# Analysis and Tuning for Active Disturbance Rejection Control

Authors: Fan Wang; Ran-Jun Wang; En-Hai Liu
Journal: Mathematical Problems in Engineering
 (2019)

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## 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.


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## Body

## 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 tuningb0 to increase the robustness of the system and failed to provide theoretical guidance for the tuning of b0 (b0 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 Section2, 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:(1)x˙1=x2,x˙2=x3,⋮x˙n=f1,x2,…,xn,wt+bu,y=x1,where y is the regulated output, xii=1,2,…,n are the system state, u is the input force, b is the high-frequency gain, and fx1,x2,…,xn,wt 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 as(2)v˙1=v2,v˙2=v3,⋮v˙n−1=vn,v˙n=λnψv1−r,v2λ,…,vnλn−1,where r is the reference input, vii=1,2,…,n are the output, and λ is the adjustable speed factor. ψv1−r,v2/λ,…,vn/λn−1 is a solution that guarantees the fast convergence from v1 to r.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:(3)z˙1=z2−β1z1−y,z˙2=z3−β2z1−y,⋮z˙n=zn+1−βnz1−y+b0u,z˙n+1=−βn+1z1−y,where zii=1,2,…,n are the outputs of the ESO; βii=1,2,…,n are the observer gains; and b0 is the nominal value of b, and bb0>0. For the sake of discussion, we assume that b>0.SEF is used to compensate the total disturbance and eliminate the residual error, and the SEF is designed as(4)u=u0−zn+1b0,u0=∑i=1nkivi−zi,where kii=1,2,…,n are the controller gains.Substituting (4) into (3), we have(5)z˙1=z2−β1z1−y,z˙2=z3−β2z1−y,⋮z˙n=−βnz1−y+u0,z˙n+1=−βn+1z1−y.From (5), we can see that zn+1 can be expressed as(6)zn+1=βn+1βizi−∫zi+1dt,1≤i≤n−1.Wheni=1, we have(7)zn+1=βn+1β1z1−∫z2dt.Thus, with a TD and an SEF, the structure of ADRC can be transformed into the form in Figure1.Figure 1
Structure of ADRC.w: total disturbance; d: sensor noise.Obviously, from Figure1, 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 as(8)z˙s=Aszs+BsKsr^+βsy,where zs=z1,z2,…,znT∈Rn,(9)As=−β110…00−β201…00⋮⋮⋮⋱⋮⋮−βn−100…10−βn−k1−k2−k3…−kn−1−kn∈Rn×n,Bs=0,0,…,b0T∈Rn,Ks=k1,k2,…,knb0∈Rn,βs=β1,β2,…,βnT∈Rn,r^=v1,v2,…,vnT∈Rn.Applying Laplace transformation to (8), we can obtain(10)Z1s=∑j=0n−1∑i=1n−j−1ki+j+1βisj+βn−jsjMsYs+1MsR^s,Z2s=∑j=1n−1∑i=2n−j−1ki+jβisj+βn−j+1sj−k1β1MsYs+β1+sMsR^s,where Ms=detsI−As=sn+∑i=0n−1βn−i+∑j=in−1kiβj−isi, in which I is the identity matrix. Ys, R^s, Z1s, and Z2s are the Laplace transforms of y, r^, z1, and z2, respectively.According to (7) and (10), we have(11)Zn+1s=βn+1β1Z1s−1sZ2s=βn+1β1β1sn−1+β1∑i=−1n−2ki+2siMsYs+β1sMsR^s=βn+1β1β1sn+β1∑i=0n−1ki+1sisMsYs+β1sMsR^s=βn+1β1M1sMsYs+β1MsR^s1s,where Zn+1s is the Laplace transform of zn+1. And it can be known from (4) that the feedback of “total disturbance” is completed by the following formula:(12)u′=zn+1b0.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, b0 can be used as a tuning parameter to improve the control performance of ADRC. However, the tuning of b0 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 b0 can reduce the integral gain; that is, the integral time constant is increased to increase the system robustness. However, if b0 is too large, the integral action will become so weak that the system will have steady-state deviation. Therefore, we must choose an eclectic b0.


## 3. System Performance Analysis

Let the set of transfer functions be(13)Ps=lmsm+⋯+l1s+l0ahsh+⋯+a1s+a0,where m and h are positive integers, and all m≤h. ADRC does not need the complete models of the controlled plant and the disturbance, but the relative order of Ps; that is, n=h−m, b=lm/ah>0, and l0≠0.


### 3.1. System Transformation

Assumption 1.
A TD can accurately track the reference input and get its derivatives [18].
Letz=z1,z2,…,zn,zn+1T. Then, (3) can be transformed into the following form:(14)z˙=Aez+βy−y^+Bu,y^=Cz,where(15)Ae=010…0001…0⋮⋮⋮⋱⋮000…1000…0∈Rn+1n+1,B=0,0,…,b0,0T∈Rn+1,C=1,0,…,0,0∈Rn+1,β=β1,β2,…,βn,βn+1T∈Rn+1.
Substituting (4) into (14), we obtain(16)z˙=Ae−βC−BKz+BKv+βy,u=Kv−z,where(17)K=k1,k2,…,kn,1b0∈Rn+1,v=v1,v2,…,vn,0T∈Rn+1.
By taking Laplace transform of (16), we have(18)sZs=Ae−βC−BKZs+BKVs+βYs,(19)Us=KVs−Zs,where Zs, Us, and Vs are the Laplace transforms of z, u, and v, respectively.
Substituting (19) into (18), we have(20)Us=1−KsI−Ae+βC+BK−1BKVs−KsI−Ae+βC+BK−1βYs.
From Assumption 1, we can see that(21)Vs=1,s,s2,…,sn−1,0TRs=FsRs,where Rs is the Laplace transform of r. Thus, (20) can be further transformed into(22)Us=1−KsI−Ae+βC+BK−1BKFsRs−KsI−Ae+βC+BK−1βYs≕1−KA−1BKFsRs−KA−1βYs≕C1sRs−C2sYs,where(23)C1s=1−KA−1BKFs,C2s=KA−1β,A=sI−Ae+βC+BK.
The 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 Figure2.Figure 2
Structure of TDF-IMC.Remark 2.
It can be shown in a similar way to Theorem 1 in [24] that an nth-order ADRC structure shown in Figure 1 can be transformed into a TDF-IMC structure in Figure 2, where P is the plant to be controlled, Pm is the nominal model of P, Q is the tracking IMC controller, Qd is the disturbance rejection IMC controller, and e is the external disturbance.
According to Figure2, we have(24)Us=Q1−PmQdRs−Qd1−PmQdYs.
Combined with (22), the following equation can be obtained:(25)Q=C11+PmC2,Qd=C21+PmC2.
With the configuration in Figure3, nine transfer functions from r,e,dT to u′,u,yT are given by(26)Q00ηQ−ηPQd−ηQdηPQη1−PmQdP−ηPQd,where(27)η=11+QdP−Pm.
If the above nine transfer functions are stable, then the closed-loop system is said to be internally stable.Figure 3
Responses of Example 1 (dash-dot:kn+1=0; solid: kn+1=15; dash: kn+1=27.78; dot: kn+1=60).


### 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 inP, P is very different from Pm. 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 Pm=P.Then, (26) can be transformed into(28)Q00Q−PQd−QdPQ1−PQdP−PQd,where Q and Qd are only related to the parameters β, K, and b0. In order to simplify parameter tuning, β and K are usually reduced to two tuning parameters as suggested in [14]: ωc, the controller bandwidth, and ωo, the extended state observer bandwidth.Substituting (25) into (28) and considering C1=C11/C12, C2=C21/C22, and P=P1/P2, we have(29)1δsP2C1100P2C11P1C21−P2C21P1C11P1C22−P1C21,where(30)δs=P2C22+P1C21,C12=C22=detA=Ms,C11=C22+detABK0KFs,C21=−detAβK0.Assumption 2.
M
=
C
22
/
s is Hurwitz.
It can be seen from equation (22) that the calculation of C22 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 C22/s is Hurwitz.Theorem 1.
Under Assumption 1 and Assumption 2, for givenb0, ωc>0, and ωo>0, the closed-loop system is said to be internally stable if and only if δs is Hurwitz.Proof of Sufficiency.
Ifδs is Hurwitz, then the nine transfer functions of (28) must be stable. For given b0, ωc>0, and ωo>0, the closed-loop system is internally stable.Proof of Necessity.
For givenb0, ωc>0, and ωo>0, there are no zero roots in C11 and C21.
And if Assumption 2 holds, neitherC1 nor C2 have unstable zeros and unstable poles that can cancel each other out; that is, C1 and C2 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 δs is Hurwitz is as follows.
Assume thatδs is not Hurwitz and one of the transfer functions of (28), such as(31)Q=P2C11P2C22+P1C21,whose numerator and denominator contain an identical factor s+s∗, and Res∗≤0. If s+s∗ exists in P2, then it must exist in C21 (i.e., s∗≠0) as Ps is a coprime polynomial. And since M=C22/s is Hurwitz, s+s∗ does not exist in P1C22; then,(32)1−PQdP=P1C22P2C22+P1C21is unstable. Similarly, if s+s∗ exists in C11, (29) will also be unstable. In fact, as long as δs 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, butsC1, sC2, and Ps are all stable. However, this situation is rare, and it is easy to assure that C22/s is Hurwitz when ωc>0 and ωo>0. If we exclude such a situation, the condition that δs 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 as(33)QP=P1C11P2C22+P1C21.
SinceC22=Ms, it is clear that the coefficient of s0 of C22 is zero, as long as C11 and C21 contain constants and no zeros of Ps are located at origin in the complex plane, and the static gain of the main channel function is only determined by the constant terms C11 and C21, but not related to C22 and Ps:(34)C11=∑j=0n+1βn+1−jsj∑i=1nkisi−1b0,(35)C21=∑j=0nsj∑i=1j+1βn+i−jkib0.Theorem 2.
Under Assumption 2, for givenb0, ωc>0, ωo>0, and all P∈℘, the closed-loop system is said to have accurate tracking performance if and only if δs is Hurwitz.Proof of Sufficiency.
Ifδs is Hurwitz, then no zeros of Ps are located at origin in the complex plane. And according to (34) and (35), we have(36)lims⟶0P1C11P2C22+P1C21=1.
According 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, δs 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 byδs, while its attenuation ability to disturbance and sensor noise is determined by pQd (as can be seen from Figure 2 and (28)).Corollary 1.
Under Assumption 2, for givenb0, ωc>0, and ωo>0, the closed-loop system is said to be internally stable if pQd is stable and C21 is Hurwitz.Proof.
From (29), we can see that(37)pQd=P1C21δs=P1C21P2C22+P1C21.
IfQd is stable and C21 is Hurwitz, then δs 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, whenn≤4, for given b0, ωc>0, and ωo>0, the closed-loop system is said to be internally stable if and only if pQd is stable.
Before proof, the following lemma is given.Lemma 1 [25].
Polynomialφs is Hurwitz if μi≤0.4655 or μi+μi+1≤0.89 (p≥5), where(38)φs=α0sp+α1sp−1+⋯+αp,μi=αiαi+3αi+1αi+2.
LetC21=φs and ωc=θωo, θ>0.Proof.
(1)
Whenn=1or2, αi are all positive, so C21 is Hurwitz.(2)
Whenn=3,(39)μ0=α0α3α1α2=4θ3ωo6+3θ2ωo66θ3ωo5+12θ2ωo5+3θωo54θ3ωo4+18θ2ωo4+12θωo4+ωo4θ3ωo7=4θ3+18θ2+12θ+124θ3+66θ2+48θ+9<4θ3+18θ2+12θ+13.664θ3+18θ2+12θ+1<13.66<0.4655.(3)
Whenn=4, we have(40)μ0<13.5<0.4655,μ1<12.52<0.4655.
Therefore, based on Lemma 1, whenn≤4, C21 is Hurwitz. Then, the necessary and sufficient condition of Corollary 2 is easy to be proved.Remark 4.
Whenn=5,(41)μ0<13.4<0.4655,μ1<12.23=0.448,μ2<11.98=0.505.
μ
1
+
μ
2 may be greater than 0.89 for ∃θ>0. In fact, it is not hard to find by calculation that as n increases, the minimum value of the upper bound of μi increases for ∀θ>0. But C21 may still be Hurwitz because Lemma 1 is only a sufficient condition. Therefore, it is necessary to determine whether C21 is Hurwitz, when n≥5.Remark 5.
According to (28) and (37), the greater the bandwidth of pQd, the stronger the antidisturbance capability of the ADRC system and the greater the influence of the sensor noise.Remark 6.
WhenC21 is Hurwitz, according to Corollary 1 and Corollary 2, the stability margin of pQd can reflect the stability margin of the ADRC system to some extent. And it can be seen from (25) that pQd can be considered a closed-loop transfer function whose open-loop transfer function is PmC2. In this way, we can use frequency-domain analysis to analyze PmC2 so as to determine the stability of the whole ADRC system.


## 3.1. System Transformation

Assumption 1.
A TD can accurately track the reference input and get its derivatives [18].
Letz=z1,z2,…,zn,zn+1T. Then, (3) can be transformed into the following form:(14)z˙=Aez+βy−y^+Bu,y^=Cz,where(15)Ae=010…0001…0⋮⋮⋮⋱⋮000…1000…0∈Rn+1n+1,B=0,0,…,b0,0T∈Rn+1,C=1,0,…,0,0∈Rn+1,β=β1,β2,…,βn,βn+1T∈Rn+1.
Substituting (4) into (14), we obtain(16)z˙=Ae−βC−BKz+BKv+βy,u=Kv−z,where(17)K=k1,k2,…,kn,1b0∈Rn+1,v=v1,v2,…,vn,0T∈Rn+1.
By taking Laplace transform of (16), we have(18)sZs=Ae−βC−BKZs+BKVs+βYs,(19)Us=KVs−Zs,where Zs, Us, and Vs are the Laplace transforms of z, u, and v, respectively.
Substituting (19) into (18), we have(20)Us=1−KsI−Ae+βC+BK−1BKVs−KsI−Ae+βC+BK−1βYs.
From Assumption 1, we can see that(21)Vs=1,s,s2,…,sn−1,0TRs=FsRs,where Rs is the Laplace transform of r. Thus, (20) can be further transformed into(22)Us=1−KsI−Ae+βC+BK−1BKFsRs−KsI−Ae+βC+BK−1βYs≕1−KA−1BKFsRs−KA−1βYs≕C1sRs−C2sYs,where(23)C1s=1−KA−1BKFs,C2s=KA−1β,A=sI−Ae+βC+BK.
The 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 Figure2.Figure 2
Structure of TDF-IMC.Remark 2.
It can be shown in a similar way to Theorem 1 in [24] that an nth-order ADRC structure shown in Figure 1 can be transformed into a TDF-IMC structure in Figure 2, where P is the plant to be controlled, Pm is the nominal model of P, Q is the tracking IMC controller, Qd is the disturbance rejection IMC controller, and e is the external disturbance.
According to Figure2, we have(24)Us=Q1−PmQdRs−Qd1−PmQdYs.
Combined with (22), the following equation can be obtained:(25)Q=C11+PmC2,Qd=C21+PmC2.
With the configuration in Figure3, nine transfer functions from r,e,dT to u′,u,yT are given by(26)Q00ηQ−ηPQd−ηQdηPQη1−PmQdP−ηPQd,where(27)η=11+QdP−Pm.
If the above nine transfer functions are stable, then the closed-loop system is said to be internally stable.Figure 3
Responses of Example 1 (dash-dot:kn+1=0; solid: kn+1=15; dash: kn+1=27.78; dot: kn+1=60).


## 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 inP, P is very different from Pm. 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 Pm=P.Then, (26) can be transformed into(28)Q00Q−PQd−QdPQ1−PQdP−PQd,where Q and Qd are only related to the parameters β, K, and b0. In order to simplify parameter tuning, β and K are usually reduced to two tuning parameters as suggested in [14]: ωc, the controller bandwidth, and ωo, the extended state observer bandwidth.Substituting (25) into (28) and considering C1=C11/C12, C2=C21/C22, and P=P1/P2, we have(29)1δsP2C1100P2C11P1C21−P2C21P1C11P1C22−P1C21,where(30)δs=P2C22+P1C21,C12=C22=detA=Ms,C11=C22+detABK0KFs,C21=−detAβK0.Assumption 2.
M
=
C
22
/
s is Hurwitz.
It can be seen from equation (22) that the calculation of C22 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 C22/s is Hurwitz.Theorem 1.
Under Assumption 1 and Assumption 2, for givenb0, ωc>0, and ωo>0, the closed-loop system is said to be internally stable if and only if δs is Hurwitz.Proof of Sufficiency.
Ifδs is Hurwitz, then the nine transfer functions of (28) must be stable. For given b0, ωc>0, and ωo>0, the closed-loop system is internally stable.Proof of Necessity.
For givenb0, ωc>0, and ωo>0, there are no zero roots in C11 and C21.
And if Assumption 2 holds, neitherC1 nor C2 have unstable zeros and unstable poles that can cancel each other out; that is, C1 and C2 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 δs is Hurwitz is as follows.
Assume thatδs is not Hurwitz and one of the transfer functions of (28), such as(31)Q=P2C11P2C22+P1C21,whose numerator and denominator contain an identical factor s+s∗, and Res∗≤0. If s+s∗ exists in P2, then it must exist in C21 (i.e., s∗≠0) as Ps is a coprime polynomial. And since M=C22/s is Hurwitz, s+s∗ does not exist in P1C22; then,(32)1−PQdP=P1C22P2C22+P1C21is unstable. Similarly, if s+s∗ exists in C11, (29) will also be unstable. In fact, as long as δs 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, butsC1, sC2, and Ps are all stable. However, this situation is rare, and it is easy to assure that C22/s is Hurwitz when ωc>0 and ωo>0. If we exclude such a situation, the condition that δs 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 as(33)QP=P1C11P2C22+P1C21.
SinceC22=Ms, it is clear that the coefficient of s0 of C22 is zero, as long as C11 and C21 contain constants and no zeros of Ps are located at origin in the complex plane, and the static gain of the main channel function is only determined by the constant terms C11 and C21, but not related to C22 and Ps:(34)C11=∑j=0n+1βn+1−jsj∑i=1nkisi−1b0,(35)C21=∑j=0nsj∑i=1j+1βn+i−jkib0.Theorem 2.
Under Assumption 2, for givenb0, ωc>0, ωo>0, and all P∈℘, the closed-loop system is said to have accurate tracking performance if and only if δs is Hurwitz.Proof of Sufficiency.
Ifδs is Hurwitz, then no zeros of Ps are located at origin in the complex plane. And according to (34) and (35), we have(36)lims⟶0P1C11P2C22+P1C21=1.
According 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, δs 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 byδs, while its attenuation ability to disturbance and sensor noise is determined by pQd (as can be seen from Figure 2 and (28)).Corollary 1.
Under Assumption 2, for givenb0, ωc>0, and ωo>0, the closed-loop system is said to be internally stable if pQd is stable and C21 is Hurwitz.Proof.
From (29), we can see that(37)pQd=P1C21δs=P1C21P2C22+P1C21.
IfQd is stable and C21 is Hurwitz, then δs 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, whenn≤4, for given b0, ωc>0, and ωo>0, the closed-loop system is said to be internally stable if and only if pQd is stable.
Before proof, the following lemma is given.Lemma 1 [25].
Polynomialφs is Hurwitz if μi≤0.4655 or μi+μi+1≤0.89 (p≥5), where(38)φs=α0sp+α1sp−1+⋯+αp,μi=αiαi+3αi+1αi+2.
LetC21=φs and ωc=θωo, θ>0.Proof.
(1)
Whenn=1or2, αi are all positive, so C21 is Hurwitz.(2)
Whenn=3,(39)μ0=α0α3α1α2=4θ3ωo6+3θ2ωo66θ3ωo5+12θ2ωo5+3θωo54θ3ωo4+18θ2ωo4+12θωo4+ωo4θ3ωo7=4θ3+18θ2+12θ+124θ3+66θ2+48θ+9<4θ3+18θ2+12θ+13.664θ3+18θ2+12θ+1<13.66<0.4655.(3)
Whenn=4, we have(40)μ0<13.5<0.4655,μ1<12.52<0.4655.
Therefore, based on Lemma 1, whenn≤4, C21 is Hurwitz. Then, the necessary and sufficient condition of Corollary 2 is easy to be proved.Remark 4.
Whenn=5,(41)μ0<13.4<0.4655,μ1<12.23=0.448,μ2<11.98=0.505.
μ
1
+
μ
2 may be greater than 0.89 for ∃θ>0. In fact, it is not hard to find by calculation that as n increases, the minimum value of the upper bound of μi increases for ∀θ>0. But C21 may still be Hurwitz because Lemma 1 is only a sufficient condition. Therefore, it is necessary to determine whether C21 is Hurwitz, when n≥5.Remark 5.
According to (28) and (37), the greater the bandwidth of pQd, the stronger the antidisturbance capability of the ADRC system and the greater the influence of the sensor noise.Remark 6.
WhenC21 is Hurwitz, according to Corollary 1 and Corollary 2, the stability margin of pQd can reflect the stability margin of the ADRC system to some extent. And it can be seen from (25) that pQd can be considered a closed-loop transfer function whose open-loop transfer function is PmC2. In this way, we can use frequency-domain analysis to analyze PmC2 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 toωc and ωo [14]. However, sometimes b0 is also adjusted as the third parameter to improve system performance because of unknown information on b and robustness requirement. But before this article, there were no specific theoretical principle and theoretical guidance for the tuning of b0. In fact, a solution to this problem was proposed in Remark 6, i.e., to solve this problem with frequency-domain analysis of PmC2.However, in practice, model information is sometimes difficult to know accurately. Therefore, a more concise method is needed to guide the tuning ofb0. According to the Routh criterion, a necessary condition for δs to be Hurwitz is that the coefficients of δs 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.Table 1
Tuning methods forb0.
P
s
b
0
1
∀
a
i
>
0
i
=
1,2
,
…
,
h but ∃li<0i=1,2,…,m
Increasingb0
2
∃
a
i
≤
0
i
=
1,2
,
…
,
h but ∀li≥0i=1,2,…,m
Decreasingb0
3
∃
a
i
<
0
i
=
1,2
,
…
,
h and ∃li<0i=1,2,…,m
More model information should be usedThe 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 zn+1, so in order to make the ADRC parameter more explicit, referring to Figure 1, βn+1/β1b0 is replaced with kn+1 as a separated adjustable parameter. In this way, zn+1 can be considered a separate integrating element, and kn+1 can be viewed as an approximate integral gain to adjust.So, the number of parameters of the ADRC can be expanded to four:ωc, the controller bandwidth; ωo, the observer bandwidth; b0; and kn+1, the approximate integral gain, where ωc and ωo are the same as in [14]. Then, (34) and (35) can be transformed into(42)C11=c2ns2n+⋯+c1sb0+k1β1kn+1,C21=qnsn+⋯+q1sb0+⋯k1β1kn+1,where c1=q1≠0 as ωc>0 and ωo>0.Obviously, (36) is still valid; that is, if δs 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 is(43)y¨=−1.41y˙+23.2Td+23.2u,where y is the output position, u is the control voltage sent to the power amplifier, and Td is the torque disturbance. The corresponding second-order ADRC is designed, and the parameters are tuned as(44)ωc=30,ωo=50,b0=40,βn+1β1b0=27.78.
For a better view of the comparison among different values ofkn+1, the four values kn+1=0,kn+1=15,kn+1=35.92,andkn+1=60 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 t=2 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 kn+1 increases, the system’s antidisturbance capability increases, and the impact of noise increases (but the increase is small). In fact, the adjustment range of kn+1 is very large, and the system becomes unstable when kn+1>300. Therefore, in terms of increasing the system’s antidisturbance capability, larger kn+1 should be chosen. In addition, when kn+1=0, 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 zn+1.Example 2.
Consider a second-order plant(45)G1=5s+1s+η,where η is unknown. In this simulation, the tuning parameters are set as(46)ωc=30,ωo=50,kn+1=βn+1β1b0.
The responses for a step reference att = 0 and a step input disturbance with an amplitude of −20 at t = 2 are shown.
Forη=5, it can be seen from Figure 4 that as b0 increases, the system’s antidisturbance capability decreases, but the effect of sensor noise also decreases. Based on Figure 5, it is clear that as b0 increases, the cutoff frequency of PmC2 is reduced; that is, the bandwidth of pQd 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 b0 can also improve the robustness of the system because the amplitude margin and phase margin of PmC2 increase. In addition, for the minimum phase system, increasing b0 will not change the phase-frequency characteristic of PmC2. But only the overall downshift of the amplitude-frequency characteristic curve is caused. Therefore, within a certain range, increasing b0 will necessarily increase the amplitude margin and phase margin.
Forη=−5 and η=−15, comparing (a) and (b) in Figures 6 or 7, we can see that the robustness and antidisturbance capability of the system decrease as b0 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 η<−15, the system of b0=7 becomes unstable; when η<−24, the system of b0=5 becomes unstable; and when η<−34, the system of b0=3 becomes unstable.Figure 4
Responses ofη=5 (solid: b0=3; dash: b0=5; dot: b0=7).Figure 5
Bode diagram ofPmC2 when η=5 (solid: b0=3; dash: b0=5; dot: b0=7).
(a)
(b)Figure 6
Responses of (a)η=−5 and (b)η=−15 (solid: b0=3; dash: b0=5; dot: b0=7).
(a)
(b)Figure 7
Bode diagram ofPmC2 when (a) η=−5 and (b) η=−15 (solid: b0=3; dash: b0=5; dot: b0=7).
(a)
(b)


## 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 ofb0. To improve the performance of ADRC, kn+1 (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 b0 and kn+1 tuning methods. Future research efforts include the extension of the results to nonlinear systems and the practical applications of the proposed approach.



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