9. MRAC Design for Affine-in-Control MIMO Systems

Transcription

1 Lectue 5 9. MRAC Desg fo Affe--Cotol MIMO Systes Readg ateal: []: Chapte 8, Secto 8.3 []: Chapte 8, Secto 8.5 I ths secto, we cosde MRAC desg fo a class of ult-put-ult-output (MIMO) olea systes whose plat dyacs s lealy paaetezed, the ucetates satsfy the so-called atchg codtos, ad f the full state s easuable, (.e., avalable o-le as the syste output). Moe specfcally, cosde the th ode MIMO syste the fo: ( ) = A+ BΛ u+ f (9.) whee R s the syste state, u R s the cotol put, B R s kow at, A R ad Λ R ae ukow atces. I addto, t s assued that Λ s dagoal, ts eleets λ ae o-egatve, ad the pa ( A, BΛ ) s cotollable. he ucetaty Λ s toduced to odel a cotol falue pheoeo. Moeove, the ukow possbly olea fucto : f R R epesets the socalled syste atched ucetaty. It s assued that the fucto ca be wtte as a lea cobato of kow bouded bass fuctos wth ukow costat coeffcets. I (9.), kow egesso vecto. f =Θ Φ (9.) Θ R s the ukow costat at, whle Φ R epesets the he cotol objectve of the MIMO tackg poble s to choose the put vecto u such that all sgals the closed-loop syste ae bouded ad the state follows the state R of a efeece odel specfed by the LI syste ef whee A R ef s Huwtz, ef ef ef ef = A + B t (9.3) B R t ef, ad R s a bouded efeece put vecto. ote that the efeece odel ad ts eteal put ( t ) ust be chose so that () t epesets a desed tajectoy that ( t ) has to follow. I othe wods, the cotol ef

2 put u eeds to be chose such that the tackg eo vecto asyptotcally teds to zeo. l t t = 0 (9.4) t If the atces A ad Λ wee kow, oe could apply the cotol law ad obta the closed-loop syste u = K + K Θ Φ (9.5) ( ) = A+ BΛ K + BΛK (9.6) Copag (9.6) wth the desed dyacs (9.3), t follows that the deal (ukow) at gas ust be chose to satsfy the so-called atchg codtos: A+ BΛ K = A BΛ K = Bef ef (9.7) Assug that the atchg codtos take place, t s easy to see that the closed-loop syste s the sae as that of the efeece odel, ad cosequetly, asyptotc (epoetal) tackg s acheved fo ay bouded efeece put sgal () t. Reak 9. Gve the atces A, B, Λ, Aef, Bef, o K, K ay est to satsfy the atchg codtos (9.7) dcatg that the cotol law (9.5) ay ot have eough stuctual fleblty to eet the cotol objectve. Ofte pactce, the stuctue of A s kow, ad the efeece odel atces Aef, B ef ae chose so that (9.7) has a soluto fo K, K. Assug that K, K (9.7) est, cosde the followg cotol law: u = K + K Θ Φ (9.8) whee,, K R K R Θ R ae the estates of the deal ukow atces K, K, Θ, espectvely. he estated atces wll be geeated o-le ad by a appopate adaptve law.

3 Substtutg (9.8) to (9.), the closed-loop syste dyacs ca be wtte. ( ) ( ) = A+ BΛ K + BΛ K Θ Θ Φ (9.9) Subtactg (9.3) fo (9.9), closed-loop dyacs of the desoal tackg eo et = t t ca be obtaed. vecto () () () ef ( ) ( ) e = A+ BΛ K + BΛ K Θ Θ Φ A B (9.0) Usg atchg codtos (9.7) futhe yelds: ef ef ef ( ( )) ( ) ( ef ef ef ) ( ) ( ) ( ef ) e = A + BΛ K K A + BΛ K K BΛ Θ Θ Φ = A e+ BΛ K K + K K Θ Θ Φ (9.) Let K = K K, K = K K, ad Θ=Θ Θ epeset the paaete estato eos. I tes of the latte, the tackg eo dyacs becoes: e = Aef e+ BΛ K + K Θ Φ (9.) Vecto ad at os Befoe poceedg ay futhe, ecall that gve a at o s defed by A= a R j, the Fobeus j A = t A A = a (9.3) F wth t the tace opeato. O the othe had, gve ay vecto p-o, the duced at o s defed by, j A p A p = sup (9.4) 0 p Collecto of Facts about vecto ad at os, (pove t). Fo vecto -o =, the duced at o s equal to the au = absolute colu su, that s: A = a aj. j = 3

4 Fo vecto -o =, the duced at o s equal to the au = sgula value of A, that s: A σ ( A) Fo vecto -o absolute ow su, that s: =. a = a, the duced at o s equal to the au A = a a. j= he duced at o satsfes: A A, ad fo ay two copatbly p p p desoed atces, A ad B, oe also has: AB A B. p p p he Fobeus o s ot a duced o of ay vecto o, but t s copatble wth the -o the sese that: A A. F Fo ay two copatbly desoed atces A ad B, the Fobeus e poduct s defed as: AB, = A B. F Accodg to the Schwatz equalty oe has: AB, = A B A B. j F F F F F Fo ay two co desoal vectos a ad b, the tace detty takes place: a b= ba. t Let Γ =Γ > 0, Γ =Γ > 0, Γ Θ =Γ Θ > 0. Gog back to aalyzg the tackg eo dyacs (9.), cosde the Lyapuov fucto caddate: (,,, Θ ) = + t( Γ + Γ + Θ ΓΘ Θ Λ) V e K K e Pe K K K K (9.5) whee P= P > 0 satsfes the algebac Lyapuov equato PA + A P= Q (9.6) ef ef fo soe Q= Q > 0. he the te devatve of V, evaluated alog the tajectoes of (9.), ca be calculated. 4

5 V e Pe e Pe t = + + K Γ K + K Γ K + Θ ΓΘ Θ Λ ( ( )) ef ep Aef e B ( K K ) = A e+ BΛ K + K Θ Φ Pe + + Λ + Θ Φ t + K Γ K + K Γ K + Θ ΓΘ Θ Λ ( ef ef ) = e A P+ PA e+ e PBΛ ( K + K Θ Φ ) t K K K K + Γ + Γ + Θ ΓΘ Θ Λ Usg (9.6), yelds: ( ) ( Θ ) V = e Qe+ e PBΛ K + t K Γ K Λ + epbλ K + t K Γ K Λ + epbλ Θ Φ + t Θ Γ ΘΛ (9.7) (9.8) Usg the tace detty, oe gets epbλ K = t K epbλ b b a a epbλ K = t K epbλ b b a a epbλ Θ Φ = t Θ Φ( epb ) Λ a b b a (9.9) Substtutg (9.9) to (9.8), futhe yelds V e Qe t K = + K e PB Γ + Λ t t K K e PB + Γ + Λ + Θ ΓΘ Θ Φ e PB Λ (9.0) Adaptve laws ae chose as follows: 5

6 K = Γ e PB K = Γ () t e PB Θ=Γ Θ Φ epb (9.) he the te-devatve of V becoes egatve se-defte. V = e Qe 0 (9.) heefoe the closed-loop eo dyacs s stable, that s the tackg eo et ad the paaete estato eos K ( t), K ( t), Θ ( t) ae bouded sgals te. heefoe, the paaete estates K ( t ), K ( t ), Θ ( t) ae also bouded. Sce ( t ) s bouded the ef () t ad ef () t ad, cosequetly the cotol put s bouded ad, hece et () ae bouded. Hece, the syste state () t s bouded u t (9.8) s bouded. he latte ples that ( t) s bouded. Futheoe, the d te devatve of V () t V = e Qe= e Qe (9.3) s bouded ad thus V () t s a ufoly cotuous fucto of te. he latte coupled V t s lowe bouded ad V ( t) 0 ples (Babalat s Lea) that wth the facts that lv () t = 0 t. hus l et () = 0 ad the MIMO tackg poble s solved. t Reak 9. (pove t) If soe of the dagoal eleets λ of the ukow dagoal at Λ ae egatve ad the sgs of all of the ae kow, the the adaptve laws K = Γ e PBsg Λ K () t e PBsg = Γ Λ Θ =Γ Θ Φ e PBsg Λ (9.4) solve the MIMO tackg poble, whee [ λ λ ] sg dag sg,, sg Λ=. 0. Atfcal eual etwoks fo Fucto Appoato Motvato. 6

7 A typcal cotol desg pocess stats wth odelg whch s bascally the pocess of costuctg a atheatcal descpto (such as a set of ODE-s) fo the physcal syste to be cotolled. ote that oe accuate odels ae ot always bette. hey ay eque uecessaly cople cotol desg ad aalyss ad oe deadg coputato. he key hee s to odel essetal effects the syste dyacs the opeatg age of teest. I addto, a good odel should also povde soe chaactezato of the odel ucetates the so-called ukow ukows, whch ca be used fo obust desg, adaptve desg, o eely sulato ad syste testg, (such as Mote Calo us). Model ucetates ae the dffeeces betwee the odel ad the eal physcal pocess. Ucetates paaetes ae called paaetc, whle the othes ae called o-paaetc ucetates. Eaple 0. Fo the odel of a cotolled ass = u the ucetaty s paaetc, whle the eglected oto dyacs, easueet ose, seso dyacs epeset the opaaetc ucetates. Eaple 0. Cosde the scala odel wth uceta dyacs = f + u, whee f kow. Suppose that = θϕ + ε = θ Φ + ε f = Paaetc Ucetaty o-paaetc s ot that s suppose that the ukow fucto f ( ) ca be appoated by a lea cobato of kow bass fuctos ϕ ad ukow costat paaetes θ. he appoato eo ε s the o-paaetc ucetaty, whle the ukow costat paaetes θ epeset the paaetc ucetaty the syste dyacs. I ode to Φ chaacteze the latte, oe eeds to be able to fd a good set of bass fuctos such that the appoato eo ε becoes sall o a copact doa. Polyoals, Foue sees epasos, sples ad feedfowad eual etwoks ca be used to appoate fuctos o copact doas. I what follows, we show how to adapt to paaetc ucetates, whle ata obustess the pesece of o-paaetc ucetates. Defto 0. Atfcal Feedfowad eual etwoks ae ult-put-ult-output systes coposed of ay te-coected olea pocessg eleets (euos) opeatg paallel. Fgues 0. ad 0. show sketches of two feedfowad -s. 7

8 4 etwok Iput 5 etwok Output Iput ode 3 6 Output Fst Hdde Laye Secod Hdde Laye Fgue 0.: Feedfowad eual etwok wth hdde layes ad 6 euos Fgue 0.: Feedfowad eual etwok wth hdde laye ad 5 euos As see fo the eaples, a atfcal feedfowad eual etwok cossts of euos ad the coectos. A block-daga of a euo s show below. Σ y Fgue 0.3: Atfcal euo Block-Daga euos, the basc pocessg eleets of -s, have two a copoets: a weghted sue a olea actvato fucto 8

9 he actvato fuctos of teest ae Radal Bass Fuctos o dge fuctos, (ofte called the sgods). Defto 0. A Radal Bass Fucto (RBF) s defed as a Gaussa: I (0.), R s the put, (, ) c ( c) W( c) c W ϕ = e = e (0.) c R s the cete, ad W W 0 ϕ, c = > s a postve-defte = ϕ to abbevate syetc at of weghts. Most ofte we wll wte ad deote a RBF whch s ceteed at the th cete. Reak 0. Othe deftos of a RBF ae avalable. Ofte the lteatue, a RBF s defed as φ = φ( c W ), whee deotes the usual Eucldea weghted o. I addto, t W s equed that φ () s tegable o R ad φ d 0. Bascally, ths type of RBF depeds oly o the weghted dstace = c betwee ts cuet put ad the W cete c. he Gaussa RBF (0.) s a eaple of ths type of actvato fucto. Othes clude: ultquadcs vese ultquadcs R ϕ = + c, c> 0 ϕ =, c> 0 ( + c ) Mcchell s heoe ϕ = ϕ be the Gaussa, the ultquadcs, o the vese ultquadcs fucto. Let Let { } whose (, ) R. he the = be a set of dstct pots tepolato at Φ, th ϕ = ϕ, s osgula. j eleet s j ( j ) Reak 0. hee s a lage class of RBF-s that s coveed by Mcchell s theoe. he theoe povdes theoetcal bass fo RBF based fucto appoato pobles. I othe wods, usg a RBF ϕ = ϕ ad a fte set of pots { } = R, t s always 9

10 such possble to appoate a lage class of fuctos f ( ) wth f = θϕ( ) that f = f, fo all { } =. Defto 0.3 A dge fucto / sgod s a olea fucto the fo: ( w b) σ = σ + (0.) whee w R σ s a olea fucto (ot ecessaly cotuous) defed R wth the followg popetes: deotes the vecto of weghts, b s a scala called the theshold, ad l σ s lσ s ( s) ( s) < < (0.3) he ost coo eaples of a dge fucto ae: a) the logstc sgod ad b) the hypebolc taget σ σ ( s) ( s) = (0.4) s + e s e = (0.5) s + e A feedfowad wth euos ts hdde laye s show Fgue 0.4. Hdde Laye of euos Iput heshold y Output Bas Fgue 0.4: Sgle-hdde-laye feedfowad wth euos 0

11 Foally, a feedfowad aps R to R, that s: y =, R, y R (0.6) Defto 0.4 A sgodal feedfowad s: = W V + + b σ ( θ) (0.7) whee W R s the at of the oute-laye weghts, σ = σ V + θ σ V + θ R () ( ) ( ) s the vecto of sgods, V R s the at of the e-laye syaptc weghts wth ts th colu deoted by V R, θ R s the vecto of thesholds, ad b R deotes the bas vecto. Defto 0.5 A feedfowad RBF s: ϕ ( C W ) ( θ ) ϕ = θ + b = b =Θ Φ ϕ ϕ ( C Θ W ) Φ (0.8) + Θ= b R s the vecto of weghts, ( ) whee ( θ ) eceptve feld, W = W > 0 s the o weghtg at, b ( ) + ϕ ϕ R C R s the cete of the th R s the bas, ad Φ = s the so-called egesso vecto, whose copoets ae the bass actvato fuctos ϕ ϕ( C ) fucto. = ad the uty W Reak 0.3 Ofte pactcal applcatos, the syetc postve-defte at W (0.8) s chose to be dagoal ad the fo:

12 W =, =,, σ whee σ epesets the wdth of the th Gaussa fucto, that s: ϕ = e C σ copoet of the egesso vecto becoes the th Φ (0.8). Most ofte, the copoets of the egesso wll be costucted usg a sotopc Gaussa fucto ϕ = e C d a whose stadad devato (.e., wdth) σ s fed accodg to the spead of the cetes C, s the ube of cetes, ad d a s the au dstace betwee the chose cetes. I ths case, the stadad devato σ of all the sotopc Gaussa RBF copoets s fed at σ = d a hs foula esues that the dvdual RBF-s ae ot too peaked o too flat. Both of these two etee codtos should be avoded. Feedfowad -s have bee show to be capable of appoatg geec classes of fuctos, cludg cotuous ad tegable oes, o a copact doa ad to wth ay toleace. hs popety of feedfowad -s s ofte efeed to as the Uvesal Appoato popety, whle the -s theselves ae ofte called the uvesal appoatos. wo elated theoes ae gve below. Uvesal Appoato heoe fo Sgodal -s, (G. Cybeko, 989) Ay cotuous fucto f : R R ca be ufoly appoated by a sglehdde-laye wth a bouded ootoe-ceasg cotuous actvato fucto ad o a copact doa X R, that s: Reak 0.4 ε > 0 WbV,,,, θ X R W σ ( V + θ) + b f ε (0.9)

13 he uvesal appoato theoe eteds to the class of L fuctos o a copact doa. I that case, t s assued that the actvato fucto s a bouded easuable sgod ad the appoato s udestood tes of the L o. Rate of Appoato heoe fo Sgodal -s, (. Bao, 993) Cosde a class of fuctos f ( ) o R fo whch thee s a Foue epesetato of the fo ω f = e f ( ω ) dω fo soe cople-valued fucto f ( ω ) fo whch ω f ( ω ) C = ω f ( ω) dω < f R R s tegable, ad defe he fo evey fucto f ( ) wth C f fte, ad evey, thee ests a sgodal of the fo (0.7), such that Reak 0.5 Fuctos wth ( ) ( C ) f L f f d C f fte ae cotuously dffeetable o R d. Moeove, the appoato eo s easued by the tegated squaed eo, ( L o), o the ball of adus. Uvesal Appoato heoe fo RBF -s, (Pak ad Sadbeg, 99) Let ϕ : R R be a tegable bouded cotuous fucto ad assue that R ϕ d 0 he fo ay cotuous fucto f ( ) ad ay ε > 0 thee s a RBF wth euos, a set of cetes { } C =, ad a coo wdth σ > 0 such that C f = θϕ =Θ Φ = σ ϕ 3

14 f ( f ) d ε O L = Copaso of sgodal ad RBF -s RBF ad sgodal -s ae uvesal appoatos RBF depeds the Eucldea dstaces betwee the put vecto ad the cetes C. Meawhle, sgodal -s deped o the e poduct of the put vecto wth the syaptc weght vectos V ad based by θ. Sgodal -s povde O ate of appoato whch does ot eplctly deped o the deso of. O the othe had, the ate of appoato fo the RBF -s s of ode O ad, cosequetly t deceases epoetally as the deso of the put vecto ceases. hs pheoeo s called the Cuse of Desoalty, (due to R. Bella). A RBF has local suppot whle a sgod does ot. he local suppot ples leag ad adaptato ablty of RBF -s. Sgodal -s adapt but do t lea. Wth specfc efeece to -s cotol, t s the ablty to epeset olea appgs, ad hece to odel olea systes, whch s the featue to be ost eadly eploted the sythess of olea cotolles. Readg ateal / Refeeces: ) F. Scasell, A.C. so, Uvesal appoato usg feedfowad eual etwoks: A suvey of soe estg ethods, ad soe ew esults, eual etwoks, vol., o., pp. 5-37, 998. ) K.J. Hut, D. Sbabao, R. Zbkowsk, P.J. Gawthop, eual etwoks fo Cotol Systes A Suvey, Autoatca, vol. 8, o. 6., pp. 083-, 99. 3) G. Cybeko, Appoato by supeposto of a sgodal fucto, Math. Cotol Sgals Systes, vol., pp , ) J. Pak, I.W. Sadbeg, Uvesal appoato usg adal-bass-fucto etwoks, eual Coputato, vol. 3, o., pp ) C. A. Mcchell, Itepolato of scatteed data: Dstace atces ad codtoally postve defte fuctos, Costuctve Appoato, vol., pp. -, MRAC fo MIMO Systes wth Ustuctued Ucetates 4

15 I secto 9, we cosdeed affe--cotol MIMO systes the fo ( ) = A+ BΛ u+ f (.) whee R s the syste state vecto, u R s the cotol put, B R s a kow at, A R ad Λ R, (a dagoal at wth postve ukow eleets), ae ukow atces. A MRAC tackg desg was caed out, assug that the atched possbly olea uceta fucto f : R R could be eactly epeseted by a RBF the fo f =Θ Φ wth costat ukow coeffcets Θ R ad a fed kow Φ R. egesso vecto I ths secto, we assue that the ukow fucto f ( ) ca be appoated usg a RBF wth kow bass fuctos, (such as Gaussas wth fed cetes). I effect, usg Uvesal Appoato popety of RBF-s, t s assued that the ukow ϕ ad appg f ( ) ca be appoated by a RBF wth fed euos deal ukow costat weghts at Θ R ε f =Θ Φ + (.) o a copact doa that s: X R, ad to wth a gve appoato toleace a 0 ε, a ε >, ε X (.3) he cotol objectve s to desg a adaptve state feedback cotolle whch guaatees boudedess of all vaables the coespodg closed-loop syste, whle tackg the state R of the desed (ope-loop asyptotcally stable) efeece odel ef ef ef ef ef = A + B t (.4) whch s tu dve by a bouded efeece sgal ( t) R. ote that the MRAC cotolle ust opeate the pesece of the syste stuctued ad ustuctued ucetates, whee the latte s epeseted by the fucto appoato eo vecto ε R, whch satsfes (.3). he MRAC soluto s chose the sae fo as Secto 9: 5

16 u = K + K Θ Φ (.5) whee,, K R K R Θ R ae the adaptve paaetes. I ode fo the soluto to ests the followg odel atchg assuptos ust hold: A+ BΛ K = A BΛ K = Bef ef (.6) I, (.6), K, K ae the deal feedback / feedfowad ga atces. ote that oly estece of the deal gas s assued, whle the kowledge s ot equed. Reak. f =Θ Φ epesets the RBF based o-le fucto appoato. I (.5), Moeove, the coespodg fucto appoato eo f the paaete estato eo Θ. lealy depeds o ( ) ε ε f = f f = Θ Θ Φ = Θ Φ Θ (.7) We ow poceed wth the developet of the adaptve laws. he tackg eo dyacs s obtaed by subtactg (.4) fo (.) ad usg the atchg codtos (.6). It yelds: ( ) ( ) ε ( ) ef ef ef ( ) ( ) ( ef ) ε e = A+ BΛ K + BΛ K Θ Θ Φ + A B ef ef ef ( A B K K ) A B ( K K ) B B ε = + Λ + Λ Λ Θ Θ Φ + Λ = A e+ BΛ K K + K K Θ Θ Φ + o, equvaletly ε (.8) e = Aef e+ BΛ K + K Θ Φ + (.9) Let P= P > 0 be the soluto of the Lyapuov equato. Cosde the Lyapuov fucto caddate PA + A P= Q, Q= Q > 0 (.0) ef ef 6

17 (,,, Θ ) = + t( Γ + Γ + Θ ΓΘ Θ Λ) V e K K e Pe K K K K (.) whee Γ =Γ > 0, Γ =Γ > 0, Γ =Γ > 0 ae the ates of adaptato, ad Θ Θ K = K K K = K K Θ = Θ Θ (.) ae the paaete estato eos. Also (.), t ( ) deotes the tace of a at. he te devatve of V alog the tajectoes of the eo dyacs (.9) s gve by: V e Pe e Pe t K = + + Γ K + K Γ K + Θ ΓΘ Θ Λ ( ( ε )) ef ep Aef e B ( K K ε ) = A e+ BΛ K + K Θ Φ + Pe + + Λ + Θ Φ + t K + Γ K + K Γ K + Θ ΓΘ Θ Λ ( ef ef ) e A P PA e ( ) = + + epbλ K + K Θ Φ + ε t K + Γ K + K Γ K + Θ ΓΘ Θ Λ (.3) Regoupg the tes ad usg (.0) yelds: V = e Qe+ e PBΛε ( Θ ) + epbλ K + t K Γ KΛ + epbλ K + t K Γ K Λ + epbλ Θ Φ + t Θ Γ ΘΛ (.4) Usg the tace detty, oe ca wte: 7

18 epbλ K = t K epbλ b b a a epbλ K = t K epbλ b b a a epbλ Θ Φ = t Θ Φ( epb ) Λ a b b a (.5) Substtutg (.5) to (.4) esults : V = e Qe+ e PBΛε t K + K e PB Γ + Λ t t K K e PB + Γ + Λ + Θ ΓΘ Θ Φ e PB Λ (.6) Reak. If adaptve laws ae chose as Secto 9, that s the the te devatve becoes: K = Γ e PB K = Γ () t e PB Θ=Γ Θ Φ epb ( λ εa ) V = e Qe+ e PBΛε λ Q e + PBΛ ε e = e Q e PBΛ Cosequetly, V < 0 outsde of the copact set the e doa: a E = e R : e PBΛ ε λ a Q Hece, oe ay attept to ague that the tackg eo vecto e s UUB. Ufotuately, sde E othg ca be sad about the adaptve gas K,, K Θ. he adaptve laws ae odfed as follows: 8

19 ( ) ( () ) ( K Φ e PB) K =Γ Poj K, e PB K =Γ Poj K, t e PB Θ=Γ Θ Poj, (.7) whee Poj (, ) deotes the Pojecto Opeato. he opeato aps two ( ) atces Ω= [ θ θ ] R ad Y = [ y y ] R to the ( ) Poj ( Ω, Y ), ad t s defed colu-wse, that s ( Y) ( ( θ y) ( θ y) ) at Poj Ω, = Poj, Poj, (.8) I patcula, the opeato copoets ae: ( θ j y j) Poj, ( θ ) f ( θ ) f ( θ j ) f j j yj y j f ( θ j), f f ( θ j) > 0 yj f ( θ j) > 0 = (.9) y j, f ot whee f ( θ j ): R R s a cove fucto. Gve θ a j the au allowable agtude of the vecto θ j, ad ε j > 0, the fucto defto s gve below. f ( θ j ) θ j θ = (.0) εθ j a j a j Key popetes ad geoetc tepetato of the Pojecto Opeato ae dscussed the et secto. Refeeces. J.J. Slote, W. L, Appled olea Cotol, Petce Hall, H.K. Khall, olea Systes, 3 d Edto, Petce Hall, ew Jesey, S. Hayk, eual etwoks: A Copehesve Foudato, d Edto, Petce Hall, ew Jesey, 99. 9