A Non-Orthogonal Projection Approach to Characterization of Almost Positive Real Systems with an Application to Adaptive Control

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1 No-Othogoal Pojectio ppoach to haacteizatio of lost Positive Real Systes with a pplicatio to daptive otol Mak Balas, Seio Mebe, EEE ad Robet Fuetes, Mebe EEE bstact: this pape we develop a vey geeal pojectio appoach to obtai ad uify ecessay ad sufficiet coditios fo akig a liea tieivaiat syste positive eal usig static output feedback. Such systes ae called lost positive Real. hese coditios ae that the ope-loop syste ust be weakly iiu phase, have positive defiite high fequecy gai ad the agially stable zeo syste aloe ust be positive eal. Whe dyaic output feedback is used the ecessay ad sufficiet coditios educe to weakly iiu phase with a positive defiite high fequecy gai. hese esults yield a stability poof fo iect Model Refeece daptive otol ad istubace acellatio usig oly alost positive eal systes. toductio: hee is substatial iteest i positive eal ad stictly positive eal systes fo vaious aspects of cotol syste desig ad aalysis, []-[3]. Hee we will addess the poble of usig output feedback to ake a syste positive eal. We coside the followig cotollable ad obsevable liea cotiuous-tie squae plat: x& = x + Bu y = x; x() = x R whee di x =,di u = di y = ` () ad B ad have full ak. Fo [8], the tasissio (o blockig) zeos of this syste ae the values of s whee B B ak < +,.. is sigula. i e M. J. Balas is with the ES epatet, Uivesity of oloado, Boulde, O US (coespodig autho phoe: ; fax: ; e-ail: ak.balas@coloado.edu). R. Fuetes is with Boeig-SVS, lbuqueque, NM US (e-ail:obet.fuetes@kitlad.af.il). he syste (, is said to be iiu phase whe B ak fo Re( ) () = + s Whe () is tue oly fo Re >, (, is weakly iiu phase; this follows a siila defiitio fo oliea systes i [9]. lso is said to be stable whe all its eigevalues lie i the ope left half-plae ad weakly stable whe they lie i the closed left half-plae. he high fequecy gai of () is B ad the syste is said to be elative degee oe whe B is osigula. We will use a static output feedback cotol law give by u = Gy + u (3) whee G is x he closed-loop tasfe fuctio fo () ad (3) is P ( s ) B whee + BG (4) ad the ope-loop vesio P(s) is give with G set to zeo. efiitio: x atioal atix tasfe fuctio P(s) is positive eal (PR) whe ll eleets of P(s) ae aalytic i the ope ight half-plae Re(s)> with oly siple poles o the iagiey axis (the esidue atix at these poles ust be positive sei-defiite), ad Re P(s) is positive sei-defiite fo Re(s)> P( s) + P whee Re P Fo the Positive Real Lea [4], it is kow that this is equivalet to P + P = Q P > (5) PB =

2 whee we have used the covetio: W > fo W syetic ad positive defiite ad W fo W syetic ad positive seidefiite. hee is little cotovesy ove this classic defiitio of PR. Howeve, whe it coes to stict positive eal, thigs ae less clea. We use the followig: efiitio: x atioal atix tasfe fuctio P(s) is stictly positive eal (SPR) whe ε > P( s ε ) is PR. We eak that, of all the defiitios of SPR, this oe is equivalet to the Kala-Yacubovic (K-Y) coditios as show i [5]: P + P = Q < P > (6) PB = his ca be see fo the Positive Real Lea shiftig to + ε i (5) ad by eplacig Q by Q + εp >. Hecefoth, whe we speak of P(s) beig SPR, we ca use this equivalece of a iial ealizatio (, satisfyig (6). We seek ecessay ad sufficiet coditios o (, fo a output feedback cotol law (3) to ake the closed-loop tasfe fuctio PR/SPR ad say that (, is alost PR/SPR. his also called Output feedback equivalece to a PR/SPR syste. Vaious authos have addessed diffeet aspects of this poble. [6] heo 3., Gao ad oaou have show that PR iplies B is positive defiite. dditioally they have show that SPR iplies: a ) B > b ) R eλ ( ) < whee λ ( )is the set of eigevalue s of c)b + ( < But ot ecessaily covesely, except fo soe special cases. [7] heo 4., Weiss, Wag, ad Speye show that, if B is positive defiite ad the ope-loop syste is iiu phase, the (3) will poduce PR. Hee we will show that these coditios ae ecessay ad sufficiet fo (, to be alost SPR. [8], Gu addesses the poble of quadatic stabilizatio via liea cotol. his is elated but ot the sae as ou poble. his o 3.7, Gu sees to pove a elated esult although it is based o a etacted o odel educed vesio of K-Y i heo. of that pape. Gu s esult also uses a scaled vesio KP (s) of the closed-loop tasfe fuctio, whee K is osigula. siila esult ca be accoplished with a odified output feedback cotolle: u = Gy + Hu whee G ad H ae x ad H is osigula takig H to be the ivese of K. We hope to claify this with a clea poof that idicates the exact fo of the feedback law (3) ad the siple stuctue of the gai G. (7) Pojectios ad Retactios: We will use a o-othogoal pojectio appoach to obtai ou esults o alost PR/SPR i a uified way. idepotet opeato P o R is a pojectio ad, fo [9] pp-, we have R = R( P) N ( P) whee R(P) is the age of P ad N(P) is the ull space of P. othogoal pojectio is oe whee N ( P) = R( P) o P = P. Give a -diesioal subspace S i R with a (possibly o-othogoal) pojectio P oto it, we will defie:w, x R, Px = Wz fo soe z R. f we let the colus of W fo a othooal basis fo S, the W W = adww = P which is S othogoal pojectio oto S. heefoe z = W Px is uique ad we call W the etactio of P (o the subspace S) oto R. hus we ca etact o pullback a pojectio ito a lowe diesioal space. lost SPR Result: We ow state the followig: heoe : (, is lost SPR with cotol law (3) if ad oly if the high fequecy gai B is positive defiite ad the ope-loop syste is iiu phase. he output Feedback Gai that does this is G ( γ ( + ) with γ > ad sufficietly lage (see eq (9)). We eed thee leae: Lea : f B is osigula the P is a (o-othogoal) pojectio oto the age of R(, alog the ull space of,n( with the copleetay pojectio, ad P P R = R( N (. Poof of Lea : oside P = ( ( = P ; hece it is a pojectio. lealy, R( P ) R( ad z = Bu R( P z = ( lso Bu = Bu = z R( P ) R( P ) = R( ). B

3 N ( P ) = N ( because N ( N ( P ) W WW = ad z N ( P ) P z = P z = z = o N ( P ) N (. W P W P W gives P is a pojectio oto R( alog N (, So but P P so it is ot othogoal pojectio. = W W = = We have R = R( P ) N ( P ) ; hece because the colus of W ae i N ( ad R = R( N (. his copletes the poof of P pojects oto N (. Futheoe Lea. Lea : f B is osigula, the W W = P + W W P B osigula W WB = ad W = [ ]. W = P + P = becausew W is othogoal pojectio oto N (. lso diect calculatio his coodiate tasfoatio puts () ito B B B WB = = y& = y + z + Bu W P B oal fo: (8) z& = y + z yields: W = [ W ] = [ ] whee the subsyste: (,, ) is called the W WW = zeo dyaics of () ad W P W P W ; W ; his copletes the poof of Lea. W ( W ( W Poof of Lea : y = x = ( x = P x oside that ad P x = x = ; y. = = lso P Futheoe ad P B = B B =. P x = W z ; z R W fo a othooal basis fo N (. he we have W W = ad the etactio: z = W P x. Now, usig x = P x + P x fo lea, we have y& = P x& = P ( P x + P x) + P Bu = ( y + ( + ( Bu = y + z + Bu d z& = W P x& = W P ( P x + P x) = W P ( = W ( whe the - colus of + W ( W z = y + his yields the oal fo (8). y y + W z ) + W P Bu ( W z ) z. hoose W. he W has a ivese W P explicitly stated as W B W.his [ ] ( Lea 3: ssue B is osigula. efie ( WW, B W W ) whee the the tasissio zeos of (, ae the eigevalues of. osequetly a) is stable if ad oly if (, is iiu phase ad b) is weakly stable if ad oly if (, is weakly iiu phase. Poof of Lea 3: Hecefoth i the iteest of savig space we will oit the poofs ad efe the eade to the expaded vesio of this pape []. Note that by escalig W with the ivese of this co-odiate tasfoatio ca poduce B = ad = [( ) ] B which will dastically siplify the poofs i [8]. Fially, ou coodiate tasfoatio poduces the so-called oal fo ad the dyaics associated with ae the zeo dyaics [9]. the poof of heoe : we ust choose: γ > λ ( Q Q Q ) (9) ax to ake ( + BG, SPR. lost PR Results: t is ow elatively easy to state ecessay coditios fo output feedback to ake 3

4 (, PR as well: heoe : f the liea syste (, ca be ade PR by output feedback (3) the (, is weakly iiu phase ad B is positive defiite, but ot covesely. o see that the covese is false, take,, [ which is aleady i B ] oal fo with the weak iiu phase popety ad B=. Let G γ < which yields γ = ad P = p > whe p >. hus P satisfies PB = but P + P ca oly be satisfied whe p = which cotadicts P >. Note that a slight chage i the above exaple, i.e., will poduce PR via output feedback (3) with p =. Yet this syste is still weakly iiu phase with B>.So it takes soethig stoge tha the weakly iiu phase coditio to get PR via output feedback. We ow give ecessay ad sufficiet coditios fo this: heoe 3: (, is alost PR with output feedback (3) if ad oly if B is positive defiite, the ope-loop syste is weakly iiu phase ad the tasfe fuctio of the zeo dyaics etacted oto its agial spectal subspace: P ( ( s is PR withu ) whee = U S V [ U U ] adv = U V V is the spectal decopositio of ito its stable subspace ( whee all eigevalues ae i the ope left half-plae) ad its agial subspace ( whee all the eigevalues ae siple ad o the iagiey axis).so = V U ; U V whee the zeo ; dyaics (, ) ae give i Lea., Note that sice B is positive defiite, scalig by it o its ivese does ot chage the PR popety. osequetly, P ( s PR ca ) eplace P PR i heo. 3. Siila esults usig passivity wee obtaied ecetly i []. yaic Output Feedback: Fially, oe ight iagie that weake ecessay ad sufficiet coditios ight suffice i heo. o heo.3 whe yaic Output Feedback is used, e. g. u = L y + L η + u & η = L y + L η + η () di( η) = ρ he siplest way to use the dyaic cotolle is to cacel all the zeo dyaics, i.e. choose L ; L L.lso G ( ; L ; ε η z will satisfy & ε = ε which is stable whe the ope-loop syste is iiu phase. With these dyaics cacelled, the closed-loop tasfe fuctio becoes = ( )( s ( + BG)) ( ). he takig P G ( γ + ), γ >, we will have P = ( )( s + γ ( which is SPR with P ( > ad Q γ i the K-Y coditios (6), as log as B is positive defiite. Howeve we ca cobie () ad () ito the well-kow Exteded Output Feedback Foulatio: & β = β + Bv x y u w = β β, w, v η η η + v v = Lw B whee, B, () ρ ρ L L, L ρ L L his is a iial ealizatio of the closed loop tasfe fuctio give by P = (s ( + BL) B, which ust be ad sufficiet coditios fo this ae (, B, ) iiu phase ad B >. But B B B = = if ad oly if B >. B ade SPR by choice of gai atix L. Now we ca apply heo. to () ad see that ecessay > Futheoe, 4

5 fo Re(s), + + ρ = ak B s ρ ρ = ak ρ B = ak = s ρ ρ ρ B ρ ρ ak + ak ρ B = ρ + ak B heefoe, is iiu phase if ad oly if (, is iiu phase. So the Exteded Output Feedback otol Law i () will equie the vey sae ecessay ad sufficiet coditios as i heo. to poduce closed-loop SPR. osequetly, static P output feedback is copletely adequate to the task! [], this sae coclusio is daw. the case of lost PR, soe thigs siplify with dyaic output feedback: heoe 4: (, is ost PR with dyaic output feedback if ad oly if the high fequecy gai B is positive defiite ad the ope-loop syste is weakly iiu phase. Note that, i [9] heo.4.7, these sae coditios ae show to ake oliea systes passive usig dyaic output feedback. Hece, ou heo.4 is a coollay of that i [9]. Howeve, ou esult uses the dyaic cotolle to cacel oly the zeo dyaics etacted oto the agially stable subspace, wheeas [9] cacels all the zeo dyaics. So ou appoach will esult i a lowe ode dyaic cotolle, i geeal. llustative Exaples: We efe the eade to [] pplicatio: daptive otol: this sectio we will apply the above esults to iect Model Refeece daptive otol [4]-[5] ad istubace Rejectio [6] to obtai a stoge stability esult tha is peseted i those papes. oside the followig fiite-diesioal liea tieivaiat syste: x& = x + Bu + Γu y = x with a pesistet distubace of the fo: u = θ φ & φ = F φ ; F agially stable φ (3) (4) whee is a vecto of kow bouded basis fuctios which ca be geeated i eal tie, e. g. siusoidal fuctios o steps geeated by the kow tie ivaiat agially stable syste F.Howeve θ is ukow. he efeece odel tackig poble of [4]-[5] ca be stated siply as the followig: e y y y t y = z (6) whee z& = F z ; F agially stable hus the plat (3) ust tack the odel (6) i the pesece of the distubace (4). his will be accoplished by the daptive otol Law: u = G z + G e + G φ (7) We itoduce the ideal tajectoies: & x = x + Bu y = x = y e y + Γu (8) x = S z + S φ whee (9) u = S z + S φ he existece of gais S ij equivalet to the followig Matchig oditios: S F = S = S S F = S = S + BS + BS + Γθ to satisfy (8)-(9) is () () We peset esults o the existece of solutios to these Matchig oditios ext. heoe 5: f B>, the a) () has uique solutios ( S, S ) if ad oly if o eigevalue of F is coo with a tasissio zeo of the ope loop syste ; 5

6 b) () has uique solutios ( S, S ) if ad F oly if o eigevalue of is coo with ay tasissio zeo of the ope loop syste. We have the followig ipoved adaptive stability esult: heoe 6: Let the plat (3) with pesistet distubace (4) be cotolled to tack the efeece odel (6) by the adaptive cotolle (7) with adaptive gais (6). f the high fequecy gai B> ad the ope-loop plat is weakly iiu phase ad the tasfe fuctio of the zeo dyaics etacted to the agial spectal subspace is positive eal, the ad the plat, G, G )., e y t output tacks the odel output ad ejects the pesistet distubace with bouded adaptive gais ( G e Note that if ( B> ad wee alost SPR (i.e. iiu phase) the the above esult would be tue. Howeve, this was show befoe i [4] ad [5]. oclusios ad Reaks: Please see [] fo these. Refeeces: ) R. Lozao, et alia, issipative Systes alysis ad otol, Spige, Lodo,. ) B. R. Baish, Necessay ad Sufficiet oditios fo Quadatic Stabilizability of a Ucetai liea Syste, J. Optiizatio theoy ad pplicatios, Vol. 46, pp , ) K. S. Naeda ad. M. asway, Stable daptive Systes, Petice-Hall, 989 4) B.. O. deso ad S. Vogpaitled, Netwok alysis ad Sythesis- Mode Syste heoy ppoach,petice-hall, ) J.. We, ie oai ad Fequecy oai oditios fo Stict positive Realess, EEE as., Vol. -33, pp , ) G. Gao ad P. oaou, Necessay ad Sufficiet oditios fo Stictly Positive Real Matices, EE Poceedigs, Vol. 37,pat G, pp , 99. 7) H. Weiss, Q. Wag, ad J. Speye, Syste haacteizatio of Positive Real Matices, EEE as., Vol. -39,pp ,994. 8) G. Gu, Stabilizability oditios of Multivaiable Ucetai Systes via Output Feedback otol, EEE as., Vol. -35, pp 95-97,99. 9). Byes,. sidoi, ad J. Willes, Passivity, Feedback Equivalece, ad the Global Stabilizatio of Miiu Phase Noliea Systes, EEE as., Vol. -36, pp 8-4, 99. ) R. Ho ad. Johso, Matix alysis, abidge Uivesity Pess, 99, heo p47. ) H. Khalil, Noliea Systes- 3 d Editio, Petice-Hall,,hapt 6 pp7-6. ). Huag, P. oaou, J. Maoulas, ad M. Safoov, desig of Stictly Positive Real systes usig ostat Output feedback, EEE tas utoatic cotol, Vol. -44, pp , ) G. Golub ad. Va Loa, Matix oputatios-3 d Editio, he Johs Hopkis Uivesity Pess, 996, heo.7..6 pp ). Bakaa ad H. Kaufa, Global Stability ad Pefoace of a daptive otol lgoith, t. J. otol, Vol. 46,986,pp ) J. We ad M. Balas, Robust daptive otol i Hilbet Space, J. Matheatical. alysis ad pplicatios, Vol 43,pp - 6,989. 6) R. Fuetes ad M. Balas, "iect daptive Rejectio of Pesistet istubaces", Joual of Matheatical alysis ad pplicatios, Vol 5, pp 8-39,. 7) G. Golub ad. Va Loa, Matix oputatios-3 d Editio, he Johs Hopkis Uivesity Pess, 996, Lea 7..5, pp ). Kailath, Liea Systes, Petice-Hall, 98 pp ). Kato, Petubatio heoy fo Liea opeatos, Spige, 966. ) M. Laso ad P. Kokotovic, O Passivatio with yaic Output Feedback, EEE as, -46, pp ,. ) M. Balas ad R. Fuetes, No-Othogoal Pojectio ppoach to haacteizatio of lost Positive Real Systes with a pplicatio to daptive otol, pepit. 6