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Research ArticleActive Disturbance Rejection Control of a Coupled-Tank System

Fayiz Abukhadra

Faculty of Engineering King Abdulaziz University Rabigh Saudi Arabia

Correspondence should be addressed to Fayiz Abukhadra fabukhadrakauedusa

Received 15 April 2018 Accepted 26 June 2018 Published 8 July 2018

Academic Editor Kamran Iqbal

Copyright copy 2018 Fayiz Abukhadra This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In process industries liquid is pumped and stored in interacting coupled tanks The liquid level in these tanks must be accuratelycontrolled This study aims to investigate the performance of the active disturbance rejection control method in controlling acoupled-tank system A mathematical model of the coupled-tank system is derived to facilitate a simulation study Assuming thatthe water level in the second tank is the only measured state an extended observer with time-varying parameters estimates thesecond state and the total disturbances of the system The system is then regulated using a time-varying feedback controller Theresults show the effectiveness of the method in improving the time domain measures and the disturbance rejection compared toother controllers

1 Introduction

In process control the liquid level control in multiple con-nected tanks performed by controlling the liquid flow is atypical nonlinear control problem present in many industrialprocesses A mathematical model of the plant is requiredto design a controller to maintain a constant level in suchtanks The mathematical model of the controlled plant canbe obtained by two techniques analytical and experimentalThe mathematical models for a coupled-tank system areobtained by applying the laws of energy conservation massconservation etc The mathematical models obtained bymeans of analytical designs are generally complex and mostoften contain nonlinear dependencies of variables The needto estimate system uncertainties using input and outputis a fundamental problem of the control theory Differentmethodologies were proposed to solve this problem Amongthem are the backstepping control strategy [1] the adaptivefuzzy proportional-integral controller [2] the neurofuzzy-slidingmode controller [3] the hybrid fuzzy inference systemthat uses artificial hydrocarbon networks at the defuzzifica-tion step or the so-called fuzzy-molecular control [4] andsecond-order sliding mode controllers (SMC) [5] A hybridsystem that combines the advantages of the robustness of thefractional control and the SMC [6] and a digital proportionalintegral controller [7] were also proposed Moreover an

observed-state feedback controller via eigenvalue assign-ment and linear-quadratic-Gaussian control were designedin discrete-time and implemented by an industrial con-troller (ie programmable logic controller) [8] Several otherresearchers reported model-based controllers as the develop-ment of an optimal PID controller for controlling the desiredliquid level using the particle swarm optimization (PSO)algorithm for optimizing the PID controller parameters [9]A static sliding mode control scheme was proposed for thesystem [10] and two different dynamic sliding mode controlschemes were proposed to reduce the chattering problemassociated with the static sliding mode control scheme

The active disturbance rejection control (ADRC) [11] isa method that does not require a complete mathematicaldescription of the system The basic idea for this methodcomprises the use of an extended observer coupled with afeedback controller in the closed-loop control The observerestimates all states of the system uncertainties and externaldisturbances (total uncertainty)The total uncertainty is con-sidered as an extended state of the system If the estimationof the observer is accurate the system to be controlled is con-verted to a simpler model because the total uncertainty iscanceled in real time The ADRC method has been success-fully applied to several practical problems [12ndash14]

The current study aims to apply the ADRC method toregulate the liquid level in the second tank of a coupled-tank

HindawiJournal of EngineeringVolume 2018 Article ID 7494085 6 pageshttpsdoiorg10115520187494085

2 Journal of Engineering

Figure 1 Schematic diagram of the coupled-tank system

system In addition this study attempts to reduce the tuningparameters of the ADRC use the time-varying parameters ofthe observer and the controller and optimize the parametersof the observer and the controller using the integral absoluteerror (IAE) as the cost function and the genetic algorithmas the optimization method To the best of the authorrsquosknowledge this is the first study to apply the ADRC methodto the problem of the coupled-tank system

The remainder of this paper is structured as followsSection 2 derives a mathematical model of the coupled-tank system and introduces the ADRC method Section 3describes the ADRC method used herein and discusses thesimulation results and Section 4 concludes this paper

2 Methods

21 Mathematical Modeling of the Coupled-Tank SystemFigure 1 shows a schematic of the coupled-tank system whichconsisted of two connected tanks A pump supplied waterinto the first tank (q)The second tankwas filled from the firsttank via a connecting pipe (q1) An outlet was located at thebottom of the second tank to change the output flow q2 Themathematical model of the coupled-tank system is nonlinear

We derive the following equation by applying the flowbalance equation for tanks 1 and 2 [10]

119889ℎ1119889119905 = 1119860 (119902 minus 1199021)

119889ℎ2119889119905 = 1119860 (1199021 minus 1199022)

(1)

In (1) q1 and q2 are defined as follows [10]

1199021 = 1198861radic2119892 (ℎ1 minus ℎ2) for ℎ1 gt ℎ21199022 = 1198862radic2119892ℎ2 for ℎ2 gt 0

(2)

where h1 and h2 are the water level in tanks 1 and 2respectively q is the inlet flow rate q1 is the flow rate fromtanks 1 to 2 A is the cross section area for both tanks a1 isthe area of the pipe connecting the two tanks a2 is the area ofthe outlet and 119892 is the constant of gravity The system can beconsidered as a single input-single output system (SISO) if theinlet flow q is selected as the input and the liquid level h2 inthe second tank is selected as the outputThe dynamic model

of the coupled tanks is described by the following equation[10]

119889ℎ1119889119905 = minus1198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 + 119902119860

119889ℎ2119889119905 = 1198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 minus 1198962radicℎ2(3)

Parameters k1 and k2 are defined as follows

1198961 = 1198861radic2119892119860

1198962 = 1198862radic2119892119860

(4)

Note that q is always positive whichmeans that the pumpcan pump water into the tank (q ge 0) At equilibrium for theconstant water level set point the derivatives with regard tothe water levels in the two tanks must be zero such that thefollowing condition can be written

119889ℎ1119889119905 = 119889ℎ2119889119905 = 0 (5)

Therefore the following algebraic relationship holdswhen (3) is used in (5)

minus1198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 + 119902119860

1198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 minus 1198962radicℎ2(6)

The equilibrium flow rate q can be calculated as follows

119902 = minus1198601198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 (7)

In the case of coupled tanks the inequality ℎ1 ge ℎ2 holdsin every operating point which implies that the terms1198961sign(ℎ1 minus ℎ2) ge 0The dynamic model can then be writtenas

119889ℎ1119889119905 = minus1198961radicℎ1 minus ℎ2 + 1198962radicℎ2dh2dt

= minusk1radicℎ1 minus ℎ2 minus 2k2radich2 + 1Au

(8)

Using the following transformation

1199091 = ℎ21199092 = minus1198961radicℎ1 + 1198962radicℎ1 minus ℎ2

(9)

Eq (8) can be written as

1 = 11990922 = 119891 (119909 119905) + 119892 (119909 119905) 119906119910 = 1199091

(10)

Journal of Engineering 3

Accordingly 119891(119909 119905) and 119892(119909 119905) in (10) have the followingform

119891 (119909 119905) = 119896111989622 ( radicℎ2radicℎ1 minus ℎ2 minusradicℎ1 minus ℎ2radicℎ2 ) + 119896122 minus 11989622

119892 (119909 119905) = 119896221198601

radicℎ1 minus ℎ2(11)

22 ActiveDisturbance RejectionControl TheADRCmethodis explained on the second-order SISO dynamical system ofthe following form

1 = 11990922 = 119891 (119909 119905) + 119889 (119905) + 119892 (119909 119905) 119906 (119905)119910 = 1199091

(12)

where 119906(119905) and 119910(119905) are the system input and output respec-tivelyThe nonlinear function119891(119909 119905) is the internal dynamicsof the system and 119889(119905) is the external disturbance Takingthe estimation value of 119892(119909 119905) as b0 (15) can be rewritten asfollows

1 = 11990922 = 1199093 + 1198870119906 (119905)3 = 119891119910 = 1199091

(13)

where the state variables 1199091 and 1199092 are the system states and1199093 = 119891 is added as an additional state representing the totaldisturbanceThe states of (13) are estimated using an extendedstate observer (ESO) The main advantage of an ESO is thatit can estimate the total uncertainties without knowledge ofthe systemrsquos mathematical model The ESO treats the totaluncertainties as a new state An ESO for the second-ordersystem is constructed as follows [15 16]

1199091 = 1199092 (119905) + 1205721119877 (119905) (1199092 (119905) minus 1199091 (119905))1199092 = 1199093 (119905) + 12057221198772 (119905) (1199092 (119905) minus 1199091 (119905)) + 1198870119906 (119905)1199091 = 12057231198773 (119905) (1199092 (119905) minus 1199091 (119905))

(14)

The time-varying function 119877(119905) has the following form

119877 (119905) = 119877119900 1 minus 119890minus1198861199051 + 119890minus119886119905 (15)

The parameter 120572119894 in (21) can be determined such that thecharacteristic polynomial

120582 (s) = 1199043 + 12057211199042 + 1205722119904 + 1205723 (16)

is HurwitzIf the observer tuning procedure is adequate the observer

states converge to the system states 1199091 997888rarr 1199091 1199092 997888rarr 1199092 and1199093 997888rarr 1199093 in finite time

Figure 2 Block diagram of the ADRC for second-order system

Table 1 Characteristic of the coupled-tank system

Gravitational rate 119892 981 cms2

Cross-sectional area of both tanks 2082 cm2

Area of the connecting pipe 11988612 058 cm2

Area of the outlet 1198862 03 cm2

The control objective is to cancel the total disturbancewhile satisfying the tracking task The total disturbance isrejected with the system input signal

119906 (119905) = minus3 (119905) + 1199060 (119905)1198870 (17)

where 1199060(119905) is a control signal from a feedback controllerSubstituting (17) in (13) and assuming an accurate estimationof the total disturbance the controlled system transforms toa double integrator

(119905) =1199093 minus 3 (119905) + 1199060 (119905) asymp 1199060 (119905) (18)

A double integrator can be controlled with any classicalcontroller design The following control law can be obtainedif a linear proportional and derivative controller is used

1199060 (119905) = 119896119901 (119903 (119905) minus 1199091 (119905)) minus 119896119889 ( 119903 (119905) minus 1199092 (119905)) (19)

where 119903(119905) and 119903(119905) are the reference signal and its derivativerespectively and 1199091 (119905) and 1199092(119905) are the estimated states ofthe plant One possible method to simplify the controllertuning is to set

119896119889 = 119877 (119905)and 119896119901 = 1198772 (119905)

4 (20)

Figure 2 shows the block diagram of the ADRC closed-loop system

3 Results and Discussion

Table 1 lists the numerical values of the parameters of thecoupled-tank system [10]

The range of the pump flow rate was limited between umin= 0 and umax = 50 [cm3s]

The Methods clearly showed that the parameters of theclosed-loop control using ADRC are 1205721 1205722 1205723 119877119900 1198870 and 119886

4 Journal of Engineering

Table 2 Comparison of the performance index measures

Performance measure Method Improvement []SMC [10] ADRC

Settling time 1131 582 48Rise time (s) 521 378 27IAE 1145 6786 41ISE 2548 1268 50ITAE 2690 1190 56

time (s)0 50 100 150

referenceresponse

0

1

2

3

4

5

6

7

8

liqui

d-le

vel (

mm

)

Figure 3 Response of the system for the 6 cm desired level

The Hurwitz characteristic polynomial is selected as followswith the poles minus44848 minus02576 + 25735i minus02576 minus 25735i

120582 (s) = 1199043 + 51199042 + 9119904 + 30 (21)

Accordingly 119877119900 1198870 and 119886 were obtained using a geneticalgorithm optimization method with the objective of mini-mizing the IAE defined as follows

119868119860119864 = int1199050|119890| 119889119905 (22)

The optimum parameters obtained are 1198770 = 7814 1198870 =030 and 119886 = 099Figure 3 shows the regulation performance of the con-

troller for a desired level of 6 cm and confirms that thecontrollers successfully regulated the water level Figure 4depicts the control signal of the ADRC Figure 5 presentsthe ESO performance in estimating the system states Theobserver accurately estimated the states The errors 1198901(119905) =1199091minus1199091 1198902(119905) = 1199092(119905)minus1199092(119905) and 1198903(119905) = 1199093(119905)minus1199093(119905) convergedto zero in less than 1 s

The following performance measures were introduced tofacilitate a comparison with the other control methods thesettling time defined as the time taken until the output finallysettles within 2 of the steady state value the rise time Tr

defined as the time taken by the output to change from 10to 90 of its final value and in addition to the IAE theintegral squared error (ISE) and the time weighted absoluteerror (ITAE) computed as follows

119868119878119864 = int11990501198902119889119905 (23)

119868119879119860119864 = int1199050119905 |119890| 119889119905 (24)

The performance of the ADRC was then compared withthat of the SMC method reported in [10] Table 2 presentsthe rise time settling time and error indices (ie IAE ISEand ITAE) for the design in [10] and the ADRC methodThe table clearly shows that the ADRC outperforms the otherdesigns in all performance measures The response of thesystem controlled by the ADRC took 582 s to settle whereasthat in the design in [10] took 113 sThe rise time of the outputresponse in the ADRC controller was 46 s whereas that of theSMC was 52 s The ADRC method resulted in a 48 smallersettling time than that in [10] Moreover the rise time was27 smaller than that in [10] The IAE ISE and ITAE were41 50 and 56 smaller than those in [10]

As a second test we tested the ADRC in a tracking testThe set point tracking test consisted of successively changingthe set point during the operation (Figure 6) The set pointchange was performed at 200 s by a magnitude of 6 cmheight in the water level Consequently the ADRC methodaccurately tracked the set point changes in thewater levelThesame parameters were used for the ADRC for the trackingexperiment

As a third test we checked the ADRC performanceagainst the input disturbance An external flow rate of 60cm3s that started at 150 s and ended at 200 s was appliedFigure 7 illustrates the closed-loop response of the ADRCcontrol and shows how fast the controller response was to thedisturbance and corrected it

4 Conclusions

In this study the ADRC approach was successfully imple-mented with the design tested by a simulation to controlthe water level in the second tank of a coupled-tank sys-tem The effectiveness of the ADRC method was verifiedthrough computer simulations The results showed that thiscontrol method can control a nonlinear system at all possibleoperating points The designed ADRC achieved the desired

Journal of Engineering 5

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

minus5

0

5

1=Ｒ1-Ｒ1Ｂ

minus20

0

20

2=Ｒ2-Ｒ2Ｂ

minus500

0

500

3=Ｒ3-Ｒ3Ｂ

Figure 4 Performance of the ESO in estimating the system states

Time (s)0 50 100 150

minus10

0

10

20

30

40

50

60

cont

rol s

igna

l u(t)

Figure 5 Control signal of the ADRC for the 6 cm desired level

referenceresponse

200 250150 35050 300100 4504000 500Time (s)

0

2

4

6

8

10

12

14

Liqu

id-le

vel (

mm

)

Figure 6 Set point tracking performance of the system

6 Journal of Engineering

referenceresponse

50 100 150 200 250 300 350 4000Time (s)

0

1

2

3

4

5

6

7

8

9

10Li

quid

-leve

l (m

m)

Figure 7 Closed-loop response of the system against the inputdisturbance

transient response with small rise and settling times Theadvantages of the ADRC are as follows (a) easiness andsimplicity in design (b) nonrequirement of a mathematicalmodel of the plant and (c) robustness against uncertaintyand disturbance Further work is anticipated in the practicalimplementation of the proposed ADRC technique

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] V Calofir V Tanasa I Fagarasan I Stamatescu N ArghiraandG Stamatescu ldquoA Backstepping ControlMethod for aNon-linear Process - Two Coupled-Tanksrdquo 2013 httpsarxivorgabs13120728

[2] S R Mahapatro B Subudhi and S Ghosh ldquoAdaptive Fuzzy PIController Design for Coupled Tank SystemAn ExperimentalValidationrdquo IFAC Proceedings Volumes vol 47 no 1 pp 878ndash881 2014

[3] A Boubakir F Boudjema and S Labiod ldquoA neuro-fuzzy-sliding mode controller using nonlinear sliding surface appliedto the coupled tanks systemrdquo International Journal of Automa-tion and Computing vol 6 no 1 pp 72ndash80 2009

[4] H Ponce P Ponce H Bastida and A Molina ldquoA novel robustliquid level controller for coupled-tanks systems using artificialhydrocarbon networksrdquo Expert Systems with Applications vol42 no 22 pp 8858ndash8867 2015

[5] MK Khan and S K Spurgeon ldquoRobustMIMOwater level con-trol in interconnected twin-tanks using second order slidingmode controlrdquo Control Engineering Practice vol 14 no 4 pp375ndash386 2006

[6] H Delavari A N Ranjbar R Ghaderi and S Momani ldquoFrac-tional order control of a coupled tankrdquoNonlinearDynamics vol61 no 3 pp 383ndash397 2010

[7] H Bastida P Ponce R Ramirez and A Molina ldquoModel andControl for Coupled Tanks Using Labviewrdquo in Proceedings ofthe 2013 International Conference on Mechatronics Electronicsand Automotive Engineering (ICMEAE) pp 127ndash133 MorelosMexico November 2013

[8] D Engules M Hot and B Alikoc ldquoLevel control of a coupled-tank system via eigenvalue assignment and LQG controlrdquo inProceedings of the 23rd Mediterranean Conference on Controland Automation MED 2015 pp 1198ndash1203 Spain June 2015

[9] H I Jaafar S Y Hussien N A Selamat et al ldquoPSO-tunedPID controller for coupled tank system via priority-based fit-ness schemerdquo in Proceedings of the INternational Conferenceon Mathematics Engineering And Industrial Applications 2014(ICoMEIA 2014) p 070032 Penang Malaysia

[10] N B Almutairi and M Zribi ldquoSliding mode control of coupledtanksrdquoMechatronics vol 16 no 7 pp 427ndash441 2006

[11] J Q Han ldquoFrom PID to active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 56 no 3 pp900ndash906 2009

[12] M PrzybyłaM Kordasz RMadonski P Herman and P SauerldquoActive Disturbance Rejection Control of a 2DOF manipulatorwith significant modeling uncertaintyrdquo Bulletin of the PolishAcademy of SciencesmdashTechnical Sciences vol 60 no 3 2012

[13] Z Chen Q Zheng and Z Gao ldquoActive disturbance rejectioncontrol of chemical processesrdquo in Proceedings of the 16th IEEEInternational Conference on Control Applications CCA 2007Part of IEEE Multi-conference on Systems and Control pp 855ndash861 Singapore October 2007

[14] X Wang An Active Disturbance Rejection Control Solutionfor Electro-hydraulic Servo Systems Cleveland State University2012

[15] D Yoo S S-T Yau and Z Q Gao ldquoOptimal fast trackingobserver bandwidth of the linear extended state observerrdquo Inter-national Journal of Control vol 80 no 1 pp 102ndash111 2007

[16] X-X Yang and Y Huang ldquoCapabilities of extended stateobserver for estimating uncertaintiesrdquo in Proceedings of theAmerican Control Conference (ACC rsquo09) pp 3700ndash3705 IEEESt Louis Mo USA June 2009

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2 Journal of Engineering

Figure 1 Schematic diagram of the coupled-tank system

system In addition this study attempts to reduce the tuningparameters of the ADRC use the time-varying parameters ofthe observer and the controller and optimize the parametersof the observer and the controller using the integral absoluteerror (IAE) as the cost function and the genetic algorithmas the optimization method To the best of the authorrsquosknowledge this is the first study to apply the ADRC methodto the problem of the coupled-tank system

The remainder of this paper is structured as followsSection 2 derives a mathematical model of the coupled-tank system and introduces the ADRC method Section 3describes the ADRC method used herein and discusses thesimulation results and Section 4 concludes this paper

2 Methods

21 Mathematical Modeling of the Coupled-Tank SystemFigure 1 shows a schematic of the coupled-tank system whichconsisted of two connected tanks A pump supplied waterinto the first tank (q)The second tankwas filled from the firsttank via a connecting pipe (q1) An outlet was located at thebottom of the second tank to change the output flow q2 Themathematical model of the coupled-tank system is nonlinear

We derive the following equation by applying the flowbalance equation for tanks 1 and 2 [10]

119889ℎ1119889119905 = 1119860 (119902 minus 1199021)

119889ℎ2119889119905 = 1119860 (1199021 minus 1199022)

(1)

In (1) q1 and q2 are defined as follows [10]

1199021 = 1198861radic2119892 (ℎ1 minus ℎ2) for ℎ1 gt ℎ21199022 = 1198862radic2119892ℎ2 for ℎ2 gt 0

(2)

where h1 and h2 are the water level in tanks 1 and 2respectively q is the inlet flow rate q1 is the flow rate fromtanks 1 to 2 A is the cross section area for both tanks a1 isthe area of the pipe connecting the two tanks a2 is the area ofthe outlet and 119892 is the constant of gravity The system can beconsidered as a single input-single output system (SISO) if theinlet flow q is selected as the input and the liquid level h2 inthe second tank is selected as the outputThe dynamic model

of the coupled tanks is described by the following equation[10]

119889ℎ1119889119905 = minus1198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 + 119902119860

119889ℎ2119889119905 = 1198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 minus 1198962radicℎ2(3)

Parameters k1 and k2 are defined as follows

1198961 = 1198861radic2119892119860

1198962 = 1198862radic2119892119860

(4)

Note that q is always positive whichmeans that the pumpcan pump water into the tank (q ge 0) At equilibrium for theconstant water level set point the derivatives with regard tothe water levels in the two tanks must be zero such that thefollowing condition can be written

119889ℎ1119889119905 = 119889ℎ2119889119905 = 0 (5)

Therefore the following algebraic relationship holdswhen (3) is used in (5)

minus1198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 + 119902119860

1198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 minus 1198962radicℎ2(6)

The equilibrium flow rate q can be calculated as follows

119902 = minus1198601198961sign (ℎ1 minus ℎ2)radic1003816100381610038161003816ℎ1 minus ℎ21003816100381610038161003816 (7)

In the case of coupled tanks the inequality ℎ1 ge ℎ2 holdsin every operating point which implies that the terms1198961sign(ℎ1 minus ℎ2) ge 0The dynamic model can then be writtenas

119889ℎ1119889119905 = minus1198961radicℎ1 minus ℎ2 + 1198962radicℎ2dh2dt

= minusk1radicℎ1 minus ℎ2 minus 2k2radich2 + 1Au

(8)

Using the following transformation

1199091 = ℎ21199092 = minus1198961radicℎ1 + 1198962radicℎ1 minus ℎ2

(9)

Eq (8) can be written as

1 = 11990922 = 119891 (119909 119905) + 119892 (119909 119905) 119906119910 = 1199091

(10)

Journal of Engineering 3

Accordingly 119891(119909 119905) and 119892(119909 119905) in (10) have the followingform

119891 (119909 119905) = 119896111989622 ( radicℎ2radicℎ1 minus ℎ2 minusradicℎ1 minus ℎ2radicℎ2 ) + 119896122 minus 11989622

119892 (119909 119905) = 119896221198601

radicℎ1 minus ℎ2(11)

22 ActiveDisturbance RejectionControl TheADRCmethodis explained on the second-order SISO dynamical system ofthe following form

1 = 11990922 = 119891 (119909 119905) + 119889 (119905) + 119892 (119909 119905) 119906 (119905)119910 = 1199091

(12)

where 119906(119905) and 119910(119905) are the system input and output respec-tivelyThe nonlinear function119891(119909 119905) is the internal dynamicsof the system and 119889(119905) is the external disturbance Takingthe estimation value of 119892(119909 119905) as b0 (15) can be rewritten asfollows

1 = 11990922 = 1199093 + 1198870119906 (119905)3 = 119891119910 = 1199091

(13)

where the state variables 1199091 and 1199092 are the system states and1199093 = 119891 is added as an additional state representing the totaldisturbanceThe states of (13) are estimated using an extendedstate observer (ESO) The main advantage of an ESO is thatit can estimate the total uncertainties without knowledge ofthe systemrsquos mathematical model The ESO treats the totaluncertainties as a new state An ESO for the second-ordersystem is constructed as follows [15 16]

1199091 = 1199092 (119905) + 1205721119877 (119905) (1199092 (119905) minus 1199091 (119905))1199092 = 1199093 (119905) + 12057221198772 (119905) (1199092 (119905) minus 1199091 (119905)) + 1198870119906 (119905)1199091 = 12057231198773 (119905) (1199092 (119905) minus 1199091 (119905))

(14)

The time-varying function 119877(119905) has the following form

119877 (119905) = 119877119900 1 minus 119890minus1198861199051 + 119890minus119886119905 (15)

The parameter 120572119894 in (21) can be determined such that thecharacteristic polynomial

120582 (s) = 1199043 + 12057211199042 + 1205722119904 + 1205723 (16)

is HurwitzIf the observer tuning procedure is adequate the observer

states converge to the system states 1199091 997888rarr 1199091 1199092 997888rarr 1199092 and1199093 997888rarr 1199093 in finite time

Figure 2 Block diagram of the ADRC for second-order system

Table 1 Characteristic of the coupled-tank system

Gravitational rate 119892 981 cms2

Cross-sectional area of both tanks 2082 cm2

Area of the connecting pipe 11988612 058 cm2

Area of the outlet 1198862 03 cm2

The control objective is to cancel the total disturbancewhile satisfying the tracking task The total disturbance isrejected with the system input signal

119906 (119905) = minus3 (119905) + 1199060 (119905)1198870 (17)

where 1199060(119905) is a control signal from a feedback controllerSubstituting (17) in (13) and assuming an accurate estimationof the total disturbance the controlled system transforms toa double integrator

(119905) =1199093 minus 3 (119905) + 1199060 (119905) asymp 1199060 (119905) (18)

A double integrator can be controlled with any classicalcontroller design The following control law can be obtainedif a linear proportional and derivative controller is used

1199060 (119905) = 119896119901 (119903 (119905) minus 1199091 (119905)) minus 119896119889 ( 119903 (119905) minus 1199092 (119905)) (19)

where 119903(119905) and 119903(119905) are the reference signal and its derivativerespectively and 1199091 (119905) and 1199092(119905) are the estimated states ofthe plant One possible method to simplify the controllertuning is to set

119896119889 = 119877 (119905)and 119896119901 = 1198772 (119905)

4 (20)

Figure 2 shows the block diagram of the ADRC closed-loop system

3 Results and Discussion

Table 1 lists the numerical values of the parameters of thecoupled-tank system [10]

The range of the pump flow rate was limited between umin= 0 and umax = 50 [cm3s]

The Methods clearly showed that the parameters of theclosed-loop control using ADRC are 1205721 1205722 1205723 119877119900 1198870 and 119886

4 Journal of Engineering

Table 2 Comparison of the performance index measures

Performance measure Method Improvement []SMC [10] ADRC

Settling time 1131 582 48Rise time (s) 521 378 27IAE 1145 6786 41ISE 2548 1268 50ITAE 2690 1190 56

time (s)0 50 100 150

referenceresponse

0

1

2

3

4

5

6

7

8

liqui

d-le

vel (

mm

)

Figure 3 Response of the system for the 6 cm desired level

The Hurwitz characteristic polynomial is selected as followswith the poles minus44848 minus02576 + 25735i minus02576 minus 25735i

120582 (s) = 1199043 + 51199042 + 9119904 + 30 (21)

Accordingly 119877119900 1198870 and 119886 were obtained using a geneticalgorithm optimization method with the objective of mini-mizing the IAE defined as follows

119868119860119864 = int1199050|119890| 119889119905 (22)

The optimum parameters obtained are 1198770 = 7814 1198870 =030 and 119886 = 099Figure 3 shows the regulation performance of the con-

troller for a desired level of 6 cm and confirms that thecontrollers successfully regulated the water level Figure 4depicts the control signal of the ADRC Figure 5 presentsthe ESO performance in estimating the system states Theobserver accurately estimated the states The errors 1198901(119905) =1199091minus1199091 1198902(119905) = 1199092(119905)minus1199092(119905) and 1198903(119905) = 1199093(119905)minus1199093(119905) convergedto zero in less than 1 s

The following performance measures were introduced tofacilitate a comparison with the other control methods thesettling time defined as the time taken until the output finallysettles within 2 of the steady state value the rise time Tr

defined as the time taken by the output to change from 10to 90 of its final value and in addition to the IAE theintegral squared error (ISE) and the time weighted absoluteerror (ITAE) computed as follows

119868119878119864 = int11990501198902119889119905 (23)

119868119879119860119864 = int1199050119905 |119890| 119889119905 (24)

The performance of the ADRC was then compared withthat of the SMC method reported in [10] Table 2 presentsthe rise time settling time and error indices (ie IAE ISEand ITAE) for the design in [10] and the ADRC methodThe table clearly shows that the ADRC outperforms the otherdesigns in all performance measures The response of thesystem controlled by the ADRC took 582 s to settle whereasthat in the design in [10] took 113 sThe rise time of the outputresponse in the ADRC controller was 46 s whereas that of theSMC was 52 s The ADRC method resulted in a 48 smallersettling time than that in [10] Moreover the rise time was27 smaller than that in [10] The IAE ISE and ITAE were41 50 and 56 smaller than those in [10]

As a second test we tested the ADRC in a tracking testThe set point tracking test consisted of successively changingthe set point during the operation (Figure 6) The set pointchange was performed at 200 s by a magnitude of 6 cmheight in the water level Consequently the ADRC methodaccurately tracked the set point changes in thewater levelThesame parameters were used for the ADRC for the trackingexperiment

As a third test we checked the ADRC performanceagainst the input disturbance An external flow rate of 60cm3s that started at 150 s and ended at 200 s was appliedFigure 7 illustrates the closed-loop response of the ADRCcontrol and shows how fast the controller response was to thedisturbance and corrected it

4 Conclusions

In this study the ADRC approach was successfully imple-mented with the design tested by a simulation to controlthe water level in the second tank of a coupled-tank sys-tem The effectiveness of the ADRC method was verifiedthrough computer simulations The results showed that thiscontrol method can control a nonlinear system at all possibleoperating points The designed ADRC achieved the desired

Journal of Engineering 5

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

minus5

0

5

1=Ｒ1-Ｒ1Ｂ

minus20

0

20

2=Ｒ2-Ｒ2Ｂ

minus500

0

500

3=Ｒ3-Ｒ3Ｂ

Figure 4 Performance of the ESO in estimating the system states

Time (s)0 50 100 150

minus10

0

10

20

30

40

50

60

cont

rol s

igna

l u(t)

Figure 5 Control signal of the ADRC for the 6 cm desired level

referenceresponse

200 250150 35050 300100 4504000 500Time (s)

0

2

4

6

8

10

12

14

Liqu

id-le

vel (

mm

)

Figure 6 Set point tracking performance of the system

6 Journal of Engineering

referenceresponse

50 100 150 200 250 300 350 4000Time (s)

0

1

2

3

4

5

6

7

8

9

10Li

quid

-leve

l (m

m)

Figure 7 Closed-loop response of the system against the inputdisturbance

transient response with small rise and settling times Theadvantages of the ADRC are as follows (a) easiness andsimplicity in design (b) nonrequirement of a mathematicalmodel of the plant and (c) robustness against uncertaintyand disturbance Further work is anticipated in the practicalimplementation of the proposed ADRC technique

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] V Calofir V Tanasa I Fagarasan I Stamatescu N ArghiraandG Stamatescu ldquoA Backstepping ControlMethod for aNon-linear Process - Two Coupled-Tanksrdquo 2013 httpsarxivorgabs13120728

[2] S R Mahapatro B Subudhi and S Ghosh ldquoAdaptive Fuzzy PIController Design for Coupled Tank SystemAn ExperimentalValidationrdquo IFAC Proceedings Volumes vol 47 no 1 pp 878ndash881 2014

[3] A Boubakir F Boudjema and S Labiod ldquoA neuro-fuzzy-sliding mode controller using nonlinear sliding surface appliedto the coupled tanks systemrdquo International Journal of Automa-tion and Computing vol 6 no 1 pp 72ndash80 2009

[4] H Ponce P Ponce H Bastida and A Molina ldquoA novel robustliquid level controller for coupled-tanks systems using artificialhydrocarbon networksrdquo Expert Systems with Applications vol42 no 22 pp 8858ndash8867 2015

[5] MK Khan and S K Spurgeon ldquoRobustMIMOwater level con-trol in interconnected twin-tanks using second order slidingmode controlrdquo Control Engineering Practice vol 14 no 4 pp375ndash386 2006

[6] H Delavari A N Ranjbar R Ghaderi and S Momani ldquoFrac-tional order control of a coupled tankrdquoNonlinearDynamics vol61 no 3 pp 383ndash397 2010

[7] H Bastida P Ponce R Ramirez and A Molina ldquoModel andControl for Coupled Tanks Using Labviewrdquo in Proceedings ofthe 2013 International Conference on Mechatronics Electronicsand Automotive Engineering (ICMEAE) pp 127ndash133 MorelosMexico November 2013

[8] D Engules M Hot and B Alikoc ldquoLevel control of a coupled-tank system via eigenvalue assignment and LQG controlrdquo inProceedings of the 23rd Mediterranean Conference on Controland Automation MED 2015 pp 1198ndash1203 Spain June 2015

[9] H I Jaafar S Y Hussien N A Selamat et al ldquoPSO-tunedPID controller for coupled tank system via priority-based fit-ness schemerdquo in Proceedings of the INternational Conferenceon Mathematics Engineering And Industrial Applications 2014(ICoMEIA 2014) p 070032 Penang Malaysia

[10] N B Almutairi and M Zribi ldquoSliding mode control of coupledtanksrdquoMechatronics vol 16 no 7 pp 427ndash441 2006

[11] J Q Han ldquoFrom PID to active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 56 no 3 pp900ndash906 2009

[12] M PrzybyłaM Kordasz RMadonski P Herman and P SauerldquoActive Disturbance Rejection Control of a 2DOF manipulatorwith significant modeling uncertaintyrdquo Bulletin of the PolishAcademy of SciencesmdashTechnical Sciences vol 60 no 3 2012

[13] Z Chen Q Zheng and Z Gao ldquoActive disturbance rejectioncontrol of chemical processesrdquo in Proceedings of the 16th IEEEInternational Conference on Control Applications CCA 2007Part of IEEE Multi-conference on Systems and Control pp 855ndash861 Singapore October 2007

[14] X Wang An Active Disturbance Rejection Control Solutionfor Electro-hydraulic Servo Systems Cleveland State University2012

[15] D Yoo S S-T Yau and Z Q Gao ldquoOptimal fast trackingobserver bandwidth of the linear extended state observerrdquo Inter-national Journal of Control vol 80 no 1 pp 102ndash111 2007

[16] X-X Yang and Y Huang ldquoCapabilities of extended stateobserver for estimating uncertaintiesrdquo in Proceedings of theAmerican Control Conference (ACC rsquo09) pp 3700ndash3705 IEEESt Louis Mo USA June 2009

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Engineering 3

Accordingly 119891(119909 119905) and 119892(119909 119905) in (10) have the followingform

119891 (119909 119905) = 119896111989622 ( radicℎ2radicℎ1 minus ℎ2 minusradicℎ1 minus ℎ2radicℎ2 ) + 119896122 minus 11989622

119892 (119909 119905) = 119896221198601

radicℎ1 minus ℎ2(11)

22 ActiveDisturbance RejectionControl TheADRCmethodis explained on the second-order SISO dynamical system ofthe following form

1 = 11990922 = 119891 (119909 119905) + 119889 (119905) + 119892 (119909 119905) 119906 (119905)119910 = 1199091

(12)

where 119906(119905) and 119910(119905) are the system input and output respec-tivelyThe nonlinear function119891(119909 119905) is the internal dynamicsof the system and 119889(119905) is the external disturbance Takingthe estimation value of 119892(119909 119905) as b0 (15) can be rewritten asfollows

1 = 11990922 = 1199093 + 1198870119906 (119905)3 = 119891119910 = 1199091

(13)

where the state variables 1199091 and 1199092 are the system states and1199093 = 119891 is added as an additional state representing the totaldisturbanceThe states of (13) are estimated using an extendedstate observer (ESO) The main advantage of an ESO is thatit can estimate the total uncertainties without knowledge ofthe systemrsquos mathematical model The ESO treats the totaluncertainties as a new state An ESO for the second-ordersystem is constructed as follows [15 16]

1199091 = 1199092 (119905) + 1205721119877 (119905) (1199092 (119905) minus 1199091 (119905))1199092 = 1199093 (119905) + 12057221198772 (119905) (1199092 (119905) minus 1199091 (119905)) + 1198870119906 (119905)1199091 = 12057231198773 (119905) (1199092 (119905) minus 1199091 (119905))

(14)

The time-varying function 119877(119905) has the following form

119877 (119905) = 119877119900 1 minus 119890minus1198861199051 + 119890minus119886119905 (15)

The parameter 120572119894 in (21) can be determined such that thecharacteristic polynomial

120582 (s) = 1199043 + 12057211199042 + 1205722119904 + 1205723 (16)

is HurwitzIf the observer tuning procedure is adequate the observer

states converge to the system states 1199091 997888rarr 1199091 1199092 997888rarr 1199092 and1199093 997888rarr 1199093 in finite time

Figure 2 Block diagram of the ADRC for second-order system

Table 1 Characteristic of the coupled-tank system

Gravitational rate 119892 981 cms2

Cross-sectional area of both tanks 2082 cm2

Area of the connecting pipe 11988612 058 cm2

Area of the outlet 1198862 03 cm2

The control objective is to cancel the total disturbancewhile satisfying the tracking task The total disturbance isrejected with the system input signal

119906 (119905) = minus3 (119905) + 1199060 (119905)1198870 (17)

where 1199060(119905) is a control signal from a feedback controllerSubstituting (17) in (13) and assuming an accurate estimationof the total disturbance the controlled system transforms toa double integrator

(119905) =1199093 minus 3 (119905) + 1199060 (119905) asymp 1199060 (119905) (18)

A double integrator can be controlled with any classicalcontroller design The following control law can be obtainedif a linear proportional and derivative controller is used

1199060 (119905) = 119896119901 (119903 (119905) minus 1199091 (119905)) minus 119896119889 ( 119903 (119905) minus 1199092 (119905)) (19)

where 119903(119905) and 119903(119905) are the reference signal and its derivativerespectively and 1199091 (119905) and 1199092(119905) are the estimated states ofthe plant One possible method to simplify the controllertuning is to set

119896119889 = 119877 (119905)and 119896119901 = 1198772 (119905)

4 (20)

Figure 2 shows the block diagram of the ADRC closed-loop system

3 Results and Discussion

Table 1 lists the numerical values of the parameters of thecoupled-tank system [10]

The range of the pump flow rate was limited between umin= 0 and umax = 50 [cm3s]

The Methods clearly showed that the parameters of theclosed-loop control using ADRC are 1205721 1205722 1205723 119877119900 1198870 and 119886

4 Journal of Engineering

Table 2 Comparison of the performance index measures

Performance measure Method Improvement []SMC [10] ADRC

Settling time 1131 582 48Rise time (s) 521 378 27IAE 1145 6786 41ISE 2548 1268 50ITAE 2690 1190 56

time (s)0 50 100 150

referenceresponse

0

1

2

3

4

5

6

7

8

liqui

d-le

vel (

mm

)

Figure 3 Response of the system for the 6 cm desired level

The Hurwitz characteristic polynomial is selected as followswith the poles minus44848 minus02576 + 25735i minus02576 minus 25735i

120582 (s) = 1199043 + 51199042 + 9119904 + 30 (21)

Accordingly 119877119900 1198870 and 119886 were obtained using a geneticalgorithm optimization method with the objective of mini-mizing the IAE defined as follows

119868119860119864 = int1199050|119890| 119889119905 (22)

The optimum parameters obtained are 1198770 = 7814 1198870 =030 and 119886 = 099Figure 3 shows the regulation performance of the con-

troller for a desired level of 6 cm and confirms that thecontrollers successfully regulated the water level Figure 4depicts the control signal of the ADRC Figure 5 presentsthe ESO performance in estimating the system states Theobserver accurately estimated the states The errors 1198901(119905) =1199091minus1199091 1198902(119905) = 1199092(119905)minus1199092(119905) and 1198903(119905) = 1199093(119905)minus1199093(119905) convergedto zero in less than 1 s

The following performance measures were introduced tofacilitate a comparison with the other control methods thesettling time defined as the time taken until the output finallysettles within 2 of the steady state value the rise time Tr

defined as the time taken by the output to change from 10to 90 of its final value and in addition to the IAE theintegral squared error (ISE) and the time weighted absoluteerror (ITAE) computed as follows

119868119878119864 = int11990501198902119889119905 (23)

119868119879119860119864 = int1199050119905 |119890| 119889119905 (24)

The performance of the ADRC was then compared withthat of the SMC method reported in [10] Table 2 presentsthe rise time settling time and error indices (ie IAE ISEand ITAE) for the design in [10] and the ADRC methodThe table clearly shows that the ADRC outperforms the otherdesigns in all performance measures The response of thesystem controlled by the ADRC took 582 s to settle whereasthat in the design in [10] took 113 sThe rise time of the outputresponse in the ADRC controller was 46 s whereas that of theSMC was 52 s The ADRC method resulted in a 48 smallersettling time than that in [10] Moreover the rise time was27 smaller than that in [10] The IAE ISE and ITAE were41 50 and 56 smaller than those in [10]

As a second test we tested the ADRC in a tracking testThe set point tracking test consisted of successively changingthe set point during the operation (Figure 6) The set pointchange was performed at 200 s by a magnitude of 6 cmheight in the water level Consequently the ADRC methodaccurately tracked the set point changes in thewater levelThesame parameters were used for the ADRC for the trackingexperiment

As a third test we checked the ADRC performanceagainst the input disturbance An external flow rate of 60cm3s that started at 150 s and ended at 200 s was appliedFigure 7 illustrates the closed-loop response of the ADRCcontrol and shows how fast the controller response was to thedisturbance and corrected it

4 Conclusions

In this study the ADRC approach was successfully imple-mented with the design tested by a simulation to controlthe water level in the second tank of a coupled-tank sys-tem The effectiveness of the ADRC method was verifiedthrough computer simulations The results showed that thiscontrol method can control a nonlinear system at all possibleoperating points The designed ADRC achieved the desired

Journal of Engineering 5

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

minus5

0

5

1=Ｒ1-Ｒ1Ｂ

minus20

0

20

2=Ｒ2-Ｒ2Ｂ

minus500

0

500

3=Ｒ3-Ｒ3Ｂ

Figure 4 Performance of the ESO in estimating the system states

Time (s)0 50 100 150

minus10

0

10

20

30

40

50

60

cont

rol s

igna

l u(t)

Figure 5 Control signal of the ADRC for the 6 cm desired level

referenceresponse

200 250150 35050 300100 4504000 500Time (s)

0

2

4

6

8

10

12

14

Liqu

id-le

vel (

mm

)

Figure 6 Set point tracking performance of the system

6 Journal of Engineering

referenceresponse

50 100 150 200 250 300 350 4000Time (s)

0

1

2

3

4

5

6

7

8

9

10Li

quid

-leve

l (m

m)

Figure 7 Closed-loop response of the system against the inputdisturbance

transient response with small rise and settling times Theadvantages of the ADRC are as follows (a) easiness andsimplicity in design (b) nonrequirement of a mathematicalmodel of the plant and (c) robustness against uncertaintyand disturbance Further work is anticipated in the practicalimplementation of the proposed ADRC technique

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

References

[1] V Calofir V Tanasa I Fagarasan I Stamatescu N ArghiraandG Stamatescu ldquoA Backstepping ControlMethod for aNon-linear Process - Two Coupled-Tanksrdquo 2013 httpsarxivorgabs13120728

[2] S R Mahapatro B Subudhi and S Ghosh ldquoAdaptive Fuzzy PIController Design for Coupled Tank SystemAn ExperimentalValidationrdquo IFAC Proceedings Volumes vol 47 no 1 pp 878ndash881 2014

[3] A Boubakir F Boudjema and S Labiod ldquoA neuro-fuzzy-sliding mode controller using nonlinear sliding surface appliedto the coupled tanks systemrdquo International Journal of Automa-tion and Computing vol 6 no 1 pp 72ndash80 2009

[4] H Ponce P Ponce H Bastida and A Molina ldquoA novel robustliquid level controller for coupled-tanks systems using artificialhydrocarbon networksrdquo Expert Systems with Applications vol42 no 22 pp 8858ndash8867 2015

[5] MK Khan and S K Spurgeon ldquoRobustMIMOwater level con-trol in interconnected twin-tanks using second order slidingmode controlrdquo Control Engineering Practice vol 14 no 4 pp375ndash386 2006

[6] H Delavari A N Ranjbar R Ghaderi and S Momani ldquoFrac-tional order control of a coupled tankrdquoNonlinearDynamics vol61 no 3 pp 383ndash397 2010

[7] H Bastida P Ponce R Ramirez and A Molina ldquoModel andControl for Coupled Tanks Using Labviewrdquo in Proceedings ofthe 2013 International Conference on Mechatronics Electronicsand Automotive Engineering (ICMEAE) pp 127ndash133 MorelosMexico November 2013

[8] D Engules M Hot and B Alikoc ldquoLevel control of a coupled-tank system via eigenvalue assignment and LQG controlrdquo inProceedings of the 23rd Mediterranean Conference on Controland Automation MED 2015 pp 1198ndash1203 Spain June 2015

[9] H I Jaafar S Y Hussien N A Selamat et al ldquoPSO-tunedPID controller for coupled tank system via priority-based fit-ness schemerdquo in Proceedings of the INternational Conferenceon Mathematics Engineering And Industrial Applications 2014(ICoMEIA 2014) p 070032 Penang Malaysia

[10] N B Almutairi and M Zribi ldquoSliding mode control of coupledtanksrdquoMechatronics vol 16 no 7 pp 427ndash441 2006

[11] J Q Han ldquoFrom PID to active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 56 no 3 pp900ndash906 2009

[12] M PrzybyłaM Kordasz RMadonski P Herman and P SauerldquoActive Disturbance Rejection Control of a 2DOF manipulatorwith significant modeling uncertaintyrdquo Bulletin of the PolishAcademy of SciencesmdashTechnical Sciences vol 60 no 3 2012

[13] Z Chen Q Zheng and Z Gao ldquoActive disturbance rejectioncontrol of chemical processesrdquo in Proceedings of the 16th IEEEInternational Conference on Control Applications CCA 2007Part of IEEE Multi-conference on Systems and Control pp 855ndash861 Singapore October 2007

[14] X Wang An Active Disturbance Rejection Control Solutionfor Electro-hydraulic Servo Systems Cleveland State University2012

[15] D Yoo S S-T Yau and Z Q Gao ldquoOptimal fast trackingobserver bandwidth of the linear extended state observerrdquo Inter-national Journal of Control vol 80 no 1 pp 102ndash111 2007

[16] X-X Yang and Y Huang ldquoCapabilities of extended stateobserver for estimating uncertaintiesrdquo in Proceedings of theAmerican Control Conference (ACC rsquo09) pp 3700ndash3705 IEEESt Louis Mo USA June 2009

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

4 Journal of Engineering

Table 2 Comparison of the performance index measures

Performance measure Method Improvement []SMC [10] ADRC

Settling time 1131 582 48Rise time (s) 521 378 27IAE 1145 6786 41ISE 2548 1268 50ITAE 2690 1190 56

time (s)0 50 100 150

referenceresponse

0

1

2

3

4

5

6

7

8

liqui

d-le

vel (

mm

)

Figure 3 Response of the system for the 6 cm desired level

120582 (s) = 1199043 + 51199042 + 9119904 + 30 (21)

119868119860119864 = int1199050|119890| 119889119905 (22)

119868119878119864 = int11990501198902119889119905 (23)

119868119879119860119864 = int1199050119905 |119890| 119889119905 (24)

4 Conclusions

Journal of Engineering 5

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

minus5

0

5

1=Ｒ1-Ｒ1Ｂ

minus20

0

20

2=Ｒ2-Ｒ2Ｂ

minus500

0

500

3=Ｒ3-Ｒ3Ｂ

Figure 4 Performance of the ESO in estimating the system states

Time (s)0 50 100 150

minus10

0

10

20

30

40

50

60

cont

rol s

igna

l u(t)

Figure 5 Control signal of the ADRC for the 6 cm desired level

referenceresponse

200 250150 35050 300100 4504000 500Time (s)

0

2

4

6

8

10

12

14

Liqu

id-le

vel (

mm

)

Figure 6 Set point tracking performance of the system

6 Journal of Engineering

referenceresponse

50 100 150 200 250 300 350 4000Time (s)

0

1

2

3

4

5

6

7

8

9

10Li

quid

-leve

l (m

m)

Figure 7 Closed-loop response of the system against the inputdisturbance

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

References

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Journal of Engineering 5

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

05 1 15 2 25 30

Time (s)

minus5

0

5

1=Ｒ1-Ｒ1Ｂ

minus20

0

20

2=Ｒ2-Ｒ2Ｂ

minus500

0

500

3=Ｒ3-Ｒ3Ｂ

Figure 4 Performance of the ESO in estimating the system states

Time (s)0 50 100 150

minus10

0

10

20

30

40

50

60

cont

rol s

igna

l u(t)

Figure 5 Control signal of the ADRC for the 6 cm desired level

referenceresponse

200 250150 35050 300100 4504000 500Time (s)

0

2

4

6

8

10

12

14

Liqu

id-le

vel (

mm

)

Figure 6 Set point tracking performance of the system

6 Journal of Engineering

referenceresponse

50 100 150 200 250 300 350 4000Time (s)

0

1

2

3

4

5

6

7

8

9

10Li

quid

-leve

l (m

m)

Figure 7 Closed-loop response of the system against the inputdisturbance

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

References

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

6 Journal of Engineering

referenceresponse

50 100 150 200 250 300 350 4000Time (s)

0

1

2

3

4

5

6

7

8

9

10Li

quid

-leve

l (m

m)

Figure 7 Closed-loop response of the system against the inputdisturbance

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

References

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

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