Linear Active Disturbance Rejection Controller Design Based onDisturbance Response Specification for a 1st Order Plant

Ryo Tanaka1†, Momoko Toyota2 and Tetsunori Koga3

1Department of Control and Information Systems Engineering, National Institute of Technology,Kurume College, Fukuoka, Japan

(Tel: +81-942-35-9372; E-mail: rtanaka@kurume-nct.ac.jp)2Advanced Engineering School, National Institute of Technology, Kurume College, Fukuoka, Japan

(E-mail: a3713mt@kurume.kosen-ac.jp)3Interdisciplinary Graduate School of Engineering Science, Kyushu University, Fukuoka, Japan

(E-mail: koga.tetsunori.828@s.kyushu-u.ac.jp)

Abstract: This paper proposed a linear active disturbance rejection control (LADRC) for the 1st order plant mathematicalmodel based on the disturbance response specification of the closed-loop system. The relationship between the disturbanceand the plant output is clarified before the peak of the disturbance response is calculated. The LADRC parameters can beuniquely determined by comparing the coefficients of the low–dimensional transfer function with the 2nd order normativemodel. Also, the effectiveness of the proposed method is obtained by performing numerical simulations and comparingthe results with the conventional PI control. Furthermore, we clarified the policy for setting up the proposed LADRC byshowing a Bode diagram of the closed–loop LADRC system.

Keywords: LADRC, linear extended state observer (LESO), disturbance response specification, Bode diagram

1. INTRODUCTIONController design approaches based on a benchmark

plant model description have been proposed [1–2].Particularly, designing a controller by specifying thepeak of the plant output for assumed disturbances hasbeen proposed [1]. Chemical plant control is one ofthe application examples of controller design based onthe disturbance response specification. The controlledobject of the chemical control is classified into a flowquantity, temperature, pressure, liquid level, etc. Mostcontrol purposes in the chemical process are disturbancerestraint. It is convenient for engineers to adjust thecontroller to reduce the influence in this disturbance ifthe magnitude of disturbance such as a change of drivingconditions or that of the outside temperature can beassumed beforehand.

PID controller is widely used in industrial fieldsto realize a more precise automatic control. ThePID controller’s main features are simplified controllerdesign, proper control performance, among others. Thecontroller is designed based on the identification of thecontrolled object. Sometimes engineers are required toredesign the controllers when the characteristics of thecontrolled object changes.

The active disturbance rejection controller (ADRC)[3–8] is introduced recently, and it ensures excellentrobustness of a plant with a modeling error and unknowndisturbance. The linear ADRC (LADRC) [5–7] is oftenapplied when the plant mathematical model can beexpressed as a linear differential equation. The LADRCconsists of a linear extended state observer (LESO)and control law and can be designed without previousknowledge of the specified controlled object. The

† Ryo Tanaka is the presenter of this paper.

objective of the LESO is to estimate the state variablesof the plant and the total disturbance, which includesplant model parameter uncertainties and disturbances andapply feedback action to the plant input. When the valueof the LESO bandwidth is sufficiently large, then theplant acts as a perfect integrator of order n within thebandwidth of the observer [8]. Under this assumption,the control law is determined by theoretically locating allpoles of the closed–loop system in a stable region.

This study determines two LADRC parameters (i.e.,the LESO bandwidth and controller one) for a 1storder plant mathematical model based on a disturbanceresponse specification. The most key idea of theproposed method is that we don’t have to redesign thecontroller even when the plant contains some modelingerrors or unexpected disturbances. Particularly, basedon the peak of the disturbance response, the proposedcontroller can be designed without previous knowledgeof the specified controlled object. When the orderof the plant mathematical model and LESO is oneand two, respectively, then the order of the transferfunction between the disturbance and plant output isthree. But, it reduced to two under the assumption thatthe LESO bandwidth is more than any other parameter.To determine the LADRC parameters, we derive thetransfer function before calculating the peak of thedisturbance response. The LADRC parameters can beuniquely determined by comparing the coefficients ofthe low–dimensional transfer function with the 2nd ordernormative model. The effectiveness of the proposedLADRC based on disturbance response specification isobtained by performing simulations and comparing theresults with the conventional PI control. Also, weclarified the policy of setting up the proposed LADRCby showing a Bode diagram of the LADRC system.

2. CONVENTIONAL PI PARAMETERTUNING BY SPECIFYING

DISTURBANCE RESPONSE

Fig. 1 Structure of the closed–loop PI control system.

This section reviews the derivation of the PI controllerdesign based on specifying the peak of the disturbanceresponse for a 1st order plant mathematical model [1].

A block diagram of the closed–loop PI control systemis shown in Fig. 1 where R(s), D(s), U(s), andY (s) represent the Laplace transforms of the reference,the plant input–side disturbance, plant input, and plantoutput, respectively.

The 1st order nominal plant mathematical modelGp(s) and its approximated Gp(s) can be expressed as

Gp(s) =Y (s)

U(s)=

Kp

Tps+ 1, (1a)

Gp(s) =1

Tqs+ 1Kp

' 1

Tqs, (1b)

where Kp > 0 and Tp > 0 are arbitrary positivecoefficients, Tq = Tp/Kp, and U(s) and Y (s) are theLaplace transforms of the input and output of the plant,respectively. The bigger Kp is, the smaller the errorbetween (1a) and (1b). Also, the mathematical model ofthe PI controller Gc(s) can be expressed as

Gc(s) = Kc

(1 +

1

Tis

), (2)

where Kc is the proportional gain coefficient, and Ti isthe integration time. Assuming the plant mathematicalmodel is given as (1b), the transfer function Gyd(s)between D(s) and Y (s) can be obtained as

Gyd(s) =Y (s)

D(s)=

TiKcs

TqTiKc

s2 + Tis+ 1. (3)

From (3), it can be seen that the order of the polynomialin the numerator and denominator is one and two,respectively. Hence, the normative model Gm(s) whichobtains the desired disturbance response is given as

Gm(s) = δs/(τs+ 1)2, (4)

where δ is the normative model gain, τ is the naturalperiod. Also, δ and τ provide arbitrary adjustableparameters to the desired disturbance response.

Comparing (3) and (4), Kc, Ti and δ can be obtainedfrom the following equations:

Kc = 2Tq/τ, (5a)Ti = 2τ, (5b)

δ =TiKc

=τ2

Tq. (5c)

We assume that the plant input–side disturbance ofmagnitude d is loaded, i.e., D(s) = d/s. By consideringwhen Gyd(s) = Gm(s), the output Y (s) can beexpressed as

Y (s) = Gyd(s)D(s) = Gm(s)D(s) =dδ

(τs+ 1)2. (6)

The plant output y(t) which can be obtained by takingthe inverse Laplace transform in (6) is given by

y(t) =dδ

τ2te−

1τ t. (7)

The peak value ypeak of y(t) can be solved by calculatingdy(t)/dt = 0, i.e.,

ypeak = y(t)|t=τ = dδ/(τe). (8)

When the peak obtained from the step response ofGm(s)matches the desired peak of the control output yset, δ is

δ = ητe, (9)

where η = yset/d is an adjustable parameter fordetermining the ratio between d and yset. From (5c) and(9), τ is given by

τ = Tqηe. (10)

By substituting (10) into (5a) and (5b), Kc and Ti can beexpressed as

Kc = 2/(ηe), (11a)Ti = 2Tqηe. (11b)

When the plant mathematical model is assumed as (1a),the transfer function Gyd(s) between D(s) and Y (s) canbe expressed as

Gyd(s) =TiKcs

TqTiKc

s2 + Ti

(1 + 1

KpKc

)s+ 1

. (12)

By substituting (11a) and (11b) into (12), Gyd(s) can beexpressed as

Gyd(s) =δs

τ2s2 + 2τζs+ 1, (13)

where

ζ = 1 +ηe

2Kp. (14)

The peak value ypeak is re–expressed as

ypeak = dδτ/ξ, (15)

where

ξ =1

2√ζ2 − 1

(ζ +

√ζ2 − 1

ζ −√ζ2 − 1

)(− ζ−

√ζ2−1

2√ζ2−1

)

−

(ζ +

√ζ2 − 1

ζ −√ζ2 − 1

)(− ζ+

√ζ2−1

2√ζ2−1

) . (16)

The derivation process of (15) can be obtained from [1].By using (8) and (15), η which indicates the ratio betweend and yset for the 1st order plant mathematical model canbe expressed as

η = yset/(deξ). (17)

Hence, the peak ratio of the 1st order plant can be derivedanalytically by substituting (17) into Kc = 2/(ηe) andTi = 2Tq ηe.

3. PROPOSED LADRC PARAMETERTUNING BY SPECIFYING

DISTURBANCE RESPONSEThis section describes how to design the LADRC

system based on a disturbance response specification. Ablock diagram of the LADRC is shown in Fig. 2.

Fig. 2 Structure of the LADRC for the 1st order plant.

3.1. Fundamental of LADRC for the 1st order plantLet the 1st order plant mathematical model Gp(s) be

(1a). The input u(t) consists of the disturbance d(t) andthe input of the LESO u(t), i.e.,

u(t) = d(t) + u(t). (18)

The inverse Laplace transform in (1a) can be summarizedas

y(t) = f(t) + cu(t), (19)

where

f(t) = − 1

Tpy(t) +

Kp

Tpd(t), (20)

and c = Kp/Tp denotes an arbitrary positive scalingfactor. When state variables x1(t) := y(t) and x2(t) :=f(t) are defined, then (19) can be rewritten as

x1(t) = x2(t) + cu(t). (21)

By taking (21) and x2(t) = f(t) into account, the matrixform can be expressed as

x(t) = Ax(t) + Bu(t) + Ef(t), (22)

where x(t) = [x1(t) x2(t)]T,

A =

[0 10 0

], B =

[c0

], E =

[01

].

The LESO for (22) can be defined as

z(t) = (A−LC)z(t) + Bu(t) + Ly(t), (23)

where z(t) = [z1(t) z2(t)]T is an estimated state vectorx(t), C = [1 0], L = [l1 l2]T is an LESO gainvector, and li(i = 1, 2) is generally determined so thatall eigenvalues of the LESO are located at the LESObandwidth −ωo, i.e., L = [2ωo ω

2o ]T. The main role

of this LESO is to estimate the total disturbance, whichincludes plant model parameter uncertainties and externaldisturbances and apply feedback action to the plant input[7]. The control law is determined as

u(t) = (v(t)− z2(t)) /c, (24a)v(t) = kc(r(t)− z1(t)), (24b)

where kc = ωc is the controller bandwidth. When theESO is well designed, i.e., f(t) = z2(t), the relationshipbetween v(t) and y(t) acts as a perfect integrator withinthe bandwidth of the observer [8]. From this assumption,kc can be selected so that the closed–loop poles areplaced at −ωc.

3.2. Proposed LADRC designTo determine the LADRC parameter ωo and kc, we

clarify the relationship between the Laplace transformof the disturbance D(s) and that of the plant outputY (s), before calculating the peak of the disturbanceresponse. By considering when R(s) = 0, then theLaplace transform in (18), (23), (24a) and (24b) can beexpressed as

U(s) = D(s) + U(s), (25a)

sZ(s) = (A−LC)Z(s) + BU(s) + LY (s), (25b)

U(s) = (V (s)− Z2(s)) /c, (25c)V (s) = −kcZ1(s). (25d)

From (25c) and (25d), U(s) becomes

U(s) = −KcZ(s), (26)

where Kc = [kc 1]/c, and Z(s) = [Z1(s) Z2(s)]T.When ωo is bigger than any other parameter (e.g., kc, c,Tp, Kp), the transfer function Gyd(s) can be calculatedfrom (25a), (25b) and (26), as

Gyd(s) =2cKps

2cTps2 +Kpωos+Kpkcωo. (27)

The derivation process of (27) is shown in Appendix A.The inverse model G−1yd (s) is given by

G−1yd (s) =kcωo2c· 1

s+ωo2c

+TpKp

s. (28)

From (4), the inverse normative model G−1m (s) is givenby

G−1m (s) =1

δ· 1

s+

2τ

δ+τ2

δs. (29)

In comparison to (28) and (29), kc, ωo and δ can beobtained as follows:

kcωo/(2c) = 1/δ, (30a)ωo/(2c) = 2τ/δ, (30b)

Tp/Kp = τ2/δ. (30c)

From δ = ητe and (30c), τ denotes

τ = ηe/c, (31)

where c = Kp/Tp. By substituting δ = ητe into (30b),ωo becomes

ωo = 4c/(ηe). (32)

By substituting (32) into (30a), kc becomes

kc = c/(2ηe). (33)

4. NUMERICAL SIMULATIONS4.1. Simulation setups

Table 1 Parameters of the proposed LADRC.

c η ωo kc η c ωo kc0.1 147.2 18.39 1.0 14.72 1.8390.5 29.43 3.679 5.0 73.58 9.197

10 1.0 14.72 1.839 0.1 10 147.2 18.392.0 7.358 0.9197 50 735.8 91.975.0 2.943 0.3679 102 1472 183.9

Table 2 Parameters of the conventional PI control.

η Kc Ti0.1 7.358 5.437× 10−2

0.5 1.472 0.27181.0 0.7358 0.54372.0 0.3679 1.0875.0 0.1472 2.718

In this section, the effectiveness of the proposedLADRC based on disturbance response specification isdemonstrated using numerical simulations. Figs. 1 and2 show block diagrams of the PI control system andLADRC, respectively. The reference r(t) is assumedto be a step signal. The step set–point is introduced att = 0 [s], and the settling time of nominal plant outputs isapproximately 2.0 [s]. The plant input–side disturbanceof magnitude −0.1 is loaded at 5.0 [s]. The 1st orderplant transfer function Gp(s) is assumed as (1a). Here,Kp = 1.0 and Tp = 0.1. The parameters of the proposedLADRC (i.e., c, η, ωo and kc) are listed in Table 1, whilethose of the conventional PI control (i.e., η, Kc and Ti)are listed in Table 2.

4.2. Simulation resultsFigs. 3 and 4 show the simulation results of the

conventional PI control, while Figs. 5 to 7 show thoseof the proposed LADRC. In Figs. 3 to 7, (a) shows areference following the performance for a step input r(t),and (b) shows an approximated disturbance suppressionperformance for a unit step disturbance d(t). In Figs. 5to 7, (c) shows a disturbance suppression performancefor d(t). Figs. 3 and 5 illustrate the simulation resultsof the output responses when the plant mathematical

(a) Reference following performance

(b) Disturbance suppression performance for (12)Fig. 3 Robustness of the conventional PI control (η =

0.1).

(a) Reference following performance

(b) Disturbance suppression performance for (12)Fig. 4 Variety of η for the conventional PI control.

model considers a modeling error. In this paper, themathematical model of a 1st order plant with modelingerror Gp(s) is defined as

Gp(s) = Kp/(αTps+ 1), (34)

where α is an error parameter. When α = 1.0, Gp(s)is a nominal plant model without modeling error, α =2.0 indicates a +100% modeling error, i.e., Gp(s) =Kp/(2.0Tps + 1), α = 0.5 indicates a −50% modelingerror, i.e., Gp(s) = Kp/(0.5Tps+ 1). From Figs. 3(a)

(a) (b) (c)Reference following performance Disturbance suppression performance for (27) Disturbance suppression performance for (A.6)

Fig. 5 Robustness of the proposed LADRC (η = 0.1, c = 10).

(a) (b) (c)Reference following performance Disturbance suppression performance for (27) Disturbance suppression performance for (A.6)

Fig. 6 Variety of c for the proposed LADRC (η = 0.1).

(a) (b) (c)Reference following performance Disturbance suppression performance for (27) Disturbance suppression performance for (A.6)

Fig. 7 Variety of η for the proposed LADRC (c = 10).

Fig. 8 Bode diagram (The blue line indicates a disturbance response of Gm(s) in (4), the green line indicates that ofGyd(s) in (A.6), and the red line indicates that of Gyd(s) in (27) ).

and 5(a), the reference following the performance of theproposed LADRC is slightly more robust than that of theconventional PI control, with respect to the plant withmodeling error and the disturbance. It is assumed thatthe excellent performance of the proposed LADRC isprovided by the basic perspective of the original LADRC,i.e., the observed signal f(t)’s feedback action to thecontrol input. In Figs. 3(b) and 4(b), it is shown that thepeak value ypeak of the disturbance response is exactlyderived from (15), i.e., (η, ypeak) = (0.1, 9.17×10−3),(0.5, 3.42 × 10−2), (1.0, 5.17 × 10−2), (2.0, 6.90 ×10−2), and (5.0, 8.56× 10−2).

Fig. 6 illustrates the simulation results of the proposedLADRC when adjusting c. From Fig. 6(a), the closerc is to Kp/Tp, the easier it is to get a better controlperformance. When c is large, the control output y(t)tends to cause some overshoot in a transient state. Asshown in Figs. 6(b) and 6(c), it is verified that the peakof the approximated disturbance response and that ofthe original disturbance response are mutually different.Particularly, when c = 10, the error between them seemsto be smallest.

Figs. 4 and 7 illustrate the simulation results ofthe conventional PI control and those of the proposedLADRC, respectively, when adjusting η. From Figs. 4(a)and 7(a), it can be obtained that the smaller η is, the fasterthe control output y(t) convergence to the target value1.0. But, when η is large, the control output y(t) tend toconverge slowly to the reference r(t) in a steady state.

Fig. 8 illustrates the Bode diagram of Gyd(s) in theLADRC system when adjusting c and η. Fig. 8 showsthat the best control performance is obtained when thegain and phase curves of the reference disturbance model,those of the true disturbance response, and those of theapproximate disturbance response approximately matchto about 100 [rad/sec] or more. In Figs. 8(c) and 8(f), itcan be seen that all shapes of the gain and phase curvesare different, even when ωo and kc are equal.

CONCLUSIONThis study proposed an approach to the LADRC

design based on disturbance response specification forthe 1st order plant mathematical model. The proposedLADRC method shows superior robustness in the plantwith modeling errors and disturbance compared toconventional PI control. The application of the LADRCmethod for 2nd and higher–order plant mathematicalmodels with experiments will be performed in the future.

REFERENCES[1] S. Nunokawa, and H. Matsuyama, “PI parameter

tuning by specifying disturbance response inchemical process control”, Transaction of ISCIE,Vol. 14, No. 2, pp. 561–573, 2001 (in Japanese).

[2] B. E.–D. Gamal, A. N. Ouda, Y. Z. El–Halwagy,and G. A. El–Nashar, “Advanced fast disturbancerejection PI controller for DC motor positioncontrol”, Proceedings of ASAT, pp. 1–12, 2015.

[3] Z. Gao, “Scaling and bandwidth–parameterizationbased controller tuning”, Proceedings of ACC, pp.4989–4996, 2003.

[4] J. Han, “From PID to active disturbance rejectioncontrol”, IEEE Transactions on IE, Vol. 56, No. 3,pp. 900–906, 2009.

[5] W. Tan, and C. Fu, “Linear activedisturbance–rejection control: analysis and tuningvia IMC”, IEEE Transactions on IE, Vol. 63, No. 4,pp. 2350–2359, 2016.

[6] C. Fu, and W. Tan, “A new method to tune linearactive disturbance rejection controller”, Proceedingsof ACC, pp. 1560–1565, 2016.

[7] R. Tanaka, and T. Koga, “An approach to linearactive disturbance rejection controller design with alinear quadratic regulator for a non–minimum phasesystem”, Proceedings of CCC, pp. 250–255, 2019.

[8] J. Tatsumi, and Z. Gao, “On the enhanced ADRCdesign with a low observer bandwidth”, Proceedingsof CCC, pp. 297–302, 2013.

APPENDIX A : DERIVATION OF (27)Let the 1st order nominal plant Gp(s), the Laplace

transform of the plant input u(t), that of u(t), that of thecontroller output v(t), and that of (23) be

Gp(s) = Kp/(1 + Tps), (A.1)

U(s) = D(s) + U(s), (A.2)

U(s) = −KcZ(s), (A.3)

sZ(s) = (A−LC)Z(s) + BU(s) + LY (s). (A.4)

From (A.2) and (A.4), U(s) can be summarized as

U(s) = D(s)−Kc·

{sI<2> − (A−BKc −LC)}−1 LY (s), (A.5)

where I<2> indicates a 2nd order identity matrix.By substituting (A.5) into (A.1), The transfer functionGyd(s) between Y (s) and D(s) is

Gyd(s) =Y (s)

D(s)=

numGyd(s)

denGyd(s), (A.6)

where numGyd(s) = cKps2 + cKp(kc + 2ωo)s, and

denGyd(s) = Tpcs3 + c(1 + Tpkc + 2Tpωo)s

2

+ (ckc + 2cωo +Kpω2o + 2Kpkcωo)s

+Kpkcω2o .

Finally, when ωo is much bigger than any other parameterin (A.6), the approximated transfer function Gyd(s) canbe obtained as

Gyd(s) '2cKpωos

2cTpωos2 +Kpω2os+Kpkcω2

o

=2cKps

2cTps2 +Kpωos+Kpkcωo. (27)