Modified active disturbance rejection control design with application to superheated steam temperature systems

Abstract

In industrial control systems, high-order inertial systems widely exist. However, how to design suitable control methods for these systems are facing many challenges, such as slow dynamic characteristics and weak anti-interference ability. This paper emphasizes the theoretical analysis and applications of modified active disturbance rejection control (MADRC) for high-order inertial systems. The stability analysis of MADRC is carried out by separating the high-order transfer function and using the dual-locus diagram method. Then, a simple and quantitative semi-empirical parameter tuning rule based on the expected adjustment time is provided. The effectiveness of the high-order inertial system with MADRC compared to other ADRCs and PIs is then verified by tracking and disturbance suppression simulations. MADRC with the proposed parameter tuning rule is applied to both primary and secondary superheated steam temperature systems of a 660 MW power plant. The running data show that the cascade control system with MADRC can obtain better control performance than the original control method, and show great application potential for more complex industrial control systems.

Keywords

modified active disturbance rejection control stability analyses the dual-locus diagram method tuning rule superheated steam temperature systems

Introduction

Active disturbance rejection control (ADRC), proposed by Prof Han,¹ is becoming an important control technique to deal with external disturbance and internal uncertainties. ADRC is an important technique for dealing with external interference and internal uncertainty. The extended state observer (ESO) is a core part of ADRC, which estimates internal uncertainties and external disturbance as total disturbance and then compensates in real-time by control actions. ADRC has been widely used in many fields.²

Compared with simple first-order systems, high-order systems^3,4 have more complex dynamic characteristics and more state variables, which is the common feature of the industrial field.⁵ The most commonly used control methods for high-order systems are proportional-integral-derivative (PID),⁶ model predictive control,⁷ adaptive control,⁸ and so on. However, when high-order systems are faced, it is not easy to analyze the control performance, implement control algorithms, and adjust their parameters.⁹

ADRC has satisfactory anti-interference ability and flexibility in various fields.¹⁰ Elbouchikhi et al.¹¹ proposed ADRC to improve the system’s resilience to load changes, power uncertainties, and faults. Wang et al.¹² proposed a method combining ADRC and vector resonance controller, which can effectively suppress harmonics and improve the anti-interference ability of the system. However, it needs to pay attention to parameter adjustment,¹³ robustness and stability analysis¹⁴ and real-time requirements in application, which should be comprehensively considered according to the characteristics and control objectives of specific systems. Therefore, on the basis of ADRC, a series of improvements are proposed. Tian et al.¹⁵ proposed an adaptive ADRC to suppress uncertain current ripples under complex conditions. Sun et al.¹⁶ presented a method combining feedforward and ADRC, applied to non-minimum phase systems, using a model-assisted observer to handle unknown disturbance and uncertainties. Wu et al.¹⁷ proposed an improved gain-scheduling based ADRC that shows promising application potential in the energy and chemical industries. Sun¹⁸ suggested a quantitative tuning method for the time-delay ADRC (TDADRC), utilizing a standard first-order plus delay (FOPTD) model. In order to improve the ability to suppress current interference, Luo¹⁹ proposed an embedded discrete-time repeated ADRC. Chen²⁰ proposed a predictor-based ADRC with delay robustness based on a wet flue gas desulfurization system. A Smith-like active disturbance rejection control (SLADRC) is proposed,²¹ its stability is analyzed theoretically, and a simple yet effective parameter tuning method is provided.

In summary, ADRC features a simple structure, but it has issues with input asynchrony in ESO and bandwidth limitation in high-order inertial systems. TDADRC simplifies the control logic and is suitable for time-delay systems, yet its application relies on an accurate mathematical model. SLADRC improves the control structure and is applicable to high-order inertial systems. However, its tracking performance and anti-disturbance capability are relatively poor.

In view of the limitations of the aforementioned control methods, as well as the problems commonly faced by high-order inertial systems, including difficulties in parameter tuning and the lack of rigorous stability analysis. This paper takes high-order inertial systems with modified active disturbance rejection control (MADRC) as the starting point, conducts in-depth research on its stability analysis, and proposes a simple, quantitative semi-empirical parameter tuning rule. The specific contributions of this paper are as follows.

A relatively novel stability theory is adopted to prove that superheated steam temperature systems (SST) controlled by MADRC can operate safely and stably.

A simple parameter tuning method is proposed, which only requires adjusting one parameter, reducing the operational complexity of SST controlled by MADRC.

The operation data indicates that control performance is satisfactory and can effectively improve the efficiency of SST controlled by MADRC.

The remaining sections of this paper are organized as follows. First, the design of MADRC is introduced. Subsequently, the system stability is derived by decomposing higher-order inertial transfer functions, and the system robustness is analyzed using maximum sensitivity ( $M_{s}$ ). Simulation results demonstrate the effectiveness of the proposed analytical approach, confirming that the control strategy achieves satisfactory tracking and disturbance rejection performance in cascaded control systems. Furthermore, practical implementation in a thermal power plant validates the feasibility of this control method for industrial applications.

MADRC design and tuning for a high-order inertial system

MADRC for a high-order system

A regular first-order ADRC schematic diagram is shown in Figure 1. State feedback control law (SFCL), the high-order inertial element $G_{p} (s)$ and ESO constitute a first-order ADRC. It can be converted to

\overset{\cdot}{y} = g (\begin{matrix} t, y, \overset{\cdot\cdot}{y}, \dots, y^{(n)}, d, w \end{matrix}) + bu

(1)

where $t$ , $d$ , $ω$ and $g (\begin{matrix} t, y, \overset{\cdot\cdot}{y}, \dots, y^{(n)}, d, w \end{matrix})$ denote the time-varying dynamic, external disturbance, internal uncertainties and the synthesis function.

Figure 1.

The block diagram of regular first-order ADRC.

$b$ is estimated by $b_{0}$ , which may be unknown, considering the actual input gain value of the equivalent first-order process. Then equation (1) can be simplified as

\overset{\cdot}{y} = f + b_{0} u

(2)

where $f = g (\begin{matrix} t, y, \overset{\cdot\cdot}{y}, \dots, y^{(n)}, d, w \end{matrix}) + (b - b_{0}) u$ is the total disturbance.

The system’s state space can be described as

{\begin{matrix} \overset{\cdot}{x} = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] x + [\begin{matrix} b_{0} \\ 0 \end{matrix}] u + [\begin{matrix} 0 \\ 1 \end{matrix}] \overset{\cdot}{f} \\ y = [\begin{matrix} 1 & 0 \end{matrix}] x \end{matrix}

(3)

where $u$ is the control signal and $x = {[\begin{matrix} x_{1} & x_{2} \end{matrix}]}^{T} = {[\begin{matrix} y & f \end{matrix}]}^{T}$ is the state vector.

The mathematical expression of the ESO is designed as

{\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} + b_{0} u + β_{1} (y - z_{1}) \\ {\overset{\cdot}{z}}_{2} = β_{2} (y - z_{1}) \end{matrix}

(4)

where $β_{1}$ and $β_{2}$ are the observer gains.

The SFCL is defined as

u = K_{c} (r - z) / b_{0}

(5)

where the input signal is $r = [\begin{matrix} r & 0 \end{matrix}]$ , observer state vector vector $z = [\begin{matrix} z_{1} & z_{2} \end{matrix}]$ and the controller gain vector is $K_{c} = [\begin{matrix} k_{p} & 1 \end{matrix}]$ .

The bandwidth parametric method proposed by Gao simplifies the design process of the controller.²² Therefore, the observer gain and the controller gain can be set as

β_{1} = 2 ω_{o}; β_{2} = ω_{o}^{2}; k_{p} = ω_{c}

(6)

where $ω_{o}$ is observer’s bandwidth and $ω_{c}$ is the controller’s bandwidth.

The block diagram of MADRC is shown in Figure 2, it is used to improve the control performance of a high-order inertial system, where $G_{cp} (s)$ is a compensation part.

Figure 2.

The block diagram of MADRC.

A class of a high-order inertial element can be defined as

G_{p} (s) = \frac{K}{{(Ts + 1)}^{n}}

(7)

where $T$ is the time constant, $K$ is the gain of this high-order system, and $n \geq$ 3.

To achieve the purpose of synchronizing the control signal applied to the ESO with the system output, it is necessary to delay the control signal, that is, to add a compensation part before entering the ESO. It is defined as

G_{cp} (s) = \frac{1}{{(Ts + 1)}^{n - 1}}

(8)

Slightly different from the regular ADRC, MADRC has no additional parameters to adjust, and is as simple and easy to implement as ADRC. The ESO portion of the MADRC transforms into the following form

{\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} + b_{0} u_{f} + β_{1} (y - z_{1}) \\ {\overset{\cdot}{z}}_{2} = β_{2} (y - z_{1}) \end{matrix}

(9)

where $u_{f}$ is the output of the compensation part, the control law is the same as the regular ADRC.

Equivalent transformation

The MADRC structure in Figure 2 needs to be converted into a two-degree-of-freedom (2-DOF) structure for subsequent stability analysis and stability analysis, as shown in Figure 3.

Figure 3.

Regular equivalent 2-DOF structure of MADRC.

By applying an appropriate Laplace transform to the function, equation (9) can be derived as follows

[\begin{matrix} z_{1} (s) \\ z_{1} (s) \end{matrix}] = [\begin{matrix} \frac{b_{0} s}{{(s + ω_{o})}^{2}} & \frac{2 ω_{o} s + ω_{o}^{2}}{{(s + ω_{o})}^{2}} \\ \frac{b_{0} ω_{o}^{2}}{{(s + ω_{o})}^{2}} & \frac{ω_{o}^{2} s}{{(s + ω_{o})}^{2}} \end{matrix}] [\begin{matrix} U_{f} (s) \\ Y (s) \end{matrix}]

(10)

Through proper transformation of the control system, the transfer function of each controller can be converted to

G_{c} (s) = \frac{k_{p} (s^{2} + 2 ω_{o} s + ω_{o}^{2})}{b_{0} \cdot A (s)}

(11)

G_{h} (s) = \frac{(2 k_{p} ω_{o} + ω_{o}^{2}) s + k_{p} ω_{o}^{2}}{k_{p} (s^{2} + 2 ω_{o} s + ω_{o}^{2})}

(12)

where polynomials $A (s)$ is

A (s) = s^{2} + 2 ω_{o} s + k_{p} {sG}_{cp} (s) + ω_{o}^{2} [1 - G_{cp} (s)]

(13)

Stability analysis

Without considering interference, equations (7) and (8) can been converted to

Y (s) = G_{p} (s) \cdot U (s) = \frac{K \cdot G_{cp} (s)}{Ts + 1} U (s) = \frac{K}{Ts + 1} U_{f} (s)

(14)

As shown in Figure 4, it is the same as a first-order control object and an equivalent 2-DOF structure of MADRC can be derived as

{\tilde{G}}_{p} (s) = \frac{K}{Ts + 1}

(15)

{\tilde{G}}_{c} (s) = \frac{k_{p} (s^{2} + 2 ω_{o} s + ω_{o}^{2}) \cdot G_{cp} (s)}{b_{0} (s^{2} + 2 ω_{o} s + ω_{o}^{2}) + (b_{0} k_{p} s - b_{0} ω_{o}^{2}) \cdot G_{cp} (s)}

(16)

{\tilde{G}}_{h} (s) = \frac{(2 k_{p} ω_{o} + ω_{o}^{2}) s + k_{p} ω_{o}^{2}}{k_{p} (s^{2} + 2 ω_{o} s + ω_{o}^{2})}

(17)

Figure 4.

Another equivalent 2-DOF structure of MADRC.

Through the above three formulas, the closed-loop characteristic equation of the system can be transformed into equation (18).

\begin{array}{l} \frac{[(2 k_{p} ω_{o} + ω_{o}^{2}) s + k_{p} ω_{o}^{2}] \cdot G_{c p} (s)}{b_{0} (s^{2} + 2 ω_{o} s + ω_{o}^{2}) + (b_{0} k_{p} s - b_{0} ω_{o}^{2}) \cdot G_{c p} (s)} \cdot \frac{K}{T s + 1} = - 1 \\ - [(2 k_{p} ω_{o} + ω_{o}^{2}) s + k_{p} ω_{o}^{2}] \cdot G_{c p} (s) \cdot K \\ = (T s + 1) \cdot [b_{0} (s^{2} + 2 ω_{o} s + ω_{o}^{2}) + (b_{0} k_{p} s - b_{0} ω_{o}^{2}) \cdot G_{c p} (s)] \\ - G_{c p} (s) \cdot {[(2 k_{p} ω_{o} + ω_{o}^{2}) s + k_{p} ω_{o}^{2}] \cdot K + (T s + 1) \cdot (b_{0} k_{p} s - b_{0} ω_{o}^{2})} \\ = (T s + 1) \cdot b_{0} (s^{2} + 2 ω_{o} s + ω_{o}^{2}) \end{array}

(18)

Based on the above equivalent transfer functions, the following stability analysis will be made.

The compensation link of the system is separated,²³ that is, equation (18) can be obtained as

- \frac{1}{{(Ts + 1)}^{n - 1}} = \frac{B (s)}{C (s) + D (s)}

(19)

where polynomials $B (s)$ , $C (s)$ and $D (s)$ are, respectively,

{\begin{matrix} B (s) = b_{0} (s^{2} + 2 ω_{o} s + ω_{o}^{2}) \cdot (Ts + 1) \\ C (s) = (b_{0} k_{p} s - b_{0} ω_{o}^{2}) \cdot (Ts + 1) \\ D (s) = K \cdot [(2 k_{p} ω_{o} + ω_{o}^{2}) s + k_{p} ω_{o}^{2}] \end{matrix}

(20)

Define $L_{1} (s)$ is the left formula and $L_{2} (s)$ is the right formula of equation (19). Accordingly, their frequency response curves in the complex plane can be expressed as $Γ_{1} (j ω)$ and $Γ_{2} (j ω)$ . The starting points of the two curves are as follows

\begin{matrix} L_{1} (0) = - 1 \\ L_{2} (0) = \frac{b_{0}}{{Kk}_{p} - b_{0}} \end{matrix}

(21)

When the frequency $ω$ increases from zero to positive infinity, $Γ_{1} (j ω)$ is a closed clockwise curve that starts at $(- 1, j 0)$ and is symmetric along the X-axis and $Γ_{2} (j ω)$ is a nonlinear curve starting at $(L_{2} (0), j 0)$ .

The following lemma and stability theorem can be derived as

Lemma 1. The poles of the closed-loop characteristic equation (19) are not on the right plane, only if $Γ_{1} (j ω)$ and $Γ_{2} (j ω)$ meet one of the following conditions.²⁴

$Γ_{1} (j ω)$ and $Γ_{2} (j ω)$ never intersect;

If there are intersections between $Γ_{1} (j ω)$ and $Γ_{2} (j ω)$ , then $Γ_{2} (j ω)$ should reach the closest intersection near the point $(- 1, j 0)$ earlier.

Lemma 1 expresses that the intersections of $Γ_{1} (j ω)$ and $Γ_{2} (j ω)$ determine the stability of equation (19). Therefore, one needs to first determine whether the initial point of $Γ_{2} (j ω)$ is inside $Γ_{1} (j ω)$ . Let’s turn $Γ_{2} (j ω)$ into a form as

L_{2} (s) = (s^{3} + \sum_{i = 0}^{2} l_{1 i} s^{i}) {(\sum_{i = 0}^{2} l_{2 i} s^{i})}^{- 1}

(22)

where all coefficients $l_{1 i}$ and $l_{2 i} (i = 0, 1, 2)$ are listed in Table 1.

Table 1.

Polynomial coefficients in equation (22).

	$i$ = 2	$i$ = 1	$i$ = 0
$l_{1 i}$	$2 ω_{o} + \frac{1}{T}$	$ω_{o}^{2} + \frac{2 ω_{o}}{T}$	$\frac{ω_{o}^{2}}{T}$
$l_{2 i}$	$k_{p}$	$\frac{k_{p}}{T} - ω_{o}^{2} + \frac{2 {Kk}_{p} ω_{o} + K ω_{o}^{2}}{b_{0} T}$	$\frac{{Kk}_{p} ω_{o}^{2}}{b_{0} T} - \frac{ω_{o}^{2}}{T}$

Since the part of $ω = - \infty ~ 0$ is symmetric concerning the real axis, it is omitted. The phase-frequency and amplitude-frequency characteristics of $L_{1} (s)$ and $L_{2} (s)$ are as follows

\begin{matrix} | L_{1} (ω) | = {(1 + T^{2} ω^{2})}^{- \frac{n - 1}{2}} \\ | L_{2} (ω) | = [R^{2} (ω) + I^{2} (ω)]^{\frac{1}{2}} \end{matrix}

(23)

\begin{matrix} ∠ L_{1} (ω) = π - (n - 1) \cdot \arctan (T ω) \\ ∠ L_{2} (ω) = {\begin{matrix} \pm \arctan | \frac{I (ω)}{R (ω)} |, R (ω) > 0 \\ \pm (π - \arctan | \frac{I (ω)}{R (ω)} |), R (ω) < 0 \end{matrix} \end{matrix}

(24)

where $R (ω)$ and $I (ω)$ are as follows

R (ω) = \frac{(- l_{12} ω^{2} + l_{10}) (- l_{22} ω^{2} + l_{20}) + (l_{11} ω - ω^{3}) (l_{21} ω)}{{(- l_{22} ω^{2} + l_{20})}^{2} + l_{21}^{2} ω^{2}}

(25)

I (ω) = \frac{(- l_{12} ω^{2} + l_{10}) (l_{21} ω) + (- l_{22} ω^{2} + l_{20}) (l_{11} ω - ω^{3})}{{(- l_{22} ω^{2} + l_{20})}^{2} + l_{21}^{2} ω^{2}}

(26)

Theorem 1: For any high-order inertial transfer function (7) controlled by MADRC, equation (19) is stable, as long as the following conditions are satisfied.

$| L_{2} (j ω) | > | L_{1} (j ω) |$ for all $ω$

Let $ω^{*} = \max {ω | | L_{2} (j ω) | = | L_{1} (j ω) |}$ . If $ω^{*}$ satisfies

{\begin{matrix} T ω^{*} < \tan (\frac{π - \arctan | \frac{I (ω^{*})}{R (ω^{*})} |}{n - 1}), & R (ω) > 0 \\ T ω^{*} < \tan (\frac{\arctan | \frac{I (ω^{*})}{R (ω^{*})} |}{n - 1}), & R (ω) < 0 \end{matrix}

(27)

Proof.

Define $K = k \frac{b_{0}}{k_{p}}$ . And $\max | L_{1} (ω) | = 1$ due to $1 + T^{2} ω^{2} \geq 1$ . Therefore $L_{2} (0) = \frac{b_{0}}{{Kk}_{p} - b_{0}} = \frac{1}{K \frac{k_{p}}{b_{0}} - 1}$ , when $k \in (1, 2)$ , $L_{2} (0) > 1$ . That means when $ω$ rises from 0 to $+ \infty$ , $Γ_{1} (j ω)$ and $Γ_{2} (j ω)$ do not intersect with each other. From lemma 1–1), it can be proved that equation (19) is stable.

If $k \notin (1, 2)$ , the initial point of $Γ_{2} (j ω)$ may be inside the $Γ_{1} (j ω)$ , both of these curves have at least one intersection. Because $ω^{*} = \max {ω ∥ L_{2} (ω) | = | L_{1} (ω) |}$ , and the intersection $L_{2} (ω^{*}) = R (ω^{*}) + jI (ω^{*})$ is the nearest intersection to the point $(- 1, j 0)$ . Hence, it can be inferred from (27) that $Γ_{2} (j ω)$ arrive $(- 1, j 0)$ earlier than $Γ_{1} (j ω)$ at all intersection point, when they exist intersections. From lemma 1–2), it can be proved that equation (19) is stable.

Under the control of MADRC, the high-order system studied in this paper exhibits no intersection points within the parameter adjustment range. This characteristic, which aligns with Lemma 1–1), will be verified in the subsequent simulation section. Reference 23 conducts an in-depth analysis of the stability conditions when intersection points emerge, providing significant support for the interpretation of Lemma 1–2).

Tuning procedure and robustness analysis

Tuning procedure and numerical simulation

MADRC has a slightly different structure than ADRC, hence, its parameters also behave differently in a high-order inertial system. This section analyzes the control performance of MADRC through simulations, so as to better summarize the parameter tuning process of MADRC.

In the parameter tuning process, parameter $b_{0}$ plays a role in controlling the stability of this system. The stable region of the system is required to be large enough so that the control parameters that meet the control needs are in the stable region. The value of $b_{0}$ can not be too large, otherwise, the control signal $u$ will be too small and the system response will change slowly. Thus, the choice of $b_{0}$ should balance the relationship between the stability and the speed of response.

In this section, the tuning method of a first-order MADRC is presented, which can be easily and conveniently applied in actual systems. As an improvement of the method presented,²² this method has a wider range of applications and is still simple.

When ESO is able to achieve $z_{2} \approx f$ , with equation (2), (4), and (5), the control system will be converted to

\overset{\cdot}{y} = f + u_{0} - z_{2} \overset{z_{2} \approx f}{\approx} u_{0} = k_{p} (r - z_{1})

(28)

Under the unit step input, the output Laplace transform of equation (28) is

Y (s) = \frac{ω_{c}}{s (s + ω_{c})} = \frac{1}{s} - \frac{1}{s + ω_{c}}

(29)

Therefore, the unit step response of the system is

y (t) = 1 - e^{- ω_{c} t}

(30)

According to the definition of setting time

| \begin{matrix} y (t_{s}) - y (\infty) \end{matrix} | = Δ

(31)

where $t_{s}$ is setting time, $y (\infty) = 1$ and $Δ = 2 %$ , one can have

t_{s} = \frac{\ln 50}{ω_{c}} \approx \frac{3.91}{ω_{c}}

(32)

With equation (32), the control parameter $ω_{c}$ can be calculated when $t_{s}$ is known, thus determining the expected dynamic characteristics of the control system. However, the actual output of the system will inevitably have a certain degree of deviation from the expectation, so one needs to retain a certain margin when selecting the expected dynamic parameters. In the actual design, one may wish to make

ω_{c} = 9 / t_{s}

(33)

To make ESO obtain better observation speed even when $ω_{o}$ value is small, this section attempts to improve the parameter selection method of ESO.

With equation (2) and (4), the mathematical relationship between $z_{2}$ and $f$ can be obtained. In the actual control process, it can be approximated that the denominator of the formula only needs to retain the first two terms to describe the transitive relationship between $z_{2}$ and $f$ .

\frac{z_{2} (s)}{f (s)} = \frac{β_{2}}{s^{2} + β_{1} s + β_{2}} \approx \frac{λ}{s + λ}

(34)

where $λ = \frac{β_{2}}{β_{1}}$ , the samller the $λ$ value is, the faster the ESO is observed.

The lag time of $z_{2}$ tracking $f$ is defined as $T_{t}$ , and $T_{t}$ is generally required to be less than $t_{s}$ when designing a controller. Based on our knowledge of first-order systems, one can get

T_{t} = \frac{4}{λ}

(35)

When $λ = ω_{o}$ /2, the value of $λ$ is restricted by $ω_{o}$ .²² From this point of view, in order to make the value of $λ$ independent of $ω_{o}$ , let

β_{1} = 2 ω_{o}, β_{2} = λ β_{1}

(36)

When the adjustment time $t_{s}$ is set to 200, in order to make $T_{t}$ less than $t_{s}$ , $λ$ must be greater than 0.02. Based on the simulation verification, $λ = 0.09$ is recommended to be more appropriate. Under the condition that $λ$ value is determined, $ω_{o}$ has less influence on ESO observation. For $ω_{o}$ and $ω_{c}$ , based on extensive application experience, their mathematical relationship is usually defined as

ω_{o} = ρ ω_{c}, ρ \in [5, 10]

(37)

To sum up, the tuning rules for the first-order MADRC controller of high-order inertial systems can be obtained as

{\begin{matrix} ω_{c} = 9 / t_{s} \\ ω_{o} = ρ ω_{c}, ρ \in [5, 10] \\ β_{1} = 2 ω_{o} \\ β_{2} = 0.09 β_{1} \\ b_{0} \in [\frac{K}{2 T}, + \infty] \end{matrix}

(38)

When the adjustment time $t_{s}$ is known, only one control parameter, $b_{0}$ , needs to be adjusted. Gradually increase the $b_{0}$ value until the system dynamic performance indicators meet the system requirements.

Numerical simulation

Without considering the inner-loop of SST, taking the control object of the outer-loop of SST as an example, it is a typical high-order inertial system, as shown in equation (7). Set $K = 0.5$ , $T = 30$ , and $n = 4$ . Its transfer function is defined as

G (s) = \frac{0.5}{{(30 s + 1)}^{4}}

(39)

Next, a simulation verification will be conducted on the stability analysis proposed above. Figure 5 presents a detailed stability analysis through graphical interpretation, considering specific numerical conditions. In this figure, the black curve represents $Γ_{1} (j ω)$ , while colored curves depicts $Γ_{2} (j ω)$ under varying parameters. As illustrated, regardless of changes in the parameters, there is no point of intersection between $Γ_{1} (j ω)$ and $Γ_{2} (j ω)$ . This observation aligns with the standard established in Lemma 1–1), which confirms the absence of any intersection between the two curves. These findings indicate that the MADRC system satisfies the necessary stability conditions, thereby reinforcing its theoretical stability.

Remark 1: A simplified setting method is proposed for MADRC.²⁵ Assuming that the adjusting time is known, the parameter setting of the controller can be completed by adjusting only one control parameter, and a satisfactory control effect can be obtained. Although the control effect may be slightly lower than that of the quantified setting method, the difference is small, and the method is easy to implement in engineering. To better analyze the control performance of MADRC, set $t_{s} = 200$ .

Figure 5.

Graphical interpretation of stability analysis.

Then, by exhaustively searching $ω_{o}$ with fixed scale coefficient $ρ$ and time constant $T$ , the system’s gain and phase margins can be obtained, as shown in Figures 6 and 7. No matter how $T$ and $ρ$ change, the system is always in a stable state. When $ω_{o}$ is less than 0.1, the gain margin increases significantly with the increase of $ω_{o}$ , and decreases with the increase of $ω_{o}$ when $ω_{o}$ is greater than 0.1. The phase margin always decreases as $ω_{o}$ increases. And when $ρ$ equals 10, the variation interval of phase margin is the smallest, so this paper chooses $ω_{o} = 10 ω_{c}$ .

Figure 6.

The gain margin and phase margin of the system with fixed $λ$ = 0.25.

Figure 7.

The gain margin and phase margin of the system with fixed $T$ = 30.

Lastly, set the initial value of $b_{0}$ as 0.011, give a unit step as input from 50 s, and add a unit step as disturbance at 500 s. Through equation (39) one can get the control performance and control signal of the system with the gradual increase of $b_{0}$ , as shown in Figure 8. The overshoot of the output signal gradually decreases as $b_{0}$ increases, and the control performance is best when $b_{0}$ is 0.017.

Figure 8.

The influences on output and control signal with different $b_{0}$ .

Robustness verification

The model of the controller is usually not accurate enough, and parameters of the target object will change with the change of operating conditions and time. Therefore, it is necessary to verify whether the robustness of the system can meet the requirements in the case of parameters perturbation.

To better analyze the robustness, a typical measure of sensitivity to process change is proposed, namely the maximum sensitivity ( $M_{s}$ ).²⁶ $M_{s}$ is as follows

M_{s} = \max_{ω} | 1 / (1 + G_{op} (j ω)) | ω \in (- \infty, + \infty)

(40)

where $G_{op} (j ω)$ is the open-loop transfer function.

By equations (9) and (39), $G_{op} (s)$ can be converted to

G_{op} (s) = E (s) \cdot F (s) \cdot G_{p} (s)

(41)

where polynomials $E (s)$ and $F (s)$ are, respectively,

E (s) = \frac{(s^{2} + 2 ω_{o} s + 0.18 ω_{o}) k_{p}}{b_{0} [(s^{2} + 2 ω_{o} s + 0.18 ω_{o}) + (k_{p} s - 0.18 ω_{o})] G_{cp} (s)}

(42)

F (s) = \frac{2 ω_{o} s + 0.18 ω_{o} + 0.18 ω_{o} s / k_{p}}{s^{2} + 2 ω_{o} s + 0.18 ω_{o}}

(43)

To verify the maximum sensitivity constraint of the control object, $T$ and $K$ in equation (28) are randomly disturbed within the range of ±15%, and the result is shown in Figure 9. It can be concluded that the maximum sensitivity constraint gradually decreases with the decrease of the time constant $T$ , and gradually increases with the increase of the gain $K$ . The value of $M_{s}$ is generally between 1.2 and 2.0, and the smaller the value of $M_{s}$ , the greater the capacity to handle system uncertainty, and vice versa. It can be seen from Figure 9 that the numerical interval of the maximum sensitivity constraint of the control system is about (1.35, 1.65), which can verify that this system has relatively good robustness.

Figure 9.

Three-dimensional data diagram of $K$ , $T$ and $M_{s}$ in the perturbed SST system.

Simulation verification and field application

The model description of SST

Figure 10 is the schematic of the cascade control system of the SST. The system is composed of a low-temperature superheater (LTS), a platen superheater (PS), a high-temperature superheater (HTS), a first desuperheater and a second desuperheater, where $C_{1}$ - $C_{4}$ represents the controllers. The steam of the water wall obtains heat from the furnace and then passes through the desuperheater and superheater. The feedback control loop of the PS and the HTS depends on the inlet temperature and outlet temperature of superheaters respectively, and both of them are cascade control loops. The HTS cascade control system and the PS cascade control system are represented by the yellow and green dashed lines in Figure 10, respectively. As shown in Figure 11, a cascade control system can be thought of as two PI controllers (PI-PI). The steam temperature is controlled by two sets of desuperheaters. Take the cascade control system of the HTS as an example, $G_{1} (s)$ involves the dynamic process from the spray water valve to the outlet temperature of the first desuperheater $y_{2}$ , and $G_{2} (s)$ is the dynamic process from $y_{2}$ to the outlet temperature of the second desuperheater $y_{1}$ . To ensure that the SST control system has not only strong anti-interference but also maintains satisfactory robustness, the outer-loop controller of the SST system applied to the secondary temperature controller is selected to use MADRC, and the inner-loop controller is selected to use ADRC (MADRC-ADRC) to optimize control performance. In Figure 11, $d_{1}$ is the disturbance of the outer-loop, $d_{2}$ is the disturbance of the inner-loop, the control signal of each controller is denoted as $u_{1} - u_{4}$ .

Figure 10.

The schematic diagram of the SST system with the cascade control.

Figure 11.

The block diagram for SST with the cascade control.

Simulation verification

To thoroughly evaluate the control performance of the MADRC-ADRC controller, a comprehensive comparison is made with several other controllers in the cascade control system. In addition to the commonly used PI-PI controller, the performance of DADRC-ADRC, Smith-like ADRC-ADRC (SLADRC-ADRC), and the standard ADRC-ADRC controllers are also simulated. This multi-faceted comparison aims to highlight the unique advantages and potential limitations of the MADRC-ADRC controller under various operating conditions. To facilitate these simulations, the transfer function of the system’s inner-loop $G_{1} (s)$ , and the outer-loop $G_{2} (s)$ are carefully defined as

{\begin{matrix} G_{1} (s) = \frac{- 1.5}{15 s + 1} \\ G_{2} (s) = \frac{0.5}{{(30 s + 1)}^{4}} \end{matrix}

(44)

Based on the tuning method described in Section IV, the parameters for the MADRC controller are computed. It should be noted that the following principles apply to the system’s parameter tuning: first, the control output should be kept as smooth as possible, with the system’s overshoot reduced to as small a level as possible; second, on the premise of meeting the first principle, the dynamic tracking speed of the system for the target signal should be improved to as fast a rate as possible. All the parameters for each of the different controllers are systematically summarized in Table 2. The results of the simulations for the SST system are presented in Figures 12 to 14, offering a detailed comparison of control performance under various conditions in this cascade control system.

Table 2.

Parameters of five control methods.

Control methods	Parameter of the outer-loop controller	Parameter of the inner-loop controller
MADRC-ADRC	$b_{02} = 0.02, ω_{o 2} = 0.152,$	$b_{01} = - 0.02, ω_{o 1} = 0.15,$
MADRC-ADRC	$ω_{c 2} = 0.038$	$ω_{c 1} = 0.01$
PI-PI	$k_{p 2} = 1, k_{i 2} = 0.012$	$k_{p 1} = - 1, k_{i 1} = - 1 / 40$
DADRC-ADRC	$b_{03} = 0.02, ω_{o 3} = 0.16,$	$b_{01} = - 0.02, ω_{o 1} = 0.15,$
DADRC-ADRC	$ω_{c 3} = 0.02, τ = 50$	$ω_{c 1} = 0.01$
SLADRC-ADRC	$b_{04} = 0.02, ω_{o 4} = 0.25,$	$b_{01} = - 0.02, ω_{o 1} = 0.15,$
SLADRC-ADRC	$ω_{c 4} = 0.02$	$ω_{c 1} = 0.01$
ADRC-ADRC	$b_{05} = 0.5, ω_{o 5} = 0.64,$	$b_{01} = - 0.02, ω_{o 1} = 0.15,$
ADRC-ADRC	$ω_{c 5} = 0.024$	$ω_{c 1} = 0.01$

Figure 12.

The control performance of the cascade control system without any disturbance.

Figure 13.

The control performance of the cascade control system with $d_{2}$ .

Figure 14.

The control performance of the cascade control system with $d_{1}$ .

Figure 12 illustrates the control performance when the input is provided without any external disturbance. The input signal is represented by the solid black line, which is followed by the system’s response. The red line, corresponding to the MADRC-ADRC controller, shows the fastest response, reaching a steady-state value more quickly than the other controllers. This demonstrates the superior speed and efficiency of the MADRC-ADRC in handling the input signal. Figure 13 demonstrates the control performance when a unit step input is applied, and an inner-loop disturbance $d_{2}$ is introduced. The system’s response under this disturbance condition is captured, showcasing the controller’s robustness to internal disturbance. Similarly, Figure 14 depicts the cascade control system’s behavior when a unit step signal is given as input, and an outer-loop disturbance $d_{1}$ is added. This scenario tests the system’s performance in the presence of external disturbance, further emphasizing the capability of each controller in dealing with real-world uncertainties. To quantitatively assess the control quality, the maximum overshoot ( $σ$ ) and the Integral of Absolute Error (IAE) values are recorded in Table 3. These metrics provide a more comprehensive comparison of the controllers’ performance, offering insights into their ability to minimize overshoot and error in the presence of both internal and external disturbance. The recorded results from Figures 12 to 14 allow for a thorough analysis of each controller’s strengths and limitations, making it possible to evaluate their effectiveness under varying operating conditions.

From the results shown in Figures 12 to 14, it is evident that the MADRC-ADRC controller in the cascade control system outperforms the other four control methods in several key aspects. Specifically, the MADRC-ADRC exhibits the smallest overshoot and demonstrates the significantly strongest ability to resist disturbance. This indicates its superior dynamic performance and robustness when compared to the other controllers under various conditions.

The statistical indices presented in Table 3 further reinforce these observations. Among all the controllers, the MADRC-ADRC achieves the smallest Integral of Absolute Error (IAE), which serves as a clear indication of its exceptional disturbance immunity. The low IAE value highlights the controller’s ability to maintain better accuracy and stability, even in the presence of external and internal disturbances. This further underscores the MADRC-ADRC’s superiority in terms of both steady-state accuracy and resilience to interference, making it the most reliable choice among the methods evaluated.

Table 3.

Data indicators of the SST system under different controllers.

Control methods	Figure 12		Figure 13		Figure 14
Control methods	$σ$ (%)	IAE	$σ$ (%)	IAE	$σ$ (%)	IAE
MADRC-ADRC	0.08	513.30	8.59	178.48	19.8	287.94
PI-PI	0.12	612.60	9.07	237.37	26.64	378.17
DADRC-ADRC	0.09	650.72	8.28	225.24	22.81	354.31
SLADRC-ADRC	0.8	596.41	8.52	204.42	20.29	316.9
ADRC-ADRC	0.92	716.95	8.33	254.78	26.86	413.19

To perform a Monte Carlo experiment for this high-order inertial system, the parameters $T$ , $K$ in equation (39) are subjected to random perturbations within a range of ±15%. The input signal is a unit step function applied at t = 10 s. In addition to this, two disturbance signals are introduced: a negative unit signal, denoted as $d_{2} (s)$ , at t = 800 s and a positive unit step signal, denoted as $d_{1} (s)$ , at t = 1600 s.

The cascade system’s performance is evaluated over a period of 2400 s. Specifically, the Integral of Absolute Error (IAE) for the setpoint, denoted as ${IAE}_{sp}$ , is computed for the first 800 s. Then, for the disturbance $d_{2} (s)$ , the Integral of Absolute Error of the unit disturbance ( ${IAE}_{ud 1}$ ) is calculated from 800 to 1600 s. Lastly, for the disturbance $d_{1} (s)$ , the IAE of the unit disturbance ( ${IAE}_{ud 2}$ ) is evaluated from 1600 to 2400 s. The IAE values represent the accumulated error over time, which provides a quantitative measure of the cascade system’s response to disturbance and setpoint changes.

In this Monte Carlo simulation, the order $n$ of $G_{2} (s)$ is randomly varied between 3 and 5. This variation introduces additional complexity into the system, which allows for the assessment of how the system’s performance behaves under different system orders. The experiment is repeated 1000 times to obtain statistically significant results, ensuring the reliability of the findings across various random perturbations.

For the analysis, data graphs illustrating ${IAE}_{sp}$ , ${IAE}_{ud 1}$ , and ${IAE}_{ud 2}$ are presented in Figure 15, providing a visual representation of the system’s error behavior over time under varying conditions. Additionally, all indicator data are systematically recorded in Table 4, which serves as a comprehensive summary of the Monte Carlo simulation results.

Figure 15.

Data recording of the perturbed cascade control system under different control methods.

Table 4.

Statistical indexes of Figure 15.

Control methods	$σ$ (%)	${IAE}_{sp}$	${IAE}_{ud 1}$	${IAE}_{ud 2}$
MADRC-ADRC	(0, 34.05)	(120.11, 234.45)	(135.77, 259.80)	(187.71, 356.08)
PI-PI	(0, 26.87)	(144.48, 252.57)	(169.55, 293.23)	(252.71, 417.26)
DADRC-ADRC	(0, 27.76)	(159.87, 261.47)	(177.55, 287.22)	(240.82, 394.21)
SPADRC-ADRC	(0, 32.33)	(142.71, 254.85)	(158.78, 275.24)	(212.10, 378.14)
ADRC-ADRC	(0, 29.24)	(175.79, 288.63)	(196.91, 336.58)	(274.19, 472.54)

From Figure 15, it can be seen that the integral error data of the cascade control system are concentrated in the lower left corner under the control of MADRC-ADRC, and the dense distribution indicates that the control method has strong robustness. At the same time, a smaller index indicates better tracking and interference suppression performance. In Table 4, the maximum values and minimum values of ${IAE}_{sp}$ , ${IAE}_{ud 1}$ , and ${IAE}_{ud 2}$ of MADRC-ADRC are the smallest, while $σ$ of MADRC-ADRC is greater than SLADRC-ADRC. Combining the above data, it can be concluded that MADRC-ADRC has the most satisfactory control performance in the cascade control system.

Field application

The potential application of MADRC-ADRC in the industrial cascade control system has been demonstrated through both simulation and experimental verification. Specifically, MADRC-ADRC was applied to both the primary and secondary SST systems of a 660 MW power plant, showcasing its effectiveness in improving system performance under complex operating conditions.

To accurately acquire the operating temperature data of the superheater and ensure the reliable execution of subsequent control algorithms in this experiment, a sampling period of 0.25 s is adopted for data sampling. On-site temperature signals are first transmitted directly to the distributed control system (DCS) via hardwiring. Subsequently, the DCS uploads the standardized temperature data to the industrial personal computer (IPC) through the Modbus TCP protocol, and the IPC finally runs the relevant control methods.

To meet the requirements of safe unit operation and precise control, clear standards for temperature measurement accuracy are established: as the core control parameter of the unit, the measurement accuracy error of the main steam temperature is strictly controlled within ± $1^{°} C$ ; the staged temperature of the superheater, which is used to monitor the heat transfer status of each section, has its measurement accuracy error controlled within ± $2^{°} C$ . In view of the actual operating condition where the inlet and outlet temperatures of the superheater are approximately $500^{°} C$ , thermocouples with high-temperature resistance, fast response speed, and excellent linearity are selected as measurement elements to achieve precise measurement within this temperature range.

For a start, Figure 16 shows the operation of the primary SST system under the two control methods for 24 h, first running MADRC-ADRC and then switching to PI-PI. When switching from MADRC-ADRC to PI-PI, a corresponding bump less transfer method is required. This switching process has no additional requirements on external conditions and can be flexibly implemented according to the actual operating conditions of the system. The core goal is to ensure the smooth transition of control variables during the switching process, and thereby guarantee the safety and reliability of the industrial control system. Specifically, before switching from MADRC-ADRC to PI-PI, this bump less transfer method requires PI-PI to first track the current operating state, calculate and initialize the initial value of its own integrator, and then execute the control method switching from MADRC-ADRC to PI-PI. This ensures that the control output remains continuous without sudden changes before and after the switch, keeping the system within a stable range. The statistical indicators in this figure are recorded in Table 5, including average deviation $σ$ , average positive deviation $σ^{+}$ , average negative deviation $σ^{-}$ , the maximum positive deviation $e^{+}$ , maximum negative deviation $e^{-}$ and variance $S^{2}$ . The load range of MADRC-ADRC is (193.05, 370.16) MW and the load range of PI-PI is (172.04, 357.44) MW. Table 5 records the error indicators of MADRC-ADRC.

Figure 16.

Field applications of the primary SST system under switching conditions.

Table 5.

Statistical indexes of Figure 16.

Control methods (side A)	$e^{+} (° C)$	$e^{-} (° C)$	$σ$ ( $° C$ )	$σ^{+} (° C)$	$σ^{-} (° C)$	$S^{2}$
MADRC-ADRC	10.90	−11.46	0.012	2.72	−2.59	11.95
PI-PI	24.15	−27.38	1.47	3.13	−3.10	42.45
Control methods (side B)	$e^{+} (° C)$	$e^{-} (° C)$	$σ$ ( $° C$ )	$σ^{+} (° C)$	$σ^{-} (° C)$	$S^{2}$
MADRC-ADRC	13.85	−11.07	0.035	2.86	−2.74	13.14
PI-PI	22.45	−26.35	0.39	3.10	−3.20	31.72

In the next part, the comparisons of the operating conditions of the primary SST system for 24 h under the MADRC-ADRC and PI-PI control methods are shown in Figures 17 and 18. And the statistical indicators in this figure are documented in Table 6. The load capacity of MADRC-ADRC is (205.40, 507.16) MW, while the load range of PI-PI is (295.30, 521.19) MW.

Figure 17.

Field applications of the primary SST system with MADRC-ADRC.

Figure 18.

Field applications of the primary SST system with PI-PI.

Table 6.

Statistical indexes of Figures 17 and 18.

Control methods (side A)	$e^{+} (° C)$	$e^{-} (° C)$	$σ$ ( $° C$ )	$σ^{+} (° C)$	$σ^{-} (° C)$	$S^{2}$
MADRC-ADRC	32.74	−15.33	0.96	4.11	−3.11	26.20
PI-PI	37.13	−27.63	1.06	8.91	−6.90	111.68
Control methods (side B)	$e^{+} (° C)$	$e^{-} (° C)$	$σ$ ( $° C$ )	$σ^{+} (° C)$	$σ^{-} (° C)$	$S^{2}$
MADRC-ADRC	24.89	−20.58	−0.30	3.22	−3.33	19.29
PI-PI	37.90	−31.58	0.73	5.82	−4.82	59.93

In the end, Figures 19 and 20 are the operating conditions of the secondary SST system for 24 h under the MADRC-ADRC and PI-PI control methods. Table 7 records their data and statistical indicators. Also, the load range of MADRC-ADRC is (205.40, 507.16) MW, and the load range of PI-PI is (295.30, 521.19) MW, because it is collected in the same time period as the working condition of the primary SST system.

Figure 19.

Field applications of the secondary SST system with MADRC-ADRC.

Figure 20.

Field applications of the secondary SST system with PI-PI.

Table 7.

Statistical indexes of Figures 19 and 20.

Control methods (side A)	$e^{+} (° C)$	$e^{-} (° C)$	$σ$ ( $° C$ )	$σ^{+} (° C)$	$σ^{-} (° C)$	$S^{2}$
MADRC-ADRC	12.11	−7.69	0.065	1.75	−1.68	5.29
PI-PI	14.82	−11.43	0.0067	2.04	−2.03	7.66
Control methods (side B)	$e^{+} (° C)$	$e^{-} (° C)$	$σ$ ( $° C$ )	$σ^{+} (° C)$	$σ^{-} (° C)$	$S^{2}$
MADRC-ADRC	11.36	−13.03	−0.29	1.43	−1.75	5.25
PI-PI	17.03	−9.85	0.049	1.81	−1.86	6.12

In Table 7, under the MADRC-ADRC control method, there is a large negative deviation on the Side B of the secondary SST system, because the setpoint suddenly changes greatly around the 16th hour of operation. In addition, under the PI-PI control method, the average deviation $σ$ of the Side A is smaller, because the largest positive and negative deviations cancel each other out, the impact of this metric on system performance is negligible. Apart from that, the statistical analysis indicates that the MADRC-ADRC generally produces more stable and efficient operating conditions for the primary SST system. The lower average deviations, smaller variance, and reduced error indicators under MADRC-ADRC suggest superior performance when compared to the PI-PI. The effectiveness of MADRC-ADRC in stabilizing system operation makes it a more favorable choice, and it is suggested that this control method could be beneficial for other complex high-order industrial systems.

Conclusions

This paper mainly studies SST controlled by MADRC, with a focus on introducing the relevant contents of stability analysis. The stability is deeply analyzed by separating higher-order transfer functions and using the dual-locus diagram method. This method proves that SST under MADRC control can operate safely and effectively. In addition, a simple and practical first-order MADRC parameter tuning method is proposed, and the robustness of SST is analyzed through the $M_{s}$ value. In addition to safely and effectively controlling SST, this method provides an effective solution for enhancing the safety and economy of power station boilers. The simulation results show that under the same working conditions, the IAE of SST controlled by MADRC can be reduced by at least 9%. MADRC-ADRC is successfully applied to the primary and secondary SST systems of a 660 MW power plant. Compared to PI-PI, SST controlled by MADRC-ADRC can reduce the maximum deviation by at least 12%, the average positive and negative deviations by 8% to 40%, and $S^{2}$ by approximately 13% to 70%. The data indicate that SST controlled by MADRC-ADRC significantly outperforms other cascade controllers in terms of tracking performance and disturbance rejection. Future research will continue to explore the potential applications of SST in adopting improved ADRCs to enhance its control performance.

Footnotes

ORCID iD

Zhenlong Wu

Ethical considerations

The research does not involve any human or animal experiments.

Consent to participate

Obtained.

Consent for publication

Obtained.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by supported by Natural Science Foundation of Henan Province under Grant No. 252300421954, Zhengzhou University Professor Team for Enterprise Innovation-Driven Development Project under Grant No. JSZLQY2022016 and Zhongyuan Scientific and Technological Innovation Leading Talent of Henan Province under Grant No. 254200510010.

Declaration of conflicting interests

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

Data availability statement

The data used to support the findings of this study are available from the corresponding author upon request.

References

Han

. From PID to active disturbance rejection control. IEEE Trans Ind Electron 2009; 56(3): 900–906.

Liu

Wang

, et al. A fast ADRC-SMC levitation control of EMS maglev train to suppress the vibration of electromagnet-track beam coupling system. Meas Control 2026; 59: 340–352.

Andrade

Normey-Rico

Raffo

. Fast embedded tube-based MPC with scaled-symmetric ADMM for high-order systems: application to load transportation tasks with UAVs. ISA Trans 2025; 156: 70–86.

Shi

, et al. A Smith-like active disturbance rejection control design for a class of overdamped processes with application to low-pressure heater system. IEEE Trans Ind Electron 2025; 72(10): 10771–10780.

Lin

Fei

, et al. A generalized PID interpretation for high-order LADRC and cascade LADRC for servo systems. IEEE Trans Ind Electron 2022; 69(5): 5207–5214.

Nguyen

. Robust integrated control for an automotive electric power steering system based on observed state variables. Meas Control 2025: 00202940251353472.

Rajasekharan

Vincent

Jacob

, et al. Dynamic consensus control of linear multi-agent systems: unmanned ground vehicle platooning. Meas Control 2026; 59: 201–216.

. Adaptive output feedback control for uncertain nonlinear systems with quantized input and output. ISA Trans 2025; 159: 226–235.

Zeng

Liu

, et al. A dual-band coil array with novel high-order circuit compensation for shielding design in EV wireless charging system. IEEE Trans Ind Electron 2024; 71(3): 2545–2555.

10.

Rojas

Cortés-Romero

Rojas

. Active disturbance rejection control based on a cascade estimator composed of reduced-order and full-order extended state observers. ISA Trans 2024; 151: 296–311.

11.

Elbouchikhi

El Moubarek

Abouloifa

, et al. Active disturbance rejection control for four-wire inverters in standalone renewable resources-based microgrid - islanded microgrids ADRC-based control. ISA Trans 2025; 156: 290–306.

12.

Wang

Zhao

Wang

, et al. Combined vector resonant and active disturbance rejection control for PMSLM current harmonic suppression. IEEE Trans Ind Inform 2020; 16(9): 5691–5702.

13.

, et al. A tuning method of active disturbance rejection control for a class of high-order processes. IEEE Trans Ind Electron 2020; 67(4): 3191–3201.

14.

Jiang

, et al. Adaptive heading control strategy for unmanned ground vehicle with variable wheelbase based on robust-active disturbance rejection control. Control Eng Pract 2024; 142: 105786.

15.

Tian

Wang

, et al. Adaptive active disturbance rejection control for uncertain current ripples suppression of PMSM drives. IEEE Trans Ind Electron 2024; 71(3): 2320–2331.

16.

Sun

Gao

, et al. Combined feedforward and model-assisted active disturbance rejection control for non-minimum phase system. ISA Trans 2016; 64: 24–33.

17.

Sui

, et al. Modified active disturbance rejection control design based on gain scheduling for selective catalytic reduction denitrification processes. IEEE Trans Ind Electron 2024; 71(12): 16441–16451.

18.

Sun

Xue

, et al. Quantitative tuning of active disturbance rejection controller for FOPTD model with application to power plant control. IEEE Trans Ind Electron 2022; 69(1): 805–815.

19.

Luo

Yang

, et al. Multiple disturbance suppression of IPMSM drives based on embedded discrete-time repetitive ADRC with optimized parameter selection. IEEE Trans Power Electron 2024; 39(5): 6052–6062.

20.

Chen

Hao

, et al. Predictor-based active disturbance rejection control of wet flue gas desulfurization system with delay robustness and simplified tuning. IEEE Trans Autom Sci Eng 2025; 22: 3731–3742.

21.

Liu

, et al. Performance analysis of improved ADRCs for a class of high-order processes with verification on main steam pressure control. IEEE Trans Ind Electron 2023; 70(6): 6180–6190.

22.

Gao

. Scaling and bandwidth-parameterization based controller tuning. In: Proceedings of the 2003 American control conference, 2003, vol. 6, pp. 4989–4996.

23.

Sun

Zhu

, et al. Robust tuning of active disturbance rejection controller for time-delay systems with application to a factual electrostatic precipitator. IEEE Trans Control Syst Technol 2022; 30(5): 2204–2211.

24.

Zhong

. Robust stability analysis of simple systems controlled over communication networks. Automatica 2003; 39(7): 1309–1312.

25.

Chen

Gao

, et al. Tuning method for second-order active disturbance rejection control. In: Proceedings of the 30th Chinese control conference, 2011, pp. 6322–6327.

26.

Han

, et al. Tuning of PID/PIDD2 controllers for integrating processes with robustness specification. ISA Trans 2023; 140: 224–236.