Received: 8 January 2021 Revised: 29 September 2021 Accepted: 30 September 2021

DOI: 10.1002/rnc.5845

S P E C I A L I S S U E A R T I C L E

An ADRC-based PID tuning rule

Sheng Zhong1,2,3 Yi Huang1,2 Lei Guo1,2

1Key Laboratory of Systems and Control,Academy of Mathematics and SystemsScience, Chinese Academy of Sciences,Beijing, People’s Republic of China2School of Mathematical Sciences,University of Chinese Academy ofSciences, Beijing, People’s Republic ofChina3Beijing Aerospace Automatic ControlInstitute, Beijing, People’s Republic ofChina

CorrespondenceYi Huang, Key Laboratory of Systems andControl, Academy of Mathematics andSystems Science, Chinese Academy ofSciences, Beijing, People’s Republic ofChina.Email: yhuang@amss.ac.cn

Funding informationNational Key R & D Program of China,Grant/Award Number: 2018YFA0703800;National Natural Science Foundation ofChina, Grant/Award Number: U20B2054

AbstractThis article puts forward an active disturbance rejection control (ADRC) basedtuning rule to the well-known proportional-integral-derivative (PID) control fora class of multi-input multi-output non-affine uncertain systems. It is provedthat the PID control with the ADRC-based tuning rule can guarantee satisfiedtracking performance, both for the transient phase and the steady state. Fur-thermore, it is illustrated that, with the new tuning rule, the PID control is ableto timely estimate and compensate for the nonlinear and coupled uncertain-ties, which explains the reason why PID control does have large-scale robustnesswith respect to the uncertainties and can achieve decoupling control with asimple structure. The theoretical results are verified by simulations.

K E Y W O R D S

active disturbance rejection control, extended state observer, MIMO non-affine uncertain system,proportional-integral-derivative

1 INTRODUCTION

It is well-known that the classical PID (proportional-integral-derivative) control, which has nearly 100 years of history,is still the most widely and successfully used method in engineering practice even though numerous advanced controltechniques have been proposed.1 As mentioned in Reference 2, more than 95% of the control loops in process controlare of PID type. However, even though the classical PID control achieves such an amazing success in practice, there isstrong evidence that PID control remain poorly understood and lots of practical PID loops are poorly tuned.3 Most tuningmethods of PID parameters are based on the experience and experiments, lacking theoretical supports.4 In fact, except fora few studies for nonlinear systems,5-7 most of the literatures on PID theory are focused on linear systems,2,8,9 causing asignificant gap between the basic theory and practical application. References 5,6 provided some necessary and sufficientconditions, which guaranteed the global convergence of the closed-loop system, on the selection of the PID parameters fora class of nonlinear uncertain systems. References 7,10 gave a 3-dimensional unbounded manifold to PID parameters fornon-affine nonlinear systems, such that the closed-loop system is globally exponentially stable. However, how to designa concrete tuning rule for PID parameters, which can guarantee strong robustness as well as good tracking performance,for nonlinear uncertain systems, is still a challenging problem.

Active disturbance rejection control (ADRC), whose basic idea is to on-line estimate and compensate for the totaldisturbance, which lumps the internal uncertainties and the external disturbances, by a well-designed extended state

Int J Robust Nonlinear Control. 2021;1–14. wileyonlinelibrary.com/journal/rnc © 2021 John Wiley & Sons Ltd. 1

2 ZHONG et al.

observer (ESO), was originally proposed by Han in 1990s.11 Because of its strong ability in dealing with a vast rangeof uncertainties/disturbances and in achieving great transient response, ADRC has become quite attractive to appliedresearchers.12-16 Stimulated by increasingly more successful applications, the theoretical research on the capability ofADRC has been gradually set in motion.17-22

Recently, a new parameter formula connecting PID control and ADRC, which is found to be beneficial for the designof both controllers, has been proposed for affine nonlinear systems with no uncertainty in the control channel.23,24 Onthe one hand, parameter formula can be used to find a quantitative lower bound for the parameter of the ADRC. On theother hand, formula provides a new and concrete tuning rule for PID parameters. The PID control, tuned by the newrule, has the power of strong robustness and excellent tracking performance. In practice, there are many systems arein the non-affine form, for example, robot manipulator systems,25 servo systems,26 flight systems,27 inverted pendulumsystems,28,29 and wind energy conversion systems.30 Control designs and analysis on multi-input multi-output (MIMO)non-affine uncertain systems are much more challenging because of the uncertainties caused by the strong coupling ofhigh-dimensional states and non-affine control input.

In this article, via generalizing the results of References 23,24, an ADRC-based tuning rule to the classical PID con-trol is proposed for MIMO non-affine uncertain systems. The analysis results on the ADRC-based PID control furtherillustrate the rationale behind the success of PID control. The rest of the article is organized as follows. In Section 2,inspired by the design of parallel reduced-order ESOs based linear ADRC, a new and quantitative tuning rule on PIDparameters is given for controlling MIMO non-affine systems. In Section 3, the analysis results show that the parallelPID controllers, tuned by the ADRC-based rule, have function of on-line estimating and compensating for the nonlin-ear and coupled uncertainties. Moreover, it is proved that, by tuning a parameter, which corresponds to the bandwidthof the ESO and will automatically bring the tuning of the parameters kp, kd, and kI , the distance between the closed-looptracking process and the ideal one in the whole time-domain can be monotonously improved. These results providethe rationale why PID control can achieve robust decoupling control and satisfied tracking performance, if it can beproperly tuned. Section 4 gives some simulations to verify the theoretical analysis. Finally, Section 5 concludes thearticle.

2 AN ADRC-BASED TUNING RULE FOR PID CONTROL

Consider the following non-affine uncertain system:{x1 = x2,

x2 = f (x1, x2,u) ,(1)

where x1 = [x11, … , x1n]T ∈ Rn and x2 = [x21, … , x2n]T ∈ Rn are the state vectors and can be measured, u =[u1, … ,un]T ∈ Rn is the control input, f (x1, x2,u) ∶ Rn × Rn × Rn → Rn is an unknown nonlinear function of the state(x1, x2) and the control input u. More specifically, the unknown function f (x1, x2,u) lies in the following functionspace:

={

f ∈ C1 (Rn × Rn × Rn,Rn) |||||||||||| 𝜕f𝜕x1

|||||||| ≤ L1,|||||||| 𝜕f𝜕x2

|||||||| ≤ L2,𝜕f𝜕u

≥ B > 0,∀ (x1, x2,u) ∈ Rn × Rn × Rn}

, (2)

where 𝜕f𝜕x1

,𝜕f𝜕x2

, and 𝜕f𝜕u

denote the n × n Jacobian matrices of f with respect to x1, x2, and u, ||⋅|| denotes the Euclidean norm,L1 and L2 are positive constants, B = diag

{b1, ..., bn

}is a n × n positive definite diagonal matrix, and C1(Rn × Rn × Rn,Rn)

denotes the space of functions from Rn × Rn × Rn to Rn, which are continuous in (x1, x2) and u, with continuous partialderivatives.

Remark 1.

i) Equation (2) gives a mathematic description for the space of unknown function f (x1, x2,u). The constants L1,L2 andmatrix B are used as priori information to quantitatively describe the uncertainties of the system. Specifically, L1 andL2 are the upper bounds for the norms of the Jacobian matrices of the unknown function f with respect to x1 and x2,respectively. B is a lower bound for the Jacobian matrix of the unknown function f with respect to u. In fact, we just

ZHONG et al. 3

need to know 𝜕f𝜕u

is positive or negative and the corresponding bound. If the matrix 𝜕f𝜕u

is diagonal dominant, it is easyto make it to be positive or negative via some simple transformation on the sign of the control signal ui. Without lossof generality, we assume 𝜕f

𝜕u≥ B > 0 in this article.

ii) The system (1) can represent many practical systems, for examples, moving bodies in R3, where x1 and x2 representthe position and velocity of the moving body, double-inverted pendulums systems,29 robot systems,31 and multi-agentsystems.32

The control objective is to make the controlled variable x1 track a given bounded dynamic reference signal vectory∗(t) =

[y∗1(t), … , y∗n(t)

]T ∈ Rn, which satisfies

limt→∞

y∗(t) = y∗∗, limt→∞

y∗(t) = 0, limt→∞

y∗(t) = 0,

where y∗(t) and y∗(t) are the first and second derivatives of y∗(t), and y∗∗ ∈ Rn is a constant vector.Based on the idea of ADRC, the uncertain system (1) can be reexpressed as follows:{

x1 = x2,

x2 = ftotal (x1, x2,u) + Bu,(3)

where ftotal (x1, x2,u) = f (x1, x2,u) − Bu represents the total disturbance of the system, including the uncertainties causedby the nonlinear coupled factors and the non-affine control input.

Treating ftotal(x1, x2,u) as an extended state of the system (1), since the state x2 is measurable, the following n parallelreduced-order ESOs can be designed:33

{��i = −𝜔o𝜉i − 𝜔2

ox2i − 𝜔obiui, 𝜉i(0) = −𝜔ox2i(0),f i = 𝜉i + 𝜔ox2i,

, i ∈ n, (4)

where f =[

f 1, … , f n

]Tis the estimation of the total disturbance ftotal with f (0) = 0, n ≜ {1, 2, … ,n} , 𝜔o is the only

parameter of the ESO (4) to be tuned.To have a satisfied transient tracking performance, the following transient process ri(t) ∈ R(i ∈ n), which is shaped

from y∗i (t)(i ∈ n) by a stable linear filter, can be designed:

ri = −2criri − c2ri(

ri − y∗i (t)), ri(0) = x1i(0), ri(0) = x2i(0), (5)

where cri > 0 is the parameter for tuning the speed of the transient process ri(t).Then, the following ADRC law

ui =kapi ei + kadi ei − f i + ri

bi, i ∈ n, (6)

can be designed to make x1i track ri(t), where ei = ri − x1i and ei = ri − x2i. The term f ibi

is used to timely compensate forthe total disturbance. kapi and kadi are the feedback parameters to be tuned.

Remark 2. In the ADRC law (4) and (6), the i th ESO (4) only uses the signals x2i and ui to estimate the i th element of thetotal disturbance ftotal (x1, x2,u) . Further, the control signal ui is generated only using the signals x1i and x2i, independentto the coupled and non-affine model ftotal. The key of the ADRC law is the performance of the ESO (4). 𝜔o, which is theonly parameter of the ESO (4) to be tuned and is named as the bandwidth of the ESO (4), characterizes the ability of theESO (4) for estimating ftotal.

Substituting (6) into (4), it can be deducted that

f i = −𝜔okadi ei − 𝜔oei − 𝜔okapi∫t

0ei(𝜏)d𝜏. (7)

4 ZHONG et al.

Thus, the ADRC law (6) can be reexpressed as

ui =(

kapi + 𝜔okadi

)ei +(

kadi + 𝜔o)

ei + 𝜔okapi∫ t0 ei(𝜏)d𝜏 + ri

bi, (8)

which turns out to be a PID control with a special tuning rule.Enlightened by the ADRC law (8), a new and concrete tuning rule for the following classical PID control

uPIDi = kpi ei + kdi ei + kIi∫t

0ei(𝜏)d𝜏 + b−1

i ri, i ∈ n, (9)

can be proposed, where ri is a feedforward term to improve the dynamic accuracy of the closed-loop system, kpi , kdi , andkIi are the controller parameters and can be determined as follows:

kpi =kapi + 𝜔okadi

bi, kdi =

kadi + 𝜔o

bi, kIi =

𝜔okapi

bi, i ∈ n. (10)

Remark 3. The ADRC-based PID control law uPIDi (9) also only uses the ith tracking errors ei and ei, which makes thePID control a simple decoupling control for the non-affine system (1).

Remark 4. Designing ri(t) aims to make the initial values of tracking error ei(0) = 0 and ei(0) = 0. Otherwise, the out-put of the ESO (4), that is, f i(t), will be related to the initial values ei(0), ei(0), and ri(t), rather than only a combinationof P–I–D three terms shown in (7). Therefore, the transient process ri(t) and the feedforward ri

(i ∈ n)

in the designof both controllers play a prominent part to deduce an inherent relationship between the PID control (9) and theADRC (6).

More meaningfully, the PID control with the new tuning rule (10) can be described as an ADRC-based PID control.The function of each PID control uPIDi can be divided into two parts:

uPIDi =

upd0i⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

kapi ei + kadi ei + ri

bi+

upidfi⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

𝜔okadi ei + 𝜔oei + 𝜔okapi∫t

0ei(𝜏)d𝜏

bi. (11)

upidfiis a suitable linear combination of the P-part 𝜔okadi ei, the D-part 𝜔oei and the I-part 𝜔okapi∫ t

0 ei(𝜏)d𝜏, which has asimilar function of the ESO (4), that is, to on-line estimate and compensate for the influence of the nonlinear and coupleduncertainties ftotal(x1, x2,u) on the ith state x2i. Another part upd0i , which consists of the simple PD feedback of the ithtracking errors ei and ei, along with a feedforward term ri, acts as assigning the dominant poles of the closed-loop system.

From above discussion, the key of upidfiis 𝜔o, which corresponds to the bandwidth of the ESO (4) and also charac-

terizes the ability of the PID control (9) for estimating ftotal (x1, x2,u). In the next section, with the aid of the conditionprovided in Reference 10 for the stabilized PID design for non-affine nonlinear systems, a quantitative lower bound to𝜔o to assure the stability of an ADRC-based PID tuning rule (9) and (10) is given. Moreover, the relationship between theperformance of the tracking transient process and 𝜔o of the PID rule (9) and (10) is developed for the MIMO non-affinesystem (1), revealing that the ADRC-based tuning rule makes the PID controller have the capability of actively estimatingand compensating for the disturbance caused by the coupled non-affine uncertainties.

3 MAIN RESULTS

First, according to the condition (3), given in Reference 10, on the stabilized PID parameters for non-affine systems, aquantitative lower bound to 𝜔o of the PID tuning rule (10) to guarantee the stability of the closed-loop system is studied.

To describe the set of 𝜔o with a clear expression, the parameters kapi and kadi are tuned as:

kapi = 𝜔2c , kadi = 2𝜔c, i ∈ n, (12)

ZHONG et al. 5

where −𝜔c is any given negative number, reflecting the poles of the error equation ëi(t) + kadi ei(t) + kapi ei(t) = 0(

i ∈ n),

which is the expected pole-placement error equation with dominant poles −𝜔c under the PID control (9)–(10) (see,Theorem 2 below).

Then, denote

Ω1 ={𝜔 ∈ R|(2𝜔c𝜔)3 + (20𝜔4

c − (L1 + 3𝜔2c )2)𝜔2 +

(8𝜔5

c − 4L1𝜔c(

L1 + 3𝜔2c))

𝜔 − 4L21𝜔

2c = 0

},

Ω2 ={𝜔 ∈ R|48𝜔4

c𝜔6 + n1𝜔

5 + n2𝜔4 + n3𝜔

3 + n4𝜔2 + n5𝜔 + n6 = 0

}, (13)

where

n1 = −8𝜔c(

L21 + L1L2𝜔c + 10L1𝜔

2c + L2

2𝜔2c + 11L2𝜔

3c − 11𝜔4

c),

n2 = −4𝜔2c(17L2

1 + 12L1L2𝜔c + 114L1𝜔2c + 13L2

2𝜔2c + 104L2𝜔

3c − 19𝜔4

c)

n3 = −4𝜔3c(56L2

1 + 25L1L2𝜔c + 232L1𝜔2c + 30L2

2𝜔2c + 159L2𝜔

3c − 64𝜔4

c),

n4 = −4𝜔4c(88L2

1 + 22L1L2𝜔c + 202L1𝜔2c + 28L2

2𝜔2c + 92L2𝜔

3c − 41𝜔4

c),

n5 = −16𝜔5c(16L2

1 + 3L1L2𝜔c + 18L1𝜔2c + 2L2

2𝜔2c + 7L2𝜔

3c − 2𝜔4

c),

n6 = −16𝜔6c(4L2

1 + 2L1𝜔2c + 2L2L1𝜔c + 𝜔4

c).

Remark 5. Since the last term of the polynomials in both Ω1 and Ω2 are negative, both polynomial equations have positivesolutions. Therefore, the two sets Ω1 and Ω2 will not be empty. Furthermore, define

kp = kpi bi = 𝜔2c + 2𝜔o𝜔c, kd = kdi bi = 2𝜔c + 𝜔o, kI = kIi bi = 𝜔o𝜔

2c , i ∈ n,

k1 = 4kIkpkd −(

3kI + L1kd

)2, k2 = 4kpk

2d −(

2kp + L2kd

)2,

k3 =(−4kIk

2d + 2k

2pkd +

(3kI + L1kd

)(2kp + L2kd

))2. (14)

Then, k1 > 0 if 𝜔o > max𝜔∈Ω1{𝜔}, and k1k2 > k3 if 𝜔o > max𝜔∈Ω2{𝜔}.

Set

𝜔∗o = max

{𝜔o1, 𝜔o2

}, (15)

where

𝜔o1 = max𝜔∈Ω1

{𝜔}, 𝜔o2 = max𝜔∈Ω2

{𝜔}. (16)

The following theorem will show that 𝜔∗o is a lower bound to the parameter 𝜔o, which can guarantee global

convergence of the closed-loop system.

Theorem 1 (Convergence). Consider the PID control with the tuning rule (9)–(10). Then, for any unknown function f ∈ with any given L1 and L2, and any given 𝜔c, if the parameter 𝜔o satisfies 𝜔o > 𝜔∗

o, the closed-loop system (1), (9), and (10)has the property:

limt→∞

x1(t) = y∗∗, limt→∞

x2(t) = 0,

for any initial value (x1(0), x2(0)) ∈ Rn × Rn and any y∗∗ ∈ Rn.

The proof of Theorem 1, which is developed based on the related proofs in Reference 10, is given in Appendix A.From Theorem 1, it shows that, with the ADRC-based tuning rule, the state x1 under PID control can converge to any

set-point y∗∗, for any initial values, as long as 𝜔o is larger than the lower bound 𝜔∗o , which is irrelevant to the initial values

and the reference signals, and can be calculated through (15). Moreover, it can be seen that the boundness of the term 𝜕f𝜕u

is not needed to guarantee the convergence of the closed-loop system.

6 ZHONG et al.

Denote ftotal =[ftotal1 , … , ftotaln

]T , the estimation error between upidfiand the uncertainty as efi = ftotali − upidfi

, ftotali =fi(x1, x2,u) − biuPIDi and the tracking error as ei(t) = ri(t) − x1i(i ∈ n). Note that ëi(t) + kadi ei(t) + kapi ei(t) = 0 is the idealtracking performance for a pure integrator system, determined by the given pole-placement via feedback parameters kapi

and kadi . Theorem 2 will show that, by increasing the parameter𝜔o > 𝜔∗o , both the estimation error efi and the performance

of tracking transient process can be improved in whole time-domain.

Theorem 2 (Decoupling and pole-placement performances). Consider the PID controlled non-affine uncertain system(1), (9), and (10), where the nonlinear unknown function f ∈ and |||||| 𝜕f

𝜕u|||||| ≤ L3, L3 is a positive constant. Then, there exists

a positive 𝜂, which is independent of 𝜔o, such that for all 𝜔o > 𝜔∗o, the closed-loop system has the following properties:

(1) |ëi(t) + kadi ei(t) + kapi ei(t)| = |efi(t)|, limt→∞

efi (t) = 0, i ∈ n,

(2) |efi(t)| ≤ (|efi(0)| − 𝜂

𝜔o

)e−𝜔ot + 𝜂

𝜔o, ∀t ≥ 0, i ∈ n.

The proof of Theorem 2 is given in Appendix B.From Theorem 2 (2), the estimation error efi of upidfi

can be divided into two parts. The first part(|efi(0)| − 𝜂

𝜔o

)e−𝜔ot

can be rapidly tuned to zero by the parameter 𝜔o. The second part 𝜂

𝜔odecreases with 𝜔o. Theorem 2 (2) illustrates that

the suitable linear combination of the three terms of P–I–D, that is, upidfi, which only utilizes the information ei and ei of

ith channel, is able to on-line estimate and compensate for the uncertainties, caused by the coupling of high-dimensionalstates and non-affine control input. This result explains the reason why PID control can achieve robust decoupling controlas well as satisfied tracking performance for MIMO non-affine system. Furthermore, Theorem 2 shows that, by increas-ing the parameter 𝜔o, the dynamic response of the closed-loop system, which only depends on the performance of theestimation error ef , can be made close to the ideal trajectory.

The properties given by Theorem 2 reveal the internal mechanisms of the ADRC-based PID control (9) and (10).Though not intuitive, the physical meanings of the parameters𝜔o, kapi , and kadi are clearer than those of the original kpi , kIi ,and kdi , that is, 𝜔o works to tune upidfi

to estimate and compensate for the influence caused by the unknown nonlinear,coupled and non-affine factors, while kapi and kadi work to assign the dominant poles.

Remark 6. Theorem 2 shows that the parameter 𝜔o characterizes the ability of the ADRC-based PID law (11) for esti-mating the total disturbance ftotal. The estimation upidfi

will be able to estimate ftotali more precisely when 𝜔o increases.Therefore, increasing 𝜔o will not cause the magnitude of the control input uPIDi to increase unboundedly, but will remainin a certain range, as long as the total disturbance ftotal is bounded.

Remark 7. Although, from Theorem 1, the condition |||||| 𝜕f𝜕u|||||| ≤ L3 is not necessary to make the closed-loop system global

convergence, it is used in Theorem 2 to guarantee the tracking performance for t ≥ 0. For practical systems, the partialderivative 𝜕f

𝜕u, which represents the gain of the control capability, is usually limited due to some physical restrictions.

Thus, the condition |||||| 𝜕f𝜕u|||||| ≤ L3 in Theorem 2 is realistic.

Remark 8. Actually, bi in the ADRC (4)–(6) can use other values, then the conditions for determine 𝜔∗o will be more

complex and its influence on the performance of the closed-loop systems should be analyzed from multiple aspects. Thisproblem has been further analyzed in References 34,35 for a kind of SISO systems. To simplify the design, in this article,the known information B is used in setting the parameters of ADRC-based PID.

4 SIMULATION

In this section, some simulations on a double-inverted pendulums connected by a torsional spring are given to demon-strate the ability of the proposed ADRC-based PID control and to verify the main results of the article.

The dynamics of the double-inverted pendulums system can be represented as follows:31

J1��1 = m1grsin𝜃1 − k𝜃1 + k𝜃2 + g1 (u1) ,J2��2 = m2grsin𝜃2 − k𝜃2 + k𝜃1 + g2 (u2) , (17)

where 𝜃1 and 𝜃2 are the angular displacements of the two pendulums, respectively. g1 (u1) and g2 (u2) are the actuators,which are represented by the hyperbolic tangent functions of control inputs, that is, gi (ui) = ui + tanh (ui) (i = 1, 2). The

ZHONG et al. 7

masses of the pendulums are m1 = 2 kg and m2 = 25 kg, the moments of inertia are J1 ∈ [0.5, 2] kg m and J2 ∈ [15, 25] kg mwith uncertainties, the constant of the connecting torsional spring is k=2 N/m, the height of pendulum is r=1 m and thegravitational acceleration is g = 9.81 m/s2.

In the simulation, 𝜃1(0) = 90◦, 𝜃2(0) = −10◦ and the reference signal is y∗(t) = [0◦, 0◦]T .Set b1 = 1

2, b2 = 1

25, and 𝜔c = 2. According to Theorem 1, it can be calculated that the lower bound of 𝜔o is 𝜔∗

o = 16.Then, the parameters of the PID (9), (10) are set as 𝜔c = 2, 𝜔o = 16, cr1 = cr2 = 3.

The response curves of the states 𝜃1 and 𝜃2 based on the PID control (9) and (10) under 10 different cases are shownin Figure 1. It can be seen that, under the ADRC-based tuning rule (10), the dynamic response of the PID controlledsystem has satisfied tracking performance with vanishing tracking error. Moreover, despite the uncertainties of the systemparameters, the response curves of the states coincide.

Figure 2 is the estimations of the coupled nonlinear uncertainties ftotal given by upidf and the I-term of the PID control(9) and (10) when J1 = 2, J2 = 25, where the blue line is the real value of ftotal, the red dash line is upidf and the yellow

(A) (B)

F I G U R E 1 (A) The response curves of the state 𝜃1; (B) the response curves of the state 𝜃2

(A) (B)

F I G U R E 2 (A) The estimations of the uncertainty ftotal1; (B) the estimations of the uncertainty ftotal2

8 ZHONG et al.

(A) (B)

F I G U R E 3 (A) The curves of ë1(t) + kad1e1(t) + kap1

e1(t); (B) the curves of ë2(t) + kad2e2(t) + kap2

e2(t)

(A) (B)

F I G U R E 4 (A) The curves of the control input u1; (B) the curves of the control input u2

line is the I-term of the PID control (9) and (10). Figure 2 shows that, compared to the single I-term, upidf can track theunknown coupled nonlinear uncertainties more quickly, indicating that parts of the P-term and the D-term also con-tribute to the estimation and compensation for the coupled nonlinear uncertainties, rather than the single I-term of PIDcontrol.

To verify the results in Theorem 2, the curves of ëi(t) + kadi ei(t) + kapi ei(t)(i = 1, 2) with a set of different 𝜔o aregiven in Figure 3 when J1 = 2, J2 = 25. It can be seen that increasing 𝜔o will lead to the decrease of |ëi(t) + kadi ei(t) +kapi ei(t)|, that is, by only increasing the parameter 𝜔o, the dynamic responses of the tracking errors ei(i = 1, 2) can beimproved.

Figure 4 is the curves of the control input ui(i = 1, 2) when 𝜔o varies, which shows that as 𝜔o increase, the controlinput will be kept within a certain range.

ZHONG et al. 9

5 CONCLUSION

In this article, the rationale behind the success of PID control is studied. First, derived from an inherent but rarely noticedrelationship between PID control and ADRC, an ADRC-based tuning rule for PID control on a class of MIMO non-affineuncertain systems is introduced. The mechanisms of the proposed ADRC-based PID control (9)–(10) are characterized bythe parameters 𝜔o, kapi , and kadi rather than traditional kpi , kIi , and kdi . 𝜔o works to tune upidfi

(11), which is a special com-bination of P–I–D three terms rather than the single I-term, to estimate and compensate for the influence caused by theunknown nonlinear, coupled and non-affine factors. A lower bound 𝜔∗

o to the parameter 𝜔o, which can be quantitativelycalculated through (15), is presented for ensuring the global convergence of the PID control based closed-loop system.Furthermore, it is illustrated that both the estimation error of upidfi

and the tracking error to the desired trajectory, whichis determined by kapi and kadi , can be improved by increasing 𝜔o > 𝜔∗

o . These results may give the reason why PID controlcan be widely used in practice, if it can be well-tuned.

This article aims to propose an ADRC—based PID tuning rule. The classical PID law is a combination of propor-tional, integral and derivative feedbacks of tracking error, which means that both the states x1(t) and x2(t) are measurable.Therefore, we consider an ADRC design, where the full-state information x1(t) and x2(t) can be obtained, in order to bein accordance with the case of the classical PID design. If, in some practical scenarios, the measurement is only x1(t),which means the initial value of x2(t) cannot be obtained beforehand, then the designs of both PID and ADRC should beadjusted such that a similar formula connecting PID and ADRC can be obtained.

Since the classical PID control is a controller with linear structure, in this article, a linear ESO based ADRC is usedto investigate the parameter formula connecting both controllers. However, the design of the ESO is not limited to thelinear structure. How to set up a corresponding relationship between nonlinear ESO based ADRC and nonlinear PID isan interesting problem, which belongs to future investigation.

ACKNOWLEDGMENTThis work was supported by the National Key R & D Program of China under Grant 2018YFA0703800 and the NationalNatural Science Foundation of China (Grant No. U20B2054).

CONFLICT OF INTERESTThe authors declare no potential conflict of interests.

DATA AVAILABILITY STATEMENTAll data generated or analyzed during this study are included in this article.

ORCIDSheng Zhong https://orcid.org/0000-0003-1866-6823

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Conference; 2006:161-166.4. Guo L. Estimation, control, and games of dynamical systems with uncertainty (in Chinese). Sci China (Inf Sci). 2020;60(9):1327-1344.5. Zhang JK, Guo L. Theory and design of PID controller for nonlinear uncertain systems. IEEE Control Syst Lett. 2019;3(3):643-648.6. Zhao C, Guo L. PID controller design for second order nonlinear uncertain systems. Sci China (Inf Sci). 2017;60(2):022201.7. Zhao C, Guo L. PID control for a class of non-affine uncertain systems. Proceedings of the 37th Chinese Control Conference; 2018; Wuhan,

China.8. Keel LH, Bhattacharyya SP. Controller synthesis free of analytical models: three term controllers. IEEE Trans Automat Contr.

2008;53(6):1353-1369.9. Ho M, Lin C. PID controller design for robust performance. IEEE Trans Automat Contr. 2003;48(8):1404-1409.

10. Zhao C, Guo L. Towards a theoretical foundation of PID control for uncertain nonlinear systems; 2010. arXiv:2010.06864.11. Han J. From PID to active disturbance rejection control. IEEE Trans Ind Electron. 2009;56(3):900-906.12. Texas Instruments. Technical Reference Manual, TMS320F28069M, TMS320F28068M InstaSPINTM-MOTION Software, Literature Num-

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19. Zhao ZL, Guo BZ. A novel extended state observer for output tracking of MIMO systems with mismatched uncertainty. IEEE Trans AutomatContr. 2017;63(1):211-218.

20. Xue W, Huang Y. Performance analysis of 2-DOF tracking control for a class of nonlinear uncertain systems with discontinuousdisturbances. Int J Robust Nonlinear Control. 2018;28(4):1456-1473.

21. Chen S, Xue W, Zhong S, Huang Y. On comparison of modified ADRCs for nonlinear uncertain systems with time delay. Sci China InfSci. 2018;61(7):70223.

22. Chen S, Huang Y. The selection criterion of nominal model in active disturbance rejection control for non-affine uncertain systems.J Franklin Inst. 2020;357:3365-3386.

23. Zhong S, Huang Y, Guo L. A parameter formula connecting PID and ADRC. Sci China (Inf Sci). 2020;63(9):192203.24. Zhong S, Huang Y, Guo L. New tuning methods of both PID and ADRC for MIMO coupled nonlinear uncertain systems. Proceedings of

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China.

How to cite this article: Zhong S, Huang Y, Guo L. An ADRC-based PID tuning rule. Int J Robust NonlinearControl. 2021;1-14. doi: 10.1002/rnc.5845

APPENDIX A. PROOF OF THEOREM 1

The following proof is based on the related proofs in Reference 10 (see, Theorem 1 and Proposition 2 there).Since 𝜕f (y∗∗,0,u)

𝜕u≥ B > 0,∀u ∈ Rn, it can be obtained that f (y∗∗, 0,u) is a global diffeomorphism on Rn. Hence, there

exists a unique u∗ ∈ Rn, such that f (y∗∗, 0,u∗) = 0.Denote ep = [e1, … , en]T ,

kp =(𝜔2

c + 2𝜔o𝜔c)

B−1, kd = (2𝜔c + 𝜔o)B−1, kI = 𝜔o𝜔2c B−1, eI(t) = ∫

t

0ep(𝜏)d𝜏 − k−1

I u∗,

ed(t) = ep(t), g (x1, x2,u) = −f (y∗∗ − x1,−x2,u + u∗) , z = kpep + kded + kIeI . (A1)

ZHONG et al. 11

Then, the closed-loop system (1), (9), and (10) turns into

⎧⎪⎨⎪⎩eI = ep,

ep = ed,

ed = g(ep, ed, z) + Δ(t),

(A2)

where Δ(t) = g(

ep + y∗∗ − r, ed − r, z + B−1r)− g(ep, ed, z) + r. Moreover, it can be obtained that

Δ(t) =𝜕g𝜕e||||(e,ed,z)

(y∗∗ − r) −𝜕g𝜕ed

||||(e,ed,z)r +

𝜕g𝜕z||||(e,ed,z)

B−1r + r, (A3)

where e = ep + 𝜃 (y∗∗ − r) , ed = ed − 𝜃r, z = z + 𝜃B−1r, 𝜃 ∈ (0, 1).Since g (x1, x2,u) = −f (y∗∗ − x1,−x2,u + u∗), it is easy to obtain that g(0, 0, 0) = 0 and

|||||||| 𝜕g𝜕x1

|||||||| ≤ L1,|||||||| 𝜕g𝜕x2

|||||||| ≤ L2,𝜕g𝜕u

≤ −B < 0,∀x1, x2,u ∈ Rn.

Denote 𝜆(eI , ep, ed) = ∫ 10 − 𝜕g

𝜕u

(ep, ed, 𝜏z

)d𝜏. It can be proved that

g(

ep, ed, z)= g(

ep, ed, 0)− 𝜆(

eI , ep, ed)

z, (A4)

and 𝜆(

eI , ep, ed) ≥ B > 0,∀eI , ep, ed ∈ Rn.

Based on the mean value theorem of integral type, g(ep, ed, 0) can be expressed as follows:

g(ep, ed, 0) ={∫

1

0

𝜕g (e, 0, 0)𝜕ep

d𝜃}

ep +

{∫

1

0

𝜕g(

ep, ed, 0)

𝜕edd𝜃

}ed ≜ a

(ep)

ep + b(

ep, ed)

ed, (A5)

where e = 𝜃ep, ed = 𝜃ed, 𝜃 ∈ [0, 1], ‖‖a(ep)‖‖ ≤ L1, ‖‖b(ep, ed)‖‖ ≤ L2.

Therefore, (A2) can be reexpressed as follows:

⎧⎪⎨⎪⎩eI = ep,

ep = ed,

ed = a(

ep)

ep + b(

ep, ed)

ed − 𝜆(

eI , ep, ed)

z + Δ(t).

(A6)

Define

kp = diag{

kp1 , kp2 , … , kpn

}, kd = diag

{kd2 , kd1 , … , kdn

}, kI = diag

{kI1 , kI2 , … , kIn

}.

To construct a Lyapunov function, the following 3n × 3n matrix is designed:

P =⎡⎢⎢⎢⎣

3k2I k−1

d 2kIkpk−1d kI

2kIkpk−1d 3k2

pk−1d kp

kI kp kd

⎤⎥⎥⎥⎦ . (A7)

First, it will be shown that the matrix P is positive definite. Denote

Pj =

⎡⎢⎢⎢⎢⎢⎣

3k2Ij

kdj

2kIjkpj

kdjkIj

2kIjkpj

kdj

3k2pj

kdjkpj

kIj kpj kdj

⎤⎥⎥⎥⎥⎥⎦, j ∈ n. (A8)

12 ZHONG et al.

It can be obtained that, when 𝜔o > 𝜔∗o , the matrix Pj, j ∈ n is positive definite. Thus, by utilizing the Laplace theorem, the

matrix P is positive definite.From Remark 5, when 𝜔o > 𝜔∗

o , k1 > 0 and k1k2 > k3 > 0. Hence, there is k2 > 0. Denote

A =⎡⎢⎢⎢⎣

0 I 00 0 I

− kI𝜆(

eI , ep, ed)

a(

ep)− kp𝜆

(eI , ep, ed

)b(

ep, ed)− kd𝜆

(eI , ep, ed

)⎤⎥⎥⎥⎦ .

Based on the deduction of proposition 2 in Reference 10, it can be easily proved that when ‖‖‖a (ep)‖‖‖ ≤ L1,

‖‖‖b (ep, ed)‖‖‖ ≤ L2

and 𝜆(eI , ep, ed) ≥ B > 0,∀eI , ep, ed ∈ Rn, there exists a constant 𝛼 > 0, such that, as long as 𝜔o > 𝜔∗o , there is PA + ATP ≤

−𝛼I3n, where I3n is the 3n × 3n unit matrix.Then, consider the following Lyapunov function:

V(

eI , ep, ed)=[

eTI eT

p eTd

]P⎡⎢⎢⎢⎣eT

I

eTp

eTd

⎤⎥⎥⎥⎦ . (A9)

It can be obtained that

𝜆min(P)‖‖‖[eT

I , eTp , eT

d]‖‖‖2 ≤ V ≤ 𝜆max(P)

‖‖‖[eTI , eT

p , eTd]‖‖‖2

,

where 𝜆min(⋅) and 𝜆max(⋅) are the minimum and maximum eigenvalues of the corresponding matrix, respectively.The time derivative of V

(eI , ep, ed

)along the trajectories of (A6) is

V(

eI , ep, ed)=[eT

I , eTp , eT

d] (

PA + ATP) [

eTI , eT

p , eTd]T + 2

[eT

I , eTp , eT

d] ⎡⎢⎢⎢⎣

kIΔkpΔkdΔ

⎤⎥⎥⎥⎦≤ −c1V + c2

√V ||Δ||, (A10)

where c0 = max{||kI|| , ||kp|| , ||kd||} , c1 = 𝛼

𝜆max(P), c2 = c0√

𝜆min(P).

Since limt→∞ r(t) = y∗∗, limt→∞ r(t) = 0 and limt→∞ r(t) = 0, it shows that limt→∞ Δ = 0. Thus, for any 𝜀 > 0, thereexists T > 0, such that for any t > T, there is

V ≤ −c1V + 𝜀. (A11)

In conclusion, limt→∞ x1(t) = y∗∗, limt→∞ x2(t) = 0. The proof of Theorem 1 is complete. □

APPENDIX B. PROOF OF THEOREM 2

It can be obtained that the dynamic equation of the tracking error ei has the following structure:{ei = edi ,

edi = −kapi ei − kadi edi − efi ,(B1)

Thus, there is |ëi(t) + kadi ei(t) + kapi ei(t)| = |efi(t)|, i ∈ n.From (A11), it can be obtained that limt→∞ eI(t) = 0, limt→∞ ep(t) = 0, limt→∞ ed(t) = 0. Therefore, limt→∞ z(t) =

0. Since limt→∞ Δ = 0, from (A6), there is limt→∞ ed(t) = 0. Based on (B1), limt→∞ efi(t) = 0. Theorem 2 (1) isobtained.

ZHONG et al. 13

Since |||||| 𝜕f𝜕u|||||| ≤ L3, from (A3), it can be deduced that Δ is bounded, that is, there exists a constant M0, such that ||Δ|| ≤

M0. Based on (A10), there is √V ≤ e

−c1 t2

√V(eI(0), ep(0), ed(0)) +

c2M0

c1

(1 − e

−c1 t2

). (B2)

Thus, when 𝜔o > 𝜔∗o , eI , ep, and ed are bounded. Consequently, ef is also bounded.

Define kap = 𝜔2c , kad = 2𝜔2 and

Q1 =⎡⎢⎢⎣(

1+kap

2kad+ kad

2kap

)I 1

2kapI

12kap

I 1+kap

2kadkapI

⎤⎥⎥⎦ .Consider the following Lyapunov function

V1(

eTp , eT

d)=[

eTp eT

d

]Q1

[eT

p

eTd

]. (B3)

The time derivative of V1(eTp , eT

d ) along the trajectories of (B1) is

V 1 = −[

eTp eT

d

][eTp

eTd

]−[

eTp eT

d

] ⎡⎢⎢⎣1

kapef

1+kap

kadkapef

⎤⎥⎥⎦ ≤ − V1

𝜆max (Q1)+

a1√

V 1||||ef ||||√

𝜆min(Q1), (B4)

where a1 =√

1k2

ap+(

1+kap

kadkap

)2.

From (B4), it can be proved that √V1 ≤ a1𝜆max(Q1)sup ||||ef ||||√

𝜆min(Q1).

Since when√

V1 >a1𝜆max(Q1)sup||ef ||√

𝜆min(Q1), there is V 1 < 0. Thus,

‖‖‖[eTp , eT

d]‖‖‖ ≤ a1𝜆max(Q1)sup ||||ef ||||

𝜆min (Q1). (B5)

Since the estimation upidf has the same function of the output f of the ESO (4). From the Equation (4), the estimationerror ef satisfies the following equation

ef = −𝜔oef + f total. (B6)

By some elementary calculations, it can be obtained that

f total = −((

𝜕f𝜕u

B−1 − I)(kad + 𝜔o) −

𝜕f𝜕x2

)ef +(

𝜕f𝜕x2

kap −(𝜕f𝜕u

B−1 − I)

kadkap

)ep

+(−

𝜕f𝜕x1

+𝜕f𝜕x2

kad +(𝜕f𝜕u

B−1 − I)(

kap − k2ad))

ed +𝜕f𝜕x1

r +𝜕f𝜕x2

r +(𝜕f𝜕u

B−1 − I)

…r . (B7)

From (B6), there is

ef = −(𝜔o +

(𝜕f𝜕u

B−1 − I)(kad + 𝜔o) −

𝜕f𝜕x2

)ef +(

𝜕f𝜕x2

kap −(𝜕f𝜕u

B−1 − I)

kadkap

)ep

+(−

𝜕f𝜕x1

+𝜕f𝜕x2

kad +(𝜕f𝜕u

B−1 − I)(

kap − k2ad))

ed +𝜕f𝜕x1

r +𝜕f𝜕x2

r +(𝜕f𝜕u

B−1 − I)

…r . (B8)

14 ZHONG et al.

Design the following Lyapunov function:

V2(ef ) =12

eTf ef . (B9)

Then, the time derivative of V2(ef ) along the trajectory of (B6) is

V 2 = −eTf

(𝜔oI +

(𝜕f𝜕u

B−1 − I)(kad + 𝜔o) −

𝜕f𝜕x2

)ef + eT

f

(𝜕f𝜕x2

kap −(𝜕f𝜕u

B−1 − I)

kadkap

)ep

+ eTf

(−

𝜕f𝜕x1

+𝜕f𝜕x2

kad +(𝜕f𝜕u

B−1 − I)(

kap − k2ad))

ed + eTf𝜕f𝜕x1

r + eTf𝜕f𝜕x2

r + eTf

(𝜕f𝜕u

B−1 − I)

…r . (B10)

Denote

a2 = L2kap + (L3 ||||B−1|||| − 1)kadkap, a3 = L1 + L2kad +(

L3 ||||B−1|||| − 1) (

kap − k2ad)

and

a4 = sup{

L1 ||r|| + L2 ||r|| + (L3 ||||B−1|||| − 1) ||||||…r ||||||} .

Since 𝜕f𝜕u

≥ B, there is

V 2 ≤ − (𝜔o − L2) ||||ef ||||2 + (a2 ||||ep|||| + a3 ||ed|| + a4) ||||ef |||| . (B11)

Denote 𝜔o1 = L2 +a1(a2+a3)𝜆max(Q1)

𝜆min(Q1) + 1. Then, it can be obtained that when 𝜔o > 𝜔o1, sup ||||ef |||| ≤ max{||||ef (0)|||| , a4

}.

Moreover, there exists a constant a5, which does not depend on 𝜔o, such that, sup ||||||f total|||||| ≤ a5. If 𝜔∗

o ≤ 𝜔o ≤ 𝜔o1, then by

Theorem 1, there exists a constant a6, such that a6 = sup𝜔o

||||||f total||||||.

Let

V3i

(efi

)= 1

2e2

fi, i = 1, … ,n. (B12)

There is

V 3i

(efi

)= −𝜔oe2

fi+ efi f totali

. (B13)

Denote 𝜂 = max {a5, a6} . Then, it can be deduced that

√V3i ≤ e−𝜔ot

√V3i

(efi(0))+√

2𝜂2𝜔o

(1 − e−𝜔ot) , (B14)

thus,

||efi|| ≤ e−𝜔ot|efi(0)| + 𝜂

𝜔o

(1 − e−𝜔ot) ≤ (|efi(0)| − 𝜂

𝜔o

)e−𝜔ot + 𝜂

𝜔o. (B15)

Theorem 2 (2) is obtained. □