research article sliding mode control for diesel engine

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 712723 11 pageshttpdxdoiorg1011552013712723

Research ArticleSliding Mode Control for Diesel Engine Air PathSubject to Matched and Unmatched Disturbances UsingExtended State Observer

Sofiane Ahmed Ali and Nicolas Langlois

IRSEEM Technopole du Madrillet BP 10024 76801 Saint-Etienne-du-Rouvray France

Correspondence should be addressed to Sofiane Ahmed Ali sofianeahmedaliesigelecfr

Received 2 August 2013 Accepted 22 September 2013

Academic Editor Xudong Zhao

Copyright copy 2013 S Ahmed Ali and N Langlois This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Thepaper develops a slidingmode controller via nonlinear disturbance observer for diesel engine air path system subject tomatchedandunmatcheddisturbancesTheproposed controller is based upon anovel nonlinear disturbance observer (NDO) structurewhichuses the concept of total disturbance estimation in order to estimate simultaneously the matched and the unmatched disturbancesin the systemThis estimation is then incorporated in a composite controller which alleviates the chattering problem andmaintainsthe nominal performance of the system in the absence of disturbances Simulations results of the proposed controller on a recentlyvalidated experimental air path diesel engine model show that the proposed methods exhibit a better performances comparing tothe baseline SMC in terms of reducing chattering and nominal performances recovery

1 Introduction

Comparing to gasoline engines diesel engines have theadvantage of producing the requested torque under anoptimal compromise between fuel consumption and givenexhaust legislation emission level To meet the requirementsof emission standards EURO V and VI the emissions ofdiesel engines particularly oxides of nitrogen (NO

119909) and

particulate matter (PM) must be controlled at every enginecycles Earlier reduction mechanism suggested that NO

119909

emissions can be reduced by increasing the intake manifoldexhaust gas recirculation (EGR) fraction and smoke can bereduced by increasing the airruel ratio (AFR) [1] The EGRand the AFR rates are controlled by the EGR and the variablegeometry turbine (VGT) actuators whose position deter-mines the amount of the EGR flow in the intake manifoldand thus controls the AFR and the EGR ratios variablesThe VGT and the EGR actuators are strongly coupled sothat conventional calibrationmapping based approacheswhich use traditional PI controllers face difficulty to producesatisfactory and robust results in terms of torque responsesengine-out emission at steady and transient state conditions

even with very time-consuming and detailed calibrationseffort

With the development of microprocessor technologyespecially digital signal processors (DSPs) electronic controlunits (ECUs) gained in terms of computational capacitiesso that sophisticated control approaches which achievesmooth fast and robust transient operations were moreand more introduced in controlling modern diesel enginesConsiderable research efforts have been dedicated to thecontrol of modern diesel engines Several controllers wereproposed in the literature for example Lyapunov controldesign [2] robust gain-scheduled controller based on a linearparameter-varying (LPV) model for turbocharged dieselengine [3ndash5] Indirect passivation [6] predictive control [7]and feedback linearization [8 9]

Most of these algorithms are control-oriented modelsthat is the control laws computed by these algorithms arebased upon a model of the diesel engine air path In thepast decade many TDE air path models were developedand presented in [2 10ndash13] In [2] the authors developed afull-seventh-order TDE air path model which describes thedynamics of several variables such as the pressures and the

2 Mathematical Problems in Engineering

oxygen mass fractions in the intake and exhaust manifoldsthe turbocharger speed and the two states of the controlsignals describing the EGR and the VGT actuator dynamicsTo simplify the control design and due to the fact thatthe oxygen mass fraction variables are difficult to measurethe seventh-order model is reduced to a third-order oneThis simplification leads to the appearance of discrepanciesbetween the description model and the real system due tothe neglected dynamics the model parametric uncertaintiesand the potential faults which can occur in the dieselengine air path Modeling uncertainties arise from sourcessuch as uncertainties on engines cartographies unmodeleddynamics and external or internal disturbances Enginefaults affect the TDE air path components For examplea leakage occurring on the EGR actuators is a fault thataffects the performance of any designed controller based onthe previous cited methods In this work both modelinguncertainties and actuator faults are considered as internaland external disturbances acting on the system via differentchannels Indeed the actuator faults naturally satisfy theso-called matching condition since they act on the controlinputs Modeling uncertainties are unmatched disturbancessince they act on different channels from the control inputand hence do not satisfy this matching condition The mainobjective of this paper is to design an efficient controllerwhich is robust facing the matched and the unmatcheddisturbances which affects the diesel engine air path

Sliding mode control [14ndash17] is a very popular strategyfor controlling uncertain nonlinear system Due to its con-ceptual simplicity and its capability to reject disturbancesandmodel uncertainties SMC strategies are widely employedin industrial applications including robotic manipulator sys-tems [18ndash20] electrical machines [21 22] power converters[23 24] and flight control systems [25 26] Despite theseattractive features themajor drawback of the traditional SMCcontrollers is that they are only insensitive to matched dis-turbance [27] while unmatched disturbance affects severelythe sliding motion and the well-known robustness of SMCdoes not hold any more To overcome this difficulty manyresearchers devoted themselves to the control design foruncertain system under mismatched disturbances

Generally speaking the matched and the unmatcheddisturbances in control systems are unmeasurable thus a dis-turbance observer is needed for the disturbance attenuationDisturbance observers can be classified into the following twocategories The first category is a model based disturbanceobserver in which the designer needs to know about themodel of the plant Many model based disturbance observerswere proposed in the literature including unknown inputobserver (UIO) [28] the disturbance observer (DOB) [2930] the perturbation observer [31] the equivalent inputdisturbance (EID) based estimation [32] and the slidingmode observer (SMO) which was primarily designed forrobust state estimation and then has been extended toestimate unknown inputs or faults by reconstructing thedisturbance using the so-called ldquoequivalent output errorinjectionrdquo [33]The second category is referred to as extendedstate observers (ESOs) [34ndash37] which is characterized bya promising features due to the least amount of plant

information needed for their implementation Comparingto model based disturbance observers (ESOs) techniquesneed only to know the system order and hence are morerobust and easy to implement from a practical point ofview However standard (ESO) observers suffer from onemajor drawback coming from the fact that they are onlyavailable for integral chain systems thus not applicable formore general class of system such as flight control system Inaddition the unmatched disturbances are no longer availablefor these standard ESO techniques Recently in [38] theauthors generalized the concept of ESO to nonintegral chainsystems subject to mismatched disturbances

To alleviate the well-known problem of chattering andretain the nominal control for the SMC in presence ofmatched and unmatched disturbances SMC disturbancebased feedforward control which combines a disturbanceobserver (DOB) with an SMC feedback control was devel-oped for matched disturbances in [39ndash43] and recently in[44] for the unmatched ones However in this work theconsidered diesel engine air path faces simultaneously thematched and the unmatched disturbances and none of theexisting disturbances observers whether it is model basedor not can estimate these disturbances simultaneously Inthis paper we propose a novel super twisting extended stateobserver (STESO) based on the concept of total disturbanceestimation and rejection via ESO introduced by the authorin [35] The structure of the proposed disturbance observercombines the advantages of the super twisting algorithm [45]which guarantee robustness and fast convergence rate of thedisturbance estimation with and the ESO concept which ischaracterized by the simplicity of its structure

The main contributions of the paper are as follows

(i) Developing a novel DOB based on the ESO frame-workwhich tracks the total disturbances in the systemin finite time

(ii) The chattering problem is significantly reduced bythe proposed SMC-STESO controller since the high-frequency switching gain is only required to bedesigned greater than the bound of the disturbanceestimation rather than that of the disturbance

(iii) Nominal performance recovery is also guaranteedby the SMC-STESO controller thanks to the STESOobserver which serves like a patch to the nominalcontroller and does not cause any adverse effects onthe system in the absence of disturbances

(iv) Compared to the works in [46ndash48] our approachexploits the benefits of the extended state observer(ESO) in the diesel engine air path control prob-lem Indeed by combining the sliding mode control(SMC)method with the super twisting extended stateobserver (STESO) the controller can achieve fastand accurate response via effective compensation forthe matched and the unmatched disturbances whichaffect the diesel engine air path

This paper is organized as follows Section 2 introducesthe TDE air path modeling Section 3 describes the faultydiesel engine air path system that we are dealing with

Mathematical Problems in Engineering 3

Variable geometry turbocharger (VGT)

Exhaust manifold (p2 T2)

Diesel engine

Exhaust gas

Compressor (PcWc)

Air

EGRcooler

Intercooler

Throttle valve

Intake manifold (p1 T1)

Figure 1 Turbocharged diesel engine

together with the assumptions required The super twistingextended state observer based sliding mode controller willbe described in Section 4 Simulation results are given inSection 5 Section 6 summarizes conclusions of our work

2 Mathematical Model of the Diesel EngineAir Path

The schematic diagram of the diesel engine is shown inFigure 1 At the top of the diagram we can see the tur-bocharger and the compressor mounted on the same shaftThe turbine delivers power to the compressor by transferringthe energy from the exhaust gas to the intake manifoldTogether the mixture of air from the compressor and theexhaust gas from the EGR valve with the injected fuel burnand producing the torque on the crank shaft

The full-order TDE model is a seventh-order one whichcontains seven states intake and exhaust manifold pressure(1199011and 119901

2) oxygen mass fractions in the intake and exhaust

manifolds (1198651and 119865

2) turbocharger speed (120596

119905119888) and the two

states describing the actuator dynamics for the two controlsignals (119906

1and 119906

2)

In order to obtain a simple control law and due to thefact that the oxygen mass fraction variables are difficult tomeasure the seventh-order model is reduced to a third-orderone [2] presented below

1= 1198961(119882119888+ 119882egr minus 119896

1198901199011) +

1

1198791

1199011

2= 1198962(1198961198901199011minus 119882egr minus 119882

119905+ 119882119891) +

2

1198792

1199012

119888=

1

120591(120578119898119875119905minus 119875119888)

(1)

where the compressor and the turbinemass flow rate (119882119888and

119882119905) are related to the compressor and the turbine power (119875

119888

and 119875119905) as follows

119882119888= 119875119888

119896119888

119901120583

1minus 1

(2)

119875119905= 119896119905(1 minus 119901

minus120583

2)119882119905 (3)

where

119896119888=

120578119888

119888119901119879119886

119896119905= 1198881199011205781199051198792

1198961=

1198771198861198791

1198811

119896119890=

120578V119873119881119889

1198771198861198791

1198962=

1198771198861198792

1198812

(4)

Notice that the real inputs are the EGR valve and theVGT valve openings The considered inputs in this case forthe sake of simplicity are 119906

1= 119882egr and 119906

2= 119882119905 which

are respectively the air flow through the EGR and the VGTvalves

Temperature sensors used for engine control have timeconstant in the order of seconds and hence are not fastenough to provide usable

1and 2signalsMoreover steady-

state values are not affected by the two terms and theirdynamics can be neglected [2] This yields to the followingsimplified model

1= 1198961(119882119888+ 119882egr minus 119896

1198901199011)

2= 1198962(1198961198901199011+ 119882119891minus 119882egr minus 119882

119905)

119888=

1

120591(120578119898119875119905minus 119875119888)

(5)

When replacing 119882119888and 119875

119905by their expressions in (2)

and (3) the simplified model can be expressed under thefollowing control-affine form

= 119891 (119909) + 1198921(119909) 1199061+ 1198922(119909) 1199062 (6)

where 119909 = (1199011 1199012 119875119888)119879 and

119891 (119909) =

[[[[[[[

[

1198961119896119888

119875119888

119901120583

1minus 1

minus 11989611198961198901199011

1198962(1198961198901199011+ 119882119891)

minus119875119888

120591

]]]]]]]

]

(7)

1198921(119909) = [

[

1198961

minus1198962

0

]

]

1198922(119909) = [

[

0

minus1198962

119870119900(1 minus 119901

minus120583

2)

]

]

(8)

with 119870119900= (120578119898120591)119896119905

We notice that the TDE model parameters (1198961 1198962

119896119888 119896119890 119896119905 120591 and 120578

119898) have been identified under steady-

state conditions (ie constant engine speed and constantfueling rate) and extensive mapping The nomenclature ofthe TDE parameters can be found in Table 1

Note that in [2] the authors proved that the setΩ definedby

Ω = (1199011 1199012 119875119888) 1 lt 119901

1lt 119875

max1

1 lt 1199012lt 119901

max2

0 lt 119875119888lt 119875

max119888

(9)

is an invariant set


Table 1 Nomenclature of the diesel engine variables

Variables Name Units1199011 Intake manifold pressure Pa

1199012 Exhaust manifold pressure Pa

119875119888 Compressor power W

119875119905 Turbine power W

119882119888 Compressor mass flow Kgs

119882119905 Turbine mass flow Kgs

119882119891 Fueling mass flow rate Kgs

120578V Engine volumetric efficiency mdash120578119888 Compressor isentropic efficiency mdash

120578119905 Turbine isentropic efficiency mdash

120578119898 Turbocharger mechanical efficiency mdash

1198811 Intake manifold volume m3

1198812 Exhaust manifold volume m3

119881119889 Engine volume cylinder m3

119879119886 Ambient temperature K

1198791 Intake manifold temperature K

1198792 Exhaust manifold temperature K

119877119886 Specific gas constant JKgK

3 Problem Formulation

31 System Fault Description In this paper we consider thediesel engine air path model ((6) (7) and (8)) which can bewritten under the following control-affine form

= 119891 (119909) + 119892 (119909) 119906 (10)

where 119909 isin R119899 119906 isin R119898 represent the state and the inputvector respectively The vector fields 119891 and columns 119892 aresupposed to satisfy the classical smoothness assumptionswith 119891(0) = 0

Assumption 1 System (10) is locally reachable (in the sense of[49 Definition 5 pp 400])

Assumption 2 A time additive actuator faults which enter thesystem in such a way that the faulty model can be written as

= 119891 (119909) + 119892 (119909) (119906 + 119865 (119909 119905)) (11)

where 119865(119909 119905) is bounded by an unknown positive constant1198631198981 that is

119865 (119909 119905) lt 1198631198981

(12)

Assumption 3 Thevector fields119891 andmatrix119892 are dependenton the systemparameters and any uncertainties in the systemparameters are reflected in them and can be expressed underthe following forms

Δ119891 = 119891 (119909) minus 1198910(119909)

Δ119892 = 119892 (119909) minus 1198920(119909)

with 1003817100381710038171003817Δ1198911003817100381710038171003817 le 119863

1199061003817100381710038171003817Δ119892

1003817100381710038171003817 le 1198631198982

(13)

where 1198910and 119892

0represent the nominal values of 119891 and 119892 and

119863119906 1198631198982

are the upper bounds of Δ119891 and Δ119892 respectively

Assumption 4 The external disturbances the unmodeleddynamics and the unknown dynamics which enters thesystem in different way from the control input can beexpressed in an uncertain part 119889 so that the global faulty-disturbed model of system (10) can be written as follows

= 119891 (119909) + 119892 (119909) (119906 + 119865 (119909 119905)) + 119889 (14)

32 Description of the Faulty Diesel Engine Air PathModel Inthis section we will present a description model of the faultydiesel engine air path The faulty disturbed model considersboth the actuator faults which affects directly the EGR andVGT actuators described in Assumption 2 and the uncertainpart 119889 Following (14) the faulty disturbed diesel engine airpath model is written as follows

1= 1198961119882119888+ 11989611199061minus 11989611198961198901199011+ 11989611198651

2= 11989621198961198901199011+ 1198962119882119891minus 11989621199061minus 11989621199062minus 1198962(1198651+ 1198652)

119888=

minus119875119888

120591+ 119870119900(1 minus 119901

minus120583

2) (1199062+ 1198652)

(15)

Clearly system (15) is under the form (14) with

119891 =(((

(

1198961119896119888

119875119888

119901120583

1minus 1

minus 11989611198961198901199011

11989621198961198901199011

minus119875119888

120591

)))

)

119892 = (

1198961

0

minus1198962

minus1198962

0 119870119900(1 minus 119901

minus120583

2)

)

119865 = (1198651

1198652

) 119889 = (0

1198962119882119891

)

(16)

Remark 5 (1198651 1198652) characterize the additive faults terms

which affects respectively the EGR and the VGT actuatorsPhysically it might represent a leakage which can occuron the EGR actuator and additive term which modifies theturbine flow and hence affects the VGT actuator

Remark 6 In this work the fueling sequence119882119891is supposed

to be unknown hence it is contained in 119889 as an externaldisturbance in (15)

33 Control Objective and Problem Statement For emissioncontrol the choice of feedback variables is an importantstep which defines the overall controller structure In thissection wewill present our proposed air path control strategywhich operates under the diesel conventional combustionmode conditions This particular mode was characterized bythe author in [48] In this work the author suggested that


for an optimal control performance compressor mass flow119882119888and exhaust pressure manifold 119901

2are suitable choice for

key output variable to be controlled By a suitable changeof coordinates the authors in [50] proposed to replace thecompressor mass flow setpoint (119882

119888119889) into an intake manifold

pressure setpoint (1199011119889

) This transformation simplifies thecontrol structure by defining new vector setpoint (119901

1119889 1199012119889

)Let us now consider system (15) and define the following twosliding variables 119878

1 1198782

1198781= 1199011minus 1199011119889

1198782= 1199012minus 1199012119889

(17)

The time derivative of 1198781 1198782along with the trajectories of the

diesel faulty system (15) leads to1198781= 1198961119882119888+ 11989611199061minus 11989611198961198901199011+ 11989611198651minus 1119889

1198782= 11989621198961198901199011+ 1198962119882119891minus 11989621199061minus 11989621199062minus 1198962(1198651+ 1198652) minus 2119889

(18)Replacing 119882

119862by its expression in (2) yields to

1198781= 1198961119896119888

119875119888

119901120583

1minus 1

+ 11989611199061minus 11989611198961198901199011+ 11989611198651minus 1119889


(19)The control objective of this paper can be stated as follows

Consider the faulty system (19) and the sliding manifolds119878 = (119878

1 1198782)119879 and find a stabilizing closed-loop control which

guarantee convergence of 119878 toward zero in finite time

4 Super Twisting Extended State ObserverBased Sliding Mode Control Design

In this section we will propose an observer based controldesign which force the sliding manifold 119878 in (19) to convergeto their reference states (119901

1119889 1199012119889

) The proposed controllercombines a sliding mode feedback with a super twistingextended state observer (STESO) which estimates the totaldisturbances whether they are matched or unmatched Toproceed the design of STESO-SMC controller the followinglemmas are required

Lemma 7 For the controlled output [119882119888 1199012] the states of the

zero dynamic 119875119888of the system (15) is stable

Proof The proof of this lemma can be found in [2]

Lemma 8 (see [51]) Assume that there exists a continuouspositive definite function 119881(119905) which satisfies the followinginequality

+ 1205811119881 (119905) + 120581

2119881120591

le 0 forall119905 gt 1199050 (20)

Then 119881(119905) converges to the equilibrium point in finite time 119905119904

119905119904le 1199050+

1

1205811(1 + 120591)

ln12058111198811minus120591

(1199050) + 1205812

1205812

(21)

where 1205811gt 0 120581

2gt 0 and 0 lt 120591 lt 1

41 SMC Control Design Consider now system (19) whichcan be rewritten under the following compact form

119878 = 119891119900(119909) + ((119892

119900(119909) + Δ119892) 119906 + 119892 (119909) 119865 (119909 119905))

+ Δ119891 + 119889 minus 119878119889

(22)

where 119878 = (1198781

1198782

) 1198910

= (1198961119900119896119888119900(119875119888119901120583119900

1minus1)minus119896

11199001198961198901199001199011

11989621199001198961198901199001199011

) 1198920

=

(11989611199000

minus1198962119900minus1198962119900

) 119865 = (1198651

1198652

) 119889 = (0

1198962119882119891) 119878119889= (1119889

2119889

)The control design will be achieved under the following

assumptions

Assumption 9 The states (1199011 1199012 119875119888) of system (19) are avail-

able for measurements at every instant

Assumption 10 The nominal control distribution functionmatrix 119892

0is nonsingular

Assumption 11 To guaranty the local reachability of system(19) the additive faults 119865

1 1198652are uniquely time dependent

System (22) contains both nominal parts 1198910 1198920which

are completely known and parametric uncertainties Δ119891 Δ119892

and external disturbance 119889 which are unknown for the TDEmodel air path In this paper model parametric uncertaintiesand disturbance are lumped together as the total distur-bances and thus the further simplified model form forsystem (22) can be written as follows

119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (23)

with 119889 = Δ119892119906 + 119892(119909)119865(119909 119905) + Δ119891 + 119889Let us now consider the following reaching law [52]

119878 = minus120574119878 minus 120582 sign 119878 (24)

where 119878 isin R2 119906 isin R2 120574 = diag[1205741 1205742] 120582 = diag[120582

1 1205822]

sign 119878 = [sign(1198781) sign(119878

2)]119879

Combining (23) and (24) yields to

minus120574119878 minus 120582 sign 119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (25)

The proposed control law 119906 is split into two parts 119906eq tocompensate for known terms and 119906

119899to compensate for the

the lumped uncertainty 119889 Thus

119906 = 119906eq + 119906119899 (26)

where

119906eq = 119892minus1

0(119909) (minus120574119878 minus 120582 sign 119878 minus 119891

0(119909) + 119878

119889)

119906119899= minus 119892

minus1

0(119909) (119889)

(27)

Hence the total control 119906 is expressed as follows

119906 = 119892minus1


0(119909) minus 119889 + 119878

119889) (28)

Note that the control 119906119899consists of the disturbance

terms 119889 which is completely unknown thus could not beimplemented into practical systems In order to obtain thedisturbances we will introduce the ESO to estimate it


42 Super Twisting ESODesign Following the general frame-work of ESO and inspired by the work in [36 53] wedesign in this section an observer which estimates the totaldisturbances 119889 in (23) Adding an extended state 119883 as thetotal disturbances 119889 to system (23) yields to the followingaugmented system

119878 = 1198910(119909) + 119892

0(119909) 119906 + 119883 minus 119878

119889

= ℎ (119905)

(29)

where ℎ(119905) is the derivative of the disturbances 119889 For system(29) we propose the following STESO

1198641= 1198851minus 119878

1= 1198852+ 1198910+ 1198920119906 minus 119878119889minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1)

2= minus 120573

2sign (119864

1)

(30)

where 1198641isin R2 119885

1isin R2 and 119885

2isin R2

1198641is the estimation error of the ESO 119885

1and 119885

2are the

observer internal states and 1205731and 120573

2are the observer gains

Define the observer error dynamics 1198641= 1198851minus119878 119864

2= 1198852minus119889

Assumption 12 There exists a constants 1205751such that

|ℎ (119905)| lt 1205751 (31)

Theorem 13 Consider the uncertain faulty augmented system(29) The STESO observer (30) ensures that the observer error[1198641 1198642] converges to zero in a finite time if the gains 120573

1 1205732

satisfy the following conditions [54]

1205731gt 0

1205732gt

61205751+ 4(12057511205731)2

2

(32)

Proof The observer error dynamic is given by

1= 1198852minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) minus 119883

2= 2minus ℎ (119905)

(33)

which yields to

1= minus 120573

1

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) + 1198642

2= minus 120573

2sign (119864

1) + 1205881

(34)

with 1205881= minusℎ(119905)

It is easy to see that system (34) is under the generalizedsuper twisting algorithm differential inclusion and it is finitetime stable (see [54]) This completes the proof

Remark 14 The class of disturbances considered in thispaper is much larger than the ones considered [44] whereit is assumed that the disturbances are bounded and theirderivatives vanish as 119905 rarr infin It may be noted that it is notrequired to know the bound 120575

1

Remark 15 We notice that the expression of 1198852in (30)

shows that 1198852can estimate (or track) the total action of

the uncertain models and the external disturbances 119889 infinite time As 119885

2is the estimation for the total action of

the unknown disturbances in the feedback 1198852is used to

compensate the disturbances

With the disturbances 119889 estimated by the STESO thecontrol law (28) is modified as follows

119906 = 119892minus1


0(119909) minus 119885

2+ 119878119889) (35)

43 Stability Analysis of Closed-Loop Dynamics Replacingthe control law (35) in (23) yields to the following closed-loopsystem

119878 = minus120574119878 minus 120582 sign 119878 + (119889 minus 1198852) (36)

which can be written as follows

119878 = minus120574119878 minus 120582 sign 119878 minus 1198642 (37)

Theorem 16 For the plant (23) with the control law (35)and the STESO observer (30) the trajectory of the closed-loopsystem (37) can be driven to the origin in finite time

Proof Consider the following Lyapunov function

119881 =1

2119878(119905)119879

119878 (119905) (38)

Taking the derivative of (38) yields to

= 119878(119905)119879

(minus120574119878 minus 120582 sign 119878 minus 1198642)

= minus

2

sum

119894=1

(1205741198941198782

119894+ 120582119894

10038161003816100381610038161198781198941003816100381610038161003816) minus 119878119879

1198642

le minus

2

sum

119894=1

1205741198941198782

119894minus

2

sum

119894=1

(120582119894minus

100381610038161003816100381611986421198941003816100381610038161003816)

10038161003816100381610038161198781198941003816100381610038161003816

(39)

Choosing 120582119894

gt |1198642119894| ensures by Lemma 8 the finite time

stability of the sliding manifolds 119878

Remark 17 Since the disturbance has been precisely esti-mated by the STESO the magnitude of the estimation error1198642119894which converges to 0 can be kept much smaller than

the magnitude of the disturbance 119889 this will alleviate thechattering phenomenon

Remark 18 (Performance Nominal Recovery) In the absenceof disturbance it is derived from the observer error dynamics(34) that is 119864

2119894rarr 0 In this case the closed-loop dynamics

(37) reduces to those of the traditional sliding mode control(TSMC) This implies that the nominal control performanceof the proposed method is retained


Table 2 Control requirements for the STESO-SMC controller

Variable Setpoint 1 Setpoint 2119882cd (Kgs) 003 004

1198752119889(Bar) 145 175

5 Simulation and Evaluation

In this section we report numerical results obtained fromthe simulation of controller (35) on the reduced third-order model developed in ((1)ndash(8)) The engine used is acommon rail direct-injection in-line-4-cylinder provided bya French manufacturer Numerical values of (120578

119905 120578119888 120578V and

120578119898) cartographies in the TDE model were provided by the

manufacturerThemodel parameters nominal values 1198961199001 1198961199002

119896119900119888 119896119900119890 119896119900119905 120591119900 120578119900119898 and 120583

119900are usually identified around some

given operating points In these simulations the parametersof the model ((1)ndash(8)) were taken from [6] that is 119896

1199001=

14391 1198961199002

= 17155 119896119900119888

= 00025 119896119900119890

= 0028 119896119900119905

=

391365 120591119900= 015 120578

119900119898= 095 and 120583

119900= 0285 A well-known

continuous approximation of the function sign(119878) is given by

Sign (119878) =119904

|119904| + 120585 (40)

This approximation is used to ensure that the sliding motionwill be in the vicinity of the line (119878 = 0) In this simulationthe approximation of the sign function has been implementedwith 120585 = 001 The simulations were conducted under thecontrol requirements defined in Table 2

In this section we mention that numerical simulationswere performed in real-time software in the loop (SIL)using the dSpace modular simulator This real-time plat-form is based on the DS-1006 board interfaced with MAT-LABSimulink software The controller (35) and the TDEthird-order model run on this platform with a sampling steptime equal to 10

minus4 s For the purpose of simulations paramet-ric uncertainties Δ119891 and actuator faults 119865 were chosen underperiodic forms (like sinus) The goal was to only prove theeffectiveness of the STESO observer and the controller (35)From a practical point of view it is very difficult to modeleither a leakage affecting the EGR and the VGT actuators orparametric uncertainties affecting the TDE air path modelWe cannot also say that the EGR and the VGT actuator faultsare periodic however and as stated in Remark 14 the onlyassumption wemade on the total disturbances 119889 is on its timederivativeswhichmust be boundedThus complex leakage orparametric uncertainties forms which can be more realisticcan be considered in practical situations

Case 1 (traditional SMC control versus STESO-SMC controlin case of matched disturbances) In this simulation wewanted to compare the performance of the STESO-SMCand the traditional SMC controllers in the case where onlymatched disturbances are considered The two controllerswere simulated under nominal parametric values (ieΔ119891 = 0

and Δ119892 = 0) and fueling sequence 119882119891

= 0

Table 3 Control gains of the TSMC and the STESO-SMC con-trollers in Case 1

Controllers Controllers parametersTSMC 120582 = 100 120574 = 250

STESO-SMC 120582 = 3 120574 = 250

20 25 30 35

0 5 10 15 20 25 30 35 40001002003004005006007

Time (s)

Time (s)0 5 10 15 40

131415161718

248 25 252 254 256172174176

248 25 2520038

0040042

Wc

(kg

s)

Wc TSMCWc STESO-SMCWc setpoint

P2

(b)

P2 TSMCP2 STESO-SMCP2 setpoint

Figure 2 Controlled variables 119882119888 1199012in Case 1

The consideredmatched disturbanceswere a time varyingadditive leakage which takes the following form

119865 (119905) = 0 times (1 1)

119879 if 119905 lt 25 sminus004 + 001 sin (02120587119905) times (1 1)

119879 if 119905 ge 25 s(41)

For the TSMCand the STESO-SMC controllers the gains 120582 120574(see Table 3) were tuned in order to get an equivalent trackingperformance for the controlled variables 119882

119888and 119901

2

Response curves of the two controlled variables119882119888 1199012are

shown in Figure 2 A brief observation from Figure 2 showsthat for 119905 lt 25 s the STESO-SMCcontroller exhibits the samelevel of tracking performance comparing to the TSMC oneWhen the faults occur it can be observed from Figure 2 thatthe proposed STESO-SMC obtains fine disturbance rejectionproperty with minimal chattering effects while the TSMCkeeps well the tracking performance of 119882

119888 1199012but pay high

price in terms of chattering due to the higher value of theswitching gain 120582 needed for the TSMC in order to counteractthe occurrence of the faults at 119905 = 25 s

This can be observed in Figure 3 where we can see thatthe control effort (119880egr 119880vgt) produced by the STESO-SMC


0 5 10 15 20 25 30 35 40

0

005

01

015

Time (s)

0 5 10 15 20 25 30 35 40

0005

01

Time (s)

minus015

minus01

minus005

minus005

UEG

R(k

gs)

UEGR TSMC (kgs)UEGR STESO-SMC (kgs)

UVG

T(k

gs)

UVGT TSMC (kgs)UVGT STESO-SMC (kgs)

Figure 3 Control effort of the EGR and the VGT actuators for theTSMC and the STESO-SMC controllers

0

5

0 5 10 15 20 25 30 35 40Time (s)

0 5 10 15 20 25 30 35 40Time (s)

010203040

minus15

minus10

minus10

minus5

d1 = k1o times F1Z2 = k1o times F1

d2 = minusk2o (F1 + F2)

Z2 = minusk2o(F1 + F2)

Figure 4 Estimation of matched disturbances via STESO in Case 1

controller is characterized by the low level of the chatteringcomparing to the TSMC one

When only matched disturbances are considered thetotal disturbance 119889 in system (23) is expressed as follows

119889 = (11989611199001198651

minus11989621199001198651minus 11989621199001198652

) (42)

Selecting appropriate values of 12057311

= 12057321

= 12057312

= 12057322

=

1000 for the STESO observer one can observe from Figure 4that it can estimate the total disturbance accurately even ifthere exist some small shocks due to the drastic changeswhich occur on the vector derivatives 119878

119889 It is important to

mention that the tracking performance and the disturbancerejection of the STESO-SMC depend on the estimation of 119889provided by the STESO

Case 2 (performance of the STESO-SMC Control in pres-ence of unmatched disturbances and matched disturbances)In this simulation we wanted to show the performanceof the STESO-SMC controller when both matched andunmatched disturbances are considered To the matcheddisturbances considered in Case 1 we assume also thefollowing unmatched disturbances

(i) Parametric variation Δ119891 takes the following form

Δ119891 = (02 sin (02120587119901

1119905)

05 sin (031205871199012119905)) (43)

(ii) Fueling sequence takes the following form

119882119891

= 0 kgh if 119905 lt 5 s7 kgh if 119905 ge 5 s

(44)

Hence the total disturbance 119889 in system (23) at differenttime is expressed as follows

For 119905 lt 5 s

119889 = (Δ1198911

Δ1198912

) (45)

For 5 lt 119905 lt 25 s

119889 = (Δ1198911

1198962119900119882119891+ Δ1198912

) (46)

For 119905 ge 25 s

119889 = (Δ1198911+ 11989611199001198651

Δ1198912+ 1198962119900119882119891minus 11989621199001198651minus 11989621199001198652

) (47)

To test the nominal performance recovery the totaldisturbances 119889 whether it is matched or unmatched isnullified at 119905 = 35 s

Figure 5 shows the response curves of the two controlledvariables 119882

119888 1199012when both matched and matched uncer-

tainties are considered In this simulation we mention thatthe control parameters 120582 120574 were kept the same as in Case 1(120582 = 3 120574 = 250) A brief observation from Figure 5 showsthat the STESO-SMC controller successfully rejects from thebeginning of the simulation the matched and the unmatcheddisturbances thanks to the action of STESO A comparisonbetween the performance of the STESO-SMC in Cases 1 and2 (see Figures 2 and 5) shows that the STESO-SMC controllerexhibits the same level of tracking performance despite thefact that in Case 2 unmatched disturbances were added tothe plant This remark can be explained by the fact that the


0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)

Wc STESO-SMCWc setpoint

14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)

P2 STESO-SMCP2 setpoint


control parameters (120582 120574) were maintained the same as it wasin Case 1 and since the STESO converges (119864

2119894rarr 0) whether

matched and unmatched disturbances are considered theclosed-loop dynamics in Cases 1 and 2 is the same

Figure 6 shows the control effort (119880egr 119880vgt) produced bythe STESO-SMC controller in Case 2 It can be observed fromFigure 6 that the STESO-SMC manages to compensate thematched and the unmatched disturbances from the begin-ning of the simulationWhenwe nullified all the disturbanceswhich affect the plant at 119905 = 35 s we could see that the twocontrols produced by the STESO-SMCandTSMC in nominalcase are the same This means that the STESO-SMC recoversthe nominal performances which validates Remark 18

Figure 7 shows the performance of the STESO in termsof estimating the total disturbances in Case 2 We keep thesame values 120573

11= 12057321

= 12057312

= 12057322

= 1000 for theSTESO in this case One can observe from Figure 7 that thereis a good tracking of the disturbances from the beginningof the simulation We conclude by noticing that the perfor-mance of the STESO-SMC controllers depend on the controlparameters (120582 120574) and the STESO gains 120573

11 12057321 12057312 and 120573

22

Each of them can be tuned separately without a compromisebetween the chattering problem and the convergence of thesliding surface Indeed to alleviate the chattering the controlparameters required to be only greater than the disturbanceserror and to ensure fast convergence of the sliding surface theSTESO gains can be chosen so that we get a fast convergenceof the disturbance error which lead to fast convergence of thesliding surface

0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045

Nominal performancerecovery


minus0005

UEG

R(k

gs)

UVG

T(k

gs)

UEGR TSMC (nominal case) (kgs)UEGR STESO-SMC with disturbances (kgs)

TSMC (nominal case) (kgs)UVGT

UVGT STESO-SMC with disturbances (kgs)

Figure 6 Control effort of the EGR and the VGT actuators for theTSMC (nominal case) and the STESO-SMC controllers in Case 2

0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances

No disturbancesUnmatched disturbances

Unmatched disturbances

minus5

minus4

minus3

minus2

minus1

Unmatched+ matcheddisturbances


d2 = Δf2 + k2oWf minus k2oF1 minus k2oF2

d1 = Δf1 + times F1k1oZ2 = Δf1 + times F1k1o

Z2 = Δf2 + k2oWf minus k2oF1 minus k2oF2

Figure 7 Estimation of matched and unmatched disturbances viaSTESO in Case 2

6 Conclusions

In this paper a novel DOB based SMC namely STESO-SMCcontroller has been proposed to attenuate the matched andthe unmatched disturbances for the purpose of controllingthe diesel engine air path The main contribution here is


the design of a new STESO observer in such a way that thesliding motion along the sliding surface can drive the statesto their desired trajectories in finite time As compared withthe traditional SMC the proposed controller has exhibitedtwo superiorities in the simulation including nominal perfor-mance recovery and chattering effect reduction

Conflict of Interests

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

Acknowledgments

The authors gratefully thank Region Haute-NormandieOSEO and FEDER for financially supporting this work andits forthcoming application to an ICE control within theframework of the VIRTUOSE Project

References

[1] L Eriksson J Wahlstrom and L Nielsen ldquoEGR-VGT controland tuning for pumping work minimization and emissioncontrolrdquo IEEE Transactions on Control Systems Technology vol18 no 4 pp 993ndash1003 2010

[2] M Jankovic M Jankovic and I Kolmanovsky ldquoConstructiveLyapunov control design for turbocharged diesel enginesrdquo IEEETransactions on Control Systems Technology vol 8 no 2 pp288ndash299 2000

[3] M Jung and K Glover ldquoCalibratable linear parameter-varyingcontrol of a turbocharged diesel enginerdquo IEEE Transactions onControl Systems Technology vol 14 no 1 pp 45ndash62 2006

[4] L Lihua W Xiukun and L Xiaohe ldquoLPV control for theair path system of diesel enginesrdquo in Proceedings of the IEEEInternational Conference on Control and Automation (ICCArsquo07) pp 873ndash878 June 2007

[5] X Wei and L del Re ldquoGain scheduled 119867infin

control for airpath systems of diesel engines using LPV techniquesrdquo IEEETransactions on Control Systems Technology vol 15 no 3 pp406ndash415 2007

[6] M Larsen M Jankovic and P V Kokotovic ldquoIndirect passiva-tion design for diesel engine modelrdquo in Proceedings of the 2000IEEE International Conference on Control Applications pp 309ndash314 2000

[7] H J Ferreau P Ortner P Langthaler L D Re and M DiehlldquoPredictive control of a real-world Diesel engine using anextended online active set strategyrdquo Annual Reviews in Controlvol 31 no 2 pp 293ndash301 2007

[8] A Plianos A Achira R Stobart N Langlois and H ChafoukldquoDynamic feedback linearization based control synthesis ofthe turbocharged Diesel enginerdquo in Proceedings of the IEEEAmerican Control Conference (ACC rsquo07) pp 4407ndash4412 July2007

[9] M Dabo N Langlois and H Chafouk ldquoDynamic feedback lin-earization applied to asymptotic tracking generalization aboutthe turbocharged diesel engine outputs choicerdquo in Proceedingsof the IEEE American Control Conference (ACC rsquo09) pp 3458ndash3463 June 2009

[10] L Guzzella and A Amstutz ldquoControl of diesel enginesrdquo IEEEControl Systems Magazine vol 18 no 5 pp 53ndash71 1998

[11] MKao and J JMoskwa ldquoTurbocharged diesel enginemodelingfor nonlinear engine control and state estimationrdquo Journal ofDynamic Systems Measurement and Control vol 117 no 1 pp20ndash30 1995

[12] I V Kolmanovsky P E Moraal M J van Nieuwstadt and AStefanopoulou ldquoIssues in modelling and control of intake flowin variable geometry turbocharged enginesrdquo in Proceedings ofthe 18th IFIP conference on system modelling and optimizationpp 436ndash445 1997

[13] P E Moraal M J van Nieuwstadt and I V KolmanovskyldquoModelling and control of a variable geometry turbochargeddiesel enginerdquo in Proceedings of the COSY Workshop EuropeanControl Conference 1997

[14] V I Utkin ldquoVariable structure systems with sliding mode asurveyrdquo IEEE Transactions on Automatic Control vol 22 no 5pp 212ndash222 1977

[15] J Y Hung W Gao and J C Hung ldquoVariable structure controlA surveyrdquo IEEE Transactions on Industrial Electronics vol 40no 1 pp 2ndash22 1993

[16] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[17] A Pisano and E Usai ldquoSliding mode control a survey withapplications in mathrdquo Mathematics and Computers in Simula-tion vol 81 no 5 pp 954ndash979 2011

[18] S Islam and X P Liu ldquoRobust sliding mode control for robotmanipulatorsrdquo IEEE Transactions on Industrial Electronics vol58 no 6 pp 2444ndash2453 2011

[19] N Chen F Song G Li X Sun and C Ai ldquoAn adaptivesliding mode backstepping control for the mobile manipulatorwith nonholonomic constraintsrdquo Communications in NonlinearScience andNumerical Simulation vol 18 no 10 pp 2885ndash28992013

[20] M L Corradini V Fossi A Giantomassi G Ippoliti S Longhiand G Orlando ldquoDiscrete time sliding mode control of roboticmanipulators development and experimental validationrdquo Con-trol Engineering Practice vol 20 pp 816ndash822 2012

[21] Z Zhang H Xu L Xu and L E Heilman ldquoSensorless directfield-oriented control of three-phase inductionmotors based onldquosliding moderdquo for washing-machine drive applicationsrdquo IEEETransactions on Industry Applications vol 42 no 3 pp 694ndash701 2006

[22] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011

[23] R Wai and L Shih ldquoDesign of voltage tracking control for DC-DC boost converter via total sliding-mode techniquerdquo IEEETransactions on Industrial Electronics vol 58 no 6 pp 2502ndash2511 2011

[24] Y Shtessel S Baev and H Biglari ldquoUnity power factor controlin three-phase ACDC boost converter using sliding modesrdquoIEEE Transactions on Industrial Electronics vol 55 no 11 pp3874ndash3882 2008

[25] B Xiao Q Hu and Y Zhang ldquoAdaptive sliding mode faulttolerant attitude tracking control for flexible spacecraft underactuator saturationrdquo IEEE Transactions on Control SystemsTechnology vol 20 pp 1605ndash1612 2012

[26] Q Hu ldquoAdaptive output feedback sliding-mode manoeuvringand vibration control of flexible spacecraft with input satura-tionrdquo IET Control Theory amp Applications vol 2 no 6 pp 467ndash478 2008


[27] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007

[28] C Johnson ldquoAccommodation of external disturbances in linearregulator and servomechanism problemsrdquo IEEE Transactionson Automatic Control vol 16 no 6 pp 635ndash644 1971

[29] W Chen ldquoDisturbance observer based control for nonlinearsystemsrdquo IEEEASME Transactions on Mechatronics vol 9 no4 pp 706ndash710 2004

[30] L Guo andWChen ldquoDisturbance attenuation and rejection forsystems with nonlinearity via DOBC approachrdquo InternationalJournal of Robust and Nonlinear Control vol 15 no 3 pp 109ndash125 2005

[31] S J Kwon and W K Chung ldquoA discrete-time design and anal-ysis of perturbation observer for motion control applicationsrdquoIEEE Transactions on Control Systems Technology vol 11 no 3pp 399ndash407 2003

[32] J She X Xin and Y Pan ldquoEquivalent-input-disturbanceapproachanalysis and application to disturbance rejection indual-stage feed drive control systemrdquo IEEE Transactions onMechatronics vol 16 no 2 pp 330ndash340 1998

[33] T Floquet C Edwards and S K Spurgeon ldquoOn sliding modeobservers for systems with unknown inputsrdquo in Proceedings ofthe International Workshop Variable Structure System pp 5ndash72006

[34] J Han ldquoExtended state observer for a class of uncertain plantsrdquoControl and Decision vol 10 no 1 pp 85ndash88 1995

[35] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[36] D Sun ldquoComments on active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 54 no 6 pp3428ndash3429 2007

[37] Y Xia P Shi G P Liu D Rees and J Han ldquoActive disturbancerejection control for uncertainmultivariable systems with time-delayrdquo IET Control Theory and Applications vol 1 no 1 pp 75ndash81 2007

[38] S Li J Yang W H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transaction on Industrial Electronics vol56 no 12 pp 4792ndash4802 2012

[39] X Wei and L Guo ldquoComposite disturbance-observer-basedcontrol and terminal sliding mode control for non-linearsystemswith disturbancesrdquo International Journal of Control vol82 no 6 pp 1082ndash1098 2009

[40] M Chen and W-H Chen ldquoSliding mode control for a classof uncertain nonlinear system based on disturbance observerrdquoInternational Journal of Adaptive Control and Signal Processingvol 24 no 1 pp 51ndash64 2010

[41] S Li K Zong and H Liu ldquoA composite speed controller basedon a second-order model of permanent magnet synchronousmotor systemrdquo IEEE Transactions on Instrumentation Measureand Control vol 33 no 5 pp 522ndash541 2011

[42] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009

[43] Y Lu and C Chiu ldquoA stability-guaranteed integral slidingdisturbance observer for systems suffering from disturbanceswith bounded first time derivativesrdquo International Journal of

Control Automation and Systems vol 9 no 2 pp 402ndash4092011

[44] J Yang S Li and X Yu ldquoSliding-mode control for systems withmismatched uncertainties via a disturbance observerrdquo IEEETransaction on Industrial Electronics vol 60 pp 160ndash169 2013

[45] L Fridman and A Levant ldquoHigher order sliding modesrdquo inSliding Mode Control in Engineering chapter 3 pp 53ndash101Marcel Dekker 2002

[46] V I Utkin H Chang I Kolmanovsky and J A CookldquoSliding mode control for variable geometry turbochargeddiesel enginesrdquo in Proceedings of the IEEE American ControlConference (ACC rsquo00) pp 584ndash588 June 2000

[47] D Upadhyay V I Utkin and G Rizzoni ldquoMultivariable controldesign for intake flow regulation of a diesel engine using slidingmoderdquo in Proceedings of the 15th Triennal IFACWorld Congresspp 1389ndash1394 2002

[48] JWang ldquoHybrid robust air-path control for diesel engines oper-ating conventional and low temperature combustion modesrdquoIEEE Transactions on Control Systems Technology vol 16 no6 pp 1138ndash1151 2008

[49] M Vidyasagar Nonlinear Systems Analysis Prentice-HallEnglewood Cliffs NJ USA 2nd edition 1993

[50] S A Ali B Nrsquodoye and N Langlois ldquoSliding mode control forturbocharged diesel enginerdquo in Proceedings of the InternationnalConference on Control and Automation (MED) pp 996ndash10012012

[51] M Zhihong and X H Yu ldquoTerminal sliding mode controlof MIMO linear systemsrdquo IEEE Transactions on Circuits andSystems I vol 44 no 11 pp 1065ndash1070 1997

[52] W Gao and J C Hung ldquoVariable structure control of nonlinearsystems A new approachrdquo IEEE Transactions on IndustrialElectronics vol 40 no 1 pp 45ndash55 1993

[53] Y Xia Z Zhu M Fu and S Wang ldquoAttitude tracking of rigidspacecraft with bounded disturbancesrdquo IEEE Transactions onIndustrial Electronics vol 58 no 2 pp 647ndash659 2011

[54] J A Moreno and M Osorio ldquoA Lyapunov approach to second-order sliding mode controllers and observersrdquo in Proceedings ofthe 47th IEEE Conference on Decision and Control (CDC rsquo08)pp 2856ndash2861 December 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


oxygen mass fractions in the intake and exhaust manifoldsthe turbocharger speed and the two states of the controlsignals describing the EGR and the VGT actuator dynamicsTo simplify the control design and due to the fact thatthe oxygen mass fraction variables are difficult to measurethe seventh-order model is reduced to a third-order oneThis simplification leads to the appearance of discrepanciesbetween the description model and the real system due tothe neglected dynamics the model parametric uncertaintiesand the potential faults which can occur in the dieselengine air path Modeling uncertainties arise from sourcessuch as uncertainties on engines cartographies unmodeleddynamics and external or internal disturbances Enginefaults affect the TDE air path components For examplea leakage occurring on the EGR actuators is a fault thataffects the performance of any designed controller based onthe previous cited methods In this work both modelinguncertainties and actuator faults are considered as internaland external disturbances acting on the system via differentchannels Indeed the actuator faults naturally satisfy theso-called matching condition since they act on the controlinputs Modeling uncertainties are unmatched disturbancessince they act on different channels from the control inputand hence do not satisfy this matching condition The mainobjective of this paper is to design an efficient controllerwhich is robust facing the matched and the unmatcheddisturbances which affects the diesel engine air path

Sliding mode control [14ndash17] is a very popular strategyfor controlling uncertain nonlinear system Due to its con-ceptual simplicity and its capability to reject disturbancesandmodel uncertainties SMC strategies are widely employedin industrial applications including robotic manipulator sys-tems [18ndash20] electrical machines [21 22] power converters[23 24] and flight control systems [25 26] Despite theseattractive features themajor drawback of the traditional SMCcontrollers is that they are only insensitive to matched dis-turbance [27] while unmatched disturbance affects severelythe sliding motion and the well-known robustness of SMCdoes not hold any more To overcome this difficulty manyresearchers devoted themselves to the control design foruncertain system under mismatched disturbances

Generally speaking the matched and the unmatcheddisturbances in control systems are unmeasurable thus a dis-turbance observer is needed for the disturbance attenuationDisturbance observers can be classified into the following twocategories The first category is a model based disturbanceobserver in which the designer needs to know about themodel of the plant Many model based disturbance observerswere proposed in the literature including unknown inputobserver (UIO) [28] the disturbance observer (DOB) [2930] the perturbation observer [31] the equivalent inputdisturbance (EID) based estimation [32] and the slidingmode observer (SMO) which was primarily designed forrobust state estimation and then has been extended toestimate unknown inputs or faults by reconstructing thedisturbance using the so-called ldquoequivalent output errorinjectionrdquo [33]The second category is referred to as extendedstate observers (ESOs) [34ndash37] which is characterized bya promising features due to the least amount of plant

information needed for their implementation Comparingto model based disturbance observers (ESOs) techniquesneed only to know the system order and hence are morerobust and easy to implement from a practical point ofview However standard (ESO) observers suffer from onemajor drawback coming from the fact that they are onlyavailable for integral chain systems thus not applicable formore general class of system such as flight control system Inaddition the unmatched disturbances are no longer availablefor these standard ESO techniques Recently in [38] theauthors generalized the concept of ESO to nonintegral chainsystems subject to mismatched disturbances

To alleviate the well-known problem of chattering andretain the nominal control for the SMC in presence ofmatched and unmatched disturbances SMC disturbancebased feedforward control which combines a disturbanceobserver (DOB) with an SMC feedback control was devel-oped for matched disturbances in [39ndash43] and recently in[44] for the unmatched ones However in this work theconsidered diesel engine air path faces simultaneously thematched and the unmatched disturbances and none of theexisting disturbances observers whether it is model basedor not can estimate these disturbances simultaneously Inthis paper we propose a novel super twisting extended stateobserver (STESO) based on the concept of total disturbanceestimation and rejection via ESO introduced by the authorin [35] The structure of the proposed disturbance observercombines the advantages of the super twisting algorithm [45]which guarantee robustness and fast convergence rate of thedisturbance estimation with and the ESO concept which ischaracterized by the simplicity of its structure

The main contributions of the paper are as follows

(i) Developing a novel DOB based on the ESO frame-workwhich tracks the total disturbances in the systemin finite time

(ii) The chattering problem is significantly reduced bythe proposed SMC-STESO controller since the high-frequency switching gain is only required to bedesigned greater than the bound of the disturbanceestimation rather than that of the disturbance

(iii) Nominal performance recovery is also guaranteedby the SMC-STESO controller thanks to the STESOobserver which serves like a patch to the nominalcontroller and does not cause any adverse effects onthe system in the absence of disturbances

(iv) Compared to the works in [46ndash48] our approachexploits the benefits of the extended state observer(ESO) in the diesel engine air path control prob-lem Indeed by combining the sliding mode control(SMC)method with the super twisting extended stateobserver (STESO) the controller can achieve fastand accurate response via effective compensation forthe matched and the unmatched disturbances whichaffect the diesel engine air path

This paper is organized as follows Section 2 introducesthe TDE air path modeling Section 3 describes the faultydiesel engine air path system that we are dealing with




Diesel engine

Exhaust gas

Compressor (PcWc)

Air

EGRcooler

Intercooler

Throttle valve










119905119888) and the two


1and 119906

2)


1= 1198961(119882119888+ 119882egr minus 119896

1198901199011) +

1

1198791

1199011

2= 1198962(1198961198901199011minus 119882egr minus 119882

119905+ 119882119891) +

2

1198792

1199012

119888=

1

120591(120578119898119875119905minus 119875119888)

(1)



119888


119882119888= 119875119888

119896119888

119901120583

1minus 1

(2)

119875119905= 119896119905(1 minus 119901

minus120583

2)119882119905 (3)

where

119896119888=

120578119888

119888119901119879119886

119896119905= 1198881199011205781199051198792

1198961=

1198771198861198791

1198811

119896119890=

120578V119873119881119889

1198771198861198791

1198962=

1198771198861198792

1198812

(4)


1= 119882egr and 119906

2= 119882119905 which





1= 1198961(119882119888+ 119882egr minus 119896

1198901199011)

2= 1198962(1198961198901199011+ 119882119891minus 119882egr minus 119882

119905)

119888=

1

120591(120578119898119875119905minus 119875119888)

(5)




= 119891 (119909) + 1198921(119909) 1199061+ 1198922(119909) 1199062 (6)

where 119909 = (1199011 1199012 119875119888)119879 and

119891 (119909) =

[[[[[[[

[

1198961119896119888

119875119888

119901120583

1minus 1

minus 11989611198961198901199011

1198962(1198961198901199011+ 119882119891)

minus119875119888

120591

]]]]]]]

]

(7)

1198921(119909) = [

[

1198961

minus1198962

0

]

]

1198922(119909) = [

[

0

minus1198962

119870119900(1 minus 119901

minus120583

2)

]

]

(8)

with 119870119900= (120578119898120591)119896119905


119896119888 119896119890 119896119905 120591 and 120578




Ω = (1199011 1199012 119875119888) 1 lt 119901

1lt 119875

max1

1 lt 1199012lt 119901

max2

0 lt 119875119888lt 119875

max119888

(9)

is an invariant set






















= 119891 (119909) + 119892 (119909) 119906 (10)




= 119891 (119909) + 119892 (119909) (119906 + 119865 (119909 119905)) (11)


119865 (119909 119905) lt 1198631198981

(12)


Δ119891 = 119891 (119909) minus 1198910(119909)

Δ119892 = 119892 (119909) minus 1198920(119909)

with 1003817100381710038171003817Δ1198911003817100381710038171003817 le 119863

1199061003817100381710038171003817Δ119892

1003817100381710038171003817 le 1198631198982

(13)

where 1198910and 119892


119863119906 1198631198982



= 119891 (119909) + 119892 (119909) (119906 + 119865 (119909 119905)) + 119889 (14)


1= 1198961119882119888+ 11989611199061minus 11989611198961198901199011+ 11989611198651


119888=

minus119875119888

120591+ 119870119900(1 minus 119901

minus120583

2) (1199062+ 1198652)

(15)


119891 =(((

(

1198961119896119888

119875119888

119901120583

1minus 1

minus 11989611198961198901199011

11989621198961198901199011

minus119875119888

120591

)))

)

119892 = (

1198961

0

minus1198962

minus1198962

0 119870119900(1 minus 119901

minus120583

2)

)

119865 = (1198651

1198652

) 119889 = (0

1198962119882119891

)

(16)













1119889 1199012119889


1 1198782

1198781= 1199011minus 1199011119889

1198782= 1199012minus 1199012119889

(17)






1198781= 1198961119896119888

119875119888

119901120583

1minus 1

+ 11989611199061minus 11989611198961198901199011+ 11989611198651minus 1119889








1119889 1199012119889






+ 1205811119881 (119905) + 120581

2119881120591

le 0 forall119905 gt 1199050 (20)


119905119904le 1199050+

1

1205811(1 + 120591)

ln12058111198811minus120591

(1199050) + 1205812

1205812

(21)

where 1205811gt 0 120581

2gt 0 and 0 lt 120591 lt 1


119878 = 119891119900(119909) + ((119892

119900(119909) + Δ119892) 119906 + 119892 (119909) 119865 (119909 119905))

+ Δ119891 + 119889 minus 119878119889

(22)

where 119878 = (1198781

1198782

) 1198910

= (1198961119900119896119888119900(119875119888119901120583119900

1minus1)minus119896

11199001198961198901199001199011

11989621199001198961198901199001199011

) 1198920

=

(11989611199000

minus1198962119900minus1198962119900

) 119865 = (1198651

1198652

) 119889 = (0

1198962119882119891) 119878119889= (1119889

2119889


assumptions




0is nonsingular






119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (23)


119878 = minus120574119878 minus 120582 sign 119878 (24)


1 1205822]

sign 119878 = [sign(1198781) sign(119878

2)]119879


minus120574119878 minus 120582 sign 119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (25)




119906 = 119906eq + 119906119899 (26)

where

119906eq = 119892minus1


0(119909) + 119878

119889)

119906119899= minus 119892

minus1

0(119909) (119889)

(27)


119906 = 119892minus1


0(119909) minus 119889 + 119878

119889) (28)





119878 = 1198910(119909) + 119892

0(119909) 119906 + 119883 minus 119878

119889

= ℎ (119905)

(29)


1198641= 1198851minus 119878

1= 1198852+ 1198910+ 1198920119906 minus 119878119889minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1)

2= minus 120573

2sign (119864

1)

(30)

where 1198641isin R2 119885

1isin R2 and 119885

2isin R2


1and 119885

2are the




2= 1198852minus119889


|ℎ (119905)| lt 1205751 (31)


1 1205732


1205731gt 0

1205732gt

61205751+ 4(12057511205731)2

2

(32)


1= 1198852minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) minus 119883

2= 2minus ℎ (119905)

(33)

which yields to

1= minus 120573

1

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) + 1198642

2= minus 120573

2sign (119864

1) + 1205881

(34)

with 1205881= minusℎ(119905)



1








119906 = 119892minus1


0(119909) minus 119885

2+ 119878119889) (35)







119881 =1

2119878(119905)119879

119878 (119905) (38)


= 119878(119905)119879


= minus

2

sum

119894=1

(1205741198941198782

119894+ 120582119894

10038161003816100381610038161198781198941003816100381610038161003816) minus 119878119879

1198642

le minus

2

sum

119894=1

1205741198941198782

119894minus

2

sum

119894=1

(120582119894minus

100381610038161003816100381611986421198941003816100381610038161003816)

10038161003816100381610038161198781198941003816100381610038161003816

(39)

Choosing 120582119894











1198752119889(Bar) 145 175



119905 120578119888 120578V and



119896119900119888 119896119900119890 119896119900119905 120591119900 120578119900119898 and 120583



1199001=

14391 1198961199002

= 17155 119896119900119888

= 00025 119896119900119890

= 0028 119896119900119905

=

391365 120591119900= 015 120578

119900119898= 095 and 120583



Sign (119878) =119904

|119904| + 120585 (40)






= 0



STESO-SMC 120582 = 3 120574 = 250

20 25 30 35

0 5 10 15 20 25 30 35 40001002003004005006007

Time (s)

Time (s)0 5 10 15 40

131415161718

248 25 252 254 256172174176

248 25 2520038

0040042

Wc

(kg

s)


P2

(b)




119865 (119905) = 0 times (1 1)


119879 if 119905 ge 25 s(41)


119888and 119901

2



119888 1199012but pay high




0 5 10 15 20 25 30 35 40

0

005

01

015

Time (s)

0 5 10 15 20 25 30 35 40

0005

01

Time (s)

minus015

minus01

minus005

minus005

UEG

R(k

gs)


UVG

T(k

gs)



0

5

0 5 10 15 20 25 30 35 40Time (s)

0 5 10 15 20 25 30 35 40Time (s)

010203040

minus15

minus10

minus10

minus5







119889 = (11989611199001198651

minus11989621199001198651minus 11989621199001198652

) (42)


= 12057321

= 12057312

= 12057322

=






Δ119891 = (02 sin (02120587119901

1119905)

05 sin (031205871199012119905)) (43)


119882119891


(44)


For 119905 lt 5 s

119889 = (Δ1198911

Δ1198912

) (45)

For 5 lt 119905 lt 25 s

119889 = (Δ1198911

1198962119900119882119891+ Δ1198912

) (46)

For 119905 ge 25 s

119889 = (Δ1198911+ 11989611199001198651

Δ1198912+ 1198962119900119882119891minus 11989621199001198651minus 11989621199001198652

) (47)






0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)


14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)








11= 12057321

= 12057312

= 12057322


11 12057321 12057312 and 120573

22


0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045



minus0005

UEG

R(k

gs)

UVG

T(k

gs)





0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances



minus5

minus4

minus3

minus2

minus1







6 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Diesel engine

Exhaust gas

Compressor (PcWc)

Air

EGRcooler

Intercooler

Throttle valve










119905119888) and the two


1and 119906

2)


1= 1198961(119882119888+ 119882egr minus 119896

1198901199011) +

1

1198791

1199011

2= 1198962(1198961198901199011minus 119882egr minus 119882

119905+ 119882119891) +

2

1198792

1199012

119888=

1

120591(120578119898119875119905minus 119875119888)

(1)



119888


119882119888= 119875119888

119896119888

119901120583

1minus 1

(2)

119875119905= 119896119905(1 minus 119901

minus120583

2)119882119905 (3)

where

119896119888=

120578119888

119888119901119879119886

119896119905= 1198881199011205781199051198792

1198961=

1198771198861198791

1198811

119896119890=

120578V119873119881119889

1198771198861198791

1198962=

1198771198861198792

1198812

(4)


1= 119882egr and 119906

2= 119882119905 which





1= 1198961(119882119888+ 119882egr minus 119896

1198901199011)

2= 1198962(1198961198901199011+ 119882119891minus 119882egr minus 119882

119905)

119888=

1

120591(120578119898119875119905minus 119875119888)

(5)




= 119891 (119909) + 1198921(119909) 1199061+ 1198922(119909) 1199062 (6)

where 119909 = (1199011 1199012 119875119888)119879 and

119891 (119909) =

[[[[[[[

[

1198961119896119888

119875119888

119901120583

1minus 1

minus 11989611198961198901199011

1198962(1198961198901199011+ 119882119891)

minus119875119888

120591

]]]]]]]

]

(7)

1198921(119909) = [

[

1198961

minus1198962

0

]

]

1198922(119909) = [

[

0

minus1198962

119870119900(1 minus 119901

minus120583

2)

]

]

(8)

with 119870119900= (120578119898120591)119896119905


119896119888 119896119890 119896119905 120591 and 120578




Ω = (1199011 1199012 119875119888) 1 lt 119901

1lt 119875

max1

1 lt 1199012lt 119901

max2

0 lt 119875119888lt 119875

max119888

(9)

is an invariant set






















= 119891 (119909) + 119892 (119909) 119906 (10)




= 119891 (119909) + 119892 (119909) (119906 + 119865 (119909 119905)) (11)


119865 (119909 119905) lt 1198631198981

(12)


Δ119891 = 119891 (119909) minus 1198910(119909)

Δ119892 = 119892 (119909) minus 1198920(119909)

with 1003817100381710038171003817Δ1198911003817100381710038171003817 le 119863

1199061003817100381710038171003817Δ119892

1003817100381710038171003817 le 1198631198982

(13)

where 1198910and 119892


119863119906 1198631198982



= 119891 (119909) + 119892 (119909) (119906 + 119865 (119909 119905)) + 119889 (14)


1= 1198961119882119888+ 11989611199061minus 11989611198961198901199011+ 11989611198651


119888=

minus119875119888

120591+ 119870119900(1 minus 119901

minus120583

2) (1199062+ 1198652)

(15)


119891 =(((

(

1198961119896119888

119875119888

119901120583

1minus 1

minus 11989611198961198901199011

11989621198961198901199011

minus119875119888

120591

)))

)

119892 = (

1198961

0

minus1198962

minus1198962

0 119870119900(1 minus 119901

minus120583

2)

)

119865 = (1198651

1198652

) 119889 = (0

1198962119882119891

)

(16)













1119889 1199012119889


1 1198782

1198781= 1199011minus 1199011119889

1198782= 1199012minus 1199012119889

(17)






1198781= 1198961119896119888

119875119888

119901120583

1minus 1

+ 11989611199061minus 11989611198961198901199011+ 11989611198651minus 1119889








1119889 1199012119889






+ 1205811119881 (119905) + 120581

2119881120591

le 0 forall119905 gt 1199050 (20)


119905119904le 1199050+

1

1205811(1 + 120591)

ln12058111198811minus120591

(1199050) + 1205812

1205812

(21)

where 1205811gt 0 120581

2gt 0 and 0 lt 120591 lt 1


119878 = 119891119900(119909) + ((119892

119900(119909) + Δ119892) 119906 + 119892 (119909) 119865 (119909 119905))

+ Δ119891 + 119889 minus 119878119889

(22)

where 119878 = (1198781

1198782

) 1198910

= (1198961119900119896119888119900(119875119888119901120583119900

1minus1)minus119896

11199001198961198901199001199011

11989621199001198961198901199001199011

) 1198920

=

(11989611199000

minus1198962119900minus1198962119900

) 119865 = (1198651

1198652

) 119889 = (0

1198962119882119891) 119878119889= (1119889

2119889


assumptions




0is nonsingular






119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (23)


119878 = minus120574119878 minus 120582 sign 119878 (24)


1 1205822]

sign 119878 = [sign(1198781) sign(119878

2)]119879


minus120574119878 minus 120582 sign 119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (25)




119906 = 119906eq + 119906119899 (26)

where

119906eq = 119892minus1


0(119909) + 119878

119889)

119906119899= minus 119892

minus1

0(119909) (119889)

(27)


119906 = 119892minus1


0(119909) minus 119889 + 119878

119889) (28)





119878 = 1198910(119909) + 119892

0(119909) 119906 + 119883 minus 119878

119889

= ℎ (119905)

(29)


1198641= 1198851minus 119878

1= 1198852+ 1198910+ 1198920119906 minus 119878119889minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1)

2= minus 120573

2sign (119864

1)

(30)

where 1198641isin R2 119885

1isin R2 and 119885

2isin R2


1and 119885

2are the




2= 1198852minus119889


|ℎ (119905)| lt 1205751 (31)


1 1205732


1205731gt 0

1205732gt

61205751+ 4(12057511205731)2

2

(32)


1= 1198852minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) minus 119883

2= 2minus ℎ (119905)

(33)

which yields to

1= minus 120573

1

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) + 1198642

2= minus 120573

2sign (119864

1) + 1205881

(34)

with 1205881= minusℎ(119905)



1








119906 = 119892minus1


0(119909) minus 119885

2+ 119878119889) (35)







119881 =1

2119878(119905)119879

119878 (119905) (38)


= 119878(119905)119879


= minus

2

sum

119894=1

(1205741198941198782

119894+ 120582119894

10038161003816100381610038161198781198941003816100381610038161003816) minus 119878119879

1198642

le minus

2

sum

119894=1

1205741198941198782

119894minus

2

sum

119894=1

(120582119894minus

100381610038161003816100381611986421198941003816100381610038161003816)

10038161003816100381610038161198781198941003816100381610038161003816

(39)

Choosing 120582119894











1198752119889(Bar) 145 175



119905 120578119888 120578V and



119896119900119888 119896119900119890 119896119900119905 120591119900 120578119900119898 and 120583



1199001=

14391 1198961199002

= 17155 119896119900119888

= 00025 119896119900119890

= 0028 119896119900119905

=

391365 120591119900= 015 120578

119900119898= 095 and 120583



Sign (119878) =119904

|119904| + 120585 (40)






= 0



STESO-SMC 120582 = 3 120574 = 250

20 25 30 35

0 5 10 15 20 25 30 35 40001002003004005006007

Time (s)

Time (s)0 5 10 15 40

131415161718

248 25 252 254 256172174176

248 25 2520038

0040042

Wc

(kg

s)


P2

(b)




119865 (119905) = 0 times (1 1)


119879 if 119905 ge 25 s(41)


119888and 119901

2



119888 1199012but pay high




0 5 10 15 20 25 30 35 40

0

005

01

015

Time (s)

0 5 10 15 20 25 30 35 40

0005

01

Time (s)

minus015

minus01

minus005

minus005

UEG

R(k

gs)


UVG

T(k

gs)



0

5

0 5 10 15 20 25 30 35 40Time (s)

0 5 10 15 20 25 30 35 40Time (s)

010203040

minus15

minus10

minus10

minus5







119889 = (11989611199001198651

minus11989621199001198651minus 11989621199001198652

) (42)


= 12057321

= 12057312

= 12057322

=






Δ119891 = (02 sin (02120587119901

1119905)

05 sin (031205871199012119905)) (43)


119882119891


(44)


For 119905 lt 5 s

119889 = (Δ1198911

Δ1198912

) (45)

For 5 lt 119905 lt 25 s

119889 = (Δ1198911

1198962119900119882119891+ Δ1198912

) (46)

For 119905 ge 25 s

119889 = (Δ1198911+ 11989611199001198651

Δ1198912+ 1198962119900119882119891minus 11989621199001198651minus 11989621199001198652

) (47)






0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)


14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)








11= 12057321

= 12057312

= 12057322


11 12057321 12057312 and 120573

22


0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045



minus0005

UEG

R(k

gs)

UVG

T(k

gs)





0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances



minus5

minus4

minus3

minus2

minus1







6 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























= 119891 (119909) + 119892 (119909) 119906 (10)




= 119891 (119909) + 119892 (119909) (119906 + 119865 (119909 119905)) (11)


119865 (119909 119905) lt 1198631198981

(12)


Δ119891 = 119891 (119909) minus 1198910(119909)

Δ119892 = 119892 (119909) minus 1198920(119909)

with 1003817100381710038171003817Δ1198911003817100381710038171003817 le 119863

1199061003817100381710038171003817Δ119892

1003817100381710038171003817 le 1198631198982

(13)

where 1198910and 119892


119863119906 1198631198982



= 119891 (119909) + 119892 (119909) (119906 + 119865 (119909 119905)) + 119889 (14)


1= 1198961119882119888+ 11989611199061minus 11989611198961198901199011+ 11989611198651


119888=

minus119875119888

120591+ 119870119900(1 minus 119901

minus120583

2) (1199062+ 1198652)

(15)


119891 =(((

(

1198961119896119888

119875119888

119901120583

1minus 1

minus 11989611198961198901199011

11989621198961198901199011

minus119875119888

120591

)))

)

119892 = (

1198961

0

minus1198962

minus1198962

0 119870119900(1 minus 119901

minus120583

2)

)

119865 = (1198651

1198652

) 119889 = (0

1198962119882119891

)

(16)













1119889 1199012119889


1 1198782

1198781= 1199011minus 1199011119889

1198782= 1199012minus 1199012119889

(17)






1198781= 1198961119896119888

119875119888

119901120583

1minus 1

+ 11989611199061minus 11989611198961198901199011+ 11989611198651minus 1119889








1119889 1199012119889






+ 1205811119881 (119905) + 120581

2119881120591

le 0 forall119905 gt 1199050 (20)


119905119904le 1199050+

1

1205811(1 + 120591)

ln12058111198811minus120591

(1199050) + 1205812

1205812

(21)

where 1205811gt 0 120581

2gt 0 and 0 lt 120591 lt 1


119878 = 119891119900(119909) + ((119892

119900(119909) + Δ119892) 119906 + 119892 (119909) 119865 (119909 119905))

+ Δ119891 + 119889 minus 119878119889

(22)

where 119878 = (1198781

1198782

) 1198910

= (1198961119900119896119888119900(119875119888119901120583119900

1minus1)minus119896

11199001198961198901199001199011

11989621199001198961198901199001199011

) 1198920

=

(11989611199000

minus1198962119900minus1198962119900

) 119865 = (1198651

1198652

) 119889 = (0

1198962119882119891) 119878119889= (1119889

2119889


assumptions




0is nonsingular






119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (23)


119878 = minus120574119878 minus 120582 sign 119878 (24)


1 1205822]

sign 119878 = [sign(1198781) sign(119878

2)]119879


minus120574119878 minus 120582 sign 119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (25)




119906 = 119906eq + 119906119899 (26)

where

119906eq = 119892minus1


0(119909) + 119878

119889)

119906119899= minus 119892

minus1

0(119909) (119889)

(27)


119906 = 119892minus1


0(119909) minus 119889 + 119878

119889) (28)





119878 = 1198910(119909) + 119892

0(119909) 119906 + 119883 minus 119878

119889

= ℎ (119905)

(29)


1198641= 1198851minus 119878

1= 1198852+ 1198910+ 1198920119906 minus 119878119889minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1)

2= minus 120573

2sign (119864

1)

(30)

where 1198641isin R2 119885

1isin R2 and 119885

2isin R2


1and 119885

2are the




2= 1198852minus119889


|ℎ (119905)| lt 1205751 (31)


1 1205732


1205731gt 0

1205732gt

61205751+ 4(12057511205731)2

2

(32)


1= 1198852minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) minus 119883

2= 2minus ℎ (119905)

(33)

which yields to

1= minus 120573

1

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) + 1198642

2= minus 120573

2sign (119864

1) + 1205881

(34)

with 1205881= minusℎ(119905)



1








119906 = 119892minus1


0(119909) minus 119885

2+ 119878119889) (35)







119881 =1

2119878(119905)119879

119878 (119905) (38)


= 119878(119905)119879


= minus

2

sum

119894=1

(1205741198941198782

119894+ 120582119894

10038161003816100381610038161198781198941003816100381610038161003816) minus 119878119879

1198642

le minus

2

sum

119894=1

1205741198941198782

119894minus

2

sum

119894=1

(120582119894minus

100381610038161003816100381611986421198941003816100381610038161003816)

10038161003816100381610038161198781198941003816100381610038161003816

(39)

Choosing 120582119894











1198752119889(Bar) 145 175



119905 120578119888 120578V and



119896119900119888 119896119900119890 119896119900119905 120591119900 120578119900119898 and 120583



1199001=

14391 1198961199002

= 17155 119896119900119888

= 00025 119896119900119890

= 0028 119896119900119905

=

391365 120591119900= 015 120578

119900119898= 095 and 120583



Sign (119878) =119904

|119904| + 120585 (40)






= 0



STESO-SMC 120582 = 3 120574 = 250

20 25 30 35

0 5 10 15 20 25 30 35 40001002003004005006007

Time (s)

Time (s)0 5 10 15 40

131415161718

248 25 252 254 256172174176

248 25 2520038

0040042

Wc

(kg

s)


P2

(b)




119865 (119905) = 0 times (1 1)


119879 if 119905 ge 25 s(41)


119888and 119901

2



119888 1199012but pay high




0 5 10 15 20 25 30 35 40

0

005

01

015

Time (s)

0 5 10 15 20 25 30 35 40

0005

01

Time (s)

minus015

minus01

minus005

minus005

UEG

R(k

gs)


UVG

T(k

gs)



0

5

0 5 10 15 20 25 30 35 40Time (s)

0 5 10 15 20 25 30 35 40Time (s)

010203040

minus15

minus10

minus10

minus5







119889 = (11989611199001198651

minus11989621199001198651minus 11989621199001198652

) (42)


= 12057321

= 12057312

= 12057322

=






Δ119891 = (02 sin (02120587119901

1119905)

05 sin (031205871199012119905)) (43)


119882119891


(44)


For 119905 lt 5 s

119889 = (Δ1198911

Δ1198912

) (45)

For 5 lt 119905 lt 25 s

119889 = (Δ1198911

1198962119900119882119891+ Δ1198912

) (46)

For 119905 ge 25 s

119889 = (Δ1198911+ 11989611199001198651

Δ1198912+ 1198962119900119882119891minus 11989621199001198651minus 11989621199001198652

) (47)






0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)


14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)








11= 12057321

= 12057312

= 12057322


11 12057321 12057312 and 120573

22


0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045



minus0005

UEG

R(k

gs)

UVG

T(k

gs)





0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances



minus5

minus4

minus3

minus2

minus1







6 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















1119889 1199012119889


1 1198782

1198781= 1199011minus 1199011119889

1198782= 1199012minus 1199012119889

(17)






1198781= 1198961119896119888

119875119888

119901120583

1minus 1

+ 11989611199061minus 11989611198961198901199011+ 11989611198651minus 1119889








1119889 1199012119889






+ 1205811119881 (119905) + 120581

2119881120591

le 0 forall119905 gt 1199050 (20)


119905119904le 1199050+

1

1205811(1 + 120591)

ln12058111198811minus120591

(1199050) + 1205812

1205812

(21)

where 1205811gt 0 120581

2gt 0 and 0 lt 120591 lt 1


119878 = 119891119900(119909) + ((119892

119900(119909) + Δ119892) 119906 + 119892 (119909) 119865 (119909 119905))

+ Δ119891 + 119889 minus 119878119889

(22)

where 119878 = (1198781

1198782

) 1198910

= (1198961119900119896119888119900(119875119888119901120583119900

1minus1)minus119896

11199001198961198901199001199011

11989621199001198961198901199001199011

) 1198920

=

(11989611199000

minus1198962119900minus1198962119900

) 119865 = (1198651

1198652

) 119889 = (0

1198962119882119891) 119878119889= (1119889

2119889


assumptions




0is nonsingular






119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (23)


119878 = minus120574119878 minus 120582 sign 119878 (24)


1 1205822]

sign 119878 = [sign(1198781) sign(119878

2)]119879


minus120574119878 minus 120582 sign 119878 = 1198910(119909) + 119892

0(119909) 119906 + 119889 minus 119878

119889 (25)




119906 = 119906eq + 119906119899 (26)

where

119906eq = 119892minus1


0(119909) + 119878

119889)

119906119899= minus 119892

minus1

0(119909) (119889)

(27)


119906 = 119892minus1


0(119909) minus 119889 + 119878

119889) (28)





119878 = 1198910(119909) + 119892

0(119909) 119906 + 119883 minus 119878

119889

= ℎ (119905)

(29)


1198641= 1198851minus 119878

1= 1198852+ 1198910+ 1198920119906 minus 119878119889minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1)

2= minus 120573

2sign (119864

1)

(30)

where 1198641isin R2 119885

1isin R2 and 119885

2isin R2


1and 119885

2are the




2= 1198852minus119889


|ℎ (119905)| lt 1205751 (31)


1 1205732


1205731gt 0

1205732gt

61205751+ 4(12057511205731)2

2

(32)


1= 1198852minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) minus 119883

2= 2minus ℎ (119905)

(33)

which yields to

1= minus 120573

1

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) + 1198642

2= minus 120573

2sign (119864

1) + 1205881

(34)

with 1205881= minusℎ(119905)



1








119906 = 119892minus1


0(119909) minus 119885

2+ 119878119889) (35)







119881 =1

2119878(119905)119879

119878 (119905) (38)


= 119878(119905)119879


= minus

2

sum

119894=1

(1205741198941198782

119894+ 120582119894

10038161003816100381610038161198781198941003816100381610038161003816) minus 119878119879

1198642

le minus

2

sum

119894=1

1205741198941198782

119894minus

2

sum

119894=1

(120582119894minus

100381610038161003816100381611986421198941003816100381610038161003816)

10038161003816100381610038161198781198941003816100381610038161003816

(39)

Choosing 120582119894











1198752119889(Bar) 145 175



119905 120578119888 120578V and



119896119900119888 119896119900119890 119896119900119905 120591119900 120578119900119898 and 120583



1199001=

14391 1198961199002

= 17155 119896119900119888

= 00025 119896119900119890

= 0028 119896119900119905

=

391365 120591119900= 015 120578

119900119898= 095 and 120583



Sign (119878) =119904

|119904| + 120585 (40)






= 0



STESO-SMC 120582 = 3 120574 = 250

20 25 30 35

0 5 10 15 20 25 30 35 40001002003004005006007

Time (s)

Time (s)0 5 10 15 40

131415161718

248 25 252 254 256172174176

248 25 2520038

0040042

Wc

(kg

s)


P2

(b)




119865 (119905) = 0 times (1 1)


119879 if 119905 ge 25 s(41)


119888and 119901

2



119888 1199012but pay high




0 5 10 15 20 25 30 35 40

0

005

01

015

Time (s)

0 5 10 15 20 25 30 35 40

0005

01

Time (s)

minus015

minus01

minus005

minus005

UEG

R(k

gs)


UVG

T(k

gs)



0

5

0 5 10 15 20 25 30 35 40Time (s)

0 5 10 15 20 25 30 35 40Time (s)

010203040

minus15

minus10

minus10

minus5







119889 = (11989611199001198651

minus11989621199001198651minus 11989621199001198652

) (42)


= 12057321

= 12057312

= 12057322

=






Δ119891 = (02 sin (02120587119901

1119905)

05 sin (031205871199012119905)) (43)


119882119891


(44)


For 119905 lt 5 s

119889 = (Δ1198911

Δ1198912

) (45)

For 5 lt 119905 lt 25 s

119889 = (Δ1198911

1198962119900119882119891+ Δ1198912

) (46)

For 119905 ge 25 s

119889 = (Δ1198911+ 11989611199001198651

Δ1198912+ 1198962119900119882119891minus 11989621199001198651minus 11989621199001198652

) (47)






0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)


14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)








11= 12057321

= 12057312

= 12057322


11 12057321 12057312 and 120573

22


0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045



minus0005

UEG

R(k

gs)

UVG

T(k

gs)





0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances



minus5

minus4

minus3

minus2

minus1







6 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119878 = 1198910(119909) + 119892

0(119909) 119906 + 119883 minus 119878

119889

= ℎ (119905)

(29)


1198641= 1198851minus 119878

1= 1198852+ 1198910+ 1198920119906 minus 119878119889minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1)

2= minus 120573

2sign (119864

1)

(30)

where 1198641isin R2 119885

1isin R2 and 119885

2isin R2


1and 119885

2are the




2= 1198852minus119889


|ℎ (119905)| lt 1205751 (31)


1 1205732


1205731gt 0

1205732gt

61205751+ 4(12057511205731)2

2

(32)


1= 1198852minus 1205731

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) minus 119883

2= 2minus ℎ (119905)

(33)

which yields to

1= minus 120573

1

10038161003816100381610038161198641100381610038161003816100381612 sign (119864

1) + 1198642

2= minus 120573

2sign (119864

1) + 1205881

(34)

with 1205881= minusℎ(119905)



1








119906 = 119892minus1


0(119909) minus 119885

2+ 119878119889) (35)







119881 =1

2119878(119905)119879

119878 (119905) (38)


= 119878(119905)119879


= minus

2

sum

119894=1

(1205741198941198782

119894+ 120582119894

10038161003816100381610038161198781198941003816100381610038161003816) minus 119878119879

1198642

le minus

2

sum

119894=1

1205741198941198782

119894minus

2

sum

119894=1

(120582119894minus

100381610038161003816100381611986421198941003816100381610038161003816)

10038161003816100381610038161198781198941003816100381610038161003816

(39)

Choosing 120582119894











1198752119889(Bar) 145 175



119905 120578119888 120578V and



119896119900119888 119896119900119890 119896119900119905 120591119900 120578119900119898 and 120583



1199001=

14391 1198961199002

= 17155 119896119900119888

= 00025 119896119900119890

= 0028 119896119900119905

=

391365 120591119900= 015 120578

119900119898= 095 and 120583



Sign (119878) =119904

|119904| + 120585 (40)






= 0



STESO-SMC 120582 = 3 120574 = 250

20 25 30 35

0 5 10 15 20 25 30 35 40001002003004005006007

Time (s)

Time (s)0 5 10 15 40

131415161718

248 25 252 254 256172174176

248 25 2520038

0040042

Wc

(kg

s)


P2

(b)




119865 (119905) = 0 times (1 1)


119879 if 119905 ge 25 s(41)


119888and 119901

2



119888 1199012but pay high




0 5 10 15 20 25 30 35 40

0

005

01

015

Time (s)

0 5 10 15 20 25 30 35 40

0005

01

Time (s)

minus015

minus01

minus005

minus005

UEG

R(k

gs)


UVG

T(k

gs)



0

5

0 5 10 15 20 25 30 35 40Time (s)

0 5 10 15 20 25 30 35 40Time (s)

010203040

minus15

minus10

minus10

minus5







119889 = (11989611199001198651

minus11989621199001198651minus 11989621199001198652

) (42)


= 12057321

= 12057312

= 12057322

=






Δ119891 = (02 sin (02120587119901

1119905)

05 sin (031205871199012119905)) (43)


119882119891


(44)


For 119905 lt 5 s

119889 = (Δ1198911

Δ1198912

) (45)

For 5 lt 119905 lt 25 s

119889 = (Δ1198911

1198962119900119882119891+ Δ1198912

) (46)

For 119905 ge 25 s

119889 = (Δ1198911+ 11989611199001198651

Δ1198912+ 1198962119900119882119891minus 11989621199001198651minus 11989621199001198652

) (47)






0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)


14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)








11= 12057321

= 12057312

= 12057322


11 12057321 12057312 and 120573

22


0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045



minus0005

UEG

R(k

gs)

UVG

T(k

gs)





0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances



minus5

minus4

minus3

minus2

minus1







6 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198752119889(Bar) 145 175



119905 120578119888 120578V and



119896119900119888 119896119900119890 119896119900119905 120591119900 120578119900119898 and 120583



1199001=

14391 1198961199002

= 17155 119896119900119888

= 00025 119896119900119890

= 0028 119896119900119905

=

391365 120591119900= 015 120578

119900119898= 095 and 120583



Sign (119878) =119904

|119904| + 120585 (40)






= 0



STESO-SMC 120582 = 3 120574 = 250

20 25 30 35

0 5 10 15 20 25 30 35 40001002003004005006007

Time (s)

Time (s)0 5 10 15 40

131415161718

248 25 252 254 256172174176

248 25 2520038

0040042

Wc

(kg

s)


P2

(b)




119865 (119905) = 0 times (1 1)


119879 if 119905 ge 25 s(41)


119888and 119901

2



119888 1199012but pay high




0 5 10 15 20 25 30 35 40

0

005

01

015

Time (s)

0 5 10 15 20 25 30 35 40

0005

01

Time (s)

minus015

minus01

minus005

minus005

UEG

R(k

gs)


UVG

T(k

gs)



0

5

0 5 10 15 20 25 30 35 40Time (s)

0 5 10 15 20 25 30 35 40Time (s)

010203040

minus15

minus10

minus10

minus5







119889 = (11989611199001198651

minus11989621199001198651minus 11989621199001198652

) (42)


= 12057321

= 12057312

= 12057322

=






Δ119891 = (02 sin (02120587119901

1119905)

05 sin (031205871199012119905)) (43)


119882119891


(44)


For 119905 lt 5 s

119889 = (Δ1198911

Δ1198912

) (45)

For 5 lt 119905 lt 25 s

119889 = (Δ1198911

1198962119900119882119891+ Δ1198912

) (46)

For 119905 ge 25 s

119889 = (Δ1198911+ 11989611199001198651

Δ1198912+ 1198962119900119882119891minus 11989621199001198651minus 11989621199001198652

) (47)






0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)


14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)








11= 12057321

= 12057312

= 12057322


11 12057321 12057312 and 120573

22


0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045



minus0005

UEG

R(k

gs)

UVG

T(k

gs)





0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances



minus5

minus4

minus3

minus2

minus1







6 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 35 40

0

005

01

015

Time (s)

0 5 10 15 20 25 30 35 40

0005

01

Time (s)

minus015

minus01

minus005

minus005

UEG

R(k

gs)


UVG

T(k

gs)



0

5

0 5 10 15 20 25 30 35 40Time (s)

0 5 10 15 20 25 30 35 40Time (s)

010203040

minus15

minus10

minus10

minus5







119889 = (11989611199001198651

minus11989621199001198651minus 11989621199001198652

) (42)


= 12057321

= 12057312

= 12057322

=






Δ119891 = (02 sin (02120587119901

1119905)

05 sin (031205871199012119905)) (43)


119882119891


(44)


For 119905 lt 5 s

119889 = (Δ1198911

Δ1198912

) (45)

For 5 lt 119905 lt 25 s

119889 = (Δ1198911

1198962119900119882119891+ Δ1198912

) (46)

For 119905 ge 25 s

119889 = (Δ1198911+ 11989611199001198651

Δ1198912+ 1198962119900119882119891minus 11989621199001198651minus 11989621199001198652

) (47)






0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)


14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)








11= 12057321

= 12057312

= 12057322


11 12057321 12057312 and 120573

22


0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045



minus0005

UEG

R(k

gs)

UVG

T(k

gs)





0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances



minus5

minus4

minus3

minus2

minus1







6 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50

50

60

60

0005001

0015002

0025003

0035004

0045

Time (s)

Time (s)0 10 20 30 40

141516171819

00398004

00402

25 2501 2502 2503 35 3502 350400298

003Wc

(kg

s)


14

145

174176

25 2502 2504 2506 35 3502 3504 3506

P2

(b)








11= 12057321

= 12057312

= 12057322


11 12057321 12057312 and 120573

22


0 10 20 30 40 50 60

00005

0010015

0020025

003

Time (s)

Time (s)0 10 20 30 40 50 60

0015002

0025003

0035004

0045



minus0005

UEG

R(k

gs)

UVG

T(k

gs)





0 10 20 30 40 50 60

012

0

0

10 20 30 40 50 60

510152025303540

Time (s)

Time (s)

No disturbances



minus5

minus4

minus3

minus2

minus1







6 Conclusions






Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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