#### Abstract

The linear active disturbance rejection control (LADRC) method is investigated in this paper. Firstly, the integral effect of the ADRC is analyzed under the premise that ADRC was transformed into a new form. Then, ADRC is changed into an internal model control (IMC) framework, and an almost necessary and sufficient condition for stability and tracking performance of the ADRC system are proposed on this basis. In addition, some useful corollaries are proposed so that the traditional open-loop frequency-domain analysis method can be applied to ADRC system stability analysis. It also provides a theoretical principle and theoretical guidance for some parameter tuning. To improve the performance of ADRC, an approximate integral gain is treated as a separated adjustable parameter according to the new structure. Furthermore, tuning of some parameters is discussed to enhance system performance. Finally, simulations are used to verify the effectiveness of proposed methods.

#### 1. Introduction

Model uncertainties and external disturbances are widely found in the industrial control system, which makes the modern control theory based on accurate models difficult to apply so that the classical PID control method still dominates the entire industrial control field. However, the classical PID control has some drawbacks, such as overshoot and limited antidisturbance capability. Therefore, PID control may not achieve the expected performance of complex systems. Active disturbance rejection control (ADRC) is a control technique proposed by Prof. Han [1, 2]. The main idea is that the model uncertainty and external disturbance are reduced to total disturbance, and the extended state observer (ESO) is used to estimate this total disturbance. Then, the state error feedback (SEF) is used to compensate the disturbance to achieve the dynamic linearization of the control system and implement the feedback control. As a practical alternative approach, the ADRC strategy has prominent performance in the control of linear and nonlinear uncertain systems, such as uncertain nonaffine-in-control nonlinear systems [3], induction motor systems [4], parallel active power filters [5], load frequency control [6], circular curved beam [7], two-mass drive systems [8], permanent magnet synchronous motors [9], diesel engines [10], buck-boost-converter/DC-motor systems [11], flywheel energy storage systems [12], and nonlinear fractional-order systems [13].

ADRC was first proposed as a nonlinear form, which generally employs nonlinear forms of ESO and SEF. Nonlinear ADRC (NLADRC) has strong antidisturbance and excellent control performance. However, the use of nonlinear functions may make the system produce some complex nonlinear behaviors, such as multiple equilibrium points, limit cycles, bifurcations, and chaos. Meanwhile, stability and performance analyses are very difficult for such nonlinear systems, and too many parameters make the tuning rather difficult, which inhibits the application of ADRC. And many theoretical issues, including its applicability in stabilization and output regulation, remain unanswered. This brings a continuous challenge in the study of the ADRC theory. Therefore, linear ADRC (LADRC) was proposed to overcome these disadvantages by Gao [14]. The previous studies on analyzing stability and performance of LADRC can be mainly classified into two groups. One is time-domain analysis [15–17], which lays a solid theoretical foundation and provides a strict theoretical support for application. And another is frequency-domain analysis [18, 19], which is significantly important so that the ADRC framework is understood using the almost universal frequency-domain analysis languages shared by practicing control engineers, including both bandwidth and stability margins.

Based on LADRC, some modified ADRC schemes were also developed to improve the control performance. In [20, 21], an ADRC is derived by effectively integrating adaptive control with ESO via the backstepping method. The adaptive law is synthesized to handle parametric uncertainties, and the remaining uncertainties are estimated by the extended state observer and then compensated in a feedforward way. In [22], ADRC for time-delay systems is considered, and a predictive ADRC is established. In [23], a modified ESO with a time-varying gain is proposed, which reduces the “peaking phenomenon” caused by the constant high gain in the traditional ESO. And in [4], a sliding-mode-based component is designed, in order to take into account disturbance estimation errors and uncertainties in the knowledge of the control gain.

Although ADRC theory is widely investigated in previous studies, there are still some problems:(1)Only sufficient condition for the stability of the ADRC system is proposed.(2)It failed to explain the reasons for tuning to increase the robustness of the system and failed to provide theoretical guidance for the tuning of ( is the nominal high-frequency gain of the plant).(3)It failed to explain when model information should be applied.

In this paper, these issues will be resolved, and some further deep research results will be presented.

In Section 2, a new structure of ADRC will be shown to illustrate its integral performance. In Section 3, the controller will be transformed into the framework of a two-degree-of-freedom (TDF) internal model control (IMC) to analyze its performance. In Section 4, tuning of some parameters will be discussed to enhance system robustness. In Section 5, simulations are used to verify the effectiveness of the proposed method. Finally, some concluding remarks are drawn in Section 6.

#### 2. ADRC Algorithm

Consider the following typical single-input-single-output (SISO) system:where is the regulated output, are the system state, is the input force, is the high-frequency gain, and is the total disturbance including both the model uncertainties and external disturbances.

ADRC generally consists of a tracking differentiator (TD), an extended state observer (ESO), and a state error feedback (SEF). A TD is often used to arrange the transition process to solve the contradiction between system speed and overshoot. The general form of a TD is described aswhere is the reference input, are the output, and is the adjustable speed factor. is a solution that guarantees the fast convergence from to .

The ESO, as the core of ADRC, is usually used to estimate system states and total disturbance, and referring to equation (1), an ESO can be usually designed in the following form:where are the outputs of the ESO; are the observer gains; and is the nominal value of , and . For the sake of discussion, we assume that .

SEF is used to compensate the total disturbance and eliminate the residual error, and the SEF is designed aswhere are the controller gains.

Substituting (4) into (3), we have

From (5), we can see that can be expressed as

When , we have

Thus, with a TD and an SEF, the structure of ADRC can be transformed into the form in Figure 1.

Obviously, from Figure 1, we can see that ESO can be considered to be made up of a state observer (SO) and an integrating element. And the SO can be described aswhere ,

Applying Laplace transformation to (8), we can obtainwhere , in which is the identity matrix. , , , and are the Laplace transforms of , , , and , respectively.

According to (7) and (10), we havewhere is the Laplace transform of . And it can be known from (4) that the feedback of “total disturbance” is completed by the following formula:

Although ADRC does not make use of integral directly, it shows that an integral is automatically generated from (12), which provides a good explanation why ADRC has a high steady-state precision. As we know, can be used as a tuning parameter to improve the control performance of ADRC. However, the tuning of is generally found by trial and error, and no theory is available to support the method [24].

*Remark 1. *With (11) and Figure 1, we can see that increasing can reduce the integral gain; that is, the integral time constant is increased to increase the system robustness. However, if is too large, the integral action will become so weak that the system will have steady-state deviation. Therefore, we must choose an eclectic .

#### 3. System Performance Analysis

Let the set of transfer functions bewhere and are positive integers, and all . ADRC does not need the complete models of the controlled plant and the disturbance, but the relative order of ; that is, , , and .

##### 3.1. System Transformation

*Assumption 1. *A TD can accurately track the reference input and get its derivatives [18].

Let . Then, (3) can be transformed into the following form:whereSubstituting (4) into (14), we obtainwhereBy taking Laplace transform of (16), we havewhere , , and are the Laplace transforms of , , and , respectively.

Substituting (19) into (18), we haveFrom Assumption 1, we can see thatwhere is the Laplace transform of . Thus, (20) can be further transformed intowhereThe above result indicates that ADRC can be equivalent to a two-degree-of-freedom internal model control (TDF-IMC), and the TDF-IMC structure is shown in Figure 2.

*Remark 2. *It can be shown in a similar way to Theorem 1 in [24] that an *n*th-order ADRC structure shown in Figure 1 can be transformed into a TDF-IMC structure in Figure 2, where is the plant to be controlled, is the nominal model of , is the tracking IMC controller, is the disturbance rejection IMC controller, and is the external disturbance.

According to Figure 2, we haveCombined with (22), the following equation can be obtained:With the configuration in Figure 3, nine transfer functions from to are given bywhereIf the above nine transfer functions are stable, then the closed-loop system is said to be internally stable.

##### 3.2. Stability and Accurate Tracking Performance

For IMC, it is difficult to model accurately. Especially when there is a nonminimum phase part or uncertainty in , is very different from . Therefore, the stability of the IMC system is generally proved by the small-gain theorem for simplicity. However, for the IMC transformed by ADRC, we can always let .

Then, (26) can be transformed intowhere and are only related to the parameters , , and . In order to simplify parameter tuning, and are usually reduced to two tuning parameters as suggested in [14]: , the controller bandwidth, and , the extended state observer bandwidth.

Substituting (25) into (28) and considering , , and , we havewhere

*Assumption 2. * is Hurwitz.

It can be seen from equation (22) that the calculation of is only related to the design parameters of ADRC and has nothing to do with the plant. According to the parameter tuning method in [14], it is easy to guarantee is Hurwitz.

Theorem 1. *Under Assumption 1 and Assumption 2, for given , , and , the closed-loop system is said to be internally stable if and only if is Hurwitz.*

*Proof of Sufficiency. *If is Hurwitz, then the nine transfer functions of (28) must be stable. For given , , and , the closed-loop system is internally stable.

*Proof of Necessity. *For given , , and , there are no zero roots in and .

And if Assumption 2 holds, neither nor have unstable zeros and unstable poles that can cancel each other out; that is, and are both coprime polynomials.

If the closed-loop system is internally stable, then the nine transfer functions of (28) must be stable, and the reason why is Hurwitz is as follows.

Assume that is not Hurwitz and one of the transfer functions of (28), such aswhose numerator and denominator contain an identical factor , and . If exists in , then it must exist in (i.e., ) as is a coprime polynomial. And since is Hurwitz, does not exist in ; then,is unstable. Similarly, if exists in , (29) will also be unstable. In fact, as long as is not Hurwitz, no matter which one of the nine transfer functions is unstable, it will cause at least one transfer function of (28) to be unstable, Q.E.D.

*Remark 3. *Theorem 1 is not able to determine internal stability when Assumption 3 does not hold, but , , and are all stable. However, this situation is rare, and it is easy to assure that is Hurwitz when and . If we exclude such a situation, the condition that is Hurwitz is not only sufficient but also necessary for internal stability. In this sense, we call it an almost necessary and sufficient condition for internal stability.

In the engineering application, the system should have better tracking performance and dynamic performance besides the requirements of the internal stability. From (28), it can be seen that the main channel transfer function can be described asSince , it is clear that the coefficient of of is zero, as long as and contain constants and no zeros of are located at origin in the complex plane, and the static gain of the main channel function is only determined by the constant terms and , but not related to and :

Theorem 2. *Under Assumption 2, for given , , , and all , the closed-loop system is said to have accurate tracking performance if and only if is Hurwitz.*

*Proof of Sufficiency. *If is Hurwitz, then no zeros of are located at origin in the complex plane. And according to (34) and (35), we haveAccording to Theorem 1, the closed-loop system is internally stable, and then the closed-loop system has accurate tracking performance.

*Proof of Necessity. *If the closed-loop system has accurate tracking performance, then (36) must be established and the closed-loop system is internally stable. According to Theorem 1, is Hurwitz, Q.E.D.

In practical applications, not only should the stability of the system be known but also its stability margin and attenuation ability to the disturbance are required. According to Theorem 1, the stability of the ADRC system is determined by , while its attenuation ability to disturbance and sensor noise is determined by (as can be seen from Figure 2 and (28)).

Corollary 1. *Under Assumption 2, for given , , and , the closed-loop system is said to be internally stable if is stable and is Hurwitz.*

*Proof. *From (29), we can see thatIf is stable and is Hurwitz, then is Hurwitz because there are no unstable zeros and unstable poles to cancel each other out, and then the closed-loop system is internally stable based on Theorem 1.

Corollary 2. *Under Assumption 2, when , for given , , and , the closed-loop system is said to be internally stable if and only if is stable.**Before proof, the following lemma is given.*

Lemma 1 [25]. *Polynomial is Hurwitz if or (), where**Let and , .*

*Proof. *(1)When , are all positive, so is Hurwitz.(2)When ,(3)When , we haveTherefore, based on Lemma 1, when , is Hurwitz. Then, the necessary and sufficient condition of Corollary 2 is easy to be proved.

*Remark 4. *When , may be greater than 0.89 for . In fact, it is not hard to find by calculation that as increases, the minimum value of the upper bound of increases for . But may still be Hurwitz because Lemma 1 is only a sufficient condition. Therefore, it is necessary to determine whether is Hurwitz, when .

*Remark 5. *According to (28) and (37), the greater the bandwidth of , the stronger the antidisturbance capability of the ADRC system and the greater the influence of the sensor noise.

*Remark 6. *When is Hurwitz, according to Corollary 1 and Corollary 2, the stability margin of can reflect the stability margin of the ADRC system to some extent. And it can be seen from (25) that can be considered a closed-loop transfer function whose open-loop transfer function is . In this way, we can use frequency-domain analysis to analyze so as to determine the stability of the whole ADRC system.

#### 4. Parameter Tuning

ADRC parameter tuning has always been an important topic because of a bunch of parameters. For practical purposes, the parameters of ADRC are simplified to and [14]. However, sometimes is also adjusted as the third parameter to improve system performance because of unknown information on and robustness requirement. But before this article, there were no specific theoretical principle and theoretical guidance for the tuning of . In fact, a solution to this problem was proposed in Remark 6, i.e., to solve this problem with frequency-domain analysis of .

However, in practice, model information is sometimes difficult to know accurately. Therefore, a more concise method is needed to guide the tuning of . According to the *Routh criterion*, a necessary condition for to be Hurwitz is that the coefficients of have the same symbol and are not zero. When the symbol of the coefficients in (13) is known, in order to guarantee the establishment of this necessary condition, the tuning methods are used, which are listed in Table 1.

The methods about how to incorporate more plant information in ADRC are described in the study of Tan and Fu [24, 26].

Moreover, it can be seen from (7) and Figure 1 that the integral function of ADRC is only related to , so in order to make the ADRC parameter more explicit, referring to Figure 1, is replaced with as a separated adjustable parameter. In this way, can be considered a separate integrating element, and can be viewed as an approximate integral gain to adjust.

So, the number of parameters of the ADRC can be expanded to four: , the controller bandwidth; , the observer bandwidth; ; and , the approximate integral gain, where and are the same as in [14]. Then, (34) and (35) can be transformed intowhere as and .

Obviously, (36) is still valid; that is, if is Hurwitz, the closed-loop system still has accurate tracking performance.

#### 5. Simulation

*Example 1. *The practical model of the simulation in this section is used in [14, 24, 27, 28]. The estimated mathematical model of the motion control system iswhere is the output position, is the control voltage sent to the power amplifier, and is the torque disturbance. The corresponding second-order ADRC is designed, and the parameters are tuned asFor a better view of the comparison among different values of , the four values are selected. The responses of the ADRC system are shown in Figure 3 for a step reference, the step disturbance with an amplitude of −10 is injected at seconds, and a 0.1% white noise in the sensor is added (this noise is added to all examples). The sampling time is 1 kHz. It is shown that as increases, the system’s antidisturbance capability increases, and the impact of noise increases (but the increase is small). In fact, the adjustment range of is very large, and the system becomes unstable when . Therefore, in terms of increasing the system’s antidisturbance capability, larger should be chosen. In addition, when , the system can accurately track the step input before *t* = 2, but after the step disturbance is injected, the accurate tracking performance of the system is lost, which shows that the integral effect of ADRC is only related to .

*Example 2. *Consider a second-order plantwhere is unknown. In this simulation, the tuning parameters are set asThe responses for a step reference at *t* = 0 and a step input disturbance with an amplitude of −20 at *t* = 2 are shown.

For , it can be seen from Figure 4 that as increases, the system’s antidisturbance capability decreases, but the effect of sensor noise also decreases. Based on Figure 5, it is clear that as increases, the cutoff frequency of is reduced; that is, the bandwidth of is reduced, which is why both the system’s antidisturbance capability and the influence of sensor noise are reduced. Further simulation results show that increasing can also improve the robustness of the system because the amplitude margin and phase margin of increase. In addition, for the minimum phase system, increasing will not change the phase-frequency characteristic of . But only the overall downshift of the amplitude-frequency characteristic curve is caused. Therefore, within a certain range, increasing will necessarily increase the amplitude margin and phase margin.

For and , comparing (a) and (b) in Figures 6 or 7, we can see that the robustness and antidisturbance capability of the system decrease as increases, which is consistent with the description of Case 2 in Table 1. In fact, it is known by the Bode stability criterion that when , the system of becomes unstable; when , the system of becomes unstable; and when , the system of becomes unstable.

**(a)**

**(b)**

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**(b)**

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#### 6. Conclusion

In this paper, the linear ADRC method was investigated. The integral effect of the ADRC was analyzed under the premise that ADRC was transformed into a new form. Then, ADRC was changed into an IMC framework, and an almost necessary and sufficient condition for stability and tracking performance of ADRC systems have been proposed on this basis. In addition, some useful corollaries have been proposed so that the traditional open-loop frequency-domain analysis method can be applied to ADRC system stability analysis. Furthermore, it also provides a theoretical principle and theoretical guidance for tuning of . To improve the performance of ADRC, (the approximate integral gain) was treated as a separated adjustable parameter according to the new structure. Finally, simulations were used to verify the effectiveness of and tuning methods. Future research efforts include the extension of the results to nonlinear systems and the practical applications of the proposed approach.

#### Data Availability

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

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (61501429).