A Dual-Loop Modified Active Disturbance Rejection Control Scheme for a High-Purity Distillation Column

Abstract

High-purity distillation columns typically give rise to multi-variable, strongly coupled nonlinear systems with substantial time delay and significant inertia. The control performance of high-purity distillation columns crucially influences the purity of the final product. Taking into account the process of a high-purity distillation column, this article puts forward a dual-loop modified active disturbance rejection control (MADRC) scheme to improve the control of product purity. During the stable operation of the distillation process, the structures of two control loops are, respectively, approximated by two linear transfer function models via open-loop experiments. Subsequently, the compensation part of the MADRC scheme is designed, respectively, for each approximate model. Furthermore, this paper employs singular perturbation theory to prove the stability of MADRC. The performance of the dual-loop MADRC scheme (MADRC) is compared with that of a proportional–integral–derivative (PID) control scheme, a cascade PID control scheme (CPID), and a regular ADRC scheme (ADRC). The simulations demonstrate that the dual-loop MADRC scheme is capable of efficiently tracking the reference value and exhibits optimal disturbance rejection capabilities. Additionally, the superiority of the dual-loop MADRC scheme is validated through Monte Carlo trials.

1. Introduction

Distillation, a thermodynamic separation process, exploits the disparate boiling points of individual components within a mixed liquid or liquid-solid system. By leveraging these differences, the low-boiling component is vaporized, followed by recondensation, thereby enabling the separation of the entire mixture. Fundamentally, distillation embodies a synthesis of evaporation and condensation processes. When compared with alternative separation techniques, including extraction, filtration, and crystallization, distillation presents a notable advantage. Specifically, it obviates the need for supplementary solvents beyond those inherent in the system components. This attribute guarantees that no novel impurities are introduced during the separation procedure. The high-purity distillation column, a tower-type gas-liquid contact distillation unit, is widely employed in chemical industry applications. The control of the distillation column plays a pivotal role in determining the purity of the final product [1,2]. Distillation columns typically give rise to multi-variable, strongly coupled nonlinear systems. Consequently, their dynamic characteristics are susceptible to perturbations and uncertainties [3,4]. Additionally, the control systems of distillation columns are confronted with substantial time delay and significant inertia [5]. These factors collectively render the control of distillation columns a challenging problem [6]. While proportional–integral–derivative (PID) control offers simplicity, well-established tuning methods, and cost-effectiveness, its performance limitations impede the production of high-purity products [7,8]. Similarly, Model Predictive Control (MPC) encounters issues such as high computational complexity, poor real-time performance, and sensitivity to model mismatch [9]. Nonlinear control suffers from restricted versatility owing to its reliance on specific function forms and demonstrates limited robustness [10]. Furthermore, cascade control requires elaborate parameter tuning and painstaking debugging, ultimately falling short in meeting the stringent purity requirements under disturbance conditions [11]. Collectively, these drawbacks underscore the pressing need for advanced control schemes.

As active disturbance rejection control (ADRC) continues to gain traction in the field of control engineering [12,13], numerous control strategies based on ADRC have been extensively applied in the control of distillation columns [14,15]. The extended state observer (ESO) in ADRC holds a crucial position. It estimates and compensates for the system’s total disturbance (including internal uncertainties and external disturbances) instantaneously. This endows ADRC to handle system nonlinearities and uncertainties with remarkable robustness [16]. However, substantial time delay and significant inertia of distillation columns cause inputs of the ESO to be out of sync, which curtails the maximum achievable value of the ESO bandwidth [17]. To mitigate the adverse impacts caused by the unsynchronized inputs of the ESO, a modified active disturbance rejection control (MADRC) has been formulated by adding a compensation part [18]. Taking into account the process of a high-purity distillation column, this paper proposes a dual-loop MADRC scheme to boost the precision of product purity control and reduce energy consumption. The dual-loop MADRC scheme operates without the need for precise models, simplifies parameter tuning, features a simple architecture, and enables easy implementation on the distributed control system (DCS) platform, thus offering an advanced solution to address the challenges in distillation column control. This paper’s primary contributions are presented in the following manner:

(1)

During the stable operation of the distillation process, open-loop experiments are carried out. The loop structure with an input, flow rate of boil-up ($V_B$), and an output, light component’s composition in bottom product ($x_B$), can be depicted approximately as a high-order inertial model. Likewise, the loop structure with an input, flow rate of reflux ($L_T$), and an output, light component’s composition in top product ($x_D$), can be depicted approximately as another high-order inertial model.

(2)

The dual-loop MADRC scheme for the high-purity distillation column is designed.

(3)

The stability of MADRC is proven by making use of the singular perturbation theory [19,20,21].

The latter part of this article is structured as follows: In Section 2, the specific details of the distillation process are explained thoroughly. Section 3 delves into the design methods of both regular ADRC and MADRC, and then theoretically proves the stability of MADRC. Section 4 makes use of the single-variable method to tune the parameters of the dual-loop MADRC controllers. In Section 5, simulations are conducted to compare the control effects of the four control schemes, which are more clearly presented through the comparison of the integral absolute error (IAE) indexes. Moreover, the superiority of the MADRC scheme is verified by Monte Carlo trials. Ultimately, Section 6 sets forth the conclusions.

2. Process Model for the Column

The high-purity distillation column capitalizes on the fact that each component within the feed mixture exhibits distinct volatility characteristics. Specifically, the vapor pressure of each component differs at a given temperature. This process facilitates the transfer of a light-boiling component from the liquid phase to the gas phase and a heavy-boiling component from the gaseous phase into the fluid phase, thereby attaining the objective of separation [22], as depicted in Figure 1.

The process model of the distillation column is composed of 41 stages. The bottom tray is a reboiler, and the top tray is a condenser.

Table 1. The parameters involved in the high-purity distillation column’s process.

Symbol	Nomenclature	Nominal Values
$n_t$	condenser stage	41
$n_f$	feed stage	21
F	feed flow rate, kmol/min	1
$z_F$	composition concentration of the feed	$0.5$
$q_F$	fluid content fraction within the feed	1
D and B	flow rates of top and bottom products, kmol/min	$0.5$
$M_D$ and $M_B$	fluid holdup of condenser and reboiler fluid holdup, kmol	$0.5$
$x_D$ and $x_B$	light component’s composition in top and bottom products	$0.99$ and $0.01$
$L_T$ and $V_B$	flow rates of reflux and boil-up, kmol/min	$2.7063$ and $3.2063$
$M_i$	fluid holdup on stage i, kmol	$0.5$
$\alpha$	relative volatility	$1.5$
$\tau$	time constant, min	$0.063$
${\lambda }_f$	steam’s influence on fluid holdup	0

summarizes the nomenclature of each process parameter and the nominal values employed in this distillation column model. The subsequent section shows the nonlinear models of every component inside the high-purity distillation column:

Reboiler:

$$\left\{\begin{array}{c}d{M}_B/dt=L_{i+1}-V_B-B\hfill \\ d\left(M_B{x}_B\right)/dt=L_{i+1}{x}_{i+1}-V_B{y}_i-B{x}_i\hfill \end{array}\right.,$$

(1)

where $i=1,M_1=M_B,x_1=x_B,$ and $V_1=V_B$. Meanwhile, $y_i$ is the gas–fluid balance equation, and its mathematical expression is presented in detail later.

Condenser:

$$\left\{\begin{array}{c}d{M}_D/dt=V_{i-1}-D-L_T\hfill \\ d\left(M_D{x}_D\right)/dt=V_{i-1}{y}_{i-1}-D{x}_i-L_T{x}_i\hfill \end{array}\right.,$$

(2)

where $i=1,V_1=V_B,x_1=x_B,M_1=M_B$.

Feed tray:

$$\left\{\begin{array}{c}d{M}_{n_f}/dt=L_{n_f+1}-L_{n_f}+V_{n_f-1}-V_{n_f}+F\hfill \\ d\left(M_{n_f}{x}_{n_f}\right)/dt=L_{n_f+1}{x}_{n_f+1}-L_{n_f}{x}_{n_f}+V_{n_f-1}{y}_{n_f-1}-V_{n_f}{y}_{n_f}+F{z}_F\hfill \end{array}\right..$$

(3)

Other distillation trays:

$$\left\{\begin{array}{c}d{M}_i/dt=L_{i+1}-L_i+V_{i-1}-V_i\hfill \\ d\left(M_i{x}_i\right)/dt=L_{i+1}{x}_{i+1}-L_i{x}_i+V_{i-1}{y}_{i-1}-V_i{y}_i\hfill \end{array}\right.,$$

(4)

where $i=2,3,\cdots ,n_f-1,n_f+1,\cdots ,n_t-1$.

Gas–fluid balance equation:

$$y_i={\displaystyle \frac{\alpha x_i}{\left(1+\left(\alpha -1\right)x_i\right)}}.$$

(5)

Fluid flow equation:

$$L_i=L_{0,i}+{\displaystyle \frac{\left(M_i-M_{0,i}\right)}{\tau }}+{\lambda }_f\left(V_{i-1}-V_{0,i-1}\right),$$

(6)

$$L_{0,i}=\left\{\begin{array}{c}{L}_{0,T},i>n_f\hfill \\ L_{0,T}+F·q_F,i\le n_f\hfill \end{array}\right..$$

(7)

Gas flow velocity equation:

$$V_i=\left\{\begin{array}{c}{V}_B,i>n_f\hfill \\ V_B+F·\left(1-q_F\right),i\le n_f\hfill \end{array}\right.,$$

(8)

where i represents the layer number concerned.

3. The Dual-Loop MADRC Scheme for the Column

The dual-loop MADRC scheme entails designing two controllers of MADRC through the adjustment of $L_T$ and $V_B$ to, respectively, control the compositions of $x_D$ and $x_B$, as depicted in Figure 2. To ensure the stability of the distillation column, the LV configuration [23] is adopted. Additionally, two proportional-integral (PI) controllers are designed: one enables D to govern $M_D$, and the other allows B to manage $M_B$. The parameters of the two PI controllers for the $M_D$ and $M_B$ are $\left\{k_{PI1}=-40,k_{iI1}=-3\right\}$ and $\left\{k_{PI2}=-40,k_{iI2}=-3\right\}$, respectively. Note that other advanced strategies like ADRC are also applicable.

3.1. The Regular Design of Linear ADRC

The concept of the frequency scale is employed in the linear ADRC controller. The nonlinear links within the nonlinear ADRC controller are simplified, and the linear structures are integrated [13]. This integration enables the establishment of a relationship between the parameters associated with the controller’s bandwidth and those of the observer. As a result, the number of parameters is decreased from nine to three. This reduction not only diminishes the complexity and debugging intricacy of the system but also enhances the practicality and reliability, as displayed in Figure 3.

A common nth-order model can be described as

$$y^{\left(n\right)}\left(t\right)=g\left(y^{\left(n-1\right)}\left(t\right),\cdots ,y\left(t\right),\omega \left(t\right)\right)+\widehat{b}\widehat{u}\left(t\right),$$

(9)

where $y\left(t\right)$, $\widehat{u}\left(t\right)$, $\omega \left(t\right)$, $g\left(·\right)$, and $\widehat{b}$ denote the output, the control signal, external disturbances, the comprehensive function, and the input gain of the system, respectively.

Define the total disturbance as

$$f=g\left(y^{\left(n-1\right)}\left(t\right),\cdots ,y\left(t\right),\omega \left(t\right)\right)+\left(\widehat{b}-{\widehat{b}}_0\right)\widehat{u}\left(t\right),$$

(10)

where f is differentiable, and ${\widehat{b}}_0$ is the estimation of $\widehat{b}$. By selecting the state variables ${\tilde{x}}_1=y\left(t\right),{\tilde{x}}_2=y^{\left(1\right)}\left(t\right),\cdots ,{\tilde{x}}_n=y^{\left(n-1\right)}\left(t\right),{\tilde{x}}_{n+1}=f$, the state equation of Equation (9) can be obtained as

$$\left\{\begin{array}{c}{\dot{\tilde{x}}}_1={\tilde{x}}_2\hfill \\ ⋮\hfill \\ {\dot{\tilde{x}}}_{n-1}={\tilde{x}}_n\hfill \\ {\dot{\tilde{x}}}_n={\tilde{x}}_{n+1}+{\widehat{b}}_0\widehat{u}\hfill \\ {\dot{\tilde{x}}}_{n+1}=\dot{f}\hfill \\ y={\tilde{x}}_1\hfill \end{array}\right.,$$

(11)

where $\tilde{X}={\left[{\tilde{x}}_1,{\tilde{x}}_2,\cdots ,{\tilde{x}}_n\right]}^T$. Thus, the state equation of linear ESO is able to be constructed as

$$\left\{\begin{array}{c}{\dot{\tilde{z}}}_1={\tilde{z}}_2+{\tilde{\beta }}_1\left(y-{\tilde{z}}_1\right)\hfill \\ ⋮\hfill \\ {\dot{\tilde{z}}}_{n-1}={\tilde{z}}_n+{\tilde{\beta }}_{n-1}\left(y-{\tilde{z}}_1\right)\hfill \\ {\dot{\tilde{z}}}_n={\tilde{z}}_{n+1}+{\tilde{\beta }}_n\left(y-{\tilde{z}}_1\right)+{\widehat{b}}_0\widehat{u}\hfill \\ {\dot{\tilde{z}}}_{n+1}={\tilde{\beta }}_{n+1}\left(y-{\tilde{z}}_1\right)\hfill \\ y={\tilde{x}}_1\hfill \end{array}\right.,$$

(12)

where ${\tilde{\beta }}_i,i=1,2,\dots ,n+1$ are the parameters of linear ESO. Relying on the bandwidth-parameterization approach [13], the succeeding conclusions can be obtained as

$$\left\{\begin{array}{c}{\tilde{\beta }}_i={\alpha }_i{\tilde{\omega }}_o^i\hfill \\ {\alpha }_i={\displaystyle \frac{\left(n+1\right)!}{i!\left(n+1-i\right)!}}\hfill \end{array}\right.,$$

(13)

where ${\tilde{\omega }}_o$ is the observer bandwidth. Therefore, it is possible to configure the characteristic equation of the ESO as

$$\begin{array}{c}{\lambda }_o\left(s\right)=s^{n+1}+{\tilde{\omega }}_o{\alpha }_1{s}^n+\dots +{\tilde{\omega }}_o^n{\alpha }_{n}s+{\tilde{\omega }}_o^{n+1}{\alpha }_{n+1}={\left(s+{\tilde{\omega }}_o\right)}^{n+1}\hfill \end{array}.$$

(14)

The above configuration places the poles of Equation (14) at the same position ${\tilde{\omega }}_o\left({\tilde{\omega }}_o>0\right)$, which not only guarantees the system’s stability but also offers a superior transition process.

The state-feedback control law (SFCL) is specified as

$$\widehat{u}={\displaystyle \frac{{\widehat{u}}_0-{\tilde{z}}_{n+1}}{{\widehat{b}}_0}}.$$

(15)

By ignoring the estimation error of ${\tilde{z}}_{n+1}$ with respect to the total disturbance f, the initial model is capable of being approximated as a series of integrals structure from

$$y^{\left(n\right)}=f-{\tilde{z}}_{n+1}+{\widehat{u}}_0\approx {\widehat{u}}_0.$$

(16)

The approximate system in Equation (16) is able to be readily governed by applying a PD controller

$${\widehat{u}}_0={\tilde{k}}_p\left(r-{\tilde{z}}_1\right)+{\tilde{k}}_{d_1}\left[r^{\left(1\right)}-{\tilde{z}}_2\right]+\cdots +{\tilde{k}}_{d_{n-1}}\left[r^{\left(n-1\right)}-{\tilde{z}}_n\right],$$

(17)

where r is the reference input of the system and $r^{\left(n-1\right)}$ is the $\left(n-1\right)$th-order derivative of r. ${\tilde{k}}_p$ and ${\tilde{k}}_{d_i},i=1,2,\cdots ,n-1$ serve as the gain parameters of the controller.

Based on the above analysis, a first-order ADRC can be designed as shown in Figure 4 if the system is of the following form:

$$y^{\left(1\right)}\left(t\right)=g\left(y\left(t\right),y^{\left(2\right)}\left(t\right),\cdots ,y^{\left(n\right)}\left(t\right),\omega \left(t\right)\right)+\widehat{b}\widehat{u}\left(t\right).$$

(18)

Define the total disturbance as

$$f=g\left(y\left(t\right),y^{\left(2\right)}\left(t\right),\cdots ,y^{\left(n\right)}\left(t\right),\omega \left(t\right)\right)+\left(\widehat{b}-{\widehat{b}}_0\right)\widehat{u}\left(t\right).$$

(19)

Then, ${\tilde{x}}_1=y,{\tilde{x}}_2=f$ are selected as state variables, and according to Equation (11), one can obtain

$$\left\{\begin{array}{c}{\dot{\tilde{x}}}_1={\tilde{x}}_2+{\widehat{b}}_0\widehat{u}\hfill \\ {\dot{\tilde{x}}}_2=\dot{f}\hfill \\ y={\tilde{x}}_1\hfill \end{array}\right..$$

(20)

Thus, the ESO of Equation (18) is able to be constructed as

$$\left\{\begin{array}{c}{\dot{\tilde{z}}}_1={\tilde{z}}_2+{\tilde{\beta }}_1\left(y-{\tilde{z}}_1\right)+{\widehat{b}}_0\widehat{u}\hfill \\ {\dot{\tilde{z}}}_2={\tilde{\beta }}_2\left(y-{\tilde{z}}_1\right)\hfill \\ y={\tilde{x}}_1\hfill \end{array}\right..$$

(21)

According to the bandwidth-parameterization method, the following conclusion can be obtained:

$$\left\{\begin{array}{c}{\tilde{\beta }}_1=2{\tilde{\omega }}_o\hfill \\ {\tilde{\beta }}_2={\tilde{\omega }}_o^2\hfill \end{array}\right..$$

(22)

And the SFCL is specified as

$$\widehat{u}={\displaystyle \frac{{\widehat{u}}_0-{\tilde{z}}_2}{{\widehat{b}}_0}}.$$

(23)

By ignoring the estimation error of ${\tilde{z}}_2$ with respect to the total disturbance f, Equation (18) becomes

$$y^{\left(1\right)}=f-{\tilde{z}}_2+{\widehat{u}}_0\approx {\widehat{u}}_0.$$

(24)

According to Equation (17), one can have

$${\widehat{u}}_0={\tilde{k}}_p\left(r-{\tilde{z}}_1\right).$$

(25)

where ${\tilde{k}}_p$ is the gain of the SFCL.

3.2. The Design of MADRC for $x_B$ and $x_D$

In this section, a MADRC based on ADRC is proposed by adding a compensation part, as shown in Figure 5. A typical high-order inertia system is defined as

$$G\left(s\right)={\displaystyle \frac{K}{{\left(Ts+1\right)}^n}},$$

(26)

where K, T, and $n,n\ge 3$ are the system gain, time constant, and system order, respectively.

The compensation part of MADRC is presented in the red box in Figure 5. In this context, the control signal u passes through the compensation part prior to reaching the ESO. The compensation part, which is chosen in accordance with Equation (26), is presented as

$$G_{cp}\left(s\right)={\displaystyle \frac{1}{{\left(Ts+1\right)}^{n-1}}}.$$

(27)

The inherent dynamic characteristics of the high-order inertia system give rise to a time lag in the output. Subsequently, $G_{cp}\left(s\right)$ synchronizes u with y before they enter the ESO, thus enabling the ESO to provide a more accurate estimate of the state of the system in Equation (26).

In this scenario, the ESO shares a similar structure to that of the regular linear ADRC. However, the control signal is substituted with the output of $G_{cp}\left(s\right)$, denoted as $u_{cp}$. Thus, the ESO of Equation (26) is able to be constructed as

$$\left\{\begin{array}{c}{\dot{\widehat{z}}}_1={\widehat{z}}_2+{\beta }_1\left(y-{\widehat{z}}_1\right)+{\tilde{b}}_0{u}_{cp}\hfill \\ {\dot{\widehat{z}}}_2={\beta }_2\left(y-{\widehat{z}}_1\right)\hfill \end{array}\right.,$$

(28)

where ${\tilde{b}}_0$ is utilized to signify the approximation of the input gain parameter. When the parameters of the ESO are suitably tuned, the output ${\widehat{z}}_2$ can track and compensate f well through the SFCL, that is

$$u={\displaystyle \frac{u_0-{\widehat{z}}_2}{{\tilde{b}}_0}},$$

(29)

and

$$u_0=k_p\left(r-{\widehat{z}}_1\right),$$

(30)

where $k_p$ is the gain of the SFCL.

It should be noted that MADRC can make use of the bandwidth-parameterization approach. Based on this, the subsequent conclusion is derived:

$$\left\{\begin{array}{c}{\beta }_1=2{\omega }_o\hfill \\ {\beta }_2={\omega }_o^2\hfill \end{array}\right..$$

(31)

Evidently, there are no extra parameters within MADRC that need tuning. MADRC adopts the advantages of ADRC, such as a simple structure and easy implementation.

According to Equation (1), the following form can be obtained as

$${\displaystyle \frac{d{x}_B}{dt}}={\displaystyle \frac{L_2\left(x_2-x_B\right)+V_B\left(x_B-y_1\right)}{M_B}},$$

(32)

where $x_B$ is directly controlled by $V_B$. In the proximity of stable operation conditions of the distillation column, at $t=100$ min, with $L_T$ kept constant, a step change of $-0.8$ times the initial value is applied to $V_B$. Dynamic data of $x_B$ are collected. For the loop with $V_B$ as the input and $x_B$ as the output, denoted as the loop $x_B$, a fitting technique is employed to establish a high-order inertial linear model for characterizing its dynamic behavior. The detailed fitting effect is shown in Figure 6. Subsequently, the loop $x_B$ is obtained as

$$G_{x_B}\left(s\right)=-{\displaystyle \frac{0.31}{{\left(3s+1\right)}^3}}.$$

(33)

It is evident that the model boasts sufficient accuracy and is well suited for the design of the compensation part $G_{cp{x}_B}\left(s\right)$. Then, the mathematical expression of $G_{cp{x}_B}\left(s\right)$ is

$$G_{cp{x}_B}\left(s\right)={\displaystyle \frac{1}{{\left(3s+1\right)}^2}}.$$

(34)

For Equation (33), the ESO is able to be constructed as

$$\left[\begin{array}{c}{\dot{\widehat{z}}}_1\\ {\dot{\widehat{z}}}_2\end{array}\right]=\left[\begin{array}{cc}-{\beta }_1& 1\\ -{\beta }_2& 0\end{array}\right]\left[\begin{array}{c}{\widehat{z}}_1\\ {\widehat{z}}_2\end{array}\right]+\left[\begin{array}{cc}{\tilde{b}}_0& {\beta }_1\\ 0& {\beta }_2\end{array}\right]\left[\begin{array}{c}{u}_{cp}\\ y\end{array}\right].$$

(35)

Then, the SFCL is selected as

$$u={\displaystyle \frac{u_0-{\widehat{z}}_2}{{\tilde{b}}_0}},$$

(36)

and

$$u_0=k_p\left(r-{\widehat{z}}_1\right).$$

(37)

Likewise, in the proximity of stable operation conditions of the distillation column, at $t=100$ min, with $V_B$ kept constant, a step change of $-0.1$ times the initial value is applied to $L_T$. Dynamic data of $x_D$ are collected. For the loop with $L_T$ as the input and $x_D$ as the output, denoted as the loop $x_D$, a fitting technique is employed to establish a high-order inertial linear model for characterizing its dynamic behavior. The detailed fitting effect is shown in Figure 7. Subsequently, the loop $x_D$ is obtained as

$$G_{x_D}\left(s\right)={\displaystyle \frac{0.245}{{\left(5.5s+1\right)}^3}}.$$

(38)

It is evident that the model boasts sufficient accuracy and is well suited for the design of the compensation part $G_{cp{x}_D}\left(s\right)$. Then, the mathematical expression of $G_{cp{x}_D}\left(s\right)$ is

$$G_{cp{x}_D}\left(s\right)={\displaystyle \frac{1}{{\left(5.5s+1\right)}^2}}.$$

(39)

The design of the ESO and the control law is the same as above.

Note that in this section, the nonlinear distillation column model is approximated as linear models in Equations (33) and (38) specifically to design the linear compensation parts in Equations (34) and (39) for the dual-loop MADRC scheme. It is crucial to emphasize that although this approximation is made for design purposes, all simulation experiments of the four control schemes presented in this paper are rigorously conducted for the original nonlinear distillation column model, as described in Section 2.

3.3. Stability Analysis of MADRC

In this subsection, the stability analysis of MADRC is conducted. According to Section 3.2, the state-space representation of the ESO of MADRC can be written as

$$\left\{\begin{array}{c}\dot{\widehat{Z}}=A\widehat{Z}+B{u}_{cp}+L\left(y-{\widehat{z}}_1\right)\hfill \\ y=C\widehat{Z}\hfill \end{array}\right..$$

(40)

where $\widehat{Z}={\left[{\widehat{z}}_1,{\widehat{z}}_2\right]}^T$, $L={\left[{\beta }_1,{\beta }_2\right]}^T={\left[2{\omega }_o,{\omega }_o^2\right]}^T$, $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, $B=\left[\begin{array}{c}{\tilde{b}}_0\\ 0\end{array}\right]$ and $C=\left[1,0\right]$.

In this paper, $\delta$ is defined as the feedback control error, and $\tilde{\rho }$ is defined as the ESO’s state estimation error [19]. Suppose the value of the reference input r remains unchanged, that is, $\dot{r}=0$. Then, the closed-loop dynamic error equation of MADRC can be written as

$$\left\{\begin{array}{c}\dot{\delta }=A_{cf}\delta +K_f\tilde{\rho }\hfill \\ \dot{\tilde{\rho }}=A_{eso}\tilde{\rho }+E\eta \hfill \end{array}\right.,$$

(41)

where $\delta =\left[{\delta }_1\right]=\left[x_1-r\right]$, ${\tilde{\rho }}_i=x_i-z_i\left(i=1,2\right)$, $\tilde{\rho }={\left[\begin{array}{cc}{\tilde{\rho }}_1& {\tilde{\rho }}_2\end{array}\right]}^T={\left[\begin{array}{cc}{x}_1-z_1& x_2-z_2\end{array}\right]}^T$, $A_{cf}=\left[-{\omega }_c\right]$, $K_f=\left[\begin{array}{cc}{\omega }_c& 1\end{array}\right]$, $A_{eso}=\left[\begin{array}{cc}-2{\omega }_o& 1\\ -{\omega }_o^2& 0\end{array}\right]$, $E=\left[\begin{array}{c}0\\ 1\end{array}\right]$ and $\eta$ is the differential of the total disturbance f.

As can be observed from Equation (41), given that the observer bandwidth ${\omega }_o$ is typically several times that of the feedback control bandwidth ${\omega }_c$, the error dynamics can be divided into two components: $\tilde{\rho }$ of the ESO with a relatively fast response, and $\delta$ with a relatively sluggish response property. This system with error dynamics featuring both rapid and sluggish components is a typical characteristic of singular perturbation systems. As a result, in this paper, the utilization of singular perturbation theory is considered to demonstrate the stability of MADRC.

To convert Equation (41) into the standard singular perturbation model, let the state estimation error $\tilde{\rho }=diag\left[\begin{array}{cc}{\omega }_o^{-1}& 1\end{array}\right]\rho$ of the ESO and the singular parameter be selected as $ϵ={\omega }_o^{-1}$, so the error dynamics of MADRC can be finally written as

$$\left\{\begin{array}{c}\dot{\delta }=A_{cf}\delta +K_{cf}\rho \hfill \\ ϵ\dot{\rho }=A_z\rho +ϵE\eta \hfill \end{array}\right.,$$

(42)

where $A_z=\left[\begin{array}{cc}-2& {\omega }_o^{-1}\\ -{\omega }_o& 0\end{array}\right]$, $K_{cf}=K_{f}diag\left[\begin{array}{cc}{\omega }_o^{-1}& 1\end{array}\right]=\left[\begin{array}{cc}{\omega }_c{\omega }_o^{-1}& 1\end{array}\right]$.

As per the singular perturbation theory, the standard singular perturbation model is

$$\left\{\begin{array}{c}\dot{p}=f\left(p,q,ϵ,t\right),p\left(t_0\right)=p^0\hfill \\ ϵ\dot{q}=h\left(p,q,ϵ,t\right),q\left(t_0\right)=q^0\hfill \end{array}\right.,$$

(43)

where $p\in R^n$ and $q\in R^m$.

By setting the singular parameter $ϵ$ to zero, one can determine the quasi-steady-state solutions $\overline{p}$ and $\overline{q}$ pertaining to the system.

$$\begin{array}{c}0=h\left(\overline{p},\overline{q},0,t\right)\Rightarrow \overline{q}=\phi \left(\overline{p},t\right),\overline{p}=f\left(\overline{p},\phi \left(\overline{p},t\right),0,t\right)\hfill \end{array}.$$

(44)

The specific solutions p and q can be represented via the two-time-scale asymptotic expansions. Here, the quasi-steady components $\overline{p}$ and $\overline{q}$ are formulated in the t-scale, while the transient elements $\widehat{p}\left(\tau \right)$ and $\widehat{q}\left(\tau \right)$ are specified in the $\tau$-scale.

$$\left\{\begin{array}{c}p=\overline{p}\left(t\right)+\widehat{p}\left(\tau \right)\hfill \\ q=\overline{q}\left(t\right)+\widehat{q}\left(\tau \right)\hfill \end{array}\right.,$$

(45)

where $\widehat{q}\left(\tau \right)$ is known as the boundary-layer subsystem, whose dynamic characteristic is

$${\displaystyle \frac{d\widehat{q}}{d\tau }}=h\left(p^0,\widehat{q}\left(\tau \right)+\overline{q}\left(t_0\right),0,t_0\right),$$

(46)

Then, the key theorem from [19] in the domain of singular perturbation theory is presented.

Assumption 1.

The balance of the boundary-layer subsystem is uniformly asymptotically stable with respect to $p^0$ and $t_0$, and the quantity $q^0-\overline{q}\left(t_0\right)$ lies within its region of attraction.

Assumption 2.

For $ϵ=0$, along the trajectories of $\overline{p}$ and $\overline{q}$, the real parts of the eigenvalues of $\partial h/\partial q$ are less than a fixed negative constant. That is, $Re\lambda \left\{\partial h/\partial q\right\}\le -c\le 0$.

Theorem 1.

Provided that Assumption 1 and Assumption 2 hold, ${\displaystyle \underset{\tau \to 0}{lim}}\widehat{q}\left(\tau \right)=0$. Then, for all $t\in \left[t_0,\infty \right]$, the approximations $p=\overline{p}\left(t\right)+O\left(ϵ\right)$ and $q=\overline{q}\left(t\right)+\widehat{q}\left(\tau \right)+O\left(ϵ\right)$ hold true. Additionally, there exists a time $t_1\ge t_0$, such that $q=\overline{q}\left(t_1\right)+O\left(ϵ\right)$.

Proof of Theorem 1.

Then, this paper examines whether the closed-loop error dynamic equation of MADRC meets the above two assumptions, aiming to prove the stability of MADRC. The closed-loop error dynamic system in Equation (42) is written in the form of Equation (43) as

$$\left\{\begin{array}{c}\dot{\delta }=A_{cf}\delta +K_{cf}\rho =f\left(\delta ,\rho ,ϵ,t\right)\delta \left(t_0\right)={\delta }^0\hfill \\ ϵ\dot{\rho }=A_z\rho +ϵE\eta =h\left(\delta ,\rho ,ϵ,t\right)\rho \left(t_0\right)={\rho }^0\hfill \end{array}\right.,$$

(47)

where $\delta \in R$ and $\rho \in R^2$.

Consequently, the quasi-steady-state solutions are

$$\begin{array}{c}\left\{\begin{array}{c}0=h\left(\overline{\delta },\overline{\rho },0,t\right)=A_z\overline{\rho }+0·E\eta \hfill \\ \dot{\overline{\delta }}=f\left(\overline{\delta },{\overline{\phi }}_i\left(\overline{\delta },t\right),0,t\right)=A_{cf}\overline{\delta }+K_{cf}·0\hfill \end{array}\right.\Rightarrow \left\{\begin{array}{c}\overline{\rho }={\overline{\phi }}_i\left(\overline{\delta },t\right)={\left[\begin{array}{cc}0& 0\end{array}\right]}^T\hfill \\ \dot{\overline{\delta }}=A_{cf}\overline{\delta }\hfill \end{array}\right.\hfill \end{array}.$$

(48)

Besides, the boundary-layer subsystem exists as

$$\begin{array}{c}{\displaystyle \frac{d\widehat{\rho }}{d\tau }}=h\left({\delta }^0,\widehat{\rho }\left(\tau \right)+\overline{\rho }\left(t_0\right),0,t_0\right)=A_z\left(\widehat{\rho }\left(\tau \right)+\overline{\rho }\left(t_0\right)\right)+0·E\eta \Rightarrow {\displaystyle \frac{d\widehat{\rho }}{d\tau }}=A_z\widehat{\rho }\left(\tau \right)\hfill \end{array}.$$

(49)

The balance points of the boundary-layer subsystem are $\left[\begin{array}{cc}{\widehat{\rho }}_1& {\widehat{\rho }}_2\end{array}\right]=\left[\begin{array}{cc}0& 0\end{array}\right]$, and the balance points are uniformly asymptotically stable. Suppose that the initial tracking error of the ESO is 0, i.e., ${\rho }^0=0$, so ${\rho }^0-\overline{\rho }\left(t_0\right)$ lies within the attraction domain of the balance points. Assumption 1 holds true.

For the singular parameter $ϵ=0$, along $\delta$ and $\overline{\rho }$, $\mathrm{Re}\lambda \left\{\partial h/\partial q\right\}=\mathrm{Re}\lambda \left\{A_z\right\}$. Thereafter, the eigenvalues of $A_z$ are calculable from

$$\begin{array}{c}\left|sI-A_z\right|=\left|\begin{array}{cc}s+2& -{\omega }_o^{-1}\\ {\omega }_o& s\end{array}\right|={\left(s+1\right)}^2\Rightarrow s_{1,2}=-1\hfill \end{array}.$$

(50)

So, $\mathrm{Re}\lambda \left\{A_z\right\}={\left[\begin{array}{cc}-1& -1\end{array}\right]}^T<0$. Assumption 2 holds true.

According to Theorem 1, $\delta =\overline{\delta }\left(t\right)+O\left(ϵ\right)$ and $\rho =\overline{\rho }\left(t\right)+O\left(ϵ\right)$ hold true in the time range $t_1\ge t_0$. And because of $\overline{\rho }\left(t\right)=0$, $\rho =O\left(ϵ\right)$ is true in $t_1\ge t_0$. For $\overline{\delta }\left(t\right)$, since $\lambda \left\{A_{cf}\right\}=-{\omega }_c<0$, $\overline{\delta }\left(t\right)$ is uniformly asymptotically stable, that is, $\underset{t\to \infty }{lim}\overline{\delta }\left(t\right)=0$, so $\underset{t\to \infty }{lim}\overline{\delta }\left(t\right)=O\left(t\right)$.

In summary, when the value of ${\omega }_o$ is relatively large and the singular parameter $ϵ={\omega }_o^{-1}$ is a small positive number, then there are bandwidth parameters ${\omega }_c$ and ${\omega }_o$, making Equation (42) controlled by MADRC uniformly asymptotically stable. □

4. The Tuning Rule of the Dual-Loop MADRC Scheme

Considering the structural disparities between the dual-loop MADRC scheme and the regular ADRC, the parameters of the former exert different influences on control performance. Accordingly, the parameter tuning for the dual-loop MADRC scheme presents its own set of specific features. To determine the optimal parameters’ range of the dual-loop MADRC scheme, the single-variable method is utilized to explore the behavior of control performance with various MADRC parameter settings. Note that the reference input of $x_D$ increases from $0.99$ to $0.995$ at $t=10$ min. At $t=100$ min, the reference input of $x_B$ decreases from $0.01$ to $0.005$ at $t=100$ min. And the disturbance is configured as follows: F is increased from 1 kmol/min to $1.2$ kmol/min at $t=200$ min. The outcomes concerning the controlling effectiveness under diverse values of ${\tilde{b}}_0$, ${\omega }_o$ and $k_p$ are illustrated within Figure 8, Figure 9 and Figure 10, respectively.

As depicted in Figure 8, when the value of ${\tilde{b}}_0$ is relatively low, it brings about a more powerful control force, but the fluctuations become more prominent. On the contrary, a relatively high value of ${\tilde{b}}_0$ causes a weaker control force and less intense fluctuations. By extensive simulation experiments, the optimal range of parameter ${\tilde{b}}_{01}$ in the loop $x_D$ is determined to be ${\tilde{b}}_{01}\in \left(0.0005,0.001\right]$, and the optimal range of parameter ${\tilde{b}}_{02}$ in the loop $x_B$ is ${\tilde{b}}_{02}\in \left[-0.001,-0.0005\right)$. Additionally, an inverse relationship between the values of ${\tilde{b}}_{01}$ and ${\tilde{b}}_{02}$ is observed, meaning their absolute values are equal while their signs are opposite.

As shown in Figure 9, when ${\omega }_o$ is small, the outputs of the loop $x_D$ and $x_B$ exhibit substantial fluctuations. Conversely, when ${\omega }_o$ is sufficiently large, the outputs exhibit negligible alterations. Given that a larger ${\omega }_o$ enhances the susceptibility to measurement noise and increases the computational stress upon the ESO, the value of ${\omega }_o$ needs to be adjusted to an appropriate value that ensures minimal output variation. Through extensive simulation experiments, the optimal range of parameter ${\omega }_{o1}$ in the loop $x_D$ is determined as ${\omega }_{o1}\in \left[15,25\right]$, and the optimal range of parameter ${\omega }_{o2}$ in the loop $x_B$ is determined as ${\omega }_{o2}\in \left[7.5,12.5\right]$. Moreover, it turns out that ${\omega }_{o1}=2{\omega }_{o2}$.

From Figure 10, when $k_p$ is relatively large, the loop $x_D$ and the loop $x_B$ show better disturbance rejection capabilities. However, the system output experiences more severe oscillations when tracking changes in the input signal. After a lot of simulation experiments, the optimal range of parameter $k_{p1}$ in the loop $x_D$ is determined as $k_{p1}\in \left(0.4,0.8\right]$, and the optimal range of parameter $k_{p2}$ in the loop $x_B$ is determined as $k_{p2}\in \left(0.75,1.5\right]$.

5. Simulation and Experiment Validation

5.1. Simulation Validations

The following analysis focuses on comparing the control performance of four control schemes under three operating conditions (A, B, and C): the dual-loop MADRC, cascade PID control (CPID), regular ADRC (ADRC), and PID control (PID). Among them, MADRC is tuned according to Section 4. ADRC is tuned by [13]. The parameters of CPID can be found in [11]. The parameters of PID are tuned via the trial-and-error method, adhering to these three principles: (1) With no overshoot or an overshoot less than $5\%$, the outputs $x_D$ and $x_B$ of the distillation column should respond as swiftly as possible. (2) The outputs $x_D$ and $x_B$ should remain free from significant oscillations. (3) PID controller parameters must exhibit adequate robustness across various operating conditions to ensure consistent control performance of the distillation column. Each controller’s tuned parameters are presented in

Table 2. The tuned parameters of four control schemes for the loop $x_D$.

Control Scheme	Tuned Parameters
MADRC	${\omega }_{o1}=20,k_{p1}=0.8,{\tilde{b}}_{01}=0.001,G_{cp{x}_B}\left(s\right)={\displaystyle \frac{1}{{\left(5.5s+1\right)}^2}}$
CPID	outer loop: ${\mathring{k}}_{p1}=0.28,{\mathring{k}}_{i1}=0.55,{\mathring{k}}_{d1}=0.56,N_1=0.33$
	inner loop: ${\mathring{k}}_{p2}=120.65,{\mathring{k}}_{i2}=81.45,{\mathring{k}}_{d2}=36.44,N_2=7.11$
ADRC	${\tilde{\omega }}_{o1}=25,{\tilde{k}}_{p1}=0.5,{\widehat{b}}_{01}=0.2$
PID	${\overline{k}}_{p1}=26.1,{\overline{k}}_{i1}=6.94,{\overline{k}}_{d1}=0$

and

Table 3. The tuned parameters of four control schemes for the loop $x_B$.

Control Scheme	Tuned Parameters
MADRC	${\omega }_{o2}=10,k_{p2}=1.5,{\tilde{b}}_{02}=-0.001,G_{cp{x}_D}\left(s\right)={\displaystyle \frac{1}{{\left(3s+1\right)}^2}}$
CPID	$k_{pB}=-153.51,k_{iB}=-92.80,k_{dB}=-35.03,N_b=2.36$
ADRC	${\tilde{\omega }}_{o2}=20,{\tilde{k}}_{p2}=0.5,{\widehat{b}}_{02}=-0.1$
PID	${\overline{k}}_{p2}=-37.5,{\overline{k}}_{i2}=-11.33,{\overline{k}}_{d2}=0$

. Note that the parameters of the four control schemes are fixed parameters and do not change with the variation in operating conditions.

Firstly, the tracking performance of the four control schemes under the three operating conditions is compared. Operating condition A: the reference input of $x_D$ increases from $0.99$ to $0.995$ at $t=10$ min, while the reference input of $x_B$ decreases from $0.01$ to $0.005$ at $t=100$ min. Operating condition B: the reference input of $x_D$ increases from $0.98$ to $0.985$ at $t=10$ min, while the reference input of $x_B$ decreases from $0.02$ to $0.015$ at $t=100$ min. Operating condition C: the reference input of $x_D$ increases from $0.985$ to $0.995$ at $t=10$ min, while the reference input of $x_B$ decreases from $0.015$ to $0.005$ at $t=100$ min. The comparison results of four control schemes are set forth in Figure 11, Figure 12 and Figure 13.

Subsequently, the disturbance rejection performance (external disturbances and model uncertainties) of four control schemes under three operating conditions is compared. Operating condition A: the reference input of $x_D$ is $0.99$, while the reference input of $x_B$ is $0.01$. Operating condition B: the reference input of $x_D$ is $0.98$, while the reference input of $x_B$ is $0.02$. Operating condition C: the reference input of $x_D$ is $0.985$, while the reference input of $x_B$ is $0.015$. The following are the settings for external disturbances and model uncertainties. Specifically, at $t=50$ min, a step disturbance with amplitude $-0.3$ is imposed on $L_T$, and at $t=200$ min, another with amplitude $0.3$ is applied to $V_B$. Model uncertainty is considered to arise from the divergence between the process-related models and the controller models, which is caused by the perturbation of system parameters. The changes in system parameters are: at $t=300$ min, $z_F$ rises from $0.5$ to $0.7$ and F grows from 1 kmol/min to $1.2$ kmol/min at $t=400$ min. The control performance of the control schemes is depicted in Figure 14, Figure 15 and Figure 16.

5.2. Discussions

As can be seen from Figure 11, Figure 12 and Figure 13, the dual-loop MADRC scheme attains the fastest equilibrium state, the minimum overshoot, and the smallest fluctuation amplitude of $x_D$ and $x_B$ under all three working conditions, demonstrating the most excellent tracking performance. Compared with the dual-loop MADRC scheme, the CPID control scheme also demonstrates a relatively high response speed, but there is always a large oscillation when $x_B$ tracks the reference input. In contrast, the PID control scheme displays the slowest response speed, the poorest tracking efficacy, and the maximum overshoot.

From Figure 14, Figure 15 and Figure 16, it can be seen that the dual-loop MADRC scheme exhibits the optimal disturbance rejection performance under all three operating conditions. The ESO and the compensation part within this scheme can estimate model uncertainties and external disturbances instantaneously and implement compensations. Consequently, the dual-loop MADRC scheme demonstrates the ability to counteract model uncertainties and external disturbances.

To make the advantages of the dual-loop MADRC scheme more prominent, the integral absolute error (IAE) indexes of four control schemes under three operating conditions are calculated. These IAE indexes are presented in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. Among them, ${\mathrm{IAE}}_{\mathrm{sp}}$ is an index of tracking performance, ${\mathrm{IAE}}_{\mathrm{ud}}$ is an index of anti-interference performance, and ${\mathrm{IAE}}_{\mathrm{sum}}$ is a comprehensive performance index of the sum of ${\mathrm{IAE}}_{\mathrm{sp}}$ and ${\mathrm{IAE}}_{\mathrm{ud}}$. As can be seen from

Table 4. IAE indexes of four control schemes under operating condition A.

Control Scheme	${\mathbf{IAE}}_{\mathbf{sp}}$		${\mathbf{IAE}}_{\mathbf{ud}}$		${\mathbf{IAE}}_{\mathbf{sum}}$
	${\mathit{x}}_{\mathit{D}}$	${\mathit{x}}_{\mathit{B}}$	${\mathit{x}}_{\mathit{D}}$	${\mathit{x}}_{\mathit{B}}$
MADRC	$0.0165$	$0.0069$	$0.0123$	$0.0050$	$0.0407$
CPID	$0.0183$	$0.0151$	$0.0060$	$0.0146$	$0.0540$
ADRC	$0.0409$	$0.0291$	$0.0374$	$0.0321$	$0.1395$
PID	$0.0939$	$0.0586$	$0.1659$	$0.1402$	$0.4586$

,

Table 5. IAE indexes of four control schemes under operating condition B.

Control Scheme	${\mathbf{IAE}}_{\mathbf{sp}}$		${\mathbf{IAE}}_{\mathbf{ud}}$		${\mathbf{IAE}}_{\mathbf{sum}}$
	${\mathit{x}}_{\mathit{D}}$	${\mathit{x}}_{\mathit{B}}$	${\mathit{x}}_{\mathit{D}}$	${\mathit{x}}_{\mathit{B}}$
MADRC	$0.0102$	$0.0050$	$0.0111$	$0.0045$	$0.0308$
CPID	$0.0153$	$0.0065$	$0.0053$	$0.0128$	$0.0399$
ADRC	$0.0259$	$0.0197$	$0.0323$	$0.0278$	$0.1057$
PID	$0.0475$	$0.0285$	$0.1604$	$0.1327$	$0.3691$

and

Table 6. IAE indexes of four control schemes under operating condition C.

Control Scheme	${\mathbf{IAE}}_{\mathbf{sp}}$		${\mathbf{IAE}}_{\mathbf{ud}}$		${\mathbf{IAE}}_{\mathbf{sum}}$
	${\mathit{x}}_{\mathit{D}}$	${\mathit{x}}_{\mathit{B}}$	${\mathit{x}}_{\mathit{D}}$	${\mathit{x}}_{\mathit{B}}$
MADRC	$0.0280$	$0.0123$	$0.0115$	$0.0047$	$0.0565$
CPID	$0.0342$	$0.0238$	$0.0056$	$0.0135$	$0.0771$
ADRC	$0.0695$	$0.0502$	$0.0346$	$0.0296$	$0.1839$
PID	$0.1396$	$0.0867$	$0.1659$	$0.1375$	$0.5297$

, the ${\mathrm{IAE}}_{\mathrm{sum}}$ of the dual-loop MADRC scheme is the smallest, indicating it has the best comprehensive control performance.

The Monte Carlo method, grounded in the law of large numbers, conducts repeated simulations by generating sequences of random numbers [24]. It transforms complex systems into stochastic process modeling and numerical computations, effectively reducing the complexity of research. For a more in-depth evaluation of the four control schemes’ robustness, this paper introduces Monte Carlo trials. Keeping the controller parameters unaltered, random perturbations of $\pm 20\%$ around the nominal values are applied to the original values of $M_D$ and $M_B$, respectively. Additionally, random perturbations at $\pm 20\%$ of their nominal values are performed on the relative volatility $\alpha$ and time constant $\tau$. Moreover, there is a gain uncertainty that varies within $\pm 20\%$ for each input channel. The disturbances and uncertainties configured in the Monte Carlo trials are meticulously designed to faithfully reproduce real operational scenarios. Note that the reference input of $x_D$ increases from $0.99$ to $0.995$ at $t=10$ min. At $t=100$ min, the reference input of $x_B$ decreases from $0.01$ to $0.005$ at $t=100$ min. The integral absolute error (IAE) indexes of four control schemes are collected from the start time to 300 min for each simulation. ${\mathrm{IAE}}_1$ functions as the evaluation index for the performance of the loop $x_D$, and the performance of the loop $x_B$ is measured by ${\mathrm{IAE}}_2$. Monte Carlo trials will be carried out randomly 800 times. The distribution of the IAE indexes for four control schemes is depicted in Figure 17. Among them, the smaller the IAE indexes are, the better the tracking and disturbance rejection capabilities will be, and the denser the distribution of the IAE indexes is, the better the performance robustness will be [18]. As clearly depicted in Figure 17, the dual-loop MADRC scheme exhibits the most concentrated distribution and the minimum values of the IAE indexes. Consequently, the dual-loop MADRC scheme can effectively maintain optimal control performance in the face of system uncertainties, thereby indicating strong application potential.

6. Conclusions

ADRC has been extensively utilized in diverse control systems. However, the performance of ADRC is suboptimal when it comes to dealing with systems characterized by significant time delays, such as high-purity distillation columns. To address this issue, this paper devises a dual-loop MADRC scheme and applies the MADRC algorithm to a high-purity distillation column. MADRC inherits the merits of ADRC and can handle the delay problem more proficiently by incorporating a compensation part. In the compensation part, the control signal is synchronized with the output of the high-purity distillation column before entering the ESO, thereby enabling the ESO to estimate the state more precisely. Furthermore, this paper utilizes singular perturbation theory to demonstrate the stability of MADRC. The control performance of the dual-loop MADRC scheme is contrasted against that of a cascade PID control scheme, a regular ADRC scheme, and a PID control scheme. The simulation results indicate that the dual-loop MADRC scheme proposed in this paper can effectively track the reference and exhibits superior disturbance rejection performance. Moreover, this is verified through Monte Carlo trials. In future research, interactions between MADRC and higher-level advanced process control techniques (such as MPC and nonlinear MPC) or real-time optimization (RTO) schemes will be investigated.