# Policies for Simultaneous Estimation and Optimization

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3 uncrtainty in th stimat of b As Π, th optimal input gos to zro h xpctd cost, for ρ =, simplifis to φ = σ (y ds ) + E = +ˆb Π ˆb As Π, th minimum xpctd cost approachs an uppr bound, which is th cost of slcting a zro input Again, small Π,,Π yild a larg xpctd cost h policy is suboptimal bcaus th slction of u dos not ta into account its ffct on Π,,Π (which ar quadratic in u ) his is, in ffct, a grdy policy: At ach tim indx, u is slctd to minimiz th immdiat xpctd cost, E((y y ds ) ), without rgard for futur costs As in, thr is no dsign for stimation, in th sns that th bnfits to b gaind from slcting inputs that ma th Π larg ar not considrd Drivativ of th xpctd cost with rspct to information An invaluabl and gnrally ovrlood fact is that, for many rgularizd or robust policis, th drivativ of th cost with rspct to th information matrix is asily computd For th rgularizd passiv larning policy with ρ =,w hav that d σ + d Π (yds ) +ˆb Πˆb (y ds ) = ˆbˆb +ˆb Πˆb 4 Prsistncy of xcitation, dithring, and maximally informativ inputs W hav sn that th cost incurrd at tim may b larg if λ min(π ) is small In othr words, if th covarianc Σ =Π is larg, ˆb is an unrliabl stimat of th systm paramtrs b, and this lads to poor prformanc h most immdiat solution is to nsur that som masur of information, such as λ min(π )=λ min(π + σ U U ), is larg his can b translatd into a rquirmnt that U b wllconditiond, which is what is usually mant by prsistncy of xcitation Informally, w want th u to span th whol input spac 4 Dithring Svral solutions hav bn proposd to satisfy th prsistncy of xcitation rquirmnt Dithring is a randomizd fasibl policy that consists of adding to th inputs som whit nois (i, normal, zro man and indpndnt random trms) his can wor wll sinc, givn a high nough nois lvl, Π will b wll-conditiond with high probability Although it has th advantag of vry simpl implmntation, dithring is obviously a sub-optimal policy, and th slction of a good nois lvl can b problmatic Exampl Considr th problm dscribd by 6 n =, ˆb =, Π =, σ =, =, Y ds = [ ] W implmnt a dithrd policy basd on rgularizd passiv larning ( ) Figur plots th xpctd cost as th varianc of th random trms (th input nois lvl) rangs from 9 to hs valus wr obtaind by Mont Carlo simulation, with runs at ach input nois lvl (th corrsponding rror bars ar also plottd) Not that this may sm countrintuitiv: h prformanc of th policy is improvd by adding indpndnt nois to th inputs As th nois lvl gos to zro, w approach th non-dithrd rgularizd passiv larning policy, which has an avrag cost of 9 (±4) At th optimal input nois lvl (slctd a postriori), th avrag cost is 4 (±9) Of cours, any practical dithring policy must slct th input nois lvl a priori, which may b difficult <φ> Figur : log σdith Expctd cost as a function of dithring lvl 4 Masuring and valuing information A mor thoughtful approach is to slct a prturbation that, for a givn lvl of control disturbanc, maximizs th information gathrd For this purpos, w nd a masur of information A list of possibl masurs, using th naming convntion from xprimnt dsign, is E-optimal: λ max (Σ ) = λ min(π ) D-optimal: log dt Π A-optimal: r (Σ ) = r Π h E-optimal and A-optimal masurs may b scald to account for th drivativ of th xpctd cost with rspct to information ( ), whil th D-optimal masur is invariant with scaling For a numrically ffctiv huristic, w would li a masur that is concav in th inputs hs masurs ar concav-quadratic, and linarizing Π in U mas thm concav his linarization can b xpctd to wor wll if th prturbations introducd for th purpos of idntification ar small W rturn to this point in 55 4 Maximally informativ inputs On approach consists of xprssing xplicitly th trad-off btwn control and information, by adding to th objctiv function an xtra trm valuing information A vry simpl xampl of such a policy is to slct th u that minimizs u Π u +(ˆb u y ds ) + ρu u γλ min(π ), and liwis for u,,u, with th appropriat updating of Π and ˆb h xtra trm mas this policy, in part, an xprimnt dsign problm As with dithring, a prturbation will b introducd in th input, what w might now call

4 an intllignt nois h factor γ> wighs th trad-off btwn idntification and control Slcting γ prsnts th sam difficultis as for slcting th dithring lvl An altrnativ approach is what has bn trmd plantfrindly idntification Although it is ssntially a solution to a diffrnt problm, plant-frindly idntification can b usd as a huristic for simultanous stimation and control h ida is to slct th maximally informativ input from within th st of inputs that p som masur of th tracing rror within a bound For a simpl xampl, w us a constraint on th absolut tracing rror (spcifid by M R) h policy is dfind by th program maximiz λ min(π ) subjct to ˆb u y ds M h bound M can b sn as th trad-off factor, with a rol similar to γ in th prvious problm Howvr, M is a mor physically maningful numbr, and should b asir to slct in practical applications h constraint on prformanc usd hr disrgards th uncrtainty in b A robust constraint can b usd in its plac (such a constraint is convx in th inputs in fact, it is a scond-ordr con constraint, s S Boyd t al []) Both ths problms ar convx if w linariz Π in u, in which cas thy ar radily solvd If this procdur linarization followd by th solution of a convx program is itratd, a (local) minimum of th non-convx problm will b rachd Not that ths huristics do not us th futur dsird outputs y ds, which w assumd nown his is part of th suboptimal natur of th huristics, and has th ffct of rducing th snsitivity of thir prformanc with rspct to th futur trajctory his rducd snsitivity may b a positiv fatur in applications whr th futur trajctory is not fully crtain 5 Optimal policy, dynamic program and approximation hs huristic approachs still lav us with som qustions, in particular about ) what masur of information to us, and ) how to dcid on th invitabl trad-off btwn th informativnss of u and th output rror xpctd to rsult from its application Roughly spaing, th answr to th scond qustion is that th trad-off should b such that th currnt loss in tracing prformanc (incurrd for th sa of informativnss) quals th total xpctd futur gains in tracing prformanc (du to improvd information about th systm) his, in turn, lads to an answr for th first qustion: h information masur should b such that it capturs th xpctd futur gain in tracing prformanc h tru solution to th problm is givn by a dynamic program, of which w will outlin th drivation his dynamic program is, howvr, hard to solv W propos an approximation which rsults in a smidfinit program 5 Optimal policy for = W assum, from hr on, ρ = Considr th simplst cas, whr = An input u is to b slctd so as to minimiz th xpctd cost φ = = E b, y y ds = E b, b u + y ds = E b, b u ßÞÐ = u Π u ßÞ Ð +(ˆb u y ds ßÞ Ð +(ˆb u y ds ßÞ Ð ) ) + σ ßÞÐ + ßÞÐ h thr trms mard can b intrprtd as ) th cost du to inaccuracy in th stimat of b, ) th cost du to dviation from th crtainty quivalnc policy, and ) th cost du to output nois h input that minimizs this function, obtaind by diffrntiating and quating to zro, is u = ψ (ˆb, Π,σ,y ds )= Π +ˆb ˆb ˆby ds Not that Π can b sn as a rgularization trm As Π bcoms small, th optimal input gos to zro h minimum xpctd cost is ) φ =(ˆb, Π,σ,y ds )=σ + (yds +ˆb Πˆb, whr w usd th matrix invrsion lmma for a ran on updat Not that φ = is convx in Σ and concav in Π For small Π, th minimum xpctd cost approachs an uppr bound, which is th cost of slcting a zro input 5 Optimal policy for = For =, th xpctd cost is φ = = E b,, y y ds + y y ds = E b,, ( b u +(ˆb u y ds )+ ) +( b u +(ˆb u y ds )+ ) = E b, ( b u +(ˆb u y ds )+ ) +E y E b,, ( b u +(ˆb u y ds )+ ) y = u Π u +(ˆb u y ds ) + σ +E y u Π u +(ˆb u y ds ) + σ whr (from ) Π =Π +σ u u, ˆb =Π Π ˆb+σ, (4) u y W usd th towr proprty of conditional xpctation, and th fact that, if y is givn, thn ˆb and u ar constants and b has zro man and covarianc Π Also, it is trivial to s that b and ar indpndnt, and b and ar indpndnt φ = is to b minimizd ovr u = ψ and u = ψ,withψ a function of y and u (both ψ and ψ ar also functions of ˆb,Π,σ,y ds and y ds, but for clarity ths paramtrs

5 will b omittd) h minimum of φ = can b found by minimizing first ovr ψ (i, finding th minimizing scond input u as a function of th first input u and output y ) o find th minimum of (4) w will nd to comput inf E y ψ (u,y ) Π ψ (u,y ) ψ (, ) +ˆb ψ (u,y ) y ds + σ = = E y inf ψ (u,y ) Π ψ (u,y ) ψ (, ) +ˆb ψ (ψ,y ) y ds + σ = E y φ =(ˆb, Π,σ,y ds ) = σ +(y ds ) E y +ˆb Πˆb (5) W conclud that th minimum xpctd cost is φ =(ˆb, Π,σ,y ds ) = inf u u Π u +(u ˆb y ds ) + σ + E y φ =(ˆb, Π,σ,y ds ) Not that w hav just drivd Bllman s principl of optimality from first principls for this particular problm h solution rquirs computing an intgral of th form E X, X N(,σ ) a X + a X + a whr a = σ 4 u Π u, a =ˆb u σ, a =+ˆb Π ˆb, X= b u + N(,σ ), σ = u Π u +σ h dnominator polynomial can b shown to b positiv for all X If an itrativ optimization procdur is to b usd, this xpctation must b valuatd numrically at ach itration Altrnativly, w will propos using a simpl approximation Exampl Considr th prvious xampl (in 4), but with a shortr horizon In particular, =, Y ds =[ ], and n, ˆb, Π,σ as bfor For a givn u, th xpctd cost φ is computd assuming that u is slctd optimally at = h xpctation in (5) is valuatd by numrical intgration Ranging ovr valus for th two ntris of u, this producs Figur h optimum is achivd at u = 998, for which th xpctd cost is φ =568 his is to b compard with th standard procdur of minimizing th xpctd squar rror at ach tim stp (i, th rgularizd passiv larning policy), which yilds u =,andφ=9 5 Approximat solution for = Considr th approximation E X a X + a X + a a With this approximation, th problm bcoms that of minimizing u Π u +(u ˆb y ds ) +σ + (yds ) +ˆb Πˆb φ u, u, Figur : Expctd cost as a function of first and scond ntry of u (with u optimal) ovr u R n Undoing th minimization ovr u,ws that this is quivalnt to minimizing f = = u Π u + u (Π + σ u u ) u +(u ˆb y ds ) +(u ˆb y ds ) +σ ovr u,u R n his approximation is quivalnt to maing th approximation ˆb ˆb in (4), which will b th motivation for an xtnsion of th approximation for any > W will ta ˆb ˆb, =,,, in th quivalnt xprssion for th xpctd cost An intuitiv dscription of this approximation is as follows First, not that th aprioridistribution of b can b dscribd by th llipsoid (x ˆb ) Π (x ˆb ) (th maximum volum st with a givn probability) Liwis, th conditional distribution of b givn y can b dscribd by th llipsoid (x ˆb ) Π (x ˆb ) h total cost will dpnd on both th cntrs (ˆb, ˆb ) and th volums (dfind by Π, Π ) of th two llipsoids From on tim indx to th nxt, with th addd nowldg of y, th cntr and volum of th llipsoid chang (s Figur ) h cntr changs randomly, and this is th trm that introducs incrasd complxity in th dynamic program (as a sid not, this random chang has a zro man normal distribution that dpnds on th inputs, and is asily computd) On th othr hand, th volum changs in a dtrministic fashion Givn th inputs, this chang in volum can b prcisly prdictd With th approximation dscribd, w ar assuming that th chang in volum is mor important in dtrmining th cost than th chang in cntr, i, w assum that th cost is much lss snsitiv to th man of th distribution than to its covarianc his is rasonabl for systms that ar not ovrdtrmind, which includs our problm ˆb Figur : Changs in th conditional distribution of b ˆb Σ Σ

6 Exampl With th sam xampl as in 5, for a givn u w comput f = Again w assum that u is slctd optimally Ranging ovr valus for th two ntris of u, this producs Figur 4 h minimum of th approximat objctiv function f = is achivd at u = h ap- proximat xpctd cost at this point is f = =997, and th tru xpctd cost is φ = =59 h prformanc dgradation rlativ to th optimal policy is 9% with th approximation, as compard to 55% with th rgularizd passiv larning policy Figur 5 plots th approximation rror as a function of u Not th small rror in th rgion whr th optimum is locatd, which sms to b a gnral fatur of this approximation With this approximation, w can rmov th nstd conditional xpctation, and group th inf oprators, so that φ inf f, u,,u with f = = u Π u + = (ˆb u y ds ) + σ Finding this minimum is not a convx program, which gratly limits our ability to solv larg scal problms in practic 55 Convx approximation (linarization of Π ) A convx approximation of th objctiv function abov can b obtaind by linarizing th information matrix in th inputs Writing U = U + U,andfor U small, f 4 Figur 4: f φ u, u, Approximation of th xpctd cost u ßÞ Ð u, ßÞ Ð ) + σ u, Figur 5: Approximation rror 54 Optimal policy and approximation for > Following th prvious analysis for = and =,andby induction on, th problm of minimizing φ can b writtn as a dynamic program h optimum is givn by φ = ϕ, with ϕ = inf u E((y y ds ) )+E(ϕ +)) =inf u u Π +(ˆb u y ds +E(ϕ +), ßÞÐ for =,,, and ϕ + = All infimums ar ovr th spac of fasibl policis, i, ovr all u masurabl σ(y,,y ) h thr mard trms can b intrprtd as ) th cost du to inaccuracy in th stimat of b, )th cost du to th prturbation introducd to improv stimation of b, and ) th cost du to output nois o ma th dynamic program tractabl w ta th sam approach as bfor, and us th approximation ˆb ˆb, ˆb ˆb, ˆb ˆb Liwis, Π U U (U) U +(U) U + U U = (U) U +U U (U) U Π + σ (U ) U + U U (U ) U = P h trm omittd is O(σ U ) It is positiv smidfinit, hnc th approximation undrvalus information W can xpct that a solution basd on this approximation will b consrvativ in th introduction of prturbations for th purpos of idntification h problm now involvs a sum of matrix fractional and quadratic trms, all of which ar convx, minimiz = u P u + = (ˆb u y ds ) whr P is as abov, and th variabls ar u,,u R n his is a matrix-fractional and scond-ordr con program, which is quivalnt to th smidfinit program minimiz = (α + β ) subjct to α (ˆb u y ds ) (ˆb u y ds, ) β u, =,, u P P =Π +σ j= =,, u j(u j) + u ju j u j(u j), =,,, whr th variabls ar α,,α,β,,β R, and u,,u R n Algorithms for solving smidfinit programs ar of polynomial complxity h complxity of solving this particular problm with an intrior-point mthod is boundd by O 7 9 n For mor on smidfinit programming s, g, Vandnbrgh and Boyd [4] 56 Algorithm A possibl practical algorithm is as follows Find a nominal input squnc u,,u È according to a simpl policy, such as minimizing = u Π u + È = (ˆb u y ds ) his amounts to solving without accounting for th bnfits of xtra information

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