Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 973903 9 pageshttpdxdoiorg1011552013973903

Research ArticleNonlinear State Space Modeling and System Identification forElectrohydraulic Control

Jun Yan Bo Li Hai-Feng Ling Hai-Song Chen and Mei-Jun Zhang

College of Field Engineering PLA University of Science and Technology Nanjing 210007 China

Correspondence should be addressed to Hai-Feng Ling hflingymailcom

Received 6 January 2013 Accepted 29 January 2013

Academic Editor Shengyong Chen

Copyright copy 2013 Jun Yan et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper deals with nonlinear modeling and identification of an electrohydraulic control system for improving its trackingperformanceWe build the nonlinear state spacemodel for analyzing the highly nonlinear system and then develop aHammerstein-Wiener (H-W) model which consists of a static input nonlinear block with two-segment polynomial nonlinearities a linear time-invariant dynamic block and a static output nonlinear block with single polynomial nonlinearity to describe it We simplify theH-Wmodel into a linear-in-parameters structure by using the key term separation principle and then use amodified recursive leastsquaremethod with iterative estimation of internal variables to identify all the unknown parameters simultaneously It is found thatthe proposedH-Wmodel approximates the actual systembetter than the independentHammersteinWiener andARXmodelsTheprediction error of the H-Wmodel is about 13 54 and 58 less than the Hammerstein Wiener and ARXmodels respectively

1 Introduction

Electrohydraulic control systems are widely used in industrydue to their unique features of small size to power ratio highnature frequency high position stiffness and low positionerror [1] However the dynamics of hydraulic systems ishighly nonlinear in nature The systems may be subjectedto nonsmooth nonlinearities due to control input saturationfriction valve overlapping and directional changes of valveopening A number of robust and adaptive control strategieshave been proposed to deal with such problems [2ndash4] butmodeling and identification of control systems remain animportant and difficult issue in most real-world applications

Linear models of electrohydraulic control systems aresimple and widely used but they assume that the hydraulicactuator always moves around an operating point [5 6]which does not accord with most real-world cases where theactuator moves in a wide range with hard nonlinearities Inthe literatureWang et al [7] analyzed the nonlinear dynamiccharacteristics of hydraulic cylinder such as nonlinear gainnonlinear spring and nonlinear friction force Jelali andSchwarz [8] identified the nonlinear models in observercanonical form of hydraulic servodrives Kleinsteuber andSepehri [9] used a polynomial abductive network modeling

technique to describe a class of hydraulic actuation systemswhich were used in heavy-duty mobile machines Yousefiet al [10] proposed the Differential Evolution algorithm toidentify the nonlinear model of a servohydraulic system withflexible load Yao et al [2] also pointed out that there weremany considerable model uncertainties such as parametricuncertainties and uncertain nonlinearities As we can seemodeling and identifying the electrohydraulic control systemas a flexible nonlinear black-box or grey-box are moreappropriate for real-world applications

In the field of nonlinear system identification the Ham-merstein and Wiener (H-W) models are widely used [11]Kwak et al [12] proposed two Hammerstein-type mod-els to identify hydraulic actuator friction dynamics TheHammerstein-type models are built by linear time-invariant(LTI) dynamic subsystems and static nonlinear (SN) elementsin a cascade structure they are able to approximate most ofthe nonlinear dynamics with an arbitrarily high accuracy andcan generate both physical insights and flexible structuresGenerally the Wiener model is supposed to represent theoutput nonlinearities and sensor nonlinearities while theHammerstein model is supposed to represent the input non-linearities and actuator nonlinearities The Hammerstein-Wiener (H-W) model which is defined as a static nonlinear

2 Mathematical Problems in Engineering

119865119891

119861119901

119865119904

119872ℎ

11987511198601 11987521198602

1198761 1198762 119909119907

From proportionalrelief valve

119875119904 119875119903

119865119897

Figure 1 Valve controlled asymmetric cylinder system

element in cascade with a linear dynamic system followed byanother static nonlinear element is adopted in this paper

The H-W model is a parameterized nonlinear model inblack-box termThere are two advantages of the H-WmodelThe first one is that only the input and output singles areused for identification of all the unknown parameters thatis no information on the internal states is needed which cansimplify the identification process and improve the predictionaccuracy by less sensors and noise The second one is thatit has a physical insight into the nonlinear characteristics ofthe actual system which is important in system analyzingmonitoring diagnosis and controller design

The rest of this paper is organized as follows Section 2presents the theoretic modeling of an electrohydraulic con-trol system Section 3 describes our H-W model in detailSection 4 proposes the iterative identification algorithm forthe H-W model Section 5 presents the experimental tests aswell as the identification results Finally Section 6 concludesthe paper

2 Theoretic Modeling

A general electrohydraulic control system is mainly com-prised of an electrohydraulic proportional valve and a valvecontrolled asymmetric cylinder In this paper we study aproportional relief valve controlled valve-cylinder system asshown in Figure 1 where ℎ is the displacement of piston119872 is the equivalent load mass 119860

1and 119860

2are the areas of

piston in the head and rod sides of cylinder 1198751and 1198752are the

pressures inside the two chambers of the cylinder 119875119904is the

supply pressure 119875119903is the pressure of return oil 119876

1and 119876

2

are the flows in and out of the cylinder 119909V is displacementof the spool valve 119861

119901is the viscous damping coefficient 119865

119891

represents nonlinear friction 119865119904represents nonlinear spring

40

30

20

10

002 04 06 08 1 12

119868119889

Pres

sure

(119901b

ar)

0Current (119868A)

(a) Dead band of the pilot relief valve

250

200

150

100

50

0minus8 minus6 minus4 minus2 0 2 4 6 8

Area ofbypassed way

NegativePositive

119888

Orifi

ce ar

ea (119860

mm

2)

Spool displacement (119909120584mm)

(b) Dead band of the main valve

Figure 2 Dead band of the electrohydraulic proportional system

force 119865119888represents viscous force and 119865

119897represents uncertain

loadModeling the system by physical laws gives us a particular

insight into the systemrsquos properties which allows us toseek the parameterized models that are flexible enough tocapture all dynamic behavior of the system [13 14] Theelectrohydraulic proportional valve is controlled directly bythe digital controller It can bemodeled as a first order transferfunction [9]

119866119897=119909V (119904)

119868V (119904)=

119896V

1 + 120591V119904 (1)

where 119896V is the gain of the electrohydraulic proportionalvalve 120591V is the time constant of the first order system119868V = 119868 minus 119868

119889is the effective current 119868 and 119868

119889are the practical

input current of the proportional relief valve and the currentto overcome dead band of the valve respectively The deadbands mainly due to the pilot relief valve and the main valveare depicted in Figure 2

The valve controlled asymmetric cylinder is shown inFigure 1 Generally its model is constructed by combiningthe flow equation of spool valve the continuity equation

Mathematical Problems in Engineering 3

of hydraulic cylinder and the force equilibrium equation ofhydraulic cylinder [2] Define the state variables as

[1199091 1199092119909311990941199095]119879

≜ [ℎ ℎ 11990111199012119909V]119879

(2)

The entire system can be modeled as the followingnonlinear state space model [15]

1= 1199092

2=11986011199093

119872minus11986021199094

119872minus119865119891(1199092)

119872

minus119865119904(1199091)

119872minus119865119888(1199092)

119872minus119865119897

119872

3= minus

12057311989011986011199092

1198811

minus120573119890(119862119894+ 119862119890) 1199093

1198811

+1205731198901198621198941199094

1198811

+1205731198901198921(119909)

1198811

1199095

4=12057311989011986021199092

1198812

+1205731198901198621198941199093

1198812

minus120573119890(119862119894+ 119862119890) 1199094

1198812

minus1205731198901198922(119909)

1198812

1199095

5=119896V

120591V

119868V minus1199095

120591V

(3)

1198921(119909) = sgn((1 + sgn (119909

5))119901119904

2minus sgn (119909

5) 1199093)

times 1198621198891119882radic

2

120588((1 + sgn (119909

5))119901119904

2minus sgn (119909

5) 1199093)

1198922(119909) = sgn((1minussgn (119909

5))119901119904

2+sgn (119909

5) 1199094)

times 1198621198892119882radic

2

120588((1minussgn (119909

5))119901119904

2+sgn (119909

5) 1199094)

(4)

where 120573119890is the effective bulk modulus119881

1and119881

2are effective

volumes of the two chambers 119862119894and 119862

119890are internal and

external leakage coefficients 119882 is the area gradient of thevalve orifice and 119862

1198891and 119862

1198892are flow discharge coefficients

of the spool valveSeveral physical phenomena have been taken into consid-

eration in the above model for example nonlinear friction119865119891 nonlinear spring force 119865

119904 viscous force 119865

119888 uncertain load

119865119897 discontinuous flow discharge 119892

119894 oil compliance internal

leakage and external leakage From the theoretic modelingof the electrohydraulic control system we can see that thesystem is a highly nonlinear system containing complexfeatures such as the dead band nonlinearity saturationsquared pressure drop and asymmetric response property

There are also some hard-to-model nonlinearities in(3) such as nonlinear friction nonlinear spring force anduncertain external disturbances So modeling this systemjust by physical laws fails to approximate the actual systemFurthermore identification of the unknown parameters in(3) is hard due to its demand on internal states measurementIn the followingwe adopt anH-Wmodel tomodel this highly

1198731(∙)119906(119896) 119908(119896) 119910(119896)119871(119911)

120584(119896)

(a) Hammerstein model

1198732 (∙)119906(119896) 119910(119896)119871(119911)

119909(119896)

120584(119896)

(b) Wiener model

1198731(∙) 1198732 (∙)119906(119896) 119908(119896) 119910(119896)119871(119911)

119909(119896)

120584(119896)

(c) Hammerstein-Wiener model

Figure 3 Hammerstein and Wiener models

nonlinear dynamic system The H-W model is a flexibleblack-box model based on the physical insight into theactual system We identify the parameters of the H-Wmodelusing the input and output signals which can simplify theidentification process and improve the prediction accuracy byless sensors and noise

3 Hammerstein-Wiener Model

The ldquouniversalrdquo nonlinear black-box methods such as neuralnetworks Volterra series and fuzzy models are widely usedto model complex nonlinear systems Most of these meth-ods can avoid unmodeled dynamics in the aforementionedmathematical model [16 17] However these models do notprovide deep insight into the nonlinear characteristics ofthe actual system which is important in system analyzingmonitoring diagnosis and controller design In comparisonthe Hammerstein-Wiener (H-W) model possesses the flexi-bility to capture all relevant nonlinear phenomena as well asthe physical insight into the actual system In this sectionwe develop an H-W model to describe the electrohydrauliccontrol system

The H-W model is composed of an internal lineardynamic block and two static nonlinear blocks it is thecombination of Hammerstein and Wiener model The Ham-merstein model is a nonlinear model with a static nonlinearblock followed by a linear dynamic block as shown inFigure 3(a) and this N-L type of model may account foractuator nonlinearities and other input nonlinear effectsThe Wiener model has linear dynamic block followed bya nonlinear block as shown in Figure 3(b) and this L-Ntype of model mainly accounts for sensor nonlinearitiesand output nonlinear effects A series combination of aHammerstein and a Wiener model yields the H-W modelas shown in Figure 3(c) and this N-L-N type of model hasboth characteristics of the Hammerstein andWiener modelsMoreover all of the three models have proved to be able toaccurately describe a wide variety of nonlinear systems in [9]

According to the nonlinearities of the abovementionedelectrohydraulic control system for example dead band

4 Mathematical Problems in Engineering

saturation nonlinear friction nonlinear spring force andasymmetric dynamics of the cylinder we describe the inputnonlinearity (119873

1) block of the models in Figure 3 by a

two-segment polynomial nonlinearities The two-segmentpolynomial nonlinearities have the advantage of describinga system whose dynamic properties differ significantly at thepositive and negative directions [18] it has less parameters tobe estimated than a single polynomial and piecewise linearmodels [19] It can be written as

119908 (119896) =

119891 (119906 (119896)) =

1199031

sum119897=0

119891119897119906119897(119896) 119906 (119896) ge 0

119892 (119906 (119896)) =

1199031

sum119897=0

119892119897119906119897(119896) 119906 (119896) lt 0

(5)

where 119891119897and 119892

119897are parameters of the polynomial function

119906(119896) is the input 119908(119896) is the output of static nonlinearfunction119873

1 and 119903

1is the degree of the polynomial function

Define the switching function as

ℎ (119906) = 0 119906 ge 0

1 119906 lt 0(6)

Then the relation between inputs 119906(119896) and outputs119908(119896) of the input nonlinear block can be written as

119908 (119896) = 119891 (119906 (119896)) + (119892 (119906 (119896)) minus 119891 (119906 (119896))) ℎ (119906 (119896))

=

1199031

sum

119897=0

119891119897119906119897(119896) +

1199031

sum

119897=0

119901119897119906119897(119896) ℎ (119906 (119896))

(7)

where 119901119897= 119892119897minus 119891119897

The difference equationmodel 119871(119911) of the linear dynamicblock is described by an extended autoregressive (ARX)model as

119860(119911minus1) 119909 (119896) = 119911

minus119899119896119861 (119911minus1)119908 (119896) + V (119896) (8)

where 119908(119896) and 119909(119896) are the input and output of the lineardynamic block respectively V(119896) is white noise 119899

119896represents

the pure delay of the system and119860(119911minus1) and 119861(119911minus1) are scalarpolynomials in the unit delay operator 119911minus1

119860(119911minus1) = 1 + 119886

1119911minus1+ sdot sdot sdot + 119886

119899119886

119911minus119899119886

119861 (119911minus1) = 1198870+ 1198871119911minus1+ sdot sdot sdot + 119887

119899119887

119911minus119899119887

(9)

The output nonlinear block 1198732is described by a single

polynomials

119910 (119896) = 119902 (119909 (119896)) =

1199032

sum

119898=1

119902119898119909119898(119896) (10)

where 119902119898

is unknown parameter 1199032is the degree of the

polynomial function 1198732 and 119910(119896) is output of the entire

system and in this paper it represents the output velocityThe H-Wmodel of the system is depicted in Figure 4

ARX

1198731 1198732

119906(119896) 119908(119896) 119910(119896)

119871(119911)

119909(119896)

Figure 4 Schematic diagram of the H-Wmodel

4 Iterative Identification Algorithm

As we know the cascade mode of the models depicted inFigure 3 leads to composite mappings for example Ham-merstein model 119871(119873

1(119906(119896))) Wiener model 119873

2(119871(119906(119896)))

H-W model 1198732(119871(1198731(119906(119896)))) Substituting the mathematic

models of each block (ie (7) (8) and (10)) into the com-posite mappings directly leads to complex models which arestrongly nonlinear in both of the variables and the unknownparameters It is not appropriate for parameter estimation[20] In the following we apply the so-called key termseparation principle to simplify the H-Wmodel into a linear-in-parameters structure and then adopt a modified recursiveleast square algorithm with internal variable estimation toestimate both of the linear and nonlinear block parameterssimultaneously

41 Key Term Separation Principle Let 119891 119892 and ℎ be one-to-one mappings defined on nonempty sets 119880119883 and 119884 as

119891 119880 997888rarr 119883

119892 119883 997888rarr 119884

ℎ = 119892 ∘ 119891 119880 997888rarr 119884

(11)

Then the composite mapping ℎ can be given by

119910 (119905) = 119892 [119909 (119905)] = 119892 [119891 [119906 (119905)]] = ℎ [119906 (119905)] (12)

Thebasic idea of key term separation principle is a formofhalf-substitution suggested in [21] Suppose 119892 be an analyticnonlinear mapping which can be rewritten into the followingadditive form

119910 (119905) = 119909 (119905) + 119866 [119909 (119905)] (13)

Which consists of the key term 119909(119905) plus the remainder ofthe originalmapping assigned as119866(sdot) Rewrite the one-to-onemapping 119891

119909 (119905) = 119891 [119906 (119905)] (14)

We substitute (13) only into the first term in the right sideof (14) and then obtain the following mapping

119910 (119905) = 119891 [119906 (119905)] + 119866 [119909 (119905)] (15)

Equations (14) and (15) describe the mapping functionℎ in a compositional way This makes the inner mappingappears both explicitly and implicitly in the outer one whichmay be helpful for parameter identification Note that thisdecomposition technique can easily be extended to a moremultilayer composite mapping

Mathematical Problems in Engineering 5

42 Modified Least Square Algorithm In this section wedecompose the H-W model into a linear-in-parametersstructure by the key term separation principle and developa modified iterative least square algorithm with internalvariables estimation to identify all the unknown parametersof the H-W model We also apply this method to theHammerstein and Wiener models

According to the key term separation principle werewrite the output nonlinear block119873

2 that is (10) as

119910 (119896) = 1199021119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896) (16)

where the internal variable 119909(119896) is separated The dynamiclinear block 119871(119911) that is (8) can be rewritten as

119909 (119896) = 1198870119908 (119896) 119911

minus119899119896 + 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896)

(17)

where the internal variable 119908(119896) is separated Now to com-plete the sequential decomposition first we substitute (7)into (17) only for 119908(119896) in the first term and then substitutethe new equation (17) into (16) only for 119909(119896) in the first termagain The final output equation of the H-Wmodel will be

119910 (119896) = 11990211198870(119891 (119906 (119896)) + 119901 (119906 (119896)) ℎ (119906 (119896))) 119911

minus119899119896

+ 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896)

(18)

As the H-W model depicted in Figure 4 consists of threesubsystems in series the parameterization of the model isnot unique because many combinations of parameters can befound [22]Therefore one parameter in at least two blocks hasto be fixed in (18) Evidently the choices 119902

1= 1 and 119887

0= 1will

simplify the model descriptionThen the H-Wmodel can bewritten as

119910 (119896) =

1199031

sum

119897=0

119891119897119906119897(119896 minus 119899

119896) +

1199031

sum

119897=0

119901119897119906119897(119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

+

119899119887

sum

119894=1

119887119894119908 (119896 minus 119899

119896minus 119894) +

119899119886

sum

119895=1

119886119895119909 (119896 minus 119895) +

1199032

sum

119898=2

119902119898119909119898(119896)

(19)

Equation (19) is linear-in-parameters for given 119906(119896) 119909(119896)and 119908(119896) it can be written in the following least squareformat

119910 (119896) = Φ119879(119896 120579) 120579 (20)

where the internal variables 119908(119896) and 119909(119896) are estimated by(7) and (17) using the preceding estimated parameters duringeach iterative process and

Φ119879= [1 119906 (119896minus119899

119896) 119906

1199031 (119896minus119899

119896)

ℎ (119906 (119896minus119899119896)) 119906 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896))

1199061199031 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896)) 119908 (119896minus119899

119896minus1)

119908 (119896 minus 119899119896minus 119894) minus119909 (119896minus1) minus119909 (119896minus119895)

1199092(119896) 119909

1199032 (119896)]

120579 = [1198910 1198911 119891

1199031

1199010 1199011 119901

1199031

1198871 119887

119899119887

1198861 119886

119899119886

1199022 119902

1199032

] 119879

(21)

Nowwe apply themodified recursive least squaremethodwith iterative estimation of the internal variable to (20) [17]Minimizing the following least square criterion [12]

= arg min120579

119899

sum

119896=1

120582119899minus119896

[119910 (119896) minus Φ119879

(119896) 120579]

2

(22)

where 120582 ≦ 1 is the forgetting factor the formulas ofthe recursive identification algorithm supplemented withinternal variable estimation are as follows

(119896) = (119896 minus 1) +

P (119896 minus 1) Φ (119896) [119910 (119896) minus Φ119879

(119896) (119896 minus 1)]

120582 + Φ119879

(119896)P (119896 minus 1) Φ (119896)

(23)

P (119896) = P (119896 minus 1)

120582minusP (119896 minus 1) Φ (119896) Φ

119879

(119896)P (119896 minus 1)

1 + Φ119879

(119896)P (119896 minus 1) Φ (119896) 120582

(24)

(119896) =

1199031

sum

119897=0

119891119897(119896 minus 1) 119906

119897(119896) +

1199031

sum

119897=0

119897(119896 minus 1) 119906

119897(119896) ℎ (119906 (119896))

(25)

(119896) = (119896 minus 119899119896) +

119899119887

sum

119894=1

119894(119896 minus 1) (119896 minus 119899

119896minus 119894)

minus

119899119886

sum

119895=1

119895(119896 minus 1) 119909 (119896 minus 119895)

(26)

Φ (119896) = [1 119906 (119896 minus 119899119896) 119906

1199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

119906 (119896 minus 119899119896) ℎ (119906 (119896 minus 119899

119896))

1199061199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

(119896 minus 119899119896minus 1) (119896 minus 119899

119896minus 119894)

minus (119896 minus 1) minus (119896 minus 119895) 2(119896)

1199032 (119896)]119879

(27)

where P(0) = 120583I I is unit matrix and 0 lt 120583 lt infin

6 Mathematical Problems in Engineering

Computer

Pressure transducer

Inclinometer

Low level controller

Electrohydraulic valve

(a)

Inclinometer

Main valve

Proportionalrelief valve

119870ℎ

Mainpump

Pilotpump

ControllerReference

input

(b)

Figure 5 Experimental prototype machine

Inclinometer

Proportionalrelief valve

120579119889+

minus

119906 Digitalcontroller

Pilot pump Main pump

119876Mainvalve

Hydraulicactuator

Mechanism 120579DA

AD

I P

Figure 6 Electrohydraulic position servocontrol system

In conclusion the iterative identification algorithm canbe presented as follwos

Step 1 Set the initial values of 119909(0) 119908(0) 119906(0) and P(0)

Step 2 Estimate the parameter (119896) by algorithm (23) andcalculate P(119896) by (24)

Step 3 Estimate the internal variables (119896) and (119896) by (25)and (26) using the recent estimates of model parameters (119896)

Step 4 Update the values of Φ(119896) by (27)

Step 5 Return to Step 2 until the parameter estimates con-verge to constant values

5 Experiment

51 Experimental Environment A hydraulic excavator wasretrofitted to be controlled by computer in our laboratory[23] Figure 5 shows the prototype machine whose manualpilot hydraulic control system was replaced by electrohy-draulic proportional control system inclinometers and pres-sure transducers were also installed on the excavator arms

for position and force servocontrol Schematic diagram of theelectrohydraulic servosystem is shown in Figure 6

52 Experimental Results In order to obtain the nonlinearcharacteristics of the system when changing the directionsand to obtain sufficient excitation we adopted a multisineinput signal which contained the frequency of 005Hz 01 Hz02Hz 04Hz and 05Hz to the identification experimentsThe sample rate was chosen to be 20Hz on the machineThe input signal and angle output were obtained from thecomputer of the experiment machine Ten groups of inputand output signals with time duration of 55 seconds weresampled in the repeated experiments the averaged measure-ment results are shown in Figure 7 Finally we calculate theoutput angle velocity by numerical differentiation

We set the parameters 119899119886= 3 119899

119887= 2 119899

119896= 8 1199031= 1199032= 3

119909(0) = 0 119908(0) = 0 119906(0) = 0 and P(0) = 106I Note that

lower forgetting factor 120582 is useful for reducing the influencesof old date while a value of 120582 close to 1 is less sensitive todisturbance Therefore we chose the forgetting factor to be120582 = 098 during the first 200 samples and 120582 = 1 otherwiseCompiling the developed iterative least square algorithm inMATLAB to identify theARXmodel containing only the119871(119911)

Mathematical Problems in Engineering 7

420

minus2

minus40 10 20 30 40 50

Input

Time (119905s)

Inpu

t (119906

v)

(a)

0 10 20 30 40 50

Angle output

2

15

1

Ang

le (r

ad)

Time (119905s)

(b)

Figure 7 Input and output signals of the identification experiment

The measured velocityPredicted by Hammerstein-Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(a) Hammerstein-Wiener prediction result

The measured velocityPredicted by Hammerstein

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(b) Hammerstein prediction result

The measured velocityPredicted by Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(c) Wiener prediction result

The measured velocityPredicted by ARX

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(d) ARX prediction result

Figure 8 Comparison results of the identified models

blockHammersteinmodel consisting of1198731and119871(119911)Wiener

model consisting of 119871(119911) and1198732 and H-Wmodel consisting

of1198731 119871(119911) and119873

2 respectively we obtain the identification

results shown in Table 1We use the identified models to predict the tracking

velocity of a general trajectory The comparative results areshown in Figures 8 and 9 They demonstrate that the H-Wand Hammerstein models which contain the input nonlinearblock with two-segment polynomial nonlinearities capturethe actual system well while the Wiener and ARX modelscannot approximate the actual system well Themean-squareerrors (MSE) of the identifiedmodels in Table 2 show that the

prediction error of the H-W model is about 13 54 and58 less than the Hammerstein Wiener and ARX modelsrespectively

6 Conclusion

This paper investigates the nonlinear modeling and identi-fication of an electrohydraulic control system We develop atheoretic state spacemodel for system analysis propose anH-W model for the highly nonlinear system based on a deeplyphysical insight into the actual system and apply a modified

8 Mathematical Problems in Engineering

05

04

03

02

01

0

minus01

minus02

minus03

0 2 4 6 8 10 12 14 16 18

Estim

atio

n re

sidua

(rad

s)

Residua of Hammerstein-WienerResidua of Hammerstein

Residua of WeinerResidua of ARX

Time (119905s)

Figure 9 Comparison of the estimation residua

Table 1 The identification results

Parameters Model typeH-W Hammerstein Wiener ARX

1198870

1 1 minus00358 minus002761198871

00349 00105 00151 001241198872

00617 00147 minus00069 005141198861

minus00309 minus00596 00069 minus006611198862

minus00598 minus00613 minus00075 minus006531198863

minus00897 minus00610 minus00081 minus006191198910

00109 00027 mdash mdash1198911

minus14619 times 10minus5 00012 mdash mdash1198912

minus00050 minus00012 mdash mdash1198913

00052 00045 mdash mdash1199010

minus00165 minus00055 mdash mdash1199011

00033 00013 mdash mdash1199012

00104 00036 mdash mdash1199013

minus00010 minus86858 times 10minus4 mdash mdash1199021

1 mdash 1 mdash1199022

00050 mdash 00013 mdash1199023

63562 times 10minus4 mdash 26895 times 10minus4 mdash

Table 2 MSE of the identified models

Errors Model typeH-W Hammerstein Wiener ARX

120575r sdot sminus1 00321 00417 01081 01218

recursive least square method with internal variables esti-mation to identify its parameters The main findings of thepaper include the follwoing (1) the proposed H-W modelsimplifies the identification procedure because it only uses theinput and output signals to identify all the parameters (2)TheH-W model containing the input nonlinear block with two-segment polynomial nonlinearities captures the actual systemverywell As shown by the comparative results the prediction

error of the H-Wmodel is about 13 54 and 58 less thanthe Hammerstein Wiener and ARX models respectivelyThe results provide a physical insight into the nonlinearcharacteristics of the actual system which is important forsystem analyzing monitoring and diagnosis Future workincludes addressing uncertain and fuzzy properties of thesystem [24 25] and extending the model for a wider rangeof equipment [26]

Acknowledgment

Thisworkwas supported in part by grants fromNational Nat-ural Science Foundation (Grant nos 51175511 and 61105073)of China

References

[1] H E Merritt Hydraulic Control System John Wiley amp SonsNew York USA 1967

[2] B Yao F Bu J Reedy and G T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[3] S Wen W Zheng J Zhu X Li and S Chen ldquoElman fuzzyadaptive control for obstacle avoidance of mobile robots usinghybrid forceposition incorporationrdquo IEEE Transactions onSystems Man and Cybernetics C vol 42 no 4 pp 603ndash6082012

[4] S Y Chen J Zhang H Zhang N M Kwok and Y F Li ldquoIntel-ligent lighting control for vision-based robotic manipulationrdquoIEEE Transactions on Industrial Electronics vol 59 no 8 pp3254ndash3263 2012

[5] Q H He P Hao and D Q Zhang ldquoModeling and parameterestimation for hydraulic system of excavatorrsquos armrdquo Journal ofCentral South University of Technology vol 15 no 3 pp 382ndash386 2008

[6] K Ziaei and N Sepehri ldquoModeling and identification ofelectrohydraulic servosrdquo Mechatronics vol 10 no 7 pp 761ndash772 2000

[7] L Wang B Wu R Du and S Yang ldquoNonlinear dynamic char-acteristics of moving hydraulic cylinderrdquo Chinese Journal ofMechanical Engineering vol 43 no 12 pp 12ndash19 2007

[8] M Jelali and H Schwarz ldquoNonlinear identification of hydraulicservo-drive systemsrdquo IEEE Control Systems Magazine vol 15no 5 pp 17ndash22 1995

[9] S Kleinsteuber and N Sepehri ldquoA polynomial network mod-eling approach to a class of large-scale hydraulic systemsrdquoComputers and Electrical Engineering vol 22 no 2 pp 151ndash1681996

[10] H Yousefi H Handroos and A Soleymani ldquoApplication of dif-ferential evolution in system identification of a servo-hydraulicsystem with a flexible loadrdquoMechatronics vol 18 no 9 pp 513ndash528 2008

[11] F Giri and E W Bai Block-Oriented Nonlinear System Identi-fication vol 404 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2010

[12] B -J Kwak A E Yagle and J A Levitt ldquoNonlinear systemidentification of hydraulic actuator friction dynamics using aHammerstein modelrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing vol 4 pp1933ndash1936 Seattle Wash USA 1998

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

119865119891

119861119901

119865119904

119872ℎ

11987511198601 11987521198602

1198761 1198762 119909119907

From proportionalrelief valve

119875119904 119875119903

119865119897

Figure 1 Valve controlled asymmetric cylinder system

element in cascade with a linear dynamic system followed byanother static nonlinear element is adopted in this paper

The H-W model is a parameterized nonlinear model inblack-box termThere are two advantages of the H-WmodelThe first one is that only the input and output singles areused for identification of all the unknown parameters thatis no information on the internal states is needed which cansimplify the identification process and improve the predictionaccuracy by less sensors and noise The second one is thatit has a physical insight into the nonlinear characteristics ofthe actual system which is important in system analyzingmonitoring diagnosis and controller design

The rest of this paper is organized as follows Section 2presents the theoretic modeling of an electrohydraulic con-trol system Section 3 describes our H-W model in detailSection 4 proposes the iterative identification algorithm forthe H-W model Section 5 presents the experimental tests aswell as the identification results Finally Section 6 concludesthe paper

2 Theoretic Modeling

A general electrohydraulic control system is mainly com-prised of an electrohydraulic proportional valve and a valvecontrolled asymmetric cylinder In this paper we study aproportional relief valve controlled valve-cylinder system asshown in Figure 1 where ℎ is the displacement of piston119872 is the equivalent load mass 119860

1and 119860

2are the areas of

piston in the head and rod sides of cylinder 1198751and 1198752are the

pressures inside the two chambers of the cylinder 119875119904is the

supply pressure 119875119903is the pressure of return oil 119876

1and 119876

2

are the flows in and out of the cylinder 119909V is displacementof the spool valve 119861

119901is the viscous damping coefficient 119865

119891

represents nonlinear friction 119865119904represents nonlinear spring

40

30

20

10

002 04 06 08 1 12

119868119889

Pres

sure

(119901b

ar)

0Current (119868A)

(a) Dead band of the pilot relief valve

250

200

150

100

50

0minus8 minus6 minus4 minus2 0 2 4 6 8

Area ofbypassed way

NegativePositive

119888

Orifi

ce ar

ea (119860

mm

2)

Spool displacement (119909120584mm)

(b) Dead band of the main valve

Figure 2 Dead band of the electrohydraulic proportional system

force 119865119888represents viscous force and 119865

119897represents uncertain

loadModeling the system by physical laws gives us a particular

insight into the systemrsquos properties which allows us toseek the parameterized models that are flexible enough tocapture all dynamic behavior of the system [13 14] Theelectrohydraulic proportional valve is controlled directly bythe digital controller It can bemodeled as a first order transferfunction [9]

119866119897=119909V (119904)

119868V (119904)=

119896V

1 + 120591V119904 (1)

where 119896V is the gain of the electrohydraulic proportionalvalve 120591V is the time constant of the first order system119868V = 119868 minus 119868

119889is the effective current 119868 and 119868

119889are the practical

input current of the proportional relief valve and the currentto overcome dead band of the valve respectively The deadbands mainly due to the pilot relief valve and the main valveare depicted in Figure 2

The valve controlled asymmetric cylinder is shown inFigure 1 Generally its model is constructed by combiningthe flow equation of spool valve the continuity equation

Mathematical Problems in Engineering 3

of hydraulic cylinder and the force equilibrium equation ofhydraulic cylinder [2] Define the state variables as

[1199091 1199092119909311990941199095]119879

≜ [ℎ ℎ 11990111199012119909V]119879

(2)

The entire system can be modeled as the followingnonlinear state space model [15]

1= 1199092

2=11986011199093

119872minus11986021199094

119872minus119865119891(1199092)

119872

minus119865119904(1199091)

119872minus119865119888(1199092)

119872minus119865119897

119872

3= minus

12057311989011986011199092

1198811

minus120573119890(119862119894+ 119862119890) 1199093

1198811

+1205731198901198621198941199094

1198811

+1205731198901198921(119909)

1198811

1199095

4=12057311989011986021199092

1198812

+1205731198901198621198941199093

1198812

minus120573119890(119862119894+ 119862119890) 1199094

1198812

minus1205731198901198922(119909)

1198812

1199095

5=119896V

120591V

119868V minus1199095

120591V

(3)

1198921(119909) = sgn((1 + sgn (119909

5))119901119904

2minus sgn (119909

5) 1199093)

times 1198621198891119882radic

2

120588((1 + sgn (119909

5))119901119904

2minus sgn (119909

5) 1199093)

1198922(119909) = sgn((1minussgn (119909

5))119901119904

2+sgn (119909

5) 1199094)

times 1198621198892119882radic

2

120588((1minussgn (119909

5))119901119904

2+sgn (119909

5) 1199094)

(4)

where 120573119890is the effective bulk modulus119881

1and119881

2are effective

volumes of the two chambers 119862119894and 119862

119890are internal and

external leakage coefficients 119882 is the area gradient of thevalve orifice and 119862

1198891and 119862

1198892are flow discharge coefficients

of the spool valveSeveral physical phenomena have been taken into consid-

eration in the above model for example nonlinear friction119865119891 nonlinear spring force 119865

119904 viscous force 119865

119888 uncertain load

119865119897 discontinuous flow discharge 119892

119894 oil compliance internal

leakage and external leakage From the theoretic modelingof the electrohydraulic control system we can see that thesystem is a highly nonlinear system containing complexfeatures such as the dead band nonlinearity saturationsquared pressure drop and asymmetric response property

There are also some hard-to-model nonlinearities in(3) such as nonlinear friction nonlinear spring force anduncertain external disturbances So modeling this systemjust by physical laws fails to approximate the actual systemFurthermore identification of the unknown parameters in(3) is hard due to its demand on internal states measurementIn the followingwe adopt anH-Wmodel tomodel this highly

1198731(∙)119906(119896) 119908(119896) 119910(119896)119871(119911)

120584(119896)

(a) Hammerstein model

1198732 (∙)119906(119896) 119910(119896)119871(119911)

119909(119896)

120584(119896)

(b) Wiener model

1198731(∙) 1198732 (∙)119906(119896) 119908(119896) 119910(119896)119871(119911)

119909(119896)

120584(119896)

(c) Hammerstein-Wiener model

Figure 3 Hammerstein and Wiener models

nonlinear dynamic system The H-W model is a flexibleblack-box model based on the physical insight into theactual system We identify the parameters of the H-Wmodelusing the input and output signals which can simplify theidentification process and improve the prediction accuracy byless sensors and noise

3 Hammerstein-Wiener Model

The ldquouniversalrdquo nonlinear black-box methods such as neuralnetworks Volterra series and fuzzy models are widely usedto model complex nonlinear systems Most of these meth-ods can avoid unmodeled dynamics in the aforementionedmathematical model [16 17] However these models do notprovide deep insight into the nonlinear characteristics ofthe actual system which is important in system analyzingmonitoring diagnosis and controller design In comparisonthe Hammerstein-Wiener (H-W) model possesses the flexi-bility to capture all relevant nonlinear phenomena as well asthe physical insight into the actual system In this sectionwe develop an H-W model to describe the electrohydrauliccontrol system

The H-W model is composed of an internal lineardynamic block and two static nonlinear blocks it is thecombination of Hammerstein and Wiener model The Ham-merstein model is a nonlinear model with a static nonlinearblock followed by a linear dynamic block as shown inFigure 3(a) and this N-L type of model may account foractuator nonlinearities and other input nonlinear effectsThe Wiener model has linear dynamic block followed bya nonlinear block as shown in Figure 3(b) and this L-Ntype of model mainly accounts for sensor nonlinearitiesand output nonlinear effects A series combination of aHammerstein and a Wiener model yields the H-W modelas shown in Figure 3(c) and this N-L-N type of model hasboth characteristics of the Hammerstein andWiener modelsMoreover all of the three models have proved to be able toaccurately describe a wide variety of nonlinear systems in [9]

According to the nonlinearities of the abovementionedelectrohydraulic control system for example dead band

4 Mathematical Problems in Engineering

saturation nonlinear friction nonlinear spring force andasymmetric dynamics of the cylinder we describe the inputnonlinearity (119873

1) block of the models in Figure 3 by a

two-segment polynomial nonlinearities The two-segmentpolynomial nonlinearities have the advantage of describinga system whose dynamic properties differ significantly at thepositive and negative directions [18] it has less parameters tobe estimated than a single polynomial and piecewise linearmodels [19] It can be written as

119908 (119896) =

119891 (119906 (119896)) =

1199031

sum119897=0

119891119897119906119897(119896) 119906 (119896) ge 0

119892 (119906 (119896)) =

1199031

sum119897=0

119892119897119906119897(119896) 119906 (119896) lt 0

(5)

where 119891119897and 119892

119897are parameters of the polynomial function

119906(119896) is the input 119908(119896) is the output of static nonlinearfunction119873

1 and 119903

1is the degree of the polynomial function

Define the switching function as

ℎ (119906) = 0 119906 ge 0

1 119906 lt 0(6)

Then the relation between inputs 119906(119896) and outputs119908(119896) of the input nonlinear block can be written as

119908 (119896) = 119891 (119906 (119896)) + (119892 (119906 (119896)) minus 119891 (119906 (119896))) ℎ (119906 (119896))

=

1199031

sum

119897=0

119891119897119906119897(119896) +

1199031

sum

119897=0

119901119897119906119897(119896) ℎ (119906 (119896))

(7)

where 119901119897= 119892119897minus 119891119897

The difference equationmodel 119871(119911) of the linear dynamicblock is described by an extended autoregressive (ARX)model as

119860(119911minus1) 119909 (119896) = 119911

minus119899119896119861 (119911minus1)119908 (119896) + V (119896) (8)

where 119908(119896) and 119909(119896) are the input and output of the lineardynamic block respectively V(119896) is white noise 119899

119896represents

the pure delay of the system and119860(119911minus1) and 119861(119911minus1) are scalarpolynomials in the unit delay operator 119911minus1

119860(119911minus1) = 1 + 119886

1119911minus1+ sdot sdot sdot + 119886

119899119886

119911minus119899119886

119861 (119911minus1) = 1198870+ 1198871119911minus1+ sdot sdot sdot + 119887

119899119887

119911minus119899119887

(9)

The output nonlinear block 1198732is described by a single

polynomials

119910 (119896) = 119902 (119909 (119896)) =

1199032

sum

119898=1

119902119898119909119898(119896) (10)

where 119902119898

is unknown parameter 1199032is the degree of the

polynomial function 1198732 and 119910(119896) is output of the entire

system and in this paper it represents the output velocityThe H-Wmodel of the system is depicted in Figure 4

ARX

1198731 1198732

119906(119896) 119908(119896) 119910(119896)

119871(119911)

119909(119896)

Figure 4 Schematic diagram of the H-Wmodel

4 Iterative Identification Algorithm

As we know the cascade mode of the models depicted inFigure 3 leads to composite mappings for example Ham-merstein model 119871(119873

1(119906(119896))) Wiener model 119873

2(119871(119906(119896)))

H-W model 1198732(119871(1198731(119906(119896)))) Substituting the mathematic

models of each block (ie (7) (8) and (10)) into the com-posite mappings directly leads to complex models which arestrongly nonlinear in both of the variables and the unknownparameters It is not appropriate for parameter estimation[20] In the following we apply the so-called key termseparation principle to simplify the H-Wmodel into a linear-in-parameters structure and then adopt a modified recursiveleast square algorithm with internal variable estimation toestimate both of the linear and nonlinear block parameterssimultaneously

41 Key Term Separation Principle Let 119891 119892 and ℎ be one-to-one mappings defined on nonempty sets 119880119883 and 119884 as

119891 119880 997888rarr 119883

119892 119883 997888rarr 119884

ℎ = 119892 ∘ 119891 119880 997888rarr 119884

(11)

Then the composite mapping ℎ can be given by

119910 (119905) = 119892 [119909 (119905)] = 119892 [119891 [119906 (119905)]] = ℎ [119906 (119905)] (12)

Thebasic idea of key term separation principle is a formofhalf-substitution suggested in [21] Suppose 119892 be an analyticnonlinear mapping which can be rewritten into the followingadditive form

119910 (119905) = 119909 (119905) + 119866 [119909 (119905)] (13)

Which consists of the key term 119909(119905) plus the remainder ofthe originalmapping assigned as119866(sdot) Rewrite the one-to-onemapping 119891

119909 (119905) = 119891 [119906 (119905)] (14)

We substitute (13) only into the first term in the right sideof (14) and then obtain the following mapping

119910 (119905) = 119891 [119906 (119905)] + 119866 [119909 (119905)] (15)

Equations (14) and (15) describe the mapping functionℎ in a compositional way This makes the inner mappingappears both explicitly and implicitly in the outer one whichmay be helpful for parameter identification Note that thisdecomposition technique can easily be extended to a moremultilayer composite mapping

Mathematical Problems in Engineering 5

42 Modified Least Square Algorithm In this section wedecompose the H-W model into a linear-in-parametersstructure by the key term separation principle and developa modified iterative least square algorithm with internalvariables estimation to identify all the unknown parametersof the H-W model We also apply this method to theHammerstein and Wiener models

According to the key term separation principle werewrite the output nonlinear block119873

2 that is (10) as

119910 (119896) = 1199021119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896) (16)

where the internal variable 119909(119896) is separated The dynamiclinear block 119871(119911) that is (8) can be rewritten as

119909 (119896) = 1198870119908 (119896) 119911

minus119899119896 + 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896)

(17)

where the internal variable 119908(119896) is separated Now to com-plete the sequential decomposition first we substitute (7)into (17) only for 119908(119896) in the first term and then substitutethe new equation (17) into (16) only for 119909(119896) in the first termagain The final output equation of the H-Wmodel will be

119910 (119896) = 11990211198870(119891 (119906 (119896)) + 119901 (119906 (119896)) ℎ (119906 (119896))) 119911

minus119899119896

+ 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896)

(18)

As the H-W model depicted in Figure 4 consists of threesubsystems in series the parameterization of the model isnot unique because many combinations of parameters can befound [22]Therefore one parameter in at least two blocks hasto be fixed in (18) Evidently the choices 119902

1= 1 and 119887

0= 1will

simplify the model descriptionThen the H-Wmodel can bewritten as

119910 (119896) =

1199031

sum

119897=0

119891119897119906119897(119896 minus 119899

119896) +

1199031

sum

119897=0

119901119897119906119897(119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

+

119899119887

sum

119894=1

119887119894119908 (119896 minus 119899

119896minus 119894) +

119899119886

sum

119895=1

119886119895119909 (119896 minus 119895) +

1199032

sum

119898=2

119902119898119909119898(119896)

(19)

Equation (19) is linear-in-parameters for given 119906(119896) 119909(119896)and 119908(119896) it can be written in the following least squareformat

119910 (119896) = Φ119879(119896 120579) 120579 (20)

where the internal variables 119908(119896) and 119909(119896) are estimated by(7) and (17) using the preceding estimated parameters duringeach iterative process and

Φ119879= [1 119906 (119896minus119899

119896) 119906

1199031 (119896minus119899

119896)

ℎ (119906 (119896minus119899119896)) 119906 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896))

1199061199031 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896)) 119908 (119896minus119899

119896minus1)

119908 (119896 minus 119899119896minus 119894) minus119909 (119896minus1) minus119909 (119896minus119895)

1199092(119896) 119909

1199032 (119896)]

120579 = [1198910 1198911 119891

1199031

1199010 1199011 119901

1199031

1198871 119887

119899119887

1198861 119886

119899119886

1199022 119902

1199032

] 119879

(21)

Nowwe apply themodified recursive least squaremethodwith iterative estimation of the internal variable to (20) [17]Minimizing the following least square criterion [12]

= arg min120579

119899

sum

119896=1

120582119899minus119896

[119910 (119896) minus Φ119879

(119896) 120579]

2

(22)

where 120582 ≦ 1 is the forgetting factor the formulas ofthe recursive identification algorithm supplemented withinternal variable estimation are as follows

(119896) = (119896 minus 1) +

P (119896 minus 1) Φ (119896) [119910 (119896) minus Φ119879

(119896) (119896 minus 1)]

120582 + Φ119879

(119896)P (119896 minus 1) Φ (119896)

(23)

P (119896) = P (119896 minus 1)

120582minusP (119896 minus 1) Φ (119896) Φ

119879

(119896)P (119896 minus 1)

1 + Φ119879

(119896)P (119896 minus 1) Φ (119896) 120582

(24)

(119896) =

1199031

sum

119897=0

119891119897(119896 minus 1) 119906

119897(119896) +

1199031

sum

119897=0

119897(119896 minus 1) 119906

119897(119896) ℎ (119906 (119896))

(25)

(119896) = (119896 minus 119899119896) +

119899119887

sum

119894=1

119894(119896 minus 1) (119896 minus 119899

119896minus 119894)

minus

119899119886

sum

119895=1

119895(119896 minus 1) 119909 (119896 minus 119895)

(26)

Φ (119896) = [1 119906 (119896 minus 119899119896) 119906

1199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

119906 (119896 minus 119899119896) ℎ (119906 (119896 minus 119899

119896))

1199061199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

(119896 minus 119899119896minus 1) (119896 minus 119899

119896minus 119894)

minus (119896 minus 1) minus (119896 minus 119895) 2(119896)

1199032 (119896)]119879

(27)

where P(0) = 120583I I is unit matrix and 0 lt 120583 lt infin

6 Mathematical Problems in Engineering

Computer

Pressure transducer

Inclinometer

Low level controller

Electrohydraulic valve

(a)

Inclinometer

Main valve

Proportionalrelief valve

119870ℎ

Mainpump

Pilotpump

ControllerReference

input

(b)

Figure 5 Experimental prototype machine

Inclinometer

Proportionalrelief valve

120579119889+

minus

119906 Digitalcontroller

Pilot pump Main pump

119876Mainvalve

Hydraulicactuator

Mechanism 120579DA

AD

I P

Figure 6 Electrohydraulic position servocontrol system

In conclusion the iterative identification algorithm canbe presented as follwos

Step 1 Set the initial values of 119909(0) 119908(0) 119906(0) and P(0)

Step 2 Estimate the parameter (119896) by algorithm (23) andcalculate P(119896) by (24)

Step 3 Estimate the internal variables (119896) and (119896) by (25)and (26) using the recent estimates of model parameters (119896)

Step 4 Update the values of Φ(119896) by (27)

Step 5 Return to Step 2 until the parameter estimates con-verge to constant values

5 Experiment

51 Experimental Environment A hydraulic excavator wasretrofitted to be controlled by computer in our laboratory[23] Figure 5 shows the prototype machine whose manualpilot hydraulic control system was replaced by electrohy-draulic proportional control system inclinometers and pres-sure transducers were also installed on the excavator arms

for position and force servocontrol Schematic diagram of theelectrohydraulic servosystem is shown in Figure 6

52 Experimental Results In order to obtain the nonlinearcharacteristics of the system when changing the directionsand to obtain sufficient excitation we adopted a multisineinput signal which contained the frequency of 005Hz 01 Hz02Hz 04Hz and 05Hz to the identification experimentsThe sample rate was chosen to be 20Hz on the machineThe input signal and angle output were obtained from thecomputer of the experiment machine Ten groups of inputand output signals with time duration of 55 seconds weresampled in the repeated experiments the averaged measure-ment results are shown in Figure 7 Finally we calculate theoutput angle velocity by numerical differentiation

We set the parameters 119899119886= 3 119899

119887= 2 119899

119896= 8 1199031= 1199032= 3

119909(0) = 0 119908(0) = 0 119906(0) = 0 and P(0) = 106I Note that

lower forgetting factor 120582 is useful for reducing the influencesof old date while a value of 120582 close to 1 is less sensitive todisturbance Therefore we chose the forgetting factor to be120582 = 098 during the first 200 samples and 120582 = 1 otherwiseCompiling the developed iterative least square algorithm inMATLAB to identify theARXmodel containing only the119871(119911)

Mathematical Problems in Engineering 7

420

minus2

minus40 10 20 30 40 50

Input

Time (119905s)

Inpu

t (119906

v)

(a)

0 10 20 30 40 50

Angle output

2

15

1

Ang

le (r

ad)

Time (119905s)

(b)

Figure 7 Input and output signals of the identification experiment

The measured velocityPredicted by Hammerstein-Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(a) Hammerstein-Wiener prediction result

The measured velocityPredicted by Hammerstein

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(b) Hammerstein prediction result

The measured velocityPredicted by Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(c) Wiener prediction result

The measured velocityPredicted by ARX

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(d) ARX prediction result

Figure 8 Comparison results of the identified models

blockHammersteinmodel consisting of1198731and119871(119911)Wiener

model consisting of 119871(119911) and1198732 and H-Wmodel consisting

of1198731 119871(119911) and119873

2 respectively we obtain the identification

results shown in Table 1We use the identified models to predict the tracking

velocity of a general trajectory The comparative results areshown in Figures 8 and 9 They demonstrate that the H-Wand Hammerstein models which contain the input nonlinearblock with two-segment polynomial nonlinearities capturethe actual system well while the Wiener and ARX modelscannot approximate the actual system well Themean-squareerrors (MSE) of the identifiedmodels in Table 2 show that the

prediction error of the H-W model is about 13 54 and58 less than the Hammerstein Wiener and ARX modelsrespectively

6 Conclusion

This paper investigates the nonlinear modeling and identi-fication of an electrohydraulic control system We develop atheoretic state spacemodel for system analysis propose anH-W model for the highly nonlinear system based on a deeplyphysical insight into the actual system and apply a modified

8 Mathematical Problems in Engineering

05

04

03

02

01

0

minus01

minus02

minus03

0 2 4 6 8 10 12 14 16 18

Estim

atio

n re

sidua

(rad

s)

Residua of Hammerstein-WienerResidua of Hammerstein

Residua of WeinerResidua of ARX

Time (119905s)

Figure 9 Comparison of the estimation residua

Table 1 The identification results

Parameters Model typeH-W Hammerstein Wiener ARX

1198870

1 1 minus00358 minus002761198871

00349 00105 00151 001241198872

00617 00147 minus00069 005141198861

minus00309 minus00596 00069 minus006611198862

minus00598 minus00613 minus00075 minus006531198863

minus00897 minus00610 minus00081 minus006191198910

00109 00027 mdash mdash1198911

minus14619 times 10minus5 00012 mdash mdash1198912

minus00050 minus00012 mdash mdash1198913

00052 00045 mdash mdash1199010

minus00165 minus00055 mdash mdash1199011

00033 00013 mdash mdash1199012

00104 00036 mdash mdash1199013

minus00010 minus86858 times 10minus4 mdash mdash1199021

1 mdash 1 mdash1199022

00050 mdash 00013 mdash1199023

63562 times 10minus4 mdash 26895 times 10minus4 mdash

Table 2 MSE of the identified models

Errors Model typeH-W Hammerstein Wiener ARX

120575r sdot sminus1 00321 00417 01081 01218

recursive least square method with internal variables esti-mation to identify its parameters The main findings of thepaper include the follwoing (1) the proposed H-W modelsimplifies the identification procedure because it only uses theinput and output signals to identify all the parameters (2)TheH-W model containing the input nonlinear block with two-segment polynomial nonlinearities captures the actual systemverywell As shown by the comparative results the prediction

error of the H-Wmodel is about 13 54 and 58 less thanthe Hammerstein Wiener and ARX models respectivelyThe results provide a physical insight into the nonlinearcharacteristics of the actual system which is important forsystem analyzing monitoring and diagnosis Future workincludes addressing uncertain and fuzzy properties of thesystem [24 25] and extending the model for a wider rangeof equipment [26]

Acknowledgment

Thisworkwas supported in part by grants fromNational Nat-ural Science Foundation (Grant nos 51175511 and 61105073)of China

References

[1] H E Merritt Hydraulic Control System John Wiley amp SonsNew York USA 1967

[2] B Yao F Bu J Reedy and G T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[3] S Wen W Zheng J Zhu X Li and S Chen ldquoElman fuzzyadaptive control for obstacle avoidance of mobile robots usinghybrid forceposition incorporationrdquo IEEE Transactions onSystems Man and Cybernetics C vol 42 no 4 pp 603ndash6082012

[4] S Y Chen J Zhang H Zhang N M Kwok and Y F Li ldquoIntel-ligent lighting control for vision-based robotic manipulationrdquoIEEE Transactions on Industrial Electronics vol 59 no 8 pp3254ndash3263 2012

[5] Q H He P Hao and D Q Zhang ldquoModeling and parameterestimation for hydraulic system of excavatorrsquos armrdquo Journal ofCentral South University of Technology vol 15 no 3 pp 382ndash386 2008

[6] K Ziaei and N Sepehri ldquoModeling and identification ofelectrohydraulic servosrdquo Mechatronics vol 10 no 7 pp 761ndash772 2000

[7] L Wang B Wu R Du and S Yang ldquoNonlinear dynamic char-acteristics of moving hydraulic cylinderrdquo Chinese Journal ofMechanical Engineering vol 43 no 12 pp 12ndash19 2007

[8] M Jelali and H Schwarz ldquoNonlinear identification of hydraulicservo-drive systemsrdquo IEEE Control Systems Magazine vol 15no 5 pp 17ndash22 1995

[9] S Kleinsteuber and N Sepehri ldquoA polynomial network mod-eling approach to a class of large-scale hydraulic systemsrdquoComputers and Electrical Engineering vol 22 no 2 pp 151ndash1681996

[10] H Yousefi H Handroos and A Soleymani ldquoApplication of dif-ferential evolution in system identification of a servo-hydraulicsystem with a flexible loadrdquoMechatronics vol 18 no 9 pp 513ndash528 2008

[11] F Giri and E W Bai Block-Oriented Nonlinear System Identi-fication vol 404 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2010

[12] B -J Kwak A E Yagle and J A Levitt ldquoNonlinear systemidentification of hydraulic actuator friction dynamics using aHammerstein modelrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing vol 4 pp1933ndash1936 Seattle Wash USA 1998

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

of hydraulic cylinder and the force equilibrium equation ofhydraulic cylinder [2] Define the state variables as

[1199091 1199092119909311990941199095]119879

≜ [ℎ ℎ 11990111199012119909V]119879

(2)

The entire system can be modeled as the followingnonlinear state space model [15]

1= 1199092

2=11986011199093

119872minus11986021199094

119872minus119865119891(1199092)

119872

minus119865119904(1199091)

119872minus119865119888(1199092)

119872minus119865119897

119872

3= minus

12057311989011986011199092

1198811

minus120573119890(119862119894+ 119862119890) 1199093

1198811

+1205731198901198621198941199094

1198811

+1205731198901198921(119909)

1198811

1199095

4=12057311989011986021199092

1198812

+1205731198901198621198941199093

1198812

minus120573119890(119862119894+ 119862119890) 1199094

1198812

minus1205731198901198922(119909)

1198812

1199095

5=119896V

120591V

119868V minus1199095

120591V

(3)

1198921(119909) = sgn((1 + sgn (119909

5))119901119904

2minus sgn (119909

5) 1199093)

times 1198621198891119882radic

2

120588((1 + sgn (119909

5))119901119904

2minus sgn (119909

5) 1199093)

1198922(119909) = sgn((1minussgn (119909

5))119901119904

2+sgn (119909

5) 1199094)

times 1198621198892119882radic

2

120588((1minussgn (119909

5))119901119904

2+sgn (119909

5) 1199094)

(4)

where 120573119890is the effective bulk modulus119881

1and119881

2are effective

volumes of the two chambers 119862119894and 119862

119890are internal and

external leakage coefficients 119882 is the area gradient of thevalve orifice and 119862

1198891and 119862

1198892are flow discharge coefficients

of the spool valveSeveral physical phenomena have been taken into consid-

eration in the above model for example nonlinear friction119865119891 nonlinear spring force 119865

119904 viscous force 119865

119888 uncertain load

119865119897 discontinuous flow discharge 119892

119894 oil compliance internal

leakage and external leakage From the theoretic modelingof the electrohydraulic control system we can see that thesystem is a highly nonlinear system containing complexfeatures such as the dead band nonlinearity saturationsquared pressure drop and asymmetric response property

There are also some hard-to-model nonlinearities in(3) such as nonlinear friction nonlinear spring force anduncertain external disturbances So modeling this systemjust by physical laws fails to approximate the actual systemFurthermore identification of the unknown parameters in(3) is hard due to its demand on internal states measurementIn the followingwe adopt anH-Wmodel tomodel this highly

1198731(∙)119906(119896) 119908(119896) 119910(119896)119871(119911)

120584(119896)

(a) Hammerstein model

1198732 (∙)119906(119896) 119910(119896)119871(119911)

119909(119896)

120584(119896)

(b) Wiener model

1198731(∙) 1198732 (∙)119906(119896) 119908(119896) 119910(119896)119871(119911)

119909(119896)

120584(119896)

(c) Hammerstein-Wiener model

Figure 3 Hammerstein and Wiener models

nonlinear dynamic system The H-W model is a flexibleblack-box model based on the physical insight into theactual system We identify the parameters of the H-Wmodelusing the input and output signals which can simplify theidentification process and improve the prediction accuracy byless sensors and noise

3 Hammerstein-Wiener Model

The ldquouniversalrdquo nonlinear black-box methods such as neuralnetworks Volterra series and fuzzy models are widely usedto model complex nonlinear systems Most of these meth-ods can avoid unmodeled dynamics in the aforementionedmathematical model [16 17] However these models do notprovide deep insight into the nonlinear characteristics ofthe actual system which is important in system analyzingmonitoring diagnosis and controller design In comparisonthe Hammerstein-Wiener (H-W) model possesses the flexi-bility to capture all relevant nonlinear phenomena as well asthe physical insight into the actual system In this sectionwe develop an H-W model to describe the electrohydrauliccontrol system

The H-W model is composed of an internal lineardynamic block and two static nonlinear blocks it is thecombination of Hammerstein and Wiener model The Ham-merstein model is a nonlinear model with a static nonlinearblock followed by a linear dynamic block as shown inFigure 3(a) and this N-L type of model may account foractuator nonlinearities and other input nonlinear effectsThe Wiener model has linear dynamic block followed bya nonlinear block as shown in Figure 3(b) and this L-Ntype of model mainly accounts for sensor nonlinearitiesand output nonlinear effects A series combination of aHammerstein and a Wiener model yields the H-W modelas shown in Figure 3(c) and this N-L-N type of model hasboth characteristics of the Hammerstein andWiener modelsMoreover all of the three models have proved to be able toaccurately describe a wide variety of nonlinear systems in [9]

According to the nonlinearities of the abovementionedelectrohydraulic control system for example dead band

4 Mathematical Problems in Engineering

saturation nonlinear friction nonlinear spring force andasymmetric dynamics of the cylinder we describe the inputnonlinearity (119873

1) block of the models in Figure 3 by a

two-segment polynomial nonlinearities The two-segmentpolynomial nonlinearities have the advantage of describinga system whose dynamic properties differ significantly at thepositive and negative directions [18] it has less parameters tobe estimated than a single polynomial and piecewise linearmodels [19] It can be written as

119908 (119896) =

119891 (119906 (119896)) =

1199031

sum119897=0

119891119897119906119897(119896) 119906 (119896) ge 0

119892 (119906 (119896)) =

1199031

sum119897=0

119892119897119906119897(119896) 119906 (119896) lt 0

(5)

where 119891119897and 119892

119897are parameters of the polynomial function

119906(119896) is the input 119908(119896) is the output of static nonlinearfunction119873

1 and 119903

1is the degree of the polynomial function

Define the switching function as

ℎ (119906) = 0 119906 ge 0

1 119906 lt 0(6)

Then the relation between inputs 119906(119896) and outputs119908(119896) of the input nonlinear block can be written as

119908 (119896) = 119891 (119906 (119896)) + (119892 (119906 (119896)) minus 119891 (119906 (119896))) ℎ (119906 (119896))

=

1199031

sum

119897=0

119891119897119906119897(119896) +

1199031

sum

119897=0

119901119897119906119897(119896) ℎ (119906 (119896))

(7)

where 119901119897= 119892119897minus 119891119897

The difference equationmodel 119871(119911) of the linear dynamicblock is described by an extended autoregressive (ARX)model as

119860(119911minus1) 119909 (119896) = 119911

minus119899119896119861 (119911minus1)119908 (119896) + V (119896) (8)

where 119908(119896) and 119909(119896) are the input and output of the lineardynamic block respectively V(119896) is white noise 119899

119896represents

the pure delay of the system and119860(119911minus1) and 119861(119911minus1) are scalarpolynomials in the unit delay operator 119911minus1

119860(119911minus1) = 1 + 119886

1119911minus1+ sdot sdot sdot + 119886

119899119886

119911minus119899119886

119861 (119911minus1) = 1198870+ 1198871119911minus1+ sdot sdot sdot + 119887

119899119887

119911minus119899119887

(9)

The output nonlinear block 1198732is described by a single

polynomials

119910 (119896) = 119902 (119909 (119896)) =

1199032

sum

119898=1

119902119898119909119898(119896) (10)

where 119902119898

is unknown parameter 1199032is the degree of the

polynomial function 1198732 and 119910(119896) is output of the entire

system and in this paper it represents the output velocityThe H-Wmodel of the system is depicted in Figure 4

ARX

1198731 1198732

119906(119896) 119908(119896) 119910(119896)

119871(119911)

119909(119896)

Figure 4 Schematic diagram of the H-Wmodel

4 Iterative Identification Algorithm

As we know the cascade mode of the models depicted inFigure 3 leads to composite mappings for example Ham-merstein model 119871(119873

1(119906(119896))) Wiener model 119873

2(119871(119906(119896)))

H-W model 1198732(119871(1198731(119906(119896)))) Substituting the mathematic

models of each block (ie (7) (8) and (10)) into the com-posite mappings directly leads to complex models which arestrongly nonlinear in both of the variables and the unknownparameters It is not appropriate for parameter estimation[20] In the following we apply the so-called key termseparation principle to simplify the H-Wmodel into a linear-in-parameters structure and then adopt a modified recursiveleast square algorithm with internal variable estimation toestimate both of the linear and nonlinear block parameterssimultaneously

41 Key Term Separation Principle Let 119891 119892 and ℎ be one-to-one mappings defined on nonempty sets 119880119883 and 119884 as

119891 119880 997888rarr 119883

119892 119883 997888rarr 119884

ℎ = 119892 ∘ 119891 119880 997888rarr 119884

(11)

Then the composite mapping ℎ can be given by

119910 (119905) = 119892 [119909 (119905)] = 119892 [119891 [119906 (119905)]] = ℎ [119906 (119905)] (12)

Thebasic idea of key term separation principle is a formofhalf-substitution suggested in [21] Suppose 119892 be an analyticnonlinear mapping which can be rewritten into the followingadditive form

119910 (119905) = 119909 (119905) + 119866 [119909 (119905)] (13)

Which consists of the key term 119909(119905) plus the remainder ofthe originalmapping assigned as119866(sdot) Rewrite the one-to-onemapping 119891

119909 (119905) = 119891 [119906 (119905)] (14)

We substitute (13) only into the first term in the right sideof (14) and then obtain the following mapping

119910 (119905) = 119891 [119906 (119905)] + 119866 [119909 (119905)] (15)

Equations (14) and (15) describe the mapping functionℎ in a compositional way This makes the inner mappingappears both explicitly and implicitly in the outer one whichmay be helpful for parameter identification Note that thisdecomposition technique can easily be extended to a moremultilayer composite mapping

Mathematical Problems in Engineering 5

42 Modified Least Square Algorithm In this section wedecompose the H-W model into a linear-in-parametersstructure by the key term separation principle and developa modified iterative least square algorithm with internalvariables estimation to identify all the unknown parametersof the H-W model We also apply this method to theHammerstein and Wiener models

According to the key term separation principle werewrite the output nonlinear block119873

2 that is (10) as

119910 (119896) = 1199021119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896) (16)

where the internal variable 119909(119896) is separated The dynamiclinear block 119871(119911) that is (8) can be rewritten as

119909 (119896) = 1198870119908 (119896) 119911

minus119899119896 + 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896)

(17)

where the internal variable 119908(119896) is separated Now to com-plete the sequential decomposition first we substitute (7)into (17) only for 119908(119896) in the first term and then substitutethe new equation (17) into (16) only for 119909(119896) in the first termagain The final output equation of the H-Wmodel will be

119910 (119896) = 11990211198870(119891 (119906 (119896)) + 119901 (119906 (119896)) ℎ (119906 (119896))) 119911

minus119899119896

+ 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896)

(18)

As the H-W model depicted in Figure 4 consists of threesubsystems in series the parameterization of the model isnot unique because many combinations of parameters can befound [22]Therefore one parameter in at least two blocks hasto be fixed in (18) Evidently the choices 119902

1= 1 and 119887

0= 1will

simplify the model descriptionThen the H-Wmodel can bewritten as

119910 (119896) =

1199031

sum

119897=0

119891119897119906119897(119896 minus 119899

119896) +

1199031

sum

119897=0

119901119897119906119897(119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

+

119899119887

sum

119894=1

119887119894119908 (119896 minus 119899

119896minus 119894) +

119899119886

sum

119895=1

119886119895119909 (119896 minus 119895) +

1199032

sum

119898=2

119902119898119909119898(119896)

(19)

Equation (19) is linear-in-parameters for given 119906(119896) 119909(119896)and 119908(119896) it can be written in the following least squareformat

119910 (119896) = Φ119879(119896 120579) 120579 (20)

where the internal variables 119908(119896) and 119909(119896) are estimated by(7) and (17) using the preceding estimated parameters duringeach iterative process and

Φ119879= [1 119906 (119896minus119899

119896) 119906

1199031 (119896minus119899

119896)

ℎ (119906 (119896minus119899119896)) 119906 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896))

1199061199031 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896)) 119908 (119896minus119899

119896minus1)

119908 (119896 minus 119899119896minus 119894) minus119909 (119896minus1) minus119909 (119896minus119895)

1199092(119896) 119909

1199032 (119896)]

120579 = [1198910 1198911 119891

1199031

1199010 1199011 119901

1199031

1198871 119887

119899119887

1198861 119886

119899119886

1199022 119902

1199032

] 119879

(21)

Nowwe apply themodified recursive least squaremethodwith iterative estimation of the internal variable to (20) [17]Minimizing the following least square criterion [12]

= arg min120579

119899

sum

119896=1

120582119899minus119896

[119910 (119896) minus Φ119879

(119896) 120579]

2

(22)

where 120582 ≦ 1 is the forgetting factor the formulas ofthe recursive identification algorithm supplemented withinternal variable estimation are as follows

(119896) = (119896 minus 1) +

P (119896 minus 1) Φ (119896) [119910 (119896) minus Φ119879

(119896) (119896 minus 1)]

120582 + Φ119879

(119896)P (119896 minus 1) Φ (119896)

(23)

P (119896) = P (119896 minus 1)

120582minusP (119896 minus 1) Φ (119896) Φ

119879

(119896)P (119896 minus 1)

1 + Φ119879

(119896)P (119896 minus 1) Φ (119896) 120582

(24)

(119896) =

1199031

sum

119897=0

119891119897(119896 minus 1) 119906

119897(119896) +

1199031

sum

119897=0

119897(119896 minus 1) 119906

119897(119896) ℎ (119906 (119896))

(25)

(119896) = (119896 minus 119899119896) +

119899119887

sum

119894=1

119894(119896 minus 1) (119896 minus 119899

119896minus 119894)

minus

119899119886

sum

119895=1

119895(119896 minus 1) 119909 (119896 minus 119895)

(26)

Φ (119896) = [1 119906 (119896 minus 119899119896) 119906

1199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

119906 (119896 minus 119899119896) ℎ (119906 (119896 minus 119899

119896))

1199061199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

(119896 minus 119899119896minus 1) (119896 minus 119899

119896minus 119894)

minus (119896 minus 1) minus (119896 minus 119895) 2(119896)

1199032 (119896)]119879

(27)

where P(0) = 120583I I is unit matrix and 0 lt 120583 lt infin

6 Mathematical Problems in Engineering

Computer

Pressure transducer

Inclinometer

Low level controller

Electrohydraulic valve

(a)

Inclinometer

Main valve

Proportionalrelief valve

119870ℎ

Mainpump

Pilotpump

ControllerReference

input

(b)

Figure 5 Experimental prototype machine

Inclinometer

Proportionalrelief valve

120579119889+

minus

119906 Digitalcontroller

Pilot pump Main pump

119876Mainvalve

Hydraulicactuator

Mechanism 120579DA

AD

I P

Figure 6 Electrohydraulic position servocontrol system

In conclusion the iterative identification algorithm canbe presented as follwos

Step 1 Set the initial values of 119909(0) 119908(0) 119906(0) and P(0)

Step 2 Estimate the parameter (119896) by algorithm (23) andcalculate P(119896) by (24)

Step 3 Estimate the internal variables (119896) and (119896) by (25)and (26) using the recent estimates of model parameters (119896)

Step 4 Update the values of Φ(119896) by (27)

Step 5 Return to Step 2 until the parameter estimates con-verge to constant values

5 Experiment

51 Experimental Environment A hydraulic excavator wasretrofitted to be controlled by computer in our laboratory[23] Figure 5 shows the prototype machine whose manualpilot hydraulic control system was replaced by electrohy-draulic proportional control system inclinometers and pres-sure transducers were also installed on the excavator arms

for position and force servocontrol Schematic diagram of theelectrohydraulic servosystem is shown in Figure 6

52 Experimental Results In order to obtain the nonlinearcharacteristics of the system when changing the directionsand to obtain sufficient excitation we adopted a multisineinput signal which contained the frequency of 005Hz 01 Hz02Hz 04Hz and 05Hz to the identification experimentsThe sample rate was chosen to be 20Hz on the machineThe input signal and angle output were obtained from thecomputer of the experiment machine Ten groups of inputand output signals with time duration of 55 seconds weresampled in the repeated experiments the averaged measure-ment results are shown in Figure 7 Finally we calculate theoutput angle velocity by numerical differentiation

We set the parameters 119899119886= 3 119899

119887= 2 119899

119896= 8 1199031= 1199032= 3

119909(0) = 0 119908(0) = 0 119906(0) = 0 and P(0) = 106I Note that

lower forgetting factor 120582 is useful for reducing the influencesof old date while a value of 120582 close to 1 is less sensitive todisturbance Therefore we chose the forgetting factor to be120582 = 098 during the first 200 samples and 120582 = 1 otherwiseCompiling the developed iterative least square algorithm inMATLAB to identify theARXmodel containing only the119871(119911)

Mathematical Problems in Engineering 7

420

minus2

minus40 10 20 30 40 50

Input

Time (119905s)

Inpu

t (119906

v)

(a)

0 10 20 30 40 50

Angle output

2

15

1

Ang

le (r

ad)

Time (119905s)

(b)

Figure 7 Input and output signals of the identification experiment

The measured velocityPredicted by Hammerstein-Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(a) Hammerstein-Wiener prediction result

The measured velocityPredicted by Hammerstein

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(b) Hammerstein prediction result

The measured velocityPredicted by Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(c) Wiener prediction result

The measured velocityPredicted by ARX

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(d) ARX prediction result

Figure 8 Comparison results of the identified models

blockHammersteinmodel consisting of1198731and119871(119911)Wiener

model consisting of 119871(119911) and1198732 and H-Wmodel consisting

of1198731 119871(119911) and119873

2 respectively we obtain the identification

results shown in Table 1We use the identified models to predict the tracking

velocity of a general trajectory The comparative results areshown in Figures 8 and 9 They demonstrate that the H-Wand Hammerstein models which contain the input nonlinearblock with two-segment polynomial nonlinearities capturethe actual system well while the Wiener and ARX modelscannot approximate the actual system well Themean-squareerrors (MSE) of the identifiedmodels in Table 2 show that the

prediction error of the H-W model is about 13 54 and58 less than the Hammerstein Wiener and ARX modelsrespectively

6 Conclusion

This paper investigates the nonlinear modeling and identi-fication of an electrohydraulic control system We develop atheoretic state spacemodel for system analysis propose anH-W model for the highly nonlinear system based on a deeplyphysical insight into the actual system and apply a modified

8 Mathematical Problems in Engineering

05

04

03

02

01

0

minus01

minus02

minus03

0 2 4 6 8 10 12 14 16 18

Estim

atio

n re

sidua

(rad

s)

Residua of Hammerstein-WienerResidua of Hammerstein

Residua of WeinerResidua of ARX

Time (119905s)

Figure 9 Comparison of the estimation residua

Table 1 The identification results

Parameters Model typeH-W Hammerstein Wiener ARX

1198870

1 1 minus00358 minus002761198871

00349 00105 00151 001241198872

00617 00147 minus00069 005141198861

minus00309 minus00596 00069 minus006611198862

minus00598 minus00613 minus00075 minus006531198863

minus00897 minus00610 minus00081 minus006191198910

00109 00027 mdash mdash1198911

minus14619 times 10minus5 00012 mdash mdash1198912

minus00050 minus00012 mdash mdash1198913

00052 00045 mdash mdash1199010

minus00165 minus00055 mdash mdash1199011

00033 00013 mdash mdash1199012

00104 00036 mdash mdash1199013

minus00010 minus86858 times 10minus4 mdash mdash1199021

1 mdash 1 mdash1199022

00050 mdash 00013 mdash1199023

63562 times 10minus4 mdash 26895 times 10minus4 mdash

Table 2 MSE of the identified models

Errors Model typeH-W Hammerstein Wiener ARX

120575r sdot sminus1 00321 00417 01081 01218

recursive least square method with internal variables esti-mation to identify its parameters The main findings of thepaper include the follwoing (1) the proposed H-W modelsimplifies the identification procedure because it only uses theinput and output signals to identify all the parameters (2)TheH-W model containing the input nonlinear block with two-segment polynomial nonlinearities captures the actual systemverywell As shown by the comparative results the prediction

error of the H-Wmodel is about 13 54 and 58 less thanthe Hammerstein Wiener and ARX models respectivelyThe results provide a physical insight into the nonlinearcharacteristics of the actual system which is important forsystem analyzing monitoring and diagnosis Future workincludes addressing uncertain and fuzzy properties of thesystem [24 25] and extending the model for a wider rangeof equipment [26]

Acknowledgment

Thisworkwas supported in part by grants fromNational Nat-ural Science Foundation (Grant nos 51175511 and 61105073)of China

References

[1] H E Merritt Hydraulic Control System John Wiley amp SonsNew York USA 1967

[2] B Yao F Bu J Reedy and G T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[3] S Wen W Zheng J Zhu X Li and S Chen ldquoElman fuzzyadaptive control for obstacle avoidance of mobile robots usinghybrid forceposition incorporationrdquo IEEE Transactions onSystems Man and Cybernetics C vol 42 no 4 pp 603ndash6082012

[4] S Y Chen J Zhang H Zhang N M Kwok and Y F Li ldquoIntel-ligent lighting control for vision-based robotic manipulationrdquoIEEE Transactions on Industrial Electronics vol 59 no 8 pp3254ndash3263 2012

[5] Q H He P Hao and D Q Zhang ldquoModeling and parameterestimation for hydraulic system of excavatorrsquos armrdquo Journal ofCentral South University of Technology vol 15 no 3 pp 382ndash386 2008

[6] K Ziaei and N Sepehri ldquoModeling and identification ofelectrohydraulic servosrdquo Mechatronics vol 10 no 7 pp 761ndash772 2000

[7] L Wang B Wu R Du and S Yang ldquoNonlinear dynamic char-acteristics of moving hydraulic cylinderrdquo Chinese Journal ofMechanical Engineering vol 43 no 12 pp 12ndash19 2007

[8] M Jelali and H Schwarz ldquoNonlinear identification of hydraulicservo-drive systemsrdquo IEEE Control Systems Magazine vol 15no 5 pp 17ndash22 1995

[9] S Kleinsteuber and N Sepehri ldquoA polynomial network mod-eling approach to a class of large-scale hydraulic systemsrdquoComputers and Electrical Engineering vol 22 no 2 pp 151ndash1681996

[10] H Yousefi H Handroos and A Soleymani ldquoApplication of dif-ferential evolution in system identification of a servo-hydraulicsystem with a flexible loadrdquoMechatronics vol 18 no 9 pp 513ndash528 2008

[11] F Giri and E W Bai Block-Oriented Nonlinear System Identi-fication vol 404 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2010

[12] B -J Kwak A E Yagle and J A Levitt ldquoNonlinear systemidentification of hydraulic actuator friction dynamics using aHammerstein modelrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing vol 4 pp1933ndash1936 Seattle Wash USA 1998

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

saturation nonlinear friction nonlinear spring force andasymmetric dynamics of the cylinder we describe the inputnonlinearity (119873

1) block of the models in Figure 3 by a

two-segment polynomial nonlinearities The two-segmentpolynomial nonlinearities have the advantage of describinga system whose dynamic properties differ significantly at thepositive and negative directions [18] it has less parameters tobe estimated than a single polynomial and piecewise linearmodels [19] It can be written as

119908 (119896) =

119891 (119906 (119896)) =

1199031

sum119897=0

119891119897119906119897(119896) 119906 (119896) ge 0

119892 (119906 (119896)) =

1199031

sum119897=0

119892119897119906119897(119896) 119906 (119896) lt 0

(5)

where 119891119897and 119892

119897are parameters of the polynomial function

119906(119896) is the input 119908(119896) is the output of static nonlinearfunction119873

1 and 119903

1is the degree of the polynomial function

Define the switching function as

ℎ (119906) = 0 119906 ge 0

1 119906 lt 0(6)

Then the relation between inputs 119906(119896) and outputs119908(119896) of the input nonlinear block can be written as

119908 (119896) = 119891 (119906 (119896)) + (119892 (119906 (119896)) minus 119891 (119906 (119896))) ℎ (119906 (119896))

=

1199031

sum

119897=0

119891119897119906119897(119896) +

1199031

sum

119897=0

119901119897119906119897(119896) ℎ (119906 (119896))

(7)

where 119901119897= 119892119897minus 119891119897

The difference equationmodel 119871(119911) of the linear dynamicblock is described by an extended autoregressive (ARX)model as

119860(119911minus1) 119909 (119896) = 119911

minus119899119896119861 (119911minus1)119908 (119896) + V (119896) (8)

where 119908(119896) and 119909(119896) are the input and output of the lineardynamic block respectively V(119896) is white noise 119899

119896represents

the pure delay of the system and119860(119911minus1) and 119861(119911minus1) are scalarpolynomials in the unit delay operator 119911minus1

119860(119911minus1) = 1 + 119886

1119911minus1+ sdot sdot sdot + 119886

119899119886

119911minus119899119886

119861 (119911minus1) = 1198870+ 1198871119911minus1+ sdot sdot sdot + 119887

119899119887

119911minus119899119887

(9)

The output nonlinear block 1198732is described by a single

polynomials

119910 (119896) = 119902 (119909 (119896)) =

1199032

sum

119898=1

119902119898119909119898(119896) (10)

where 119902119898

is unknown parameter 1199032is the degree of the

polynomial function 1198732 and 119910(119896) is output of the entire

system and in this paper it represents the output velocityThe H-Wmodel of the system is depicted in Figure 4

ARX

1198731 1198732

119906(119896) 119908(119896) 119910(119896)

119871(119911)

119909(119896)

Figure 4 Schematic diagram of the H-Wmodel

4 Iterative Identification Algorithm

As we know the cascade mode of the models depicted inFigure 3 leads to composite mappings for example Ham-merstein model 119871(119873

1(119906(119896))) Wiener model 119873

2(119871(119906(119896)))

H-W model 1198732(119871(1198731(119906(119896)))) Substituting the mathematic

models of each block (ie (7) (8) and (10)) into the com-posite mappings directly leads to complex models which arestrongly nonlinear in both of the variables and the unknownparameters It is not appropriate for parameter estimation[20] In the following we apply the so-called key termseparation principle to simplify the H-Wmodel into a linear-in-parameters structure and then adopt a modified recursiveleast square algorithm with internal variable estimation toestimate both of the linear and nonlinear block parameterssimultaneously

41 Key Term Separation Principle Let 119891 119892 and ℎ be one-to-one mappings defined on nonempty sets 119880119883 and 119884 as

119891 119880 997888rarr 119883

119892 119883 997888rarr 119884

ℎ = 119892 ∘ 119891 119880 997888rarr 119884

(11)

Then the composite mapping ℎ can be given by

119910 (119905) = 119892 [119909 (119905)] = 119892 [119891 [119906 (119905)]] = ℎ [119906 (119905)] (12)

Thebasic idea of key term separation principle is a formofhalf-substitution suggested in [21] Suppose 119892 be an analyticnonlinear mapping which can be rewritten into the followingadditive form

119910 (119905) = 119909 (119905) + 119866 [119909 (119905)] (13)

Which consists of the key term 119909(119905) plus the remainder ofthe originalmapping assigned as119866(sdot) Rewrite the one-to-onemapping 119891

119909 (119905) = 119891 [119906 (119905)] (14)

We substitute (13) only into the first term in the right sideof (14) and then obtain the following mapping

119910 (119905) = 119891 [119906 (119905)] + 119866 [119909 (119905)] (15)

Equations (14) and (15) describe the mapping functionℎ in a compositional way This makes the inner mappingappears both explicitly and implicitly in the outer one whichmay be helpful for parameter identification Note that thisdecomposition technique can easily be extended to a moremultilayer composite mapping

Mathematical Problems in Engineering 5

42 Modified Least Square Algorithm In this section wedecompose the H-W model into a linear-in-parametersstructure by the key term separation principle and developa modified iterative least square algorithm with internalvariables estimation to identify all the unknown parametersof the H-W model We also apply this method to theHammerstein and Wiener models

According to the key term separation principle werewrite the output nonlinear block119873

2 that is (10) as

119910 (119896) = 1199021119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896) (16)

where the internal variable 119909(119896) is separated The dynamiclinear block 119871(119911) that is (8) can be rewritten as

119909 (119896) = 1198870119908 (119896) 119911

minus119899119896 + 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896)

(17)

where the internal variable 119908(119896) is separated Now to com-plete the sequential decomposition first we substitute (7)into (17) only for 119908(119896) in the first term and then substitutethe new equation (17) into (16) only for 119909(119896) in the first termagain The final output equation of the H-Wmodel will be

119910 (119896) = 11990211198870(119891 (119906 (119896)) + 119901 (119906 (119896)) ℎ (119906 (119896))) 119911

minus119899119896

+ 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896)

(18)

As the H-W model depicted in Figure 4 consists of threesubsystems in series the parameterization of the model isnot unique because many combinations of parameters can befound [22]Therefore one parameter in at least two blocks hasto be fixed in (18) Evidently the choices 119902

1= 1 and 119887

0= 1will

simplify the model descriptionThen the H-Wmodel can bewritten as

119910 (119896) =

1199031

sum

119897=0

119891119897119906119897(119896 minus 119899

119896) +

1199031

sum

119897=0

119901119897119906119897(119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

+

119899119887

sum

119894=1

119887119894119908 (119896 minus 119899

119896minus 119894) +

119899119886

sum

119895=1

119886119895119909 (119896 minus 119895) +

1199032

sum

119898=2

119902119898119909119898(119896)

(19)

Equation (19) is linear-in-parameters for given 119906(119896) 119909(119896)and 119908(119896) it can be written in the following least squareformat

119910 (119896) = Φ119879(119896 120579) 120579 (20)

where the internal variables 119908(119896) and 119909(119896) are estimated by(7) and (17) using the preceding estimated parameters duringeach iterative process and

Φ119879= [1 119906 (119896minus119899

119896) 119906

1199031 (119896minus119899

119896)

ℎ (119906 (119896minus119899119896)) 119906 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896))

1199061199031 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896)) 119908 (119896minus119899

119896minus1)

119908 (119896 minus 119899119896minus 119894) minus119909 (119896minus1) minus119909 (119896minus119895)

1199092(119896) 119909

1199032 (119896)]

120579 = [1198910 1198911 119891

1199031

1199010 1199011 119901

1199031

1198871 119887

119899119887

1198861 119886

119899119886

1199022 119902

1199032

] 119879

(21)

Nowwe apply themodified recursive least squaremethodwith iterative estimation of the internal variable to (20) [17]Minimizing the following least square criterion [12]

= arg min120579

119899

sum

119896=1

120582119899minus119896

[119910 (119896) minus Φ119879

(119896) 120579]

2

(22)

where 120582 ≦ 1 is the forgetting factor the formulas ofthe recursive identification algorithm supplemented withinternal variable estimation are as follows

(119896) = (119896 minus 1) +

P (119896 minus 1) Φ (119896) [119910 (119896) minus Φ119879

(119896) (119896 minus 1)]

120582 + Φ119879

(119896)P (119896 minus 1) Φ (119896)

(23)

P (119896) = P (119896 minus 1)

120582minusP (119896 minus 1) Φ (119896) Φ

119879

(119896)P (119896 minus 1)

1 + Φ119879

(119896)P (119896 minus 1) Φ (119896) 120582

(24)

(119896) =

1199031

sum

119897=0

119891119897(119896 minus 1) 119906

119897(119896) +

1199031

sum

119897=0

119897(119896 minus 1) 119906

119897(119896) ℎ (119906 (119896))

(25)

(119896) = (119896 minus 119899119896) +

119899119887

sum

119894=1

119894(119896 minus 1) (119896 minus 119899

119896minus 119894)

minus

119899119886

sum

119895=1

119895(119896 minus 1) 119909 (119896 minus 119895)

(26)

Φ (119896) = [1 119906 (119896 minus 119899119896) 119906

1199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

119906 (119896 minus 119899119896) ℎ (119906 (119896 minus 119899

119896))

1199061199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

(119896 minus 119899119896minus 1) (119896 minus 119899

119896minus 119894)

minus (119896 minus 1) minus (119896 minus 119895) 2(119896)

1199032 (119896)]119879

(27)

where P(0) = 120583I I is unit matrix and 0 lt 120583 lt infin

6 Mathematical Problems in Engineering

Computer

Pressure transducer

Inclinometer

Low level controller

Electrohydraulic valve

(a)

Inclinometer

Main valve

Proportionalrelief valve

119870ℎ

Mainpump

Pilotpump

ControllerReference

input

(b)

Figure 5 Experimental prototype machine

Inclinometer

Proportionalrelief valve

120579119889+

minus

119906 Digitalcontroller

Pilot pump Main pump

119876Mainvalve

Hydraulicactuator

Mechanism 120579DA

AD

I P

Figure 6 Electrohydraulic position servocontrol system

In conclusion the iterative identification algorithm canbe presented as follwos

Step 1 Set the initial values of 119909(0) 119908(0) 119906(0) and P(0)

Step 2 Estimate the parameter (119896) by algorithm (23) andcalculate P(119896) by (24)

Step 3 Estimate the internal variables (119896) and (119896) by (25)and (26) using the recent estimates of model parameters (119896)

Step 4 Update the values of Φ(119896) by (27)

Step 5 Return to Step 2 until the parameter estimates con-verge to constant values

5 Experiment

51 Experimental Environment A hydraulic excavator wasretrofitted to be controlled by computer in our laboratory[23] Figure 5 shows the prototype machine whose manualpilot hydraulic control system was replaced by electrohy-draulic proportional control system inclinometers and pres-sure transducers were also installed on the excavator arms

for position and force servocontrol Schematic diagram of theelectrohydraulic servosystem is shown in Figure 6

52 Experimental Results In order to obtain the nonlinearcharacteristics of the system when changing the directionsand to obtain sufficient excitation we adopted a multisineinput signal which contained the frequency of 005Hz 01 Hz02Hz 04Hz and 05Hz to the identification experimentsThe sample rate was chosen to be 20Hz on the machineThe input signal and angle output were obtained from thecomputer of the experiment machine Ten groups of inputand output signals with time duration of 55 seconds weresampled in the repeated experiments the averaged measure-ment results are shown in Figure 7 Finally we calculate theoutput angle velocity by numerical differentiation

We set the parameters 119899119886= 3 119899

119887= 2 119899

119896= 8 1199031= 1199032= 3

119909(0) = 0 119908(0) = 0 119906(0) = 0 and P(0) = 106I Note that

lower forgetting factor 120582 is useful for reducing the influencesof old date while a value of 120582 close to 1 is less sensitive todisturbance Therefore we chose the forgetting factor to be120582 = 098 during the first 200 samples and 120582 = 1 otherwiseCompiling the developed iterative least square algorithm inMATLAB to identify theARXmodel containing only the119871(119911)

Mathematical Problems in Engineering 7

420

minus2

minus40 10 20 30 40 50

Input

Time (119905s)

Inpu

t (119906

v)

(a)

0 10 20 30 40 50

Angle output

2

15

1

Ang

le (r

ad)

Time (119905s)

(b)

Figure 7 Input and output signals of the identification experiment

The measured velocityPredicted by Hammerstein-Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(a) Hammerstein-Wiener prediction result

The measured velocityPredicted by Hammerstein

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(b) Hammerstein prediction result

The measured velocityPredicted by Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(c) Wiener prediction result

The measured velocityPredicted by ARX

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(d) ARX prediction result

Figure 8 Comparison results of the identified models

blockHammersteinmodel consisting of1198731and119871(119911)Wiener

model consisting of 119871(119911) and1198732 and H-Wmodel consisting

of1198731 119871(119911) and119873

2 respectively we obtain the identification

results shown in Table 1We use the identified models to predict the tracking

velocity of a general trajectory The comparative results areshown in Figures 8 and 9 They demonstrate that the H-Wand Hammerstein models which contain the input nonlinearblock with two-segment polynomial nonlinearities capturethe actual system well while the Wiener and ARX modelscannot approximate the actual system well Themean-squareerrors (MSE) of the identifiedmodels in Table 2 show that the

prediction error of the H-W model is about 13 54 and58 less than the Hammerstein Wiener and ARX modelsrespectively

6 Conclusion

This paper investigates the nonlinear modeling and identi-fication of an electrohydraulic control system We develop atheoretic state spacemodel for system analysis propose anH-W model for the highly nonlinear system based on a deeplyphysical insight into the actual system and apply a modified

8 Mathematical Problems in Engineering

05

04

03

02

01

0

minus01

minus02

minus03

0 2 4 6 8 10 12 14 16 18

Estim

atio

n re

sidua

(rad

s)

Residua of Hammerstein-WienerResidua of Hammerstein

Residua of WeinerResidua of ARX

Time (119905s)

Figure 9 Comparison of the estimation residua

Table 1 The identification results

Parameters Model typeH-W Hammerstein Wiener ARX

1198870

1 1 minus00358 minus002761198871

00349 00105 00151 001241198872

00617 00147 minus00069 005141198861

minus00309 minus00596 00069 minus006611198862

minus00598 minus00613 minus00075 minus006531198863

minus00897 minus00610 minus00081 minus006191198910

00109 00027 mdash mdash1198911

minus14619 times 10minus5 00012 mdash mdash1198912

minus00050 minus00012 mdash mdash1198913

00052 00045 mdash mdash1199010

minus00165 minus00055 mdash mdash1199011

00033 00013 mdash mdash1199012

00104 00036 mdash mdash1199013

minus00010 minus86858 times 10minus4 mdash mdash1199021

1 mdash 1 mdash1199022

00050 mdash 00013 mdash1199023

63562 times 10minus4 mdash 26895 times 10minus4 mdash

Table 2 MSE of the identified models

Errors Model typeH-W Hammerstein Wiener ARX

120575r sdot sminus1 00321 00417 01081 01218

recursive least square method with internal variables esti-mation to identify its parameters The main findings of thepaper include the follwoing (1) the proposed H-W modelsimplifies the identification procedure because it only uses theinput and output signals to identify all the parameters (2)TheH-W model containing the input nonlinear block with two-segment polynomial nonlinearities captures the actual systemverywell As shown by the comparative results the prediction

error of the H-Wmodel is about 13 54 and 58 less thanthe Hammerstein Wiener and ARX models respectivelyThe results provide a physical insight into the nonlinearcharacteristics of the actual system which is important forsystem analyzing monitoring and diagnosis Future workincludes addressing uncertain and fuzzy properties of thesystem [24 25] and extending the model for a wider rangeof equipment [26]

Acknowledgment

Thisworkwas supported in part by grants fromNational Nat-ural Science Foundation (Grant nos 51175511 and 61105073)of China

References

[1] H E Merritt Hydraulic Control System John Wiley amp SonsNew York USA 1967

[2] B Yao F Bu J Reedy and G T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[3] S Wen W Zheng J Zhu X Li and S Chen ldquoElman fuzzyadaptive control for obstacle avoidance of mobile robots usinghybrid forceposition incorporationrdquo IEEE Transactions onSystems Man and Cybernetics C vol 42 no 4 pp 603ndash6082012

[4] S Y Chen J Zhang H Zhang N M Kwok and Y F Li ldquoIntel-ligent lighting control for vision-based robotic manipulationrdquoIEEE Transactions on Industrial Electronics vol 59 no 8 pp3254ndash3263 2012

[5] Q H He P Hao and D Q Zhang ldquoModeling and parameterestimation for hydraulic system of excavatorrsquos armrdquo Journal ofCentral South University of Technology vol 15 no 3 pp 382ndash386 2008

[6] K Ziaei and N Sepehri ldquoModeling and identification ofelectrohydraulic servosrdquo Mechatronics vol 10 no 7 pp 761ndash772 2000

[7] L Wang B Wu R Du and S Yang ldquoNonlinear dynamic char-acteristics of moving hydraulic cylinderrdquo Chinese Journal ofMechanical Engineering vol 43 no 12 pp 12ndash19 2007

[8] M Jelali and H Schwarz ldquoNonlinear identification of hydraulicservo-drive systemsrdquo IEEE Control Systems Magazine vol 15no 5 pp 17ndash22 1995

[9] S Kleinsteuber and N Sepehri ldquoA polynomial network mod-eling approach to a class of large-scale hydraulic systemsrdquoComputers and Electrical Engineering vol 22 no 2 pp 151ndash1681996

[10] H Yousefi H Handroos and A Soleymani ldquoApplication of dif-ferential evolution in system identification of a servo-hydraulicsystem with a flexible loadrdquoMechatronics vol 18 no 9 pp 513ndash528 2008

[11] F Giri and E W Bai Block-Oriented Nonlinear System Identi-fication vol 404 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2010

[12] B -J Kwak A E Yagle and J A Levitt ldquoNonlinear systemidentification of hydraulic actuator friction dynamics using aHammerstein modelrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing vol 4 pp1933ndash1936 Seattle Wash USA 1998

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

42 Modified Least Square Algorithm In this section wedecompose the H-W model into a linear-in-parametersstructure by the key term separation principle and developa modified iterative least square algorithm with internalvariables estimation to identify all the unknown parametersof the H-W model We also apply this method to theHammerstein and Wiener models

According to the key term separation principle werewrite the output nonlinear block119873

2 that is (10) as

119910 (119896) = 1199021119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896) (16)

where the internal variable 119909(119896) is separated The dynamiclinear block 119871(119911) that is (8) can be rewritten as

119909 (119896) = 1198870119908 (119896) 119911

minus119899119896 + 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896)

(17)

where the internal variable 119908(119896) is separated Now to com-plete the sequential decomposition first we substitute (7)into (17) only for 119908(119896) in the first term and then substitutethe new equation (17) into (16) only for 119909(119896) in the first termagain The final output equation of the H-Wmodel will be

119910 (119896) = 11990211198870(119891 (119906 (119896)) + 119901 (119906 (119896)) ℎ (119906 (119896))) 119911

minus119899119896

+ 119911minus119899119896 [119861 (119911

minus1) minus 1198870]119908 (119896)

+ [1 minus 119860 (119911minus1)] 119909 (119896) +

1199032

sum

119898=2

119902119898119909119898(119896)

(18)

As the H-W model depicted in Figure 4 consists of threesubsystems in series the parameterization of the model isnot unique because many combinations of parameters can befound [22]Therefore one parameter in at least two blocks hasto be fixed in (18) Evidently the choices 119902

1= 1 and 119887

0= 1will

simplify the model descriptionThen the H-Wmodel can bewritten as

119910 (119896) =

1199031

sum

119897=0

119891119897119906119897(119896 minus 119899

119896) +

1199031

sum

119897=0

119901119897119906119897(119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

+

119899119887

sum

119894=1

119887119894119908 (119896 minus 119899

119896minus 119894) +

119899119886

sum

119895=1

119886119895119909 (119896 minus 119895) +

1199032

sum

119898=2

119902119898119909119898(119896)

(19)

Equation (19) is linear-in-parameters for given 119906(119896) 119909(119896)and 119908(119896) it can be written in the following least squareformat

119910 (119896) = Φ119879(119896 120579) 120579 (20)

where the internal variables 119908(119896) and 119909(119896) are estimated by(7) and (17) using the preceding estimated parameters duringeach iterative process and

Φ119879= [1 119906 (119896minus119899

119896) 119906

1199031 (119896minus119899

119896)

ℎ (119906 (119896minus119899119896)) 119906 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896))

1199061199031 (119896minus119899

119896) ℎ (119906 (119896minus119899

119896)) 119908 (119896minus119899

119896minus1)

119908 (119896 minus 119899119896minus 119894) minus119909 (119896minus1) minus119909 (119896minus119895)

1199092(119896) 119909

1199032 (119896)]

120579 = [1198910 1198911 119891

1199031

1199010 1199011 119901

1199031

1198871 119887

119899119887

1198861 119886

119899119886

1199022 119902

1199032

] 119879

(21)

Nowwe apply themodified recursive least squaremethodwith iterative estimation of the internal variable to (20) [17]Minimizing the following least square criterion [12]

= arg min120579

119899

sum

119896=1

120582119899minus119896

[119910 (119896) minus Φ119879

(119896) 120579]

2

(22)

where 120582 ≦ 1 is the forgetting factor the formulas ofthe recursive identification algorithm supplemented withinternal variable estimation are as follows

(119896) = (119896 minus 1) +

P (119896 minus 1) Φ (119896) [119910 (119896) minus Φ119879

(119896) (119896 minus 1)]

120582 + Φ119879

(119896)P (119896 minus 1) Φ (119896)

(23)

P (119896) = P (119896 minus 1)

120582minusP (119896 minus 1) Φ (119896) Φ

119879

(119896)P (119896 minus 1)

1 + Φ119879

(119896)P (119896 minus 1) Φ (119896) 120582

(24)

(119896) =

1199031

sum

119897=0

119891119897(119896 minus 1) 119906

119897(119896) +

1199031

sum

119897=0

119897(119896 minus 1) 119906

119897(119896) ℎ (119906 (119896))

(25)

(119896) = (119896 minus 119899119896) +

119899119887

sum

119894=1

119894(119896 minus 1) (119896 minus 119899

119896minus 119894)

minus

119899119886

sum

119895=1

119895(119896 minus 1) 119909 (119896 minus 119895)

(26)

Φ (119896) = [1 119906 (119896 minus 119899119896) 119906

1199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

119906 (119896 minus 119899119896) ℎ (119906 (119896 minus 119899

119896))

1199061199031 (119896 minus 119899

119896) ℎ (119906 (119896 minus 119899

119896))

(119896 minus 119899119896minus 1) (119896 minus 119899

119896minus 119894)

minus (119896 minus 1) minus (119896 minus 119895) 2(119896)

1199032 (119896)]119879

(27)

where P(0) = 120583I I is unit matrix and 0 lt 120583 lt infin

6 Mathematical Problems in Engineering

Computer

Pressure transducer

Inclinometer

Low level controller

Electrohydraulic valve

(a)

Inclinometer

Main valve

Proportionalrelief valve

119870ℎ

Mainpump

Pilotpump

ControllerReference

input

(b)

Figure 5 Experimental prototype machine

Inclinometer

Proportionalrelief valve

120579119889+

minus

119906 Digitalcontroller

Pilot pump Main pump

119876Mainvalve

Hydraulicactuator

Mechanism 120579DA

AD

I P

Figure 6 Electrohydraulic position servocontrol system

In conclusion the iterative identification algorithm canbe presented as follwos

Step 1 Set the initial values of 119909(0) 119908(0) 119906(0) and P(0)

Step 2 Estimate the parameter (119896) by algorithm (23) andcalculate P(119896) by (24)

Step 3 Estimate the internal variables (119896) and (119896) by (25)and (26) using the recent estimates of model parameters (119896)

Step 4 Update the values of Φ(119896) by (27)

Step 5 Return to Step 2 until the parameter estimates con-verge to constant values

5 Experiment

51 Experimental Environment A hydraulic excavator wasretrofitted to be controlled by computer in our laboratory[23] Figure 5 shows the prototype machine whose manualpilot hydraulic control system was replaced by electrohy-draulic proportional control system inclinometers and pres-sure transducers were also installed on the excavator arms

for position and force servocontrol Schematic diagram of theelectrohydraulic servosystem is shown in Figure 6

52 Experimental Results In order to obtain the nonlinearcharacteristics of the system when changing the directionsand to obtain sufficient excitation we adopted a multisineinput signal which contained the frequency of 005Hz 01 Hz02Hz 04Hz and 05Hz to the identification experimentsThe sample rate was chosen to be 20Hz on the machineThe input signal and angle output were obtained from thecomputer of the experiment machine Ten groups of inputand output signals with time duration of 55 seconds weresampled in the repeated experiments the averaged measure-ment results are shown in Figure 7 Finally we calculate theoutput angle velocity by numerical differentiation

We set the parameters 119899119886= 3 119899

119887= 2 119899

119896= 8 1199031= 1199032= 3

119909(0) = 0 119908(0) = 0 119906(0) = 0 and P(0) = 106I Note that

lower forgetting factor 120582 is useful for reducing the influencesof old date while a value of 120582 close to 1 is less sensitive todisturbance Therefore we chose the forgetting factor to be120582 = 098 during the first 200 samples and 120582 = 1 otherwiseCompiling the developed iterative least square algorithm inMATLAB to identify theARXmodel containing only the119871(119911)

Mathematical Problems in Engineering 7

420

minus2

minus40 10 20 30 40 50

Input

Time (119905s)

Inpu

t (119906

v)

(a)

0 10 20 30 40 50

Angle output

2

15

1

Ang

le (r

ad)

Time (119905s)

(b)

Figure 7 Input and output signals of the identification experiment

The measured velocityPredicted by Hammerstein-Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(a) Hammerstein-Wiener prediction result

The measured velocityPredicted by Hammerstein

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(b) Hammerstein prediction result

The measured velocityPredicted by Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(c) Wiener prediction result

The measured velocityPredicted by ARX

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(d) ARX prediction result

Figure 8 Comparison results of the identified models

blockHammersteinmodel consisting of1198731and119871(119911)Wiener

model consisting of 119871(119911) and1198732 and H-Wmodel consisting

of1198731 119871(119911) and119873

2 respectively we obtain the identification

results shown in Table 1We use the identified models to predict the tracking

velocity of a general trajectory The comparative results areshown in Figures 8 and 9 They demonstrate that the H-Wand Hammerstein models which contain the input nonlinearblock with two-segment polynomial nonlinearities capturethe actual system well while the Wiener and ARX modelscannot approximate the actual system well Themean-squareerrors (MSE) of the identifiedmodels in Table 2 show that the

prediction error of the H-W model is about 13 54 and58 less than the Hammerstein Wiener and ARX modelsrespectively

6 Conclusion

This paper investigates the nonlinear modeling and identi-fication of an electrohydraulic control system We develop atheoretic state spacemodel for system analysis propose anH-W model for the highly nonlinear system based on a deeplyphysical insight into the actual system and apply a modified

8 Mathematical Problems in Engineering

05

04

03

02

01

0

minus01

minus02

minus03

0 2 4 6 8 10 12 14 16 18

Estim

atio

n re

sidua

(rad

s)

Residua of Hammerstein-WienerResidua of Hammerstein

Residua of WeinerResidua of ARX

Time (119905s)

Figure 9 Comparison of the estimation residua

Table 1 The identification results

Parameters Model typeH-W Hammerstein Wiener ARX

1198870

1 1 minus00358 minus002761198871

00349 00105 00151 001241198872

00617 00147 minus00069 005141198861

minus00309 minus00596 00069 minus006611198862

minus00598 minus00613 minus00075 minus006531198863

minus00897 minus00610 minus00081 minus006191198910

00109 00027 mdash mdash1198911

minus14619 times 10minus5 00012 mdash mdash1198912

minus00050 minus00012 mdash mdash1198913

00052 00045 mdash mdash1199010

minus00165 minus00055 mdash mdash1199011

00033 00013 mdash mdash1199012

00104 00036 mdash mdash1199013

minus00010 minus86858 times 10minus4 mdash mdash1199021

1 mdash 1 mdash1199022

00050 mdash 00013 mdash1199023

63562 times 10minus4 mdash 26895 times 10minus4 mdash

Table 2 MSE of the identified models

Errors Model typeH-W Hammerstein Wiener ARX

120575r sdot sminus1 00321 00417 01081 01218

recursive least square method with internal variables esti-mation to identify its parameters The main findings of thepaper include the follwoing (1) the proposed H-W modelsimplifies the identification procedure because it only uses theinput and output signals to identify all the parameters (2)TheH-W model containing the input nonlinear block with two-segment polynomial nonlinearities captures the actual systemverywell As shown by the comparative results the prediction

error of the H-Wmodel is about 13 54 and 58 less thanthe Hammerstein Wiener and ARX models respectivelyThe results provide a physical insight into the nonlinearcharacteristics of the actual system which is important forsystem analyzing monitoring and diagnosis Future workincludes addressing uncertain and fuzzy properties of thesystem [24 25] and extending the model for a wider rangeof equipment [26]

Acknowledgment

Thisworkwas supported in part by grants fromNational Nat-ural Science Foundation (Grant nos 51175511 and 61105073)of China

References

[1] H E Merritt Hydraulic Control System John Wiley amp SonsNew York USA 1967

[2] B Yao F Bu J Reedy and G T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[3] S Wen W Zheng J Zhu X Li and S Chen ldquoElman fuzzyadaptive control for obstacle avoidance of mobile robots usinghybrid forceposition incorporationrdquo IEEE Transactions onSystems Man and Cybernetics C vol 42 no 4 pp 603ndash6082012

[4] S Y Chen J Zhang H Zhang N M Kwok and Y F Li ldquoIntel-ligent lighting control for vision-based robotic manipulationrdquoIEEE Transactions on Industrial Electronics vol 59 no 8 pp3254ndash3263 2012

[5] Q H He P Hao and D Q Zhang ldquoModeling and parameterestimation for hydraulic system of excavatorrsquos armrdquo Journal ofCentral South University of Technology vol 15 no 3 pp 382ndash386 2008

[6] K Ziaei and N Sepehri ldquoModeling and identification ofelectrohydraulic servosrdquo Mechatronics vol 10 no 7 pp 761ndash772 2000

[7] L Wang B Wu R Du and S Yang ldquoNonlinear dynamic char-acteristics of moving hydraulic cylinderrdquo Chinese Journal ofMechanical Engineering vol 43 no 12 pp 12ndash19 2007

[8] M Jelali and H Schwarz ldquoNonlinear identification of hydraulicservo-drive systemsrdquo IEEE Control Systems Magazine vol 15no 5 pp 17ndash22 1995

[9] S Kleinsteuber and N Sepehri ldquoA polynomial network mod-eling approach to a class of large-scale hydraulic systemsrdquoComputers and Electrical Engineering vol 22 no 2 pp 151ndash1681996

[10] H Yousefi H Handroos and A Soleymani ldquoApplication of dif-ferential evolution in system identification of a servo-hydraulicsystem with a flexible loadrdquoMechatronics vol 18 no 9 pp 513ndash528 2008

[11] F Giri and E W Bai Block-Oriented Nonlinear System Identi-fication vol 404 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2010

[12] B -J Kwak A E Yagle and J A Levitt ldquoNonlinear systemidentification of hydraulic actuator friction dynamics using aHammerstein modelrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing vol 4 pp1933ndash1936 Seattle Wash USA 1998

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Computer

Pressure transducer

Inclinometer

Low level controller

Electrohydraulic valve

(a)

Inclinometer

Main valve

Proportionalrelief valve

119870ℎ

Mainpump

Pilotpump

ControllerReference

input

(b)

Figure 5 Experimental prototype machine

Inclinometer

Proportionalrelief valve

120579119889+

minus

119906 Digitalcontroller

Pilot pump Main pump

119876Mainvalve

Hydraulicactuator

Mechanism 120579DA

AD

I P

Figure 6 Electrohydraulic position servocontrol system

In conclusion the iterative identification algorithm canbe presented as follwos

Step 1 Set the initial values of 119909(0) 119908(0) 119906(0) and P(0)

Step 2 Estimate the parameter (119896) by algorithm (23) andcalculate P(119896) by (24)

Step 3 Estimate the internal variables (119896) and (119896) by (25)and (26) using the recent estimates of model parameters (119896)

Step 4 Update the values of Φ(119896) by (27)

Step 5 Return to Step 2 until the parameter estimates con-verge to constant values

5 Experiment

51 Experimental Environment A hydraulic excavator wasretrofitted to be controlled by computer in our laboratory[23] Figure 5 shows the prototype machine whose manualpilot hydraulic control system was replaced by electrohy-draulic proportional control system inclinometers and pres-sure transducers were also installed on the excavator arms

for position and force servocontrol Schematic diagram of theelectrohydraulic servosystem is shown in Figure 6

52 Experimental Results In order to obtain the nonlinearcharacteristics of the system when changing the directionsand to obtain sufficient excitation we adopted a multisineinput signal which contained the frequency of 005Hz 01 Hz02Hz 04Hz and 05Hz to the identification experimentsThe sample rate was chosen to be 20Hz on the machineThe input signal and angle output were obtained from thecomputer of the experiment machine Ten groups of inputand output signals with time duration of 55 seconds weresampled in the repeated experiments the averaged measure-ment results are shown in Figure 7 Finally we calculate theoutput angle velocity by numerical differentiation

We set the parameters 119899119886= 3 119899

119887= 2 119899

119896= 8 1199031= 1199032= 3

119909(0) = 0 119908(0) = 0 119906(0) = 0 and P(0) = 106I Note that

lower forgetting factor 120582 is useful for reducing the influencesof old date while a value of 120582 close to 1 is less sensitive todisturbance Therefore we chose the forgetting factor to be120582 = 098 during the first 200 samples and 120582 = 1 otherwiseCompiling the developed iterative least square algorithm inMATLAB to identify theARXmodel containing only the119871(119911)

Mathematical Problems in Engineering 7

420

minus2

minus40 10 20 30 40 50

Input

Time (119905s)

Inpu

t (119906

v)

(a)

0 10 20 30 40 50

Angle output

2

15

1

Ang

le (r

ad)

Time (119905s)

(b)

Figure 7 Input and output signals of the identification experiment

The measured velocityPredicted by Hammerstein-Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(a) Hammerstein-Wiener prediction result

The measured velocityPredicted by Hammerstein

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(b) Hammerstein prediction result

The measured velocityPredicted by Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(c) Wiener prediction result

The measured velocityPredicted by ARX

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(d) ARX prediction result

Figure 8 Comparison results of the identified models

blockHammersteinmodel consisting of1198731and119871(119911)Wiener

model consisting of 119871(119911) and1198732 and H-Wmodel consisting

of1198731 119871(119911) and119873

2 respectively we obtain the identification

results shown in Table 1We use the identified models to predict the tracking

velocity of a general trajectory The comparative results areshown in Figures 8 and 9 They demonstrate that the H-Wand Hammerstein models which contain the input nonlinearblock with two-segment polynomial nonlinearities capturethe actual system well while the Wiener and ARX modelscannot approximate the actual system well Themean-squareerrors (MSE) of the identifiedmodels in Table 2 show that the

prediction error of the H-W model is about 13 54 and58 less than the Hammerstein Wiener and ARX modelsrespectively

6 Conclusion

This paper investigates the nonlinear modeling and identi-fication of an electrohydraulic control system We develop atheoretic state spacemodel for system analysis propose anH-W model for the highly nonlinear system based on a deeplyphysical insight into the actual system and apply a modified

8 Mathematical Problems in Engineering

05

04

03

02

01

0

minus01

minus02

minus03

0 2 4 6 8 10 12 14 16 18

Estim

atio

n re

sidua

(rad

s)

Residua of Hammerstein-WienerResidua of Hammerstein

Residua of WeinerResidua of ARX

Time (119905s)

Figure 9 Comparison of the estimation residua

Table 1 The identification results

Parameters Model typeH-W Hammerstein Wiener ARX

1198870

1 1 minus00358 minus002761198871

00349 00105 00151 001241198872

00617 00147 minus00069 005141198861

minus00309 minus00596 00069 minus006611198862

minus00598 minus00613 minus00075 minus006531198863

minus00897 minus00610 minus00081 minus006191198910

00109 00027 mdash mdash1198911

minus14619 times 10minus5 00012 mdash mdash1198912

minus00050 minus00012 mdash mdash1198913

00052 00045 mdash mdash1199010

minus00165 minus00055 mdash mdash1199011

00033 00013 mdash mdash1199012

00104 00036 mdash mdash1199013

minus00010 minus86858 times 10minus4 mdash mdash1199021

1 mdash 1 mdash1199022

00050 mdash 00013 mdash1199023

63562 times 10minus4 mdash 26895 times 10minus4 mdash

Table 2 MSE of the identified models

Errors Model typeH-W Hammerstein Wiener ARX

120575r sdot sminus1 00321 00417 01081 01218

recursive least square method with internal variables esti-mation to identify its parameters The main findings of thepaper include the follwoing (1) the proposed H-W modelsimplifies the identification procedure because it only uses theinput and output signals to identify all the parameters (2)TheH-W model containing the input nonlinear block with two-segment polynomial nonlinearities captures the actual systemverywell As shown by the comparative results the prediction

error of the H-Wmodel is about 13 54 and 58 less thanthe Hammerstein Wiener and ARX models respectivelyThe results provide a physical insight into the nonlinearcharacteristics of the actual system which is important forsystem analyzing monitoring and diagnosis Future workincludes addressing uncertain and fuzzy properties of thesystem [24 25] and extending the model for a wider rangeof equipment [26]

Acknowledgment

Thisworkwas supported in part by grants fromNational Nat-ural Science Foundation (Grant nos 51175511 and 61105073)of China

References

[1] H E Merritt Hydraulic Control System John Wiley amp SonsNew York USA 1967

[2] B Yao F Bu J Reedy and G T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[3] S Wen W Zheng J Zhu X Li and S Chen ldquoElman fuzzyadaptive control for obstacle avoidance of mobile robots usinghybrid forceposition incorporationrdquo IEEE Transactions onSystems Man and Cybernetics C vol 42 no 4 pp 603ndash6082012

[4] S Y Chen J Zhang H Zhang N M Kwok and Y F Li ldquoIntel-ligent lighting control for vision-based robotic manipulationrdquoIEEE Transactions on Industrial Electronics vol 59 no 8 pp3254ndash3263 2012

[5] Q H He P Hao and D Q Zhang ldquoModeling and parameterestimation for hydraulic system of excavatorrsquos armrdquo Journal ofCentral South University of Technology vol 15 no 3 pp 382ndash386 2008

[6] K Ziaei and N Sepehri ldquoModeling and identification ofelectrohydraulic servosrdquo Mechatronics vol 10 no 7 pp 761ndash772 2000

[7] L Wang B Wu R Du and S Yang ldquoNonlinear dynamic char-acteristics of moving hydraulic cylinderrdquo Chinese Journal ofMechanical Engineering vol 43 no 12 pp 12ndash19 2007

[8] M Jelali and H Schwarz ldquoNonlinear identification of hydraulicservo-drive systemsrdquo IEEE Control Systems Magazine vol 15no 5 pp 17ndash22 1995

[9] S Kleinsteuber and N Sepehri ldquoA polynomial network mod-eling approach to a class of large-scale hydraulic systemsrdquoComputers and Electrical Engineering vol 22 no 2 pp 151ndash1681996

[10] H Yousefi H Handroos and A Soleymani ldquoApplication of dif-ferential evolution in system identification of a servo-hydraulicsystem with a flexible loadrdquoMechatronics vol 18 no 9 pp 513ndash528 2008

[11] F Giri and E W Bai Block-Oriented Nonlinear System Identi-fication vol 404 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2010

[12] B -J Kwak A E Yagle and J A Levitt ldquoNonlinear systemidentification of hydraulic actuator friction dynamics using aHammerstein modelrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing vol 4 pp1933ndash1936 Seattle Wash USA 1998

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

420

minus2

minus40 10 20 30 40 50

Input

Time (119905s)

Inpu

t (119906

v)

(a)

0 10 20 30 40 50

Angle output

2

15

1

Ang

le (r

ad)

Time (119905s)

(b)

Figure 7 Input and output signals of the identification experiment

The measured velocityPredicted by Hammerstein-Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(a) Hammerstein-Wiener prediction result

The measured velocityPredicted by Hammerstein

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(b) Hammerstein prediction result

The measured velocityPredicted by Wiener

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(c) Wiener prediction result

The measured velocityPredicted by ARX

06

04

02

0

minus02

minus040 2 4 6 8 10 12 14 16 18

Ang

le v

eloci

ty (r

ads

)

Time (119905s)

(d) ARX prediction result

Figure 8 Comparison results of the identified models

blockHammersteinmodel consisting of1198731and119871(119911)Wiener

model consisting of 119871(119911) and1198732 and H-Wmodel consisting

of1198731 119871(119911) and119873

2 respectively we obtain the identification

results shown in Table 1We use the identified models to predict the tracking

velocity of a general trajectory The comparative results areshown in Figures 8 and 9 They demonstrate that the H-Wand Hammerstein models which contain the input nonlinearblock with two-segment polynomial nonlinearities capturethe actual system well while the Wiener and ARX modelscannot approximate the actual system well Themean-squareerrors (MSE) of the identifiedmodels in Table 2 show that the

prediction error of the H-W model is about 13 54 and58 less than the Hammerstein Wiener and ARX modelsrespectively

6 Conclusion

This paper investigates the nonlinear modeling and identi-fication of an electrohydraulic control system We develop atheoretic state spacemodel for system analysis propose anH-W model for the highly nonlinear system based on a deeplyphysical insight into the actual system and apply a modified

8 Mathematical Problems in Engineering

05

04

03

02

01

0

minus01

minus02

minus03

0 2 4 6 8 10 12 14 16 18

Estim

atio

n re

sidua

(rad

s)

Residua of Hammerstein-WienerResidua of Hammerstein

Residua of WeinerResidua of ARX

Time (119905s)

Figure 9 Comparison of the estimation residua

Table 1 The identification results

Parameters Model typeH-W Hammerstein Wiener ARX

1198870

1 1 minus00358 minus002761198871

00349 00105 00151 001241198872

00617 00147 minus00069 005141198861

minus00309 minus00596 00069 minus006611198862

minus00598 minus00613 minus00075 minus006531198863

minus00897 minus00610 minus00081 minus006191198910

00109 00027 mdash mdash1198911

minus14619 times 10minus5 00012 mdash mdash1198912

minus00050 minus00012 mdash mdash1198913

00052 00045 mdash mdash1199010

minus00165 minus00055 mdash mdash1199011

00033 00013 mdash mdash1199012

00104 00036 mdash mdash1199013

minus00010 minus86858 times 10minus4 mdash mdash1199021

1 mdash 1 mdash1199022

00050 mdash 00013 mdash1199023

63562 times 10minus4 mdash 26895 times 10minus4 mdash

Table 2 MSE of the identified models

Errors Model typeH-W Hammerstein Wiener ARX

120575r sdot sminus1 00321 00417 01081 01218

recursive least square method with internal variables esti-mation to identify its parameters The main findings of thepaper include the follwoing (1) the proposed H-W modelsimplifies the identification procedure because it only uses theinput and output signals to identify all the parameters (2)TheH-W model containing the input nonlinear block with two-segment polynomial nonlinearities captures the actual systemverywell As shown by the comparative results the prediction

error of the H-Wmodel is about 13 54 and 58 less thanthe Hammerstein Wiener and ARX models respectivelyThe results provide a physical insight into the nonlinearcharacteristics of the actual system which is important forsystem analyzing monitoring and diagnosis Future workincludes addressing uncertain and fuzzy properties of thesystem [24 25] and extending the model for a wider rangeof equipment [26]

Acknowledgment

Thisworkwas supported in part by grants fromNational Nat-ural Science Foundation (Grant nos 51175511 and 61105073)of China

References

[1] H E Merritt Hydraulic Control System John Wiley amp SonsNew York USA 1967

[2] B Yao F Bu J Reedy and G T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[3] S Wen W Zheng J Zhu X Li and S Chen ldquoElman fuzzyadaptive control for obstacle avoidance of mobile robots usinghybrid forceposition incorporationrdquo IEEE Transactions onSystems Man and Cybernetics C vol 42 no 4 pp 603ndash6082012

[4] S Y Chen J Zhang H Zhang N M Kwok and Y F Li ldquoIntel-ligent lighting control for vision-based robotic manipulationrdquoIEEE Transactions on Industrial Electronics vol 59 no 8 pp3254ndash3263 2012

[5] Q H He P Hao and D Q Zhang ldquoModeling and parameterestimation for hydraulic system of excavatorrsquos armrdquo Journal ofCentral South University of Technology vol 15 no 3 pp 382ndash386 2008

[6] K Ziaei and N Sepehri ldquoModeling and identification ofelectrohydraulic servosrdquo Mechatronics vol 10 no 7 pp 761ndash772 2000

[7] L Wang B Wu R Du and S Yang ldquoNonlinear dynamic char-acteristics of moving hydraulic cylinderrdquo Chinese Journal ofMechanical Engineering vol 43 no 12 pp 12ndash19 2007

[8] M Jelali and H Schwarz ldquoNonlinear identification of hydraulicservo-drive systemsrdquo IEEE Control Systems Magazine vol 15no 5 pp 17ndash22 1995

[9] S Kleinsteuber and N Sepehri ldquoA polynomial network mod-eling approach to a class of large-scale hydraulic systemsrdquoComputers and Electrical Engineering vol 22 no 2 pp 151ndash1681996

[10] H Yousefi H Handroos and A Soleymani ldquoApplication of dif-ferential evolution in system identification of a servo-hydraulicsystem with a flexible loadrdquoMechatronics vol 18 no 9 pp 513ndash528 2008

[11] F Giri and E W Bai Block-Oriented Nonlinear System Identi-fication vol 404 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2010

[12] B -J Kwak A E Yagle and J A Levitt ldquoNonlinear systemidentification of hydraulic actuator friction dynamics using aHammerstein modelrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing vol 4 pp1933ndash1936 Seattle Wash USA 1998

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

05

04

03

02

01

0

minus01

minus02

minus03

0 2 4 6 8 10 12 14 16 18

Estim

atio

n re

sidua

(rad

s)

Residua of Hammerstein-WienerResidua of Hammerstein

Residua of WeinerResidua of ARX

Time (119905s)

Figure 9 Comparison of the estimation residua

Table 1 The identification results

Parameters Model typeH-W Hammerstein Wiener ARX

1198870

1 1 minus00358 minus002761198871

00349 00105 00151 001241198872

00617 00147 minus00069 005141198861

minus00309 minus00596 00069 minus006611198862

minus00598 minus00613 minus00075 minus006531198863

minus00897 minus00610 minus00081 minus006191198910

00109 00027 mdash mdash1198911

minus14619 times 10minus5 00012 mdash mdash1198912

minus00050 minus00012 mdash mdash1198913

00052 00045 mdash mdash1199010

minus00165 minus00055 mdash mdash1199011

00033 00013 mdash mdash1199012

00104 00036 mdash mdash1199013

minus00010 minus86858 times 10minus4 mdash mdash1199021

1 mdash 1 mdash1199022

00050 mdash 00013 mdash1199023

63562 times 10minus4 mdash 26895 times 10minus4 mdash

Table 2 MSE of the identified models

Errors Model typeH-W Hammerstein Wiener ARX

120575r sdot sminus1 00321 00417 01081 01218

recursive least square method with internal variables esti-mation to identify its parameters The main findings of thepaper include the follwoing (1) the proposed H-W modelsimplifies the identification procedure because it only uses theinput and output signals to identify all the parameters (2)TheH-W model containing the input nonlinear block with two-segment polynomial nonlinearities captures the actual systemverywell As shown by the comparative results the prediction

error of the H-Wmodel is about 13 54 and 58 less thanthe Hammerstein Wiener and ARX models respectivelyThe results provide a physical insight into the nonlinearcharacteristics of the actual system which is important forsystem analyzing monitoring and diagnosis Future workincludes addressing uncertain and fuzzy properties of thesystem [24 25] and extending the model for a wider rangeof equipment [26]

Acknowledgment

Thisworkwas supported in part by grants fromNational Nat-ural Science Foundation (Grant nos 51175511 and 61105073)of China

References

[1] H E Merritt Hydraulic Control System John Wiley amp SonsNew York USA 1967

[2] B Yao F Bu J Reedy and G T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[3] S Wen W Zheng J Zhu X Li and S Chen ldquoElman fuzzyadaptive control for obstacle avoidance of mobile robots usinghybrid forceposition incorporationrdquo IEEE Transactions onSystems Man and Cybernetics C vol 42 no 4 pp 603ndash6082012

[4] S Y Chen J Zhang H Zhang N M Kwok and Y F Li ldquoIntel-ligent lighting control for vision-based robotic manipulationrdquoIEEE Transactions on Industrial Electronics vol 59 no 8 pp3254ndash3263 2012

[5] Q H He P Hao and D Q Zhang ldquoModeling and parameterestimation for hydraulic system of excavatorrsquos armrdquo Journal ofCentral South University of Technology vol 15 no 3 pp 382ndash386 2008

[6] K Ziaei and N Sepehri ldquoModeling and identification ofelectrohydraulic servosrdquo Mechatronics vol 10 no 7 pp 761ndash772 2000

[7] L Wang B Wu R Du and S Yang ldquoNonlinear dynamic char-acteristics of moving hydraulic cylinderrdquo Chinese Journal ofMechanical Engineering vol 43 no 12 pp 12ndash19 2007

[8] M Jelali and H Schwarz ldquoNonlinear identification of hydraulicservo-drive systemsrdquo IEEE Control Systems Magazine vol 15no 5 pp 17ndash22 1995

[9] S Kleinsteuber and N Sepehri ldquoA polynomial network mod-eling approach to a class of large-scale hydraulic systemsrdquoComputers and Electrical Engineering vol 22 no 2 pp 151ndash1681996

[10] H Yousefi H Handroos and A Soleymani ldquoApplication of dif-ferential evolution in system identification of a servo-hydraulicsystem with a flexible loadrdquoMechatronics vol 18 no 9 pp 513ndash528 2008

[11] F Giri and E W Bai Block-Oriented Nonlinear System Identi-fication vol 404 of Lecture Notes in Control and InformationSciences Springer Berlin Germany 2010

[12] B -J Kwak A E Yagle and J A Levitt ldquoNonlinear systemidentification of hydraulic actuator friction dynamics using aHammerstein modelrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing vol 4 pp1933ndash1936 Seattle Wash USA 1998

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

[13] Y J Zheng S Y Chen Y Lin and W L Wang ldquoBio-Inspired optimization of sustainable energy systems a reviewrdquoMathematical Problems in Engineering vol 2013 Article ID354523 12 pages 2013

[14] C Cattani R Badea S Chen and M Crisan ldquoBiomedicalsignal processing and modeling complexity of living systemsrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 298634 2 pages 2012

[15] B Li J Yan A X Liu Y H Zeng and G Guo ldquoNonlinearidentification of excavatorrsquos electro-hydraulic servo systemrdquoTransactions of the Chinese Society For Agricultural Machineryvol 43 no 4 pp 20ndash25 2012

[16] J Ljung System IdentificationTheory for the User Prentice HallPress New York NY USA 1999

[17] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[18] J Voros ldquoIterative algorithm for parameter identification ofHammerstein systems with two-segment nonlinearitiesrdquo IEEETransactions onAutomatic Control vol 44 no 11 pp 2145ndash21491999

[19] J Voros ldquoRecursive identification of Hammerstein systemswith discontinuous nonlinearities containing dead-zonesrdquoIEEE Transactions on Automatic Control vol 48 no 12 pp2203ndash2206 2003

[20] J Voros ldquoAn iterative method for Hammerstein-Wiener sys-tems parameter identificationrdquo Journal of Electrical Engineeringvol 55 no 11-12 pp 328ndash331 2004

[21] J Voros ldquoIdentification of nonlinear dynamic systems usingextended Hammerstein and Wiener modelsrdquo Control Theoryand Advanced Technology vol 10 no 4 part 2 pp 1203ndash12121995

[22] E-W Bai ldquoA blind approach to theHammerstein-Wienermodelidentificationrdquo Automatica vol 38 no 6 pp 967ndash979 2002

[23] J Yan B Li Q Z Tu G Gang and Y H Zeng ldquoAutomatizationof excavator and study of its auto-controlrdquo in Proceedings ofthe 3rd International Conference on Measuring Technology andMechatronics Automation pp 604ndash609 Shanghai China 2011

[24] Y J Zheng andH F Ling ldquoEmergency transportation planningin disaster relief supply chain management a cooperative fuzzyoptimization approachrdquo Soft Computing 2013

[25] Y J Zheng and S Y Chen ldquoCooperative particle swarm opti-mization for multiobjective transportation planningrdquo AppliedIntelligence 2013

[26] Y Zheng S Chen and H Ling ldquoEfficient multi-objective tabusearch for emergency equipment maintenance scheduling indisaster rescuerdquo Optimization Letters vol 7 no 1 pp 89ndash1002013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of