Fractional Order Modeling of Human Operator Behavior with Second Order Controlled Plant and Experiment Research

1.7 MB·

by  Jiacai Huang

IEEE/CAAJOURNALOFAUTOMATICASINICA,VOL.X,NO.X,XX 1
Fractional Order Modeling of Human Operator
Behavior with Second Order Controlled Plant and
Experiment Research
∗
Jiacai Huang, Yangquan Chen , Senior member, IEEE, Haibin Li, Xinxin Shi,
6 Abstract—Modeling human operator’s dynamic plays a very in certain closed loop control systems and proposed a quasi-
1 important role in the manual closed-loop control system, and linear mathematical model for the human operator, and the
0 it is an active research area for several decades. Based on the proposedmodelis composedof two components-adescribing
2 characteristics of human brain and behaviour, a new kind of
function and remnant [3]. In [4], the rms-error performance
fractionalordermathematicalmodelforhumanoperatorinSISO
n systemsisproposed.Comparedwiththetraditionalmodelsbased of a human operator in a simple closed-loop control system
a on the commonly used quasi-linear transfer function method was measured and compared with the performance of an
J
or the optimal control theory method, the proposed fractional ‘optimum’ linear controller, the comparison results showed
8 ordermodelhassimplestructurewithonlyfewparameters, and
thatthehumanoperatorperformaboutaswellasahighlycon-
each parameter has explicit physical meanings. The actual data
strainedoptimumlinearcontroller.In[5] thehumanoperators
] andexperimentresultswiththesecond-ordercontrolledelement
Y illustrate the effectiveness of the proposed method. were considered as a monitor and controller of multidegree
S of freedom system, and the experiment results showed that
Index Terms—Fractional order modeling, fractional calculus,
. the human operators are in fact random sampling device and
s human operator, human in the loop, second order controlled
c plant nearly ideal observers, meanwhile individual operator may
[ havefixedpatternsofscanningforashortperiodsandchange

1 I. INTRODUCTION the patterns from time to time, and different human operators
v THE modeling of human operator is still an open prob- have different patterns.
8 In 1965, McRUER[6] studied the human pilot dynamics in
lem. In manual closed-loop control system, the accurate
7
compensatory system and proposed human pilot models with
7 mathematical model of human operator is very importantand
different controlled element, and the experiments results val-
1 providescriteriatothecontrollerdesignofthemanualcontrol
0 system. The human operatoris a very complexsystem whose idated the proposed models. In 1967, McRUER summarized
. the current state of the quasi-linear pilot models, including
1 behaviour range includes not only skilled control tasks, but
experimentaldataandequationsofdescribingfunctionmodels
0 alsoinstinctiveandemotionalreactions,suchasthoseresulting
6 from pain or fear. for compensatory, pursuit, periodic, and multiloop control
1 situations [7]. In [8], the deficiencies of the existing quasi-
For decades, modelinghuman operator’sdynamichas been
: linear pilot models have been analyzed and then some new
v an active research area.The earliest study that considered
i the human operator as a linear servomechanism is Tustin in analyticalapproachesfromautomaticcontroltheoryhavebeen
X
proposed to estimate pilot response characteristics for novel
1947[1], who proposed that the main part of the operator’s
r situations.
a behaviour might be described by an ’appropriate linear law’,
In [9], based on the assumption that the operator behaves
despitetheamplitudenonlinearvariationsandhaphazardfluc-
as an optimal controller and information processor subject
tuations. In 1948, reference [2] studied the human operator
to the operators inherent physical limitations, a mathematical
as an engineering system, and proposed the following theory
model of the instrument-monitoring behavior of the human
of the human operator in control system: the human operator
operator was developed. In [10], an adaptive model with
behaves basically as an intermittent correction servo which
variable structure was presented to describe the behavior of
consistsofballisticmovement,moreovertherearesomecoun-
the human operator in response to sudden changes in plant
teracting processing tending to make controls seem continu-
dynamics and transient disturbances. In [11], a pilot model
ous.In1959,McRUERconsideredtheroleofhumanelements
based on Kalman filtering and optimal control was given
∗Corresponding author. Email:[email protected] which provides for estimation of the plant state variables,
Jiacai Huang, Haibin Li and Xinxin Shi are with School of Automation,
the forcing functions, the time delay, and the neuromuscular
Nanjing Institute of Technology, Nanjing 211167, China. e-mail: huangjia-
[email protected] lag. The remnant which is an important component of the
Yangquan Chen is with School of Engineering, University of California, quasi-linear model for the human operator was discussed in
CA95343,USA.e-mail: [email protected]
in [12], and a model for remnant was postulated in which
*This work was supported by National Natural Science Foundation
(NNSF) of China under Grant(61104085), Natural Science Foundation of remnantis considered to arise from an equivalentobservation
Jiangsu Province (BK20151463, BK20130744), Innovation Foundation of noisevectorwhosecomponentsarelinearlyindependentwhite
NJIT(CKJA201409,CKJB201209), sponsored by Jiangsu Qing Lan Project,
noiseprocesses.In[13]and[14],amathematicalmodelofthe
andtheJiangsuGovernmentScholarship forOverseas Studies(JS-2012-051)
Manuscript received August31,2015;revised***,***. human as a feedback controller was developed using optimal
2 IEEE/CAAJOURNALOFAUTOMATICASINICA,VOL.X,NO.X,XX
control and estimation theory. Where a and t are the limits of the operation, λ is the order
From 70s to the early 21st century, the problem of human oftheoperation,andgenerallyλ∈Randλcanbeacomplex
operator modeling has been widely studied and a lot of new number.
achievements emerged [15-28]. The three most used definitions for the general fractional
In recent years, with the new situation and different ap- differentiationandintegrationaretheGrunwald-Letnikov(GL)
plication, the modeling human operator’s dynamic is still an definition[36], the Riemann-Liouville(RL) definitionand the
activeresearcharea.In[29],atwo-stepmethodusingwavelets Caputo definition [37].
and a windowed maximum likelihood estimation method was The GL definition is given as
proposed for the estimation of a time-varying pilot model
[t−a]
parameters. In [30] the human control model in teleoperation h λ
Dλf(t)= lim h−λ (−1)j f(t−jh) (2)
rendezvousonthebasisofhumaninformationprocessingwas a t h→0 j
studied, and the longitudinal and lateral control models for Xj=0 (cid:18) (cid:19)
the human operator were presented based on phase plane where [·] means the integer part, h is the calculus step, and
controlmethodandfuzzycontrolmethod.In[31],areviewof λ
= λ! is the binomial coefficient.
pilot model used for flight control system design that focuses j j!(λ−j)!
specifically on physiological and manual control aspects was (cid:18)The(cid:19)RL definition is given as
presented. 1 dn t f(τ)
catFioonr,athheumcoarnre-icnt-ntehses-looofpsusychstesmystienmssafdeetpye-cnrditsicnalotapopnlliy- aDtλf(t)= Γ(n−λ)dtn Za (t−τ)λ−n+1dτ (3)
on the autonomous controller, but also on the actions of where n−1<λ<n and Γ(·) is the Gamma function.
the human controller. In [32] a formalism for human-in- The Caputo definition is given as
the-loop control systems was presented which focuses on
1 t fn(τ)
the problem of synthesizing a semi-autonomous controller Dλf(t)= dτ (4)
from high-level temporal specification that expect occasional a t Γ(n−λ)Za (t−τ)λ−n+1
human intervention for correct operation. In [33] the three where n−1<λ<n.
different approaches (Engineering, Physiology and Applied Having zero initial conditions, the Laplace transformation
ExperimentalPsychology)tothestudyofhumanoperatorhave of the RL definition for a fractional order λ is given by
beendiscussed,andtheimportanceofthestudyingthehuman L Dλf(t) = sλF(s), where F(s) is Laplace transforma-
a t
operator has been pointed out. In [34] the accurate control of tion of f(t).
human arm movementin machine-human cooperative control (cid:8) (cid:9)
of GTAW process was studied and an adaptive ANFIS model
III. REVIEWOF THEQUASI-LINEAR MODELSFORHUMAN
was proposed to model the intrinsic nonlinear and time-
OPERATOR
varyingcharacteristicofthehumanwelderresponse,atlastthe
Thequasi-lineartransferfunctionisaneffectivemethodfor
human control experimental results verified that the proposed
the modeling of human operator, and the quasi-linear models
controller was able to track varying set-points and is robust
have been found to be useful for the analysis of closed loop
under measurement and input disturbances.
compensatory behaviour in the manual control system. For a
The existing models for human operator are complicated
simple compensatory manual control system, the functional
and established by integer order calculus. In this paper, based
block diagram is shown as Fig.1, where i(t) is the system
on the characteristics of human brain and behaviour,the frac-
input, e(t) is the system error, c(t) is the human operator
tional order human operator model is proposed and validated
output, m(t) is the system output.
by the actual data.
II. FRACTIONAL ORDERCALCULUS
Fractional calculus has been known since the development +
of the integer order calculus, but for a long time it has been
consideredasasolemathematicalproblem.Inrecentdecades,
fractional calculus has become an interesting topic among
system analysis and control fields due to its long memory
Fig. 1: Functional block diagram of the manual control
characteristic [35], [36], [37], [38], [39], [40].
system
Fractional calculus is a generalization of integer order
integration and differentiation to non-integer order ones.Let
For the above compensatory manual control system, the
symbol Dλ denotes the fraction order fundamentaloperator,
a t generalizedformofthequasi-linearmodelforhumanoperator
defined as follows [35]:
was proposed as follow [3], [6], [7], [8]:
ddtλλ, R(λ)>0; C(s) τLs+1 e−Ls
Y (s)= =K (5)
P1 p
Dλ =∆ Dλ = 1, R(λ)=0; (1) E(s) τIs+1 τNs+1
a t  Where C(s) and E(s) are the Laplace transform of c(t)
t(dτ)−λ, R(λ)<0. and e(t) respectively, τL and τI represent the equalisation
 Ra
SHELLetal.:BAREDEMOOFIEEETRAN.CLSFORJOURNALS 3
characteristics of human operator, L and τ represent the
N "
reaction time and neuromuscular delay of human operator
respectively, K represents the human operators gain which
p
is dependant on the task and the operators adaptive ability. %!
Theparametersintheabovetransferfunctionareadjustableas 654
54
neededtomakethesystemoutputfollowtheforcingfunction, (*.(
i.e.,theparameters,asadjusted,reflecttheoperatorseffortsto 3,4)
make the overall system (including himself) stable and the 2
$ %!
errorsmall. The quasi-linearmodelof Eq.(5) hasbeenwidely
quoted by further research.
Based on the human operator model described by Eq.(5), $"
! " "! #
themathematicalmodelofthemanualcontrolsystemisshown
&’()*+,)-./0,1
in Fig.2.
(b) Systemoutputm(t)
"
%!
Fig. 2: Themathematicalmodelofthe manualcontrolsystem .5
)55
(*)
In reference [15], a detail research was made to the com- 3,4)
pensatory manual control system which as shown in Fig.1, 2
in which the forcing function (i.e. the system input) i(t) is a $ %!
randomappearingsignal,andin thehumanoperatingprocess,
the error e(t) and human output c(t) can be obtained. By
$"
studying the relationship between the error e(t) and human ! " "! #
&’()*+,)-./0,1
outputc(t), themathematicalmodelsforhumanoperatorwith
respect to controlled elements was proposed [15]. (c) Systemerrore(t)
Because the second order controlled element includes
not only those which reflect the particular nature, but also
represent the classic and representative about model, in this 1
paperwetakesecondordercontrolledelementasasexample,
which is described as follow:
0.5
K 1 0/
Yc(s)= c , T = ,Kc =1, (6) 0/,
s(Ts+1) 3 (
(-$’ 0
thenthesysteminputi(t),thesystemoutputm(t),thesystem #-./
,
error e(t), the human operator output c(t) and the lag of the (
-0.5
operator output C(s) were recorded as Fig.3(a)-Fig.3(e).
s+3
-1
" 0 5 10 15 20
!"#$%&#’()*&+
%! (d) Humanoperatoroutputc(t)
64
5
/
(*’’
) e(t), a great similarity can be seen, so the following transfer
23,4 function between c(t) and e(t) was proposed in [15] :
$ %!
C(s) 1
Y (s)= =K (s+ )e−Ls =K (s+3)e−Ls, (7)
$" P2 E(s) p T p
! " "! #
&’()*+,)-./0,1
where K is the human operator’s gain; L is the time delay
p
(a) Systeminputi(t) of human operator, which is about L=0.16s.
Based on the human operator model described by Eq.(7),
Fromtheaboveexperimentresult,whenthelagoftheoperator themathematicalmodelofthemanualcontrolsystemisshown
output c(t) (i.e. fig.3(d))is compared with the system error in Fig.4.
4 IEEE/CAAJOURNALOFAUTOMATICASINICA,VOL.X,NO.X,XX
1 (1) For human brain, the later the thing happens, the
clearer the memory is. On the contrary, the earlier, the
poorer.Inotherwords,thehumanbrainhashighermemory
0.5
level for the newer things, and lower memory level for the
older things.
-.+ 0 (2) During the human action, there exist dead-time in the
’,%& nervoussystem,includingthedead-timefromtheretinatothe
brain, and the dead time from the brain to the muscle.
-0.5
(3) The human muscle has the viscoelastic property.
Fromtheabovefacts,itcanbeconcludedthatthedynamics
-1 of the human operators brain is most like a kind of fractional
0 5 10 15 20
order integral or derivative which exhibits a long memory
!"#$%&#’()*&+
characteristics, and so the human operator can be seen as a
(e) Thelagofoperator output,i.e. c fractionalorder controllerwith time delay, then in this paper
s+3
the fractional order model for human operator in SISO
Fig. 3: Manual control system response, systems is proposed as follow:
Y (s)= Kc ,with T = 1,K =1
c s(Ts+1) 3 c
C(s) K e−Ls
Y (s)= = p ,α∈R. (8)
P3 E(s) sα
Whereαisthefractionalorderwhichdescribesthedynam-
ics of the human operator, and α can be positive or negative;
K is the human operator’s gain; L is the total time delay of
p
humanoperator,includingthedead-timeinthenervoussystem
Fig. 4: Themathematicalmodelofthe manualcontrolsystem from the retina to the brain, and the dead time in the nervous
system from the brain to the muscle. In real system, the α
and other parameters can be obtained by online or off-line
IV. FRACTIONAL ORDER MATHEMATICALMODELFOR identification.
HUMAN OPERATORBEHAVIOR Based on fractional order model of the human operator
described by Eq.(8), the mathematical model of the manual
In the existing research, the human operator models are
control system is shown in Fig.6.
established based on the integer order calculus. In fact, the
human body is a high nonlinear servomechanism, the control
task is completed through the cooperation of the eyes, the
brain/nervous system, the muscle and the hands, as shown in
Fig.5.
Fig. 6:The mathematicalmodelof the manualcontrolsystem
In the following section, the effectiveness of the proposed
fractional order model for human operator will be validated.
V. MODELVALIDATION WITH ACTUAL DATA
Inthis section,the off-lineverificationandcomparisonwill
be done to the traditional mathematical models described by
Eq.(5) and Eq.(7), and the new proposed fractional order
model described by Eq.(8). In the model verification process,
thebest fitparametersforthe abovethreemodelshavebeen
Fig. 5: The control structure of a human operator
obtained by the fminsearch function with actual data taken
from reference [15], and the following cost function, i.e. the
LetusconsiderthemanualcontrolsystemshowninFig.1,in
root mean square error(RMSE) is used,
whichthehumanoperatorisshowninfig.3.Inthissystem,the
human operator controls the machine by hands to follow the T (m (t)−m(t))2
target.Theeyesactasasensor,thebrainactsascontrollerand J = 0 model , (9)
s T
sendsthenervoussystemsignaltothearmandhandtofollow R
the target. The muscles of the arm and hand are employed wherem(t) isthe actualoutputof themanualcontrolsystem,
as power actuators. Meanwhile the human has the following m (t) is the model output of the manual control system
model
characteristic[1], [32]: by using the human operator model and the actual input (i.e.
SHELLetal.:BAREDEMOOFIEEETRAN.CLSFORJOURNALS 5
as shown in Fig.(2),Fig.(4) and Fig.(6)), T is the operating value between 0 to 0.4, and the gain K gets the fixed value
p
time period of human operator. of1,3 and5.ForeachK andL, theα isvariedfrom−0.95
p
In order to get the best fit parameters of each model, the to −0.05with 0.05step length.The scan results are shownin
following searching criteria is adopted. Fig.(7) to Fig.(11).
Case 1: When the human operator model is described by (1) Whenthe gain ofthe humanoperatoris K =1, the
p
Eq.(8), i.e. the proposedfractionalorder model, the searching RMSEscanresultforeachL isshowninFig.(7),andthe3-D
criteria is RMSE scan result to different α and L is shown in Fig.(8).
FromFig.(7)andFig.(8)itisclearthat:(a)thecorrespondingα
α∗,K∗,L∗ = min (J). (10)
p best fit α∈R;Kp,L∈R+ totheminimumRMSEisfractional;(b)whenthetimedelayL
(cid:8) (cid:9) gets bigger valve, the corresponding minimum RMSE is also
In this case, the fractional order differentiation/integration
bigger.
symbol 1 is implemented by the Grunwald-Letnikov(GL)
sα (2) Whenthe gain ofthe humanoperatoris K =3, the
definition described as Eq.(2). p
RMSE scan result for each L is shown in Fig.(9),from which
Case 2: When the human operator model is described as
it can be seen that: (a)the corresponding α to the minimum
Eq.(7), i.e. the traditional model, the searching criteria is
RMSE is fractional; (b)when the time delay L gets smaller
K∗,L∗ = min (J). (11) valve, the corresponding minimum RMSE is bigger, this is
p best fit Kp,L∈R+ because the human gain gets the bigger value in this case.
(cid:8) (cid:9)
Case 3: When the human operator model is described as (3) When the gain of the human operator gets the value
Eq.(5), i.e. the traditional model, the searching criteria is K = 5 or K = 7, the RMSE scan results for each L are
p p
showninFig.(10)andFig.(11)respectively.Fromthefiguresit
T∗,T∗,T∗,K∗,L∗ = min (J). (12)
L I N p best fit TL,TI,TN,Kp,L∈R+ canbeseenthatthecorrespondingαtotheminimumRMSEis
(cid:8) (cid:9) fractional.MeanwhileastheKp getsthe bigvalueinthistwo
A. The minimum RMSE and best fit parameters for each cases,Fig.(10)andFig.(11)onlyshowtheRMSEtoL=0.05,
models andtheRMSEtootherL(whichisgreaterthan0.05seconds)
is too large to be shown in the figures.
Using the above searching criteria Eq.(10)− Eq.(12), the
minimum RMSE and the corresponding best fit parameters
(
value for each model are obtained as shown in Table 1. *+!"
)#)
From Table 1, it is obvious that the proposed fractional order 01"#") 231!
model described by Eq.(8) has the smallest RMSE, and the 01"#!
corresponding order of the model is α = −0.4101. This ) 01"#!)
meansthatcomparedwith the traditionalmodel,the proposed 01"#’
fractional order model described by Eq.(8) is the best fit 01"#’)
/
model for describing the human operator behavior, in other -.&#) 01"#(
01"#()
word, the human operator is a fractional order system. ,
01"#&
TABLE I: best fit parameters value and RMSE for each &
model
Model Parameters Values (#)
! "#$ "#% "#& "#’ "
YP3(s)=Kpseα−Ls RαKM∗p∗SE -004...04401010321 Fig. 7: The RMSE scan result to+ different α with fixed L,
L∗(sec) 0.117
and K =1
RMSE 0.0018 p
YP2(s)=Kp(s+3)e−Ls Kp∗ 7.994
L∗(sec) 0.014
RMSE 0.0024
K∗ 1.7298 VI. EXPERIMENT RESEARCH
p
YP1(s)=(KTpI(sT+L1s)+(T1N)es−+L1s) TTTLI∗∗∗ 100...8111664226 wilIlnbtehidsonseecbtiaosne,dtohnethhuemQauna-nins-etrhSe-RloVo0p2cRoonttarroyl SexerpveorimBaesnet
LN∗(sec) 0.006 unit. The experiment platform is shown in Fig.12, which is
composed of a human operator, a steering wheel, a torque
sensor,amotor,acomputerinstalledwithQuanser/Matlabreal
time software and QPIDe data acquisition card. The steering
B. The RMSE of the proposed fractional order model to
wheel is fixed with the torque sensor which is mounted on
different α and L the desk. The voltage output of the torque sensor is power
In this section, the RMSE of the proposedmodeldescribed amplified and transferred to the motor. The motor works on
by Eq.(8) to different α, L and K will be scanned. Because voltage to position control mode, and the encoder on the
p
the time delay and gain of human operator have finite range, motoroffersahighresolutionof4096countsperrevolutionin
so in this scanning process, the time delay L gets some fixed quadrature mode(1024 line per revolution). The QPIDe card
6 IEEE/CAAJOURNALOFAUTOMATICASINICA,VOL.X,NO.X,XX
()!" ’ ’#*+,!" *
()!" ’
$#% 452( 12"#"3
$#$
’#’
$
$
&#0 ’#!
. &#$ &#/ 0
,- ./ ’
+ &#& -
&
&#% !#)
’#$
"#& &
!#$
"
"#% ’#0
"#$
* " ! ’#/ !#( ! "#$ "#% "#& "#’ "
,
Fig. 8: The 3-D RMSE scan result to different α and L, and Fig. 11: The RMSE scan result to different α with fixed L,
Kp =1 and Kp =7
(
)*!"
(#’ computer, and the human operator observes the system error
230( /0"#"1 andappliesa forcearoundthe steeringwheel, andso controls
( /0"#!
the motor’s position to follow the system input. The block
/0"#!1
diagram of the human-in-the-loopcontrol system is shown in
’#$
Fig.13.
’#%
.
-
,
+ ’#&
’#’
’
!#$
! "#$ "#% "#& "#’ "
*
Fig. 9: The RMSE scan result to different α with fixed L,
and K =3
p
,
*+!"
’#’
4523 12"#"3 Fig. 12: The human-in-the-loopcontrol experiment platform
’#!
’
0
./!#) (cid:16)(cid:16) (cid:72)
- (cid:76)
(cid:16)
!#$
(cid:80)
!#(
!#%
! "#$ "#% "#& "#’ " (cid:70)
+
Fig. 10: The RMSE scan result to different α with fixed L, (cid:3134) (cid:3135)
and K =5
p
Fig. 13: The block diagram of the human-in-the-loopcontrol
experiment
samples the voltage output of the torque sensor together with
theencoderoutput.Intheexperiment,thesysteminput,output In the experiment, the motor works on position control
anderrorinformationareallshownonthedisplayscreenofthe mode, in this case it is a second order system and its transfer
SHELLetal.:BAREDEMOOFIEEETRAN.CLSFORJOURNALS 7
function is described as follow:
"
K 60.2362
Y (s)= = , (13)
c s(τs+1) s(s+39.37) !*%
where K = 1.53rad/s/V, τ = 0.0254s. In this experiment,
56 !
tLhe=ti0m.3es,daenladythoefstyhsetehmuminapnutoip(te)r,astyosr’tsemdeolauytpuistmte(stt)e,dsyasbtoemut 3:/0:;
error e(t) and operator output c(t) are real time recorded as -/.::. *%
.
shown in Fig.14 to Fig.17. 819
7
% ) *%
)!
! " # $ % & ’ (
+,-./01.234516
45
:8./;< Fig. 16: The system erroerxpoefrtihmeehnutman-in-the-loopcontrol
,.+39+
-
708
6
!*%
!
)%
! " # $ % & ’ (
*+,-./0-123405 =6
Fig. 14: The system input of the human-in-the-loopcontrol 8<;/0 *%
experiment 3<;2
39/
:;
.9
8
7
&
) *%
$
)!
45 " ! " # $ % & ’ (
<
98./; +,-./01.234516
:
98, Fig. 17: The human operator output(1V=4N.m)
2
,.
-
708 )"
6
Eq.(8)isthebest fitmodelfordescribingthehumanoperator
)$
behavior, in other word, the human operator is a fractional
ordersystem. Thisresultis consistentwith theresultobtained
)&
! " # $ % & ’ (
in section IV.
*+,-./0-123405
Fig. 15: The system output of the human-in-the-loopcontrol
TABLE II: best fit parameters value and RMSE for each
experiment
model (L=0.3s)
Model Parameters Values
A. The minimum RMSE and best fit parameters for each RMSE 3.751×10−3
models YP3(s)=Kpseα−Ls αK∗p∗ -00..73684733
Using the experiment data and the searching criteria RMSE 4.172×10−3
Eq.(10)−Eq.(12),theminimumRMSEandthecorresponding YP2(s)=Kp(s+3)e−Ls Kp∗ 0.6099
RMSE 4.036×10−3
best fit parameters value for each model are obtained as K∗ 1.078
p
sphroopwonseidnfrTaacbtiloenaIIl.oFrdreormmToadbelledeIIs,criitbeids boybvEiqo.u(s8)thhaats tthhee YP1(s)=(KTpI(sT+L1s)+(T1N)es−+L1s) TTTLI∗∗∗ 000...107408800114
smallest RMSE, and the corresponding order of the model is N
α=−0.3873. This means that compared with the traditional
model, the proposed fractional order model described by
8 IEEE/CAAJOURNALOFAUTOMATICASINICA,VOL.X,NO.X,XX
B. The models parameters to different L
%
Ingeneral,thetimedelayofhumanoperatorvariesinsmall
$!&
range, so in this section the proposed fractional order model
described by Eq.(8) and the conventional model described
$
by Eq.(5) will be considered, and the models parameters
distribution to different human time delay L will be scanned.
#!&
As the time delay of human operator has finite range, so in 3
2
thisscanningprocessthetimedelayLvariesfrom0.01to0.6 #
with 0.01 step length. The scan results are shown in Fig.(18)
to Fig.(21). "!&
"
( !"&
( !# !&
!" !# !$ !% !& !’
()*+,-./0+1
( !#&
Fig. 20: The gain K distribution of human operator to
p
( !$
different L using the conventional model described by Eq.(5)
( !$&
$
( !%
#
( !%& )
(
"
( !&

!" !# !$ !% !& !’
( !&&
!" !# !$ !% !& !’
"
)*+,-./01,2
&
Fig. 18: The fractional order α distribution of human (+
operator to different L using the proposed model described *&
!" !# !$ !% !& !’
by Eq.(8) "&
"
5 &
(

0.8 *&
!" !# !$ !% !& !’
),-./0123.4
Fig. 21: The T ,T , T distributions of human operator to
0.75 L I N
different L using the conventional model described by Eq.(5)
+
*
0.7
view,the proposedfractionalordermodeldescribedbyEq.(8)
is suitable to describe the human operator behavior.
0.65
VII. CONCLUSIONS
In this paper, based on the characteristics of human brain
and behaviour, the fractional order mathematical model for
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
human operator is proposed. Based on the actual data, the
!"#$%&’(#)
modelsverificationshavebeendone,andthe best fitparam-
Fig. 19: The gain Kp distribution of human operator to eters for the proposed model and the traditional models have
different L using the proposed model described by Eq.(8) beenobtained.Theverificationresults showthatthe proposed
fractional order model described by Eq.(8) is the best fit
Fig.(18) and Fig.(19) show that the distributions of α model for describing the human operator behavior, in other
and K of the proposed fraction order model are smooth, words, the human operator is a fractional order in such a
p
meanwhile as the time delay L decreases, the fractionalorder system. The experiment results also provide the correctness
α tend to negative increase. Fig.(20) and Fig.(21) show that of the above conclusion.
the parametersK , T , T andT of the conventionalmodel The proposed fractional order model described by Eq.(8)
p L I N
describedbyEq.(5)fluctuateinlargescale.Fromthispointof for human operator behavior not only has small RMSE, but
SHELLetal.:BAREDEMOOFIEEETRAN.CLSFORJOURNALS 9
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ACKNOWLEDGMENT
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theFirstJoint. IEEE,2:1009vol.2,1999.
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