Research Article Fractional-Order Generalized Predictive Control: Application …downloads.hindawi.com/journals/mpe/2013/895640.pdf · 2019-07-31 · Research Article Fractional-Order

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 895640 10 pageshttpdxdoiorg1011552013895640

Research ArticleFractional-Order GeneralizedPredictive Control Application for Low-Speed Control ofGasoline-Propelled Cars

M Romero1 A P de Madrid1 C Mantildeoso1 V Milaneacutes2 and B M Vinagre3

1 Escuela Tecnica Superior de Ingenierıa Informatica UNED Juan del Rosal 16 28040 Madrid Spain2 California PATH University of California at Berkeley Richmond CA 94804-4698 USA3 Industrial Engineering School University of Extremadura Avenida de Elvas sn 06071 Badajoz Spain

Correspondence should be addressed to M Romero mromerosccunedes

Received 9 November 2012 Accepted 22 January 2013

Academic Editor Clara Ionescu

Copyright copy 2013 M Romero et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

There is an increasing interest in using fractional calculus applied to control theory generalizing classical control strategies as thePID controller and developing new ones with the intention of taking advantage of characteristics supplied by this mathematicaltool for the controller definition In this work the fractional generalization of the successful and spread control strategy knownas model predictive control is applied to drive autonomously a gasoline-propelled vehicle at low speeds The vehicle is a CitroenC3 Pluriel that was modified to act over the throttle and brake pedals Its highly nonlinear dynamics are an excellent test bed forapplying beneficial characteristics of fractional predictive formulation to compensate unmodeled dynamics and external disturba-nces

1 Introduction

Fractional calculus can be defined as a generalization ofderivatives and integrals to noninteger orders allowing cal-culations such as deriving a function to real or complexorder [1 2] Although this branch of mathematical analysisbegan 300 years ago when Liebniz and LrsquoHopital discussedthe possibility that 119899 could be a fraction 12 for 119899th derivative119889119899

119910119889119909119899 it was really developed at the beginning of the

19th century by Liouville Riemann Letnikov and othermathematicians [3]

Fractional-order operators are commonly represented by119863120572 that stands for 120572-th-order derivative Negative values

of 120572 correspond to fractional-order integrals 119863minus120572 equiv 119868120572

These operators can be evaluated using two general frac-tional definitions Riemann-Liouville (RL) and Grunwald-Letnikov (GL) Both definitions continuous and discreteare equivalent for a wide class of functions which appear inreal physical and engineering applications [1] In this workdiscrete domain will be exclusively considered Hence in the

following the GL definition (1) will be used to implementfractional operators

119863120572

119891(119905)119905=119896ℎ

= limℎrarr0

ℎminus120572

infin

sum

119895=0

(minus1)119895

(

120572

119895)119891 (119896ℎ minus 119895ℎ) 120572 isin R

(1)

where120572 is the fractional order of the derivative or integral h isthe differential incrementmdashclose to zeromdash and 119895 varies from0 toinfin due to the infinite memory of fractional operators

In order to describe the dynamical behaviour of systemsthe Laplace transform is often used Expression (2) gives theLaplace transform of theGL definition under zero initial con-ditionsNevertheless the discretization of (2) does not lead toa transfer function with a limited number of coefficients in z[4] Thus the so-called short memory principle [1] is appliedwhich means taking into account the behaviour only in therecent past that corresponds to a n-term truncated series

2 Mathematical Problems in Engineering

119905 minus 2 119905 minus 1 119905 + 1198731 119905 + 1198732119905

Figure 1 Model-based predictive control analogy

paying a penalty in the form of some inaccuracy [5]

119871 119863plusmn120572

119891 (119905) = 119904plusmn120572

119865 (119904) forall120572 isin R (2)

Nowadays this mathematical tool is more and more used incontrol theory to enhance the system performance Typicalfractional-order controllers include the CRONE control [6]and the PI120582D120583 controller [7 8] Advanced control systemstrategies have also been generalized fractional optimal con-trol [9ndash11] fractional fuzzy adaptive control [12] fractionalnonlinear control [13] fractional iterative learning control[14] and fractional predictive control the latter knownas fractional-order generalized predictive control (FGPC)which was initially proposed in [15]

Model predictive control (MPC) is an advanced processcontrolmethodology inwhich a dynamicalmodel of the plantis used to predict and optimize the future behaviour of theprocess over a time interval [16ndash18] At each present time tMPC generates a set of future control signals 119906(119905 + 119896 | 119905)

based on the prediction of future process outputs 119910(119905 + 119896 | 119905)

within the time window defined by 1198731(minimum costing

horizon) 1198732(maximum costing horizon) and 119873

119906(control

horizon) (With this notation 119909(119905 + 119896 | 119905) stands for the valueof 119909 at time 119905 + 119896 predicted at time t) However only the firstelement of the control sequence 119906(119905 | 119905) is applied to the sys-tem input When the next measurement becomes available(present time equal to 119905 + 1) the previous procedure isrepeated to find new predicted future process outputs 119910(119905 +

1 + 119896 | 119905 + 1) and calculate the corresponding system input119906(119905+1 | 119905+1)with prediction time windowsmoving forwardfor this reason this kind of control is also known as recedinghorizon control (RHC) Figure 1 depicts the analogy betweenpredictive control and a car driver who calculates the carmanoeuvre following a receding horizon strategy [16]

MPC has become an industrial standard that has beenwidely adopted during the last 30 years With over 2000industrial installations this control method is currently themost implemented for process plants [19] It was originallydeveloped tomeet the specialized control needs of petroleumrefineries [20 21] MPC technology can now be found ina wide variety of application areas such as chemicals [2223] solar power plants [24] agriculture [25] or clinicalanaesthesia supply [26] Recent developments related toMPCcan be found in [27 28]

Generalized predictive control (GPC) [29 30] is one ofthe most representative MPC formulations Its fractional-order counterpart FGPC uses a real-order fractional costfunction to combine the characteristics of fractional calculusand predictive control into a versatile control strategy [31ndash33]

On the other hand driver-assistance systems have beena topic of active research during the last decades They areintended to reduce traffic accidents and traffic congestions[34ndash37] Open-loop cruise control (CC) systems are a well-known class of driver-assistance systems based on control-ling the throttle pedal that reduces driver workload andimprove vehicle safety [38]

Nowadays the tedious task of driving in traffic jamsrepresents an unresolved issue in the automotive sector[39] because commercial vehicles exhibit highly nonlineardynamics due to the behaviour of the vehicle engine at verylow speedTherefore it constitutes one of themost importantcontrol challenges of the automotive sector [40] Recentlyapproaches to resolve this problem have been studied bothusing experimental scaled-down vehicles [41] and usingcommercial vehicles [42 43]

In this paper an application of FGPC to the velocity con-trol of a mass-produced car at very low speeds is describedThe goal is to highlight the beneficial characteristics ofFGPC to compensate unmodeled dynamics and externaldisturbances using the proposed tuning methodThese char-acteristics were shown up in [32] where the lateral control ofan autonomous vehicle is carried out by FGPC in the presenceof sensor noise and the effect of the communication network

The remainder of this paper is organized as followsSection 2 summarizes the fundamentals of fractional predic-tive control methodology Section 3 includes the descriptionof the experimental vehicle presents the design and tuningof the fractional predictive control and shows the results ofthe experimental trial including a comparison with integer-order GPC controllers Finally Section 4 draws the mainconclusions of this work

2 Controller Formulation

The GPC control law is obtained by minimizing possiblysubject to a set of constraints the cost function

119869GPC (Δ119906 119905)=

1198732

sum

119896=1198731

120574119896(119903 (119905 + 119896)minus119910 (119905+ 119896))

2

+

119873119906

sum

119896=1

120582119896Δ119906(119905+119896 minus 1)

2

(3)

where 119903 is the reference y is the output u is the controlsignal 120574

119896and 120582

119896are nonnegative weighting elements Δ is

the increment operator and it is assumed that u(t) remainsconstant from time instant 119905 + 119873

119906(1 le 119873

119906le 1198732) [29 30]

For the sake of simplicity in the notation (sdot | 119905) is omittedsince all expressions are referred to the present time t

Outputs are predicted making use of a CARIMA modelto describe the system dynamics

119860(119911minus1

) 119910 (119905) = 119861 (119911minus1

) 119906 (119905) +

119879119888(119911minus1

)

Δ

120585 (119905) (4)

where 119861(119911minus1

) and 119860(119911minus1

) are the numerator and denom-inator of the model transfer function respectively 120585(t)represents uncorrelated zero-mean white noise and 119879

119888(119911minus1

)

is a (pre)filter to improve the system robustness rejectingdisturbance and noise [44 45]

Mathematical Problems in Engineering 3

119903(119905) 119890(119905) 119910(119905)

119889(119905)

119906(119905)+ + +

minus

119879119888119878119888

119861

119860

119878119888Δ119877119888

Figure 2 Closed-loop equivalent control schema

Using model (4) the future system outputs 119910(119905 + 119896) arepredicted as 119910 = 119910

119862+ 119910119865 where 119910

119862mdashforced responsemdashis the

part of the future output that depends on the future controlactions Δ119906 (with 119910

119862= 119866 sdot Δ119906 and 119866 the matrix of the step

response coefficients of the model) and 119910119865mdashfree responsemdash

is the part of the future output that does not depend on Δu(ie the evolution of the process exclusively due to its presentstate) [29]

When no constraints are defined the minimization of (3)leads to a linear time invariant (LTI) control law that can beprecomputed in advance

FGPC generalizes the GPC cost function (3) making useof the so-called fractional-order definite integration operator120572

119868119887

119886(sdot) [15 46 47] (see the appendix)

119869FGPC (Δ119906 119905) =120572

1198681198732

1198731

[119890 (119905)]2

+120573

119868119873119906

1[Δ119906 (119905minus1)]

2

forall120572 120573isinR

(5)

where 119890 equiv 119903 minus 119910 is the error This cost function has beendiscretized with sampling period Δ119905 and evaluated using(A2)

The FGPC cost function has an equivalent matrix form

119869FGPC (Δ119906 119905) ≃ 1198901015840

Γ (120572 Δ119905) 119890 + Δ1199061015840

Λ (120573 Δ119905) Δ119906 (6)

where Γ andΛ are infinite-dimensional square real weightingmatrices which depend by construction on 120572 and 120573 respec-tively

Γ equiv Δ119905120572 diag (sdot sdot sdot 119908

119899119908119899minus1

sdot sdot sdot 11990811199080) (7)

with119908119895= 120596119895minus120596119895minus119899

119899 = 1198732minus1198731120596119897= (minus1)

119897

(minus120572

119897) and120596

119897= 0

for all 119897 lt 0

Λ equiv Δ119905120573 diag (sdot sdot sdot 119908

119873119906minus1

119908119873119906minus2

sdot sdot sdot 11990811199080) (8)

with119908119895= 120596119895minus120596119895minus119899

119899 = 119873119906minus 1 120596

119897= (minus1)

119897

(minus120573

119897) and 120596

119897= 0

for all 119897 lt 0In absence of constraints the minimization of this cost

function leads to a LTI control law similar to the one of GPCwhose equivalent closed-loop schema is shown in Figure 2See [46 48] and the references therein for details

119877119888and 119878

119888are the controller polynomials obtained from

the model polynomials 119860 and 119861 and the controller parame-ters 119873

1 119873119906 1198732 120572 and 120573 and 119889 stand for disturbance From

schema it is easy to obtain

119877119888Δ119906 (119905) = 119879

119888119903 (119905) minus 119878

119888119910 (119905) (9)

The value of polynomials 119877119888and 119878

119888is obtained using the

expressions (10)Φ and 119865 are two polynomials obtained from

the resolution of two Diophantine equations See [16ndash18] formore details

119877119888(119911minus1

) =

119879119888(119911minus1

) + sum1198732

119894=1198731

119896119894Φ119894

sum1198732

119894=1198731

119896119894119911minus1198732+119894

119878119888(119911minus1

) =

sum1198732

119894=1198731

119896119894119865119894

sum1198732

119894=1198731

119896119894119911minus1198732+119894

(10)

In GPC the weighting sequences 120574119896and 120582

119896are controller

parameters defined by the user However in FGPC thesesequences are obtained from the optimization process itselfand depend on the fractional integration orders 120572 (7) and 120573

(8) as well as the controller horizonsTuning GPC and FGPC means setting the horizon

parameters (1198731 119873119906 1198732) together with the weighting

sequences 120574119896and 120582

119896for GPC and 120572 and 120573 for FGPC

respectively This task is critical because closed-loop stabilitydepends on this choice In GPC some thumb rules are usuallyaccepted [29] In FGPC these thumb rules are also adequatefor choosing the horizons [15 46]

A FGPC-tuning method was proposed in [49] Basedon optimization the objective is the system to fulfil phasemargin sensitivity functions and some other robustnessspecifications (This tuning method has already been usedto tune fractional-order PI120582D120583 controllers successfully [50ndash52]) In order to keep the dimension of the optimizationproblem low it is assumed that the horizon parameters(1198731 119873119906 1198732) are given (for instance following the thumb-

rules previously announced) and only the two unknownparameters the fractional orders 120572 and 120573 are used in theoptimization process Thus the function FMINCON of theMATLAB optimization toolbox [53] can be used to solve thecorresponding optimization problem

3 Experimental Application

In this section we present a practical application of FGPCWe describe its design tuning and practical performance onthe longitudinal speed control of a commercial vehicle

31 Experimental Vehicle The vehicle used for the experi-mental phase is a convertible Citroen C3 Pluriel (Figure 3)which is equipped with automatic driving capabilities bymeans of hardware modifications to permit autonomousactions on the accelerator and brake pedals These modifica-tions let the controllerrsquos outputs steer the vehiclersquos actuators

The carrsquos throttle is handled by an analog signal thatrepresents the pressure on the pedal generated by an analogcard The action over the throttle pedal is transformed intotwo analogue valuesmdashone of them twice the othermdashbetween0 and 5V A switch has been installed on the dashboard tocommute between automatic throttle control and originalthrottle circuit

The brakersquos automation has been done taking into accountthat its action is critical In case of a failure of any of theautonomous systems the vehicle can be stopped by humandriver intervention So an electrohydraulic braking system


Figure 3 Commercial Citroen C3 prototype vehicle

is mounted in parallel with the original one permitting tocoexist the two braking system independently More detailsabout throttle and brake automation can be found in [54 55]

Concerning the on-board sensor systems a real-timekinematic-differential global positioning system (RTK-DGPS) that gives vehicle position with a 1 centimeterprecision and an inertial unit (IMU) to improve thepositioning when GPS signal fails are used to obtain thevehiclersquos true positionThe carrsquos actual speed and accelerationare obtained from a differential hall effect sensor and apiezoelectric sensor respectively These values are acquiredvia controller area network bus (CAN) and provide thenecessary information to the control algorithm which isrunning in real-time in the on-board control unit (OCU)generating the control actions to govern the actuators

For the purpose of this work the gearbox is always infirst gear forcing the car to move at low speed The samplinginterval was fixed by the parameters of GPS at 200msTherefore the frequency of actions on the pedals is set to 5HzUsing these settings the OCU can approximately perform anaction every metre at a maximum speed of 20 kmh

32 Identification of the Longitudinal Dynamics Due to thegasoline-propelled vehicle dynamics at very low speeds arehighly nonlinear and finding an exact dynamical model forthe vehicle is not an easy task Nevertheless as we have seenpreviously fractional predictive controller needs a CARIMAmodel of the plant to make the predictions Therefore anidentification process has to be carried out despite inevitableuncertainties and circuit perturbations

Since the vehicle always remains in first gear restrictingits speed at less than 20 kmh and acting a high engine brakeforce the identification process is only fulfilled for the throttlepedal Taking the brake pedal effect into account leads us to ahybrid control strategy that is not the purpose of this paper

The experimental vehicle response is shown in Figure 4(solid line) where the vehicle has been subjected to sev-eral speed changes by means of successive throttle pedalactuations (In Figure 4 the action of the brake pedal isalso depicted but is not taken into consideration in the

identification process it has been used for the purpose ofreturning to the initial speed 0 kmh)

Themodel of the vehicle is obtained by means of an iden-tification process using the MATLAB Identification Toolbox[56] considering a normalized inputmdashin the interval (0 1)mdashfor the throttle pedal and the sampling time of GPS fixed at200ms

119866(119911minus1

) =

51850119911minus4

1 minus 07344119911minus1

minus 02075119911minus2 (11)

The time-domain model validation is depicted in Figure 4 Itis observable that model (11) captures the vehicle dynamicsreasonably good (dash line) in comparison with the exper-imental data (solid line) despite environment and circuitperturbations

33 Controller Design This section describes the controllerdesign for the longitudinal speed control of the vehicledescribed previously Transfer function (11) constitutes thestarting point in the controller tuning where beneficial char-acteristics of fractional predictive formulation will be used tocompensate unmodeled dynamics and external disturbances

Other practical requirements have to be taken intoaccount during the design process (1) The car response hasto be smooth to guarantee that its acceleration is less thanplusmn2ms2 the maximum acceptable acceleration for standingpassengers [57] (2)Control action 119906 is normalized and has tobe in the interval [0 1] where negative values are not allowedas they mean brake actions

Firstly the horizons are chosen to capture the loopdominant dynamics We have taken a time window of 2seconds ahead defined by 119873

1= 1 and 119873

2= 10 which is

appropriated in a heavy traffic scene (low speed) A widertime window supposes an increment of 119873

2that would lead

to a system with an excessively slow response On the otherhand we have also considered the control horizon 119873

119906=

2 which represents an agreement between system responsespeed and comfort of the vehiclersquos occupants It is well-knownthat larger values of 119873

119906produce tighter control actions [16]

that could even make the system unstableMoreover we have used a prefilter119879

119888(z minus1) to improve the

system robustness against the model-process mismatch andthe disturbance rejection In [44] a guideline is given

119879119888(119911minus1

) = (1 minus 120588119911minus1

)

1198731

(12)

where 120588 is recommended to be close to the dominant pole of(11)

Thus the chosen prefilter has the following expression

119879119888(119911minus1

) = 1 minus 09119911minus1

(13)

Once the controller horizons and the prefilter are chosen theobjective of the optimization process is finding the pair (120572 120573)that fulfils some specified robustness criteria In our case weshall impose the following

(i) Maximize the phasemargin (no specification is set onthe gain margin)


0 50 100 150 200 250

0

01

02

Time (s)

ThrottleBrake

Actu

ator

s (minus1

1)

minus01

minus02

(a)

0 50 100 150 200 250Time (s)

0

5

10

15

20

Spee

d (k

mh

)

Real speedSimulated speed

(b)

Figure 4 Experimental vehicle response and time-domain model validation

Gai

n m

argi

n (d

B)

120573 03

1020

0minus10minus20minus30minus40minus50minus60minus70minus80

33

22

11

00

minus1minus1

minus2 minus2minus3 minus3minus4 minus4

Gain margin

120572 120573

120572 minus21Gain 1058

Figure 5 FGPC gain margin versus 120572 and 120573

(ii) Sensitivity function |119878(119895120596)|le minus30 dB for120596le 001 rads

(iii) Complementary sensitivity function |119879(119895120596)| le 0 dBfor 120596 ge 01 rads

(Phase margin maximization guarantees smooth systemoutput and robustness sensitivity functions constraints givegood noise and disturbance rejection)

In order to initialize the optimization algorithm an initialseed (120572

0 1205730) is needed Figures 5 and 6 depict the closed loop

magnitude and phase margins respectively in the interval120572 120573 isin [minus3 3] We select 120572

0= minus21 and 120573

0= 03 for their

corresponding good gain and phase marginsThe optimization process has been carried out in an

interval of 20minus30 seconds using a PC computer with IntelCore 2 Duo T9300 25GHz running MATLAB 2007a Thesolution to the optimization problem is 120572lowast = minus22456 and120573lowast

= 29271 for which the weighting sequences Γ and Λ

are given in (14) with a phase margin of 7676∘ (and a gainmargin of 1551 dB) The controller sensitivity functions meetthe design specifications as it is depicted in Figure 7

Γ = diag (minus369671 minus00406 minus00683 minus01273 minus01273

minus07881 minus07881 514711 828442 369411)

Λ = diag (00173 00090)

(14)

012 0 1 23

30

10203040506070

Phas

e mar

gin

(deg

)

Phase margin

120572120573minus1

minus1

minus2minus2

minus3 minus3minus4 minus4

120573 03120572 minus21Phase 6054

Figure 6 FGPC phase margin versus 120572 and 120573

34 Experimental Results The experimental trial was accom-plished at the Centre for Automation and Robotics (CARjoint research centre by the Spanish Consejo Superior deInvestigaciones Cientıficas and the Universidad Politecnicade Madrid) private driving circuit using the Citroen C3Pluriel described previously The circuit has been designedwith scientific purposes and represents an inner-city areawith straight-road segments bends and so on Figure 8shows an aerial sight

To validate the proposed controller various target speedchanges were set each 25 seconds trying to keep the speederror close to zero Moreover the automatic gearbox wasalways in first gear avoiding any effect of gear changes andforcing the car to move at low speed Figure 9 depicts theresponses of the vehicle both actualmdashreal timemdash(dot line)and simulated (dash-dot line) The FGPC controller accom-plished all practical requirements which were set previouslyThe vehicle response is stable smooth and reasonably goodin comparison with its simulation It is important to remarkthat the positive reference changes are faster than the negativeoneThis ismainly due to the fact that the brakingmanoeuvrehas to be achieved by the engine brake force and it is affectedby the slope of the circuit

With respect to the comfort of the vehiclersquos occupantsit is observable that vehicle acceleration always remains (in


0

20Sensitivity analysis

Frequency (rads)

119878(d

B)

minus60

minus40

minus20

10minus3 10minus2 10minus1 100 101 102

(a)

0

minus20

minus10

119879(d

B)

Frequency (rads)


(b)

Figure 7 Sensitivity functions

Figure 8 Private driving circuit at CAR

absolute value) below the maximum acceptable accelerationrequirement 2ms2 It is due to the soft action over the throt-tle vehicle actuator satisfying the comfort driving requisites

For comparison purposes we have also tested the per-formance of several GPCs which were tuned using the samehorizons (119873

1= 1119873

119906= 2 and119873

2= 10) and prefilter 119879

119888(13)

as FGPCIn practice in GPC it is commonly assumed that the

weighting sequences are constant that is 120574119896= 120574 and 120582

119896=

120582 Under this assumption it has not been possible to finda GPC controller that fulfils the robustness criteria usingand equivalent optimization method (The set of dynamicsthat can be found with constant weights is much smallerthan in the case of FGPC Furthermore trying to optimizea GPC controller in the general case (120574

119896 120582119896) would lead to

an optimization problem with an extremely high dimensionOn the other hand in the case of FGPC one has to optimizeonly two parameters 120572 and 120573 and this automatically leads tononconstant weighting sequences recall that GPC and FGPCcontrollers share a common LTI expression as was pointedout in Section 2 [49])

0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)

Reference

Experimental FGPCSimulation FGPC

(a)

0 10 20 30 40 50 60 70 80Time (s)

0

01

02

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

00204

minus02minus04


Acce

lera

tion

(ms2)

(c)

Figure 9 FGPC controller performance

For this reason we have tuned several GPC controllerswith different constant weighting sequences 120574 and 120582 Specifi-cally 120582 isin 10

minus6

10minus1

101

105

and 120574 = 1 (as the variation of120574 does not affect the system dynamics considerably)

Using these settings we have obtained two GPC con-trollers that in practice turned out to be unstable althoughthey were stable in simulation These controllers correspondto 120582 = 10

minus6 and 120582 = 10minus1 (labelled Experimental GPC

1 and Experimental GPC 2 in Figure 10 resp) Thus theywere not able to compensate unmodeled dynamics and circuitperturbations

On the other hand GPC controllers for 120582 = 101 and 120582 =

105 (labelled Experimental GPC 3 and Experimental GPC 4

in Figure 11 resp) were stable in practice It is well-knownthat higher values of 120582 give rise to smooth control actionsincreasing the closed loop system robustness [16] Howeveran excessively high value of120582 couldmake the system responsetoo slow It would mean in practice that our car could notstop in time and it would probably crash into the front car

To quantify these results we shall compare the prin-cipal control quality indicators for the stable realizations(GPC 3 GPC 4 and FGPC) speed error (reference speedmdashexperimental speed) softness of the control action andaccelerationThe last ones require to calculate the fast fouriertransform (FFT) to estimate them


0 2 4 6 8 10 12 14 16 180

10

20

Time (s)

Spee

d (k

mh

)

Reference

Experimental GPC 2Experimental GPC 1

(a)

0 2 4 6 8 10 12 14 16 18Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 2 4 6 8 10 12 14 16 18Time (s)

012


Acce

lera

tion

(ms2)

(c)

Figure 10 Unstable GPC controllers Action over the throttle hasbeen limited to [0minus05] for passengers safety during the experimen-tal trial

It is well known that FFT (15) is an efficient algorithm tocompute the discrete fourier transform (DFT)F

119880119896= F (119906

119896) =

119873minus1

sum

119894=0

119906119896119890(2120587119873119896

119894)

119896 = 0 119873 minus 1 (15)

where 119906119896is the control action or acceleration value at time

119905119896and 119873 the length of these signals FFT yields the signal

sharpness by means of a frequency spectrum analysis of thesampled signal

In order to get a good indicator of the overall controlaction and acceleration signals with robustness to outliers wehave used the median of sequence 119880

119896

119875 (119880119896le ) ge

1

2

and 119875 (119880119896ge ) ge

1

2

(16)

The following widely used statistics parameters have beenused to evaluate the speed error

(i) mean

119890 =

1

119873

119873minus1

sum

119894=0

119890119894 (17)

0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)

ReferenceExperimental GPC 3Experimental GPC 4

(a)

0 10 20 30 40 50 60 70 80Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

005

115


minus05

Acce

lera

tion

(ms2)

(c)

Figure 11 Stable GPC controllers

(ii) standard deviation

120590 = radic1

119873

119873minus1

sum

119894=0

(119890119894minus 119890)2

(18)

(iii) root mean square error

RMSE = radic1

119873

119873minus1

sum

119894=0

1198902

119894 (19)

where 119890119896is the speed error at time 119905

119896 Moreover we have also

used the median All of these control quality indicators are reflected in

Table 1One observes (see Figures 9 and 11) that the speed changes

of GPC 4 and FGPC are slower than the response of GPC3 so they need more time to reach the steady state afterspeed changes This is reflected in Table 1 where in termsof speed error all statistics parameters of GPC 3 are betterthan the GPC 4 and FGPC ones However it presents verypoor values in the control action and acceleration indicatorsdue to the very large fluctuations of these signals as wecan see graphically in Figure 11 This undesirable behaviour


Table 1 Comparison of stable controllers

Contr Speed error Control action AccelerationMean St dev Median RMSE FFT median FFT median

GPC 3 00474 21340 25539 21317 04397 19320GPC 4 03955 25186 63434 25461 00149 10562FGPC 04604 24119 58174 24523 00044 01652

compromises seriously the comfort of standing passengersbordering on the maximum acceptable acceleration 2ms2Furthermore it could injurey the throttle actuator due to itscontinuous and aggressive fluctuations in the control action

The FGPC controller shows the best behaviour in thesteady state without overshoot and presenting the best valuesin terms of the softness of the control action and accelerationdue to the precise parameters tuning carried out by theoptimization method FGPC takes advantage of its diversityof responses (varying the fractional orders 120572 and 120573) tomeet the design specifications and to improve the systemrobustness against the model-process mismatch

4 Conclusions

The longitudinal control of a gasoline-propelled vehicle at lowspeeds (common situation in traffic jams) constitutes one ofthemost important topics in the automotive sector due to thehighly nonlinear dynamics that the vehicle presents in thissituation

In this paper the fractional predictive control strategyFGPC has been used to solve this problem Taking advantageof its beneficial characteristics and its tuningmethod to com-pensate un-modeled dynamics a FGPC controller has beendesigned which has achieved closed loop stability followingthe changes in the velocity reference Moreover practicalrequirements to guarantee standing passengers comfort havebeen also achieved by means of the appropriate parameterschoice carried out by the optimization-based tuning in spiteof inevitable uncertainties and circuit perturbations

Finally the comparison between the fractional predictivecontrol strategy FGPC and its integer-order counterpartGPC has shown that the task of finding the correct settingfor the weighting sequences 120574

119896and 120582

119896is crucial In FGPC

the fractional orders 120572 and 120573 allow us to find them keepingthe dimension of the optimization problem low since onlytwo parameters have been optimized

Appendix

Fractional-Order Definite Integral Operator

The fractional-order definite integral of function f (x) withininterval [a b] has the following expression [47]

120572

119868119887

119886119891 (119909) equiv int

119887

119886

[1198631minus120572

119891 (119909)] 119889119909 120572 119886 119887 isin R (A1)

Using the GL definition (1) assuming that1198631minus120572[119891(119909)] = 0 thefractional-order definite integrator operator 120572119868119887

119886(sdot) has the

following discretized expression with a sampling period Δt

120572

119868119887

119886119891 (119909) = Δ119909

120572

119882

1015840

119891 (A2)

where

119882 = (sdot sdot sdot 119908119887119908119887minus1

sdot sdot sdot 119908119899+1

119908119899

sdot sdot sdot 11990811199080)

1015840

119891 = ( sdot sdot sdot 119891 (0) 119891 (Δ119909) sdot sdot sdot 119891 (119886 minus Δ119909) 119891 (119886)

sdot sdot sdot 119891 (119887 minus Δ119909) 119891 (119887) )

1015840

(A3)

with 119908119895= 120596119895minus 120596119895minus119899

119899 = 119887 minus 119886 120596119897= (minus1)

119897

(minus120572

119897) and 120596

119897= 0

for all 119897 lt 0

Conflicts of Interest

The authors report no actual or potential conflict of interestsin relation to this manuscript

Acknowledgments

The authors wish to acknowledge the economical support ofthe Spanish Distance Education University (UNED) underProject reference PROY29 (Proyectos Investigacion 2012)and the AUTOPIA Program of the Center for Automationand Robotics UPM-CSIC

References

[1] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[2] K B Oldham and J SpanierThe Fractional Calculus vol 111 ofMathematics in Science and Engineering Academic Press NewYork NY USA 1974

[3] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[4] B M Vinagre C A Monje and A J Calderon ldquoFractionalorder systems and fractional order control actionsrdquo in Proceed-ings of the 41st Conference on Decision and Control TutorialWorkshop 2 Fractional Calculus Applications in AutomaticControl and Robotics Las Vegas Nev USA 2002

[5] I Podlubny ldquoNumerical solution of ordinary fractional dif-ferential equations by the fractional difference methodrdquo inAdvances in Difference Equations (Veszprem 1995) pp 507ndash515Gordon and Breach Amsterdam The Netherlands 1997


[6] A Oustaloup BMathieu and P Lanusse ldquoTheCRONE controlof resonant plants application to a flexible transmissionrdquoEuropean Journal of Control vol 1 pp 113ndash121 1995

[7] I Podlubny ldquoFractional-order systems and 119875119868120582119863120583-controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999

[8] I Petras ldquoThe fractional-order controllersrdquo Journal of ElectricalEngineering vol 50 pp 284ndash288 1999

[9] O P Agrawal ldquoA general formulation and solution scheme forfractional optimal control problemsrdquo Nonlinear Dynamics vol38 no 1ndash4 pp 323ndash337 2004

[10] O P Agrawal and D Baleanu ldquoA Hamiltonian formulationand a direct numerical scheme for fractional optimal controlproblemsrdquo Journal of Vibration and Control vol 13 no 9-10 pp1269ndash1281 2007

[11] C Tricaud and Y Q Chen ldquoSolving fractional order optimalcontrol problms in RIOTS 95mdasha general purpose optimalcontrol problems solverrdquo in Proceedings of the 3rd IFAC Work-shop on Fractional Differentiation and Its Applications AnkaraTurkey 2008

[12] M O Efe ldquoFractional fuzzy adaptive sliding-mode control of a2-DOF direct-drive robot armrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 38 no 6 pp 1561ndash1570 2008

[13] I S Jesus J T Machado and R S Barbosa ldquoFractional ordernonlinear control of heat systemrdquo in Proceedings of the 3rd IFACWorkshop on Fractional Differentiation and Its ApplicationsAnkara Turkey 2008

[14] Y Li Y Chen and H-S Ahn ldquoFractional-order iterativelearning control for fractional-order linear systemsrdquo AsianJournal of Control vol 13 no 1 pp 54ndash63 2011

[15] M Romero A P de Madrid C Manoso and R HernandezldquoGeneralized predictive control of arbitrary real orderrdquo in NewTrends in Nanotechnology and Fractional Calculus ApplicationsD Baleanu Z B Guvenc and J A T Machado Eds pp 411ndash418 Springer Dordrecht The Netherlands 2009

[16] E F Camacho and C Bordons Model Predictive ControlSpringer New York NY USA 2nd edition 2004

[17] J A Rossiter Model Based Predictive Control A PracticalApproach CRC Press New York NY USA 2003

[18] J M Maciejowski Predictive Control with Constraints PrenticeHall New York NY USA 2002

[19] J Qin and T Badgwell ldquoAn overview of industrial model pre-dictive control technologyrdquo in Proceedings of the InternationalConference on Chemical Process J C Kantor C E Garcia andB Carnahan Eds vol 93 of AIChE Symposium Series pp 232ndash256 1997

[20] C R Cutler and B L Ramaker ldquoDynamic matrix controlmdashacomputer control algorithmrdquo in Proceedings of Joint AutomaticControl Conference San Francisco Calif USA 1980

[21] W L Luyben Ed Practical Distillation Control Van NostrandReinhold New York NY USA 1992

[22] T Alvarez M Sanzo and C de Prada ldquoIdentification and con-strained multivariable predictive control of chemical reactorsrdquoin Proceedings of the IEEE Conference on Control Applicationspp 663ndash664 Albany NY USA September 1995

[23] J M Martın Sanchez and J Rodellar Adaptive PredictiveControl From the Concepts to Plant Optimization Prentice HallUpper Saddle River NJ USA 1996

[24] E F Camacho and M Berenguel ldquoApplication of generalizedpredictive control to a solar power plantrdquo in Proceedings of

the IEEE Conference on Control Applications pp 1657ndash1662Glasgow UK August 1994

[25] F Han C Zuo W Wu J Li and Z Liu ldquoModel predictivecontrol of the grain drying processrdquo Mathematical Problems inEngineering vol 2012 Article ID 584376 12 pages 2012

[26] D A Linkens andMMahfouf ldquoGeneralized predictive control(GPC) in clinical anaesthesiardquo in Advances in Model-BasedPredictive Control D W Clarke Ed pp 429ndash445 OxfordUniversity Press Oxford UK 1994

[27] H Yang and S Li ldquoA data-driven bilinear predictive controllerdesign based on subspacemethodrdquoAsian Journal of Control vol13 no 2 pp 345ndash349 2011

[28] X-H Chang and G-H Yang ldquoFuzzy robust constrained modelpredictive control for nonlinear systemsrdquo Asian Journal ofControl vol 13 no 6 pp 947ndash955 2011

[29] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive control Part I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987

[30] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralizedpredictive control Part II Extensions and interpretationsrdquoAutomatica vol 23 no 2 pp 149ndash160 1987

[31] M Romero I Tejado B M Vinagre and A P de MadridldquoPosition and velocity control of a servo by using GPC ofarbitrary real orderrdquo in New Trends in Nanotechnology andFractional Calculus Applications D Baleanu Z B Guvenc andJ A T Machado Eds pp 369ndash376 Springer Dordrecht TheNetherlands 2009

[32] M Romero I Tejado J I Suarez B M Vinagre and A P DeMadrid ldquoGPC strategies for the lateral control of a networkedAGVrdquo in Proceedings of the 5th International Conference onMechatronics (ICM rsquo09) Malaga Spain April 2009

[33] I Tejado M Romero B M Vinagre A P de Madrid and YQ Chen ldquoExperiences on an internet link characterization andnetworked control of a smart wheelrdquo International Journal ofBifurcation and Chaos vol 22 no 4 2012

[34] R Bishop ldquoA survey of intelligent vehicle applications world-widerdquo in Proceedings of the IEEE Intelligent Vehicles Symposiumpp 25ndash30 Dearborn Mich USA 2000

[35] G Marsden M McDonald and M Brackstone ldquoTowardsan understanding of adaptive cruise controlrdquo TransportationResearch Part C vol 9 no 1 pp 33ndash51 2001

[36] A Vahidi andA Eskandarian ldquoResearch advances in intelligentcollision avoidance and adaptive cruise controlrdquo IEEE Transac-tions on Intelligent Transportation Systems vol 4 no 3 pp 132ndash153 2003

[37] B Van Arem C J G Van Driel and R Visser ldquoThe impactof cooperative adaptive cruise control on traffic-flow character-isticsrdquo IEEE Transactions on Intelligent Transportation Systemsvol 7 no 4 pp 429ndash436 2006

[38] T Aono and T Kowatari ldquoThrottle-control algorithm forimproving engine response based on air-intake model andthrottle-response modelrdquo IEEE Transactions on Industrial Elec-tronics vol 53 no 3 pp 915ndash921 2006

[39] N B Hounsell B P Shrestha J Piao and M McDonaldldquoReview of urban traffic management and the impacts of newvehicle technologiesrdquo IET Intelligent Transport Systems vol 3no 4 pp 419ndash428 2009

[40] S Moon I Moon and K Yi ldquoDesign tuning and evaluationof a full-range adaptive cruise control system with collisionavoidancerdquo Control Engineering Practice vol 17 no 4 pp 442ndash455 2009


[41] L Cai A B Rad and W L Chan ldquoAn intelligent longitudinalcontroller for application in semiautonomous vehiclesrdquo IEEETransactions on Industrial Electronics vol 57 no 4 pp 1487ndash1497 2010

[42] V Milanes J Villagra J Godoy and C Gonzalez ldquoComparingfuzzy and intelligent PI controllers in stop-and-gomanoeuvresrdquoIEEE Transactions on Control Systems Technology vol 20 no 3pp 770ndash778 2011

[43] I Tejado V Milanes J Villagra J Godoy H HosseinNia andB M Vinagre ldquoLow speed control of an autonomous vehicle byusing a fractional PIrdquo in Proceedings of the 18th World CongressInternational Federation Automatic Control pp 15025ndash15030Milano Italy 2011

[44] T-W Yoon and D W Clarke ldquoObserver design in receding-horizon predictive controlrdquo International Journal of Control vol61 no 1 pp 171ndash191 1995

[45] C Manoso A P de Madrid M Romero and R HernandezldquoGPC with structured perturbations the influence of prefilter-ing and terminal equality constraintsrdquo ISRNAppliedMathemat-ics vol 2012 Article ID 623484 12 pages 2012

[46] M Romero A P de Madrid C Manoso and B M VinagreldquoFractional-order generalized predictive control formulationand some propertiesrdquo in proceedings of the 11th Interna-tional Conference on Control Automation Robotics and Vision(ICARCV rsquo10) pp 1495ndash1500 Singapore December 2010

[47] M Romero A P deMadrid and BM Vinagre ldquoArbitrary real-order cost functions for signals and systemsrdquo Signal Processingvol 91 no 3 pp 372ndash378 2011

[48] M Romero A P de Madrid C Manoso and B M VinagreldquoA survey of fractional-order generalized predictive controlrdquo inProceedings of the 51st Conference on Decision and Control pp6867ndash6872 Maui Hawaii USA 2012

[49] M Romero I Tejado A P de Madrid and B M VinagreldquoTuning predictive controllers with optimization applicationto GPC and FGPCrdquo in Proceedings of the 18th World WorldCongress International FederationAutomatic Control pp 10824ndash10829 Milano Italy 2011

[50] CAMonje A J Calderon BMVinagre Y Chen andV FeliuldquoOn fractional PI120582 controllers some tuning rules for robustnessto plant uncertaintiesrdquoNonlinear Dynamics vol 38 no 1ndash4 pp369ndash381 2004

[51] Y Q Chen H Dou B M Vinagre and C A Monje ldquoA robusttuning method for fractional order PI controllersrdquo in Proceed-ings of the 2nd IFACWorkshop on Fractional Differentiation andIts Applications Porto Portugal 2006

[52] C A Monje B M Vinagre V Feliu and Y Chen ldquoTuningand auto-tuning of fractional order controllers for industryapplicationsrdquo Control Engineering Practice vol 16 no 7 pp798ndash812 2008

[53] Mathworks Inc ldquoMatlab optimization toolbox userrsquos guiderdquo2007

[54] V Milanes C Gonzalez J E Naranjo E Onieva and Tde Pedro ldquoElectro-hydraulic braking system for autonomousvehiclesrdquo International Journal of Automotive Technology vol11 no 1 pp 89ndash95 2010

[55] V Milanes D F Llorca B M Vinagre C Gonzalez and MA Sotelo ldquoClavileno evolution of an autonomous carrdquo in Pro-ceedings of the 13th IEEE International Intelligent TransportationSystems Madeira Portugal 2010

[56] Mathworks Inc ldquoMatlab identification toolbox userrsquos guiderdquo2007

[57] BECHTEL ldquoCompendium of executive summaries from themaglev system concept definition final reportsrdquo Tech RepUS Department of Transportation 1993 httpntlbtsgovDOCSCEShtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

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


119905 minus 2 119905 minus 1 119905 + 1198731 119905 + 1198732119905

Figure 1 Model-based predictive control analogy

paying a penalty in the form of some inaccuracy [5]

119871 119863plusmn120572

119891 (119905) = 119904plusmn120572

119865 (119904) forall120572 isin R (2)

Nowadays this mathematical tool is more and more used incontrol theory to enhance the system performance Typicalfractional-order controllers include the CRONE control [6]and the PI120582D120583 controller [7 8] Advanced control systemstrategies have also been generalized fractional optimal con-trol [9ndash11] fractional fuzzy adaptive control [12] fractionalnonlinear control [13] fractional iterative learning control[14] and fractional predictive control the latter knownas fractional-order generalized predictive control (FGPC)which was initially proposed in [15]

Model predictive control (MPC) is an advanced processcontrolmethodology inwhich a dynamicalmodel of the plantis used to predict and optimize the future behaviour of theprocess over a time interval [16ndash18] At each present time tMPC generates a set of future control signals 119906(119905 + 119896 | 119905)

based on the prediction of future process outputs 119910(119905 + 119896 | 119905)

within the time window defined by 1198731(minimum costing

horizon) 1198732(maximum costing horizon) and 119873

119906(control

horizon) (With this notation 119909(119905 + 119896 | 119905) stands for the valueof 119909 at time 119905 + 119896 predicted at time t) However only the firstelement of the control sequence 119906(119905 | 119905) is applied to the sys-tem input When the next measurement becomes available(present time equal to 119905 + 1) the previous procedure isrepeated to find new predicted future process outputs 119910(119905 +

1 + 119896 | 119905 + 1) and calculate the corresponding system input119906(119905+1 | 119905+1)with prediction time windowsmoving forwardfor this reason this kind of control is also known as recedinghorizon control (RHC) Figure 1 depicts the analogy betweenpredictive control and a car driver who calculates the carmanoeuvre following a receding horizon strategy [16]

MPC has become an industrial standard that has beenwidely adopted during the last 30 years With over 2000industrial installations this control method is currently themost implemented for process plants [19] It was originallydeveloped tomeet the specialized control needs of petroleumrefineries [20 21] MPC technology can now be found ina wide variety of application areas such as chemicals [2223] solar power plants [24] agriculture [25] or clinicalanaesthesia supply [26] Recent developments related toMPCcan be found in [27 28]

Generalized predictive control (GPC) [29 30] is one ofthe most representative MPC formulations Its fractional-order counterpart FGPC uses a real-order fractional costfunction to combine the characteristics of fractional calculusand predictive control into a versatile control strategy [31ndash33]

On the other hand driver-assistance systems have beena topic of active research during the last decades They areintended to reduce traffic accidents and traffic congestions[34ndash37] Open-loop cruise control (CC) systems are a well-known class of driver-assistance systems based on control-ling the throttle pedal that reduces driver workload andimprove vehicle safety [38]

Nowadays the tedious task of driving in traffic jamsrepresents an unresolved issue in the automotive sector[39] because commercial vehicles exhibit highly nonlineardynamics due to the behaviour of the vehicle engine at verylow speedTherefore it constitutes one of themost importantcontrol challenges of the automotive sector [40] Recentlyapproaches to resolve this problem have been studied bothusing experimental scaled-down vehicles [41] and usingcommercial vehicles [42 43]

In this paper an application of FGPC to the velocity con-trol of a mass-produced car at very low speeds is describedThe goal is to highlight the beneficial characteristics ofFGPC to compensate unmodeled dynamics and externaldisturbances using the proposed tuning methodThese char-acteristics were shown up in [32] where the lateral control ofan autonomous vehicle is carried out by FGPC in the presenceof sensor noise and the effect of the communication network

The remainder of this paper is organized as followsSection 2 summarizes the fundamentals of fractional predic-tive control methodology Section 3 includes the descriptionof the experimental vehicle presents the design and tuningof the fractional predictive control and shows the results ofthe experimental trial including a comparison with integer-order GPC controllers Finally Section 4 draws the mainconclusions of this work

2 Controller Formulation

The GPC control law is obtained by minimizing possiblysubject to a set of constraints the cost function

119869GPC (Δ119906 119905)=

1198732

sum

119896=1198731

120574119896(119903 (119905 + 119896)minus119910 (119905+ 119896))

2

+

119873119906

sum

119896=1

120582119896Δ119906(119905+119896 minus 1)

2

(3)

where 119903 is the reference y is the output u is the controlsignal 120574

119896and 120582

119896are nonnegative weighting elements Δ is

the increment operator and it is assumed that u(t) remainsconstant from time instant 119905 + 119873

119906(1 le 119873

119906le 1198732) [29 30]

For the sake of simplicity in the notation (sdot | 119905) is omittedsince all expressions are referred to the present time t

Outputs are predicted making use of a CARIMA modelto describe the system dynamics

119860(119911minus1

) 119910 (119905) = 119861 (119911minus1

) 119906 (119905) +

119879119888(119911minus1

)

Δ

120585 (119905) (4)

where 119861(119911minus1

) and 119860(119911minus1

) are the numerator and denom-inator of the model transfer function respectively 120585(t)represents uncorrelated zero-mean white noise and 119879

119888(119911minus1

)

is a (pre)filter to improve the system robustness rejectingdisturbance and noise [44 45]


119903(119905) 119890(119905) 119910(119905)

119889(119905)

119906(119905)+ + +

minus

119879119888119878119888

119861

119860

119878119888Δ119877119888



119862+ 119910119865 where 119910








119868119887


119869FGPC (Δ119906 119905) =120572

1198681198732

1198731

[119890 (119905)]2

+120573

119868119873119906

1[Δ119906 (119905minus1)]

2


(5)



119869FGPC (Δ119906 119905) ≃ 1198901015840

Γ (120572 Δ119905) 119890 + Δ1199061015840

Λ (120573 Δ119905) Δ119906 (6)



119899119908119899minus1

sdot sdot sdot 11990811199080) (7)

with119908119895= 120596119895minus120596119895minus119899

119899 = 1198732minus1198731120596119897= (minus1)

119897

(minus120572

119897) and120596

119897= 0

for all 119897 lt 0


119873119906minus1

119908119873119906minus2

sdot sdot sdot 11990811199080) (8)

with119908119895= 120596119895minus120596119895minus119899

119899 = 119873119906minus 1 120596

119897= (minus1)

119897

(minus120573

119897) and 120596

119897= 0



119877119888and 119878





119877119888Δ119906 (119905) = 119879

119888119903 (119905) minus 119878

119888119910 (119905) (9)





119877119888(119911minus1

) =

119879119888(119911minus1

) + sum1198732

119894=1198731

119896119894Φ119894

sum1198732

119894=1198731

119896119894119911minus1198732+119894

119878119888(119911minus1

) =

sum1198732

119894=1198731

119896119894119865119894

sum1198732

119894=1198731

119896119894119911minus1198732+119894

(10)






sequences 120574119896and 120582




















119866(119911minus1

) =

51850119911minus4


minus 02075119911minus2 (11)





1= 1 and 119873

2= 10 which is


2that would lead


119906=






119879119888(119911minus1

) = (1 minus 120588119911minus1

)

1198731

(12)



119879119888(119911minus1

) = 1 minus 09119911minus1

(13)




0 50 100 150 200 250

0

01

02

Time (s)

ThrottleBrake

Actu

ator

s (minus1

1)

minus01

minus02

(a)

0 50 100 150 200 250Time (s)

0

5

10

15

20

Spee

d (k

mh

)


(b)


Gai

n m

argi

n (d

B)

120573 03

1020


33

22

11

00

minus1minus1


Gain margin

120572 120573

120572 minus21Gain 1058








0= minus21 and 120573

0= 03 for their






minus07881 minus07881 514711 828442 369411)

Λ = diag (00173 00090)

(14)

012 0 1 23

30

10203040506070

Phas

e mar

gin

(deg

)

Phase margin

120572120573minus1

minus1

minus2minus2


120573 03120572 minus21Phase 6054






0


Frequency (rads)

119878(d

B)

minus60

minus40

minus20


(a)

0

minus20

minus10

119879(d

B)

Frequency (rads)


(b)





1= 1119873

119906= 2 and119873


119888(13)



119896=


119896 120582119896) would lead to


0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

01

02

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

00204

minus02minus04


Acce

lera

tion

(ms2)

(c)



minus6

10minus1

101

105










0 2 4 6 8 10 12 14 16 180

10

20

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 2 4 6 8 10 12 14 16 18Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 2 4 6 8 10 12 14 16 18Time (s)

012


Acce

lera

tion

(ms2)

(c)



119880119896= F (119906

119896) =

119873minus1

sum

119894=0

119906119896119890(2120587119873119896

119894)

119896 = 0 119873 minus 1 (15)





119896

119875 (119880119896le ) ge

1

2

and 119875 (119880119896ge ) ge

1

2

(16)


(i) mean

119890 =

1

119873

119873minus1

sum

119894=0

119890119894 (17)

0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

005

115


minus05

Acce

lera

tion

(ms2)

(c)



120590 = radic1

119873

119873minus1

sum

119894=0

(119890119894minus 119890)2

(18)


RMSE = radic1

119873

119873minus1

sum

119894=0

1198902

119894 (19)









GPC 3 00474 21340 25539 21317 04397 19320GPC 4 03955 25186 63434 25461 00149 10562FGPC 04604 24119 58174 24523 00044 01652



4 Conclusions




119896and 120582



Appendix



120572

119868119887

119886119891 (119909) equiv int

119887

119886

[1198631minus120572

119891 (119909)] 119889119909 120572 119886 119887 isin R (A1)




120572

119868119887

119886119891 (119909) = Δ119909

120572

119882

1015840

119891 (A2)

where



119908119899


1015840



1015840

(A3)

with 119908119895= 120596119895minus 120596119895minus119899

119899 = 119887 minus 119886 120596119897= (minus1)

119897

(minus120572

119897) and 120596

119897= 0

for all 119897 lt 0



Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119903(119905) 119890(119905) 119910(119905)

119889(119905)

119906(119905)+ + +

minus

119879119888119878119888

119861

119860

119878119888Δ119877119888



119862+ 119910119865 where 119910








119868119887


119869FGPC (Δ119906 119905) =120572

1198681198732

1198731

[119890 (119905)]2

+120573

119868119873119906

1[Δ119906 (119905minus1)]

2


(5)



119869FGPC (Δ119906 119905) ≃ 1198901015840

Γ (120572 Δ119905) 119890 + Δ1199061015840

Λ (120573 Δ119905) Δ119906 (6)



119899119908119899minus1

sdot sdot sdot 11990811199080) (7)

with119908119895= 120596119895minus120596119895minus119899

119899 = 1198732minus1198731120596119897= (minus1)

119897

(minus120572

119897) and120596

119897= 0

for all 119897 lt 0


119873119906minus1

119908119873119906minus2

sdot sdot sdot 11990811199080) (8)

with119908119895= 120596119895minus120596119895minus119899

119899 = 119873119906minus 1 120596

119897= (minus1)

119897

(minus120573

119897) and 120596

119897= 0



119877119888and 119878





119877119888Δ119906 (119905) = 119879

119888119903 (119905) minus 119878

119888119910 (119905) (9)





119877119888(119911minus1

) =

119879119888(119911minus1

) + sum1198732

119894=1198731

119896119894Φ119894

sum1198732

119894=1198731

119896119894119911minus1198732+119894

119878119888(119911minus1

) =

sum1198732

119894=1198731

119896119894119865119894

sum1198732

119894=1198731

119896119894119911minus1198732+119894

(10)






sequences 120574119896and 120582




















119866(119911minus1

) =

51850119911minus4


minus 02075119911minus2 (11)





1= 1 and 119873

2= 10 which is


2that would lead


119906=






119879119888(119911minus1

) = (1 minus 120588119911minus1

)

1198731

(12)



119879119888(119911minus1

) = 1 minus 09119911minus1

(13)




0 50 100 150 200 250

0

01

02

Time (s)

ThrottleBrake

Actu

ator

s (minus1

1)

minus01

minus02

(a)

0 50 100 150 200 250Time (s)

0

5

10

15

20

Spee

d (k

mh

)


(b)


Gai

n m

argi

n (d

B)

120573 03

1020


33

22

11

00

minus1minus1


Gain margin

120572 120573

120572 minus21Gain 1058








0= minus21 and 120573

0= 03 for their






minus07881 minus07881 514711 828442 369411)

Λ = diag (00173 00090)

(14)

012 0 1 23

30

10203040506070

Phas

e mar

gin

(deg

)

Phase margin

120572120573minus1

minus1

minus2minus2


120573 03120572 minus21Phase 6054






0


Frequency (rads)

119878(d

B)

minus60

minus40

minus20


(a)

0

minus20

minus10

119879(d

B)

Frequency (rads)


(b)





1= 1119873

119906= 2 and119873


119888(13)



119896=


119896 120582119896) would lead to


0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

01

02

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

00204

minus02minus04


Acce

lera

tion

(ms2)

(c)



minus6

10minus1

101

105










0 2 4 6 8 10 12 14 16 180

10

20

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 2 4 6 8 10 12 14 16 18Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 2 4 6 8 10 12 14 16 18Time (s)

012


Acce

lera

tion

(ms2)

(c)



119880119896= F (119906

119896) =

119873minus1

sum

119894=0

119906119896119890(2120587119873119896

119894)

119896 = 0 119873 minus 1 (15)





119896

119875 (119880119896le ) ge

1

2

and 119875 (119880119896ge ) ge

1

2

(16)


(i) mean

119890 =

1

119873

119873minus1

sum

119894=0

119890119894 (17)

0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

005

115


minus05

Acce

lera

tion

(ms2)

(c)



120590 = radic1

119873

119873minus1

sum

119894=0

(119890119894minus 119890)2

(18)


RMSE = radic1

119873

119873minus1

sum

119894=0

1198902

119894 (19)









GPC 3 00474 21340 25539 21317 04397 19320GPC 4 03955 25186 63434 25461 00149 10562FGPC 04604 24119 58174 24523 00044 01652



4 Conclusions




119896and 120582



Appendix



120572

119868119887

119886119891 (119909) equiv int

119887

119886

[1198631minus120572

119891 (119909)] 119889119909 120572 119886 119887 isin R (A1)




120572

119868119887

119886119891 (119909) = Δ119909

120572

119882

1015840

119891 (A2)

where



119908119899


1015840



1015840

(A3)

with 119908119895= 120596119895minus 120596119895minus119899

119899 = 119887 minus 119886 120596119897= (minus1)

119897

(minus120572

119897) and 120596

119897= 0

for all 119897 lt 0



Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















119866(119911minus1

) =

51850119911minus4


minus 02075119911minus2 (11)





1= 1 and 119873

2= 10 which is


2that would lead


119906=






119879119888(119911minus1

) = (1 minus 120588119911minus1

)

1198731

(12)



119879119888(119911minus1

) = 1 minus 09119911minus1

(13)




0 50 100 150 200 250

0

01

02

Time (s)

ThrottleBrake

Actu

ator

s (minus1

1)

minus01

minus02

(a)

0 50 100 150 200 250Time (s)

0

5

10

15

20

Spee

d (k

mh

)


(b)


Gai

n m

argi

n (d

B)

120573 03

1020


33

22

11

00

minus1minus1


Gain margin

120572 120573

120572 minus21Gain 1058








0= minus21 and 120573

0= 03 for their






minus07881 minus07881 514711 828442 369411)

Λ = diag (00173 00090)

(14)

012 0 1 23

30

10203040506070

Phas

e mar

gin

(deg

)

Phase margin

120572120573minus1

minus1

minus2minus2


120573 03120572 minus21Phase 6054






0


Frequency (rads)

119878(d

B)

minus60

minus40

minus20


(a)

0

minus20

minus10

119879(d

B)

Frequency (rads)


(b)





1= 1119873

119906= 2 and119873


119888(13)



119896=


119896 120582119896) would lead to


0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

01

02

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

00204

minus02minus04


Acce

lera

tion

(ms2)

(c)



minus6

10minus1

101

105










0 2 4 6 8 10 12 14 16 180

10

20

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 2 4 6 8 10 12 14 16 18Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 2 4 6 8 10 12 14 16 18Time (s)

012


Acce

lera

tion

(ms2)

(c)



119880119896= F (119906

119896) =

119873minus1

sum

119894=0

119906119896119890(2120587119873119896

119894)

119896 = 0 119873 minus 1 (15)





119896

119875 (119880119896le ) ge

1

2

and 119875 (119880119896ge ) ge

1

2

(16)


(i) mean

119890 =

1

119873

119873minus1

sum

119894=0

119890119894 (17)

0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

005

115


minus05

Acce

lera

tion

(ms2)

(c)



120590 = radic1

119873

119873minus1

sum

119894=0

(119890119894minus 119890)2

(18)


RMSE = radic1

119873

119873minus1

sum

119894=0

1198902

119894 (19)









GPC 3 00474 21340 25539 21317 04397 19320GPC 4 03955 25186 63434 25461 00149 10562FGPC 04604 24119 58174 24523 00044 01652



4 Conclusions




119896and 120582



Appendix



120572

119868119887

119886119891 (119909) equiv int

119887

119886

[1198631minus120572

119891 (119909)] 119889119909 120572 119886 119887 isin R (A1)




120572

119868119887

119886119891 (119909) = Δ119909

120572

119882

1015840

119891 (A2)

where



119908119899


1015840



1015840

(A3)

with 119908119895= 120596119895minus 120596119895minus119899

119899 = 119887 minus 119886 120596119897= (minus1)

119897

(minus120572

119897) and 120596

119897= 0

for all 119897 lt 0



Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250

0

01

02

Time (s)

ThrottleBrake

Actu

ator

s (minus1

1)

minus01

minus02

(a)

0 50 100 150 200 250Time (s)

0

5

10

15

20

Spee

d (k

mh

)


(b)


Gai

n m

argi

n (d

B)

120573 03

1020


33

22

11

00

minus1minus1


Gain margin

120572 120573

120572 minus21Gain 1058








0= minus21 and 120573

0= 03 for their






minus07881 minus07881 514711 828442 369411)

Λ = diag (00173 00090)

(14)

012 0 1 23

30

10203040506070

Phas

e mar

gin

(deg

)

Phase margin

120572120573minus1

minus1

minus2minus2


120573 03120572 minus21Phase 6054






0


Frequency (rads)

119878(d

B)

minus60

minus40

minus20


(a)

0

minus20

minus10

119879(d

B)

Frequency (rads)


(b)





1= 1119873

119906= 2 and119873


119888(13)



119896=


119896 120582119896) would lead to


0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

01

02

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

00204

minus02minus04


Acce

lera

tion

(ms2)

(c)



minus6

10minus1

101

105










0 2 4 6 8 10 12 14 16 180

10

20

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 2 4 6 8 10 12 14 16 18Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 2 4 6 8 10 12 14 16 18Time (s)

012


Acce

lera

tion

(ms2)

(c)



119880119896= F (119906

119896) =

119873minus1

sum

119894=0

119906119896119890(2120587119873119896

119894)

119896 = 0 119873 minus 1 (15)





119896

119875 (119880119896le ) ge

1

2

and 119875 (119880119896ge ) ge

1

2

(16)


(i) mean

119890 =

1

119873

119873minus1

sum

119894=0

119890119894 (17)

0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

005

115


minus05

Acce

lera

tion

(ms2)

(c)



120590 = radic1

119873

119873minus1

sum

119894=0

(119890119894minus 119890)2

(18)


RMSE = radic1

119873

119873minus1

sum

119894=0

1198902

119894 (19)









GPC 3 00474 21340 25539 21317 04397 19320GPC 4 03955 25186 63434 25461 00149 10562FGPC 04604 24119 58174 24523 00044 01652



4 Conclusions




119896and 120582



Appendix



120572

119868119887

119886119891 (119909) equiv int

119887

119886

[1198631minus120572

119891 (119909)] 119889119909 120572 119886 119887 isin R (A1)




120572

119868119887

119886119891 (119909) = Δ119909

120572

119882

1015840

119891 (A2)

where



119908119899


1015840



1015840

(A3)

with 119908119895= 120596119895minus 120596119895minus119899

119899 = 119887 minus 119886 120596119897= (minus1)

119897

(minus120572

119897) and 120596

119897= 0

for all 119897 lt 0



Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0


Frequency (rads)

119878(d

B)

minus60

minus40

minus20


(a)

0

minus20

minus10

119879(d

B)

Frequency (rads)


(b)





1= 1119873

119906= 2 and119873


119888(13)



119896=


119896 120582119896) would lead to


0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

01

02

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

00204

minus02minus04


Acce

lera

tion

(ms2)

(c)



minus6

10minus1

101

105










0 2 4 6 8 10 12 14 16 180

10

20

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 2 4 6 8 10 12 14 16 18Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 2 4 6 8 10 12 14 16 18Time (s)

012


Acce

lera

tion

(ms2)

(c)



119880119896= F (119906

119896) =

119873minus1

sum

119894=0

119906119896119890(2120587119873119896

119894)

119896 = 0 119873 minus 1 (15)





119896

119875 (119880119896le ) ge

1

2

and 119875 (119880119896ge ) ge

1

2

(16)


(i) mean

119890 =

1

119873

119873minus1

sum

119894=0

119890119894 (17)

0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

005

115


minus05

Acce

lera

tion

(ms2)

(c)



120590 = radic1

119873

119873minus1

sum

119894=0

(119890119894minus 119890)2

(18)


RMSE = radic1

119873

119873minus1

sum

119894=0

1198902

119894 (19)









GPC 3 00474 21340 25539 21317 04397 19320GPC 4 03955 25186 63434 25461 00149 10562FGPC 04604 24119 58174 24523 00044 01652



4 Conclusions




119896and 120582



Appendix



120572

119868119887

119886119891 (119909) equiv int

119887

119886

[1198631minus120572

119891 (119909)] 119889119909 120572 119886 119887 isin R (A1)




120572

119868119887

119886119891 (119909) = Δ119909

120572

119882

1015840

119891 (A2)

where



119908119899


1015840



1015840

(A3)

with 119908119895= 120596119895minus 120596119895minus119899

119899 = 119887 minus 119886 120596119897= (minus1)

119897

(minus120572

119897) and 120596

119897= 0

for all 119897 lt 0



Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 180

10

20

Time (s)

Spee

d (k

mh

)

Reference


(a)

0 2 4 6 8 10 12 14 16 18Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 2 4 6 8 10 12 14 16 18Time (s)

012


Acce

lera

tion

(ms2)

(c)



119880119896= F (119906

119896) =

119873minus1

sum

119894=0

119906119896119890(2120587119873119896

119894)

119896 = 0 119873 minus 1 (15)





119896

119875 (119880119896le ) ge

1

2

and 119875 (119880119896ge ) ge

1

2

(16)


(i) mean

119890 =

1

119873

119873minus1

sum

119894=0

119890119894 (17)

0 10 20 30 40 50 60 70 8005

1015

Time (s)

Spee

d (k

mh

)


(a)

0 10 20 30 40 50 60 70 80Time (s)

0

05

Con

trol a

ctio

n [0

-1]

(b)

0 10 20 30 40 50 60 70 80Time (s)

005

115


minus05

Acce

lera

tion

(ms2)

(c)



120590 = radic1

119873

119873minus1

sum

119894=0

(119890119894minus 119890)2

(18)


RMSE = radic1

119873

119873minus1

sum

119894=0

1198902

119894 (19)









GPC 3 00474 21340 25539 21317 04397 19320GPC 4 03955 25186 63434 25461 00149 10562FGPC 04604 24119 58174 24523 00044 01652



4 Conclusions




119896and 120582



Appendix



120572

119868119887

119886119891 (119909) equiv int

119887

119886

[1198631minus120572

119891 (119909)] 119889119909 120572 119886 119887 isin R (A1)




120572

119868119887

119886119891 (119909) = Δ119909

120572

119882

1015840

119891 (A2)

where



119908119899


1015840



1015840

(A3)

with 119908119895= 120596119895minus 120596119895minus119899

119899 = 119887 minus 119886 120596119897= (minus1)

119897

(minus120572

119897) and 120596

119897= 0

for all 119897 lt 0



Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












GPC 3 00474 21340 25539 21317 04397 19320GPC 4 03955 25186 63434 25461 00149 10562FGPC 04604 24119 58174 24523 00044 01652



4 Conclusions




119896and 120582



Appendix



120572

119868119887

119886119891 (119909) equiv int

119887

119886

[1198631minus120572

119891 (119909)] 119889119909 120572 119886 119887 isin R (A1)




120572

119868119887

119886119891 (119909) = Δ119909

120572

119882

1015840

119891 (A2)

where



119908119899


1015840



1015840

(A3)

with 119908119895= 120596119895minus 120596119895minus119899

119899 = 119887 minus 119886 120596119897= (minus1)

119897

(minus120572

119897) and 120596

119897= 0

for all 119897 lt 0



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
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Fractional-Order Generalized Predictive Control: Application …downloads.hindawi.com/journals/mpe/2013/895640.pdf · 2019-07-31 · Research Article Fractional-Order