Research Article Adaptive Quasi-Sliding Mode …downloads.hindawi.com/journals/mpe/2013/693685.pdfResearch Article Adaptive Quasi-Sliding Mode Control for ... e motor speed of a buck

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 693685 12 pageshttpdxdoiorg1011552013693685

Research ArticleAdaptive Quasi-Sliding Mode Control forPermanent Magnet DC Motor

Fredy E Hoyos1 Alejandro Rincoacuten2 John Alexander Taborda3

Nicolaacutes Toro4 and Fabiola Angulo4

1 Tecnologico de Antioquia Institucion Universitaria Facultad de Ingenierıa Grupo de Investigacion en Ingenierıa de Software delTecnologico de Antioquia (GIISTA) Calle 78B No 72A-220 Bloque 6 Campus Robledo Medellın 050040 Colombia

2 Facultad de Ingenierıa y Arquitectura Universidad Catolica de Manizales Cr 23 No 60-63 Manizales 170002 Colombia3 Facultad de Ingenierıa Ingenierıa Electronica Universidad del Magdalena Santa Marta Colombia4Departamento de Ingenierıa Electrica Electronica y Computacion Facultad de Ingenierıa y Arquitectura Universidad Nacional deColombia Sede Manizales Percepcion y Control Inteligente Bloque Q Campus La Nubia Manizales 170003 Colombia

Correspondence should be addressed to Fabiola Angulo fangulogunaleduco

Received 24 June 2013 Revised 23 August 2013 Accepted 6 September 2013

Academic Editor Guanghui Sun

Copyright copy 2013 Fredy E Hoyos et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The motor speed of a buck power converter and DC motor coupled system is controlled by means of a quasi-sliding scheme Thefixed point inducting control technique and the zero average dynamics strategy are used in the controller design To estimate theload and friction torques an online estimator computed by the least mean squares method is used The control scheme is testedin a rapid control prototyping system which is based on digital signal processing for a dSPACE platform The closed loop systemexhibits adequate performance and experimental and simulation results match

1 Introduction

Permanent Magnet DC (PMDC) motors are currently usedas variable speed drive systems for aerospace automotiveindustrial and automation applications because of their highpower density and low maintenance cost [1] However theproper functioning of these systems has important challengesrelated to modeling dynamical analysis control and imple-mentation aspects Nonlinear phenomena as bifurcations andchaos have been observed in many classes of motor drivesystems Therefore uncertainties and disturbances should betackled [2]

Sliding mode control appears to be effective for PMDCmotor control since it is widely reported as a powerfulapproach for robust control issues [3 4] It is fast andmakes itpossible to tackle the effect of both unknown varying param-eters and disturbances The basic idea of the sliding controlmethod is to design a control law so that the trajectoriesof the tracking error and its time derivatives converge to aproperly defined manifold called sliding surface and remain

thereinThe tracking problem is translated into enforcing thedynamical system to converge towards the sliding surfacedefined by 119904(119905) = 0 where 119904(119905) is a linear function of thetracking error and its time derivatives The sliding surfaceis rendered attractive and invariant by the control law Acommon assumption is that there are unknown varyingparameters or disturbancesThere are two possibilities [5] (i)assuming that the bounds of such uncertainties are knownand (ii) using a parameter updating mechanism in order totackle the uncertain bounds Ideal sliding motion only existsfor infinite switching frequency Thus from a practical pointof view undesired chattering phenomenon is present [1]Some research projects on slidingmode control are discussedbelow

In [6] an adaptive robust sliding nonlinear controller isformulated for a three-phase induction motor drive on thebasis of a space-svector modulation scheme The goal is tocontrol the torque and stator flux The motor model is fifthorder based on a stationary frame with two axes and thestator currents and stator fluxes are the state variables It is

2 Mathematical Problems in Engineering

assumed that stator and rotor resistances may be varying anduncertain The controller takes into account such variationthe stator and rotor resistances are online updated on thebasis of the stator measured currents and voltages

In [7 8] a sliding mode controller is developed for asensorless induction motor (IM) drive using the principlesof direct torque control and space-vector pulse-width mod-ulation (PWM) The goal is to control the rotor speed andstator flux (torque and flux) by means of two sliding modecontrollers Sliding mode observers are used for torque andflux estimation employing two motor models The inverter iscontrolled using space-vector PWM

In [9] an adaptive sliding mode controller is designedfor a two-mass IM drive without mechanical sensors withan elastic joint The goal is to control the motor speed Thespeed is notmeasured whereas the stator current and the DClink voltage of the motor are The control structure consistsof the rotor flux control loop and the speed control loop (i)in the flux control loop an MRAS type estimator is used forestimation of rotor speed and flux based on the current andvoltages of the IM and (ii) in the speed control loop thereis an inner control loop where the electromagnetic torqueof the IM is regulated In addition a fuzzy neural networkis used with online tuning of its connective weights

In [10] an adaptive robust sliding controller is formulatedfor SISO nonlinear systems with implementation on a linearmotor It is assumed that (i) the nonlinear part can belinearly parameterized and (ii) there are unknown varyingparameters and disturbances whose bounds are known Adiscontinuous projection type updated law is used in order toguarantee the bounded nature of the parameter updates Thecontrol scheme is applied to an X-Y table driven by a linearmotorThe control design gives as result a signum type signalbut a saturation type function is proposed to replace it

Although the basic approach of the sliding mode con-trol is intended for a continuous control input there areapplications to two-value control input For instance in[11] a sliding control scheme is designed for discrete pulsemodulated converters An integral term is incorporated inthe definition of the error function 119878 in order to achieve thedesired steady state behavior Simulations and experimentsconfirm the improvement of the steady state behavior Themain features of such control scheme are (i) the idea is thatthe error function 119878 converges to zero (ii) the control designis based on the properties of the time derivative of the errorfunction 119878 and (iii) the commutation frequency is not fixed

The drawbacks of sliding mode control are (cf [5 610]) (i) the control gain is large when the defined param-eter bounds are large (ii) the control input may exhibitchattering which may excite high frequency unmodeledmodes of the system and (iii) the input chattering maylead to high power consumption and wear of mechani-cal components (cf [12 13]) There have been significantworks focused on the reduction or fixing of the commu-tation frequency in order to reduce chattering mainly inthe field of power electronics The current sliding controlschemes with fixed frequency lead to significantly flawedconvergence of the system variables towards the slidingsurface

Quasi-sliding mode control emerged as an alternativesolution to the drawbacks of sliding control retaining theinherent advantages of its operation Indeed the undesirablechattering and high frequency switching of the control signalare avoided The controller design aims at achieving systemconvergence towards a boundary layer instead of the slidingsurface as mentioned in [14 pages 335 336] In the case ofa continuous control input the boundary layer implies theuse of a saturation function in the control law instead ofa discontinuous signal (see [10 15]) The main features areas follows (Ci) the main idea is that the system convergestowards a region so that |119878| le 120576 where 120576 is a small user-defined positive constant (Cii) the control law is designedto achieve an adequate negative nature of the time derivativeof the Lyapunov function or equivalently adequate signumproperties of the time derivative of the error function (Ciii)the commutation frequency is not fixed

Thus it is important to formulate a control law thatachieves desired system performancewhile tackling the effectof uncertainties or disturbances To develop such controllerin this work two different controllers were used the zeroaverage dynamics (ZAD) technique and the fixed pointinducting control (FPIC) The ZAD technique working witha second loop of control (FPIC) results in a quasi-slidingcontroller that has proven to be effective and robust touncertainties and disturbances

The theoretical basis of the ZAD technique design wasexplained by Fossas et al in [14] in 2000 In the ZAD designthe definition of the duty cycle is such that the averagedynamics of the error function is zero for one measuringperiod This idea leads to a fixed switching frequency withconvergence of the system towards the boundary layercompars [14 pages 341ndash343] In [16] a piece-wise linearapproximation of the error function was proposed in orderto simplify the computation of the duty cycle Neverthelessthe ZAD strategy is not as robust as sliding control Forthis reason the ZAD-FPIC strategy was developed in [16]as an improvement of the ZAD strategy Indeed The ZAD-FPIC duty cycle is the weighted sum of (i) the ZAD dutycycle and (ii) a feed-forward component defined as the steadystate of the ZAD duty cycle The ZAD-FPIC strategy allowsfurther improvement of the convergence properties of theZAD technique The theoretical basis of the FPIC design wasproposed for the first time in [16] and experimentally provenin [17 18] The main features of the ZAD-FPIC strategy areas follows (i) the control design is not aimed at obtainingnegativeness or positiveness properties of the time derivativeof the error function instead a linear combination of theerror and its derivative is forced to have zero average (ii)ZAD-FPIC scheme improves the properties of the ZADtechnique by defining a new duty cycle that combines theZAD duty cycle with its steady state or with its expected dutycycle in each iteration thus stability is guaranteed in a widerrange of the parameter values and (iii) the closed loop systemis more robust with this controller

In this paper an adaptive ZAD-FPIC strategy is formu-lated for motor speed control The system involves a buckpower converter a PMDC motor and a dSPACE platformThe load torque and the friction torque are considered

Mathematical Problems in Engineering 3

unknown so that they lead to an uncertain parameter Theuncertain parameter is estimated by means of a least meansquares (LMS) mechanism which is formulated and testedon the real systemThe ZAD-FPIC scheme is formulated andexperimentally tested Bifurcation diagrams are developedfor the adaptive ZAD-FPIC control system for differentvalues of the controller parameters Numerical and experi-mental diagrams match The control parameters are definedaccording to these curvesThemain differences of the presentpaper with respect to closely related papers on ZAD-FPICdesign for instance [16 17] are as follows (i) ZAD-FPICtechnique is proven for the first time in a high order system(PMDC motor) In previous works ZAD-FPIC technique isonly applied to second order systems (ii) in previous worksparameters have been assumed as constant or measured Inthis work the effect of parameter estimation on the closedloop performance is taken into account in experiments andsimulations and (iii) the effect of the values of the covariancematrix on the parameter estimation performance is analyzedusing simulations and experiments

The paper is organized as follows In Section 2 theproposed system and hardware considerations as well asthe mathematical model of the system are presented InSection 3 the control techniques and load torque estimationare defined In Section 4 numerical and experimental resultsare shown Finally in Section 5 conclusions are presented

2 Physical and Modelling Considerations

In this section the physical system and the mathematicalmodel are presented The physical considerations includehardware characteristics whereas the mathematical model isa relatively simple dynamical model

21 Physical System Figure 1(a) shows the block diagram ofthe system under study whereas Figure 1(b) shows the basiccircuit considered in this work According to Figure 1(a)this system is divided into two major groups The firstone is composed of all hardware parts This group includesmechanical and electronic components The second oneinvolves software developments Software is implemented ina dSPACE platform where the acquisition signals and thecontrol techniques implementations are performed

The hardware is composed of a PMDCmotor with a ratedpower of 250W rated voltage of 42V rated DC current of6A and maximum speed of 4000 RPM For motor speedacquisition a 1000 pulse per turn encoder was used Themotor is fed by a buck power converter (see Figure 1(b))To measure the capacitor voltage (120592

119888) the inductor current

(119894119871) and the armature current (119894

119886) series resistances were

used The switch gate (MOSFET) is driven by the PWMcontrollerThe PWM signal and theMOSFET are coupled viafast optocouplers HCPL-J312

The digital part was developed in the control- anddevelopment-card dSPACE DS1104 where the control tech-niques were implementedThis card is programmed from theMatlabSimulink platform and it has a graphical interfacecalled ControlDeskThe sampling rate of all the variables was

set at 6 kHz The state variables 120592119888 119894119871 and 119894

119886were digitalized

with 12-bit resolution while the controlled variable 119882119898was

sensed by an encoder with 28-bit resolution The parametersassociated with the buck converter and the controller as wellas themotor parameters119861 119869eq 119896119905 119896119890119879fric119877119886 119871

119886 are constant

and are passed to the control block by the userinterfaceThe load torque 119879

119871is variable and will be online estimated

The performance and robustness of the system will be testedby varying one parameter At each sampling time 119905 = 119896119879the controller computes the duty cycle and the equivalentPWM signal to control the switch Because of the physicalcharacteristics of the system the control action at samplingtime 119905 = 119896119879 is computed according to the values of thestate variables at the previous sampling time 119905 = (119896 minus 1)119879This restriction must be taken into account in the controllerdesign

22 Mathematical Model Figure 1(b) shows a basic diagramof the system The buck power converter is used to feed theDC-motor The system is of variable structure because onoffswitch positions produce different topologies Particularlywhen the buck converter works in a continuous conductionmode (CCM) two systems are achieved When the switch isON the system is represented by

[

[

[

[

119898

119894119886

120592119888

119894119871

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

minus119861

119869eq

119896119905

119869eq0 0

minus119896119890

119871119886

minus119877119886

119871119886

1

119871119886

0

0

minus1

119862

0

1

119862

0 0

minus1

119871

minus (119903119904

+ 119903119871)

119871

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

119882119898

119894119886

120592119888

119894119871

]

]

]

]

+

[

[

[

[

[

[

[

[

minus (119879fric + 119879119871)

119869eq0

0

119864

119871

]

]

]

]

]

]

]

]

(1)

This equation can be expressed in a compact form by

= 1198601119909 + 1198611 (2)

where the state variables are the motor speed 119882119898

= 1199091 the

armature current 119894119886

= 1199092 the capacitor voltage 120592

119888= 1199093 and

the inductor current 119894119871

= 1199094 When the switch is OFF the

system is represented by

[

[

[

[

119898

119894119886

120592119888

119894119871

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

minus119861

119869eq

119896119905

119869eq0 0

minus119896119890

119871119886

minus119877119886

119871119886

1

119871119886

0

0

minus1

119862

0

1

119862

0 0

minus1

119871

minus (119903119871)

119871

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

119882119898

119894119886

120592119888

119894119871

]

]

]

]


Digital PWMcontrollerdSPACEsimulink PWM

ZAD-FPICcontroller

AD

ADParameters

120592c iLsensor

sensor

DCmotorGate Buck

converter

HardwareSoftware(dSPACE)

Ks123

Wm ia

(a) Block diagram of the proposed system

rs rLS L

Vfd

iLia

Ra

La120592cC

minus

minus

minusminus

+ ++

E

keWm(t)TemTfric +

+

TL

JL

LoadDC motorBuck converter

Wm

Diode

(b) Electrical circuit for the buck-motor system

Figure 1 Schematic diagrams of implementation and physical system

+

[

[

[

[

[

[

[

[

minus (119879fric + 119879119871)

119869eq0

0

minus119881fd119871

]

]

]

]

]

]

]

]

(3)

and in a compact form as

= 1198602119909 + 1198612 (4)

In all cases the output of the system is 119882119898 which corre-

sponds to the variable to be controlled 119896119890is the voltage

constant (Vrads) 119871119886is the armature inductance (H) 119877

119886

is the armature resistance (Ω) 119861 is the viscous frictioncoefficient (Nsdotmrads) 119869eq is the moment of inertia (kgsdotm2)119896119905is the motor torque constant (NsdotmA) 119879fric is the friction

torque (Nsdotm) 119879119871is the load torque (Nsdotm)The parameters 119862

and119871 are the capacitance and inductance of the converter theparasitic resistance 119903

119904is equal to the sumof the source internal

resistance and MOSFET resistance the resistance 119903119871is equal

to the sum of the resistance of the coil and the resistance usedfor measuring the current 119864 is the power source to feed thebuck power converter when the switch is ON (see (1)) 119881fd isthe required voltage for the diode to be on (see (3)) Equation(3) is valid until 119894

119871= 0 Only CCM and centered PWM are

considered in this work (see Figure 2(a)) so that the systemoperates as follows

=

1198601119909 + 1198611

si 119896119879 le 119905 lt 119896119879 +

119889119879

2

1198602119909 + 1198612

si 119896119879 +

119889119879

2

le 119905 lt 119896119879 + 119879 minus

119889119879

2

1198601119909 + 1198611

si 119896119879 + 119879 minus

119889119879

2

le 119905 lt 119896119879 + 119879

(5)

where 119896 represents the 119896th iteration119879 is the sampling periodand 119889 isin [0 1] is the duty cycle which will be computed inSection 3 and it corresponds to the time ratio between theON position switch and the sampling time 119879

3 Control Goal and Control Strategies

The objective of controlling a DC-motor is to maintain themotor speed in a fixed reference value defined by the userThe motor is fed by a power converter which in turn iscontrolled by a PWMTherefore the voltage of the converteris chosen as control input Thus the controller computesthe duty cycle to be applied to the buck converter in such away that the system output (119882

119898) is regulated This section

is organized as follows Section 31 shows the definition ofthe error function and its time derivative Section 32 showsthe formulation of the ZAD control strategy Section 33shows the formulation of the ZAD-FPIC control strategySection 34 shows the definition of the LMS mechanism forestimation of the unknown parameter

31 Definition of the Error Function and Its Time DerivativeTo define the error function it is necessary to define theoutput error and its time derivatives The output error isdefined as

119890 (119905) = 119882119908ref (119905) minus 119882

119898(119905) (6)

where 119882119908ref(119905) is the reference value for the motor speed

and 119898

(119905) is obtained from (1) or (3) depending on the timeinstant The derivative of (6) is given by

119890 (119905) = 119908ref (119905) minus

119898(119905) (7)

At the sampling time 119905 = 119896119879 (6) and (7) are written as

119890 (119896119879) = 119882119908ref (119896119879) minus 119882

119898(119896119879) (8)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879) (9)

Equations (7) and (9) indicate that the derivative of the outputerror at the sampling time 119905 = 119896119879 is computed in two steps inthe first step the derivative of the error is computed in timedomain and in the second step the corresponding samplingvalues of the states are assigned to the expressionThe signals

119890(119896119879) and 119890(119896119879) have the same interpretations With theaim of formulating a quasi-sliding controller and using thefact that the motor is a fourth-order system an auxiliary


E

kT t

dT2

T

V

dT2

(k + 1)TminusVfd

(a) Centered PWM

s

kT t

s1

s2

middot middot middot

s+

s+

sminus

t1

t2

spwl (t)

kT + T

(b) Piece-wise linear function 119904pwl

Figure 2 Schemes of the PWM and the quasi-sliding surface

third-order function involving 119890(119905) 119890(119905) and higher orderderivatives is defined as

119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (10)

Constants 1198961199041 1198961199042 and 119896

1199043are control parameters which

should be adjusted by the designer to satisfy stability require-ments At 119905 = 119896119879 (10) can be written as

119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (11)

The ZAD strategy requires the computation of signal 119904 and itsevaluation at 119905 = 119896119879 The derivative of (10) is

119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (12)

Evaluated at time 119896119879 the previous equation is written as

119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (13)

where 119890(119896119879) is defined in (9) whereas 119890(119896119879) 119890(119896119879) and

119890

(119896119879) are obtained by differentiating (7) and replacing 119905 by119896119879as follows

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) =

119882119908ref (119896119879) minus

119882119898

(119896119879)

(14)

In turn the derivatives 119898

(119896119879) 119898

(119896119879) and

119882119898

(119896119879) areobtained by differentiating (1) or (3) depending on therequired signal

Remark 1 The error function 119904(119896119879) and its time derivative119904(119896119879) are defined in (11) and (13) The signals required fortheir computations are the output error (8) and the first andhigher order derivatives of the output error that is 119890(119896119879)

119890(119896119879) 119890(119896119879) and

119890 (119896119879) which are defined in (9) and (14)

Remark 2 The signals 119898

(119896119879) 119898

(119896119879) and

119882119898

(119896119879) areobtained by differentiating expressions (1) or (3) so that theircomputation involves the values of the state variables at thesampling time 119905 = 119896119879 and matrices 119860

1and 119861

1of (2) in the

case that the switch is ON or matrices 1198602and 119861

2of (4) in the

case that the switch is OFF

Remark 3 The error function 119904(119896119879) and its time derivative119904(119896119879) depend on the values of the state variables at thesampling time 119905 = 119896119879

32 ZAD Strategy The ZAD strategy is a quasi-sliding con-trol strategy widely studied for second order systems [14 16]The basic idea of this technique is to compute the duty cycleso that the average dynamics of the error function is zero inone time period A complete description of the ZAD strategyand its application to control buck power converters can befound in [16] Contrary to the sliding mode control withhysteresis band this technique allows the a priori definitionof the switching frequency

In this paper the average dynamics of the error functionis solved analytically using the assumption that the errorfunction is piece-wise linear as it is illustrated in Figure 2(b)As a result in average the error function is zero in eachtime period so that the system is forced to evolve in a closeneighborhood of the sliding surface The tasks of the ZADcontrol design are (cf [14 pages 338 339]) as follows (i)define the error function in terms of the output error (ii)assume that the error function 119904 behaves as a piece-wiselinear function and define this function and (iii) develop thetime integral of the error function using the piece-wise lineardefinition and (iv) solve the resulting expression for the ZADduty cycle 119889

119896

According to the previous steps the average dynamicsare defined as a time integral of the error function 119904(119896119879)

(11) in one time period using the assumption that the errorfunction is piece-wise linear namely 119904pwl The piece-wiselinear function is presented in Figure 2(b) where (i) when119896119879 le 119905 lt 119905

1the switch is ON and the system model is

given by (1) when 1199051

le 119905 le 1199052the switch is OFF and

the model of the system is given by (3) when 1199052

le 119905 lt

(119896 + 1)119879 the switch turns ON again and the system modelis given by (1) (ii) 119904

+and 119904minusare the slopes of 119904pwl when the

switch is ON and OFF respectively and (iii) 1199041and 1199042are

the values of 119904pwl at some specific instants which correspondto switching instants Other simplifications are introduced inthe definition of 119904pwl namely (i) the slopes of the first andthird parts of the function are the same denoted as 119904

+ and

(ii) all slopes can be computed using the data at the beginningof the sampling time Considering these simplifications thepiece-wise linear function 119904pwl is defined as

119904pwl (119905) =

119904 (119896119879) + (119905 minus 119896119879) 119904+

(119896119879) if 119896119879 le 119905 lt 1199051

1199041

(119896119879) + (119905 minus 1199051) 119904minus

(119896119879) if 1199051

le 119905 le 1199052

1199042

(119896119879) + (119905 minus 1199052) 119904+

(119896119879) if 1199052

lt 119905 le (119896 + 1) 119879

(15)


where 1199051

= 119896119879 + 1198631198961198792 and 119905

2= (119896 + 1)119879 minus 119863

1198961198792

(similar to Figure 2(a)) signals 119904(119896119879) 119904+

(119896119879) and 119904minus

(119896119879)

are computed according to Remarks 1 and 2 The signal119904(119896119879) is computed using (11) and (1) The signal 119904

+(119896119879) is

computed using (13) and matrices 1198601and 119861

1(2) The signal

119904minus

(119896119879) is computed using (13) and matrices 1198602and 119861

2(4)

1199041(119896119879) = 119904(119896119879) + 119863

1198961198792 119904+

(119896119879) and 1199042(119896119879) = 119904

1(119896119879) + 119879(1 minus

119863119896) 119904minus

(119896119879) Dependence with respect to 119896 has been explicitlyintroduced into the equations with the aim of clarifying thatthese derivatives change each sampling time 119863

119896119879 is the time

the switch is ON and 119863119896

isin [0 1] It can be noticed that119904pwl(119905) depends on 119863

119896and (1) and (3) The idea of the ZAD

strategy is to design the duty cycle 119863119896 such that the average

of signal 119904pwl(119905) is zero during one time periodThis conditionis expressed as

int

(119896+1)119879

119896119879

119904pwl (119905) 119889119905 = 0 (16)

The above integral is solved for 119863119896

119863119896|119896

=

2119904 (119896119879) + 119879 119904minus

(119896119879)

119879 ( 119904minus

(119896119879) minus 119904+

(119896119879))

(17)

where the notation 119896|119896 is used to emphasize that the dutycycle value depends on the state variables measured at thesampling time 119896119879 However in the experimental setup it hasbeen experimentally confirmed that the control action expe-riences one period delay so that it is necessary to compute119863119896using the values acquired in the previous sampling time

In this case

119863119896|(119896minus1)

=

2119904 ((119896 minus 1) 119879) + 119879 119904minus

((119896 minus 1) 119879)

119879 ( 119904minus

((119896 minus 1) 119879) minus 119904+

((119896 minus 1) 119879))

(18)

Finally the ZAD duty cycle is given by

119889119896

=

1 if 119863119896|(119896minus1)

gt 1

119863119896|(119896minus1)

if 0 le 119863119896|(119896minus1)

le 1

0 if 119863119896|(119896minus1)

lt 0

(19)

where (119896 minus 1) means a one-period delay in all the measuredvalues and 119889

119896is the duty cycle that should be applied to the

system between 119896119879 and (119896 + 1)119879

Remark 4 Theduty cycle119889119896(19) is saturated between 0 and 1

because 119889119896is the ratio between the time the switch is ON and

the time period119879 For duty cycle values outside of this specificrange the duty cycle is saturated and in the next period thecontroller computes the duty cycle again with the new valuesof the states

Remark 5 The ZAD duty cycle 119889119896defined in (19) involves

the state variables values at the sampling time 119905 = (119896 minus 1)119879matrices 119860

1and 119861

1of (2) andmatrices 119860

2and 119861

2of (4)The

signals necessary for its computation are (i) the signal119863119896|(119896minus1)

(18) (ii) 119904((119896 minus 1)119879) which is computed using (11) where119890((119896minus1)119879) 119890((119896minus1)119879) 119890((119896minus1)119879) and 119890((119896minus1)119879) are obtainedaccording to Remarks 1 and 2 (iii) 119904

+((119896 minus 1)119879) which is

computed using (13) where 119890((119896minus1)119879) 119890((119896minus1)119879) 119890((119896minus1)119879)

and

119890 ((119896 minus 1)119879) are obtained according to Remarks 1 and 2using matrices 119860

1and 119861

1of (2) (iv) 119904

minus((119896 minus 1)119879) which is

computed in a similar way as 119904+

((119896 minus 1)119879) but using matrices1198602and 119861

2of (4)

Remark 6 Depending on the values of 119896119904119894associated with

119904(119870119879) (11) and its derivatives the ZAD duty cycle 119889119896induces

the convergence of the error function 119904pwl(119905) towards a closeneighborhood of 119904(119905) = 0 this convergence is achievedthrough the fulfillment of the ZAD condition (16) Never-theless this convergence property is not as strong as theconvergence of sliding mode control and the ZAD strategyis not as robust as sliding control Even more ZAD controlstrategy may lead to unstable closed loop system when thereARE period delays The system analyzed in this work hasone period delay Experimental and numerical results notpresented here show that the states of the converter (120592

119888

and 119894119871) never stabilize at a fixed point For this reason

the ZAD-FPIC strategy which was developed in [16] as animprovement of the ZAD strategy is added to ZAD controlZAD-FPIC controller has proven to be effective and robust to(i) stabilize the closed loop system and (ii) achieve adequateperformance of the controlled system

33 FPIC Control Technique As mentioned in Remark 6 theZAD strategy is not as robust as sliding mode control andit implies severe degradation of the closed loop performancewhen there is a time delay Therefore the ZAD-FPIC is usedto control the system

The duty cycle of the ZAD-FPIC strategy is based on (i)the ZAD duty cycle (119889

119896) defined in (19) (ii) a feedforward

component defined as 119889lowast which corresponds to the steady

state value of 119889119896 Indeed the new duty cycle is a weighted

sum of the duty cycles 119889119896and 119889

lowast The theoretical basisof FPIC design was proposed for the first time in [16]and experimentally proven in [17 18] The mathematicaldefinition of the steady state value of the ZAD duty cyclethat is 119889

lowast is based on the discrete time representation ofthe system as explained below The system described inSection 21with theZADduty cycle119889

119896(19) can be represented

as 119909(119896+1) = 119891(119909(119896) 119889119896) As 119889

119896is a function of the states then

119909(119896 + 1) can be expressed as a function of states 119909(119896) that is119909(119896+1) = 119891(119909(119896))This equation can be solved to find out thesteady states values To compute 119889

lowast these steady state valuesare replaced in the expression of 119889

119896 The following theorem

defines the unsaturated ZAD-FPIC duty cycle as a weightedsum of the ZAD duty cycle 119889

119896and its steady state 119889

lowast

Theorem 7 (FPICTheorem) Consider the following discrete-time system

119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (20)

where 119909(119896) isin R119899 119891 R119899+1 rarr R119899 119889119896

isin R is the dutycycle that depends on the system states The fixed point of theabove system is given by 119909

lowast

= 119891(119909lowast

119889lowast

) where 119909lowast and 119889

lowast arethe steady state values of 119909(119896) and 119889

119896 respectively Assume that

the fixed point (119909lowast

119889lowast

) is stable so that the modulus of each


eigenvalue is less than one that is |120582119894(119869)| lt 1 for all 119894 where

1205821 120582

119899are the eigenvalues of the jacobian 119869

1

1198691

=

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816(119909lowast 119889lowast)

(21)

Then the choice

119889119896

=

119889119896

+ 119873119889lowast

119873 + 1

(22)

with high enough 119873 achieves the asymptotic convergence ofthe system (20) to the fixed point (119909

lowast

119889lowast

) independently of thevalue of 119889

119896

Remark 8 As parameter 119896119904119894

varies the system can turnunstable but the steady state does not change Evenmore thisresult is the same when one period delay is considered

Remark 9 Note that (22) does not change the fixed point of(20)

Theproof is presented in the appendixThe value of119873 canbe found by means of a numerical analysis 119889

119896refers to the

duty cycle computed using the ZAD-technique (19) and 119889lowast

corresponds to the steady state value of the controlled systemHowever for PWM controlled systems 119889

lowast can be computedby sensing the input voltage (Vin) at the beginning of theperiod In this way it is possible to find out the expected dutycycle in that period and 119889

lowast is replaced by the expected dutycycle in the current period Particularly in the case 119864 ≫ 119881fdwhich is the normal case the expected duty cycle will be

119889lowast

=

VrefVin

(23)

Taking into account the above theorem and the easy compu-tation of 119889

lowast the final duty cycle applied to the motor systemdescribed by (5) is expressed as

119889ZAD-FPIC = 119889 =

1 if 119889119896

gt 1

119889119896

if 119889119896

isin [0 1]

0 if 119889119896

lt 0

(24)

Remark 10 The ZAD-FPIC duty cycle (119889ZAD-FPIC) is definedin (24) and the signals and equations required for its com-putation are the unsaturated duty cycle 119889

119896(22) the ZAD

duty cycle 119889119896(19) the signal 119889

lowast (23) and the value of theconstant 119873 In addition the ZAD duty cycle (19) is computedaccording to Remark 5

Remark 11 Equations (22) and (24) indicate that the ZAD-FPIC duty cycle 119889ZAD-FPIC is a weighted sum of (i) the ZADduty cycle 119889

119896(19) and (ii) the steady state of the ZAD duty

cycle or the expected duty cycle in the current period that is119889lowast (23)

34 Motor Load Torque Estimator The control scheme to beformulated relies on the known parameter values of the plantmodel but the load torque (119879

119871) and the friction torque (119879fric)

exhibit an unknown time varying behavior Therefore thissubsection shows the formulation of the online estimationmechanism for 119879

119871+ 119879fric using the LMS technique Since the

sampling rate is relatively high the estimation mechanismis formulated in continuous time The LMS is a simplifiedversion of the recursive least squares (RLS) technique as isdiscussed below The RLS method for systems in continu-ous time aims at minimizing the weighted integral of thedifference between the actually observed and the computedvalues (see [19 20]) Consider a continuous time system If thesystem is of first order a first order filter given by (25) mustbe introduced where 119888

119891is a user-defined positive constant

(cf [19]) Consider the following

119867119891

=

119888119891

119901 + 119888119891

(25)

The parameterization of the system leads to

119865 = 120593⊤

120579 (26)

where 119865 is the ldquooutputrdquo vector and 120593 is the regression vectorBoth of them are known and vary with time whereas 120579 isthe parameter vector which contains the unknown systemparametersThe formulation of the RLSmethod assumes thatthe unknown parameters contained in 120579 are constant whichmakes it possible to guarantee the convergence propertiesDespite this assumption the RLS method also achieves ade-quate performance in real life systems where the parametersusually vary with time (see [19 21 22]) This can be handledby properly defining the forgetting factor The RLS estimatoris described by the following equations (cf [19 20])

119889120579

119889119905

= 119875120593120576 (27)

120576 = 119865 minus 120593⊤

120579 (28)

119889119875

119889119905

= 120572119875 minus 119875120593120593⊤

119875 (29)

where 119875 120579 119865 and 120593 vary with time 120572 is the forgetting

factor and 120579 is the estimate of 120579 This estimator achieves the

asymptotic convergence of the estimate 120579 towards the real

parameter vector 120579 if the ldquopersistent excitationrdquo condition isfulfilled (see [20]) There are several simplified versions ofthe RLS method that render the computation of the matrix 119875

simpler but the cost is a slower convergence of the parameterestimate

120579 (see [19])The LMS algorithm considers a constantscalar value for the matrix 119875 (see [19 23 24]) that is

119875 = 120574 (30)

where 120574 is a positive scalar constant Thus the LMS estimate120579 is provided by (27) (28) and (30) with 119865(119905) and 120593(119905)

defined on the basis of parameterization (26) To apply theLMSmethod to the estimation of uncertain parameters of themotormodel the first differential equation given in (1) is usedand it is given by

119898

(119905) =

minus119861119882119898

(119905)

119869eq+

119896119905119894119886

(119905)

119869eqminus

(119879fric + 119879119871)

119869eq (31)


[Parameters]

T

d

d

dg s

L

L TLWm

Wm

ia

ia

fcn fcnfcn

[Wmref ] Wmref

P

iL

Vc

x1

x2Duty cycle

PWMC

Triangular

Quantizer10 bits

ZAD-FPIC

Torque estimator

rL i

s

i

L

C

E

Diode

Unit delay QuantizerN bits

Zero-order

Parameter

Mechanicalmodel

motor

hold

+

+

+

+

minus i+minus

minusminusv

1

z

rs + rMKe lowast Wm

keWm

WmPmotor

iafcn

Vc

Ra iaLa

1T

Figure 3 ZAD-FPIC controller simulation block

By applying the first order filter (25) to the above equation itis obtained (see [19])

119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905)

=

minus119861119867119891

119882119898

(119905)

119869eq+

119896119905119867119891

119894119886

(119905)

119869eqminus

119867119891

(119879fric + 119879119871)

119869eq

(32)

By defining 119865 120593 and 120579 as

119865 (119905) = 119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905) +

119861119867119891

119882119898

(119905)

119869eqminus

119896119905119867119891

119894119886

(119905)

119869eq

120593 (119905) = minus

1

119869eq119867119891

120579 = (119879fric + 119879119871)

(33)

Equation (32) can be written as

119865 (119905) = 120593 (119905) 120579 (34)

where 120579 is the unknown parameter

Remark 12 The estimate 120579 for the uncertain parameter 120579 =

119879fric + 119879119871is provided by (27) (28) and (30) with 119865(119905) and

120593(119905) defined in (33) 119888119891being a user-defined positive constant

Equation (30) is considered instead of (29) in order to avoidcomplex computation of the matrix 119875

4 Numerical and Experimental Results

In view of Remark 6 the ZAD-FPIC strategy with duty cycle(119889ZAD-FPIC) given by (24) is considered for implementationon the real system However there are two major problemsassociated with the implementation First the values of thestate variables at the beginning of each period are requiredfor the computation of the duty cycle Thus the sampledsignals and PWM are synchronized using a trigger signal atthe beginning of each switching period Second the ZAD-FPIC strategy relies on known values of the friction torqueand load torque which may be uncertain Indeed manyvalues of the plant model parameters both the DC-motorand converter parameters are used by the controller Thusthe load and friction torque term is estimated online by theLMS mechanism whereas the remaining parameters wereestablished previously The uncertain term 120579 = 119879fric + 119879

119871is

estimated as 120579 provided by the LMS estimation mechanism

according to Remark 12 of Section 34Figure 3 shows the block diagram for the ZAD-FPIC

controller simulation benchmark with parameter estimationThe parameter estimation and ZAD-FPIC blocks use theequations mentioned in Remark 10

Several tests performed on the motor system and notshown here proved that the sensitivity of the closed loopsystem with respect to parameter 119896

1199043is significant but

sensitivity with respect to 1198961199041and 119896

1199042is low Therefore 119896

1199043

is chosen as a variable control parameter and bifurcationparameter whereas 119896

1199041and 119896

1199042are fixed control parameters


0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3

(a) Simulation results and output of the system

0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d

(b) Simulation results and duty cycle

0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3

(c) Experimental results and output of the system

0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d

(d) Experimental results and duty cycle

Figure 4 Bifurcation diagram of the output (119882119898) and duty cycle (119889) of the controlled system varying the bifurcation parameter 119870

1198783

Other parameter values are shown inTable 1 To ease the visu-alization of the controller parameters they will be expressedin dimensionless form and they are called 119870

1198781 1198701198782 and 119870

1198783

The relationships between dimensionless parameters (1198701198781

1198701198782 and 119870

1198783) and control parameters (119896

1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783

119871119862radic119871119862Figure 4 shows the bifurcation diagrams of the system

output (119882119898) and the duty cycle 119889 when the parameter 119870

1198783is

varied In this experiment119879fric+119879119871

= 00284Nsdotm and119873 = 1Figures 4(a) and 4(b) show simulation results while Figures4(c) and 4(d) show experimental results It is important tonote that even in the cases where the duty cycle does nothave a fixed frequency speed regulation is reached with avery low error Fixed frequency is lost for 119870

1198783⪅ 15 and

the converter works in chaotic way In the interval 15 lt

1198701198783

lt 30 the converter works with a fixed switchingfrequency but chaotic motion is present in the converterThe chaotic motion cannot be noticed in the speed becausethe motor has a slow response For 119870

1198783gt 30 the system

operation is very close to fixed frequency without chaoticmotion The output error in chaotic regime is greater thanthe output error in stable region (see Figures 4(a) and 4(c))The shapes of the closed loop signals of the experiment for1198701198783

gt 30 are due tomeasurement noise delays uncertaintiesand unmodeled dynamics From a practical point of view

the chaotic behavior of the duty cycle does not affect thespeed regulation However fixed switching frequency has theadvantage that the filtering of harmonics is easier than thatof chaotic behavior For this reason it is better to designthe controller in such a way that all the states are stable andoperate in a fixed point

Figure 5 shows time evolution for the output 119882119898

andestimated load torque (119879fric + 119879

119871) when the following values

are considered 119882119898ref = 400 rads 119870

1198783= 80 and 119873 =

15 The torque changes from (119879fric + 119879119871) = 004Nsdotm to

(119879fric + 119879119871) = 00715Nsdotm at 119905 = 05 s The motor speed (119882

119898)

follows the reference value (119882119898ref = 400) with an error value

below 025 only when the load torque estimator is used (seeFigures 5(a) and 5(c)) Figures 5(b) and 5(d) show estimatednumerical and experimental load torque values respectively

Figure 6 shows the estimated load torque value whenthe adaptive constant 120574 is varied For small values of 120574 thevelocity of convergence from the estimated parameter to thereal parameter is slow but oscillations are absent This is aresult of the parameter estimation theory

5 Conclusions

In this work ZAD-FPIC technique has been applied to a highorder dynamical system for the first time In particular it


0 02 04 06 08 1390

395

400

405

Time (s)

ZAD-FPIC controller with load estimatorZAD-FPIC controller without load estimator

0 02 04 06 08 1Time (s)


Wm

(rad

s)

(a) Simulation results

0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001

(b) Simulation results

0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)

(c) [Experimental results

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001

(d) Experimental results

Figure 5 Evolution of 119882119898and estimated torque (119879fric + 119879

119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001

(b) Experimental results

Figure 6 Dynamics of estimator when 120574 is varied


Table 1 Parameters for simulation and experiment

Parameter Value119864 Input voltage 40035V119903119904 Internal resistance of the source and

MOSFET 084Ω

119881fd Diode forward voltage 11 V119871 Inductance 2473mH119903119871 Internal resistance of the inductor 1695Ω

119862 Capacitance 4627 120583F119877119886 Armature resistance 27289Ω

119871119886 Armature inductance 117mH

119861 Viscous friction coefficient 0000138Nsdotmrads119869eq Moment of inertia 0000115 kgsdotm2

119896119905 Motor torque constant 00663NsdotmA

119896119890 Voltage constant 00663Vrads

119879fric Friction torque 00284Nsdotm119879119871 Load torque Variable Nsdotm

119882119898ref Reference speed 400 rads

119873 FPIC control parameter 1 or 15119865119888 Switching frequency 6 kHz

119865119904 Sampling frequency 6 kHz

1119879 119901 1 Delay time 1666 120583s1198701198781

1198701198782 Control parameters Equal to 2

1198701198783 Bifurcation parameters Variable

was designed and applied successfully to a coupled motor-power converter system Simulations and experiments wereperformed and they showed significant agreement

The quasi-sliding control technique designed in this workallows to choose the switching frequency of the controllerThis switching frequency is given by the sampling period andonce the sampling period is defined the system works in afixed switching frequency regime

The stability of the closed loop system was analyzedusing bifurcation diagrams and stable behavior and chaoticbehavior were observed For all cases the regulation errorwas lower than 05 However when the system operates ina stable zone (119870

1198783gt 30) the regulation error is lower than

015 In addition the parameter estimator handles the effectof uncertainty on the load torque and friction torque so thatoutput regulation is achieved even in the case that the load ischanged

Appendix

Proof of Theorem 7

The system (20) with the new manipulated input (22) isdescribed by the following equation

119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)

From (22) it follows that if 119873 is high enough then 119889119896

= 119889lowast

so that (119909lowast

119889lowast

) is still a fixed point of the system (A1) TheJacobian matrix of system (A1) around the fixed point is

119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)

using (21) and (22) yield

119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)

where 1198691is defined in (21) From the theorem of continuity

it is follow that if 119873 is high enough the eigenvalues ofmatrix 119869 approximate the eigenvalues of matrix 119869

1 Since the

eigenvalues of 1198691are inside the unit circle the system (A1) is

locally stable (see [17 page 02])

Acknowledgments

The authors wish to thank the research groups GREDyP andPCI This project was partially supported by UniversidadNacional de Colombia Manizales Project no 16074 Vicer-rectorıa de Investigacion and DIMA

References

[1] C F Huang J S Lin T L Liao C Y Chen and J J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011

[2] TWang KWang andN Jia ldquoChaos control and hybrid projec-tive synchronization of a novel chaotic systemrdquo MathematicalProblems in Engineering vol 2011 Article ID 452671 13 pages2011

[3] H T Yau J S Lin and N S Pai ldquoRobust controller designfor modified projective synchronization of chen-lee chaoticsystems with nonlinear inputsrdquo Mathematical Problems inEngineering vol 2009 Article ID 649401 10 pages 2009

[4] L Boutat-Baddas J P Barbot D Boutat and R TauleigneldquoSlidingmode observers and observability singularity in chaoticsynchronizationrdquo Mathematical Problems in Engineering vol2004 no 1 pp 11ndash31 2004

[5] A C Huang and Y S Kuo ldquoSliding control of non-linearsystems containing time-varying uncertainties with unknownboundsrdquo International Journal of Control vol 74 no 3 pp 252ndash264 2001

[6] R Yazdanpanah J Soltani and G R Arab Markadeh ldquoNon-linear torque and stator flux controller for induction motordrive based on adaptive input-output feedback linearization andslidingmode controlrdquo Energy Conversion andManagement vol49 no 4 pp 541ndash550 2008

[7] C Lascu and A M Trzynadlowski ldquoCombining the princi-ples of sliding mode direct torque control and space-vectormodulation in a high-performance sensorless AC driverdquo IEEETransactions on Industry Applications vol 40 no 1 pp 170ndash1772004


[8] C Lascu I Boldea and F Blaabjerg ldquoDirect torque control ofsensorless induction motor drives a sliding-mode approachrdquoIEEE Transactions on Industry Applications vol 40 no 2 pp582ndash590 2004

[9] TOrlowska-KowalskaMDybkowski andK Szabat ldquoAdaptivesliding-mode neuro-fuzzy control of the two-mass inductionmotor drive without mechanical sensorsrdquo IEEE Transactions onIndustrial Electronics vol 57 no 2 pp 553ndash564 2010

[10] J Wu D Pu and H Ding ldquoAdaptive robust motion control ofSISOnonlinear systemswith implementation on linearmotorsrdquoMechatronics vol 17 no 4-5 pp 263ndash270 2007

[11] G Venkataramanan and D M Divan ldquoDiscrete time integralsliding mode control for discrete pulse modulated convertersrdquoin Proceedings of the 21st Annual IEEE Power Electronics Special-ists Conference (PESC rsquo90) pp 67ndash73 San Antonio Tex USAJune 1990

[12] A Leva L Piroddi M Di Felice A Boer and R PaganinildquoAdaptive relay-based control of household freezers with on-offactuatorsrdquoControl Engineering Practice vol 18 no 1 pp 94ndash1022010

[13] M D Felice L Piroddi A Leva and A Boer ldquoAdaptivetemperature control of a household refrigeratorrdquo in Proceedingsof the American Control Conference (ACC rsquo09) pp 889ndash894 StLouis Mo USA June 2009

[14] E Fossas R Grino and D Biel ldquoQuasi-sliding control basedon pulse width modulation zero averaged dynamics and the L2normrdquo in Advances in Variable Structure Systems X Yu and JXu Eds pp 335ndash344 World Scientific 2000

[15] X S Wang C Y Su and H Hong ldquoRobust adaptive controlof a class of nonlinear systems with unknown dead-zonerdquoAutomatica vol 40 no 3 pp 407ndash413 2004

[16] F Angulo Analisis de la dinamica de convertidores electronicosde potencia usando PWM basado en promediado cerode la dinamica del error (ZAD) [PhD dissertation]Universitat Politecnica de Catalunya Barcelona Spain2004 httpwwwtdxcathandle108035941

[17] F Angulo G Olivar J Taborda and F Hoyos ldquoNonsmoothdynamics and FPIC chaos control in a DC-DC ZAD-strategypower converterrdquo in Proceedings of the 6th EUROMECH Non-linear Oscillations Conference (ENOC rsquo08) pp 2392ndash2401 SaintPetersburg Russia June-July 2008

[18] F E Hoyos D Burbano F Angulo G Olivar N Toro and J ATaborda ldquoEffects of quantization delay and internal resistancesin digitally ZAD-controlled buck converterrdquo International Jour-nal of Bifurcation andChaos inApplied Sciences andEngineeringvol 22 no 10 Article ID 1250245 9 pages 2012

[19] K J Astrom and B Wittenmark Adaptive Control Addison-Wesley Boston Mass USA 2nd edition 1995

[20] J Slotine and W Li Applied Nonlinear Control Prentice HallEnglewood Cliffs NJ USA 1991

[21] J Seok ldquoHybrid adaptive optimal control of anaerobic fluidizedbed bioreactor for the de-icing waste treatmentrdquo Journal ofBiotechnology vol 102 no 2 pp 165ndash175 2003

[22] G P Liu and S Daley ldquoOptimal-tuning nonlinear PID controlof hydraulic systemsrdquo Control Engineering Practice vol 8 no 9pp 1045ndash1053 2000

[23] B Widrow P Mantey L Griffiths and B Goode ldquoAdaptiveantenna systemsrdquo Proceedings of the IEEE vol 55 no 12 pp2143ndash2159 1967

[24] C Mays ldquoThe relationship of algorithms used with adjustablethreshold elements to differential equationsrdquo IEEE Transactionson Electronic Computers vol 14 no 1 pp 62ndash63 1965

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Differential EquationsInternational Journal of

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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


assumed that stator and rotor resistances may be varying anduncertain The controller takes into account such variationthe stator and rotor resistances are online updated on thebasis of the stator measured currents and voltages

In [7 8] a sliding mode controller is developed for asensorless induction motor (IM) drive using the principlesof direct torque control and space-vector pulse-width mod-ulation (PWM) The goal is to control the rotor speed andstator flux (torque and flux) by means of two sliding modecontrollers Sliding mode observers are used for torque andflux estimation employing two motor models The inverter iscontrolled using space-vector PWM

In [9] an adaptive sliding mode controller is designedfor a two-mass IM drive without mechanical sensors withan elastic joint The goal is to control the motor speed Thespeed is notmeasured whereas the stator current and the DClink voltage of the motor are The control structure consistsof the rotor flux control loop and the speed control loop (i)in the flux control loop an MRAS type estimator is used forestimation of rotor speed and flux based on the current andvoltages of the IM and (ii) in the speed control loop thereis an inner control loop where the electromagnetic torqueof the IM is regulated In addition a fuzzy neural networkis used with online tuning of its connective weights

In [10] an adaptive robust sliding controller is formulatedfor SISO nonlinear systems with implementation on a linearmotor It is assumed that (i) the nonlinear part can belinearly parameterized and (ii) there are unknown varyingparameters and disturbances whose bounds are known Adiscontinuous projection type updated law is used in order toguarantee the bounded nature of the parameter updates Thecontrol scheme is applied to an X-Y table driven by a linearmotorThe control design gives as result a signum type signalbut a saturation type function is proposed to replace it

Although the basic approach of the sliding mode con-trol is intended for a continuous control input there areapplications to two-value control input For instance in[11] a sliding control scheme is designed for discrete pulsemodulated converters An integral term is incorporated inthe definition of the error function 119878 in order to achieve thedesired steady state behavior Simulations and experimentsconfirm the improvement of the steady state behavior Themain features of such control scheme are (i) the idea is thatthe error function 119878 converges to zero (ii) the control designis based on the properties of the time derivative of the errorfunction 119878 and (iii) the commutation frequency is not fixed

The drawbacks of sliding mode control are (cf [5 610]) (i) the control gain is large when the defined param-eter bounds are large (ii) the control input may exhibitchattering which may excite high frequency unmodeledmodes of the system and (iii) the input chattering maylead to high power consumption and wear of mechani-cal components (cf [12 13]) There have been significantworks focused on the reduction or fixing of the commu-tation frequency in order to reduce chattering mainly inthe field of power electronics The current sliding controlschemes with fixed frequency lead to significantly flawedconvergence of the system variables towards the slidingsurface

Quasi-sliding mode control emerged as an alternativesolution to the drawbacks of sliding control retaining theinherent advantages of its operation Indeed the undesirablechattering and high frequency switching of the control signalare avoided The controller design aims at achieving systemconvergence towards a boundary layer instead of the slidingsurface as mentioned in [14 pages 335 336] In the case ofa continuous control input the boundary layer implies theuse of a saturation function in the control law instead ofa discontinuous signal (see [10 15]) The main features areas follows (Ci) the main idea is that the system convergestowards a region so that |119878| le 120576 where 120576 is a small user-defined positive constant (Cii) the control law is designedto achieve an adequate negative nature of the time derivativeof the Lyapunov function or equivalently adequate signumproperties of the time derivative of the error function (Ciii)the commutation frequency is not fixed

Thus it is important to formulate a control law thatachieves desired system performancewhile tackling the effectof uncertainties or disturbances To develop such controllerin this work two different controllers were used the zeroaverage dynamics (ZAD) technique and the fixed pointinducting control (FPIC) The ZAD technique working witha second loop of control (FPIC) results in a quasi-slidingcontroller that has proven to be effective and robust touncertainties and disturbances

The theoretical basis of the ZAD technique design wasexplained by Fossas et al in [14] in 2000 In the ZAD designthe definition of the duty cycle is such that the averagedynamics of the error function is zero for one measuringperiod This idea leads to a fixed switching frequency withconvergence of the system towards the boundary layercompars [14 pages 341ndash343] In [16] a piece-wise linearapproximation of the error function was proposed in orderto simplify the computation of the duty cycle Neverthelessthe ZAD strategy is not as robust as sliding control Forthis reason the ZAD-FPIC strategy was developed in [16]as an improvement of the ZAD strategy Indeed The ZAD-FPIC duty cycle is the weighted sum of (i) the ZAD dutycycle and (ii) a feed-forward component defined as the steadystate of the ZAD duty cycle The ZAD-FPIC strategy allowsfurther improvement of the convergence properties of theZAD technique The theoretical basis of the FPIC design wasproposed for the first time in [16] and experimentally provenin [17 18] The main features of the ZAD-FPIC strategy areas follows (i) the control design is not aimed at obtainingnegativeness or positiveness properties of the time derivativeof the error function instead a linear combination of theerror and its derivative is forced to have zero average (ii)ZAD-FPIC scheme improves the properties of the ZADtechnique by defining a new duty cycle that combines theZAD duty cycle with its steady state or with its expected dutycycle in each iteration thus stability is guaranteed in a widerrange of the parameter values and (iii) the closed loop systemis more robust with this controller

In this paper an adaptive ZAD-FPIC strategy is formu-lated for motor speed control The system involves a buckpower converter a PMDC motor and a dSPACE platformThe load torque and the friction torque are considered

















119886 are constant





[

[

[

[

119898

119894119886

120592119888

119894119871

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

minus119861

119869eq

119896119905

119869eq0 0

minus119896119890

119871119886

minus119877119886

119871119886

1

119871119886

0

0

minus1

119862

0

1

119862

0 0

minus1

119871

minus (119903119904

+ 119903119871)

119871

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

119882119898

119894119886

120592119888

119894119871

]

]

]

]

+

[

[

[

[

[

[

[

[

minus (119879fric + 119879119871)

119869eq0

0

119864

119871

]

]

]

]

]

]

]

]

(1)


= 1198601119909 + 1198611 (2)


= 1199091 the



119888= 1199093 and




[

[

[

[

119898

119894119886

120592119888

119894119871

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

minus119861

119869eq

119896119905

119869eq0 0

minus119896119890

119871119886

minus119877119886

119871119886

1

119871119886

0

0

minus1

119862

0

1

119862

0 0

minus1

119871

minus (119903119871)

119871

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

119882119898

119894119886

120592119888

119894119871

]

]

]

]



ZAD-FPICcontroller

AD

ADParameters

120592c iLsensor

sensor

DCmotorGate Buck

converter


Ks123

Wm ia


rs rLS L

Vfd

iLia

Ra

La120592cC

minus

minus

minusminus

+ ++

E

keWm(t)TemTfric +

+

TL

JL


Wm

Diode



+

[

[

[

[

[

[

[

[

minus (119879fric + 119879119871)

119869eq0

0

minus119881fd119871

]

]

]

]

]

]

]

]

(3)


= 1198602119909 + 1198612 (4)




119886









=

1198601119909 + 1198611

si 119896119879 le 119905 lt 119896119879 +

119889119879

2

1198602119909 + 1198612

si 119896119879 +

119889119879

2

le 119905 lt 119896119879 + 119879 minus

119889119879

2

1198601119909 + 1198611

si 119896119879 + 119879 minus

119889119879

2

le 119905 lt 119896119879 + 119879

(5)







119890 (119905) = 119882119908ref (119905) minus 119882

119898(119905) (6)


and 119898


119890 (119905) = 119908ref (119905) minus

119898(119905) (7)


119890 (119896119879) = 119882119908ref (119896119879) minus 119882

119898(119896119879) (8)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879) (9)




E

kT t

dT2

T

V

dT2

(k + 1)TminusVfd

(a) Centered PWM

s

kT t

s1

s2


s+

s+

sminus

t1

t2

spwl (t)

kT + T




119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (10)

Constants 1198961199041 1198961199042 and 119896



119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (11)


119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (12)


119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (13)


119890


119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) =

119882119908ref (119896119879) minus

119882119898

(119896119879)

(14)


(119896119879) 119898

(119896119879) and

119882119898



119890(119896119879) 119890(119896119879) and



(119896119879) 119898

(119896119879) and

119882119898


1and 119861

1of (2) in the


2of (4) in the





119896







le 119905 lt





+ and


119904pwl (119905) =

119904 (119896119879) + (119905 minus 119896119879) 119904+

(119896119879) if 119896119879 le 119905 lt 1199051

1199041

(119896119879) + (119905 minus 1199051) 119904minus

(119896119879) if 1199051

le 119905 le 1199052

1199042

(119896119879) + (119905 minus 1199052) 119904+

(119896119879) if 1199052

lt 119905 le (119896 + 1) 119879

(15)


where 1199051

= 119896119879 + 1198631198961198792 and 119905

2= (119896 + 1)119879 minus 119863

1198961198792


(119896119879) and 119904minus

(119896119879)


+(119896119879) is


1(2) The signal

119904minus


2(4)

1199041(119896119879) = 119904(119896119879) + 119863

1198961198792 119904+

(119896119879) and 1199042(119896119879) = 119904

1(119896119879) + 119879(1 minus

119863119896) 119904minus


119896119879 is the time






int

(119896+1)119879

119896119879

119904pwl (119905) 119889119905 = 0 (16)


119863119896|119896

=

2119904 (119896119879) + 119879 119904minus

(119896119879)

119879 ( 119904minus

(119896119879) minus 119904+

(119896119879))

(17)


In this case

119863119896|(119896minus1)

=

2119904 ((119896 minus 1) 119879) + 119879 119904minus

((119896 minus 1) 119879)

119879 ( 119904minus

((119896 minus 1) 119879) minus 119904+

((119896 minus 1) 119879))

(18)


119889119896

=

1 if 119863119896|(119896minus1)

gt 1

119863119896|(119896minus1)

if 0 le 119863119896|(119896minus1)

le 1

0 if 119863119896|(119896minus1)

lt 0

(19)









1and 119861


2and 119861

2of (4)The



+((119896 minus 1)119879) which is


and


1and 119861

1of (2) (iv) 119904




2of (4)




119888












as 119909(119896+1) = 119891(119909(119896) 119889119896) As 119889







lowast


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (20)



lowast

= 119891(119909lowast

119889lowast





119889lowast




1205821 120582


1

1198691

=

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816(119909lowast 119889lowast)

(21)

Then the choice

119889119896

=

119889119896

+ 119873119889lowast

119873 + 1

(22)


lowast

119889lowast


119896





119896refers to the





119889lowast

=

VrefVin

(23)



119889ZAD-FPIC = 119889 =

1 if 119889119896

gt 1

119889119896

if 119889119896

isin [0 1]

0 if 119889119896

lt 0

(24)


119896(22) the ZAD













119867119891

=

119888119891

119901 + 119888119891

(25)


119865 = 120593⊤

120579 (26)


119889120579

119889119905

= 119875120593120576 (27)

120576 = 119865 minus 120593⊤

120579 (28)

119889119875

119889119905

= 120572119875 minus 119875120593120593⊤

119875 (29)







119875 = 120574 (30)



119898

(119905) =

minus119861119882119898

(119905)

119869eq+

119896119905119894119886

(119905)

119869eqminus

(119879fric + 119879119871)

119869eq (31)


[Parameters]

T

d

d

dg s

L

L TLWm

Wm

ia

ia

fcn fcnfcn

[Wmref ] Wmref

P

iL

Vc

x1

x2Duty cycle

PWMC

Triangular

Quantizer10 bits

ZAD-FPIC

Torque estimator

rL i

s

i

L

C

E

Diode


Zero-order

Parameter

Mechanicalmodel

motor

hold

+

+

+

+

minus i+minus

minusminusv

1

z

rs + rMKe lowast Wm

keWm

WmPmotor

iafcn

Vc

Ra iaLa

1T



119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905)

=

minus119861119867119891

119882119898

(119905)

119869eq+

119896119905119867119891

119894119886

(119905)

119869eqminus

119867119891

(119879fric + 119879119871)

119869eq

(32)


119865 (119905) = 119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905) +

119861119867119891

119882119898

(119905)

119869eqminus

119896119905119867119891

119894119886

(119905)

119869eq

120593 (119905) = minus

1

119869eq119867119891

120579 = (119879fric + 119879119871)

(33)


119865 (119905) = 120593 (119905) 120579 (34)








119871is








1199043


1199041and 119896



0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d


0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d



1198783


1198781 1198701198782 and 119870

1198783


1198701198782 and 119870


1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783



1198783is



1198783⪅ 15 and


1198701198783










1198783= 80 and 119873 =



119898)




5 Conclusions



0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























119886 are constant





[

[

[

[

119898

119894119886

120592119888

119894119871

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

minus119861

119869eq

119896119905

119869eq0 0

minus119896119890

119871119886

minus119877119886

119871119886

1

119871119886

0

0

minus1

119862

0

1

119862

0 0

minus1

119871

minus (119903119904

+ 119903119871)

119871

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

119882119898

119894119886

120592119888

119894119871

]

]

]

]

+

[

[

[

[

[

[

[

[

minus (119879fric + 119879119871)

119869eq0

0

119864

119871

]

]

]

]

]

]

]

]

(1)


= 1198601119909 + 1198611 (2)


= 1199091 the



119888= 1199093 and




[

[

[

[

119898

119894119886

120592119888

119894119871

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

minus119861

119869eq

119896119905

119869eq0 0

minus119896119890

119871119886

minus119877119886

119871119886

1

119871119886

0

0

minus1

119862

0

1

119862

0 0

minus1

119871

minus (119903119871)

119871

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

119882119898

119894119886

120592119888

119894119871

]

]

]

]



ZAD-FPICcontroller

AD

ADParameters

120592c iLsensor

sensor

DCmotorGate Buck

converter


Ks123

Wm ia


rs rLS L

Vfd

iLia

Ra

La120592cC

minus

minus

minusminus

+ ++

E

keWm(t)TemTfric +

+

TL

JL


Wm

Diode



+

[

[

[

[

[

[

[

[

minus (119879fric + 119879119871)

119869eq0

0

minus119881fd119871

]

]

]

]

]

]

]

]

(3)


= 1198602119909 + 1198612 (4)




119886









=

1198601119909 + 1198611

si 119896119879 le 119905 lt 119896119879 +

119889119879

2

1198602119909 + 1198612

si 119896119879 +

119889119879

2

le 119905 lt 119896119879 + 119879 minus

119889119879

2

1198601119909 + 1198611

si 119896119879 + 119879 minus

119889119879

2

le 119905 lt 119896119879 + 119879

(5)







119890 (119905) = 119882119908ref (119905) minus 119882

119898(119905) (6)


and 119898


119890 (119905) = 119908ref (119905) minus

119898(119905) (7)


119890 (119896119879) = 119882119908ref (119896119879) minus 119882

119898(119896119879) (8)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879) (9)




E

kT t

dT2

T

V

dT2

(k + 1)TminusVfd

(a) Centered PWM

s

kT t

s1

s2


s+

s+

sminus

t1

t2

spwl (t)

kT + T




119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (10)

Constants 1198961199041 1198961199042 and 119896



119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (11)


119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (12)


119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (13)


119890


119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) =

119882119908ref (119896119879) minus

119882119898

(119896119879)

(14)


(119896119879) 119898

(119896119879) and

119882119898



119890(119896119879) 119890(119896119879) and



(119896119879) 119898

(119896119879) and

119882119898


1and 119861

1of (2) in the


2of (4) in the





119896







le 119905 lt





+ and


119904pwl (119905) =

119904 (119896119879) + (119905 minus 119896119879) 119904+

(119896119879) if 119896119879 le 119905 lt 1199051

1199041

(119896119879) + (119905 minus 1199051) 119904minus

(119896119879) if 1199051

le 119905 le 1199052

1199042

(119896119879) + (119905 minus 1199052) 119904+

(119896119879) if 1199052

lt 119905 le (119896 + 1) 119879

(15)


where 1199051

= 119896119879 + 1198631198961198792 and 119905

2= (119896 + 1)119879 minus 119863

1198961198792


(119896119879) and 119904minus

(119896119879)


+(119896119879) is


1(2) The signal

119904minus


2(4)

1199041(119896119879) = 119904(119896119879) + 119863

1198961198792 119904+

(119896119879) and 1199042(119896119879) = 119904

1(119896119879) + 119879(1 minus

119863119896) 119904minus


119896119879 is the time






int

(119896+1)119879

119896119879

119904pwl (119905) 119889119905 = 0 (16)


119863119896|119896

=

2119904 (119896119879) + 119879 119904minus

(119896119879)

119879 ( 119904minus

(119896119879) minus 119904+

(119896119879))

(17)


In this case

119863119896|(119896minus1)

=

2119904 ((119896 minus 1) 119879) + 119879 119904minus

((119896 minus 1) 119879)

119879 ( 119904minus

((119896 minus 1) 119879) minus 119904+

((119896 minus 1) 119879))

(18)


119889119896

=

1 if 119863119896|(119896minus1)

gt 1

119863119896|(119896minus1)

if 0 le 119863119896|(119896minus1)

le 1

0 if 119863119896|(119896minus1)

lt 0

(19)









1and 119861


2and 119861

2of (4)The



+((119896 minus 1)119879) which is


and


1and 119861

1of (2) (iv) 119904




2of (4)




119888












as 119909(119896+1) = 119891(119909(119896) 119889119896) As 119889







lowast


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (20)



lowast

= 119891(119909lowast

119889lowast





119889lowast




1205821 120582


1

1198691

=

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816(119909lowast 119889lowast)

(21)

Then the choice

119889119896

=

119889119896

+ 119873119889lowast

119873 + 1

(22)


lowast

119889lowast


119896





119896refers to the





119889lowast

=

VrefVin

(23)



119889ZAD-FPIC = 119889 =

1 if 119889119896

gt 1

119889119896

if 119889119896

isin [0 1]

0 if 119889119896

lt 0

(24)


119896(22) the ZAD













119867119891

=

119888119891

119901 + 119888119891

(25)


119865 = 120593⊤

120579 (26)


119889120579

119889119905

= 119875120593120576 (27)

120576 = 119865 minus 120593⊤

120579 (28)

119889119875

119889119905

= 120572119875 minus 119875120593120593⊤

119875 (29)







119875 = 120574 (30)



119898

(119905) =

minus119861119882119898

(119905)

119869eq+

119896119905119894119886

(119905)

119869eqminus

(119879fric + 119879119871)

119869eq (31)


[Parameters]

T

d

d

dg s

L

L TLWm

Wm

ia

ia

fcn fcnfcn

[Wmref ] Wmref

P

iL

Vc

x1

x2Duty cycle

PWMC

Triangular

Quantizer10 bits

ZAD-FPIC

Torque estimator

rL i

s

i

L

C

E

Diode


Zero-order

Parameter

Mechanicalmodel

motor

hold

+

+

+

+

minus i+minus

minusminusv

1

z

rs + rMKe lowast Wm

keWm

WmPmotor

iafcn

Vc

Ra iaLa

1T



119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905)

=

minus119861119867119891

119882119898

(119905)

119869eq+

119896119905119867119891

119894119886

(119905)

119869eqminus

119867119891

(119879fric + 119879119871)

119869eq

(32)


119865 (119905) = 119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905) +

119861119867119891

119882119898

(119905)

119869eqminus

119896119905119867119891

119894119886

(119905)

119869eq

120593 (119905) = minus

1

119869eq119867119891

120579 = (119879fric + 119879119871)

(33)


119865 (119905) = 120593 (119905) 120579 (34)








119871is








1199043


1199041and 119896



0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d


0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d



1198783


1198781 1198701198782 and 119870

1198783


1198701198782 and 119870


1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783



1198783is



1198783⪅ 15 and


1198701198783










1198783= 80 and 119873 =



119898)




5 Conclusions



0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ZAD-FPICcontroller

AD

ADParameters

120592c iLsensor

sensor

DCmotorGate Buck

converter


Ks123

Wm ia


rs rLS L

Vfd

iLia

Ra

La120592cC

minus

minus

minusminus

+ ++

E

keWm(t)TemTfric +

+

TL

JL


Wm

Diode



+

[

[

[

[

[

[

[

[

minus (119879fric + 119879119871)

119869eq0

0

minus119881fd119871

]

]

]

]

]

]

]

]

(3)


= 1198602119909 + 1198612 (4)




119886









=

1198601119909 + 1198611

si 119896119879 le 119905 lt 119896119879 +

119889119879

2

1198602119909 + 1198612

si 119896119879 +

119889119879

2

le 119905 lt 119896119879 + 119879 minus

119889119879

2

1198601119909 + 1198611

si 119896119879 + 119879 minus

119889119879

2

le 119905 lt 119896119879 + 119879

(5)







119890 (119905) = 119882119908ref (119905) minus 119882

119898(119905) (6)


and 119898


119890 (119905) = 119908ref (119905) minus

119898(119905) (7)


119890 (119896119879) = 119882119908ref (119896119879) minus 119882

119898(119896119879) (8)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879) (9)




E

kT t

dT2

T

V

dT2

(k + 1)TminusVfd

(a) Centered PWM

s

kT t

s1

s2


s+

s+

sminus

t1

t2

spwl (t)

kT + T




119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (10)

Constants 1198961199041 1198961199042 and 119896



119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (11)


119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (12)


119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (13)


119890


119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) =

119882119908ref (119896119879) minus

119882119898

(119896119879)

(14)


(119896119879) 119898

(119896119879) and

119882119898



119890(119896119879) 119890(119896119879) and



(119896119879) 119898

(119896119879) and

119882119898


1and 119861

1of (2) in the


2of (4) in the





119896







le 119905 lt





+ and


119904pwl (119905) =

119904 (119896119879) + (119905 minus 119896119879) 119904+

(119896119879) if 119896119879 le 119905 lt 1199051

1199041

(119896119879) + (119905 minus 1199051) 119904minus

(119896119879) if 1199051

le 119905 le 1199052

1199042

(119896119879) + (119905 minus 1199052) 119904+

(119896119879) if 1199052

lt 119905 le (119896 + 1) 119879

(15)


where 1199051

= 119896119879 + 1198631198961198792 and 119905

2= (119896 + 1)119879 minus 119863

1198961198792


(119896119879) and 119904minus

(119896119879)


+(119896119879) is


1(2) The signal

119904minus


2(4)

1199041(119896119879) = 119904(119896119879) + 119863

1198961198792 119904+

(119896119879) and 1199042(119896119879) = 119904

1(119896119879) + 119879(1 minus

119863119896) 119904minus


119896119879 is the time






int

(119896+1)119879

119896119879

119904pwl (119905) 119889119905 = 0 (16)


119863119896|119896

=

2119904 (119896119879) + 119879 119904minus

(119896119879)

119879 ( 119904minus

(119896119879) minus 119904+

(119896119879))

(17)


In this case

119863119896|(119896minus1)

=

2119904 ((119896 minus 1) 119879) + 119879 119904minus

((119896 minus 1) 119879)

119879 ( 119904minus

((119896 minus 1) 119879) minus 119904+

((119896 minus 1) 119879))

(18)


119889119896

=

1 if 119863119896|(119896minus1)

gt 1

119863119896|(119896minus1)

if 0 le 119863119896|(119896minus1)

le 1

0 if 119863119896|(119896minus1)

lt 0

(19)









1and 119861


2and 119861

2of (4)The



+((119896 minus 1)119879) which is


and


1and 119861

1of (2) (iv) 119904




2of (4)




119888












as 119909(119896+1) = 119891(119909(119896) 119889119896) As 119889







lowast


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (20)



lowast

= 119891(119909lowast

119889lowast





119889lowast




1205821 120582


1

1198691

=

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816(119909lowast 119889lowast)

(21)

Then the choice

119889119896

=

119889119896

+ 119873119889lowast

119873 + 1

(22)


lowast

119889lowast


119896





119896refers to the





119889lowast

=

VrefVin

(23)



119889ZAD-FPIC = 119889 =

1 if 119889119896

gt 1

119889119896

if 119889119896

isin [0 1]

0 if 119889119896

lt 0

(24)


119896(22) the ZAD













119867119891

=

119888119891

119901 + 119888119891

(25)


119865 = 120593⊤

120579 (26)


119889120579

119889119905

= 119875120593120576 (27)

120576 = 119865 minus 120593⊤

120579 (28)

119889119875

119889119905

= 120572119875 minus 119875120593120593⊤

119875 (29)







119875 = 120574 (30)



119898

(119905) =

minus119861119882119898

(119905)

119869eq+

119896119905119894119886

(119905)

119869eqminus

(119879fric + 119879119871)

119869eq (31)


[Parameters]

T

d

d

dg s

L

L TLWm

Wm

ia

ia

fcn fcnfcn

[Wmref ] Wmref

P

iL

Vc

x1

x2Duty cycle

PWMC

Triangular

Quantizer10 bits

ZAD-FPIC

Torque estimator

rL i

s

i

L

C

E

Diode


Zero-order

Parameter

Mechanicalmodel

motor

hold

+

+

+

+

minus i+minus

minusminusv

1

z

rs + rMKe lowast Wm

keWm

WmPmotor

iafcn

Vc

Ra iaLa

1T



119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905)

=

minus119861119867119891

119882119898

(119905)

119869eq+

119896119905119867119891

119894119886

(119905)

119869eqminus

119867119891

(119879fric + 119879119871)

119869eq

(32)


119865 (119905) = 119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905) +

119861119867119891

119882119898

(119905)

119869eqminus

119896119905119867119891

119894119886

(119905)

119869eq

120593 (119905) = minus

1

119869eq119867119891

120579 = (119879fric + 119879119871)

(33)


119865 (119905) = 120593 (119905) 120579 (34)








119871is








1199043


1199041and 119896



0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d


0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d



1198783


1198781 1198701198782 and 119870

1198783


1198701198782 and 119870


1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783



1198783is



1198783⪅ 15 and


1198701198783










1198783= 80 and 119873 =



119898)




5 Conclusions



0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










E

kT t

dT2

T

V

dT2

(k + 1)TminusVfd

(a) Centered PWM

s

kT t

s1

s2


s+

s+

sminus

t1

t2

spwl (t)

kT + T




119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (10)

Constants 1198961199041 1198961199042 and 119896



119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (11)


119904 (119905) = 119890 (119905) + 1198961199041

119890 (119905) + 1198961199042

119890 (119905) + 1198961199043

119890 (119905) (12)


119904 (119896119879) = 119890 (119896119879) + 1198961199041

119890 (119896119879) + 1198961199042

119890 (119896119879) + 1198961199043

119890 (119896119879) (13)


119890


119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) = 119908ref (119896119879) minus

119898(119896119879)

119890 (119896119879) =

119882119908ref (119896119879) minus

119882119898

(119896119879)

(14)


(119896119879) 119898

(119896119879) and

119882119898



119890(119896119879) 119890(119896119879) and



(119896119879) 119898

(119896119879) and

119882119898


1and 119861

1of (2) in the


2of (4) in the





119896







le 119905 lt





+ and


119904pwl (119905) =

119904 (119896119879) + (119905 minus 119896119879) 119904+

(119896119879) if 119896119879 le 119905 lt 1199051

1199041

(119896119879) + (119905 minus 1199051) 119904minus

(119896119879) if 1199051

le 119905 le 1199052

1199042

(119896119879) + (119905 minus 1199052) 119904+

(119896119879) if 1199052

lt 119905 le (119896 + 1) 119879

(15)


where 1199051

= 119896119879 + 1198631198961198792 and 119905

2= (119896 + 1)119879 minus 119863

1198961198792


(119896119879) and 119904minus

(119896119879)


+(119896119879) is


1(2) The signal

119904minus


2(4)

1199041(119896119879) = 119904(119896119879) + 119863

1198961198792 119904+

(119896119879) and 1199042(119896119879) = 119904

1(119896119879) + 119879(1 minus

119863119896) 119904minus


119896119879 is the time






int

(119896+1)119879

119896119879

119904pwl (119905) 119889119905 = 0 (16)


119863119896|119896

=

2119904 (119896119879) + 119879 119904minus

(119896119879)

119879 ( 119904minus

(119896119879) minus 119904+

(119896119879))

(17)


In this case

119863119896|(119896minus1)

=

2119904 ((119896 minus 1) 119879) + 119879 119904minus

((119896 minus 1) 119879)

119879 ( 119904minus

((119896 minus 1) 119879) minus 119904+

((119896 minus 1) 119879))

(18)


119889119896

=

1 if 119863119896|(119896minus1)

gt 1

119863119896|(119896minus1)

if 0 le 119863119896|(119896minus1)

le 1

0 if 119863119896|(119896minus1)

lt 0

(19)









1and 119861


2and 119861

2of (4)The



+((119896 minus 1)119879) which is


and


1and 119861

1of (2) (iv) 119904




2of (4)




119888












as 119909(119896+1) = 119891(119909(119896) 119889119896) As 119889







lowast


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (20)



lowast

= 119891(119909lowast

119889lowast





119889lowast




1205821 120582


1

1198691

=

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816(119909lowast 119889lowast)

(21)

Then the choice

119889119896

=

119889119896

+ 119873119889lowast

119873 + 1

(22)


lowast

119889lowast


119896





119896refers to the





119889lowast

=

VrefVin

(23)



119889ZAD-FPIC = 119889 =

1 if 119889119896

gt 1

119889119896

if 119889119896

isin [0 1]

0 if 119889119896

lt 0

(24)


119896(22) the ZAD













119867119891

=

119888119891

119901 + 119888119891

(25)


119865 = 120593⊤

120579 (26)


119889120579

119889119905

= 119875120593120576 (27)

120576 = 119865 minus 120593⊤

120579 (28)

119889119875

119889119905

= 120572119875 minus 119875120593120593⊤

119875 (29)







119875 = 120574 (30)



119898

(119905) =

minus119861119882119898

(119905)

119869eq+

119896119905119894119886

(119905)

119869eqminus

(119879fric + 119879119871)

119869eq (31)


[Parameters]

T

d

d

dg s

L

L TLWm

Wm

ia

ia

fcn fcnfcn

[Wmref ] Wmref

P

iL

Vc

x1

x2Duty cycle

PWMC

Triangular

Quantizer10 bits

ZAD-FPIC

Torque estimator

rL i

s

i

L

C

E

Diode


Zero-order

Parameter

Mechanicalmodel

motor

hold

+

+

+

+

minus i+minus

minusminusv

1

z

rs + rMKe lowast Wm

keWm

WmPmotor

iafcn

Vc

Ra iaLa

1T



119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905)

=

minus119861119867119891

119882119898

(119905)

119869eq+

119896119905119867119891

119894119886

(119905)

119869eqminus

119867119891

(119879fric + 119879119871)

119869eq

(32)


119865 (119905) = 119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905) +

119861119867119891

119882119898

(119905)

119869eqminus

119896119905119867119891

119894119886

(119905)

119869eq

120593 (119905) = minus

1

119869eq119867119891

120579 = (119879fric + 119879119871)

(33)


119865 (119905) = 120593 (119905) 120579 (34)








119871is








1199043


1199041and 119896



0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d


0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d



1198783


1198781 1198701198782 and 119870

1198783


1198701198782 and 119870


1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783



1198783is



1198783⪅ 15 and


1198701198783










1198783= 80 and 119873 =



119898)




5 Conclusions



0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 1199051

= 119896119879 + 1198631198961198792 and 119905

2= (119896 + 1)119879 minus 119863

1198961198792


(119896119879) and 119904minus

(119896119879)


+(119896119879) is


1(2) The signal

119904minus


2(4)

1199041(119896119879) = 119904(119896119879) + 119863

1198961198792 119904+

(119896119879) and 1199042(119896119879) = 119904

1(119896119879) + 119879(1 minus

119863119896) 119904minus


119896119879 is the time






int

(119896+1)119879

119896119879

119904pwl (119905) 119889119905 = 0 (16)


119863119896|119896

=

2119904 (119896119879) + 119879 119904minus

(119896119879)

119879 ( 119904minus

(119896119879) minus 119904+

(119896119879))

(17)


In this case

119863119896|(119896minus1)

=

2119904 ((119896 minus 1) 119879) + 119879 119904minus

((119896 minus 1) 119879)

119879 ( 119904minus

((119896 minus 1) 119879) minus 119904+

((119896 minus 1) 119879))

(18)


119889119896

=

1 if 119863119896|(119896minus1)

gt 1

119863119896|(119896minus1)

if 0 le 119863119896|(119896minus1)

le 1

0 if 119863119896|(119896minus1)

lt 0

(19)









1and 119861


2and 119861

2of (4)The



+((119896 minus 1)119879) which is


and


1and 119861

1of (2) (iv) 119904




2of (4)




119888












as 119909(119896+1) = 119891(119909(119896) 119889119896) As 119889







lowast


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (20)



lowast

= 119891(119909lowast

119889lowast





119889lowast




1205821 120582


1

1198691

=

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816(119909lowast 119889lowast)

(21)

Then the choice

119889119896

=

119889119896

+ 119873119889lowast

119873 + 1

(22)


lowast

119889lowast


119896





119896refers to the





119889lowast

=

VrefVin

(23)



119889ZAD-FPIC = 119889 =

1 if 119889119896

gt 1

119889119896

if 119889119896

isin [0 1]

0 if 119889119896

lt 0

(24)


119896(22) the ZAD













119867119891

=

119888119891

119901 + 119888119891

(25)


119865 = 120593⊤

120579 (26)


119889120579

119889119905

= 119875120593120576 (27)

120576 = 119865 minus 120593⊤

120579 (28)

119889119875

119889119905

= 120572119875 minus 119875120593120593⊤

119875 (29)







119875 = 120574 (30)



119898

(119905) =

minus119861119882119898

(119905)

119869eq+

119896119905119894119886

(119905)

119869eqminus

(119879fric + 119879119871)

119869eq (31)


[Parameters]

T

d

d

dg s

L

L TLWm

Wm

ia

ia

fcn fcnfcn

[Wmref ] Wmref

P

iL

Vc

x1

x2Duty cycle

PWMC

Triangular

Quantizer10 bits

ZAD-FPIC

Torque estimator

rL i

s

i

L

C

E

Diode


Zero-order

Parameter

Mechanicalmodel

motor

hold

+

+

+

+

minus i+minus

minusminusv

1

z

rs + rMKe lowast Wm

keWm

WmPmotor

iafcn

Vc

Ra iaLa

1T



119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905)

=

minus119861119867119891

119882119898

(119905)

119869eq+

119896119905119867119891

119894119886

(119905)

119869eqminus

119867119891

(119879fric + 119879119871)

119869eq

(32)


119865 (119905) = 119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905) +

119861119867119891

119882119898

(119905)

119869eqminus

119896119905119867119891

119894119886

(119905)

119869eq

120593 (119905) = minus

1

119869eq119867119891

120579 = (119879fric + 119879119871)

(33)


119865 (119905) = 120593 (119905) 120579 (34)








119871is








1199043


1199041and 119896



0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d


0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d



1198783


1198781 1198701198782 and 119870

1198783


1198701198782 and 119870


1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783



1198783is



1198783⪅ 15 and


1198701198783










1198783= 80 and 119873 =



119898)




5 Conclusions



0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205821 120582


1

1198691

=

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816(119909lowast 119889lowast)

(21)

Then the choice

119889119896

=

119889119896

+ 119873119889lowast

119873 + 1

(22)


lowast

119889lowast


119896





119896refers to the





119889lowast

=

VrefVin

(23)



119889ZAD-FPIC = 119889 =

1 if 119889119896

gt 1

119889119896

if 119889119896

isin [0 1]

0 if 119889119896

lt 0

(24)


119896(22) the ZAD













119867119891

=

119888119891

119901 + 119888119891

(25)


119865 = 120593⊤

120579 (26)


119889120579

119889119905

= 119875120593120576 (27)

120576 = 119865 minus 120593⊤

120579 (28)

119889119875

119889119905

= 120572119875 minus 119875120593120593⊤

119875 (29)







119875 = 120574 (30)



119898

(119905) =

minus119861119882119898

(119905)

119869eq+

119896119905119894119886

(119905)

119869eqminus

(119879fric + 119879119871)

119869eq (31)


[Parameters]

T

d

d

dg s

L

L TLWm

Wm

ia

ia

fcn fcnfcn

[Wmref ] Wmref

P

iL

Vc

x1

x2Duty cycle

PWMC

Triangular

Quantizer10 bits

ZAD-FPIC

Torque estimator

rL i

s

i

L

C

E

Diode


Zero-order

Parameter

Mechanicalmodel

motor

hold

+

+

+

+

minus i+minus

minusminusv

1

z

rs + rMKe lowast Wm

keWm

WmPmotor

iafcn

Vc

Ra iaLa

1T



119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905)

=

minus119861119867119891

119882119898

(119905)

119869eq+

119896119905119867119891

119894119886

(119905)

119869eqminus

119867119891

(119879fric + 119879119871)

119869eq

(32)


119865 (119905) = 119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905) +

119861119867119891

119882119898

(119905)

119869eqminus

119896119905119867119891

119894119886

(119905)

119869eq

120593 (119905) = minus

1

119869eq119867119891

120579 = (119879fric + 119879119871)

(33)


119865 (119905) = 120593 (119905) 120579 (34)








119871is








1199043


1199041and 119896



0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d


0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d



1198783


1198781 1198701198782 and 119870

1198783


1198701198782 and 119870


1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783



1198783is



1198783⪅ 15 and


1198701198783










1198783= 80 and 119873 =



119898)




5 Conclusions



0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[Parameters]

T

d

d

dg s

L

L TLWm

Wm

ia

ia

fcn fcnfcn

[Wmref ] Wmref

P

iL

Vc

x1

x2Duty cycle

PWMC

Triangular

Quantizer10 bits

ZAD-FPIC

Torque estimator

rL i

s

i

L

C

E

Diode


Zero-order

Parameter

Mechanicalmodel

motor

hold

+

+

+

+

minus i+minus

minusminusv

1

z

rs + rMKe lowast Wm

keWm

WmPmotor

iafcn

Vc

Ra iaLa

1T



119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905)

=

minus119861119867119891

119882119898

(119905)

119869eq+

119896119905119867119891

119894119886

(119905)

119869eqminus

119867119891

(119879fric + 119879119871)

119869eq

(32)


119865 (119905) = 119888119891

119882119898

(119905) minus 119888119891

119867119891

119882119898

(119905) +

119861119867119891

119882119898

(119905)

119869eqminus

119896119905119867119891

119894119886

(119905)

119869eq

120593 (119905) = minus

1

119869eq119867119891

120579 = (119879fric + 119879119871)

(33)


119865 (119905) = 120593 (119905) 120579 (34)








119871is








1199043


1199041and 119896



0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d


0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d



1198783


1198781 1198701198782 and 119870

1198783


1198701198782 and 119870


1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783



1198783is



1198783⪅ 15 and


1198701198783










1198783= 80 and 119873 =



119898)




5 Conclusions



0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 35 40390

395

400

405

410W

m(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d


0 5 10 15 20 25 30 35 40390

395

400

405

410

Wm(rads)

KS3


0 5 10 15 20 25 30 35 400

02

04

06

08

1

KS3

d



1198783


1198781 1198701198782 and 119870

1198783


1198701198782 and 119870


1199041 1198961199042 and 119896

1199043) are

1198961199041

= 1198701198781

radic119871119862 1198961199042

= 1198701198782

119871119862 and 1198961199043

= 1198701198783



1198783is



1198783⪅ 15 and


1198701198783










1198783= 80 and 119873 =



119898)




5 Conclusions



0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

0 02 04 06 08 1Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001


0 02 04 06 08 1390

395

400

405

Time (s)


0 02 04 06 08 1Time (s)


Wm

(rad

s)


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 0000001



119871)

0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001


0 02 04 06 08 1002

003

004

005

006

007

008

Time (s)

Estim

ated

(TL+T

fric

) (Nmiddotm

)

120574 = 00000001

120574 = 0000001

120574 = 000001






MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












MOSFET 084Ω











1119879 119901 1 Delay time 1666 120583s1198701198781








Appendix

Proof of Theorem 7


119909 (119896 + 1) = 119891 (119909 (119896) 119889119896) (A1)


= 119889lowast


119889lowast


119869 =

120597119891

120597119909

10038161003816100381610038161003816100381610038161003816119909lowast119889lowast

+

120597119891

120597119889119896

120597119889119896

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A2)


119869 = 1198691

+

120597119891

120597119889119896

1

119873 + 1

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

119869 = 1198691

+

1

119873 + 1

120597119891

120597119889119896

120597119889119896

120597119909

1003816100381610038161003816100381610038161003816100381610038161003816119909lowast119889lowast

(A3)



1 Since the



Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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