Holdup in drums	HD = 10.0 kg mole

The potential manipulated variables of the distillation process are reflux (F_R) and reboiler flow rate (F_V), whereas distillate (X_D) and bottom composition (X_B) are the controlled variables here. The disturbance variable is the feed composition (X_F).

The outputs are recorded by conducting experiment on each input variables. From the process reaction curve, the transfer functions are obtained for the distillation column is described by the following equation

X_{D} (s) = \frac{0.0747 e^{- 3 s}}{12 s + 1} F_{R} (s) - \frac{0.0667 e^{- 2 s}}{15 s + 1} F_{V} (s) + \frac{0.7 e^{- 5 s}}{14.4 s + 1} X_{F} (s)

(14)

X_{B} (s) = \frac{0.1173 e^{- 3.3 s}}{11.7 s + 1} F_{R} (s) - \frac{0.1253 e^{- 2 s}}{10.2 s + 1} F_{V} (s) + \frac{1.3 e^{- 5 s}}{12 s + 1} X_{F} (s)

(15)

This nonlinear system has three input variables and two output variables. Hence, the distillation column is defined as 2-by-3 over defined non-square systems.

The same system can be treated as square system whose transfer function matrix (equation 16) is described as

(\begin{matrix} X_{D} (s) \\ X_{B} (s) \end{matrix}) = (\begin{matrix} \frac{0.0747 e^{- 3 s}}{12 s + 1} & \frac{- 0.0667 e^{- 2 s}}{15 s + 1} \\ \frac{0.1173 e^{- 3.3 s}}{11.7 s + 1} & \frac{- 0.1253 e^{- 2 s}}{10.2 s + 1} \end{matrix}) (\begin{matrix} F_{R} (s) \\ F_{V} (s) \end{matrix})

(16)

Proposed methodology

Algorithm

Step 1: Obtain the off-diagonal closed loop transfer function of under defined and over defined non-square MIMO systems.

Step 2: Conduct the relay feedback test for these systems.

Step 3: Perform modeling to produce simulated ideal relay responses using sequential method.

Step 4: Validate the simulated response with the theoretical ideal relay responses using the analytical expressions.

Step 5: Estimate the parameters using the boundary conditions and landmark points.

Step 6: Control configuration selection for appropriate input output pairing.

Step 7: Closed loop studies for the under defined and over defined non-square MIMO systems.

4.2. Flowchart

Modeling and validation for non-linear chemical processes using relay feedback test

As shown in Figure 12, consider n-by-n MIMO systems with decentralized PI controller.

The general modified transfer function of non-square MIMO systems represented as 2-by-2 square MIMO systems is described by

(\begin{matrix} y_{1} \\ y_{2} \end{matrix}) = (\begin{matrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix}) (\begin{matrix} m_{1} \\ m_{2} \end{matrix})

(17)

In auto tuning procedure the initial relationship between y and m is simply as

{(\frac{y_{1}}{m_{1}})}_{OL} = G_{11}

(18)

where the subscript OL denotes the open-loop. When the loop is closed sequentially, the closed-loop relationship between y and m becomes complicated. Therefore, it is considered that all the individual transfer functions of 2-by-2 MIMO systems are of FOPDT types.

In this case, the off-diagonal closed-loop transfer function can be found as

G_{12, CL} = \frac{G_{12}}{1 + G_{C 1} G_{11}}

(19)

G_{21, CL} = \frac{G_{21}}{1 + G_{C 2} G_{22}}

(20)

The above Equations (19) and (20) give interaction measure in closed-loop sense and the MIMO auto tuning can be done by sequential tuning procedures. Consider a 2-by-2 MIMO system with a known pairing (y₁- m₁); (y₂- m₂) under decentralized control. Then, the sequential auto tuning approach has several advantages; firstly, it makes the problem simple. The reason is that the proposed approach treats the MIMO system as a sequence of SISO systems where the relay feedback system is proven to be useful and reliable. Secondly, it operates in an efficient manner and, thirdly, it is a more accurate approach in terms of identification.

The procedure for MIMO auto tuning is described as follows.

As shown in Figure 13(a), initially a relay is placed between y₁ and m₁, while loop 1 is on manual mode. Following the relay feedback test, a controller for loop 1 can be designed from the ultimate gain and ultimate frequency.

The next step, as shown in Figure 13(b), is to perform relay feedback test between y₃ and m₂, while loop 1 is on auto mode. A controller can be designed for loop 2 following the relay feedback test.

As shown in Figure 13(c), once the controller on loop 2 is set on auto mode, another relay feedback test is formed between y₁ and m₁. In general, a new set of tuning constants is found for the controller in loop 1 and this procedure is repeated until the controller parameter converges. Typically, the controller parameter converges in 3–4 relay feedback tests in the case of 2-by-2 MIMO systems.

Modeling of non-square MIMO systems using ideal relay feedback test

The procedure for obtaining mathematical model of MIMO systems are explained in detail by Sujatha and Panda (2012). The ideal relay response for equation (14) is assumed to be formed by n-number of small step changes. At the first instant, (after synchronizing input with output by time shift) the response can be described as

y_{1} = k_{21} [1 - e^{- t / τ_{21}}] - \frac{k_{22} K_{C 2}}{τ_{I 2}} [1 + (τ_{I 2} - τ_{22}) e^{- t / τ_{22}}] y_{1} (t - 1)

(21)

Where, the term y₁(t-1) is one step ahead prediction of y₁.

At the second instant, the time is reset to zero at the initial point. The step response (relay output) can be given by: (i.e. introducing a time shift by D amount in equation (21))

y_{2} = k_{21} [(1 - e^{- \frac{t + D_{21}}{τ_{21}}}) - 2 (1 - e^{- \frac{t}{τ_{21}}})] - \frac{k_{22} K_{C 2}}{τ_{I 2}} [(1 + (τ_{I 2} - τ_{22}) e^{- \frac{t + D_{22}}{τ_{22}}}) - 2 (1 + (τ_{I 2} - τ_{22}) e^{- \frac{t}{τ_{22}}})] y_{2} (t - 1)

(22)

where $D_{21}$ and $D_{22}$ denote the time delay of the individual transfer functions of the system.

Equation (22) can be simplified as follows

y_{2} = k_{21} {[1 - 2] - e^{- \frac{t}{τ_{21}}} [e^{- \frac{D_{21}}{τ_{21}}} - 2]} - \frac{k_{22} K_{C 2}}{τ_{I 2}} {[1 - 2] + ((τ_{I 2} - τ_{22}) e^{- \frac{t}{τ_{22}}} [e^{- \frac{D_{22}}{τ_{22}}} - 2])} y_{2} (t - 1)

(23)

The relay response at the third interval lags by an amount D₂₁+P_u/2 and D₂₂+P_u/2 from an input of I and II parts: Thus, the above equation can be given as

\begin{array}{l} y_{3} = k_{21} [(1 - e^{- \frac{t + D_{21} + \frac{P_{u}}{2}}{τ_{21}}}) - 2 (1 - e^{- \frac{t + \frac{P_{u}}{2}}{τ_{21}}}) + 2 (1 - e^{- \frac{t}{τ_{21}}})] - ​ \\ ​ - \frac{k_{22} K_{C 2}}{τ_{I 2}} [(1 + (τ_{I 2} - τ_{22}) e^{- \frac{t + D_{22} + \frac{P_{u}}{2}}{τ_{22}}}) - 2 (1 + (τ_{I 2} - τ_{22}) e^{- \frac{t + \frac{P_{u}}{2}}{τ_{22}}}) \\ + 2 (1 + (τ_{I 2} - τ_{22}) e^{- \frac{t}{τ_{22}}})] y_{3} (t - 1) \end{array}

(24)

Equation (24) can be easily simplified as

\begin{matrix} y_{3} = k_{21} {[1 - 2 + 2] - e^{- \frac{t}{τ_{21}}} [e^{- \frac{D_{21} + \frac{P_{u}}{2}}{τ_{21}}} - 2 e^{- \frac{\frac{P_{u}}{2}}{τ_{21}}} + 2]} - \frac{k_{22} K_{C 2}}{τ_{I 2}} {[1 - 2 + 2] + ((τ_{I 2} - τ_{22}) e^{- \frac{t}{τ_{22}}} [e^{- \frac{D_{22} + \frac{P_{u}}{2}}{τ_{22}}} - 2 e^{- \frac{\frac{P_{u}}{2}}{τ_{22}}} + 2])} y_{3} (t - 1) \end{matrix}

(25)

It can be seen from the above equation that the terms in the RHS (i.e. y_n = y₁+y₂+y₃+…… is the sum of the step responses at t₁, t₂, t₃….) are slowly forming a series. As the time tends to infinity the response becomes stabilized and it can be described as y_n = y₁+y₂+y₃+……. In general, the ultimate properties are found out in 3–4 iterations, which are enough to get converged properties of ultimate parameters (K_u and P_u)

\begin{matrix} y_{n} = k_{21} {[1 - 2 + 2 - . . . . .] - e^{- \frac{t}{τ_{21}}} [e^{- \frac{D_{21} + \sum_{n = 1}^{n - 1} \frac{P_{u}}{2}}{τ_{21}}} - 2 e^{- \frac{\sum_{n = 1}^{n - 1} \frac{P_{u}}{2}}{τ_{21}}} + . . . . . + 2 e^{- \frac{\frac{P_{u}}{2}}{τ_{21}}} - 2]} - \\ - \frac{k_{22} K_{C 2}}{τ_{I 2}} {[1 - 2 + 2 - . . . .] + ((τ_{I 2} - τ_{22}) e^{- \frac{t}{τ_{22}}} [e^{- \frac{D_{22} + \sum_{n = 1}^{n - 1} \frac{P_{u}}{2}}{τ_{22}}} - 2 e^{- \frac{\sum_{n = 1}^{n - 1} \frac{P_{u}}{2}}{τ_{22}}} + . . . . . + 2 e^{- \frac{\frac{P_{u}}{2}}{τ_{22}}} - 2])} \\ [y_{1} (t - 1) + y_{2} (t - 1) + y_{3} (t - 1) + . . . . . . + y_{n} (t - 1)] \end{matrix}

(26)

Let $r_{1} = e^{\frac{- P_{u}}{2 τ_{21}}}$ and $r_{2} = e^{\frac{- P_{u}}{2 τ_{22}}}$

In equation (20), RHS has four parts

$y_{n} = k_{21}$ {First part – Second part} - $\frac{k_{22} K_{C 2}}{τ_{I 2}}$ {Third part + Fourth part}

The RHS of the above series becomes

First part = [1-2+2-2+…. upto $\infty$ ] = 1

Similar way, Third part = 1

Second part = $2 [e^{- \frac{D_{21}}{τ_{21}}} {r_{1}}^{n} - 2 {r_{1}}^{n - 1} + 2 {r_{1}}^{n - 2} - . . . . + 2 r_{1} - 2]$

This second part can be substituted into following series

Second part = $2 [1 - r_{1} + {r_{1}}^{2} - {r_{1}}^{3} + . . . .] = \frac{2}{1 + r_{1}} = \frac{2}{1 + e^{- \frac{P_{u}}{2 τ_{21}}}}$

Similar way, Fourth part = $\frac{2}{1 + r_{2}} = \frac{2}{1 + e^{- \frac{P_{u}}{2 τ_{22}}}}$

Thus, the generalized analytical expressions for equation (20) using ideal relay feedback test for 2-by-2 MIMO system is given by

y_{n} = k_{21} (1 - e^{- \frac{t}{τ_{21}}} (\frac{2}{1 + e^{- \frac{P_{u}}{2 τ_{21}}}})) - \frac{k_{22} K_{C 2}}{τ_{I 2}} (1 + (τ_{I 2} - τ_{22}) e^{- \frac{t}{τ_{22}}} (\frac{2}{1 + e^{- \frac{P_{u}}{2 τ_{22}}}})) y_{n} (t - 1)

(27)

The term y_n(t-1) in Equation (27) is one step ahead prediction of y_n.

Similarly, the generalized analytical expressions for equation (19) using ideal relay feedback test for 2-by-2 MIMO system is given by

y_{n} = k_{12} (1 - e^{- \frac{t}{τ_{12}}} (\frac{2}{1 + e^{- \frac{P_{u}}{2 τ_{12}}}})) - \frac{k_{11} K_{C 1}}{τ_{I 1}} (1 + (τ_{I 1} - τ_{11}) e^{- \frac{t}{τ_{11}}} (\frac{2}{1 + e^{- \frac{P_{u}}{2 τ_{11}}}})) y_{n} (t - 1)

(28)

The term y_n(t-1) in equation (28) is one step ahead prediction of y_n.

Model validation for 2-By-2 MIMO systems

The analytical expressions (developed in equations (21) and (22)) provide the mathematical models for undesirable relay responses of MIMO system. These theoretical equations are validated for non-square systems having 2-by-2 MIMO structure using ideal relay feedback tests for the period from t=0 to t=P_U/2, respectively.

From Figure 14, it can be seen that the validation of the derived mathematical models where theoretical response of 3-by-2 non-square MIMO systems matches exactly with experimental one for ideal relay feedback tests.

Similarly, the model validation of all the subsystems of the example taken for over defined non-square MIMO systems are shown in Figures 15, 16 and 17, respectively.

From the above figures, it is clear that undesired relay responses obtained after simulating the benchmark under defined and over defined non-square MIMO systems are compared with its original ideal relay responses and it is validated.

The model validation is done for all the nonlinear benchmark chemical processes described in Section 3 by simulation using MATLAB.

Figure 18 shows the model validation curve for CSTR chemical process and it is observed that the theoretical response curve matches exactly with the simulated response curve.

The model validation is conducted for the pH process and it is found that the theoretical curve is approximately following the simulated curve as depicted in Figure 19.

For the distillation column, the ideal relay response for both theoretical and experimental are validated and it is represented in Figure 20 depicting the model validation.

From these responses, it is clear that the experimental ideal response matches the theoretical ideal relay response for the nonlinear chemical processes.

Identification of parameters of 2-By-2 MIMO systems

The parameters of nonlinear chemical processes are estimated as given in Sujatha et al. (2013a) by formulating and applying the following boundary conditions to equations (27) and (28).

{(y_{m})}_{t = 0} = - a

(29)

{(y_{m})}_{t = \frac{P_{u}}{2}} = + a

(30)

{(y_{m})}_{t = \frac{P_{u}}{2} - D_{21}} = 0

(31)

{(y_{m})}_{t = \frac{P_{u}}{2} - D^{*}} = 0

(32)

where m represents the model, a represents amplitude and D^* for apparent dead time.

Applying the boundary conditions from equations (29) to (32) in the generalized analytical expression for 2-by-2 MIMO process, one can obtain the unknown parameters $k_{21}, τ_{21}, k_{22}, τ_{22}$ , $k_{12}, τ_{12}, k_{11}, τ_{11}$ by solving the equations.

The steps involved to estimate the parameters of 2-by-2 MIMO system:

From the undesired relay response, find out the dead times D₂₁ and D₁₂ from initial part of response.

Similarly, from the desired relay response obtain the dead times D₂₂ and D₁₁ from initial part of response.

Solve the equations simultaneously by applying the boundary conditions, to find unknown parameters k₂₁, τ₂₁, k₂₂, τ₂₂, k₁₂, τ₁₂, k₁₁, τ₁₁ of 2-by-2 MIMO system.

Tables 2 and 3 show the comparison between the true and estimated parameters of non-square MIMO systems and also error between these two values.

Table 2.

Comparison of true and estimated parameters of under defined non-square MIMO systems.

Parameters	Actual value	Estimated value	% Error
k ₁₁	0.66	0.67	0.015
τ₁₁	6.7	6.6	0.014
k ₁₂	−0.0049	−0.0048	0.020
τ₁₂	9.06	9.07	0.001
k ₃₁	−33.68	−33.67	0.0002
τ₃₁	8.15	8.12	0.0002
k ₃₂	0.86	0.85	0.003
τ₃₂	6.6	6.5	0.011

Table 3.

Comparison of true and estimated parameters of over defined non-square MIMO systems.

Parameters	Actual value	Estimated value	% Error
Subsystem 1:
k ₁₁	0.5	0.6	0.2
τ₁₁	3	3.1	0.033
k ₁₂	0.07	0.06	0.142
τ₁₂	2.5	2.45	0.02
k ₂₁	0.004	0.0039	0.025
τ₂₁	1.5	1.48	0.013
k ₂₂	−0.003	−0.0027	0.1
τ₂₂	1	0.98	0.02
Subsystem 2:
k ₁₁	0.5	0.6	0.2
τ₁₁	3	2.96	0.013
k ₁₃	0.04	0.038	0.05
τ₁₃	2.8	2.77	0.01
k ₂₁	0.004	0.0037	0.075
τ₂₁	1.5	1.48	0.013
k ₂₃	−0.001	−0.0009	0.1
τ₂₃	1.6	1.58	0.012
Subsystem 3:
k ₁₂	0.07	0.06	0.142
τ₁₂	2.5	2.45	0.02
k ₁₃	0.04	0.038	0.01
τ₁₃	2.8	2.77	0.01
k ₂₂	−0.003	−0.0027	0.1
τ₂₂	1	0.98	0.02
k ₂₃	−0.001	−0.0009	0.1
τ₂₃	1.6	1.58	0.012

Tables 4, 5 and 6 show the error between the true and estimated parameters of benchmark nonlinear chemical processes. This method of parameter estimation is efficient in identifying the parameters of non-square MIMO processes.

Table 4.

CSTR process parameter estimation.

Parameters	Actual value	Estimated value	% Error
k ₁₁	0.022	0.021	0.045
τ₁₁	15	14.7	0.02
D ₁₁	0.33	0.33	0
k ₁₂	5	4.9	0.02
τ₁₂	21	20.8	0.009
D ₁₂	0.33	0.33	0
k ₂₁	0.0056	0.0057	−0.01
τ₂₁	21	20.9	0.004
D ₂₁	6	6	0
k ₂₂	5.9	5.88	0.003
τ₂₂	21	20.98	0.0009
D ₂₂	0.33	0.33	0

Table 5.

Parameter estimation of pH process.

Parameters	Actual value	Estimated value	% Error
k ₁₁	−0.32	−0.318	0.00625
τ₁₁	2.36	2.358	0.0008
D ₁₁	0.8	0.8	0
k ₁₂	0.32	0.317	0.0093
τ₁₂	2.03	2.017	0.006
D ₁₂	0.4	0.4	0
k ₂₁	0.42	0.418	0.004
τ₂₁	3.32	3.319	−0.9
D ₂₁	0.4	0.4	0
k ₂₂	0.41	0.409	0.002
τ₂₂	2.07	2.06	0.004
D ₂₂	0.1	0.1	0

Table 6.

Estimation of parameters in distillation process.

Parameters	Actual value	Estimated value	% Error
k ₁₁	0.0747	0.0745	0.002
τ₁₁	12	11.9	0.008
D ₁₁	3	3	0
k ₁₂	−0.0667	−0.0666	0.001
τ₁₂	15	14.9	0.006
D ₁₂	2	2	0
k ₂₁	0.1173	0.1172	0.0001
τ₂₁	11.7	11.68	0.001
D ₂₁	3.3	3.3	0
k ₂₂	−0.1253	−0.1251	0.001
τ₂₂	10.2	10.1	0.009
D ₂₂	2	2	0

Data from undesired closed-loop containing information on interactions (loop) are used for identifying the unknown process parameters. Parameter estimation algorithms using ultimate properties and land mark points are formulated to identify model parameters. The method is efficient for identifying MIMO process parameters.

Control configuration selection of non-square systems

Conventional method of loop pairing for under defined non-square MIMO systems

The modified (square) subsystem’s steady-state gain matrix is obtained as

k = (\begin{matrix} - 0.66 & - 0.0049 \\ - 33.68 & 0.87 \end{matrix})

(33)

which leads to the RGA = (\begin{matrix} 1.4 & - 0.4 \\ - 0.4 & 1.4 \end{matrix})

(34)

The recommended input/ output pairing scheme is:

y₁– m₁; overhead reflux to control overhead composition

y₃– m₂; reboiler steam pressure to control Tray #19 temperature

The strategy is to choose a square subsystem by dropping off the excess number of output variables on the basis of economic importance; the subsequent analysis is the same as for square systems.

Proposed method of loop pairing under defined system

The under defined systems have two inputs (m₁ and m₂) and three outputs (y₁, y₂ and y₃) and are squared by leaving y₂ and expressed as 2x2 square MIMO systems. The square system that have two inputs (m₁ and m₂) and three outputs (y₁ and y₃) are considered. The proposed method of loop pairing is done by comparing the area under undesired responses for both diagonal and off-diagonal pairing, as given by Sujatha et al. (2013b). The pair that offers minimum area is chosen as a judicious loop pairing.

Case 1:

(y₁-m₁); (y₃-m₂) pairing

Figure 21 and Figure 22 show the closed loop undesired responses for diagonal pairing and the area under the curve is obtained in Table 7.

Case2:

(y₁-m₂); (y₃-m₁) pairing

Table 7.

Comparison of areas under the disturbance/ load responses for OR column.

Area under the disturbance / load response
(y₁- m₁); (y₃- m₂) pairing		(y₁- m₂); (y₃- m₁) pairing
$(\frac{y_{3}}{m_{1}})$	124.6624	$(\frac{y_{3}}{m_{2}})$	0.0019
$(\frac{y_{1}}{m_{2}})$	0.0042	$(\frac{y_{1}}{m_{1}})$	281.7845

The closed loop response for off diagonal pairing is shown in Figure 23 and Figure 24 and its corresponding area is tabulated in Table 7.

From Table 7, it is clear that the desirable pairing for OR column is (y₁− m₁); (y₃− m₂) pairing.

Conventional method of loop pairing for over defined non-square MIMO systems

RGA: In this case, we have an excess of input variables, and therefore we can actually achieve arbitrary control of the fewer output variables in more than one way. The strategy for loop pairing is:

First determine, from the given process model, all the $(\begin{matrix} m \\ n \end{matrix})$ possible square subsystems.

Obtain the RGAs for each of these square subsystems.

Examine these RGAs and pick the best subsystem on the basis of the overall character of its RGA (in terms of how close it is to the ideal RGA).

Having thus determined the best subsystem, use its RGA to determine which input variable within this subsystem to pair with which output variable.

Next, RGA for each subsystem are obtained and the results are as follows.

For subsystem 1

RGA = (\begin{matrix} 0.843 & 0.157 \\ 0.157 & 0.843 \end{matrix})

(35)

For subsystem 2

RGA = (\begin{matrix} 0.758 & 0.242 \\ 0.242 & 0.758 \end{matrix})

(36)

For subsystem 3

RGA = (\begin{matrix} - 1.4 & 2.4 \\ 2.4 & - 1.4 \end{matrix})

(37)

Based on inspection of these three RGAs, it appears that Subsystem 1 will offer us the best possible control, because its RGA is the closest to the ideal situation; it is somewhat better than Subsystem 2, and far superior to Subsystem 3.

Having thus decided on Subsystem 1, the implication is that the input variables m₁ and m₂ are to be used for control and m₃ is to be redundant. The final task is to pair the variables of this subsystem, an easy exercise. The best possible loop pairing within this subsystem is y₁– m₁ and y₂– m₂.

This example has shown that for under defined systems, the strategy is to choose a square subsystem by dropping off the excess number of output variables on the basis of economic importance; the subsequent analysis is the same as for square systems.

Proposed method of loop pairing – over defined system

The pairing for over defined systems having three inputs and two outputs can be treated as three subsystems having two inputs and two outputs and the method used for pairing of the square system is extended.

Subsystem 1:

Case 1: (y₁-m₁); (y₃-m₂) pairing

Subsystem 1 closed loop responses are plotted in Figures 25 and 26 and its area is given in Table 8.

Case 2: (y₂-m₁); (y₁-m₂) pairing

Table 8.

Comparison of areas under the load responses for subsystem1.

Area under the disturbance / load response
(y₁-m₁); (y₂-m₂) pairing		(y₂-m₁);(y₁-m₂) pairing
$(\frac{y_{2}}{m_{1}})$	0.0036	$(\frac{y_{2}}{m_{2}})$	0.0637
$(\frac{y_{1}}{m_{2}})$	15.7309	$(\frac{y_{1}}{m_{1}})$	0.8989

The undesired responses for off diagonal pairing are obtained in Figures 27 and 28 and its area is also given in Table 8.

Hence, desirable pairing for the subsystem 1 is (y₂-m₁); (y₁-m₂).

Subsystem 2:

Case 1: (y₁–m₁); (y₂-m₃) pairing

For the diagonal pairing, the closed loop undesired responses are obtained in Figure 29 and Figure 30, respectively.

Case 2: (y₁-m₃); (y₂-m₁) pairing

The closed loop undesired responses for off diagonal pairing for subsystem 2 is obtained in Figure 31 and Figure 32.

From Table 9, is clear that minimum area is obtained only for (y₁-m₃); (y₂-m₁) pairing. Hence, desirable pairing for the subsystem 2 is (y₁-m₃); (y₂-m₁).

Table 9.

Comparison of areas under the load responses for subsystem 2.

Area under the disturbance / load response
(y₁-m₁); (y₂-m₃) pairing		(y₁-m₃);(y₂-m₁) pairing
$(\frac{y_{2}}{m_{1}})$	0.0065	$(\frac{y_{2}}{m_{3}})$	0.0658
$(\frac{y_{1}}{m_{3}})$	24.3136	$(\frac{y_{1}}{m_{1}})$	2.4242

Similarly, the same procedure can be extended for subsystem 3. From subsystem 1 and 2, it is clear that for this particular system, the recommended input-output pairing is (y₁-m₃); (y₂-m₁).

Conventional method of loop pairing for non-linear chemical process

From Table 10, it is clear that from RGA matrix for all the benchmark process, the desirable input-output pair is (y₁-m₁); (y₂-m₂).

Table 10.

Conventional loop pairing method- relative gain array.

S.No	Process	Relative gain array
1	CSTR	$(\begin{matrix} 1.2750 & - 0.2750 \\ - 0.2750 & 1.2750 \end{matrix})$
2	pH	$(\begin{matrix} 0.4940 & 0.5060 \\ 0.5060 & 0.4940 \end{matrix})$
3	Distillation column	$(\begin{matrix} 6.0937 & - 5.0937 \\ - 5.0937 & 6.0937 \end{matrix})$

Proposed loop pairing for the non-linear chemical processes

The judicious choice of loop pairing is based on comparing the areas under the closed loop undesired responses for all combinations of input-output pairing. The pair that offers minimum area is chosen as a judicious choice of loop pairing.

Loop pairing of CSTR process

The loop pairing in CSTR process is obtained by assuming diagonal pairing and off diagonal pairing, the closed loop response are obtained and shown in Figures 33, 34, 35 and 36, respectively. The area under the closed loop undesired responses are calculated and its values are tabulated in Table 11 for comparison. The correct pairing is chosen based on the minimum area obtained after comparison.

Case 1: Diagonal pairing: (y₁–m₁); (y₂-m₂) pairing

Table 11.

Proposed loop pairing – Comparison of areas under the load responses.

CSTR process
(y₁-m₁); (y₂-m₂) pairing		(y₂-m₁);(y₁-m₂) pairing
$(\frac{y_{2}}{m_{1}})$	0.0036	$(\frac{y_{2}}{m_{2}})$	0.0637
$(\frac{y_{1}}{m_{2}})$	15.7309	$(\frac{y_{1}}{m_{1}})$	0.8989
pH Process
$(\frac{y_{2}}{m_{1}})$	0.3333	$(\frac{y_{2}}{m_{2}})$	0.2539
$(\frac{y_{2}}{m_{1}})$	0.8328	$(\frac{y_{2}}{m_{2}})$	1.0930
Distillation column
$(\frac{y_{2}}{m_{1}})$	51.0823	$(\frac{y_{2}}{m_{2}})$	29.0406
$(\frac{y_{1}}{m_{2}})$	26.0268	$(\frac{y_{1}}{m_{1}})$	45.7813

Figure 33 represents the closed loop undesired response, y₂ of CSTR process by assuming the diagonal pairing when there is change in input m₁ while m₂=0.

For CSTR process, the closed-loop undesired response, y₁ when there is change in m₂ for diagonal pairing of CSTR process is represented in Figure 34.

Case 2: Off-diagonal pairing: (y₂-m₁); (y₁-m₂) pairing

Figure 35 shows that for CSTR process, the closed loop undesired response y₂ for the change in m₂ by assuming off-diagonal pairing.

The closed loop undesired response y₁ for the change in m₁ for off diagonal pairing in CSTR process is represented in Figure 36.

Loop pairing of pH process

For the pH process, closed loop undesired responses are obtained for both diagonal and off-diagonal pairing as shown in Figures 37, 38, 39 and 40. The areas obtained under these curves are given in Table 11 and the same is compared to obtain the desired input-output pair based on minimum value.

Case 1: Diagonal pairing: (y₁–m₁); (y₂-m₂) pairing

Figure 37 represents the closed loop undesired response y₂ for the change in m₁ for diagonal pairing of pH process.

For diagonal pairing in pH process, closed loop undesired response y₁ for change in m₂ is shown in Figure 38.

Case 2: Off diagonal pairing (y₂-m₁); (y₁-m₂) pairing

For the pH process, the closed loop undesired response y₂ for change in m₂ is represented in Figure 39 for off-diagonal pairing.

For off diagonal pairing in pH process, Figure 40 represents the closed-loop undesired response y₁ for the change in m₁.

Loop pairing of distillation column

Similar to CSTR and pH process, the pairing for distillation column are also carried out to choose the correct input-output pairing.

Case 1: Diagonal pairing:

Closed loop undesired response for diagonal pairing of distillation column is represented in Figure 41.

Figure 42 shows that closed loop response of distillation column for diagonal pairing.

Case 2: Off- diagonal pairing

For the distillation column, closed loop undesired response y2 for the change in m2 for off diagonal pairing is shown in Figure 43.

Figure 44 represents that the closed loop undesired response y1 for off diagonal pairing of distillation column

Similarly, Figures 41, 42 and Figures 43, 44 represent undesired responses for distillation column and its area is given in Table 11.

From Table 11, for all benchmark nonlinear chemical processes, it is clear that the minimum area is obtained only for (y₁-m₁); (y₂-m₂) pairing. Hence, desirable pairing for all these processes is (y₁-m₁); (y₂-m₂).

Closed loop controller tuning Internal Model Control-Proportional Integral Derivative (IMC-PID) laurent tuning rule

IMC-PI controller parameters are derived by equating the closed loop response to desired closed-loop response involving user defined tuning parameter, λ. Thus, the IMC-PI controller parameter by Panda (2009) for FOPDT processes are computed using the following expressions

K_{C} = \frac{τ_{I}}{k_{p} (λ + D_{p})}; τ_{I} = (τ_{p} + β) + \frac{{D_{p}}^{2}}{2 (λ + D_{p})}

(38)

Closed loop response of under-defined non-square MIMO system

The parameters of multi-loop controllers using IMC for SISO system has been extended to MIMO processes are listed in the following Table 12.

Table 12.

Controller parameters for under defined non-square MIMO system.

S.no	Process	IMC design (proposed)		Existing
		$K_{C}$	$τ_{I}$	$K_{C}$	$τ_{I}$
1	OR	1.5862 3.3905	7.349 6.928	1.28 2.24	19.3 7.78

The simulation results given in Figure 45 shows that IMC design provides better performance.

Thus, the ISE values are compared for proposed and existing method for the under defined systems are tabulated in Table 13.

Table 13.

Comparison of ISE values for proposed and existing method.

ISE	Proposed	Existing	Proposed	Existing
OR process (Loop 1)			OR process (Loop 2)
y1	92.7380	77.9130	−0.0042	–0.0125
y2	−117.2097	−196.0389	99.5450	96.0808

Closed loop response of over defined non-square MIMO system

The controller parameters of subsystem 1, subsystem 2 and subsystem 3 are listed in Table 14.

Table 14.

Controller parameters of over defined non-square MIMO systems.

S.no	Process	IMC design (proposed)		ETF design (existing)
		$K_{C}$	$τ_{I}$	$K_{C}$	$τ_{I}$
1	Over defined non-square MIMO systems	Subsystem 1		Subsystem 1
		7.75–655.5784	3.101.0620	1.4–37.939	3–37.939
		Subsystem 2		Subsystem 2
		7.75–1587.1	3.11.7141	1.4–37.939	3–37.939
		Subsystem 3		Subsystem 3
		46.1738–1587.1	2.61811.7141	1.4–37.939	3–37.939

The closed-loop step responses for subsystem 1 and subsystem 2 of overdefined nonsquare MIMO systems are shown in Figures 46 and 47, better performance can be obtained by designing controllers using IMC. Similarly, the closed loop step responses for subsystem 3 can also be obtained.

Table 15 represents the comparison of ISE values for over defined MIMO systems.

Table 15.

ISE values for over defined systems.

ISE	Proposed	Existing	Proposed	Existing
Subsystem 1 (Loop 1)			Subsystem 1 (Loop 2)
y1	99.3258	96.3884	−15.7309	–84.2697
y2	0.0036	0.0592	99.5450	92.5960
Subsystem 2 (Loop 1)			Subsystem 2 (Loop 2)
y1	99.3939	96.7468	−24.3138	−129.2055
y2	0.0065	0.1591	99.1819	80.1018

Closed loop response of nonlinear chemical process

Table 16 represents the controller parameters for the nonlinear chemical processes

Table 16.

Controller parameters of over defined non-square MIMO systems.

S.no	Process	IMC design (proposed)		Existing
		$K_{C}$	$τ_{I}$	$K_{C}$	$τ_{I}$
1	Non-linear chemical process	CSTR
		210.09220.8058	15.391421.5370	1519.3722.683663	0.660.66
		pH
		−3.714010.1143	2.56712.1315	−5.01294.8705	2.4330.833
		Distillation column
		21.2464–15.992	12.855610.8254	10.4–6.8	9.06.1

The closed loop responses for these non-linear chemical processes are shown in Figures 48, 49 and 50, respectively.

ISE values for nonlinear chemical processes are tabulated in Table 17.

Table 17.

ISE values for nonlinear chemical processes.

ISE	Proposed	Existing	Proposed	Existing
CSTR (Loop 1)			CSTR (Loop 2)
y1	26.0516	28.4723	3.2532	1.2865
y2	1.2917	0.0793	24.6814	29.6728
pH (Loop 1)			pH (Loop 2)
y1	98.9330	99.2508	1.0930	0.7675
y2	−0.2539	−0.2061	99.7461	99.7940
	Distillation (Loop 1)		Distillation (Loop 2)
y1	451.0618	430.9493	−45.7813	−64.5298
y2	−29.0406	−38.0413	467.4473	457.2946

Stability analysis of non-square systems

The characteristic equation of any system, closed loop or open loop, is the equation that you get when you take the denominator of the transfer function describing the system and set it equal to zero. The resulting N^th– order polynomial equation in s has N roots, and these dictate the stability, damping and speed of response of the system.

For closed loop systems by Luyben (1996), the denominator of the transfer functions in the closed loop servo and load transfer function matrices gives the closed-loop characteristic equation. The denominator is to be $Det [I + G_{C} G_{P}] = 0$ , in which a scalar N^th order polynomial in s. Therefore, the closed loop characteristic equation for a multivariable process is

Det [I + G_{C} G_{P}] = 0

(39)

This is the most important equation in multivariable control. It applies for any type of controller, diagonal (multi loop SISO) or full multivariable controller. If any of the roots of this equation are in the right half of the s plane, the system is closed loop unstable.

The closed loop characteristic equation for a 2x2 process with a diagonal feedback controller

G_{p} = [\begin{matrix} G_{P 11} & G_{P 12} \\ G_{P 21} & G_{P 22} \end{matrix}] G_{C} = [\begin{matrix} K_{C 1} & 0 \\ 0 & K_{C 2} \end{matrix}]

(40)

Det [I + G_{C} G_{P}] = 0

Det [[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] + [\begin{matrix} K_{C 1} & 0 \\ 0 & K_{C 2} \end{matrix}] [\begin{matrix} G_{P 11} & G_{P 12} \\ G_{P 21} & G_{P 22} \end{matrix}]] = 0

Det [[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] + [\begin{matrix} K_{C 1} G_{P 11} & K_{C 1} G_{P 12} \\ K_{C 2} G_{P 21} & K_{C 2} G_{P 22} \end{matrix}]] = 0

(41)

Det [\begin{matrix} (1 + K_{C 1} G_{P 11}) & K_{C 1} G_{P 12} \\ K_{C 2} G_{P 21} & (1 + K_{C 2} G_{P 22}) \end{matrix}] = 0

(1 + K_{C 1} G_{P 11}) (1 + K_{C 2} G_{P 22}) - K_{C 2} G_{P 21} K_{C 1} G_{P 12} = 0

1 + K_{C 1} G_{P 11} + K_{C 2} G_{P 22} + K_{C 1} K_{C 2} (G_{P 11} G_{P 22} - G_{P 12} G_{P 21}) = 0

The closed loop characteristic equation depends on the tuning of both feedback controllers.

Thus, the stability analysis of non-square MIMO systems are listed in Table 18. It is observed that the system is stable.

Table 18.

Stability analysis of non-square MIMO systems.

S.No	System	Roots of the closed loop characteristic equation		Analysis
1	Under defined system	S=-0.132		Stable
2	Over defined system
	Subsystem 1	S1=-0.356, S2=-0.824		Stable
	Subsystem 2	S1=-0.346, S2=-0.642		Stable
	Subsystem 3	S=-0.25		Stable
3	Non-linear chemical processes
3	CSTR		S=0	Stable
	pH		S=-0.5	Stable
	Distillation column		S1=-0.077, S2=-0.086	Stable

Conclusion

In this study, two types of non-square systems namely, underdefined and overdefined systems are modelled and controlled. The estimated parameters are obtained using the landmark points from the model. The control configuration selection has been done by capturing and comparing the area under the closed loop undesired responses for both diagonal and off-diagonal pairing. The pairing is decided based on area comparison, which gives the minimum area under the curve. The input-output pairing based on the proposed methods is carried out for both under defined and over defined non-square MIMO systems. Closed loop studies also carried out using IMC-based PI controller. The proposed method of modeling and control can also be extended to overdefined non-square MIMO systems with non minimum phase.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Vijay Sujatha

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