A Mthemtil Theory of Co-Design Andre Censi 1 rxiv:1512.08055v7 [s.lo] 12 Ot 2016 Astrt One of the hllenges of modern engineering, nd rootis in prtiulr, is designing omplex systems, omposed of mny susystems, rigorously nd with optimlity gurntees. This pper introdues theory of o-design tht desries design prolems, defined s tuples of funtionlity spe, implementtion spe, nd resoures spe, together with fesiility reltion tht reltes the three spes. Design prolems n e interonneted together to rete o-design prolems, whih desrie possily reursive o-design onstrints mong susystems. A o-design prolem indues fmily of optimiztion prolems of the type find the miniml resoures needed to implement given funtionlity ; the solution is n ntihin (Preto front) of resoures. A speil lss of o-design prolems re Monotone Co-Design Prolems (MCDPs), for whih funtionlity nd resoures re omplete prtil orders nd the fesiility reltion is monotone nd Sott ontinuous. The indued optimiztion prolems re multi-ojetive, nononvex, nondifferentile, nonontinuous, nd not even defined on ontinuous spes; yet, there exists omplete solution. The ntihin of miniml resoures n e hrterized s lest fixed point, nd it n e omputed using Kleene s lgorithm. The omputtion needed to solve o-design prolem n e ounded y funtion of grph property tht quntifies the interdependene of the suprolems. These results mke us muh more optimisti out the prolem of designing omplex systems in rigorous wy. I. INTRODUCTION ONE of the gret engineering hllenge of this entury is deling with the design of omplex systems. A omplex system is omplex euse its omponents nnot e deoupled; otherwise, it would e just (simple) produt of simple systems. The design of omplex system is omplited euse of the o-design onstrints, whih re the onstrints tht one susystem indues on nother. This pper is n ttempt towrds formlizing nd systemtilly solving the prolem of odesign of omplex systems with reursive design onstrints. Rooti systems s the prototype of omplex systems: Rootis is the prototypil exmple of field tht inludes heterogeneous multi-domin o-design onstrints. The design of rooti system involves the hoie of physil omponents, suh s the tutors, the sensors, the power supply, the omputing units, the network links, et. Not less importnt is the hoie of the softwre omponents, inluding pereption, plnning, nd ontrol modules. All these omponents indue odesign onstrints on eh other. Eh physil omponent hs SWAP hrteristis suh s its shpe (whih must ontined somewhere), weight (whih dds to the pylod), power (whih needs to e provided y something else), exess het (whih must e dissipted somehow), et. Anlogously, the softwre omponents hve similr o-design onstrints. For exmple, plnner needs stte estimte. An estimtor provides stte estimte, nd requires the dt from sensor, whih requires the presene of sensor, whih requires power. Everything s money to uy or develop or liense. Andre Censi <ensi@mit.edu> is with the Lortory for Informtion nd Deision Systems (LIDS) t the Msshusetts Institute of Tehnology. o-design prolem design prolem design prolem strtion funtionlity to provide design prolem miniml resoures required Figure 1. A design prolem is reltion tht reltes the implementtions ville to the funtionlity provided nd the resoures required, oth represented s prtilly ordered sets. A o-design prolem is the interonnetion of two or more design prolems. An edge in o-design digrm like in the figure represent o-design onstrint: the resoures required y the first design prolem re lower ound for the funtionlity to e provided y the seond. The optimiztion prolem to e solved is: find the solutions tht re miniml in resoures usge, given lower ound on the funtionlity to e provided. Wht mkes system design prolems non trivil is tht the onstrints might e reursive. This is form of feedk in the prolem of design (Fig. 1). For exmple, ttery provides power, whih is used y tutors to rry the pylod. A lrger ttery provides more power, ut it lso inreses the pylod, so more power is needed. Extremely interesting trde-offs rise when onsidering onstrints etween the mehnil system nd the emodied intelligene. For ontrol, typilly etter stte estimte sves energy in the exeution, ut requires etter sensors (whih inrese the nd the pylod) or etter omputtion (whih inreses the power onsumption). Contriution: A Prinipled Theory of Co-Design: This pper desries theory to del with ritrrily omplex o-design prolems in prinipled wy. A design prolem is defined s tuple of funtionlity spe, implementtion spe, nd resoures spe, plus the two mps tht relte n implementtion to funtionlity provided nd resoures required. A design prolem defines fmily of optimiztion prolems of the type find the miniml resoures needed to implement given funtionlity. A o-design prolem is n interonnetion of design prolems ording to n ritrry grph struture, inluding feedk onnetions. Monotone Co-Design Prolems (MCDPs) re the omposition of design prolems for whih oth funtionlity nd resoures re omplete prtil orders, nd the reltion etween funtionlity implemented nd resoures needed is monotone (order-preserving) nd Sott ontinuous. The first min result in this pper (Theorem 1 on pge 9) is tht the lss of MCDPs is losed with respet to interonnetion. The seond min result (Theorem 2 on pge 10) is tht there exists systemti proedure to solve n MCDP, ssuming there is proedure to solve the primitive design prolems. The solution of n MCDP Preto front, or ntihin of miniml resoures n e found y solving lest fixed point itertion in the spe of ntihins. The omplexity of this itertion depends on the struture of the o-design digrm. This pper is generliztion of previous work [1], where
2 the interonnetion ws limited to one yle. A onferene version of this work ppered s [2]. Outline: Se. II rells neessry kground out prtil orders. Se. III defines o-design prolems. Se. IV ontins rief sttement of results. Se. V desries omposition opertors for design prolems. Se. VI shows how ny interonnetion of design prolems n e desried using three omposition opertors (, prllel, feedk). Se. VII desries the invrine of monotoniity property tht is preserved y the omposition opertors. Se. VIII desries solution lgorithms for MCDPs. Se. IX shows numeril exmples. Se. X disusses relted work. Note in (1) the use of insted of, whih might seem more nturl. This hoie will mke things esier lter. In the poset UP, UP, the top is the empty set, nd the ottom is the entire poset P. Order on ntihins: The upper losure opertor mps suset of poset to n upper set. Definition 4 (Upper losure). The opertor mps suset to the smllest upper set tht inludes it: : P(P) UP, S {y P : x S : x y}. II. BACKGROUND We will use si fts out order theory. Dvey nd Priestley [3] nd Romn [4] re possile referene texts. Let P, P e prtilly ordered set (poset), whih is set P together with prtil order P ( reflexive, ntisymmetri, nd trnsitive reltion). The prtil order P is written s if the ontext is ler. If poset hs lest element, it is lled ottom nd it is denoted y P. If the poset hs mximum element, it is lled top nd denoted s P. Chins nd ntihins: A hin x y z... is suset of poset in whih ll elements re omprle. An ntihin is suset of poset in whih no elements re omprle. This is the mthemtil onept tht formlizes the ide of Preto front. Definition 1 (Antihins). A suset S P is n ntihin iff no elements re omprle: for x, y S, x y implies x = y. Cll AP the set of ll ntihins in P. By this definition, the empty set is n ntihin: AP. Definition 2 (Width nd height of poset). width(p) is the mximum rdinlity of n ntihin in P nd height(p) is the mximum rdinlity of hin in P. Miniml elements: Upperse Min will denote the miniml elements of set. The miniml elements re the elements tht re not dominted y ny other in the set. Lowerse min denotes the lest element, n element tht domintes ll others, if it exists. (If min S exists, then Min S = {min S}.) The set of miniml elements of set re n ntihin, so Min is mp from the power set P(P) to the ntihins AP: Min: P(P) AP, S {x S : (y S) (y x) (x = y) }. Mx nd mx re similrly defined. Upper sets: An upper set is suset of poset tht is losed upwrd. Definition 3 (Upper sets). A suset S P is n upper set iff x S nd x y implies y S. Cll UP the set of upper sets of P. By this definition, the empty set is n upper set: UP. Lemm 1. UP is poset itself, with the order given y A UP B A B. (1) Figure 2. A = Min U A U U = A By using the upper losure opertor, we n define n order on ntihins using the order on the upper sets (Fig. 2). Lemm 2. AP is poset with the reltion AP defined y A AP B A B. In the poset AP, AP, the top is the empty set: AP =. If ottom for P exists, then the ottom for AP is the singleton ontining only the ottom for P: AP = { P }. Monotoniity nd fixed points: We will use Kleene s theorem, elerted result tht is used in disprte fields. It is used in omputer siene for defining denottionl semntis (see, e.g., [5]). It is used in emedded systems for defining the semntis of models of omputtion (see, e.g., [6]). Definition 5 (Direted set). A set S P is direted if eh pir of elements in S hs n upper ound: for ll, S, there exists S suh tht nd. Definition 6 (Completeness). A poset is direted omplete prtil order (DCPO) if eh of its direted susets hs supremum (lest of upper ounds). It is omplete prtil order (CPO) if it lso hs ottom. Exmple 1 (Completion of R + to R + ). The set of rel numers R is not CPO, euse it lks ottom. The nonnegtive rels R + = {x R x 0} hve ottom = 0, however, they re not DCPO euse some of their direted susets do not hve n upper ound. For exmple, tke R +, whih is suset of R +. Then R + is direted, euse for eh, R +, there exists = mx{, } R + for whih nd. One wy to mke R +, CPO is y dding n rtifiil top element, y defining R + R + { }, nd extending the prtil order so tht for ll R +. Two properties of mps tht will e importnt re monotoniity nd the stronger property of Sott ontinuity. Definition 7 (Monotoniity). A mp f : P Q etween two posets is monotone iff x P y implies f(x) Q f(y). Definition 8 (Sott ontinuity). A mp f : P Q etween DCPOs is Sott ontinuous iff for eh direted suset D P, the imge f(d) is direted, nd f(sup D) = sup f(d).
3 Remrk 1. Sott ontinuity implies monotoniity. Remrk 2. Sott ontinuity does not imply topologil ontinuity. A mp from the CPO R +, to itself is Sott ontinuous iff it is nonderesing nd left-ontinuous. For exmple, the eiling funtion x x is Sott ontinuous (Fig. 3). Exmple 2 (Motor design). Suppose we need to hoose motor for root from given set. The funtionlity of motor ould e prmetrized y torque nd speed. The resoures to onsider ould inlude the [$], the [g], the input voltge [V], nd the input urrent [A]. The mp exe : I F ssigns to eh motor its funtionlity, nd the mp evl : I R ssigns to eh motor the resoures it needs (Fig. 14). Figure 3. A fixed point of f : P P is point x suh tht f(x) = x. Definition 9. A lest fixed point of f : P P is the minimum (if it exists) of the set of fixed points of f:. lfp(f) = min {x P : f(x) = x}. (2) The equlity in (2) n e relxed to. The lest fixed point need not exist. Monotoniity of the mp f plus ompleteness is suffiient to ensure existene. Lemm 3 ([3, CPO Fixpoint Theorem II, 8.22]). If P is CPO nd f : P P is monotone, then lfp(f) exists. With the dditionl ssumption of Sott ontinuity, Kleene s lgorithm is systemti proedure to find the lest fixed point. Lemm 4 (Kleene s fixed-point theorem [3, CPO fixpoint theorem I, 8.15]). Assume P is CPO, nd f : P P is Sott ontinuous. Then the lest fixed point of f is the supremum of the Kleene sent hin f( ) f(f( )) f (n) ( ). Figure 5. funtionlity speed [rd/s] torque [Nm] implementtions resoures [$] [g] voltge [V] urrent [A] Exmple 3 (Chssis design). Suppose we need to hoose hssis for root (Fig. 6). The implementtion spe I ould e the set of ll hssis tht ould ever e designed (in se of theoretil nlysis), or just the set of hssis ville in the tlogue t hnd (in se of prtil design deision). The funtionlity of hssis ould e formlized s the ility to trnsport ertin pylod [g] nd t given speed [m/s]. More refined funtionl requirements would inlude mneuverility, the rgo volume, et. The resoures to onsider ould e the [$] of the hssis; the totl ; nd, for eh motor to e pled in the hssis, the required speed [rd/s] nd torque [Nm]. Figure 6. funtionlity pylod [g] veloity [m/s] implementtions ll hssis resoures [$] totl [g] motor speed [m/s] motor torque [Nm] III. CO-DESIGN PROBLEMS The si ojets onsidered in this pper re design prolems, of whih severl lsses will e investigted. We strt y defining design prolem with implementtion, whih is tuple of funtionlity spe, implementtion spe, nd resoures spe, together with two mps tht desrie the fesiility reltions etween these three spes (Fig. 4). Definition 10. A design prolem with implementtion (DPI) is tuple F, R, I, exe, evl where: F is poset, lled funtionlity spe; R is poset, lled resoures spe; I is set, lled implementtion spe; the mp exe: I F, mnemonis for exeution, mps n implementtion to the funtionlity it provides; the mp evl: I R, mnemonis for evlution, mps n implementtion to the resoures it requires. hssis ville t ServoCity.om 1) Querying DPI: A DPI is model tht indues fmily of optimiztion prolems, of the type Given lower ound on the funtionlity f, wht re the implementtions tht hve miniml resoures usge? (Fig. 7). Prolem 1. Given f F, find the implementtions in I tht relize the funtionlity f (or higher) with miniml resoures, or provide proof tht there re none: using i I, Min R r, (3) s.t. r = evl(i), f F exe(i). Figure 4. funtionlity implementtions resoures Figure 7. funtionlity implementtions resoures
4 Remrk 3 (Miniml vs lest solutions). Note the use of Min R in (3), whih indites the set of miniml (nondominted) elements ording to R, rther thn min R, whih would presume the existene of lest element. In ll prolems in this pper, the gol is to find the optiml trde-off of resoures ( Preto front ). So, for eh f, we expet to find n ntihin R AR. We will see tht this formliztion llows n elegnt wy to tret multi-ojetive optimiztion. The lgorithm to e developed will diretly solve for the set R, without resorting to tehniques suh s slriztion, nd therefore is le to work with ritrry posets, possily disrete. Remrk 4 (Dul formultion). In n entirely symmetri fshion, we ould fix n upper ound on the resoures usge, nd then mximize the funtionlity provided (Fig. 8). The formultion is entirely dul, in the sense tht it is otined from (3) y swpping Min with Mx, F with R, nd exe with evl. Figure 8. using i I, Mx F f, s.t. f = exe(i), r R evl(i). funtionlity implementtions resoures 2) The funtionlity-to-miniml resoures mp h: It is useful to lso desrie design prolem s mp from funtionlity to sets of resoures tht strts over implementtions. (A useful nlogy is the stte spe representtion vs the trnsfer funtion representtion of liner time-invrint system: the stte spe representtion is riher, ut we only need the trnsfer funtion to hrterize the input-output response.) Definition 11. Given DPI F, R, I, exe, evl, define the mp h : F AR tht ssoites to eh funtionlity f the ojetive funtion of Prolem 1, whih is the set of miniml resoures neessry to relize f: h : F AR, f Min R {evl(i) (i I) (f exe(i))}. If ertin funtionlity f is infesile, then h(f) =. Figure 9. (4) Figure 10. speed torque By onstrution, h is monotone (Def. 7), whih mens tht f 1 F f 2 h(f 1 ) AR h(f 2 ), where AR is the order on ntihins defined in Lemm 2. Monotoniity of h mens tht if the funtionlity f is inresed the ntihin of resoures will go up in the poset of ntihins AR, nd t some point it might reh the top of AR, whih is the empty set, mening tht the prolem is not fesile. 3) Co-design prolems: A grphil nottion will help resoning out omposition. A DPI is represented s ox with nf green edges nd nr red edges (Fig. 11). Figure 11. funtionlity funtionlity design prolem resoure resoure This mens tht the funtionlity nd resoures spes n e ftorized in nf nd nr omponents: F = nf i=1 π if i, R = nr j=1 π jr, where π i represents the projetion to the i-th omponent. If there re no green (respetively, red) edges, then nf (respetively, nr) is zero, nd F (respetively, R) is equl to 1 = { }, the set ontining one element, the empty tuple. These o-design digrms re not to e onfused with signl flow digrms, in whih the oxes represent oriented systems nd the edges represent signls. A o-design prolem will e defined s multigrph of design prolems. Grphilly, one is llowed to onnet only edges of different olor. This interonnetion is indited with the symol in rounded ox (Fig. 12). Figure 12. The semntis of the interonnetion is tht the resoures required y the first DPI re provided y the seond DPI. This is prtil order inequlity onstrint of the type r 1 f 2. Definition 12. A Co-Design Prolem with Implementtion (CDPI) is tuple F, R, V, E, where F nd R re two posets, nd V, E is multigrph of DPIs. Eh node v V is DPI v = F v, R v, I v, exe v, evl v. An edge e E is tuple e = v 1, i 1, v 2, j 2, where v 1, v 2 V re two nodes nd i 1 nd j 2 re the indies of the omponents of the funtionlity nd resoures to e onneted, nd it holds tht π i1 R v1 = π j2 F v2 (Fig. 13). Figure 13. Exmple 4. In the se of the motor design prolem, the mp h ssigns to eh pir of speed, torque the hievle trde-off of,, nd other resoures (Fig. 10). The ntihins re depited s ontinuous urves, ut they ould lso e omposed y finite set of points. A CDPI is equivlent to DPI with n implementtion spe I tht is suset of the produt v V I v, nd ontins only the tuples tht stisfy the o-design onstrints. An implementtion tuple i v V I v elongs to I iff it respets
5 ll funtionlity resoures onstrints on the edges, in the sense tht, for ll edges hhv1, i1 i, hv2, j2 ii in E, it holds tht yle in the o-design grph extr pylod hssis design prolem πi1 evlv1 (πv1 i) πj2 exev2 (πv2 i). The posets F, R for the entire CDPI re the produts of the funtionlity nd resoures of the nodes tht remin unonneted. For node v, let UFv nd URv e the set of unonneted funtionlities nd resoures. Then F nd R for the CDPI re defined s the produt of the Q unonneted funtionlity Q nd resoures of ll DPIs: F = v V j UFv πj F v nd Q Q R = π R. The mps exe, evl return the i v v V i URv vlues of the unonneted funtionlity nd resoures: Y Y exe : i 7 πj exev (πv i), veloity Y Y totl Figure 18. This formlism mkes it esy to strt wy the detils in whih we re not interested. One digrm like Fig. 18 is otined, we n drw ox round it nd onsider the strted prolem (Fig. 19). hssis + motor o-design prolem veloity πi evlv (πv i). voltge urrent speed v V j UFv evl : i 7 motor design prolem torque Figure 19. v V i URv extr pylod totl voltge urrent Exmple 5. Consider the o-design of hssis (Exmple 3) plus motor (Exmple 2). The design prolem for motor hs speed Let us finish ssemling our root. A motor needs motor nd torque s the provided funtionlity (wht the motor must ontrol ord. The funtionl requirements re the (pek) output provide), nd,, voltge, nd urrent s the required urrent nd the output voltge rnge (Fig. 20). resoures (Fig. 14). speed [rd/s] Figure 14. torque [Nm] motor design prolem output urrent [A] [$] [g] voltge [V] urrent [A] Figure 20. output voltges [V] motor ontroller ord [$] [g] input voltge [V] input urrent [A] For the hssis (Fig. 15), the provided funtionlity is prme- The funtionlity for power supply ould e prmeterized y terized y the of the pylod nd the pltform veloity. the output urrent, the output voltges, nd the pity. The The required resoures inlude the, totl, nd wht resoures ould inlude nd (Fig. 21). the hssis needs from its motor(s), suh s speed nd torque. output urrent [A] pylod [g] Figure 15. [$] totl [g] required motor speed [rd/s] required motor torque [Nm] hssis design prolem veloity [m/s] output voltges [V] Figure 21. power supply unit pity [J] [$] [g] Reltions suh s urrent voltge power required nd yle in the o-design grph The two design prolem n e onneted t the edges for power endurne energy required n e modeled y torque nd speed (Fig. 16). The semntis is tht the motor trivil multiplition DPI (Fig. 22). needs to hve t lest the given torque nd speed. torque speed pylod Figure 16. veloity hssis design prolem speed voltge [V] motor design prolem voltge urrent totl Figure 22. We n onnet these DPs to otin o-design prolem with funtionlity voltge, urrent, endurne nd resoures -V nd (Fig. 23). voltge rnge (V -V Resoures n e summed together using trivil DP orresponding to the mp h : hf 1, f 2 i 7 {f 1 + f 2 } (Fig. 17). Figure 17. [V] [A] Figure 23. totl power [W] urrent [A] MCB [V] [A] PSU [$] [g] [J] endurne A o-design prolem might ontin reursive o-design onstrints. For exmple, if we set the pylod to e trnsported to e the sum of the motor plus some extr pylod, Drw ox round the digrm, nd ll it MCB+PSU ; then yle ppers in the grph (Fig. 18). interonnet it with the hssis+motor digrm in Fig. 24. -V -V -V -V voltge rnge (V -V)
6 extr pylod Figure 24. veloity endurne extr power hssis + motor [V] [A] [g] [$] MCB + PSU We n further strt wy the digrm in Fig. 24 s moility+power CDPI, s in Fig. 25. The formlism llows to onsider nd s independent resoures, mening tht we wish to otin the Preto frontier for the miniml resoures. Of ourse, one n lwys redue everything to slr ojetive. For exmple, onversion from to exists nd it is lled shipping. Depending on the destintion, the onversion ftor is etween $0.5/ls, using USPS, to $10k/ls for sending your root to low Erth orit. Figure 25. veloity mission time extr pylod extr power moility + power [$] [$] [$] shipping Exmples from the literture: Mny reent works in rootis nd neighoring fields tht del with minimlity nd resoure onstrints n e inorported in this frmework. Exmple 6. Svorenov et l. [7] onsider joint sensor sheduling nd ontrol synthesis prolem, in whih root n deide to not perform sensing to sve power, given performne ojetives on the proility of rehing the trget nd the proility of ollision. The method outputs Preto frontier of ll possile operting points. This n e st s design prolem with funtionlity equl to the proility of rehing the trget nd (the inverse of) the ollision proility, nd with resoures equl to the tution power, sensing power, nd sensor ury. [$] Figure 27. e r / g t gt [ er e e g [ e Other exmples in miniml rootis: Mny works hve sought to find miniml designs for roots, nd n e understood s hrterizing the reltion etween the poset of tsks nd the poset of physil resoures, whih is the produt of sensing, tution, nd omputtion resoures, plus other nonphysil resoures, suh s prior knowledge (Fig. 28). Given tsk, there is miniml ntihin in the resoures poset tht desries the possile trde-offs (e.g., ompensting lousier sensors with more omputtion). Figure 28. t t s t t t t ps The poset struture rises nturlly: for exmple, in the sensor lttie [9], sensor domintes nother if it indues finer prtition of the stte spe. Similr dominne reltions n e defined for tution nd omputtion. O Kne nd Lvlle [10] define root s union of rooti primitives, where eh primitive is n strtion for set of sensors, tutors, nd ontrol strtegies tht n e used together (e.g., ompss plus ontt sensor llow to drive North until wll is hit ). The effet of eh primitive is modeled s n opertor on the root s informtion spe. It is possile to work out wht re the miniml omintions of rooti primitives (miniml ntihin) tht re suffiient to perform tsk (e.g., glol loliztion), nd desrie dominne reltion (prtil order) of primitives. Other works hve foused on minimizing the omplexity of the ontroller. Egerstedt [11] studies the reltion etween the omplexity of the environment nd notion of minimum desription length of ontrol strtegies, whih n e tken s proxy for the omputtion neessry to perform the tsk. Sotto [12] studies the reltion etween the performne of visul tsk, nd the miniml representtion tht is needed to perform tht tsk. The hope is tht the theory of o-design presented in this pper will help to integrte ll this previous work in the sme theoretil nd quntittive frmework. Figure 26. t pp 1 pp s IV. PROBLEM STATEMENT AND SUMMARY OF RESULTS Given n ritrry grph of design prolems, nd ssuming we know how to solve eh prolem seprtely, we sk whether we n solve the o-design prolem optimlly. Exmple 7. Nrdi et l. [8] desrie enhmrking system for visul SLAM tht provides the empiril hrteriztion of the monotone reltion etween the ury of the visul SLAM solution, the throughput [frmes/s] nd the energy for omputtion [J/frme]. The implementtion spe is the produt of lgorithmi prmeters, ompiler flgs, nd rhiteture hoies, suh s the numer of GPU ores tive. This is n exmple of design prolem whose funtionlity-resoures mp needs to e experimentlly evluted. Prolem 2. Suppose tht we re given CDPI F, R, V, E, nd tht we n evlute the mp h v for ll v V. Given required funtionlity f F, we wish to find the miniml resoures in R for whih there exists fesile implementtion vetor tht mkes ll su-prolems fesile t the sme time nd ll o-design onstrints stisfied; or, if none exist, provide ertifite of infesiility. In other words, given the mps {h v, v V} for the suprolems, one needs to evlute the mp h : F AR for the entire CDPI (Fig. 29).
7? Figure 29. The rest of the pper will provide solution to Prolem 2, under the ssumption tht ll the DPIs inside the CDPI re monotone, in the sense of Def. 13. Definition 13. A DPI F, R, I, exe, evl is monotone if 1) The posets F, R re omplete prtil orders (Def. 6). 2) The mp h is Sott ontinuous (Def. 8). Cll Monotone Co-Design Prolems (MCDPs) the set of CDPIs for whih ll suprolems respet the onditions in Def. 13. I will show two min results: 1) A modeling result (Theorem 1 on pge 9) sys tht the lss of MCDPs is losed with respet to ritrry interonnetions. Therefore, given o-design digrm, suh s the one in Fig. 25, if we know tht eh design prolem is n MCDP, we n onlude tht the digrm represents n MCDP s well. 2) An lgorithmi result (Theorem 2 on pge 10) sys tht the funtionlity-resoures mp h for the entire MCDP hs n expliit expression in terms of the mps {h v, v V} for the suprolems. If there re yles in the o-design digrm, the mp h involves the solution of lest fixed point eqution in the spe of ntihins. This eqution n e solved using Kleene s lgorithm to find the ntihin ontining ll miniml solutions t the sme time. Approh: The strtegy to otin these results onsists in reduing n ritrry interonnetion of design prolems to onsidering only finite numer of omposition opertors (, prllel, nd feedk). Se. V defines these omposition opertors. Se. VI shows how to turn grph into tree, where eh juntion is one of the three opertors. Given the tree representtion of n MCDPs, we will e le to give indutive rguments to prove the results. Expressivity of MCDPs: The results re signifint euse MCDPs indue rih fmily of optimiztion prolems. We re not ssuming, let lone strong properties like onvexity, even weker properties like differentiility or ontinuity of the onstrints. In ft, we re not even ssuming tht funtionlity nd resoures re ontinuous spes; they ould e ritrry disrete posets. (In tht se, ompleteness nd Sott ontinuity re trivilly stisfied.) Moreover, even ssuming topologil ontinuity of ll spes nd mps onsidered, MCDPs re strongly not onvex. Wht mkes them nononvex is the possiility of introduing feedk interonnetions. To show this, I will give n exmple of 1- dimensionl prolem with ontinuous h for whih the fesile set is disonneted. () () Figure 30. One feedk onnetion nd topologilly ontinuous h re suffiient to indue disonneted fesile set. Exmple 8. Consider the CDPI in Fig. 30. The miniml resoures M AR re the ojetives of this optimiztion prolem: using f, r F = R, M =. Min r, r h(f), r f. The fesile set Φ F R is the set of funtionlity nd resoures tht stisfy the onstrints r h(f) nd r f: Φ = { f, r F R : (r h(f)) (r f)}. (5) The projetion P of Φ to the funtionlity spe is: P = {f f, r Φ}. In the slr se (F = R = R +, ), the mp h: F AR is simply mp h: R + R +. The set P of fesile funtionlity is desried y P = {f R + : h(f) f}. (6) Fig. 30 shows n exmple of ontinuous mp h tht gives disonneted fesile set P. Moreover, P is disonneted under ny order-preserving nonliner re-prmetriztion. V. COMPOSITION OPERATORS FOR DESIGN PROBLEMS This setion defines hndful of omposition opertors for design prolems. Lter, Se. VI will prove tht ny o-design prolem n e desried in terms of suset of these opertors. Definition 14 (). The omposition of two DPIs dp 1 = F 1, R 1, I 1, exe 1, evl 1 nd dp 2 = F 2, R 2, I 2, exe 2, evl 2, for whih F 2 = R 1, is where: (dp 1, dp 2 ). = F 1, R 2, I, exe, evl, I = { i 1, i 2 I 1 I 2 evl 1 (i 1 ) R1 exe 2 (i 2 )}, exe : i 1, i 2 exe 1 (i 1 ), evl : i 1, i 2 evl 2 (i 2 ). Figure 31. Definition 15 (pr). The prllel omposition of two DPIs dp 1 = F 1, R 1, I 1, exe 1, evl 1 nd dp 2 = F 2, R 2, I 2, exe 2, evl 2 is pr(dp 1, dp 2 ). = F 1 F 2, R 1 R 2, I 1 I 2, exe, evl,
8 where: exe : i 1, i 2 exe 1 (i 1 ), exe 2 (i 2 ), (7) evl : i 1, i 2 evl 1 (i 1 ), evl 2 (i 2 ). Figure 35. Figure 32. Definition 16 (loop). Suppose dp is DPI with ftored funtionlity spe F 1 R: dp = F 1 R, R, I, exe 1, exe 2, evl. Then we n define the DPI loop(dp) s loop(dp). = F 1, R, I, exe 1, evl, where I I limits the implementtions to those tht respet the dditionl onstrint evl(i) exe 2 (i): I = {i I : evl(i) exe 2 (i)}. This is equivlent to losing loop round dp with the onstrint f 2 r (Fig. 33). evl R Figure 33. The opertor loop is symmetri euse it ts on design prolem with 2 funtionlities nd 1 resoures. We n define symmetri feedk opertor loop s in Fig. 34, whih n e rewritten in terms of loop, using the onstrution in Fig. 34. Figure 34. () () A symmetri opertor loop n e defined in terms of loop. A o-produt (see, e.g., [13, Setion 2.4]) of two design prolems is design prolem with the implementtion spe I = I 1 I 2, nd it represents the exlusive hoie etween two possile lterntive fmilies of designs. Definition 17 (Co-produt). Given two DPIs with sme funtionlity nd resoures dp 1 = F, R, I 1, exe 1, evl 1 nd dp 2 = F, R,, I 2, exe 2, evl 2, define their o-produt s where dp 1 dp 2. = F, R, I1 I 2, exe, evl, exe : { exe 1 (i), if i I 1, i 2 (i), if i I 2, exe { evl : i evl 1 (i), if i I 1, evl 2 (i), if i I 2. (8) VI. DECOMPOSITION OF MCDPS This setion shows how to desrie n ritrry interonnetion of design prolems using only three omposition opertors. More preisely, for eh CDPI with set of toms V, there is n equivlent one tht is uilt from /pr/loop pplied to the set of toms V plus some extr plumling (identities, multiplexers). Equivlene: The definition of equivlene elow ensures tht two equivlent DPIs hve the sme mp from funtionlity to resoures, while one of the DPIs n hve slightly lrger implementtion spe. Definition 18. Two DPIs F, R, I 1, exe 1, evl 1 nd F, R, I 2, exe 2, evl 2 re equivlent if there exists mp ϕ : I 2 I 1 suh tht exe 2 = exe 1 ϕ nd evl 2 = evl 1 ϕ. Pluming: We lso need to define trivil DPIs, whih serve s pluming. These n e uilt y tking mp f : F R nd lifting it to the definition of DPI. The implementtion spe of trivil DPI is opy of the funtionlity spe nd there is 1-to-1 orrespondene etween funtionlity nd implementtion. Definition 19 (Trivil DPIs). Given mp f : F R, we n lift it to define trivil DPI Triv(f) = F, R, F, Id F, f, where Id F is the identity on F. Proposition 1. Given CDPI F, R, V, E, we n find n equivlent CDPI otined y pplying the opertors pr//loop to set of toms V tht ontins V plus set of trivil DPIs. Furthermore, one instne of loop is suffiient. Proof: We show this onstrutively. We will temporrily remove ll yles from the grph, to e retthed lter. To do this, find n r feedk set (AFS) F E. An AFS is set of edges tht, when removed, remove ll yles from the grph (see, e.g., [14]). For exmple, the CDPI represented in Fig. 36 hs miniml AFS tht ontins the edge (Fig. 36). r feedk set () () () Figure 36. An exmple o-design digrm with three nodes V = {,, }, in whih miniml r feedk set is { }. Id Find wek topologil ordering of V. Then the grph V, E Id r feedk set removed Remove the AFS F from E to otin the redued edge set E = E\F. The resulting grph V, E does not hve yles, nd n e written s -prllel grph, y pplying the opertors pr nd from set of nodes V. The nodes V will ontin V, plus some extr onnetors tht re trivil DPIs. n e written s the of V sugrphs, eh ontining one node of V. In the exmple, the wek topologil ordering is,, nd there re three sugrphs (Fig. 37).
9 Figure 37. Eh sugrph n e desried s the prllel interonnetion of node v V nd some extr onnetors. Id For exmple, the seond sugrph in the grph n e written s the prllel interonnetion of node nd the identity Triv(Id) (Fig. 38). Figure 38. Id dded identity After this is done, we just need to lose the loop round the edges in the AFS F to otin CDPI tht is equivlent to the originl one. Suppose the AFS F ontins only one edge. Then one instne of the loop opertor is suffiient (Fig. 39). In this exmple, the tree representtion (Fig. 39) is loop(((, pr(id, )), ). Proposition 2. If dp 1 nd dp 2 re monotone (Def. 13), then lso the omposition pr(dp 1, dp 2 ) is monotone. Proof: We need to refer to the definition of pr in Def. 15 nd hek the onditions in Def. 13. If F 1, F 2, R 1, R 2 re CPOs, then F 1 F 2 nd R 1 R 2 re CPOs s well. From Def. 11 nd (7) we know h n e written s h : F 1 F 2 A(R 1 R 2 ) f 1, f 2 Min R { evl 1 (i 1 ), evl 2 (i 2 ) ( i 1, i 2 I 1 I 2 ) ( f 1, f 2 exe 1 (i 1 ), exe 2 (i 2 ) )}. All terms ftorize in the two omponents, giving: h: f 1, f 2 Min R 1 { evl 1(i 1) (i I 1) (f 1 exe 1(i 1))} whih redues to Min R 2 { evl 2(i 2) (i I 2) (f 2 exe 2(i 2))}, h: f 1, f 2 h 1 (f 1 ) h 2 (f 2 ). (9) Id Id () () Figure 39. Tree representtion for the o-design digrm in Fig. 36. If the AFS ontins multiple edges, then, insted of losing one loop t time, one n n lwys rewrite multiple nested Id Id loops s only one loop y tking the produt of the edges. For exmple, digrm like the one in Fig. 40 n e rewritten s Fig. 40. This onstrution is nlogous to the onstrution used for the nlysis of proess networks [6] (nd ny other onstrut involving tred monoidl tegory). Therefore, it is possile to desrie n ritrry grph of design prolems using only one instne of the loop opertor. The mp h is Sott ontinuous iff h 1 nd h 2 re [15, Lemm II.2.8]. Proposition 3. If dp 1 nd dp 2 re monotone (Def. 13), then lso the omposition (dp 1, dp 2 ) is monotone. Proof: From the definition of (Def. 14), the semntis of the interonnetion is ptured y this prolem: using r 1, f 2 R 1, r 2 R 2, Min R2 r 2, h : f 1 s.t. r 1 h 1 (f 1 ), (10) r 1 R1 f 2, r 2 h 2 (f 2 ). The sitution is desried y Fig. 41. The point f 1 is fixed, nd thus h(f 1 ) is fixed ntihin in R 1. For eh point r 1 h(f 1 ), we n hoose f 2 r 1. For eh f 2, the ntihin h 2 (f 2 ) tres the solution in R 2, from whih we n hoose r 2. Figure 41. () () Figure 40. If there re nested loops in o-design digrm, they n e rewritten s one loop, y tking the produt of the edges. VII. MONOTONICITY AS COMPOSITIONAL PROPERTY The first min result of this pper is n invrine result. Theorem 1. The lss of MCDPs is losed with respet to interonnetion. Proof: Prop. 1 hs shown tht ny interonnetion of design prolems n e desried using the three opertors pr,, nd loop. Therefore, we just need to hek tht monotoniity in the sense of Def. 13 is preserved y eh opertor seprtely. This is done elow in Prop. 2 4. Beuse h 2 is monotone, h 2 (f 2 ) is minimized when f 2 is minimized, hene we know tht the onstrint r 1 f 2 will e tight. We n then onlude tht the ojetive does not hnge introduing the onstrint r 1 = f 2. The prolem is redued to: using f 2 R 1, r 2 R 2, Min R2 r 2, h : f 1 s.t. f 2 h 1 (f 1 ), r 2 h 2 (f 2 ). (11) Minimizing r 2 with the only onstrint eing r 2 h 2 (f 2 ), nd with h 2 (f 2 ) eing n ntihin, the solutions re ll nd
10 only h 2 (f 2 ). Hene the prolem is redued to using f 2 R 1, h : f 1 Min R2 h 2 (f 2 ), s.t. f 2 h 1 (f 1 ). The solution is simply h : f 1 Min R2 f 2 h 1(f 1) (12) h 2 (f 2 ). (13) This mp is Sott ontinuous euse it is the omposition of Sott ontinuous mps. Proposition 4. If dp is monotone (Def. 13), so is loop(dp). Proof: The digrm in Fig. 33 implies tht the mp h loop(dp) n e desried s: h loop(dp) : F 1 AR, (14) using r, f 2 R, Min R r, f 1 (15) s.t. r h dp (f 1, f 2 ), r R f 2. Denote y h f1 the mp h dp with the first element fixed: h f1 : f 2 h dp (f 1, f 2 ). Rewrite r h dp (f 1, f 2 ) in (14) s r h f1 (f 2 ). (16) Let r e fesile solution, ut not neessrily miniml. Beuse of Lemm 5, the onstrint (16) n e rewritten s {r} = h f1 (f 2 ) r. (17) Beuse f 2 r, nd h f1 is Sott ontinuous, it follows tht h f1 (f 2 ) AR h f1 (r). Therefore, y Lemm 6, we hve {r} AR h f1 (r) r. (18) This is reursive ondition tht ll fesile r must stisfy. Let R AR e n ntihin of fesile resoures, nd let r e generi element of R. Tutologilly, rewrite R s the miniml elements of the union of the singletons ontining its elements: R = Min {r}. (19) R r R Sustituting (18) in (19) we otin (f Lemm 7) R AR Min h f1 (r) r. (20) R r R [Converse: It is lso true tht if n ntihin R stisfies (20) then ll r R re fesile. The onstrint (20) mens tht for ny r 0 R on the left side, we n find r 1 in the right side so tht r 0 R r 1. The point r 1 needs to elong to one of the sets of whih we tke the union; sy tht it omes from r 2 R, so tht r 1 h f1 (r 2 ) r 2. Summrizing: r 0 R : r 1 : (r 0 R r 1 ) ( r 2 R: r 1 h f1 (r 2 ) r 2 ). (21) Beuse r 1 h f1 (r 2 ) r 2, we n onlude tht r 1 r 2, nd therefore r 1 R r 2, whih together with r 0 R r 1, implies r 0 R r 2. We hve onluded tht there exist two points r 0, r 2 in the ntihin R suh tht r 0 R r 2 ; therefore, they re the sme point: r 0 = r 2. Beuse r 0 R r 1 R r 2, we lso onlude tht r 1 is the sme point s well. We n rewrite (21) y using r 0 in ple of r 1 nd r 2 to otin r 0 R : r 0 h f1 (r 0 ), whih mens tht r 0 is fesile resoure.] We hve onluded tht ll ntihins of fesile resoures R stisfy (20), nd onversely, if n ntihin R stisfies (20), then it is n ntihin of fesile resoures. Eqution (20) is reursive onstrint for R, of the kind with the mp Φ f1 defined y Φ f1 (R) AR R, Φ f1 : AR AR, (22) R Min h f1 (r) r. R r R If we wnt the miniml resoures, we re looking for the lest ntihin: min AR { R AR: Φ f1 (R) AR R }, whih is equl to the lest fixed point of Φ f1. Therefore, the mp h loop(dp) n e written s h loop(dp) : f 1 lfp(φ f1 ). (23) Lemm 8 shows tht lfp(φ f1 ) is Sott ontinuous in f 1. Lemm 5. Let A e n ntihin in P. Then A {} = A. Lemm 6. For A, B AP, nd S P, A AR B implies A S AR B S. Lemm 7. For A, B, C, D AP, A AR C nd B AR D implies A B AR C D. Lemm 8. Let f : P Q Q e Sott ontinuous. For eh x P, define f x : y f(x, y). Then f : x lfp(f x ) is Sott ontinuous. Proof: Dvey nd Priestly [3] leve this s Exerise 8.26. A proof is found in Gierz et l. [15, Exerise II-2.29]. VIII. SOLUTION OF MCDPS The seond min result is tht the mp h of MCDP hs n expliit expression in terms of the mps h of the suprolems. Theorem 2. The mp h for n MCDP hs n expliit expression in terms of the mps h of its suprolems, defined reursively using the rules in Tle I. Tle I RECURSIVE EXPRESSIONS FOR h dp = (dp 1, dp 2 ) h = h 1 h 2 prllel dp = pr(dp 1, dp 2 ) h = h 1 h 2 feedk dp = loop(dp 1 ) h = h 1 o-produt dp = dp 1 dp 2 h = h 1 h 2
11 Proof: These expressions were derived in the proofs of Prop. 2 4. The opertors,,, re defined in Def. 20 23. Definition 20 (Series opertor ). For two mps h 1 : F 1 AR 1 nd h 2 : F 2 AR 2, if R 1 = F 2, define h 1 h 2 : F 1 AR 2, f 1 Min R2 s h 1(f) h 2 (s). Definition 21 (Prllel opertor ). For two mps h 1 : F 1 AR 1 nd h 2 : F 2 AR 2, define h 1 h 2 : (F 1 F 2 ) A(R 1 R 2 ), (24) f 1, f 2 h 1 (f 1 ) h 2 (f 2 ), where is the produt of two ntihins. Definition 22 (Feedk opertor ). For h : F 1 R AR, define h : F 1 AR, f 1 lfp ( Ψ h f 1 ), (25) The digrm ontins three primitive DPIs: dp 1, dp 2 (used twie), nd dp 3. Their h mps re: h 1 : N N N AN, f 1, f 2, f 3 {f 1 + f 2 + f 3 }, h 2 : N AN, f { f }, h 3 : N A(N N), f {, N N : + = f}. The tree deomposition (Fig. 42) orresponds to the expression dp = loop((pr(dp 2, dp 2 ), (dp 1, dp 3 ))). (28) Consulting Tle I, from (28) one otins n expression for h: h = ((h 2 h 2 ) h 1 h 3 ). (29) This prolem is smll enough tht we n write down n expliit expression for h. By sustituting in (29) the definitions given in Def. 20 22, we otin tht evluting h() mens finding the lest fixed point of mp Ψ : where Ψ h f 1 is defined s h : lfp(ψ ). Ψ h f 1 : AR AR, R Min h(f 1, r) r. (26) R r R Definition 23 (Coprodut opertor ). For h 1, h 2 : F AR, define h 1 h 2 : F AR, f Min R (h 1 (f) h 2 (f)). A. Exmple: Optimizing over the nturl numers This is the simplest exmple tht n show two interesting properties of MCDPs: 1) the ility to work with disrete posets; nd 2) the ility to tret multi-ojetive optimiztion prolems. Consider the fmily of optimiztion prolems indexed y N: { Min N N x, y, s.t. x + y x + (27) y +. One n show tht this optimiztion prolem is n MCDP y produing o-design digrm with n equivlent semntis, suh s the one in Fig. 42. The mp Ψ : A(N N) A(N N) n e otined from (26) s follows: Ψ : R Min x, y x,y R {, N 2 : ( + x + y + )}. Kleene s lgorithm is the itertion R k+1 = Ψ (R k ) strting from R 0 = A(N N) = { 0, 0 }. For = 0, the sequene onverges immeditely: R 0 = { 0, 0 } = h(0). For = 1, the sequene onverges t the seond step: R 0 = { 0, 0 }, R 1 = { 0, 1, 1, 0 } = h(1). For = 2, the sequene onverges t the fourth step; however, some solutions (in old) onverge sooner: R 0 = { 0, 0 }, R 1 = { 0, 2, 1, 1, 2, 0 }, R 2 = { 0, 4, 2, 2, 4, 0 }, R 3 = { 0, 4, 3, 3, 4, 0 } = h(2). Figure 42. () () Co-design digrm equivlent to (27) nd its tree representtion. The next vlues in the sequene re: h(3) = { 0, 6, 3, 4, 4, 3, 6, 0 }, h(4) = { 0, 7, 3, 6, 4, 4, 6, 3, 7, 0 }. Fig. 43 shows the sequene for = 20.
12 () () (e) (d) (f) The urrent ntihin We know this re is unfesile. Mye - there might e miniml solutions here. These points re miniml solutions. Don t re - Not neessrily fesile, ut we know tht there re no fesile points tht re not dominted y the miniml points lredy found. Figure 43. Kleene sent to solve the prolem (27) for = 20. The sequene onverges in five steps to R 5 = R. Gurntees of Kleene sent: Solving n MCDP with yles redues to omputing Kleene sent sequene R k. At eh instnt k we hve some dditionl gurntees. For ny finite k, the resoures elow R k (the set R \ R k,) re infesile. (In Fig. 43, those re olored in red.) If the itertion onverges to non-empty ntihin R, the ntihin R divides R in two. Below the ntihin, ll resoures re infesile. However, ove the ntihin (purple re), it is not neessrily true tht ll points re fesile, euse there might e holes in the fesile set, s in Exmple 8. Note tht this method does not ompute the entire fesile set, ut rther only the miniml elements of the fesile set, whih might e muh esier to ompute. Finlly, if the sequene onverges to the empty set, it mens tht there re no solutions. The sequene R k n e onsidered ertifite of infesiility. B. Complexity of the solution 1) Complexity of fixed point itertion: Consider first the se of n MCDP tht n e desried s dp = loop(dp 0 ), where dp 0 is n MCDP tht is desried only using the nd pr opertors. Suppose tht dp 0 hs resoure spe R. Then evluting h for dp is equivlent to omputing lest fixed point itertion on the spe of ntihins AR. This llows to give worst-se ounds on the numer of itertions. Proposition 5. Suppose tht dp = loop(dp 0 ) nd dp 0 hs resoure spe R 0 nd evluting h 0 tkes t most omputtion. Then we n otin the following ounds for the lgorithm s resoures usge: memory O(width(R 0 )) numer of steps O(height(AR 0 )) totl omputtion O(width(R 0 ) height(ar 0 ) ) Proof: The memory utiliztion is ounded y width(r 0 ), euse the stte is n ntihin, nd width(r 0 ) is the size of the lrgest ntihin. The itertion hppens in the spe AR 0, nd we re onstruting n sending hin, so it n tke t most height(ar 0 ) steps to onverge. Finlly, in the worst se the mp h 0 needs to e evluted one for eh element of the ntihin for eh step. These worst se ounds re strit. Exmple 9. Consider solving dp = loop(dp 0 ) with dp 0 defined y h 0 :, x x + 1 with x N. Then the lest fixed point eqution is equivlent to solving min{x: Ψ(x) x} with Ψ : x x + 1. The itertion R k+1 = Ψ(R k ) onverges to in height(n) = ℵ 0 steps. Remrk 5. Mking more preise lims requires dditionl more restritive ssumptions on the spes involved. For exmple, without dding metri on R, it is not possile to otin properties suh s liner or qudrti onvergene. Remrk 6 (Invrine to re-prmeteriztion). All the results given in this pper re invrint to ny order-preserving reprmeteriztion of ll the vriles involved. 2) Relting omplexity to the grph properties: Prop. 5 ove ssumes tht the MCDP is lredy in the form dp = loop(dp 0 ), nd reltes the omplexity to the poset R 0. Here we relte the results to the grph struture of n MCDP. Tke n MCDP dp = F, R, V, E. To put dp in the form dp = loop(dp 0 ) ording to the proedure in Se. VI, we need to find n r feedk set (AFS) of the grph V, E. Given AFS F E, then the resoure spe R 0 for dp 0 suh tht dp = loop(dp 0 ) is the produt of the resoures spes long the edges: R 0 = e F R e. Now tht we hve reltion etween the AFS nd the omplexity of the itertion, it is nturl to sk wht is the optiml hoie of AFS whih, so fr, ws left s n ritrry hoie. The AFS should e hosen s to minimize one of the performne mesures in Prop. 5. Of the three performne mesures in Prop. 5, the most fundmentl ppers to e width(r 0 ), euse tht is lso n upper ound on the numer of distint miniml solutions. Hene we n ll it design omplexity of the MCDP. Definition 24. Given grph V, E nd leling of eh edge e E with poset R e, the design omplexity DC( V, E ) is defined s DC( V, E ) = min width( R e ). (30) F is n AFS e F In generl, width nd height of posets re not dditive with respet to produts; therefore, this prolem does not redue to ny of the known vrints of the minimum r feedk set prolem, in whih eh edge hs weight nd the gol is to minimize the sum of the weights.
Identity 0.00133 pr R[J] R[J] Identity 0.35211 USD/Wh R[] R[] pr R[J] R[] R[J] R[] R[] R[] GeneriUnry(<uilt-in funtion eil>) R[J] R[J*USD/Wh] pr R[J]R[J] R[] R[J]R[J*USD/Wh] R[] Identity R[J]R[] R[g]R[USD]R[] R[J]R[] R[J]R[] MuxMp:R[J]R[]"R[]((R[J]R[J])R[]) Identity R[J]R[] R[g]R[USD]R[] R[] R[] (R[J]R[J])R[] (R[J]R[J*USD/Wh])R[] R[J]R[] R[]((R[J]R[J])R[]) pr (R[J]R[J])R[] R[J]R[J*USD/Wh] (R[J]R[J*USD/Wh])R[] R[J]R[J*USD/Wh] R[]((R[J]R[J])R[]) R[](R[J]R[J*USD/Wh]) (R[J]R[J*USD/Wh])R[] R[J](R[J*USD/Wh]R[]) R[]((R[J]R[J])R[]) R[g]R[USD]R[] MuxMp:(R[J]R[J*USD/Wh])R[]"R[J](R[J*USD/Wh]R[]) MuxMp:R[](R[J]R[J*USD/Wh])"(R[]R[J])R[J*USD/Wh] Identity R[](R[J]R[J*USD/Wh]) R[g]R[USD]R[] R[](R[J]R[J*USD/Wh]) (R[]R[J])R[J*USD/Wh] R[J](R[J*USD/Wh]R[]) R[J]R[J*USD/Wh] pr R[J] R[J] R[J*USD/Wh]R[] R[J*USD/Wh] ProdutN Identity (R[]R[J])R[J*USD/Wh] R[g]R[USD]R[] pr R[]R[J] R[J*USD/Wh] R[]R[J] R[USD] J*USD/Wh-to-USD (R[]R[J])R[J*USD/Wh] (R[]R[J])R[USD] (R[]R[J])R[USD] R[g]R[USD]R[] (R[]R[J])R[USD] (R[]R[USD])R[J] MuxMp:(R[]R[J])R[USD]"(R[]R[USD])R[J] Identity pr R[]R[USD] R[]R[USD] 0.00513 kg/wh R[J] R[g] R[J] R[J*kg/Wh] (R[]R[USD])R[J] R[g]R[USD]R[] (R[]R[USD])R[J] (R[]R[USD])R[g] MuxMp:(R[]R[USD])R[g]"R[g]R[USD]R[] R[J*kg/Wh] R[g] J*kg/Wh-to-g (R[]R[USD])R[g] R[g]R[USD]R[] speifi speifi 13 3) Considering reltions with infinite rdinlity: This nlysis shows the limittions of the simple solution presented so fr: it is esy to produe exmples for whih width(r 0 ) is infinite, so tht one needs to represent ontinuum of solutions. [$] pity [J] [g] () Interfe of ttery design prolem. Exmple 10. Suppose tht the pltform to e designed must trvel distne d [m], nd we need to hoose the endurne T [s] nd the veloity v [m/s]. The reltion mong the quntities is d T v. This is design prolem desried y the mp h : R + AR + R +, d { T, v R + R + : d = T v}. For eh vlue of d, there is ontinuum of solutions. One pproh to solving this prolem would e to disretize the funtionlity F nd the resoures R y smpling nd/or orsening. However, smpling nd orsening mkes it hrd to mintin ompleteness nd onsisteny. One effetive pproh, outside of the sope of this pper, tht llows to use finite omputtion is to pproximte the design prolem itself, rther thn the spes F, R, whih re left s possily infinite. The si ide is tht n infinite ntihin n e ounded from ove nd ove y two ntihins tht hve finite numer of points. This ide leds to n lgorithm tht, given presried omputtion udget, n ompute n inner nd outer pproximtion to the solution ntihin [16]. () MCDPL ode equivlent to equtions (31) (33). missions 0.00200 eil(r[]->r[]) [J] splitter [J] [J] [J] pity [J] 0.13333 USD/Wh 0.03333 kg/wh IX. EXTENDED NUMERICAL EXAMPLES [J*USD/Wh] [J*kg/Wh] This exmple onsiders the hoie of different ttery tehnologies for root. The gols of this exmple re: 1) to show how design prolems n e omposed; 2) to show how to define hrd onstrints nd preedene etween resoures to e minimized; 3) to show how even reltively simple models n give very omplex trde-offs surfes; nd 4) to introdue MCDPL, forml lnguge for the desription of MCDPs. Lnguge nd interpreter/solver: MCDPL is modeling lnguge to desrie MCDPs nd their ompositions. It is inspired y CVX nd disiplined onvex progrmming [17]. MCDPL is even more disiplined thn CVX; for exmple, multiplying y negtive numer is syntx error. The figures re generted y PyMCDP, n interpreter nd solver for MCDPs, whih implements the tehniques desried in this pper. The softwre nd mnul re ville t http://mdp.mit.edu. Model of ttery: The hoie of ttery n e modeled s DPI (Fig. 44) with funtionlities pity [J] nd numer of missions nd with resoures [kg], [$] nd, defined s the numer of times tht the ttery needs to e repled over the lifetime of the root. Eh ttery tehnology is desried y the three prmeters speifi energy, speifi, nd lifetime (numer of yles): ρ. = speifi energy [Wh/kg], α. = speifi [Wh/$],. = ttery lifetime [# of yles]. [J*USD/Wh] [J*USD/Wh] [USD] [J*USD/Wh] [J*kg/Wh] unit onversion [g] () Co-design digrm generted y PyMCDP from ode in pnel (). (d) Tree representtion using pr/ of digrm in pnel (). Figure 44. Pnel () shows the o-design digrm generted from the ode in (). Pnel (d) shows tree representtion (, prllel) for the digrm. The edges show the types of funtionlity nd resoures. The leves re leled with the Python lss used internlly y the interpreter PyMCDP.