Behavio Reseach Methods, Instuments, & Comutes 1990, 22 (4), 349-359 - METHODS & DESIGNS Seadsheet analysis of a hieachical contol system model of behavio RICHARD S. MARKEN Aeosace Cooation, Los Angeles, Califonia The behavio of a hieachy of contol systems can be simulated with an electonic seadsheet. Each contol system is a column of thee cells eesenting the efeence signal, ecetual signal, and outut vaiable of the system. All of the contol systems ae closed-loo, the inut to each system being a function of its outut. The cicula efeences in the seadsheet ae esolved though iteative ecalculation. When the aametes of each contol system (amlification and slowing factos) ae set to aoiate values, all contol systems in the hieachy continue to match ecetual signals with efeence signals. A thee-level hieachy with fou systems at each level is simulated in the seadsheet. The seadsheet model makes it ossible to obseve the dynamic behavio of the contol systems as they coect fo the effects of envionmental distubance and changes in highe ode efeence signals. It is ossible to "eoganize" the system by changing the ecetions contolled by systems at diffeent levels of the hieachy. The use can also test to detemine the vaiables being contolled by the system. Powes (1973, 1989) has oosed a hieachical contol system model of the uosive behavio of oganisms. In this ae, a method of imlementing the model on an electonic seadsheet will be descibed. The seadsheet system makes it elatively easy to exloe the model's behavio with a esonal comute. A seadsheet imlementation was selected because many (ehas most) esonal comute uses have access to seadsheet softwae and ae familia with its use. Moeove, the matix design of the seadsheet ovides an excellent fomat fo eesenting a contol system hieachy. The seadsheet model is designed to be a self-instuctional system fo those who want to lean how contol systems wok, but it could also be used as a eseach tool. The behavio of an aoiately designed vesion of the model could be comaed to that of human o animal subjects in exeiments like those descibed by Maken (1986, 1989) and Boubon (1989). HIERARCHICAL CONTROL SYSTEM MODEL Basic Contol System The comonents of a basic contol system ae shown in Figue 1. The sensoconvets an inut vaiable, i, into a ecetual signal,. The comaato subtacts the ecetual signal fom a efeence signal,, to oduce an eo signal, e. The amlifie convets the eo signal Coesondence shouldbeaddessed to RichadS. Maken,Aeosace Cooation, Mail StationMII076,.o. Box92957, Los Angeles, CA 90009. into an outut vaiable, o. (Note: Signals ae quantities that vay inside the contol system; vaiables ae quantities that vay outside the contol system.) What constitutes an inut and an outut vaiable deends on the location of this basic system in a contol hieachy. If the system is at the lowest level ofthe hieachy (as is shown in Figue 1), then inut and outut ae hysical vaiables in the envionment. Ifthe contol system is highe in the hieachy, then inut and outut ae signals coming fom and going to lowe level contol systems; the lowe level systems ae the "envionment" of the highe level systems. Regadless of thei osition in the hieachy, all contol systems ae designed to do the same thing-kee the inut vaiable, i, in a edetemined state secified by the efeence signal,. The oblem of contol aises because the value of the inut vaiable is affected by system oututs as well as distubances, d. A distubance is any extenal influence on the inut vaiable that is not caused by the system itself. When set u oely, a contol system oduces oututs that counteact the effects of distubances on the inut. The inut vaiable, which is maintained at a value that coesonds to that secified by the efeence signal, is called the contolled vaiable. The value of the inut that coesonds to the setting of the efeence signal is the efeence state of the contolled vaiable. The efeence state is constant if the efeence signal is constant, and it vaies if the efeence signal vaies. Howeve, constant o vaying, the contolled vaiable is ket in the efeence state, continuouslyotected fom the effects of distubance by the outut of the contol system. 349 Coyight 1990 Psychononic Society, Inc.
350 MARKEN senso - - t-- d --... i.. ~----- e Figue 1. A basic contol system. A concete examle of a basic contol system is the light intensity contol system of the eye (sometimes called the "uillay eflex"). The inut vaiable, i, is the light intensity and the outut vaiable, 0, is the size of the uil. The senso (etina) convets light intensity into a ecetual signal. The comaato convets the diffeence between the ecetual signal and a efeence signal (fom highe levels of the bain) into an eo signal. The amlifie convets the eo signal into muscle tensions, which incease o decease the size of the uil (the outut vaiable). The size of the uil affects the intensity of the light incident on the etina, so that the outut of the system affects the inut. Since the inut also affects the outut, via the contol system, thee is a closed loo of cause and effect. The intensity of light at the etina is the contolled vaiable. Light intensity will be ket in some efeence state, desite distubances such as vaiations in the intensity of the light souce. Distubances ae ecisely counteed by vaiations in uil size (outut), which can be seen if you vay the intensity of a light while looking in the mio. Highe Level Systems In a hieachy ofcontol systems, the lowest level systems ae connected to the extenal envionment; inut and outut ae hysical vaiables. Highe level contol systems eceive inuts fom and send oututs to these lowe level systems. Inuts to highe level systems ae ecetual signals fom the lowe level systems, and oututs fom highe level systems become the efeence signals of the lowe level systems. The elationshi between thee levels of contol in a contol hieachy is shown in Figue 2. Note that systems at all levels ae closed-loo; thee is a connection, though the envionment, fom outut to inut and, though the contol system, fom inut to outut. The oututs of highe level systems ass though one o moe layes of lowe ode systems befoe enteing the envionment. The inuts to highe level systems ass fom the envionment though one o moe layes of ecetual ocessing befoe becoming highe ode ecetual signals. The senso comonent of a highe level contol system tansfoms one o moe ecetual inuts fom lowe level systems into a single ecetual signal. The natue of the tansfomation can, in incile, be quite comlex, making the ecetual signal a measue of vaiation in some abstact asect of the envionment. Fo examle, the senso might comute a ecetual signal that is a weighted sum of seveal lowe level ecetual signals. This weighted sum is the vaiable that is contolled by the system. The efeence signal secifies the efeence state fo this weighted sum (actually, fo the ecetual signal that eesents the weighted sum). The system contols this ecetual signal by vaying its oututs, which become the efeence signals oflowe level systems. These efeence signals tell the lowe ode contol systems what to eceive, not what to do; efeence signals ae secifications fo inut, not outut. The highe level systems contol thei inuts by secifying the level of inut to be eceived by lowe ode systems. A Woking Model The hieachical contol model is designed to oduce uosive behavio like that seen in living oganisms. This behavio is difficult to visualize in a static eesentation of the model, like that in Figue 2. It is not obvious that contol systems at all levels of the hieachy can achieve thei goals simultaneously, even in the esence of andomly vaying distubances. The static eesentation can even lead to misconcetions about the caabilities of a contol hieachy. Fowle and Tuvey (1978), fo examle, claimed that a two-level contol hieachy, simila to the one in Figue 2, could not achieve two diffeent goals simultaneously. Unfotunately, mistakes like this, made by authoitative authos, have led othe eseaches to eject contol system models befoe thei caabilities have been exloed. The seadsheet model descibed in this ae was develoed, in at, to emedy this oblem. Thee is a lage liteatue on the theoy of contol systems, but it can be difficult to undestand the behavio of these systems by looking at the equations that descibe them. Those who want to lean the caabilities of a contol system model of behavio, esecially those who ae not mathematically sohisticated, would benefit fom a woking, dynamic simulation of contol system behavio. A seadsheet imlementation of this simulation has seveal advantages ove conventional languages. Fist, the matix layout of the seadsheet is well suited fo the dislay of a hieachical contol model. Second, the modula design of the seadsheet makes it elatively easy to change the model (adding levels o adding systems to existing levels).
Level 3 Level 2 seno comaa1o Level! senso senso --f-- o seno - -f -- d ~ i.-4----- o seno --f-- ----.~ i +-4---- o --..~ i +-4----- o --..~ i +-4----- Figue 2. A thee-level hieachy of contol systems. en ""0 :::c m :>oen ::c m ṃ...j n o~ :::c o-' ~ oom -' ~ VI
352 MARKEN Thid, it is easy to change model aametes by tying a new value into an aoiate cell of the seadsheet. Finally, the seadsheet is a "ogamming envionment" that is familia to most uses of esonal comutes. THE SPREADSHEET CONTROL HIERARCHY A contol system can be eesented in an electonic seadsheet as a column of thee cells, as is shown in Figue 3. Each cell contains a fomula that comutes the value of a signal o vaiable in the contol system. The to, middle, and bottom cells contain fomulas fo the efeence signal, ecetual signal, and outut vaiable, esectively. The fomulas fo the efeence and ecetual signals diffe, deending on the level of the contol system in the contol hieachy. Howeve, the comutation of the outut vaiable is the same fo all contol systems. Outut Signal Calculation The outut signal is ootional to the diffeence between the efeence and ecetual signal. Algebaically, the fomula fo the outut signal can be witten as a diffeence equation as follows: o(t+ 1) = o(t)+s[g(-)-o(t)], whee 0 is the outut signal at time t; sand g ae constants eesenting slowing and gain factos, esectively; is the efeence signal; and is the ecetual signal. The equivalent seadsheet vesion of the fomula, witten in Lotus 1 2 3 (LeBlond & Cobb, 1985) is +O+SLOW*(GAIN * (R-P)-O). The seadsheet fomula is contained in Cell 0, making it self-efeential. 1 The amlified eo signal is accumulated into Cell 0 (outut). The eo signal is the diffeence between efeence, R, and ecetual, P, signals (R - P). The amlification facto is in the cell called GAIN. The ate at which the amlified eo is accumulated into o is detemined by the slowing facto in the cell named SLOW. The accumulated outut in Cell 0 is equivalent to the time integal of the eo signal. The integation efomed in eal nevous systems is not efect; thee is some loss o "leakage" fom the integato ove time. This leakage is catued by the loss due to slowing in the equation fo the outut function. The aoiate values of GAIN and SLOW deend on the level of the basic contol system in the hieachy. SLOW detemines the "seed of esonse" of the contol system. In a hieachy of contol systems, highe level systems cannot esond moe aidly than lowe level systems. This means that the value of SLOW fo highe ode systems must be the same as o smalle than that fo lowe level systems; the highe level systems should accumulate outut moe slowly than lowe level systems do. The GAIN of the system detemines its sensitivity to eo; the highe the value in GAIN, the geate the outut e unit eo. High GAIN means ecise contol, but systems with high GAIN must esond moe slowly than systems with low GAIN. The values ofgain and SLOW must be invesely elated. Pecetual Signal Calculation Powes (1973) has suggested that the ecetual signals at diffeent levels of a contol hieachy eesent diffeent classes of ecetual vaiables. The lowest level ecetual signals eesent only the intensity ofthe inut. This level of the hieachy deals with the outside wold only in tems of magnitude of the inut incident on the sensos. The next level of ecetion is sensationa function of seveal intensities. Going u fom sensations, thee ae configuations (combinations of sensations), tansitions (temoal changes in configuations), events (a sequence of changing configuations), elationshis (eceived elationshi-logical, statistical, causal, etc.-between seaate events), categoies, ogams, inciles, and system concets (see Powes, 1989,. 190-208). In the seadsheet model, thee ae only thee levels ofecetion-intensity (Level 1), sensation (Level 2), and elationshi (Level 3). Level 1. The fomula fo the Level 1 (intensity) ecetual signal is = i, whee is the ecetual signal and i is an envionmental inut vaiable. The seadsheet vesion of this fomula is just +I in the ecetual signal cell of the Level 1 contol system. The fomula eesents the senso as a efect linea tansduce that tansfoms an inut vaiable, located in Cell I, into a ecetual signal. Setting the Level 1 ecetual signal to the value in Cell I imlies that the ecetual signal is an exact measue of the hysical vaiable. A moe ealistic model of the senso might eesent the ecetual signal as a logaithmic o exonential function ofthe inut vaiable. This can be done by using a seadsheet function, such as @LOG(I), to comute the Level 1 ecetual signal. These equations must take account of the fact that I can Cell Values Cell Fomulas R( j,i )1_---'--17 +E4-D4 P(j,i) 6.004 @INDEX(PW,l,PW21)*Pll+@INDEX(PW,2,PW21)*P12... O( j,i) 41.63 +021 +SLOW*(GAIN*(R21-P21)-021) Figue 3. The basic contol system as imlemented in the seadsheet.
SPREADSHEET CONTROL MODEL 353 PWMatix 1 1 1 1 1 2 1 1 1-1 3 1 1-1 1 4 1 1-1 -1 5 1-1 1 1 6 1-1 1-1 7 1-1 -1 1 Rov 8 1-1 -1-1 Labels 9-1 1 1 1 10-1 1 1-1 11-1 1-1 1 12-1 1-1 -1 13-1 -1 1 1 14-1 -1 1-1 15-1 -1-1 1 16-1 -1-1 -1 Figue 4. The ecetual weirhtinr matix, PW. be negative. The model has been successfully un using a logaithmic tansfom: P = log(i) i > 0, whee P is the ecetual signal and i is the inut vaiable. The coesonding seadsheet fomula is: @IF(I >O,@LOG(I)*IO,-IO). The log of I is multilied by 10 to incease the dynamic ange of the ecetual signal. If the hysical signal is less than o equal to 0, the ecetual signal becomes an abitay (and faily lage) negative numbe, -10. Level 2. The Level 2 ecetual signal is a weighted sum of fou Level I ecetions. The weights ae selected fom a set of 16 in a matix called PW (an abbeviation fo ecetual weights). The PW matix is shown in Figue 4. The weights assigned to a Level I ecetion ae eithe 1.0 o -1.0, eesenting an excitatoy o inhibitoy connection of the Level I ecetual signal to the Level 2 senso. Each Level 2 senso uses weights fom a diffeent ow of the PW matix. The numbes identifying the ows of PW that ae used as weights by Level 2, Systems I though 4, ae stoed in ointe cells named PW21 though PW24, esectively. PW21 contains a ointe (a numbe between I and 16) to the ow of the PW matix that contains the weights used by Level 2, System I; PW22 contains a ointe to the ow of the PW matix that contains the weights used by Level 2, System 2, and so foth. The following fomula was used to comute the ecetual signal fo Level 2, System I: P = l:wlipi, whee the sum is ove the i = I to 4 Level I systems, Wli is the weight assigned to Level I, Pecetual Signal i by Level 2, System I, and Pi is Level I, Pecetual Signal i. The coesonding seadsheet fomula fo this ecetual signal is @INDEX(PW,I,PW21)*PII + @INDEX(PW,2,PW21)*PI2 + @INDEX(PW,3,PW21)*PI3 + @INDEX(PWA,PW21)*PI4. The fomula uses the @INDEX function to access the aoiate ow of weights in the PW matix. Cell P II is the ecetual signal fom Level I, System I; Cell P 12 is the ecetual signal fom Level I, System 2, and so on. Cell PW21 contains a numbe coesonding to the ow in PW to be used in comuting the ecetual signal fo Level 2, System I. The statement @INDEX(PW.I,PW21)*Pll multilies Level I, Pecetual Signal I by the ecetual weight in ow PW21, column I of Matix PW. If the value in PW21 is 3, then the weight comes fom column I of the thid ow of Matix PW. By changing the values in Cell PW21, it is ossible to change the ecetual weights (and, thus, the natue of the ecetion) comuted by this Level 2 system. Level 3. The Level 3 ecetual signals ae logical functions of one o moe Level 2 ecetions. The logical exession fo the ecetual signal comuted by Level 3, System I is if P21 > PH then P31 = I else P31 = -I, whee P21 and PH ae the ecetual signals of Level 2, Systems I and 2, esectively, and P31 is the ecetual signal of Level 3, System I. The seadsheet eesentation of this fomula is @IF (P21 >P22,I,-I),
354 MARKEN and it is located in Cell P31. P21 and P22 ae ecetual signals comuted by Level 2, Systems 1 and 2, esectively. The value of the function is 1 if P21 is geate than P22; othewise, it is -1. Othe systems at Level 3 also use the @IF function to detemine whethe o not othe elationshis (>, <', =, <» hold between vaious Level 2 ecetions. Refeence Signal Calculation Oututs fom seveal highe level systems ae combined to fom the efeence signal fo a single lowe level system. The contibution of a aticula highe level system to the efeence signal of a lowe level system deends on how that lowe level system's ecetual signal contibutes to the ecetual signal of the highe level system (Powes, 1979a). The basic ule is that the sign of the highe level system's contibution to the lowe level system's efeence signal must be the same as the sign of the lowe level system's contibution to the highe level system's ecetual signal. Level 1. The efeence signal fo each Level 1 system is a weighted combination of the oututs fom the Level 2 systems. The @INDEX function is again used to detemine the weights assigned to the highe level system's contibution to each Level 1 efeence signal. The following fomula was used to comute the efeence signal fo Level 1, System I: whee the sum is ove the i = I to 4 Level 2 systems, Wit is the weight assigned to Level 1, Pecetual Signal 1 by Level 2, System i, and OJ is the Level 2 outut vaiable fom System i. The coesonding seadsheet fomula fo this efeence signal is @INDEX(PW,1,PW21)*021 + @INDEX(PW, 1,PW22)*022 @INDEX(PW, 1,PW23)*023 + @INDEX(PW,1,PW24)*024. The cells 021 though 024 ae the oututs of Level 2, Systems 1 though 4, esectively. Cells PW21 though PW24 contain the PW ow indexes of the weights given to the Level 1, System 1 ecetual signal by Level 2, Systems 1 though 4, esectively. The efeence signals fo othe Level 1 systems use the same fomula; only the second agument of the @INDEX function (l in this case) is changed-to 2 fo System 2, 3 fo System 3, and 4 fo System 4. Level 2. The efeence signals fo the Level 2 systems deend on how the Level 2 ecetual signal was used in the comutation of the Level 3 ecetual signal. Level 3 ecetual signals eesent elationshis between Level 2 ecetions. In ode to eseve negative feedback, the following ule was used to detemine how the Level 3 outut vaiables contibute to the Level 3 efeence signals: If an incease in the Level 2 ecetion oduces an incease in the Level 3 ecetion, then the sign of the Level 3 outut connection to the Level 2 efeence signal is ositive; if an incease in the Level 2 ecetion oduces a decease in the Level 3 ecetion, then the sign of the Level 3 outut connection to the Level 2 efeence signal is negative. Fo examle, if the Level 3 ecetion is @IF(P21>P22,1,-I), an incease in ecetion P21 (Level 2, System 1) will tend to make the Level 3 ecetion incease (towad 1). Thus, the Level 3 outut contibution to the Level 2, System 1 efeence signal is ositive. Level 3. The Level 3 efeence signal values ae enteed as constants by the use. In a eal system, these signals would be genetically detemined (if the hieachy had only thee levels) o ovided by even highe level systems that eceive vaiables that ae functions of the Level 3 ecetual signals. Any value can be enteed fo the Level 3 efeence signals, but, because of the way in which the Level 3 ecetual signals ae comuted, it only makes sense to ente values of 1 o -1. The Comlete Hieachy The comlete seadsheet model of a contol system hieachy is shown in Figue 5. Thee ae thee levels of contol systems with fou systems at each level. The fist two columns of the dislay contain the values of GAIN and SLOW fo systems at each level. Note that the highe level systems (towad the to of the seadsheet) have smalle values fo SLOW than the lowe level systems, making the highe level systems esond moe slowly than lowe level systems. Because the highe level systems esond moe slowly, thei GAIN can be lage. The thid column of the seadsheet contains the labels of the signals and vaiables in contol systems at each level of the hieachy. R, P, and 0 identify the efeence signal, ecetual signal, and outut vaiable, esectively. Numbes in aentheses identify the level and system numbe of each signal and vaiable. Thus, R(2,i) identifies the efeence signal fo Level 2, System i (the system identification numbes, i, which go fom I to 4, ae found in the to ow of the seadsheet). The numbes in the fou cells to the ight of each identifying label ae the momentay values of the signals and vaiables in the fou contol systems at each of the thee levels of the hieachy. The column labeled "aveage eo" gives the aveage value of the eo signal at each level of the hieachy. Each aveage is taken ove the fou eo signals at one level of the hieachy. A single hoizontal line seaates the contol system hieachy fom the hysical envionment. The fist ow of fou numbes below this line (labeled "inut vaiable") eesent stimulation at the sensos of the Level 1 contol systems. This stimulation is caused by hysical vaiables that ae outside the contol hieachy-the ow 1abeled "distubances" -as well as by actions of the contol hieachy itself (the Level 1 oututs). The fomula fo the inut vaiable fo Level 1, System i is i li = Oli+dj, whee i li is the inut to Level 1, System i, 0li is the out-
Sys1em (i) 1 2 3 4 SPREADSHEET CONTROL MODEL 355 Delay Gain R(3,i) -1 1 1 1 Aveage Level 3 P(3,i) -1 1 1 1 Eo 0.00 le-os 500 0(3,i) -11.6 38.58 58.58 74.21 R(2,i) -11.6 50.19 20 15.63 Aveage Level 2 P(2,i) -11.8 49.97 20.17 15.63 Eo 0.14 le-04 150 0(2,i) 21.86 17.98 0.53 21.25 R(1,i) 3.88 19.12 17.9-16.8 Aveage Level 1 P(l,i) 3.87 19.1 17.9-16.8 Eo 0.01 0.001 70 0(1,i) 53.87-69.1 67.89-33.2 Inut Vaiable I 3.87 19.11 17.89-16.8 Distubance D -50 50-50 50 Test Ve.ne.ble 12 Welghts -1 1-1 -1 Stability Facto 327.2 Behavio 19.49 Figue 5. Dislay of a thee-level hieachy of contol systems imlemented in the seadsheet, ut of that system, and d, is the distubance to Inut i. The seadsheet vesion of this fomula, contained in the aoiate inut vaiable cell, is +0 +D. The inut vaiable is a esult of the combined effects of a distubance, D, and a Level I system outut, O. Fo examle, the inut vaiable might be the intensity of sound at the sensoy suface. The value of this vaiable deends on the intensity of sound souces in the envionment (D) as well as on the actions of the behaving systems (movements elative to the sound souces). A oximal inut can be influenced by moe than one distubance and by moe than one system outut. The seadsheet model will wok in these cases as long as at least one of the oututs affecting the inut vaiable comes fom the system contolling that inut. The at of the seadsheet below the ow of distubance values is used fo testing the model. The use of the "test vaiable," "weights," "stability facto," and "behavio " cells is exlained below. It should be noted that this model is not meant to eesent behavio in a aticula situation. The goal is to show that a woking hieachical model can be constucted. Once the basic inciles of the hieachical model ae gased, it should be ossible to develo a hieachical model of some secific behavio, such as "ointing at a taget" o "lifting a glass." RUNNING THE MODEL The following descition of the behavio of the model is best undestood if the eade follows along with the seadsheet model u and unning.? The comleted hieachical contol model consists of the equations defining the efeence signal, ecetual signal, and outut vaiable fo each of the fou contol systems at each level of the hieachy. The numeical values fo GAIN and SLOW, eesenting gain and slowing factos fo each level of the model, must also be set. Some values that oduce stable behavio ae shown in Figue 5. Changing Goals The values of the highest level efeence signals (Level 3) ae the ultimate "goals" that the hieachy of contol systems is woking towad; all the actions of the hieachy ae aimed at oducing the ecetions secified by these Level 3 efeences. Figue 5 shows these efeence values set to -I, I, I, and I. Because of the way in which the elationshi ecetions ae defined, Level 3, System I (with a efeence signal value of -I) will ty to kee P21 less than o equal to P22; Level 3, System 2 will ty to kee P22 geate than P23; Level 3, System 3 will ty to kee P23 geate than P24; and Level 3, System 4 will ty to kee P24 equal to a constant. By changing the settings of the Level 3 efeence signals, it is ossible to change the behavio of the entie hieachy of contol systems. Since the Level 3 efeence signals only make sense as I o -I, the highest level goals of the system can be changed by enteing a new set of Level 3 efeences, with I and -I assigned to diffeent Level 3 systems. The ecetual goals of the Level 3 systems (as secified by the efeence signals to the Level 3 systems) must be achieved in the extenal wold, as eceived at lowe
356 MARKEN levels of the hieachy. The state of the extenal wold is detemined by the value of the envionmental distubances. Any set of fou numbes can be enteed as the distubances. In Figue 5, the distubances ae set to -50, 50, -50, and 50. Once the distubance values ae set, the model can be un by stating the ecalculation ocess, which is initiated in Lotus by essing function key F9. Recalculation iteatively calculates values fo all the equations in the model; each iteation can be thought of as a change in the state of the entie hieachy that occus evey faction of a second. The ecalculation ocess mimics the dynamic behavio of the contol hieachy. Ifthe model is set u coectly, each iteation of ecalculation will bing the contol hieachy close to satisfying the goal of getting all ecetual signals matching all efeence signals. Reaching this goal can take a lage numbe of iteations, esecially if you ae stating the model "fom scatch, " so that thee ae initially vey lage eo signals to be educed. If each iteation is thought of as eesenting a bief slice of time (say,. I sec), then the eo signals ae educed athe quickly. Temoal Resolution By setting the numbe of iteations executed e ecalculation, you can detemine how "fine gained" a look you get at the dynamic behavio of the hieachy. Fo examle, if you set the numbe of iteations to I e ecalculation, then, each time you ess F9, you will see the eo eduction esulting fom each iteation of the contol ocess. If you set the numbe of iteations to 50, then you will see the eo eduction that esults afte evey 50 iteations of the contol ocess. A small numbe of iteations e calculation is obably best when you ae fist unning the model, because it lets you get a good look at the "dynamic" changes in ecetions, efeences, and oututs in the entie hieachy. Each ess of the ecalculation key (F9) oduces a eiod of behavio aimed at binging ecetual signals into coesondence with efeence signals. Ultimately, this is done by vaying the Level I oututs. When unning the model, notice that the lowe ode systems bing thei ecetual signals into coesondence with thei efeence signals almost immediately. Notice also how the Level 2 systems contol thei ecetions by vaying the efeence signals going to the Level I systems. As the un ogesses, the aveage eo at each level of contol will oscillate between high and low values. But the system should eventually stabilize, eaching a steady state with vey little aveage eo at each level of the contol hieachy. Distubance Resistance When the system eaches a steady state, all eos ae at the minimum that can be oduced given the cuent state of envionment (defined by the values of the fou distubances). The system has geneated oututs that, when combined with the evailing distubances, oduce values fo the inut vaiables that satisfy the efeence secifications fo ecetions in all systems at all levels of the contol hieachy. The contol hieachy can continue to match these ecetions with efeence signals even as distubances vay. In the eal wold, distubances vay constantly, due to changes in the system elative to its envionment (as when the system moves elative to a light souce) o due to indeendent changes in the envionment (as when a light dims). The contol hieachy kees all ecetions at thei efeence levels by vaying its oututs aoiately. You can watch the system solve this oblem by tying in new values fo the distubances once the system has eached a faily steady state. The system will quickly (afte a few ecalculations) alte all oututs as is necessay to bing ecetions back into line with efeence signals. This execise shows that a hieachy of contol systems contols its ecetual inuts, not its oututs. The efeence signals cause the ecetual signals (via the closed contol loos) to take on aticula values. Thus, efeence signals ae secifications fo inut, not outut. Oututs deend mainly on envionmental distubances, although some of the outut is used to move inut vaiables to new values when thee is a change in the efeence signal. The model shows why it would be easy fo an outside obseve to inteet the behavio of a contol system in stimulus-esonse o inut-outut tems. Distubances ae the only inuts that can be vaied by an agent extenal to the model. Changes to the distubances lead to damatic changes in the outut of the model. Indeed, the changes in outut ae fa moe noticeable than changes in inut, which ae negligible. Even though these stimulus-esonse elationshis ae damatic and inteesting, they exist because the system is tying to kee vaious inut vaiables in efeence states. Contolling Behavio The contol system model can be used to show how an outside obsevecan contol the behavio of a contol system by maniulating distubances. The celllabeled "behavio" in Figue 5 contains a numbe that is a function of the oututs of the system. The numbe eesents a behavioal vaiable (such as a baess o ating esonse) that could be teated as a deendent vaiable in a standad sychology exeiment. These behavioal vaiables ae a function of many individual system oututs (such as muscle tensions and limb movements). You can do exeiments to see how distubances affect this behavioal vaiable. Just ente values fo the distubances, ess F9 to have the system calculate a esonse, and see the esulting behavio. The distubances can be thought of as stimuli that ae maniulated by the exeimente. They ae the indeendent vaiables in you exeiments. Once you have a easonable idea of how each of the fou distubances affects behavio, you can contol the behavio (that is, bing it to some desied value) by selecting the aoiate dis-
SPREADSHEET CONTROL MODEL 357 tubances. The model shows that this aoach to the contol of behavio woks as long the system doesn't change its highe ode goals. Ty changing the Level 3 efeence signal values and see what haens to the system's behavio. This change in behavio would be inteeted as "andom eo" in standad sychological eseach. The andomness, howeve, is only aaent. It esults fom failue to notice that the system is contolling ecetual vaiables elative to vaying intenal efeences. Conflict and Reoganization Once you see how the contol hieachy woks "as is," you can do some exeiments to see the effect of changes in the hieachy on its behavio. One simle exeiment can be done athe quickly; just change the values of SLOW and GAIN at each level. See what haens when systems at any level esond too aidly (lage value of SLOW) o with too much GAIN. Test the effect of making SLOW the same at each level of the hieachy. See if the GAIN can be the same at all levels. Some of these changes will cause the hieachy to become unstable; the values of signals and vaiables will oveflow thei fomat limits. If this haens, just ead the old vesion of the seadsheet back into the woksace. A aticulaly damatic exeiment involves changing the way one of the Level 2 systems comutes its ecetual signal fom Level 1 ecetions. This is done by changing the efeence to the ow in matix PW fo one of the Level 2 ecetual signal cells. A maco has been witten (invoked by essing ALT R) that automatically changes all fou Level 2 ecetual comutations by andomly selecting new ecetual weights fom the PW matix. Suddenly, the system eceives the wold in a new way. If the new ecetions ae elatively indeendent of one anothe, thee will be no oblem and the hieachy will be able to use the Level 1 systems to contol them. If, howeve, the new ecetions ae simila to one anothe, thee will be conflict. The contol systems will find it imossible to find a set of efeences fo the Level 1 systems that satisfy the efeences fo all the Level 2 ecetions. When you tinke with the aametes and ecetions of the hieachical contol model, you ae laying the ole that Powes (1973) attibutes to the eoganization system. The eoganization system monitos the status of the contol hieachy; it obseves, fo examle, whethe the aveage eo at each level is inceasing o deceasing. The goal of the eoganization system (which is a "neta" contol system) is to kee the oveall level of eo in the contol hieachy small (efeably at zeo). If eo at any level inceases (o is too lage in the fist lace), the eoganization system acts on the contol hieachy by changing contol aametes and/o the way systems eceive the wold. Reoganization does not ty to get the contol hieachy to eceive a aticula asect of the envionment; it does not even know what the contol systems ae eceiving. The eoganization system only knows if thee is "too much" o "just the ight amount" (zeo) of eo in each contol system. Thus, the eoganizing system acts vey much like you do when you ae tinkeing with the contol hieachy. You may not know what the contol systems ae tying to eceive, but you do know (fom the aveage eo) when things ae going wong. The eoganization system is the leaning system of the contol hieachy. You can make leaning at of the seadsheet model by making the eoganization ocess automatic. This can be done by building a contol system that eceives the aveage eo at some level (say, Level 2) of the hieachy and comaes this ecetion to a efeence signal value fixed at zeo. The diffeence is conveted into an outut which affects the value of GAIN o SLOW of the systems at the aoiate level (Level 2, in this case) of the hieachy. Once you get this eoganization system to wok, you will have an adative hieachical contol system-a system that can change its own aametes if it finds itself living in a aticulaly difficult envionment. The develoment of an automatic eoganization system will not be tivial. An attemt to build an efficient eoganization system fo the model is cuently in ogess. The Test fo ContoUed Vaiables If oganisms ae oganized as a hieachy of contol systems-and thee is consideable evidence that this is the case (Albus, 1981; Maken & Powes, 1989; Mille, Galante, & Pibam, 1960)-then we can undestand thei behavio only by leaning what vaiables they ae contolling. The seadsheet model shows that this is not an easy task. Even in a simle envionment consisting of only fou envionmental vaiables (the distubances), the system is contolling many ecetual asects of this envionment. An obseve can see the system affect many diffeent asects of its envionment. You can see, fo examle, that the system is affecting the values of the inut vaiables (which ae in the envionment of the system and the obseve). But it is not easy to see that the system is contolling vaious linea combinations of the inut vaiables. No is it easy to see that the system is contolling elationshis between diffeent linea combinations of the inuts. Thee is a method fo detemining whethe o not the system is contolling a aticula function of a set of envionmental vaiables. It is called the test fo the contolled vaiable (Powes, 1979b). A vesion of this test can be caied out on the seadsheet contol model. You can test whethe o not the system is contolling a aticula linea combination of the inut vaiables. At the bottom of the seadsheet model in Figue 5 is a celllabeled "test vaiable." The numbe in this cell indicates the ow of the PW matix (ow 12 in this case) that will be used as the hyothesized linea weighting of the inut vaiables that is being contolled. The fou weights associated with this test vaiable ae shown in the ow of cells labeled "weights." To test whethe o not this vaiable is contolled, we oduce a sequence of distubances to the fou
358 MARKEN inut vaiables. Ifthe test vaiable is contolled, the distubances will have fa less of an effect on this vaiable than would be exected. The exected effect of the distubances is measued in tems of the exected vaiance of the test vaiable. Ifthe test vaiable is not contolled, its exected vaiance is equal to the sum of the vaiances of the distubances and system oututs. If the test vaiable is contolled, its actual vaiance will be consideably less than exected. The atio of exected to obseved vaiance is called the stability facto. It is a measue of the system's ability to stabilize (contol) an envionmental vaiable against distubance. The stability facto is shown at the bottom of the seadsheet model (Figue 5). Its value will be close to I (exected equals actual vaiance) if the test vaiable is not contolled; the test vaiable is not stabilized. Its value will be close to zeo ifthe test vaiable is actively destabilized; the system inceases the vaiance of the test vaiable. The lage the value of the stability facto, the moe likely it is that the hyothesized contolled vaiable is, indeed, contolled (stabilized against distubance). In the esent case, even test vaiables that ae not contolled will have a stability facto that is consideably geate than 1.0 (often close to 150.(0), because these test vaiables ae vey simila to the contolled vaiables. To do the test oely, seveal diffeent vaiables should be tested. A contolled vaiable will have a stability facto that is two to thee times lage than the stability facto fo an uncontolled vaiable. The stability facto is comuted fo a sequence of 15 sets of distubances by unning a maco that is initiated by essing ALT S. Afte each set of distubances is alied, the model goes though seveal iteations (the default is 50) in ode to bing the inut vaiables to thei efeence states. The actual vaiance of the test vaiable used in the comutation of the stability facto is based on the steady state values of the inut vaiables. Afte a tansient distubance, the contol system equies seveal iteations of calculation to each a steady state. DISCUSSION A method of imlementing a hieachical contol system model on an electonic seadsheet has been descibed in this ae. The seadsheet model makes it ossible to exloe the behavio of a hieachy of contol systems in some detail. A numbe of exeiments ae descibed that can be done to see how the model esonds to changes in the envionment and in its own highe level efeence signals. The seadsheet imlementation makes it elatively easy to change vaious asects of the model, such as the contol aametes and ecetual functions. It is also ossible to exand the model, by adding levels of contol o by adding systems at each level. Alication to an Exeiment The seadsheet model descibed above simulates the behavio of an oganism that is contolling fou intensities (Level I), fou sensations that ae linea combinations of these intensities (Level 2), and fou elationshis (Level 3). It is not a model of behavio in a aticula exeiment. But it is ossible to augment the model to simulate behavio in an exeiment. Fo examle, a vesion of the seadsheet model, with only two levels and two systems at each level, can oduce coodinated behavio like that obseved in exeiments descibed by Maken (1986) and Boubon (1989). The inut and outut values of the seadsheet model can be comaed to the values obseved when subjects contol vaiables such as the osition of and distance between lines on a video dislay sceen. Seadsheet Modeling The electonic seadsheet ovides an excellent envionment fo quickly imlementing and dislaying the esults of woking models of behavio. It was ossible to wite and debug the thee-level hieachical contol model in less than an hou, a ocess that took thee times longe with a moe conventional ogamming envionment (PASCAL with a fast comile). Moeove, once the seadsheet vesion of the model was comleted, it was easy to change (fo examle, by adding systems o levels). It is much moe difficult to make such changes in the conventional language vesion of the ogam. One of the main advantages of the seadsheet aoach to modeling is the eady-made dislay fomat. When develoing the contol hieachy model, you do not need to wite code to dislay model values at aoiate locations on the sceen. One of the main disadvantages of the seadsheet aoach is that the code is un inteetively; the seed of execution of the seadsheet model is faily slow. This would be a significant disadvantage if the goal of the contol system model wee to simulate behavio in the context of continuously changing envionmental distubances, such as those used in many manual tacking exeiments. Simulation ofbehavio in these cicumstances is best left to moe oweful, comiled vesions of the contol hieachy. REFERENCES ALBUS, J. (1981). Bains, behavioand obotics. Petesboough, NH: Byte Books. BoURBON, W. T. (1989). A contol-theoy analysis of intefeence duing social tacking. In W. Heshbege (Ed.), Volitional action: Conation and contol. Amstedam: Noth-Holland. FOWLER, C., & TURVEY, M. (1978).Skillacquisition: An eventaoach with secial efeence to seaching fo the otimum of a function of seveal vaiables. In G. Stelmach (Ed.), Infomation ocessing in motocontoland leaning (. 1-39). New Yok: Academic Pess. LEBwND, G. T., & COBB, D. F. (1985). Using 1-2-3. Indianaolis, IN: Que Cooation. MARKEN, R. S. (1986). Pecetual oganization ofbehavio: A hieachical contol model of coodinated action. Jounalof Exeimental Psychology: Human Pecetion & Pefomance, 12,67-76. MARKEN, R. S. (1989). Behavioin the fist degee. In W. Heshbege (Ed.), Volitional action: Conation and contol. Amstedam: Noth Holland. MARKEN, R. S., & POWERS, W. T. (1989). Levels of intention in be-
SPREADSHEET CONTROL MODEL 359 havio. In W. Heshbege (&I.), Volitional action: Conation and contol. Amstedam: Noth-Holland. MILLER, G. A., GALANTER, E., '" PRtBRAM, K. H. (1960). Plans and the stuctue ofbehavio. New Yok: Holt, Rinehat & Winston. POWERS, W. T. (1973). Behavio: The contol ofecetion. Chicago: Aldine. POWERS, W. T. (l979a, August). The natue of obots: Pat 3. A close look at human behavio. Byte,. 94-116. POWERS, W. T. (l979b, Setembe). The natue of obots: Pat 4. Looking fo contolled vaiables, Byte,. 96-112. POWERS, W. T. (1989). Living contol systems. Gavel Switch, KY: Contol Systems Gou. NOTES I. Fo claity, seadsheet cells ae efeed to by functional names (e.g., 0 and R fo "outut vaiable" and "efeence signal" cells, esectively) athe than seadsheet coodinates (e.g., A15 and A12). 2. The model is available fom the autho as a Lotus 1'2'3 woksheet. The woksheet will be sent uon eceit of a fomatted 5'.4- o 3'h-in. double density o high density disk in a eusable maile with etun ostage. (Manuscit eceived Ail 13, 1990; evision acceted fo ublication July 16, 1990.)