research article neural network based active disturbance...

Research ArticleNeural Network Based Active Disturbance RejectionControl of a Novel Electrohydraulic Servo System forSimultaneously Balancing and Positioning byIsoactuation Configuration

Qiang Gao1 Yuanlong Hou1 Kang Li1 Zhan Sun2 Chao Wang1 and Runmin Hou1

1School of Mechanical Engineering Nanjing University of Science and Technology Nanjing 210014 China2Institute of North Automatic Control Technology Taiyuan 030006 China

Correspondence should be addressed to Qiang Gao gaoq0916sinacom

Received 12 June 2015 Revised 1 September 2015 Accepted 2 September 2015

Academic Editor Mario Terzo

Copyright copy 2016 Qiang Gao et al 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

To satisfy the lightweight requirements of large pipe weapons a novel electrohydraulic servo (EHS) system where the hydrauliccylinder possesses three cavities is developed and investigated in the present study In the EHS system the balancing cavity of theEHS is especially designed for active compensation for the unbalancing force of the system whereas the two driving cavities areemployed for positioning and disturbance rejection of the large pipe Aiming at simultaneously balancing and positioning of theEHS system a novel neural network based active disturbance rejection control (NNADRC) strategy is developed In the NNADRCthe radial basis function (RBF) neural network is employed for online updating of parameters of the extended state observer (ESO)Thereby the nonlinear behavior and external disturbance of the system can be accurately estimated and compensated in real timeThe efficiency and superiority of the system are critically investigated by conducting numerical simulations showing that muchhigher steady accuracy as well as system robustness is achieved when comparing with conventional ADRC control system Itindicates that the NNADRC is a very promising technique for achieving fast stable smooth and accurate control of the novelEHS system

1 Introduction

Large pipe weapons arms of heavy machineries industrialrobotics and space-borne manipulators possess a sort oflong and heavy linkages which have high ratio of lengthto diameter and time-varying unbalancing torque inducedby the misalignment between its gravity center and thecorresponding trunnion The unbalancing torque servingas a sort of strong disturbance highly deteriorates controlperformances of the system [1 2]

Currently there are two kinds of system balancingstrategies namely the external force based system balanc-ing and the internal driving based system balancing Theexternal methods mainly depended on adopting properbalance weights or adopting balance machinery and the

static unbalancing components can be well compensatedPractically the unbalancing part including partial static anddynamic unbalancing components was treated as externaldisturbance and further compensated during active controlof the system Regarding the large diameter of the pipethe unbalancing components were heavy highly challengingthe control system for positioning [3 4] Moreover theextra machinery for balancing strongly increased weightsand costs of mechanical systems being difficult to meet thelightweight requirements With the internal driving basedsystem balancing methods the most directly way was touse the driving forces for system balancing [5 6] Generallythe currently adopted method based on direct driving wasessentially a feedforward inverse compensation strategy andit cannot achieve real-time balancing It was obvious that not

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 4921095 9 pageshttpdxdoiorg10115520164921095

2 Shock and Vibration

enough driving forces can be provided by the motor servosystem in terms of the heavy pipes Another kind of adaptivebalancing method combining the hydraulic accumulator andbalancingmachinery was recently developed in [7] Howeveronly partial unbalancing forces can be compensated duringthe process Also this method cannot be applied for activerejection of the unbalancing forces during the positioning

As discussed above active compensation for the unbal-ancing components in pipe positioning is still an outstandingissue In the present study a novel electrohydraulic servo(EHS) system where the hydraulic cylinder possesses threecavities is developed In the EHS system the balancing cavityof the EHS is especially designed for active compensationfor the unbalancing force of the system whereas the drivingcavity is employed for real-time positioning and disturbancerejection of the large pipe Bymeans of the hydraulic cylinderwith three cavities simultaneous control and disturbancerejection can be achieved by using only one driving sourcenamely isoactuation greatly reducing weights of weaponswith heavy pipes However there are certain extremelycomplicated segments with strong nonlinearities in guncontrol systems including time-varying parameters inducedby varying working conditions random external appliedloads and complex friction forces between the cannonand trunnion All these nonlinear behaviors add difficultiesin achieving high control performances (both static anddynamic) blocking improvements of working performancesof the gun control system In terms of nonlinear controlconsiderable work has been done based on certain types ofadaptive control strategies such as fuzzy control adaptivesliding mode robust control and adaptive equivalent dis-turbance compensation control [4ndash7] All these nonlinearcontrol methods can significantly improve the uncertaintiesof the control system in terms of its tolerance and robustnessbut only at the expense of the positioning accuracy andresponse speed

The recently developed active disturbance rejection con-trol (ADRC) is an efficient nonlinear digital control strategywhich regards the unmodeled part and external disturbanceas overall disturbance of the controlled system [8ndash10] TheADRC mainly consists of a tracking differentiator (TD) anextended state observer (ESO) and a nonlinear state errorfeedback (NLSEF) control law In this ADRC approach theprocesses with higher orders uncertainties and unmodeleddynamics are viewed as lower-ordered systems with generaldisturbances meanwhile the general disturbances are esti-mated by ESO and are actively compensated [11ndash20] Thereare a large number of adjustable parameters in the ADRCand choosing proper parameters can contribute to excellentperformances of the control system To optimally get the sys-tem parameters the global optimization strategy based off-line tune methods [11 12] and the artificial intelligent basedonline tune methods [13ndash16] were developed Generally theoff-line tune methods are highly dependent on identifiedphysical model of the controlled system and the obtainedparameters cannot adapt to variable working conditionsTheonline updating method can adjust control parameters to getoptimal control performance in real time In [13] the fuzzycontrol scheme was introduced in the ADRC to estimate the

states and accordingly update the compensation factors ofthe ESO In [14] the diagonal recurrent neural network wasintroduced to realize online tuning of the NLSEF In termsof parameters of the NLSEF the compensation gains in ESOhighly affect estimation performance of the control systemespecially the estimation accuracy of overall disturbancesThereby the backpropagation neural network (BPNN) wasadopted in [15] to tune the gains in ESO However the BPNNsuffers easily trapping in local minimum and slow convergentspeed

Motivated by this the radial biases function (RBF) neuralnetwork is introduced in an improved ADRC to solve thepractical issue in control of the newly developed EHS systemwith three hydraulic cavities for simultaneous balancing andpositioning constructing the novel neural network basedactive disturbance rejection control (NNADRC) strategyFour adjustable compensation gains in the ESO are onlineupdated by adopting the RBF neural network

2 The Isoactuation ElectrohydraulicServo System

21 System Configuration and Working Principle Figure 1illustrates the schematic of the isoactuation EHS systemfor simultaneous balancing and positioning As shown inFigure 1 the EHS system mainly consists of a positioningcontroller a variable capacity pump a proportional servovalve a rotary transformer a RDC modular a balancingcontroller a constant displacement pump a proportionalpressure-reducing valve a pressure sensor and an actuationhydraulic cylinderThe actuation hydraulic cylinder has threecavities namely the upper lower and balancing cavity Acombination of the upper and lower cavities is adopted forsystem actuation

In practice the balancing controller compares and cal-culates the deviation between the measured and desiredpressure in the balancing cavity The control signal forthe proportional pressure-reducing valve is obtained basedon the deviation to accurately control the pressure in thebalancing cavity accordingly cancelling the load and weighttorque on the system As for the positioning the deviationbetween the measured and desired positions of the pipeis similarly calculated and accordingly a control signal isgenerated for the proportional servo valve to control the flowrate and direction of the hydraulic fluid for the upper andlower cavities Thereby position of the pipe can be accuratelycontrolled by adopting the feedback control strategy

22 Modelling the EHS System Assume the state variables are119909 = [1199091

1199092

1199093]

119879 with 1199091

= 120579 1199092

=

120579 and 1199093

=

120579 thestate space equation for the isoactuation EHS system can bedetermined by

1= 1199092

2= 1199093

3= 119891 (119909) + 119892119906

1+ 119889 (119905)

(1)

Shock and Vibration 3

Lower cavity

Proportionalrelief valve

Upper cavity

Balance cavity

Quantitative pump

Proportionalservo valve

Variable pump

Pressure sensor

Balance controller

Positioning controllerResolver RDC

module

120579

A

B

O

Figure 1 Schematic of the isoactuation EHS system

where

119891 (119909) = minus

120573119890119862119905119866

1198691198810

1199091minus

120573119890

1198691198810

(1198632

119890+ 119862119905119861119898

+

1198810

120573119890

119866)1199092

minus

120573119890

1198691198810

(119869119862119905+

1198810

120573119890

119861119898)1199093

119892 =

120573119890

1198691198810

1198631198901198701198861198701

119889 (119905) =

120573119890119862119905

1198691198810

119879119871+

1

119869

119879119871

|119889 (119905)| le Const

(2)

where 120579 denotes the actual position of the pipe 1199061denotes the

output signal of the positioning controller 120573119890is the effective

bulkmodulus of elasticity119862119905is the overall leakage coefficient

119866 represents the load elastic stiffness 119869 is the rotationalinertia 119881

0is the volume of a cavity 119863

119890is the equivalent

surface total displacement 119861119898

is the viscoelastic dampingcoefficient 119870

1represents the current-flow rate amplification

ratio 119870119886is the gain of the servo amplifier and 119879

119871represents

the external loads Practically the overall leakage coefficientthe viscoelastic damping coefficient the equivalent surfacetotal displacement and the external loads vary with respectto working conditions showing strong nonlinear behavior ofthe working system

23 Controlling Principle of the EHS System

231 Principle of Balancing Control In (1) the external load119879119871denotes a combination of unbalancing torque external

disturbance torque and launch impact torque To compen-sate for these external disturbances a novel active actuationconfiguration based on a hydraulic cylinder with three cavi-ties is proposed for simultaneous positioning and balancingThe basic principle can be summarized as follows the weighttorque varies with respect to the rotation position 120579 whichcan be expressed by 119879

119866(120579) With a specified arm of force 119897(120579)

and the acting area 119860 the required pressure 119875119866(119904) provided

by the balancing cavity can be accordingly determinedTo actively balance the weight torque the pressure in thebalancing cavity is controlled by means of the proportionalpressure-reducing valve Thereby the required torque forbalancing the weight torque can be achieved

Since the working frequency of the proportionalpressure-reducing valve is much higher than the naturalfrequency of its actuation it can be simply regarded as aproportional element Thus the relationship between thecontrol signal 119880

2(119904) and output flow rate 119876

1(119904) of the valve

can be expressed by

1198761(119904) = 119870

211987011990211198802(119904) (3)

where 1198702is the voltage-displacement coefficient and 119870

1199021is

the flow rate gain for the valve


The theoretical pressure in the balancing cavity can bedetermined by

119875119901(119904) =

1

(1198811120573119890) 119904 + 119862

1198711

[1198761199041

(119904) minus 1198761198711

(119904) minus 1198761(119904)]

=

1

(1198811120573119890) 119904 + 119862

1198711

[1198761199041

(119904) minus

1

119860119897 (120579)

119904120579 (119904)

minus 119870211987011990211198802(119904)]

(4)

where 1198761199041(119904) and 119876

1198711(119904) denote the flow rate of the constant

displacement pump and the flow rate into the balancingcavity 119881

1is the volume between the output end of the pump

the input end of the valve and the balancing cavity and 1198621198711

is the leakage coefficient of the balancing cavity

232 Principle of Positioning Control The feedback controlscheme is employed for the positioning Since there arecertain nonlinear components and unbalanced torque in thesystem the ESO is further employed for the estimation ofthese unbalanced components the control signal is thenobtained by following the nonlinear state error feedback con-trol law with consideration of the disturbance compensation

3 Design of the Controllers

31 Balancing Controller The typical PID controller isadopted for the balancing of the gun control system Assumethat the desired and practical pressures in the balancing cavityare 119875119866(119905) and 119875

119901119888(119905) respectively the command signal for the

balancing system can be determined by

1199062(119905) = 119880

119898minus [119896119901119890119901(119905) + 119896

119894int

119905

0

119890119901(119905) 119889119905 + 119896

119889

119889119890119901(119905)

119889119905

]

119890119901(119905) = 119875

119866(119905) minus 119875

119901119888(119905)

(5)

where 119880119898

denotes the maximum output voltage for thecontrol and 119896

119901 119896119894 and 119896

119889are proportional integral and

differential coefficients of the PID controller

32 The Improved ADRC Controller To better track thetrajectory of the control system an improved ADRC con-trol strategy is developed To suppress the inherent chatterphenomenon a novel nonlinear function nfal(sdot) featuringsmooth switching behavior is employed to replace the con-ventionally adopted nonlinear function fal(sdot) which is a keycomponent of the ADRC controller Besides the radial biasesfunction (RBF) neural network is adopted to online updatethe compensation gains of the ESO in the ADRC

321 Configuration of the Control System Schematic of theADRC is illustrated in Figure 2 it mainly consists of a TD anESO and a NLSEF control law In this ADRC approach theprocesses with higher orders uncertainties and unmodeleddynamics are viewed as lower-ordered systems with generaldisturbances which are further estimated by ESO and areactively compensated

With the TD process it is employed for transient pro-cess and command signal generation Fast tracking withoutovershoots as well as suspension of rapid fluctuations of thecontrol signal when the presetting parameters are suddenlychanged can be achieved by employing the TDThe definitionof the TD process can be expressed by

V1(119896 + 1) = V

1(119896) + ℎV

2(119896)

V2(119896 + 1) = V

2(119896) + ℎV

3(119896)

V3(119896 + 1) = V

3(119896) + ℎ119891119904119905

119891119904119905 = minus119903 (119903 (119903 (V1minus 120579119889) + 3V

2) + 3V

3)

(6)

where 120579119889(119905) denotes the desired trajectory of the control

system 119903 and ℎ determine the speed and the step lengthrespectively and V

1(119905) V2(119905) and V

3(119905) track 120579

119889(119905)

120579119889(119905) and

120579119889(119905) respectivelyWith the ESO the change rate and total disturbance of

the error can be real-time estimated using ESO from thecontrol variables and the position errors of the input signalThereby dynamic compensation for the disturbances canbe conducted properly using estimated total disturbancesBy introducing the novel nonlinear function nfal(sdot) theimproved third-order discrete ESO can be configured asfollows

119890 (119896) = 1199111(119896) minus 120579 (119896)

1199111(119896 + 1) = 119911

1(119896) + ℎ [119911

2(119896) minus 120573

01119890 (119896)]

1199112(119896 + 1)

= 1199112(119896) + ℎ [119911

3(119896) minus 120573

02nfal (119890 (119896) 119888

1 1198871 1205741)]

1199113(119896 + 1)

= 1199113(119896)

+ ℎ [1199114(119896) minus 120573

03nfal (119890 (119896) 119888

1 1198872 1205741) + 1198870119906 (119896)]

1199114(119896 + 1) = 119911

4(119896) minus ℎ120573

04nfal (119890 (119896) 119888

1 1198873 1205741)

(7)

where 12057301 12057302 12057303 and 120573

04are adjustable compensation

gains and 1198861 1198862 1198863 1205751 and 119887

0are design parameters 119911

1(119905)

1199112(119905) 1199113(119905) and 119911

4(119905) are the estimate outputs of the ESO

process respectively tacking 120579(119905) 120579(119905)

120579(119905) and the estimatedtotal disturbance

The nonlinear function nfal(sdot) can be expressed by

nfal (119890 (119896) 119888 119887 120574) = 119887 arc tan119888 (119890 (119896) minus 120574) minus 120583

0

1205872 minus 1205830

1205830= arc tan (119888 (0 minus 120574))

(8)

where V determines the shape of the operator 119887 determinesthe range of the operator and 120574 determines the center of theoperator

Essentially the NLSEF controller is a nonlinear PDcontroller nonlinear combination of the error componentsderiving from the outputs of the TD and the state estimation


120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0

TL

120579(t)

z1(t)

z2(t)

z3(t)

z4(t)

ESO

Plant

minus

minus

minus

minus

⨂

⨂⨂

⨂

Figure 2 Schematics of ADRC

120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0 TL

120579(t)

z1(t)z2(t)

z3(t)

z4(t)

ESO RBFNN

Plant

12057301120573021205730312057304

minus

minus

minus

minus⨂

⨂

⨂

⨂

Figure 3 The diagram of ADRC based on RBF neural network

from the ESO is properly designed to construct the commandsignal 119906

119900(119905) The third-order discrete governing law of the

NLSEF can be determined by

1198901(119896 + 1) = V

1(119896 + 1) minus 119911

1(119896 + 1)

1198902(119896 + 1) = V

2(119896 + 1) minus 119911

2(119896 + 1)

1198903(119896 + 1) = V

3(119896 + 1) minus 119911

3(119896 + 1)

1199060(119896 + 1) = 120573

1nfal (119890

1(119896 + 1) 119888

0 1198874 1205740)

+ 1205732nfal (119890

2(119896 + 1) 119888

0 1198875 1205740)

+ 1205733nfal (119890

3(119896 + 1) 119888

0 1198876 1205740)

(9)

where 1205731 1205732 and 120573

3represent the control gains respectively

and119886411988651198866 and120575

0are the parameters for the design process

By combining the estimated disturbance through theESO the actual control variables applied to the actuator ofthe control system can be obtained as follows

1199061(119896 + 1) =

[1199060(119896 + 1) minus 119911

4(119896 + 1)]

1198870

(10)

where b0is the compensation factor

322 Adaptive Updating of the ESO Overall there are fourimportant parameters in the ESO that govern the controlaccuracy and system robustness of the whole control system

namely 12057301 12057302 12057303 and 120573

04 To achieve optimal control

during the whole working process an online updating of thefour parameters based on RBFNN is developed in the presentstudy The configuration of the adaptive control system isillustrated in Figure 3 As shown in Figure 3 a three-layerRBFNN is adopted where 119890

1(119905) 1198902(119905) 1198903(119905) and 120579(119905) are

serving as the input nodes while 12057301 12057302 12057303 and 120573

04are

serving as the output nodes The number of node in hiddenlayer is chosen as 6

The goal of adaptive adjustment of the control parametersis to seek for a control signal that canminimize the differencebetween the process output and the desired output The per-formance criterion employed in this paper for the parameterupdating is defined by

119864 (119896) =

1

2

[120579119889(119896) minus 120579 (119896)]

2

=

1

2

1198902(119896) (11)

The output of each hidden node in the RBFNN can beobtained as follows

120593119894(119896) = exp(minus

1003817100381710038171003817119883 minus 119862

119894

1003817100381710038171003817

2

21198872

119894

) (12)

where 119883 = [1198901 1198902 1198903 120579]119879 denotes the input vector of the

NN 119862119894= [1198881198941 1198881198942 1198881198943 1198881198944]119879 denotes the center vector of the 119894th

hidden node and 119887119894is the width of the radial basis function

of this node


0 1 2 3 4 5 6 7 8 9 10minus01

minus005

0

005

01

015

02

025

03

035

04

Time (s)

Posit

ion

trac

king

erro

r (ra

d)

NNADRC tracking signal errorADRC tracking signal error

(a)

minus6

minus4

minus2

0

2

4

6

Posit

ion

trac

king

cont

rol v

olta

ge (V

)

NNADRC control voltageADRC control voltage

0 1 2 3 4 5 6 7 8 9 10Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

100

120

Time (s)

Squa

re w

ave d

istur

banc

ez 4

ADRC z4

NNADRC z4

Disturbance

(c)

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus200

minus100

0

100

200

300

400

500

600

700

800

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)

Figure 4 Step response of the control system (a) errors of position signal with square wave disturbance (b) position tracking control voltage(c) square wave disturbance and the estimated z

4 and (d) torque errors of position tracking

Thus output of the RBFNN can be determined by

1205730119897(119896) =

6

sum

119894=1

120596119897119894120593119894(119896) (13)

where 120596119897119894represents the connection weight between the 119894th

(119894 = 1 2 3 4) hidden node and the 119897th output node

The update law for the weights of the RBFNN yields thefollowing

120596119897119894(119896) = 120596

119897119894(119896 minus 1) + Δ120596

119897119894(119896) + 120588Δ120596

119897119894(119896 minus 1)

Δ120596119897119894(119896) = minus120578

120597119864 (119896)

120597120596119897119894(119896)

= minus120578

120597119864 (119896)

120597120579 (119896)

120597120579 (119896)

1205971205730119897(119896)

1205971205730119897(119896)

120597120596119897119894(119896)

= 120578119890 (119896)

120597120579 (119896)

1205971205730119897(119896)

120593119894(119896)

(14)


where 120588 and 120578 are the momentum and learning factors ofthis RBFNN respectively It is to be noticed that the term120597120579(119896)120597120573

0119897(119896) is unknown Thereby it will be replaced by

sgn119897(119896) in practice

4 Simulation Results and Discussion

To demonstrate the effectiveness and superiority of theimproved ADRC controller for the EHS system with isoactu-ation configuration numerical simulation on step responsesand constant velocity tracking of the system was conducted

In the simulation parameters for the electrohydraulicservo systemwere 119869 = 155times10

5 Kg sdotm2119862119905= 15times10

minus13(m3 sdot

sminus1)Pa 119861119898

= 135 times 105N sdotm(rad sdot sminus1) 119881

0= 0011781m3

120573119890= 700MPa and 119879

119885= 128 times 10

5Nm With the balancingcontroller three coefficients for the PID controller were set as119896119901= 4times10

minus5 119896119894= 23times10

minus6 and 119896119889= 312times10

minus7 Parametersfor the TD were ℎ = 001 and 119903 = 012 and those for theNLSEF are set as 120573

1= 095 120573

2= 106 120573

3= 173 119888

0= 025

1198874

= 25 1198875

= 20 1198876

= 15 and 1205740

= 001 As for the ESOthe design parameters were set as 119888

1= 05 119887

1= 25 119887

2= 20

1198873

= 15 and 1205741

= 001 while the adjustable compensationgains were online updated and the initial values were set as12057301

= 150 12057302

= 40 12057303

= 40 and 12057304

= 500

41 Step Response Positioning errors of the EHS system instep response obtained by both ADRC and the improvedNNADRC controllers are illustrated in Figure 4(a) No over-shoots are observed for the two control systems and theresponse time for the two control systems at which thepositioning errors are within plusmn00005236 rad (asymp plusmn05mil) isabout 2545 s The steady state error is about 27 times 10minus4 radTo further evaluate system robustness external disturbancesfeaturing square wave with an amplitude of 100KN sdot m wereadded in the control system from 119905 = 5 s to 119905 = 6 s From thepositioning errors in Figure 4(a) the maximum deviationsfor ADRC and NNADRC induced by the disturbances are0055 rad and 0005 rad respectively This indicates that theNNADRCpossessesmuchhigher robustness than theADRC

The command signal and the estimated total disturbance(z4) during the step responses are further illustrated in

Figures 4(b) and 4(c) As shown in Figure 4(b) much smoothestimation is obtained by the NNADRC controller showingstrong suspension capacity of the chatter phenomenon byadopting the improved nonlinear function nfal(sdot) In addi-tionmuchmore accurate estimation of the total disturbancesis achieved in the NNADRC system which may attribute tothe real-time updating capacity of the RBFNNTheweightingand balancing torques in the NNADRC system are illustratedin Figure 4(d) a deviation of the torque is about 50Nm Itsuggests that the output torque of the balancing cavity finelyagrees with the weighting torque by adopting the balancingcontroller being capable for active balance of the systemweighting torque

To further investigate system nonlinearity induced per-formance variation of the control system parameter per-turbation was adopted to mimic the system state deviationscaused by the nonlinear effects Step responses of the ADRC

0 1 2 3 4 5 6 7 8 9 100

005

01

015

02

025

03

035

04

045

05

Time (s)

Posit

ion

signa

l (ra

d)

Target signalNNARDC position signalARDC position signal

Figure 5 ARDC and NNARDC positioning results with 30change of 119869

andNNADRC systems are illustrated in Figure 5 with respectto a variation of the rotational inertia J up to 30 As shownin Figure 5 the overshoot of the ADRC system is about04147 rad (367) and the time for entering the positioningerror tolerance is about 565 s In contrast the overshoot ofthe NNADRC system is only about 01152 rad (106) whichis only about 288 of that of the ADRC system and theentering time is about 311 s The parameter perturbationsimulation indicates that the NNADRC system has highrobustness against parameter perturbation of the controlsystem

42 Constant Velocity Tracking Constant velocity trackingwith V = 08727 radsdotsminus1 was simulated to further investigateperformances of the control system A harmonic disturbancewith amplitude and frequency of 119860 = 20KN sdot m and119891 = 05Hz is added to simulate external disturbanceThe resulting tracking errors obtained by both ADRC andthe NNADRC are illustrated in Figure 6(a) As shown inFigure 6(a) the maximum tracking error in the steady statefor the NNADRC is about 00011 rad which is only about19 that obtained by ADRC Similarly the command voltagesand estimated total disturbances for the two control systemsare respectively illustrated in Figures 6(b) and 6(c) Strongfluctuations of the command and estimated signals especiallyat the initial stage are observed in the ADRC control systemwhile those of the NNADRC are much smoother Thisindicates that the NNADRC well outperform the ADRCcontrol strategy The weighting and balancing torques inthe NNADRC system are illustrated in Figure 6(d) Slightharmonic fluctuation of the torques is observed and themaximumunbalanced torque is only about 90NmThe resultsuggests that the developed EHS system is effective and very


minus4

minus2

0

2

4

6

8

10

12

14

Spee

d tr

acki

ng er

ror (

rad)

0 1 2 3 4 5 6 7 8 9 10Time (s)


times10minus3

(a)

minus4

minus2

0

2

4

6

8

Spee

d tr

acki

ng co

ntro

l vol

tage

(V)

0 1 2 3 4 5 6 7 8 9 10Time (s)


(b)

minus30

minus20

minus10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10Time (s)

ADRC z4

NNADRC z4

Disturbance

Sinu

soid

al d

istur

banc

ez 4

(c)

0 1 2 3 4 5 6 7 8 9 10

minus100

0

100

200

300

400

500

600

700

Time (s)

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)

Figure 6 Response of the control system with constant speed tracking (a) Errors of speed signal with sinusoidal disturbance (b) speedtracking control voltage (c) sinusoidal disturbance and the estimated z

4 and (d) torque errors of speed tracking

promising for simultaneous balancing and positioning oflarge pipes

5 Conclusions

To satisfy the lightweight requirements of large pipe weaponsa novel electrohydraulic servo (EHS) system where thehydraulic cylinder possesses three cavities is developed andinvestigated in the present study In the EHS system thebalancing cavity of the EHS is especially designed for active

compensation for the unbalancing force of the systemwhereas the driving cavity is employed for the positioningand disturbance rejection of the large pipe By adopting theisoactuation configuration a more compact and lightweightgun control system can be achieved

Aiming at simultaneous balancing and positioning of theEHS system a novel neural network based active disturbancerejection control (NNADRC) strategy is developed In theNNADRC the radial basis function (RBF) neural networkis employed for online updating parameters of the extendedstate observer (ESO) By online updating of the estimator


gains of the ESO the nonlinear behavior and externaldisturbance of the system can be accurately estimated andcompensated in real time

The efficiency and superiority of the control system arecritically investigated by conducting numerical simulationsWhen suffering square disturbance in step response thepositioning error obtained by NNADRC is only 10 of thatobtained by ADRC and the overshoot and entering time oftheNNADRCare about 288 and 55of them inADRC sys-temwhen encountering parameter perturbation respectivelyMeanwhile when suffering harmonic disturbance in constantspeed tracking the steady accuracy is enhanced by 9 timesIn addition the inherent chatter phenomenon in ADRC canalso be well suspended by adopting the improved nonlinearfunction scheme in the NNADRC

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The work described in this paper was supported by theNational Natural Science Foundation of China (51305205)

References

[1] D J Purdy ldquoComparison of balance and out of balance mainbattle tank armamentsrdquo Shock and Vibration vol 8 no 3-4 pp167ndash174 2001

[2] V Marcopeli M Ng and C Wells ldquoRobust control design forthe elevation axis stabilization of the M256E1 long gunrdquo Reportof Defense Technical Information Center of USA DefenseTechnical Information Center Fort Belvoir Va USA 2001

[3] Q Gao Z Sun G L Yang R Hou L Wang and Y HouldquoA novel active disturbance rejection-based control strategyfor a gun control systemrdquo Journal of Mechanical Science andTechnology vol 26 no 12 pp 4141ndash4148 2012

[4] C Ma and X Zhang ldquoSimulation of contamination preventionfor optical window in laser ignition systems of large-calibergunsrdquo Journal of Applied Mechanics vol 78 no 5 Article ID051014 7 pages 2011

[5] O Gumusay Intelligent Stabilization Control of Turret Subsys-tems under Disturbances form Unstructured Terrain MiddleEast Technical University 2006

[6] T Karayumak Modeling and stabilization control of a mainbattle tank [PhD thesis] Middle East Technical UniversityAnkara Turkey 2011

[7] C W Han The Robust Control and Application of ArtilleryPosition Servo System Xirsquoan Jiaotong University 2002

[8] J Q Han ldquoAuto disturbances rejection controller and itrsquosapplicationsrdquoControl andDecision vol 13 no 1 pp 19ndash23 1998

[9] J Q Han ldquoFrom PID technique to active disturbances rejectioncontrol techniquerdquo Control and Decision vol 9 no 3 pp 13ndash182002

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

[11] W Y Chen F L Chu and S Z Yan ldquoStepwise optimal design ofactive disturbances rejection vibration controller for intelligenttruss structure based on adaptive genetic algorithmrdquo Journal ofMechanical Engineering vol 46 no 7 pp 74ndash81 2010

[12] H-P Ren and F Zhu ldquoOptimal design of speed adaptivedisturbance rejection controller for brushless DC motor basedon immune clonal selection algorithmsrdquo Electric Machines andControl vol 14 no 9 pp 69ndash74 2010

[13] X-Q Liu L Tang and L-T Zhu ldquoThree-motor synchronouscontrol system based on fuzzy active disturbances rejectioncontrolrdquo Electric Machines and Control vol 17 no 4 pp 104ndash109 2013

[14] G-L Qiao C-N Tong and Y-K Sun ldquoStudy on mould leveland casting speed coordination control based on ADRC withDRNN optimizationrdquoActa Automatica Sinica vol 33 no 6 pp641ndash648 2007

[15] X-HQi J Li and S-T Han ldquoAdaptive active disturbance rejec-tion control and its simulation based on BP neural networkrdquoActa Armamentarii vol 34 no 6 pp 776ndash782 2013

[16] K C Li ldquoAdaptive auto disturbance rejection control forservo systems based on the RBF neural networkrdquo ElectricalAutomation vol 32 no 2 pp 23ndash25 2010

[17] Z Y LiuW LiuM Y Fu et al ldquoTrace-keeping of an air cushionvehicle based on an auto disturbance rejection controller witha recurrent networks modelrdquo Journal of Harbin EngineeringUniversity vol 33 no 3 pp 283ndash288 2012

[18] X T Li B Zhang J H SunDMao X Bai andH Shen ldquoADRCbased on disturbance frequency adaptive of aerial photoelectri-cal stabilized platformrdquo Infrared and Laser Engineering vol 43no 5 pp 1574ndash1581 2014

[19] Y-Y Ma S-J Tang J Guo and J Shi ldquoHigh angle of attackcontrol system design based on ADRC and fuzzy logicrdquo SystemsEngineering and Electronics vol 35 no 8 pp 1711ndash1716 2013

[20] Y A Feng X P Zhu and Z Zhou ldquoAdaptation cascade activedisturbance rejection controller for flexible flying wing UAVattitude controlrdquo Infrared and Laser Engineering vol 43 no 5pp 1594ndash1599 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



enough driving forces can be provided by the motor servosystem in terms of the heavy pipes Another kind of adaptivebalancing method combining the hydraulic accumulator andbalancingmachinery was recently developed in [7] Howeveronly partial unbalancing forces can be compensated duringthe process Also this method cannot be applied for activerejection of the unbalancing forces during the positioning

As discussed above active compensation for the unbal-ancing components in pipe positioning is still an outstandingissue In the present study a novel electrohydraulic servo(EHS) system where the hydraulic cylinder possesses threecavities is developed In the EHS system the balancing cavityof the EHS is especially designed for active compensationfor the unbalancing force of the system whereas the drivingcavity is employed for real-time positioning and disturbancerejection of the large pipe Bymeans of the hydraulic cylinderwith three cavities simultaneous control and disturbancerejection can be achieved by using only one driving sourcenamely isoactuation greatly reducing weights of weaponswith heavy pipes However there are certain extremelycomplicated segments with strong nonlinearities in guncontrol systems including time-varying parameters inducedby varying working conditions random external appliedloads and complex friction forces between the cannonand trunnion All these nonlinear behaviors add difficultiesin achieving high control performances (both static anddynamic) blocking improvements of working performancesof the gun control system In terms of nonlinear controlconsiderable work has been done based on certain types ofadaptive control strategies such as fuzzy control adaptivesliding mode robust control and adaptive equivalent dis-turbance compensation control [4ndash7] All these nonlinearcontrol methods can significantly improve the uncertaintiesof the control system in terms of its tolerance and robustnessbut only at the expense of the positioning accuracy andresponse speed

The recently developed active disturbance rejection con-trol (ADRC) is an efficient nonlinear digital control strategywhich regards the unmodeled part and external disturbanceas overall disturbance of the controlled system [8ndash10] TheADRC mainly consists of a tracking differentiator (TD) anextended state observer (ESO) and a nonlinear state errorfeedback (NLSEF) control law In this ADRC approach theprocesses with higher orders uncertainties and unmodeleddynamics are viewed as lower-ordered systems with generaldisturbances meanwhile the general disturbances are esti-mated by ESO and are actively compensated [11ndash20] Thereare a large number of adjustable parameters in the ADRCand choosing proper parameters can contribute to excellentperformances of the control system To optimally get the sys-tem parameters the global optimization strategy based off-line tune methods [11 12] and the artificial intelligent basedonline tune methods [13ndash16] were developed Generally theoff-line tune methods are highly dependent on identifiedphysical model of the controlled system and the obtainedparameters cannot adapt to variable working conditionsTheonline updating method can adjust control parameters to getoptimal control performance in real time In [13] the fuzzycontrol scheme was introduced in the ADRC to estimate the

states and accordingly update the compensation factors ofthe ESO In [14] the diagonal recurrent neural network wasintroduced to realize online tuning of the NLSEF In termsof parameters of the NLSEF the compensation gains in ESOhighly affect estimation performance of the control systemespecially the estimation accuracy of overall disturbancesThereby the backpropagation neural network (BPNN) wasadopted in [15] to tune the gains in ESO However the BPNNsuffers easily trapping in local minimum and slow convergentspeed

Motivated by this the radial biases function (RBF) neuralnetwork is introduced in an improved ADRC to solve thepractical issue in control of the newly developed EHS systemwith three hydraulic cavities for simultaneous balancing andpositioning constructing the novel neural network basedactive disturbance rejection control (NNADRC) strategyFour adjustable compensation gains in the ESO are onlineupdated by adopting the RBF neural network

2 The Isoactuation ElectrohydraulicServo System

21 System Configuration and Working Principle Figure 1illustrates the schematic of the isoactuation EHS systemfor simultaneous balancing and positioning As shown inFigure 1 the EHS system mainly consists of a positioningcontroller a variable capacity pump a proportional servovalve a rotary transformer a RDC modular a balancingcontroller a constant displacement pump a proportionalpressure-reducing valve a pressure sensor and an actuationhydraulic cylinderThe actuation hydraulic cylinder has threecavities namely the upper lower and balancing cavity Acombination of the upper and lower cavities is adopted forsystem actuation

In practice the balancing controller compares and cal-culates the deviation between the measured and desiredpressure in the balancing cavity The control signal forthe proportional pressure-reducing valve is obtained basedon the deviation to accurately control the pressure in thebalancing cavity accordingly cancelling the load and weighttorque on the system As for the positioning the deviationbetween the measured and desired positions of the pipeis similarly calculated and accordingly a control signal isgenerated for the proportional servo valve to control the flowrate and direction of the hydraulic fluid for the upper andlower cavities Thereby position of the pipe can be accuratelycontrolled by adopting the feedback control strategy

22 Modelling the EHS System Assume the state variables are119909 = [1199091

1199092

1199093]

119879 with 1199091

= 120579 1199092

=

120579 and 1199093

=

120579 thestate space equation for the isoactuation EHS system can bedetermined by

1= 1199092

2= 1199093

3= 119891 (119909) + 119892119906

1+ 119889 (119905)

(1)


Lower cavity


Upper cavity

Balance cavity

Quantitative pump


Variable pump

Pressure sensor

Balance controller


module

120579

A

B

O


where

119891 (119909) = minus

120573119890119862119905119866

1198691198810

1199091minus

120573119890

1198691198810

(1198632

119890+ 119862119905119861119898

+

1198810

120573119890

119866)1199092

minus

120573119890

1198691198810

(119869119862119905+

1198810

120573119890

119861119898)1199093

119892 =

120573119890

1198691198810

1198631198901198701198861198701

119889 (119905) =

120573119890119862119905

1198691198810

119879119871+

1

119869

119879119871

|119889 (119905)| le Const

(2)











119871represents











can be expressed by

1198761(119904) = 119870

211987011990211198802(119904) (3)


1199021is




119875119901(119904) =

1

(1198811120573119890) 119904 + 119862

1198711

[1198761199041

(119904) minus 1198761198711

(119904) minus 1198761(119904)]

=

1

(1198811120573119890) 119904 + 119862

1198711

[1198761199041

(119904) minus

1

119860119897 (120579)

119904120579 (119904)

minus 119870211987011990211198802(119904)]

(4)

where 1198761199041(119904) and 119876











1199062(119905) = 119880

119898minus [119896119901119890119901(119905) + 119896

119894int

119905

0

119890119901(119905) 119889119905 + 119896

119889

119889119890119901(119905)

119889119905

]

119890119901(119905) = 119875

119866(119905) minus 119875

119901119888(119905)

(5)

where 119880119898


119901 119896119894 and 119896






V1(119896 + 1) = V

1(119896) + ℎV

2(119896)

V2(119896 + 1) = V

2(119896) + ℎV

3(119896)

V3(119896 + 1) = V

3(119896) + ℎ119891119904119905

119891119904119905 = minus119903 (119903 (119903 (V1minus 120579119889) + 3V

2) + 3V

3)

(6)



1(119905) V2(119905) and V

3(119905) track 120579

119889(119905)

120579119889(119905) and



119890 (119896) = 1199111(119896) minus 120579 (119896)

1199111(119896 + 1) = 119911

1(119896) + ℎ [119911

2(119896) minus 120573

01119890 (119896)]

1199112(119896 + 1)

= 1199112(119896) + ℎ [119911

3(119896) minus 120573

02nfal (119890 (119896) 119888

1 1198871 1205741)]

1199113(119896 + 1)

= 1199113(119896)

+ ℎ [1199114(119896) minus 120573

03nfal (119890 (119896) 119888

1 1198872 1205741) + 1198870119906 (119896)]

1199114(119896 + 1) = 119911

4(119896) minus ℎ120573

04nfal (119890 (119896) 119888

1 1198873 1205741)

(7)

where 12057301 12057302 12057303 and 120573


gains and 1198861 1198862 1198863 1205751 and 119887


1(119905)

1199112(119905) 1199113(119905) and 119911






0

1205872 minus 1205830

1205830= arc tan (119888 (0 minus 120574))

(8)




120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0

TL

120579(t)

z1(t)

z2(t)

z3(t)

z4(t)

ESO

Plant

minus

minus

minus

minus

⨂

⨂⨂

⨂


120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0 TL

120579(t)

z1(t)z2(t)

z3(t)

z4(t)

ESO RBFNN

Plant

12057301120573021205730312057304

minus

minus

minus

minus⨂

⨂

⨂

⨂





1198901(119896 + 1) = V

1(119896 + 1) minus 119911

1(119896 + 1)

1198902(119896 + 1) = V

2(119896 + 1) minus 119911

2(119896 + 1)

1198903(119896 + 1) = V

3(119896 + 1) minus 119911

3(119896 + 1)

1199060(119896 + 1) = 120573

1nfal (119890

1(119896 + 1) 119888

0 1198874 1205740)

+ 1205732nfal (119890

2(119896 + 1) 119888

0 1198875 1205740)

+ 1205733nfal (119890

3(119896 + 1) 119888

0 1198876 1205740)

(9)

where 1205731 1205732 and 120573


and119886411988651198866 and120575



1199061(119896 + 1) =

[1199060(119896 + 1) minus 119911

4(119896 + 1)]

1198870

(10)



namely 12057301 12057302 12057303 and 120573



1(119905) 1198902(119905) 1198903(119905) and 120579(119905) are


04are



119864 (119896) =

1

2

[120579119889(119896) minus 120579 (119896)]

2

=

1

2

1198902(119896) (11)


120593119894(119896) = exp(minus

1003817100381710038171003817119883 minus 119862

119894

1003817100381710038171003817

2

21198872

119894

) (12)




of this node


0 1 2 3 4 5 6 7 8 9 10minus01

minus005

0

005

01

015

02

025

03

035

04

Time (s)

Posit

ion

trac

king

erro

r (ra

d)


(a)

minus6

minus4

minus2

0

2

4

6

Posit

ion

trac

king

cont

rol v

olta

ge (V

)


0 1 2 3 4 5 6 7 8 9 10Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

100

120

Time (s)

Squa

re w

ave d

istur

banc

ez 4

ADRC z4

NNADRC z4

Disturbance

(c)

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus200

minus100

0

100

200

300

400

500

600

700

800

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




1205730119897(119896) =

6

sum

119894=1

120596119897119894120593119894(119896) (13)




120596119897119894(119896) = 120596

119897119894(119896 minus 1) + Δ120596

119897119894(119896) + 120588Δ120596

119897119894(119896 minus 1)

Δ120596119897119894(119896) = minus120578

120597119864 (119896)

120597120596119897119894(119896)

= minus120578

120597119864 (119896)

120597120579 (119896)

120597120579 (119896)

1205971205730119897(119896)

1205971205730119897(119896)

120597120596119897119894(119896)

= 120578119890 (119896)

120597120579 (119896)

1205971205730119897(119896)

120593119894(119896)

(14)








5 Kg sdotm2119862119905= 15times10

minus13(m3 sdot

sminus1)Pa 119861119898


0= 0011781m3

120573119890= 700MPa and 119879

119885= 128 times 10


minus5 119896119894= 23times10

minus6 and 119896119889= 312times10


1= 095 120573

2= 106 120573

3= 173 119888

0= 025

1198874

= 25 1198875

= 20 1198876

= 15 and 1205740


1= 05 119887

1= 25 119887

2= 20

1198873

= 15 and 1205741


= 150 12057302

= 40 12057303

= 40 and 12057304

= 500





0 1 2 3 4 5 6 7 8 9 100

005

01

015

02

025

03

035

04

045

05

Time (s)

Posit

ion

signa

l (ra

d)






minus4

minus2

0

2

4

6

8

10

12

14

Spee

d tr

acki

ng er

ror (

rad)

0 1 2 3 4 5 6 7 8 9 10Time (s)


times10minus3

(a)

minus4

minus2

0

2

4

6

8

Spee

d tr

acki

ng co

ntro

l vol

tage

(V)

0 1 2 3 4 5 6 7 8 9 10Time (s)


(b)

minus30

minus20

minus10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10Time (s)

ADRC z4

NNADRC z4

Disturbance

Sinu

soid

al d

istur

banc

ez 4

(c)

0 1 2 3 4 5 6 7 8 9 10

minus100

0

100

200

300

400

500

600

700

Time (s)

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




5 Conclusions









Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Lower cavity


Upper cavity

Balance cavity

Quantitative pump


Variable pump

Pressure sensor

Balance controller


module

120579

A

B

O


where

119891 (119909) = minus

120573119890119862119905119866

1198691198810

1199091minus

120573119890

1198691198810

(1198632

119890+ 119862119905119861119898

+

1198810

120573119890

119866)1199092

minus

120573119890

1198691198810

(119869119862119905+

1198810

120573119890

119861119898)1199093

119892 =

120573119890

1198691198810

1198631198901198701198861198701

119889 (119905) =

120573119890119862119905

1198691198810

119879119871+

1

119869

119879119871

|119889 (119905)| le Const

(2)











119871represents











can be expressed by

1198761(119904) = 119870

211987011990211198802(119904) (3)


1199021is




119875119901(119904) =

1

(1198811120573119890) 119904 + 119862

1198711

[1198761199041

(119904) minus 1198761198711

(119904) minus 1198761(119904)]

=

1

(1198811120573119890) 119904 + 119862

1198711

[1198761199041

(119904) minus

1

119860119897 (120579)

119904120579 (119904)

minus 119870211987011990211198802(119904)]

(4)

where 1198761199041(119904) and 119876











1199062(119905) = 119880

119898minus [119896119901119890119901(119905) + 119896

119894int

119905

0

119890119901(119905) 119889119905 + 119896

119889

119889119890119901(119905)

119889119905

]

119890119901(119905) = 119875

119866(119905) minus 119875

119901119888(119905)

(5)

where 119880119898


119901 119896119894 and 119896






V1(119896 + 1) = V

1(119896) + ℎV

2(119896)

V2(119896 + 1) = V

2(119896) + ℎV

3(119896)

V3(119896 + 1) = V

3(119896) + ℎ119891119904119905

119891119904119905 = minus119903 (119903 (119903 (V1minus 120579119889) + 3V

2) + 3V

3)

(6)



1(119905) V2(119905) and V

3(119905) track 120579

119889(119905)

120579119889(119905) and



119890 (119896) = 1199111(119896) minus 120579 (119896)

1199111(119896 + 1) = 119911

1(119896) + ℎ [119911

2(119896) minus 120573

01119890 (119896)]

1199112(119896 + 1)

= 1199112(119896) + ℎ [119911

3(119896) minus 120573

02nfal (119890 (119896) 119888

1 1198871 1205741)]

1199113(119896 + 1)

= 1199113(119896)

+ ℎ [1199114(119896) minus 120573

03nfal (119890 (119896) 119888

1 1198872 1205741) + 1198870119906 (119896)]

1199114(119896 + 1) = 119911

4(119896) minus ℎ120573

04nfal (119890 (119896) 119888

1 1198873 1205741)

(7)

where 12057301 12057302 12057303 and 120573


gains and 1198861 1198862 1198863 1205751 and 119887


1(119905)

1199112(119905) 1199113(119905) and 119911






0

1205872 minus 1205830

1205830= arc tan (119888 (0 minus 120574))

(8)




120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0

TL

120579(t)

z1(t)

z2(t)

z3(t)

z4(t)

ESO

Plant

minus

minus

minus

minus

⨂

⨂⨂

⨂


120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0 TL

120579(t)

z1(t)z2(t)

z3(t)

z4(t)

ESO RBFNN

Plant

12057301120573021205730312057304

minus

minus

minus

minus⨂

⨂

⨂

⨂





1198901(119896 + 1) = V

1(119896 + 1) minus 119911

1(119896 + 1)

1198902(119896 + 1) = V

2(119896 + 1) minus 119911

2(119896 + 1)

1198903(119896 + 1) = V

3(119896 + 1) minus 119911

3(119896 + 1)

1199060(119896 + 1) = 120573

1nfal (119890

1(119896 + 1) 119888

0 1198874 1205740)

+ 1205732nfal (119890

2(119896 + 1) 119888

0 1198875 1205740)

+ 1205733nfal (119890

3(119896 + 1) 119888

0 1198876 1205740)

(9)

where 1205731 1205732 and 120573


and119886411988651198866 and120575



1199061(119896 + 1) =

[1199060(119896 + 1) minus 119911

4(119896 + 1)]

1198870

(10)



namely 12057301 12057302 12057303 and 120573



1(119905) 1198902(119905) 1198903(119905) and 120579(119905) are


04are



119864 (119896) =

1

2

[120579119889(119896) minus 120579 (119896)]

2

=

1

2

1198902(119896) (11)


120593119894(119896) = exp(minus

1003817100381710038171003817119883 minus 119862

119894

1003817100381710038171003817

2

21198872

119894

) (12)




of this node


0 1 2 3 4 5 6 7 8 9 10minus01

minus005

0

005

01

015

02

025

03

035

04

Time (s)

Posit

ion

trac

king

erro

r (ra

d)


(a)

minus6

minus4

minus2

0

2

4

6

Posit

ion

trac

king

cont

rol v

olta

ge (V

)


0 1 2 3 4 5 6 7 8 9 10Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

100

120

Time (s)

Squa

re w

ave d

istur

banc

ez 4

ADRC z4

NNADRC z4

Disturbance

(c)

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus200

minus100

0

100

200

300

400

500

600

700

800

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




1205730119897(119896) =

6

sum

119894=1

120596119897119894120593119894(119896) (13)




120596119897119894(119896) = 120596

119897119894(119896 minus 1) + Δ120596

119897119894(119896) + 120588Δ120596

119897119894(119896 minus 1)

Δ120596119897119894(119896) = minus120578

120597119864 (119896)

120597120596119897119894(119896)

= minus120578

120597119864 (119896)

120597120579 (119896)

120597120579 (119896)

1205971205730119897(119896)

1205971205730119897(119896)

120597120596119897119894(119896)

= 120578119890 (119896)

120597120579 (119896)

1205971205730119897(119896)

120593119894(119896)

(14)








5 Kg sdotm2119862119905= 15times10

minus13(m3 sdot

sminus1)Pa 119861119898


0= 0011781m3

120573119890= 700MPa and 119879

119885= 128 times 10


minus5 119896119894= 23times10

minus6 and 119896119889= 312times10


1= 095 120573

2= 106 120573

3= 173 119888

0= 025

1198874

= 25 1198875

= 20 1198876

= 15 and 1205740


1= 05 119887

1= 25 119887

2= 20

1198873

= 15 and 1205741


= 150 12057302

= 40 12057303

= 40 and 12057304

= 500





0 1 2 3 4 5 6 7 8 9 100

005

01

015

02

025

03

035

04

045

05

Time (s)

Posit

ion

signa

l (ra

d)






minus4

minus2

0

2

4

6

8

10

12

14

Spee

d tr

acki

ng er

ror (

rad)

0 1 2 3 4 5 6 7 8 9 10Time (s)


times10minus3

(a)

minus4

minus2

0

2

4

6

8

Spee

d tr

acki

ng co

ntro

l vol

tage

(V)

0 1 2 3 4 5 6 7 8 9 10Time (s)


(b)

minus30

minus20

minus10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10Time (s)

ADRC z4

NNADRC z4

Disturbance

Sinu

soid

al d

istur

banc

ez 4

(c)

0 1 2 3 4 5 6 7 8 9 10

minus100

0

100

200

300

400

500

600

700

Time (s)

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




5 Conclusions









Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119875119901(119904) =

1

(1198811120573119890) 119904 + 119862

1198711

[1198761199041

(119904) minus 1198761198711

(119904) minus 1198761(119904)]

=

1

(1198811120573119890) 119904 + 119862

1198711

[1198761199041

(119904) minus

1

119860119897 (120579)

119904120579 (119904)

minus 119870211987011990211198802(119904)]

(4)

where 1198761199041(119904) and 119876











1199062(119905) = 119880

119898minus [119896119901119890119901(119905) + 119896

119894int

119905

0

119890119901(119905) 119889119905 + 119896

119889

119889119890119901(119905)

119889119905

]

119890119901(119905) = 119875

119866(119905) minus 119875

119901119888(119905)

(5)

where 119880119898


119901 119896119894 and 119896






V1(119896 + 1) = V

1(119896) + ℎV

2(119896)

V2(119896 + 1) = V

2(119896) + ℎV

3(119896)

V3(119896 + 1) = V

3(119896) + ℎ119891119904119905

119891119904119905 = minus119903 (119903 (119903 (V1minus 120579119889) + 3V

2) + 3V

3)

(6)



1(119905) V2(119905) and V

3(119905) track 120579

119889(119905)

120579119889(119905) and



119890 (119896) = 1199111(119896) minus 120579 (119896)

1199111(119896 + 1) = 119911

1(119896) + ℎ [119911

2(119896) minus 120573

01119890 (119896)]

1199112(119896 + 1)

= 1199112(119896) + ℎ [119911

3(119896) minus 120573

02nfal (119890 (119896) 119888

1 1198871 1205741)]

1199113(119896 + 1)

= 1199113(119896)

+ ℎ [1199114(119896) minus 120573

03nfal (119890 (119896) 119888

1 1198872 1205741) + 1198870119906 (119896)]

1199114(119896 + 1) = 119911

4(119896) minus ℎ120573

04nfal (119890 (119896) 119888

1 1198873 1205741)

(7)

where 12057301 12057302 12057303 and 120573


gains and 1198861 1198862 1198863 1205751 and 119887


1(119905)

1199112(119905) 1199113(119905) and 119911






0

1205872 minus 1205830

1205830= arc tan (119888 (0 minus 120574))

(8)




120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0

TL

120579(t)

z1(t)

z2(t)

z3(t)

z4(t)

ESO

Plant

minus

minus

minus

minus

⨂

⨂⨂

⨂


120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0 TL

120579(t)

z1(t)z2(t)

z3(t)

z4(t)

ESO RBFNN

Plant

12057301120573021205730312057304

minus

minus

minus

minus⨂

⨂

⨂

⨂





1198901(119896 + 1) = V

1(119896 + 1) minus 119911

1(119896 + 1)

1198902(119896 + 1) = V

2(119896 + 1) minus 119911

2(119896 + 1)

1198903(119896 + 1) = V

3(119896 + 1) minus 119911

3(119896 + 1)

1199060(119896 + 1) = 120573

1nfal (119890

1(119896 + 1) 119888

0 1198874 1205740)

+ 1205732nfal (119890

2(119896 + 1) 119888

0 1198875 1205740)

+ 1205733nfal (119890

3(119896 + 1) 119888

0 1198876 1205740)

(9)

where 1205731 1205732 and 120573


and119886411988651198866 and120575



1199061(119896 + 1) =

[1199060(119896 + 1) minus 119911

4(119896 + 1)]

1198870

(10)



namely 12057301 12057302 12057303 and 120573



1(119905) 1198902(119905) 1198903(119905) and 120579(119905) are


04are



119864 (119896) =

1

2

[120579119889(119896) minus 120579 (119896)]

2

=

1

2

1198902(119896) (11)


120593119894(119896) = exp(minus

1003817100381710038171003817119883 minus 119862

119894

1003817100381710038171003817

2

21198872

119894

) (12)




of this node


0 1 2 3 4 5 6 7 8 9 10minus01

minus005

0

005

01

015

02

025

03

035

04

Time (s)

Posit

ion

trac

king

erro

r (ra

d)


(a)

minus6

minus4

minus2

0

2

4

6

Posit

ion

trac

king

cont

rol v

olta

ge (V

)


0 1 2 3 4 5 6 7 8 9 10Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

100

120

Time (s)

Squa

re w

ave d

istur

banc

ez 4

ADRC z4

NNADRC z4

Disturbance

(c)

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus200

minus100

0

100

200

300

400

500

600

700

800

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




1205730119897(119896) =

6

sum

119894=1

120596119897119894120593119894(119896) (13)




120596119897119894(119896) = 120596

119897119894(119896 minus 1) + Δ120596

119897119894(119896) + 120588Δ120596

119897119894(119896 minus 1)

Δ120596119897119894(119896) = minus120578

120597119864 (119896)

120597120596119897119894(119896)

= minus120578

120597119864 (119896)

120597120579 (119896)

120597120579 (119896)

1205971205730119897(119896)

1205971205730119897(119896)

120597120596119897119894(119896)

= 120578119890 (119896)

120597120579 (119896)

1205971205730119897(119896)

120593119894(119896)

(14)








5 Kg sdotm2119862119905= 15times10

minus13(m3 sdot

sminus1)Pa 119861119898


0= 0011781m3

120573119890= 700MPa and 119879

119885= 128 times 10


minus5 119896119894= 23times10

minus6 and 119896119889= 312times10


1= 095 120573

2= 106 120573

3= 173 119888

0= 025

1198874

= 25 1198875

= 20 1198876

= 15 and 1205740


1= 05 119887

1= 25 119887

2= 20

1198873

= 15 and 1205741


= 150 12057302

= 40 12057303

= 40 and 12057304

= 500





0 1 2 3 4 5 6 7 8 9 100

005

01

015

02

025

03

035

04

045

05

Time (s)

Posit

ion

signa

l (ra

d)






minus4

minus2

0

2

4

6

8

10

12

14

Spee

d tr

acki

ng er

ror (

rad)

0 1 2 3 4 5 6 7 8 9 10Time (s)


times10minus3

(a)

minus4

minus2

0

2

4

6

8

Spee

d tr

acki

ng co

ntro

l vol

tage

(V)

0 1 2 3 4 5 6 7 8 9 10Time (s)


(b)

minus30

minus20

minus10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10Time (s)

ADRC z4

NNADRC z4

Disturbance

Sinu

soid

al d

istur

banc

ez 4

(c)

0 1 2 3 4 5 6 7 8 9 10

minus100

0

100

200

300

400

500

600

700

Time (s)

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




5 Conclusions









Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0

TL

120579(t)

z1(t)

z2(t)

z3(t)

z4(t)

ESO

Plant

minus

minus

minus

minus

⨂

⨂⨂

⨂


120579d(t)TD

1(t)

2(t)

3(t)

e1(t)

e2(t)

e3(t)NLSEF

u0(t) u1(t)

1b0 b0 TL

120579(t)

z1(t)z2(t)

z3(t)

z4(t)

ESO RBFNN

Plant

12057301120573021205730312057304

minus

minus

minus

minus⨂

⨂

⨂

⨂





1198901(119896 + 1) = V

1(119896 + 1) minus 119911

1(119896 + 1)

1198902(119896 + 1) = V

2(119896 + 1) minus 119911

2(119896 + 1)

1198903(119896 + 1) = V

3(119896 + 1) minus 119911

3(119896 + 1)

1199060(119896 + 1) = 120573

1nfal (119890

1(119896 + 1) 119888

0 1198874 1205740)

+ 1205732nfal (119890

2(119896 + 1) 119888

0 1198875 1205740)

+ 1205733nfal (119890

3(119896 + 1) 119888

0 1198876 1205740)

(9)

where 1205731 1205732 and 120573


and119886411988651198866 and120575



1199061(119896 + 1) =

[1199060(119896 + 1) minus 119911

4(119896 + 1)]

1198870

(10)



namely 12057301 12057302 12057303 and 120573



1(119905) 1198902(119905) 1198903(119905) and 120579(119905) are


04are



119864 (119896) =

1

2

[120579119889(119896) minus 120579 (119896)]

2

=

1

2

1198902(119896) (11)


120593119894(119896) = exp(minus

1003817100381710038171003817119883 minus 119862

119894

1003817100381710038171003817

2

21198872

119894

) (12)




of this node


0 1 2 3 4 5 6 7 8 9 10minus01

minus005

0

005

01

015

02

025

03

035

04

Time (s)

Posit

ion

trac

king

erro

r (ra

d)


(a)

minus6

minus4

minus2

0

2

4

6

Posit

ion

trac

king

cont

rol v

olta

ge (V

)


0 1 2 3 4 5 6 7 8 9 10Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

100

120

Time (s)

Squa

re w

ave d

istur

banc

ez 4

ADRC z4

NNADRC z4

Disturbance

(c)

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus200

minus100

0

100

200

300

400

500

600

700

800

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




1205730119897(119896) =

6

sum

119894=1

120596119897119894120593119894(119896) (13)




120596119897119894(119896) = 120596

119897119894(119896 minus 1) + Δ120596

119897119894(119896) + 120588Δ120596

119897119894(119896 minus 1)

Δ120596119897119894(119896) = minus120578

120597119864 (119896)

120597120596119897119894(119896)

= minus120578

120597119864 (119896)

120597120579 (119896)

120597120579 (119896)

1205971205730119897(119896)

1205971205730119897(119896)

120597120596119897119894(119896)

= 120578119890 (119896)

120597120579 (119896)

1205971205730119897(119896)

120593119894(119896)

(14)








5 Kg sdotm2119862119905= 15times10

minus13(m3 sdot

sminus1)Pa 119861119898


0= 0011781m3

120573119890= 700MPa and 119879

119885= 128 times 10


minus5 119896119894= 23times10

minus6 and 119896119889= 312times10


1= 095 120573

2= 106 120573

3= 173 119888

0= 025

1198874

= 25 1198875

= 20 1198876

= 15 and 1205740


1= 05 119887

1= 25 119887

2= 20

1198873

= 15 and 1205741


= 150 12057302

= 40 12057303

= 40 and 12057304

= 500





0 1 2 3 4 5 6 7 8 9 100

005

01

015

02

025

03

035

04

045

05

Time (s)

Posit

ion

signa

l (ra

d)






minus4

minus2

0

2

4

6

8

10

12

14

Spee

d tr

acki

ng er

ror (

rad)

0 1 2 3 4 5 6 7 8 9 10Time (s)


times10minus3

(a)

minus4

minus2

0

2

4

6

8

Spee

d tr

acki

ng co

ntro

l vol

tage

(V)

0 1 2 3 4 5 6 7 8 9 10Time (s)


(b)

minus30

minus20

minus10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10Time (s)

ADRC z4

NNADRC z4

Disturbance

Sinu

soid

al d

istur

banc

ez 4

(c)

0 1 2 3 4 5 6 7 8 9 10

minus100

0

100

200

300

400

500

600

700

Time (s)

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




5 Conclusions









Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5 6 7 8 9 10minus01

minus005

0

005

01

015

02

025

03

035

04

Time (s)

Posit

ion

trac

king

erro

r (ra

d)


(a)

minus6

minus4

minus2

0

2

4

6

Posit

ion

trac

king

cont

rol v

olta

ge (V

)


0 1 2 3 4 5 6 7 8 9 10Time (s)

(b)

0 1 2 3 4 5 6 7 8 9 10minus20

0

20

40

60

80

100

120

Time (s)

Squa

re w

ave d

istur

banc

ez 4

ADRC z4

NNADRC z4

Disturbance

(c)

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus200

minus100

0

100

200

300

400

500

600

700

800

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




1205730119897(119896) =

6

sum

119894=1

120596119897119894120593119894(119896) (13)




120596119897119894(119896) = 120596

119897119894(119896 minus 1) + Δ120596

119897119894(119896) + 120588Δ120596

119897119894(119896 minus 1)

Δ120596119897119894(119896) = minus120578

120597119864 (119896)

120597120596119897119894(119896)

= minus120578

120597119864 (119896)

120597120579 (119896)

120597120579 (119896)

1205971205730119897(119896)

1205971205730119897(119896)

120597120596119897119894(119896)

= 120578119890 (119896)

120597120579 (119896)

1205971205730119897(119896)

120593119894(119896)

(14)








5 Kg sdotm2119862119905= 15times10

minus13(m3 sdot

sminus1)Pa 119861119898


0= 0011781m3

120573119890= 700MPa and 119879

119885= 128 times 10


minus5 119896119894= 23times10

minus6 and 119896119889= 312times10


1= 095 120573

2= 106 120573

3= 173 119888

0= 025

1198874

= 25 1198875

= 20 1198876

= 15 and 1205740


1= 05 119887

1= 25 119887

2= 20

1198873

= 15 and 1205741


= 150 12057302

= 40 12057303

= 40 and 12057304

= 500





0 1 2 3 4 5 6 7 8 9 100

005

01

015

02

025

03

035

04

045

05

Time (s)

Posit

ion

signa

l (ra

d)






minus4

minus2

0

2

4

6

8

10

12

14

Spee

d tr

acki

ng er

ror (

rad)

0 1 2 3 4 5 6 7 8 9 10Time (s)


times10minus3

(a)

minus4

minus2

0

2

4

6

8

Spee

d tr

acki

ng co

ntro

l vol

tage

(V)

0 1 2 3 4 5 6 7 8 9 10Time (s)


(b)

minus30

minus20

minus10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10Time (s)

ADRC z4

NNADRC z4

Disturbance

Sinu

soid

al d

istur

banc

ez 4

(c)

0 1 2 3 4 5 6 7 8 9 10

minus100

0

100

200

300

400

500

600

700

Time (s)

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




5 Conclusions









Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















5 Kg sdotm2119862119905= 15times10

minus13(m3 sdot

sminus1)Pa 119861119898


0= 0011781m3

120573119890= 700MPa and 119879

119885= 128 times 10


minus5 119896119894= 23times10

minus6 and 119896119889= 312times10


1= 095 120573

2= 106 120573

3= 173 119888

0= 025

1198874

= 25 1198875

= 20 1198876

= 15 and 1205740


1= 05 119887

1= 25 119887

2= 20

1198873

= 15 and 1205741


= 150 12057302

= 40 12057303

= 40 and 12057304

= 500





0 1 2 3 4 5 6 7 8 9 100

005

01

015

02

025

03

035

04

045

05

Time (s)

Posit

ion

signa

l (ra

d)






minus4

minus2

0

2

4

6

8

10

12

14

Spee

d tr

acki

ng er

ror (

rad)

0 1 2 3 4 5 6 7 8 9 10Time (s)


times10minus3

(a)

minus4

minus2

0

2

4

6

8

Spee

d tr

acki

ng co

ntro

l vol

tage

(V)

0 1 2 3 4 5 6 7 8 9 10Time (s)


(b)

minus30

minus20

minus10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10Time (s)

ADRC z4

NNADRC z4

Disturbance

Sinu

soid

al d

istur

banc

ez 4

(c)

0 1 2 3 4 5 6 7 8 9 10

minus100

0

100

200

300

400

500

600

700

Time (s)

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




5 Conclusions









Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus4

minus2

0

2

4

6

8

10

12

14

Spee

d tr

acki

ng er

ror (

rad)

0 1 2 3 4 5 6 7 8 9 10Time (s)


times10minus3

(a)

minus4

minus2

0

2

4

6

8

Spee

d tr

acki

ng co

ntro

l vol

tage

(V)

0 1 2 3 4 5 6 7 8 9 10Time (s)


(b)

minus30

minus20

minus10

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10Time (s)

ADRC z4

NNADRC z4

Disturbance

Sinu

soid

al d

istur

banc

ez 4

(c)

0 1 2 3 4 5 6 7 8 9 10

minus100

0

100

200

300

400

500

600

700

Time (s)

Posit

ion

trac

king

torq

ue er

ror (

Nm

)

(d)




5 Conclusions









Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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