predictive active disturbance rejection control for processes with time delay

Predictive active disturbance rejection control for processes withtime delay

Qinling Zheng n, Zhiqiang GaoCenter for Advanced Control Technologies, Department of Electrical and Computer Engineering, Cleveland State University, Cleveland, OH 44115, United States

a r t i c l e i n f o

Article history:Received 14 June 2013Received in revised form23 September 2013Accepted 30 September 2013This paper was recommended forpublication by Didier Theilliol

Keywords:Predicative active disturbancerejection controlTime delayDisturbance rejectionNoise attenuationBoiler controlChemical reactor control

a b s t r a c t

Active disturbance rejection control (ADRC) has been shown to be an effective tool in dealing with realworld problems of dynamic uncertainties, disturbances, nonlinearities, etc. This paper addresses itsexisting limitations with plants that have a large transport delay. In particular, to overcome the delay, theextended state observer (ESO) in ADRC is modified to form a predictive ADRC, leading to significantimprovements in the transient response and stability characteristics, as shown in extensive simulationstudies and hardware-in-the-loop tests, as well as in the frequency response analysis. In this research, itis assumed that the amount of delay is approximately known, as is the approximated model of the plant.Even with such uncharacteristic assumptions for ADRC, the proposed method still exhibits significantimprovements in both performance and robustness over the existing methods such as the dead-timecompensator based on disturbance observer and the Filtered Smith Predictor, in the context of somewell-known problems of chemical reactor and boiler control problems.

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1. Introduction

Active disturbance rejection control (ADRC) was first conceivedby Han in 1995 [2], 1998 [3], and was fully articulated in 2009 [1].Han demonstrated with his penetrating insight and computersimulations, and later validated by others in various theoreticalstudies [4–6], that both the unknown plant dynamics and theexternal disturbances can be accurately estimated in real time,based on the input-output signals of the plant. Such estimation isused to reduce the plant to a canonical, integral form where acontrol solution is readily available. In doing so, a highly effectivesolution was born for processes that are nonlinear, time-varying,and full of uncertainties, both internal and external. With suchuniqueness in design concept, ADRC provides excellent solutionsto many pressing engineering problems; see for example [7–12].

As a new control design framework, ADRC solutions are stillgrowing and they are not without limitations. For example, muchof the success of ADRC has been achieved with systems with littleor no dead time [13], and processes with long time delays stillpose a great challenge. This paper strives to meet this challenge.

Processes with time delay are difficult to control because delayintroduces extra phase lag leading to reduced stability margins

and putting a stringent limit on the bandwidth. Therefore, specificanalysis techniques and design methods must be developed toadequately address the presence of delays. Concerned with thisvery problem in the context of ADRC, Han suggested the followingthree methods [1]: (1) Transfer function approximation of thedelay; (2) Input prediction; and (3) Output prediction.

In the first method, first-order lag or Padé approximation can beused to approximate the delay term, essentially treating the processas a higher order system without delay [14]. With this method, bothstability robustness and performance are improved for a multi-variable process with time delay in the input. However, the controllerbandwidth and observer bandwidth are still quite limited, leading tosluggish transient and disturbance rejection response.

The second method of predicting the control signal is not easilydone, unless the accurate model information and future set pointinformation are given, making it quite limited as a practical solution.

In this paper we adopt a strategy based on the third methodsuggested by Han, by using a prediction method to obtain the delay-less output feedback, similar to the idea of the Smith Predictor (SP),leading to a predictive ADRC solution that is able to handle longtime-delays in the process. Such approach overlaps somewhat withthe existing literature on disturbance-observer based control designfor processes with a time delay. A disturbance-observer in the Dead-time compensators (DTC) is presented in [15] and analyzed in [16,17],showing that the method is applicable to both stable and unstablesystems. The modified versions of this structure were presented in

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0019-0578/$ - see front matter Published by Elsevier Ltd. on behalf of ISAhttp://dx.doi.org/10.1016/j.isatra.2013.09.021

n Corresponding author. Tel.: þ1 2165335285.E-mail addresses: [email protected], [email protected] (Q. Zheng).

Please cite this article as: Zheng Q, Gao Z. Predictive active disturbance rejection control for processes with time delay. ISA Transactions(2013), http://dx.doi.org/10.1016/j.isatra.2013.09.021i

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[18–21], where more complex algorithms and tuning rules aredescribed to deal with integral plants with delay. The main distin-guishing factor between these methods and the proposed one iswhether or not the disturbance model is required: the existingmethods do, the proposed doesn't. Another difference is that theADRC based approach is more tolerant of the uncertainties in bothplant dynamics and external disturbances, as shown in this paper.

The rest of the paper is organized as follows. First, the conceptof ADRC is introduced in Section 2, the proposed structure isintroduced in Section 3, and then some numerical examples aretested in Section 4. Stability analysis using classical control theoryin frequency domain is given in Section 5, followed by someapplication examples in Section 6. Finally, concluding remarks aregiven in Section 7.

2. Active disturbance rejection

Active disturbance rejection is a unique design concept thataims to accommodate not only external disturbances but alsounknown internal dynamics in a way that control design can becarried out in the absence of a detailed mathematical model, asmost classical and modern design methods require. To illustratethe basic idea, consider an ADRC design for a second order systemwithout time delay described as:

€yþa1 _yþa0y¼ bðuþwÞ; ð1Þwhere u and y are the input and output of the plant, respectively.The external disturbance is w. a1, a0, and b are system parameters.In the ADRC design, (1) is rewritten as:

€y¼ buþ f ; ð2Þwhere f ¼ bw�a1 _y�a0y. The function f ðUÞ is a general nonlinear,time-varying dynamic representing the total disturbance includingboth internal (unknown dynamics) and external (disturbance)uncertainties. The key idea is to extend the state definition of f ,which allows us to estimate it in real time using the state observer.

With the state vector defined as X ¼ x1 x2 x3� �T ¼ y _y f

h iT,

(2) is rewritten in the state space form as:

_X ¼ AXþBuþE_f

y¼ CX;

(ð3Þ

where A¼0 1 00 0 10 0 0

264

375, B¼

0b

0

264

375, C ¼

100

264

375T

and E¼001

264

375.

Note that the total disturbance f ðU Þis defined as the extendedstate x3 that is augmented to the original second order system.According to (3), a linear extended state observer (ESO) isconstructed as:

_Z ¼ AZþBuþL y� y� �

y¼ CZ

(; ð4Þ

where Z ¼ z1 z2 z3� �T is the observer state vector which pro-

vides an estimation of the system state vector X. L¼ l1 l2 l3� �T

is the observer gain vector; y is the estimation of output. Mostimportantly, the ESO provides z3, an estimation of the totaldisturbance. The idea of ADRC, as in Fig. 1, is to actively estimatef ðUÞ and then cancel it with the control signal, thereby reducingthe problem to controlling an integral plant.

The ADRC control law is given as:

u¼ ðk1ðÞr�z1þk2ð_r�z2Þ�z3Þ=bo ð5Þwhere k1 and k2 are controller gains, r is the reference signal, andbo is the estimation of b.

In a plant with input time delay lp, the estimated output andthe plant output, which is delayed, were not synchronized in ESOand Eq. (4) becomes:

_ZðtÞ ¼ AZðtÞþBuðtÞþL½y t� lp� �� yðtÞ �

yðtÞ ¼ CZðtÞ

(ð6Þ

More generally, consider a plant that is single-input single-output, nth-order, nonlinear, uncertain, and with time delay; itsinput uðtÞ and output yðtÞ are governed by:

yðnÞ ¼ f ðy; _y;…; y n�1ð Þ;w;uÞþb� uðt� lpÞ ð7Þ

where f ðU Þ is again the total disturbance to be estimated andcancelled. The state space form of (7) is:

_Xðt� lpÞ ¼ AgXðt� lpÞþBguðt� lpÞþEg _f

yðt� lpÞ ¼ CgXðt� lpÞ

(ð8Þ

with:

X ¼

X1

X2

⋮Xn

266664

377775; Ag ¼

0 1 00 0 1⋮ ⋱ 0

…⋯⋱

00⋮

⋮ ⋱ ⋱ ⋱ 10 0 ⋯ ⋯ 0

26666664

37777775; Bg ¼

0⋮0b

0

26666664

37777775;

Eg ¼

0⋮01

26664

37775; Cg ¼ 1 0 ⋯ 0

h i

To obtain the estimation of total disturbance f ðU Þ using the ESOin its standard form, we have:

_ZðtÞ ¼ AgZþBguðtÞþLðy t� lp� �� yðtÞÞ

yðtÞ ¼ CgZ

(ð9Þ

where Z ¼ z1 z2 … znþ1h iT

. Clearly the observer error

yðt� lpÞ� yðtÞ is misaligned time wise, which can lead to ESOinstability.

For the plant with no delay, i.e. lp ¼ 0, the standard designof ADRC can be stated as follows. Define ω0 as the observerbandwidth, the ESO gain vector L can be chosen as:

L¼ β1ω0 β2ω20 … βnþ1ω

nþ10

h iT; ð10Þ

such that the polynomial snþ1þβ1snþ⋯þβnsþβnþ1 is Hurwitz.

Note that the ESO gains in (10) can be chosen so that all itseigenvalues are placed at �ω0, the observer bandwidth, which

makes ESO easy to tune. With a well-tuned ESO, znþ1 ¼ f ðU Þ � f ðU Þ,the control law

u¼ ð� f ðU Þþu0Þ=bo ð11Þ

Fig. 1. Active disturbance rejection control scheme.

Q. Zheng, Z. Gao / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





reduces the original plant to the cascade integral form of

yðnÞ ¼ f ðU Þ� f ðU Þþu0 � u0 ð12Þ

which is now linear time invariant (LTI) and can be easilycontrolled with, say,

u0 ¼ k1ðyd�z1Þþk2ð _yd�z2Þþ⋯þknðyðnÞd �znÞ ð13Þ

where yd is the desired value of y. To make the controller easy totune, the controller gains ½k1; k2;…; kn� can be chosen to place allthe closed-loop poles at �ωc , whereωc is defined as the controllerbandwidth.

Note that the standard ESO design is still able to provide acomparatively accurate estimation of f ðU Þ for a small time delay.However, as time delay increases, ESO will deteriorate and even-tually become unstable. The proposed method described belowaddress this concern.

3. Proposed method

We noticed above the mismatch in ESO of the estimated andthe real plant outputs, leading to an inaccurate estimation of thetotal disturbance. Understandably the ADRC loses its stabilitymargin quickly when the process has a significant time delay. Asa remedy, we proposed a new structure, as shown in Fig. 2,predicative ADRC (PADRC) for the purpose of obtaining a delay-less estimation of the total disturbance. It is assumed the amountdelay is approximately known, as is the approximated model ofthe plant. We hope to demonstrate that even with such assump-tions ADRC will still exhibit significant improvements in bothperformance and robustness over the existing methods describedin Section 1.

The main difference between the basic ADRC structure (asshown in Fig. 1) and PADRC (as shown in Fig. 2) is that the modelinformation is used to obtain the predicted output yp whichreplaces yðt� lpÞ in the ESO as:

_ZðtÞ ¼ AgZðtÞþBguðtÞþLðypðtÞ� y tð ÞÞy tð Þ ¼ CgZ tð Þ

(ð14Þ

The predicted output, ypðtÞ, is the output of the predictor, thestructure of which is shown in Fig. 3. Let the model of the plant begiven as:

_zmðt� lmÞ ¼ AmZmðt� lmÞþBmuðt� lmÞym t� lmð Þ ¼ CmZm t� lmð Þ

(ð15Þ

where ðAm; Bm; CmÞ are matrices of coefficients of appropriatedimensions and lm is the approximated value of the real delay lp.Let PTD be the transfer function form of (15) and P be the sametransfer function as PTD except its delay, i.e. P is the transfer

function of

_ZðtÞ ¼ AmZðtÞþBmuðtÞym tð Þ ¼ CmZ ðtÞ

(ð16Þ

Note that ym t� lmð Þ is the output of PTD and ym tð Þ is the outputof P. Based on the idea from the Smith Predictor (SP), the predictedoutput can be constructed as:

ypðtÞ ¼ y t� lp� ��ym t� lmð Þþym tð Þ ð17Þ

In the proposed control structure of Fig. 2, by first cancelling thedelay in the output based on the given model of the plant and theidea of SP, the two inputs to the ESO, the u tð Þ and the predictedoutput yp tð Þ, are synchronized. This particular ADRC with outputprediction is associated with a particular method of output predic-tion. Other methods of prediction can be used in its place and willbe investigated in the future. Collectively we denote all modifica-tions of the basic ADRC employing the predicted input and/oroutput of the plant as Predicative ADRC, or simply PADRC. In thispaper, we focus on using the basic idea of SP as the mechanism ofprediction. And we denote this unique form of PADRC as SPADRC,the advantages of which are demonstrated next.

4. Numerical examples

In order to demonstrate the numerical behavior of SPADRC, wepresent a controller design solution for a first order plus dead-timeplant (FOPDT). Simulation comparisons between the widely usedSmith Predictor with PI controller (SPPI) and SPADRC are shownin parts A and B. Furthermore, part C also presents an SPADRCapplication for a system without time delay.

4.1. Robustness test

It is well known that the original SP is very sensitive to aninaccurate estimation of the plant, which will make the systemoscillate or even become unstable. However, ADRC is very robustagainst parameter variations, disturbances, and noise. In this part,robustness of SPADRC against parameter variations will be tested.The widely used Smith Predictor with PI controller (SPPI) is chosenas the basis for comparison.

Given the plant of the form:

P sð Þ ¼ KTsþ1

e� lps ð18Þ

Define Km ¼ 0:1, Tm ¼ 0:1 (s), and lm ¼ 5 (s) as the estimatedvalues of model parameters that correspond to K, T, and lp, respec-tively. We proceed to compare the second-order SPADRC withSPPI. First, tests are done when dead-time of the process is chang-ing, as shown in Fig. 4. Using the parameterization techniqueproposed in [24], parameters of SPADRC are selected as follows:ωo ¼ 36 rad=s

� �; ωc ¼ 12 ðrad=sÞ, and b0 ¼ 10. To have a fairFig. 2. Predictor active disturbance rejection control scheme.

x' = Ax+Bu y = Cx+Du

x' = Ax+Bu y = Cx+Du

Fig. 3. Predictor scheme.

Q. Zheng, Z. Gao / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





comparison, SPPI is tuned as kp¼12 and ki¼30, so that its stepresponse is very close to SPADRC when the model information isaccurate. Fig. 3 shows the output and input of the plant under ofSPPI and SPADRC, respectively, when there is a 6% mismatchbetween lm and lp. Note that the SPADRC has much less variationsthan the SPPI which cannot handle such mismatch beyond 710%;the SPADRC, however, has a much larger stability margin andkeeps the system under control for up to 740% of dead-timemismatch.

4.2. Disturbance rejection test

In order to demonstrate the disturbance rejection character-istics of SPADRC in a challenging way, we select the plant as

P sð Þ ¼ e� 5s

ð1þ sÞ3 but its model as Pm sð Þ ¼ e� 6s

ð1þ2sÞ. In presence of sensor

noise and model uncertainty, the disturbance rejection is testedwhen the system is engaged in set-point tracking. For comparison,the existing disturbance observer based dead-time compensator(DOBDTC) method is used as a benchmark. Fig. 5a shows thecomplete 2DOF DOBDTC control structure, however, is not appro-priate for implementation. So a modified scheme is used instead asshown in Fig. 5b.

The parameters for SPADRC are selected as follows: ωo ¼1:2 rad=s

� �; ωc ¼ 0:3 rad=s

� �; b0 ¼ 1=2, and lm ¼ 6 sð Þ. The control

structure VðsÞ and FðsÞ of DOBDTC are given by [26]:

V sð Þ ¼ 6þ2Tvð Þsþ1ðTvsþ1Þ2

; F sð Þ ¼ 1 ð19Þ

where Tv ¼ 3 sð Þ is used.First, the input disturbance rejection ability is tested in simula-

tion. Fig. 6 shows the response of both DOBDTC and SPADRC forthe case of step-type disturbance. Note that SPADRC recoversfaster and has less oscillation to the step-type disturbance. Fig. 7shows the response for the case of sine-type disturbance (ampli-tude of 1, frequency of 50 rad/s) where the responses of DOBDTCand SPADRC are very close to each other. Moreover, if disturbances

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

Time (Seconds)

Out

put

DT=4.7 SPADRCDT=4.7 SPPIDT=5.3 SPADRCDT=5.3 SPPI

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

Time (Seconds)

Inpu

t

Fig. 4. Step response when DT is changing.

Fig. 5. Control structure of 2DOF DOBDTC with a disturbance observer and a feed-forward action: (a) structure for analysis; (b) structure for implementation.

0 50 100 150 200 250 300 350 400 450 500-4

-2

0

2

Time (Seconds)

Out

put

Setpoint

DOBDTC

SPADRC

0 50 100 150 200 250 300 350 400 450 500-6

-4

-2

0

2

Time (Seconds)

Inpu

t

Disturbance

DOBDTC

SPADRC

Fig. 6. Plant output, set-point, and control input for the case of step-typedisturbance.

0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

Time (Seconds)

Out

put

Setpoint

DOBDTC

SPADRC

0 50 100 150 200 250 300 350 400 450 500-4

-2

0

2

Time (Seconds)

Inpu

t

Disturbance

DOBDTC

SPADRC

Fig. 7. Plant output, set-point, and control input for the case of sinusoidaldisturbance.






are measurable, a feed-forward SPADRC can provide significantimprovements in disturbance attenuation in a direct fashion.

Next, the sensor noise attenuation ability is tested and shownin Fig. 8. In this case, the measurement noise is generatedby MATLAB/SIMULINK “band-limited white noise” with noisepower¼0.001, sample time¼0.1 sð Þ, seed¼23,341. Although theoutput response is similar between DOBDTC and SPADRC, thecontrol signal of SPADRC is much less sensitive to sensor noises,which means more energy saving and less wear and tear.

4.3. SPADRC for system without time delay

Eq. (20) shows a high-order process combined with the eighth-order inherent-type load disturbance [27]:

Y sð Þ ¼ 1

sþ1ð Þ5U sð Þþ 1

sþ1ð Þ8Ud sð Þ ð20Þ

A step change of Δh¼ 1 at 0 s is assigned to be the set pointand an inherent-type load disturbance with a magnitude of -0.1 isadded to the process at 60 s. Simulation results are shown in Fig. 9.Regular ADRC control results are used for comparison, where“ADRC1” has the same transient response of SPADRC at the first10 s but with obviously larger overshoot. “ADRC2” has no

overshoot, but with sluggish transient response. Tuning para-meters for the three controllers are listed in Table 1. Comparedto regular ADRC, the SPADRC has the advantage for this type ofproblem having fast transient without overshoot. This is becausesuch a plant can be reasonably approximated as a first ordersystem with a time delay.

5. Frequency analysis of SPADRC

The results in Section 4 show that SPADRC is a robust controllercapable of fast set-point tracking and strong disturbance rejection.This section will use the FOPDT plant to analyze the controller'sstability in frequency domain.

To analyze the robustness, consider a family of plants PðsÞ suchthat P sð Þ ¼ Pm sð Þ½1þδPðsÞ�, where Pm indicate the measurementof the plant dynamics. The multiplicative uncertainty, δPðjωÞ,verifies:

δPðjωÞ�� rδP ωð Þ; 8ω40 ð21Þ

where δPðωÞ is the multiplicative norm-bound uncertainty [24].The characteristic equation for PðsÞ is:

1þCðsÞPm sð ÞþCðsÞPm sð ÞδP sð Þ ¼ 0 ð22ÞConsidering that the nominal system is stable, the robust

stability condition for the proposed controller is:

δP ðωÞodPðsÞ ¼ 1þCðjωÞPmðjωÞ��

jCðjωÞj ; 8ω40 ð23Þ

So for the FOPDT plant, with the SPADRC parameters chosen asωo ¼ 3�ωc , robust stability condition of the plant can be shownas in (24). With an accurate estimation of the high frequency gainb0ffi Km

Tm, Eq. (24) can be rewritten as (25).

dP sð Þ ¼ Km

Tmsþ1þb0

sðsþ7ωcÞωc

2ð15sþ9ωcÞ

�� ð24Þ

dP sð Þffi Km

Tm

�� Tm

Tmsþ1þ sðsþ7ωcÞ

15ωc2ðsþ0:6ωcÞ

�� ð25Þ

To further analyze the robustness of SPADRC, classical fre-quency analysis methods are used to compare to the frequencyresponse of SPPI. Fig. 10 shows the Bode diagram of both SPPI andSPADRC using the plant Gp sð Þe� lps ¼ 1

sþa e� lps, when the plant is

known exactly (i.e. when a¼10, lp¼5 (s)). In the Bode diagram, themagnitude plot crosses 0 dB while the phase plot crosses �1801n(where n¼1, 2, 3, ⋯) axis several times resulting in many “GainMargins (GM)” and “Phase Margins (PM)”. Among those, only theminimum margin is necessary to determine the real stabilitymargin. In Fig. 9, round points were used to mark all the margins.The leftmost blue and red points are the minimum stabilitymargins of SPPI and SPADRC. The minimum stability margins foreach response from part A of Section 4 are shown in Table 2 whichremarkably shows that the SPADRC is much more tolerant to dead-time changes in the plant than SPPI.

Since Nyquist criterion is both sufficient and necessary condi-tion for stability and can be applied to systems defined by non-rational functions and non-minimal phase system. Furthermore, itcan also provide a graphic algorithm to compute the delay margin.

0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

Time (Seconds)

Out

put

Setpoint

DOBDTC

SPADRC

0 50 100 150 200 250 300 350 400 450 500-3

-2

-1

0

1

Time (Seconds)

Inpu

t

DOBDTC

SPADRC

Fig. 8. Plant output, set-point, and control input with measurement noise.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (Second)

Out

put

setpointADRC1SPADRCADRC2

Fig. 9. Response for the system without time delay.

Table 1Tuning parameters of SPADRC and ADRC.

Controller b0 ωo ωc lm

SPADRC 200 3� 5 5 1.8ADRC1 200 3� 5 5 —

ADRC2 200 3� 3:7 3.7 —






So we will analyze the stability in frequency domain utilizingNyquist criterion. Figs. 11 and 12 show the Nyquist diagrams ofboth SPPI and SPADRC with the plant Gp sð Þe� lps. However, whentransformed to 2-DOF, the SPADRC will contain an integrator,adding a half circle with the radius of infinity to the right halfplane of the Nyquist diagram, which are not shown in the figures.

Fig. 11 shows us that when dead-time decreases gradually, theNyquist plot will move from the right half-plane to the left part ofthe S-plane. We can see that when dead-time equals to 4.7, theNyquist plot is about to encircle (�1, j0), which is consistent withthe SPPI simulation result from Section 4, part A. When dead-timedecreases by 10%, the plot crosses the line of RefGðjωÞg ¼ �1,causing the system to lose stability. Fig. 12 shows when the plant iscontrolled by SPADRC, the Nyquist plot remains flat and is pushedto the right half of the S-plane when the DT is within the range of3.2 to 6.8. We can see that when the dead-time increases to 6.8 ordecreases to 3.2, the Nyquist plot is about to encircle (�1, j0),which is also consistent with the SPADRC limits of the dead-timechanging range ð740%Þ that we get from simulation test in part Aof Section 4.

In the above Nyquist diagrams, when lp decreases graduallyfrom 5, one of the margin points will rotate firstly to (�1, 0) point,which is called lower boundary gain crossover point (LBGCP). Thesystem is critically stable and the step response in time domain iscontinuously oscillatory. The dead-time of the process for this timeis called lower boundary of delay margin (LBDM) denoted as τlow.

If lp continues to decrease, the intersection between the real axisand the Nyquist curve is left of the (�1, 0) point resulting in anunstable closed-loop system. Similarly, when lp increase graduallyfrom 5, the point firstly rotate to (�1, 0) is called upper boundarygain crossover point (UBGCP). The dead-time of the process forthis time is called upper boundary of delay margin (UBDM)denoted as τup [25]. The delay margin of SPADRC and SPPI arealso shown in Table III and from which we can see that the stablerange of dead-time for the plant 1

sþ10 e�5s is much larger when

applying SPADRC than SPPI.

In part C of Section 4, applying SPADRC to a system withouttime delay can effectively compensate overshoot. An interestingphenomenon has been found that by increasing the estimation ofdead-time (lm) in the process helps to compensate overshoot. Toexplain this phenomenon in frequency domain, Fig. 13 presentsthe Bode diagram of plant Gp sð Þ ¼ 1

sþ10 when applying SPADRCcontroller. When lm¼0, SPADRC is equivalent to a regular ADRC;when lm increases gradually, we can see the phase lead corre-spondingly. The Bode diagram shows that SPADRC's phaseresponse from higher values of lm will cause a larger phase leadand margin; however, the crossover frequency decreases onlyslightly. Therefore, if absolutely no overshoot is tolerated in asystem (even for one without time delay), and the overshoot isdifficult for a regular controller to cope with, SPADRC can be

Bode Diagram

Frequency (rad/sec)10-2 10-1 100 101 102 103 104

-2880

-2160

-1440

-720

0

Pha

se (d

eg)

-100

-50

0

50

Mag

nitu

de (d

B)

SPPISPADRC

MinimumStabilityMargin

Fig. 10. Open-loop loop gain Bode diagram.

Table 2Stable margin of SPADRC AND SPPI.

1sþa e

� lps SPADRC SPPI

a lp GM dB (rad/s) PM deg (rad/s) GM dB (rad/s) PM deg (rad/s)

10 4.7 2.2243 64.3988 1.21917 �23.8719(0.5694) (0.1659) (5.9509) (4.7333)

5 2.2036 61.5425 2.0213 60.0382(0.5156) (0.1659) (0.5986) (0.1994)

5.3 2.1537 58.7173 1.2947 22.4576(0.4709) (0.1659) (6.2513) (4.9893)

LBDM τlow 3.2602 4.6952UBDM τup 6.9815 5.2974

Nyquist Diagram

Real Axis

Imag

inar

y A

xis

-10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20SPPI DT=4.0

SPPI DT=4.7

SPPI DT=5.0

Fig. 11. SPPI Nyquist Diagram with different dead-times.

-1.5 -1 -0.5 0 0.5 1 1.5-20

-15

-10

-5

0

5

10

15

20Nyquist Diagram

Real Axis

Imag

inar

y A

xis

SPADRC DT=3.2SPADRC DT=4.0SPADRC DT=5.0SPADRC DT=6.0SPADRC DT=6.8

Fig. 12. SPADRC Nyquist Diagram with different dead-times.






applied to compensate the overshoot easily. From practical experi-ence with many loops typical to the process industries, a standardADRC with a properly tuned SP performs better than an ADRCwithout SP, even when the model used in the SP is of lower orderthan the true behavior of the loop because lm in the model cancompensate well for the order reduction.

6. Case studies

In this section, two industrial concentration control applica-tions are considered. Case A shows comparative simulation studyof SPADRC and a modified SP structure from a recent work foundin the literature; case B shows a comparative test between thesimulation and experimental results. The oxygen concentrationcontrol problem in a boiler system.

6.1. Chemical reactor concentration control

The chemical reactor concentration control problem is ana-lyzed in this case study. The problemwas studied in [22,23], wherea Filtered Smith Predictor (FSP) and Generalized Predictor (GP) arecompared and analyzed. This case study is used to show thatSPADRC can effectively reduce the noise effect, and meanwhile, itis a very robustness controller.

The dynamics of the chemical reactors with non-ideal mixingcan be described by the Cholettle model:

dCdt

¼ F tð ÞV

Ci tð Þ�C tð Þ½ �� k1C tð Þk2C tð Þþ1� �2 ð26Þ

where C(t) is the output concentration, CiðtÞ is the input concen-tration, F(t) is the input flow, V is the reactor volume andk1 ¼ 10 l=s, k2 ¼ 10 l=mol and V ¼ 1 l are parameters. Thisreactor has an unstable operating point for F ¼ 0:0333 l=s andCi ¼ 3:288mol=l at C ¼ 1:316 mol=l.

The linearized concentration control process model at the operat-

ing point is given by sð Þ ¼ 3:433e� 20s

103�1 s� 1. The SPADRC parameters are

selected as ωo ¼ 0:09 rad=s� �

; ωc ¼ 0:06 rad=s� �

; b0 ¼ 3:433103:1 ¼

0:033, and lm ¼ 20 sð Þ. The control structures of FSP and GP aredefined in [22]. To improve noise attenuation and to guarantee robust

stability, λ¼ 160 is used in GP and Fr zð Þ ¼ 0:03535 z2ðz�0:9968Þz�0:995ð Þðz�0:85Þ2 is

used in the FSP strategy.The new controllers were simulated in the following four

scenarios: (i) nominal model with a step disturbance, (ii) 30% lessdead-time estimation error (lm¼14 sð Þ) with a step disturbance,

(iii) 30% more dead-time estimation error (lm¼26 sð Þ) with a stepdisturbance and (iv) nominal model with measurement noisewhich is generated by means of Simulink “band-limited whitenoise” with Ts¼0.5 sð Þ, noise power¼0.1 and seed¼0. Since thereis not a significant difference between the proposed GP and FSPstrategies in the simulations without noise [22], we only comparethe performance between FSP and SPADRC in scenarios (i), (ii), and(iii), then compare all three controllers in scenario (iv). Simulationresults are shown in Figs. 14–17, respectively.

The result of the nominal case comparison shows very similarperformance between FSP and SPADRC, see Fig. 14. Although the

-100

-50

0

50

Mag

nitu

de (d

B)

10-2 10-1 100 101 102 103-180

-135

-90

-45

0

Pha

se (d

eg)

Bode Diagram

Frequency (rad/sec)

lm = 0lm = 1lm = 2

Fig. 13. Applying SPADRC to a plant without dead-time. 0 200 400 600 8000

2

4

6

y(t) Set-point

SPADRCFSP

0 200 400 600 800-5

0

5

10

Time (s)u(

t)

SPADRCFSP

Fig. 14. Nominal case performance comparison.

0 200 400 600 8000

2

4

6

y(t) Set-point

SPADRCFSP

0 200 400 600 800-5

0

5

10

Time (s)

u(t)

SPADRCFSP

Fig. 15. Perturbed system (30% less DT estimation error) performance comparison.

0 200 400 600 8000

2

4

6

y(t) Set-point

SPADRCFSP

0 200 400 600 800-5

0

5

10

Time (s)

u(t)

SPADRCFSP

Fig. 16. Perturbed system (30% more DT estimation error) performancecomparison.






control signal differs in steady-state, the absolute values are nearlythe same. However, when the dead-time estimation is off by 30%,the output of SPADRC is much smoother than that of the FSPsystem as shown in Figs. 15 and 16. Particularly, when the dead-time estimation is 30% higher than the actual dead-time (Fig. 16),the FSP suffers from considerable oscillation in the output andcontrol signal. When noise is introduced, the control signal of GPrequires a much greater effort in order to produce similar outputperformances as FSP and SPADRC whereas the latter two are onlyslightly affected by the noise as shown in Fig. 17. Based upon theseresults, SPADRC shows robustness and good disturbance rejectionwhile providing greater resilience against noise.

6.2. Oxygen concentration control

The oxygen concentration control problem in a boiler system asshown in Fig. 18 is studied in this case study. The process ischaracterized by high nonlinearity and large time delay. Thenonlinearity mainly comes from the trigonometric relationshipsin valves and dampers containing rotating objects to determineflow rates. Large time delay is due to the reason that as the gasenters the exhaust flue, it takes time for it to travel to an oxygensensor located inside it.

In this case study, SPADRC control strategy will be used tohandle the significant dead-time delays expected within the boilerwhile meeting the requirement of fast reaction to changes in heatdemand and adapting to deviations in either the controlledcharacteristics or the unknown internal variables within the boiler.So despite the complex model, a first-order-plus-dead-time modelyðsÞ ¼ 0:2075

81:9sþ1 e�5suðsÞ is used in the SPADRC structure based upon

an expected model structure with parameters adjusted to outexperiment setup [28].

The controller is running on the UPAC programmable logiccontroller running the OpenPCS software which supports the IEC61131-3 industry standard supporting many different PLC devicesincluding the UPAC controller, see Fig. 19. The UPAC's analog inputsand outputs were connected to a PCI DAC1602-16 real-time inter-face PC card. This interface card provides 16 analog inputs and2 analog outputs, and each of which are converted to or from a16-bit digital signal. The software for the card allows connectionsto the MATLAB/Simulink software.

The test results are shown in Fig. 20. The figure shows that thesimulation and test results match well, and the demand signal canbe tracked accurately and quickly. The implementation of theproposed method on an industrial platform is demonstrated.

7. Conclusion

In this paper a predictive ADRC structure is proposed as asolution to the problem of disturbance rejection in industrialprocesses with transport delay. It addressed the limitations ofADRC as applied to systems with transport delay and the longstanding weakness in the industry standard solution, the SmithPredictor. Simulation studies are carried out to show that the

0 200 400 600 8000

2

4

6y(

t)

0 200 400 600 800-10

0

10

Time (s)

u(t)

Set-pointGPFSPSPADRC

GPFSPSPADRC

Fig. 17. Nominal case with noise performance comparison.

Fig. 18. Oxygen concentration in boiler system.

Fig. 19. Experiment test controller.

0 100 200 300 400 500 600-20

0

20

40

60

80

Time (Seconds)

O2

Flow

O2 DemandSPADRC-simulationSPADRC-experiment

0 100 200 300 400 500 600

0

20

40

60

Time (Seconds)

Fan

Dam

per P

ositi

on SPADRC-simulationSPADRC-experiment

Fig. 20. Oxygen concentration and control signal.






proposed structure is simple to use, intuitive to understand, andeffective in dealing with delays while maintaining good transientresponse and robustness. Significant improvements over existingmethods are observed.

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