Automatica. Invertibility of switched nonlinear systems. Aneel Tanwani, Daniel Liberzon. a b s t r a c t. 1. Introduction

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1 Automatca 46 (2010) Contents lsts avalable at ScenceDrect Automatca journal homepage: wwwelsevercom/locate/automatca Invertblty o swtched nonlnear systems Aneel Tanwan, Danel Lberzon Coordnated Scence Laboratory, Unversty o Illnos at Urbana-Champagn, USA a r t c l e n o a b s t r a c t Artcle hstory: Receved 16 February 2009 Receved n revsed orm 26 January 2010 Accepted 18 July 2010 Avalable onlne 31 August 2010 Keywords: Invertblty Swtched nonlnear systems Swtch-sngular pars Structure algorthm Ths artcle addresses the nvertblty problem or swtched nonlnear systems ane n controls The problem s concerned wth reconstructng the nput and swtchng sgnal unquely rom gven output and ntal state We extend the concept o swtch-sngular pars, ntroduced recently, to nonlnear systems and develop a ormula or checng the gven state and output orm a swtch-sngular par A necessary and sucent condton or the nvertblty o swtched nonlnear systems s gven, whch requres the nvertblty o ndvdual subsystems and the nonexstence o swtch-sngular pars When all the subsystems are nvertble, we present an algorthm or ndng swtchng sgnals and nputs that generate a gven output n a nte nterval when there s a nte number o such swtchng sgnals and nputs Detaled examples are ncluded to llustrate these newly developed concepts 2010 Elsever Ltd All rghts reserved 1 Introducton Swtched systems reer to dynamcal systems wth dscrete swtchng events Ther evoluton s descrbed by a collecton o dynamcal subsystems, together wth a swtchng sgnal, that speces an actve subsystem at each tme nstant Examples nclude swtchng power converters, networs wth swtchng topologes, and arcrat wth derent thrust modes Also, swtchng control technques, especally n the adaptve context, have been shown to acheve stablty and mproved transent response (see Lberzon, 2003, Chapter 6) Because o ther utlty n modelng and control desgn, swtched systems have been a ocus o ongong research and several results related to stablty, controllablty, observablty, and nput-to-state stablty o such systems have been publshed; see Lberzon (2003) or reerences More recently, Vu and Lberzon (2008) ntroduced the problem o nvertblty o swtched lnear systems In ths paper, we extend ther methodology to study the problem o nvertblty o contnuous-tme swtched nonlnear systems, whch s concerned wth ndng the condtons on the subsystems to guarantee unque recovery o the swtchng sgnal and the nput rom the ntal state and the output The problem statement s analogous to the classcal nvertblty problem or Ths wor was supported by NSF under grant ECCS The materal n ths paper was partally presented at the IEEE Conerence on Decson and Control December 9 11, 2008, Cancun, Mexco Ths paper was recommended or publcaton n revsed orm by Assocate Edtor Hendr Njmejer under the drecton o Edtor Andrew R Teel Correspondng author Tel: ; ax: E-mal addresses: tanwan2@llnosedu (A Tanwan), lberzon@llnosedu (D Lberzon) nonswtched systems In act, or every control system wth an output, we have an nput output map and the queston o let (resp rght) nvertblty s, roughly speang, that o the njectvty (surjectvty) o ths map System nvertblty problems are o great mportance rom a theoretcal and practcal vewpont and have been studed extensvely or ty years, ater beng poneered by Brocett and Mesarovc (1965) For nonswtched lnear systems, the algebrac crteron or nvertblty and the constructon o nverse systems were gven by Slverman (1969), and also by San and Massey (1969) The systematc study o nvertblty or nonswtched nonlnear systems began wth Hrschorn, who rst studed the sngle-nput sngle-output (SISO) case (see Hrschorn, 1979b), and then generalzed Slverman s structure algorthm to multplenput multple-output (MIMO) nonlnear systems (see Hrschorn, 1979a) Sngh (1981) then moded the algorthm to cover a larger class o systems Isdor and Moog (1988) used ths algorthm to calculate zero-output constraned dynamcs and reduced nverse system dynamcs The algorthm s also closely related to the dynamc extenson algorthm used to solve the dynamc state eedbac nput output decouplng problem (see Njmejer & van der Schat, 1990, Sectons 82 and 113) Geometrc methods have been studed by Njmejer (1982) A hgher-level nterpretaton gven by a lnear-algebrac ramewor, whch also establshes lns between these algorthms and the geometrc approach, s presented by D Benedetto, Grzzle, and Moog (1989) We also recommend a useul survey on varous nvertblty technques by Responde (1990) The problem o nvertblty or swtched lnear systems was ntroduced very recently by Vu and Lberzon (2008) where the authors used Slverman s structure algorthm to ormulate condtons /$ see ront matter 2010 Elsever Ltd All rghts reserved do:101016/jautomatca

2 A Tanwan, D Lberzon / Automatca 46 (2010) or the nvertblty o swtched systems wth contnuous dynamcs The problem o nvertblty or dscrete-tme swtched lnear systems has been dscussed by Mlleroux and Daaouz (2007) and Sundaram and Hadjcosts (2006) but there, the authors assume that the swtchng sequence s nown and nd the correspondng nput In ths paper, we mae no such assumpton and adopt an approach smlar to Vu and Lberzon (2008), to study the nvertblty problem or contnuous-tme swtched nonlnear systems, ane n controls, usng Sngh s nonlnear structure algorthm 1 The concept o sngular pars, conceved by Vu and Lberzon (2008), s extended to nonlnear systems; although, n ths paper, such pars are termed as swtch-sngular pars to avod conlct wth the sngulartes o ndvdual nonlnear subsystems Even though the orm o the man result (nvertblty o subsystems plus no swtch-sngular pars) and the concepts presented n ths artcle are essentally smlar to those gven by Vu and Lberzon (2008), the man contrbuton o ths paper les n the techncal detals o developng and checng the condtons or nvertblty o nonlnear systems In partcular, the use o the nonlnear structure algorthm, possblty o nte escape tmes, and the exstence o sngulartes n state space and output set requre more careul analyss and techncal rgor as compared to the lnear case As s the case n the classcal settng o nonswtched systems, we start wth an output and an ntal state, but here there s a set o dynamc models and we wsh to recover the swtchng sgnal n addton to the nput In the context o hybrd systems, recoverng the swtchng sgnal s equvalent to the mode dentcaton or hybrd systems or the observablty o the dscrete state varable (locaton), whch has been studed by Babaal and Pappas (2005), Vdal, Chuso, and Soatto (2002) and Vdal, Chuso, Soatto, and Sastry (2003) Hence, the nverson o swtched systems can also be thought o dong the mode detecton and nput recovery smultaneously Consequently, the basc dea or solvng the nvertblty problem s to rst do the mode dentcaton by utlzng the relatonshp among the outputs and the states o the subsystems, and then use the nonlnear structure algorthm or the correspondng subsystem to recover the nput For the case when subsystems are lnear, Slverman s structure algorthm seems to be the most convenent tool to ormulate nvertblty condtons whch leads to a smple and elegant ran test or checng the exstence o swtch-sngular pars, but n nonlnear systems t s hard to acheve such a level o generalty For ths reason, we start wth the SISO case to hghlght the techncal dcultes n movng rom lnear to nonlnear systems Dscussng the SISO case rst also helps n understandng the concepts behnd the ormula derved or vercaton o swtch-sngular pars The paper s organzed as ollows Secton 2 contans the dentons o nvertblty and the ormal problem statement The man result on let nvertblty s presented n Secton 3 We then gve a characterzaton o swtch-sngular pars and the constructon o nverse systems n Secton 4 An algorthm or output generaton s gven n Secton 5 along wth an example We conclude the artcle wth some remars on urther research drectons 2 Prelmnares In ths secton, we develop the requred notatons and provde some bacground on nvertblty o nonswtched nonlnear systems Based on that, we develop the denton or the nvertblty o swtched nonlnear systems ollowed by the ormal problem statement to whch we see soluton n the paper 1 A related problem s dscussed by Chab, Boutat, Banal, and Kratz (2007) but t doesn t ollow the same theoretcal approach we do, and nstead uses a heurstc approach wth the purpose o studyng a specc applcaton 21 Nonswtched nonlnear systems The dynamcs o a square nonlnear system, ane n controls, are gven by m ẋ = (x) + G(x)u = (x) + g Γ := (x)u, (1) =1 y = h(x) where x M, an n-dmensonal real connected smooth manold, or example R n ;, g are smooth vector elds on M, and h : M R m s a smooth uncton Admssble nput sgnals are locally essentally bounded, Lebesgue measurable unctons u : [t 0, ) R m I the two nputs der on a set o measure zero, e u 1 (t) = u 2 (t) almost everywhere (ae), then they are consdered to be equal We use the notaton u [t0,t) to denote the nput u over the tme nterval [t 0, T); and Γ x0 (u) denotes the state trajectory generated by (1) ater applyng the nput u wth ntal condton We start o by revewng classcal dentons o nvertblty or such systems For that, consder the nput output map H x0 : U Y or some nput uncton space U and the correspondng output uncton space Y H x0 maps an nput u( ) to the output y( ) generated by the system drven by u( ) wth an ntal condton Snce the state trajectores o nonlnear systems may exhbt nte escape tmes, an nput u [t0, ) may not have a well dened mage n the output space, over the nterval [t 0, ), under ths map For ths reason, we only consder nputs over a nte nterval [t 0, T), whch s the maxmal nterval o the exstence o a soluton, such that H x0 (u [t0,t)) = y [t0,t) always exsts and s well-dened Invertblty 2 o the dynamcal system (1) bascally reers to the njectvty o the map H x0 Beore gvng a ormal denton, let us loo at an example rst Example 1 Consder a nonswtched nonlnear system wth two nputs and two outputs, ẋ1 x1 u 1 y1 x1 ẋ 2 = x 3 u 1, =, M = R 3 y ẋ 3 u 2 x 2 2 We then have ẏ 1 = x 1 u 1, (2a) ÿ 2 = x 3ÿ 1 ẏ 1 ẏ 2 + ẏ 1 u 2 x 1 (2b) It ollows that u 1 can be recovered unquely rom ẏ 1 x 1 0, and u 2 can be recovered unquely rom ÿ 2 ẏ 1 0 and x 1 0 The pont x 1 = 0 and ẏ 1 = 0 are the sngulartes n the state space and the output space, respectvely Let M α := {x R 3 x 1 0}; Y s = {z R 2 z 1 = 0}, and Y s := {y : [t 0, T) R 2 ẏ(t) Y s or almost all t [t 0, t 0 + δ) [t 0, T), where δ > 0 s arbtrary} In words, Y s ncludes those outputs whch reman n a sngular set or some tme The complement o Y s s gven by Y α := {y : [t 0, T) R 2 ẏ(t) Y s 1 or almost all t [t 0, t 0 + ε) and some ε > 0} I the system s drven by a class o nputs u such that the resultng moton Γ x0 (u) M α ae and H x0 (u) Y α, then there s a one-to-one relaton between the output and nput sgnals provded ther domans are restrcted to [t 0, t 0 + ε) In summary, the nput can be recovered unquely usng the nowledge o output, ts dervatves and possbly some states as long as the output and state trajectores do not ht some sngulartes We now proceed to the ormal denton o nvertblty or nonswtched systems 2 Throughout the paper, nvertblty reers to the let nvertblty

3 1964 A Tanwan, D Lberzon / Automatca 46 (2010) Denton 1 Fx an output set Y and consder an arbtrary nterval [t 0, T) The system (1) s nvertble at a pont := x(t 0 ) M over Y or every y [t0,t) Y, the equalty H x0 (u 1[t0,T)) = H x0 (u 2[t0,T)) = y [t0,t) mples that ε > 0 such that u 1[t0,t 0 +ε) = u 2[t0,t 0 +ε) The system s strongly nvertble at a pont t s nvertble or each x N( ), where N s some open neghborhood o The system s strongly nvertble there exsts an open and dense submanold M α such that M α, the system s strongly nvertble at As llustrated n Example 1, a system s nvertble at or the class o nputs u( ) such that along the trajectory o the system (1), the resultng moton x( ), y( ) does not ht any sngulartes It s entrely possble that the state trajectory or the output hts sngularty at a tme nstant t 0 +ε wth 0 < ε < T t 0, thus mang t mpossble to recover u unquely beyond t 0 +ε; ths explans why we requre dstnct nputs over arbtrarly small tme domans n Denton 1 In the most general constructon o nverse systems as the one gven by Sngh (1981), there exsts a set o sngular outputs Y s such that the system s not nvertble or y Y s ; and ts complement Y α := Y \ Y s s the set o all outputs on whch the system s strongly nvertble Also, n general, the nverses o nonlnear dynamcal systems are not dened on the entre state space I the vector elds (x), g(x) and the output uncton h(x) are analytc, then the sngular ponts are reduced to a closed and nowhere dense set comprsng zeros o certan analytc unctons Under these assumptons, the system s nvertble then there exsts an open and dense subset o M on whch the dynamcs o a nonlnear system are nvertble; that subset s called the nverse submanold and s denoted by M α All these notons wll be developed ormally n Secton 4 Usng Denton 1, nvertblty at s equvalent to sayng that u 1[t0,t 0 +ε) u 2[t0,t 0 +ε) or all ε (0, T t 0 ) mples that H x0 (u 1[t0,T)) H x0 (u 2[t0,T)) Ths noton was captured by Hrschorn (1979a) Our denton s essentally the same as one consdered by Hrschorn n the sense that both notons address the njectvty o an nput output map The derence les n the act that Hrschorn consdered a class o analytc nonlnear systems wth analytc nputs and Y s =, an empty set In that case, the system s nvertble and the state trajectory starts rom a nonsngular set, t s possble to recover nputs on a small nterval but because o analytcty, we contnue to recover nputs unquely even ater httng sngularty; or two analytc nputs are derent on a subnterval then they are derent everywhere, otherwse ther derence (an analytc uncton) would have an nnte number o zeros on a nte nterval In ths paper though, we consder nonanalytc systems drven by nputs that are not necessarly analytc, so the nput recovery can only be guaranteed over small tme ntervals only We wll now generalze ths noton o local nvertblty to the swtched systems 22 Swtched nonlnear systems In the paper we wll consder swtched nonlnear systems, ane n controls, that have the ollowng structure: m ẋ = Γ σ : σ (x) + G σ (x)u = σ (x) + (g ) σ (x)u, (3) =1 y = h σ (x) where σ : [t 0, T) P s the swtchng sgnal that ndcates the actve subsystem at every tme, P s some nte ndex set, and p, G p, h p, where p P, dene the dynamcs o ndvdual subsystems The state space M s a connected real smooth manold o dmenson n, or example R n ; p, (g ) p are real smooth vector elds on M, and h p : M R m s a smooth uncton A swtchng sgnal s a pecewse constant and everywhere rght-contnuous uncton that has a nte number o dscontnutes at τ, whch we call swtchng tmes, on every bounded tme nterval Denote by σ p [t 0,T) the constant swtchng sgnal over the nterval [t 0, T) such that σ p (t) := p P, t [t 0, T) We assume that all the subsystems are equdmensonal, they lve n the same state space M, and that there s no state jump at swtchng tmes For any ntal state, swtchng sgnal σ ( ), and any admssble nput u( ), a soluton o (3) always exsts (n Carathéodory sense) and s unque, provded the low o every subsystem s well-dened or the tme nterval durng whch t s actve, e, the state trajectores do not blow up n nte tme In act, ths assumpton results n absolutely contnuous state trajectores (see Sontag, 1998) Denote by [t 0, T) the maxmal nterval o exstence o soluton, so that the outputs are well-dened on [t 0, T) Snce the swtchng sgnals are rght-contnuous, the outputs are also rght-contnuous (note that, n general, h (x) h j (x), or j) and whenever we tae the dervatve o an output, we assume t s the rght dervatve For p P, denote by Γ p,x0 (u) the trajectory o the correspondng subsystem wth the ntal state and the nput u, and the correspondng output by Γ O p, (u) We wll use F pc to denote the space o pecewse rghtcontnuous unctons 3 and F n to denote the subset o F pc whose elements are n tmes derentable between two consecutve dscontnutes Lewse, F AC denotes the subset o F pc whose elements are absolutely contnuous between two consecutve dscontnutes Fnally, we use or the concatenaton o two sgnals In case o swtched systems (3), the map H x0 has an augmented doman; that s, now we have a (swtchng sgnal nput)-output map H x0 : S U Y, where S s a swtchng sgnal set, U s the nput space, and Y s the output space Let us rst extend the denton o nvertblty to swtched systems Denton 2 Fx an output set Y and consder an arbtrary nterval [t 0, T) A swtched system s nvertble at a pont over Y or every y [t0,t) Y, the equalty H x0 (σ 1[t0,T), u 1[t0,T)) = H x0 (σ 2[t0,T), u 2[t0,T)) = y [t0,t) mples that ε > 0 such that σ 1[t0,t 0 +ε) = σ 2[t0,t 0 +ε) and u 1[t0,t 0 +ε) = u 2[t0,t 0 +ε) A swtched system s strongly nvertble at a pont t s nvertble at each x N( ), where N s some open neghborhood o A swtched system s strongly nvertble there exsts an open and dense submanold M α o M such that M α, the system s strongly nvertble at For lnear swtched systems, as dscussed by Vu and Lberzon (2008), all the notons n Denton 2 concde and a system s termed nvertble the nput and swtchng sgnal could be recovered unquely or all The nvertblty property ormulated n Denton 2 may al to hold n two ways: (a) ether because there exst two derent nputs u 1 and u 2 that yeld the same output or (b) because there exst two derent swtchng sgnals σ 1 ( ) and σ 2 ( ) that yeld the same output The rst case reers to the noton o classcal nvertblty as already explaned n Denton 1 and Secton 21 To address the second possblty, we need the concept o swtch-sngular pars whch reers to the ablty o more than one subsystem to produce a segment o the desred output startng rom the same ntal condton The ormal denton s gven below: 3 By pecewse rght-contnuous unctons, we mean that there s a nte number o jump dscontnutes n any nte nterval; the uncton s contnuous n between any two consecutve dscontnutes; and the uncton s contnuous rom the rght at dscontnutes To avod excessve rgdness, we wll use the term pecewse contnuous throughout the paper, and t s understood that pecewse contnuous means pecewse rght-contnuous

4 A Tanwan, D Lberzon / Automatca 46 (2010) Denton 3 Consder M and y Y p Y q on some tme nterval [t 0, T), where p, q P, p q The par (, y) s a swtch-sngular par o the two subsystems Γ p, Γ q there exst u 1, u 2 and ε > 0 such that Γ O p, (u 1[t0,t 0 +ε)) = Γ O q, (u 2[t0,t 0 +ε)) = y [t0,t 0 +ε) I all subsystems are lnear, = 0 and y 0 always orm a swtch-sngular par regardless o the dynamcs o the subsystems Ths s because u 0 and any swtchng sgnal wll produce y 0, that s, H 0 (σ, 0) = 0 σ, and thereore H 0 s not njectve the zero uncton belongs to Y In nonlnear systems, ths s not the case n general and all swtch-sngular pars are solely determned by the subsystem dynamcs As stated earler and wll be ormally proved below, the swtched system s not nvertble Y contans outputs that orm swtch-sngular pars wth Thus, there exst any swtch-sngular pars, we have to restrct the output set Y, nstead o lettng Y be the set o all possble concatenatons o nonsngular output trajectores Next, we use the concept o swtch-sngular pars to study the nvertblty problem o swtched systems Snce Denton 2 contans derent varants o nvertblty, we start o wth the weaest o them all, e, nvertblty o a swtched system at a pont In partcular, we are nterested n solvng the ollowng undamental problem: Fnd a sutable set Y and a condton on the subsystems such that the system s nvertble at over Y An abstract characterzaton o the set Y and constrants on subsystem dynamcs whch guarantee nvertblty are gven n Secton 3 under Theorem 1; Corollares 1 and 2 then characterze the set Y more explctly (dependng on the requred varant o nvertblty) Later n Secton 4, we gve mathematcal ormulae (Lemmas 1 through 5) or checng the abstract condtons gven n Secton 3 3 Characterzaton o nvertblty In ths secton, we descrbe the output set Y used n Denton 2 and gve condtons on the subsystem dynamcs so that the swtched system s nvertble or some sets S, U, and Y Restrctng the outputs to le n Y mples a set o restrctons on the set o allowable nputs, but an explct characterzaton o such nputs s not always obtanable That s why we do not explctly specy what the nput sets U and S are, but nstead specy the set Y and then U wll be the correspondng set whch, together wth S, generates Y For all p P, let Y p be the set o smooth outputs 4 that can be generated by Γ p, and let Y all be the set o all the possble concatenatons o all elements o Y p, p P Due to the exstence o certan sngular outputs (or whch the system s not nvertble), we see nvertblty at a xed pont over a subset Y α Y all Theorem 1 Consder the swtched system (3) and an output set Y α Y all The swtched system s nvertble at M over Y α and only each subsystem Γ p s nvertble at over Y α Y p and or all y Y α, the pars (, y) are not swtch-sngular pars o Γ p, Γ q or all p q, p, q P Proo (Necessty) We show that any o the subsystems s not nvertble at or there exst swtch-sngular pars (, y), then the swtched system s not nvertble Suppose that a subsystem Γ p, p P, s not nvertble at over Y α Y p, then there exsts y [t0,t) Y α Y p such that Γ O p, (u 1[t0,T)) = Γ O p, (u 2[t0,T)) = y [t0,t) or some u 1, u 2 and ε 4 Ths assumpton can be relaxed dependng upon the system under consderaton, see Remars 4 and 5 n Secton 4 or detals (0, T t 0 ), u 1 u 2 on [t 0, t 0 + ε) Ths mples that H x0 (σ p [t 0,T), u 1[t 0,T)) = H x0 (σ p [t 0,T), u 2[t 0,T)) = y [t0,t) and thus, Denton 2 mples that the swtched system s not nvertble at over Y α For necessty o the second condton, suppose that y Y α Y p Y q, so that (, y) s a swtch-sngular par o Γ p, Γ q, p q Ths means that both subsystems, even though nvertble at, can produce ths output over the nterval [t 0, t 0 + ε) [t 0, T), ε > 0 Consequently, u 1[t0,T), u 2[t0,T) (possbly same) such that Γ O p, (u 1[t0,T)) = Γ O q, (u 2[t0,T)) = y [t0,t) Hence, we have H x0 (σ p [t 0,T), u 1[t 0,T)) = H x0 (σ q [t 0,T), u 2[t 0,T)) = y [t0,t); that s, the premage o y s not unque as σ p σ q on [t 0 + t 0 + ε), ε (0, T t 0 ) Ths mples that the swtched system s not nvertble at or gven Y α Sucency: Suppose that or the gven M, there exst some nputs u 1, u 2 and swtchng sgnals σ 1, σ 2 such that H x0 (σ 1, u 1 ) = H x0 (σ 2, u 2 ) = y Y α over [t 0, T) Snce (, y) s not a swtchsngular par, there exsts ε 1 such that σ 1 (t) = σ 2 (t) = p, t [t 0, t 0 + ε 1 ) 5 and y [t0,t 0 +ε 1 ) Y p Snce Γ p s nvertble at, ε 2 < ε 1 such that u 1[t0,t 0 +ε 2 ) = u 2[t0,t 0 +ε 2 ) and Γ O p, (u 1[t0,t 0 +ε 2 )) = Γ O p, (u 2[t0,t 0 +ε 2 )) = y [t0,t 0 +ε 2 ) Lettng ε = mn{ε 1, ε 2 }, t then ollows rom Denton 2 that the swtched system s nvertble at over Y α In the proo o the sucency part, the swtched system s strongly nvertble at or the sgnals whose doman s restrcted to the nterval [t 0, t 0 + ε), where t 0 + ε s the tme nstant at whch the state trajectory or the output enters the sngular set I the output y loses contnuty over the nterval [t 0, t 0 + ε) because o swtchng, then (σ [t0,t 0 +ε), u [t0,t 0 +ε)) = (σ [t0,τ 1 ), u [t0,τ 1 )) (σ [τ,t 0 +ε), u [τ,t 0 +ε)), where s the total number o swtches n the nterval [t 0, t 0 +ε) and τ, = 1,,, are the swtchng nstants Let us now consder a renement o Theorem 1 by characterzng the set Y α For all p P, let Y s p be the set o sngular outputs o Γ p or whch Γ p s not nvertble (see Example 1 and Secton 42, or Sngh (1981)), and let Y α = p Y p \ Y s p be the set o outputs on whch Γ p s nvertble at Dene Y s := p P Y s p as the collecton o all sngular outputs and let Y all be the set o outputs generated by all the possble concatenatons o all elements o Y p, p P ; Fnally, dene Y α := Y all \ Y s as a set o outputs over whch we see nvertblty We now have the ollowng moded verson o Theorem 1 Corollary 1 The swtched system s nvertble at over the set Y α and only the pars (, y) are not swtch-sngular pars o Γ p and Γ q, or all y Y α, or all p q, p, q P Proo By the applcaton o Theorem 1, the desred result s obtaned by showng that Γ p, p P, s nvertble at over the set Y α Y p By constructon, Y p = Y α p Ys p and Yα Y s p = ; usng these two equaltes, t s easy to see that Y α Y p Y α p As each subsystem Γ p s nvertble at over Y α p, t ollows that each subsystem Γ p s, n partcular, nvertble at over the output set Y α Y p Corollary 2 Consder the swtched system (3) and an output set Y α Y The swtched system s strongly nvertble at M over Y α and only each subsystem Γ p s strongly nvertble at 5 Ths argument can also be proved n another way: t wll be shown later that the ponts n state space that orm swtch-sngular pars are actually a zero set o smooth nonlnear equatons Thus, does not orm a swtch-sngular par wth y then there exsts a neghborhood N( ) such that x N( ), (x, y) s not a swtchsngular par As there are no swtch-sngular pars n N( ), ε 1 > 0 such that σ 1[t0,t 0 +ε 1 ) = σ 2[t0,t 0 +ε 1 )

5 1966 A Tanwan, D Lberzon / Automatca 46 (2010) over Y α Y p and there exsts a neghborhood N( ) such that or all x N( ), y Y α, the pars (x, y) are not swtch-sngular pars o Γ p, Γ q or all p q, p, q P Proo (Necessty) I the swtched system s strongly nvertble at, then N( ) such that the swtched system s nvertble at every x N( ) over Y α Let N( ) := N( ) By Theorem 1, each subsystem s nvertble at every x N( ), hence strongly nvertble at, and there does not exst any swtch-sngular pars (x, y), or all x N( ), y Y α Sucency: I each subsystem s strongly nvertble at, e, N p ( ) such that Γ p s nvertble at every x N p ( ), then N α := p P N p s an open set on whch all subsystems are nvertble I we dene N := N α N, then the swtched system s nvertble at every x N( ) over Y α and hence by Theorem 1, strongly nvertble at For the strong nvertblty o the swtched system on an open and dense subset, assume that the vector elds p, (g ) p and the output uncton h p are analytc Under these assumptons, a subsystem Γ p s strongly nvertble, then M α p denotes the nverse submanold o Γ p Corollary 3 The swtched system (3) s strongly nvertble, wth nverse submanold M α M, over an output set Y α Y and only each subsystem s strongly nvertble over Y α Y p and the subsystem dynamcs are such that the pars (, y) are not swtchsngular pars o Γ p, Γ q or all p q, p, q P, or every M α, and every y Y α Proo (Necessty) I the swtched system s strongly nvertble, then t s strongly nvertble at every M α over Y α By Corollary 2, each subsystem s strongly nvertble at every M α, and hence strongly nvertble wth nverse submanold M α Furthermore, there does not exst any swtch-sngular pars (, y), M α, y Y α Sucency: Under the gven hypothess, there exsts an nverse submanold M α p such that Γ p s strongly nvertble at every M α p over Yα Y p, or all p P Dene M α := p P Mα p, then M α s an open and dense subset o M because t s a nte ntersecton o open and dense subsets Under relatve topology, M α s a submanold Snce each subsystem Γ p s strongly nvertble at every M α over Y α Y p and there exst no swtch-sngular pars, applcaton o Corollary 2 mples that the swtched system s strongly nvertble at every M α over Y α In essence, Theorem 1, and the related corollares state that the nvertblty o subsystems n a certan sense mples the nvertblty o the swtched system n a smlar sense provded there are no swtch-sngular pars between the states and the outputs consdered Beore concludng ths secton, a couple o remars are n order Remar 1 For the swtched system (3), all the subsystems are globally nvertble n addton to the hypothess o Corollary 3, that s, M α = M and Y s =, then t s possble to recover the nputs and swtchng sgnals unquely over the tme nterval [t 0, T) Also note that T may be arbtrarly large the state trajectores do not exhbt nte escape tme Remar 2 I a subsystem has more nputs than outputs, then t cannot be (let) nvertble On the other hand, t has more outputs than nputs, then some outputs are redundant (as ar as the tas o recoverng the nput s concerned) Thus, the case o nput and output dmensons beng equal s, perhaps, the most nterestng case 4 Checng nvertblty In ths secton, we address the computatonal aspect o the concepts ntroduced n prevous sectons and develop algebrac crtera or checng the nvertblty o swtched systems The rst condton n Theorem 1 ass or nvertblty o subsystems and s vered by the structure algorthm To put everythng nto perspectve, we provde approprate bacground related to the nvertblty o nonswtched systems and use t to develop the concept o unctonal reproducblty To chec (, y) s a swtch-sngular par, we develop a ormula usng the unctonal reproducblty crtera o nonswtched systems Ater veryng the nvertblty o subsystems and nonexstence o swtch-sngular pars, we wll be able to construct a swtched nverse system that recovers the orgnal nput and swtchng sgnal unquely 41 Sngle-nput sngle-output (SISO) systems We start o wth the case when all the subsystems are SISO because t gves more nsght nto computatons and helps understand the concepts whch we wll later generalze to multvarable systems To ths end, consder a SISO nonlnear system ane n controls (1) wth m = 1 and assume t has a relatve degree r at (see Isdor, 1995), e, a neghborhood N( ) such that L g L h(x) = 0, x N(), = 0,, r 1 and 1 h(x)) h( ) 0, where L (L h(x) = (x) and L 0 x h(x) = h(x) To chec the subsystem s nvertble or not, ollowng Hrschorn (1979b), we rst derve an explct expresson or the nput u n terms o the output y by computng the dervatves o y as ollows: y(t) = h(x(t)) ẏ(t) = L h(x(t)) (4a) (4b) y (r) (t) = L r h(x(t)) + L gl r 1 h(x(t))u(t) (4c) From the last equaton, we can derve an expresson or u(t): u(t) = Lr h(x(t)) h(x(t)) + 1 h(x(t)) y(r) (t) (5) Hence, u can be determned explctly n terms o the measured output y, and state x On substtutng the expresson or u rom (5) n Eq (1), one gets the dynamcs or the nverse system: ż = (z) + g(z) Lr h(z) h(z) + 1 h(z) y(r), u = Lr h(z) h(z) + 1 h(z) y(r) (6) The dynamcs o ths nverse subsystem evolve on the set M α := {z M h(z) 0} M α s open and dense, g, h are analytc Snce the nverse system dynamcs are drven by y (r) ( ) whch satses Eq (4c), t s not hard to see that the state trajectores o the nverse system satsy the derental equaton o the orgnal system (1) where the nput has just been replaced by a uncton o y So the nverse system s ntalzed wth the same ntal condton as that o the plant, then both o the systems ollow exactly the same trajectory Ths dscusson motvates the ollowng result: Lemma 1 A SISO system s strongly nvertble at the system has a nte relatve degree r at

6 A Tanwan, D Lberzon / Automatca 46 (2010) Remar 3 The condton gven n Lemma 1 or strong nvertblty at a pont s only sucent, and not necessary As an example, consder ẋ = 1 + xu, y = x, x R, = 0; no relatve degree at, but the system s strongly nvertble at because the trajectory mmedately leaves the sngularty In general, ths occurs when the rst uncton o the sequence L g h(x), L g L h(x),, L g L h(x) whch s not dentcally zero (n a neghborhood o ) has a zero exactly at the pont x = A result somewhat smlar to Lemma 1 appears n Hrschorn (1979b, Theorem 21), where the author gves a necessary and sucent condton or strong nvertblty o a SISO system but consders only analytc systems wth a slghtly derent noton o relatve degree Remar 4 For SISO systems, the nput u appears n the r-th dervatve o the output (4) Thus the derentablty/smoothness o u wll not aect the exstence o rst r 1 dervatves o y I u : [t 0, T) R s a locally essentally bounded, Lebesgue measurable uncton, then y (r) ( ) exsts almost everywhere and y (r 1) ( ) s absolutely contnuous (see Sontag, 1998) So or SISO nonlnear nonswtched systems, U s dened as the space o unctons whch are locally essentally bounded and Lebesgue measurable; and Y α s the set o correspondng outputs We now turn to the concept o unctonal reproducblty, whch n broad terms means the ablty to ollow a gven reerence sgnal Ths concept wll help us study the exstence o swtch-sngular pars We loo at the condtons under whch a system can produce the desred output y d over some nterval [t 0, T) startng rom a partcular ntal state To be precse, gven the system (1) wth m = 1 and ntal state, we want to nd out there exsts a control u such that Γ O (u) = y d The ollowng result was gven by Hrschorn (1979b): Lemma 2 I the system (1), wth m = 1 and x(t 0 ) =, has a relatve degree r < at, then there exsts a control nput u such that Γ O (u) = y d and only y () d (t 0) = L h() = 0, 1,, r 1 (7) Ths result s easy to comprehend by loong at the expressons or the output dervatves (4) As control u(t) does not drectly aect y () (t), or = 1,, r 1, ther values at t 0 are determned by the ntal state Substtutng u(t) = Lr h(x(t)) h(x(t)) + 1 h(x(t)) y(r) d (t) (8) n (4c) gves y (r) (t) = y (r) d (t) Usng (7), repeated ntegraton yelds y(t) = y d (t) We can now easly chec or the swtch-sngular pars among Γ p, Γ q wth relatve degrees r p, r q respectvely, where p, q P Lemma 3 For SISO swtched systems, (, y) s a swtch-sngular par o two subsystems Γ p and Γ q and only y Y p Y q and y y (r κ 1) (t 0 ) = h κ ( ), κ = p, q (9) h κ ( ) L r κ 1 κ The example below llustrates the use o these concepts Example 2 Consder a SISO swtched system wth two modes x1 + x 2 0 ẋ = x Γ p := u, M = R 3 x 1 x 2 x 2 y = x 1 x2 0 ẋ = x Γ q := 2 x u, M = R 3 x 2 x 2 y = 2x 1 I Γ p s actve, then ẏ = x 1 + x 2 ; Γ q s actve, then ẏ = 2x 2 Both subsystems have relatve degree 2 on R 3 whch can be vered by tang second dervatve o the output I there exsts x R 3 whch orms a swtch-sngular par wth y Y p Y q, then the ollowng equalty must be satsed x1 = x 1 + x 2 2x1 2x 2 whch gves x 1 = x 2 = 0 Ths state constrant yelds y = ẏ = 0 I we let Y α := y : [t 0, T) R ẏ [t0,t) F AC y(t) and ẏ(t) 0 or almost all t [t 0, T), then there exsts no swtch-sngular par between R 3 and y Y α Theorem 1 and Lemma 1 ner that the swtched system generated by {Γ p, Γ q } s strongly nvertble wth nverse submanold R 3 over Y α Alternatvely, 0 then (, y) s not a swtch-sngular par or any y and the swtched system s strongly nvertble wth nverse submanold R 3 \{0} over Y all For general swtched nonlnear systems, t s hard to chec or the exstence o swtch-sngular pars To see ths, consder the system (3) wth m = 1 For smplcty, assume that P = {p, q} and the subsystems Γ p, Γ q have equal relatve degrees, e, r p = r q =: r Lemma 3 states that Γ p, Γ q have a swtch-sngular par (, y) and only ŷ = H p ( ) = H q ( ) (10) where ŷ = (y, ẏ,, y (r 1) ) T and H κ = (h p, L p h p,, L r 1 p h p ) T, κ = {p, q} To see there exst any swtch-sngular pars between two subsystems, one s nterested n solvng H p ( ) = H q ( ) or ; that s, that orms swtch-sngular par actually les n the soluton space o r-nonlnear equatons where each equaton tsel nvolves unctons o an n-dmensonal varable As t s hard to tal about the solutons o nonlnear equatons n general, nvestgaton nto more constructve condtons or checng o swtch-sngular pars s a topc o ongong research Nonetheless, n the case o SISO swtched blnear systems, the nonlnear equatons n (9) become lnear and the tas o checng the exstence o swtch-sngular pars between two subsystems s comparatvely easer, as llustrated below Example 3 Consder a swtched system wth SISO blnear subsystems, havng the dynamcs o the orm ẋ = A σ (t) x + B σ (t) xu, y = C σ (t) x (11) where σ (t) = p P, x R n, A p, B p R n n, C p R 1 n Also, u(t), y(t) R I some mode p P s actve over a tme nterval, then at any tme t n that nterval, the expresson or the dervatves o output s y(t) = C p x(t), ẏ(t) = C p A p x(t),

7 1968 A Tanwan, D Lberzon / Automatca 46 (2010) y (rp 1) (t) = C p A r p 1 p x(t), y (r p) (t) = C p A r p p x(t) + C p A r p 1 p B p xu(t) (12) where r p denotes the relatve degree o subsystem p I we ntroduce the notatons y(t) C p ẏ(t) C p A p ŷ p (t) := y (r p 1)(t) and Z p := C p A r p 1 p, then based on the unctonal reproducblty crtera, an output y [t0,t 0 +ε) can be produced by a subsystem p and only ŷ p (t 0 ) = Z p x(t 0 ) Consequently, two subsystems p, q can produce a gven segment o output on an nterval [t 0, t 0 + ε), then we wll have ŷp (t 0 ) = x(t ŷ q (t 0 ) Z 0 ) (13) q Ths s equvalent to sayng that [ ] Ip ŷ(t I 0 ) = x(t 0 ) (14) q Z q where ŷ := y, ẏ,, y (r 1)T, r := max{r p, r q }, and or κ = {p, q}, I κ s an r κ r matrx whose jth element s 1 = j and 0 otherwse Thus, the exstence o swtch-sngular pars n case o SISO blnearswtched systems mples that the ntersecton o Ip Ip range spaces o and s not empty Snce and are I q Z q I q Z q both lnear operators actng on lnear subspaces, the zero vector s always n ther range space Thus, an dentcally zero output always orms a swtch sngular par wth the ernel o ; that Z q s, er Z q, 0 orms a swtch-sngular par or such systems That s the trval case; or the nontrval case we chec Ip I q and Z q have a nontrval common range space So, there exsts a nonzero output that orms a swtch-sngular par wth some state at tme t, then ŷ(t) Ip range range, or equvalently Z q Ip Z ran p I q Z q < Ip ran I q + ran Z q In other words, all the subsystems n (11) are nvertble and r := max p P r p <, then or all x(t 0 ) := R n and y Y α := {y y (r 1) F AC and ŷ [t0,t 0 +ε) 0, or some ε > 0}, the pars (, y) are not swtch-sngular pars o Γ p, Γ q, and only the ollowng ran condton holds: [ ] [ ] [ ] Ip Z ran p Ip = ran + ran (15) I q Z q I q Z q I q or all p q, p, q P such that Y p Y q {0} Ths condton s smlar to the one gven n Vu and Lberzon (2008, Lemma 3) or checng the exstence o swtch-sngular pars n swtched lnear systems The common ramewor n both cases s the appearance o lnear equatons when tang the dervatves o the outputs, whch maes t easer to derve the ran condtons We now have a toolset to chec the nvertblty condtons gven n Theorem 1 I these condtons are satsed and the swtched system s strongly nvertble, a swtched nverse system can be constructed to recover the nput and swtchng sgnal σ rom gven output and ntal state For the swtched nverse system, dene the ndex nverson uncton Σ 1 : M α Y α P as: Σ 1 (, y) = p : y Y p and y () (t 0 ) = L p h p ( ) (16) where = 0, 1,, r p 1, t 0 s the ntal tme o y, and = x(t 0 ) The uncton Σ 1 s well-dened snce p s unque by the act that there are no swtch-sngular pars The exstence o p s guaranteed because t s assumed that y Y α s an output The dynamcs o the nverse swtched system Γσ 1 are: σ (t) = Σ 1 (z(t), y [t,t+ε) ), y (r σ ) L r σ σ ż = σ (z) + g σ (z) h σ (z), L g σ Lr σ 1 h σ σ (z) u(t) = y(r σ ) (t) L r σ σ h σ (z(t)) L g σ Lr σ 1 h σ σ (z(t)) wth the ntal condton z(t 0 ) = The notaton ( ) σ denotes the object n the parenthess calculated or the subsystem wth ndex σ (t) The ntal condton σ (t 0 ) determnes the ntal actve subsystem at the ntal tme t 0, rom whch tme onwards, the actve subsystem ndexes and the nput as well as the state are determned unquely and smultaneously 42 Multple-nput multple-output (MIMO) systems For multple-nput multple-output (MIMO) nonlnear systems ane n controls (1), one uses the structure algorthm to compute the nverse When a system s nvertble, the structure algorthm, or Sngh s nverson algorthm, allows us to express the nput as a uncton o the output, ts dervatves and possbly some states The structure algorthm: Ths verson o the algorthm closely ollows the constructon gven by D Benedetto et al (1989), whch s a slghtly moded verson o the algorthm by Sngh (1981) Step 1: Calculate ẏ = L h(x) + L G h(x)u = h [ (x) + G(x)u] x and wrte t as ẏ =: a 1 (x) + b 1 (x)u Dene s 1 := ran b 1 (x), whch s the ran o b 1 (x) n some neghborhood o, denoted as N 1 ( ) Permute, necessary, the components o the output so that the rst s 1 rows o b 1 (x) are lnearly dependent Decompose y as ỹ1 ã1 (x) + b1 (x)u ẏ = = ŷ 1 â 1 (x) + ˆb1 (x)u where ỹ 1 conssts o the rst s 1 rows o ẏ Snce the last m s 1 rows o b 1 (x) are lnearly dependent upon the rst s 1 rows, there exsts a matrx F 1 (x) such that ỹ 1 = ã 1 (x) + b1 (x)u, ŷ 1 = ĥ 1 (x, ỹ 1 ) = â 1 (x) + F 1 (x)( ỹ 1 ã 1 (x)) (17) where the last equaton s ane n ỹ 1 Fnally, set B1 (x) := b1 (x) Step + 1: Suppose that n steps 1 through, ỹ 1,, ỹ () have been dened so that ỹ 1 = ã 1 (x) + b1 (x)u, ỹ () ŷ () = ã (x, {ỹ (j) 1 1, j }) + b (x, {ỹ (j) 1 1, j 1})u, = ĥ (x, {ỹ (j) 1, j }), ŷ()

8 A Tanwan, D Lberzon / Automatca 46 (2010) where all the expressons on the rght-hand sde are ratonal unctons o ỹ (j) Suppose also that the matrx B := [ bt,, 1 bt ]T (vertcal stacng o the lnearly ndependent rows obtaned at each step) has ull ran equal to s n N ( ) Then calculate ŷ (+1) = ĥ [ (x) + G(x)u] + x and wrte t as ŷ (+1) =1 ĥ j= ỹ (j) ỹ (j+1) = a +1 (x, {ỹ (j) 1, j + 1}) + b +1 (x, {ỹ (j) 1, j })u (18) Dene B +1 := [ B T, bt +1 ]T, and s +1 := ran B +1 Permute, necessary, the components o ŷ (+1) so that the rst s +1 rows o B +1 are lnearly ndependent Decompose ŷ (+1) as ŷ (+1) = ỹ(+1) +1 ŷ (+1) +1 where ỹ (+1) +1 conssts o the rst (s +1 s ) rows Snce the last rows o B +1 (x, {ỹ (j) 1, j }) are lnearly dependent on the rst s +1 rows, we can wrte ỹ 1 = ã 1 (x) + b1 (x)u, ỹ (+1) +1 = ã +1 (x, {ỹ (j) + b+1 (x, {ỹ (j) 1, j + 1}) 1, j })u, ŷ (+1) +1 = ĥ +1 (x, {ỹ (j) 1 + 1, j + 1}) where once agan everythng s ratonal n ỹ (j) Fnally, set B+1 := [ B T, bt +1 ]T, whch has ull ran equal to s +1 locally End o Step + 1 By constructon, s 1 s 2 m I or some nteger α we have s α = m, then the algorthm termnates and the system s strongly nvertble at We call α the relatve order 6 o the system The closed orm expresson or u s derved rom the α- th step o the structure algorthm, whch gves an nvertble matrx Bα := [ bt,, 1 bt α ] T havng ull ran equal to m n a neghborhood N α ( ) =: N( ), namely, u = B 1 α ỹ 1 ỹ (α) α ã 1 ã α =: B 1 α [Ỹ α Ã α ] (19) Note that the entres o the matrx Bα are ratonal unctons o the dervatves o the output and there may exst an output or whch the ran o Bα drops We denote by Y s the values o the output and ts dervatves, evaluated at a tme nstant t, or whch the ran o Bα (x, y(t)) s less than m, whle x N( ) We can now ormally dene the sets Y s and Y α or a subsystem as ollows: Y s := {y : [t 0, T) R 2 y(t) Y s or almost all t [t 0, t 0 + δ) [t 0, T), where δ > 0 s arbtrary}, and Y α := {y : [t 0, T) R 2 y(t) Y s or almost all t [t 0, t 0 + ε) and some ε > 0} In other words, Y s ncludes those outputs or whch the matrx Bα s not nvertble and Y α s ts complement Hence, we wor wth u such that Γ O (u) Y s Comparng to the 6 The term was coned by Hrschorn (1979a) and s weaer than the noton o vector relatve degree Parallel to the termnology used n lnear system theory, Njmejer and Schumacher (1985) show that α s the hghest order o zeros at nnty SISO case, we had Bα = h(x) whch s a uncton o the state only and thus there exsts no sngular output or SISO systems Another useul class o systems or whch Y s = was dscussed by Hrschorn (1979a) As was the case n SISO systems, substtuton o the expresson or u rom (19) n (1) gves the dynamcs o the nverse system These dynamcs are dened on the set M α := {x M ran Bα (x, y(t)) = m, y(t) Y s }, whch s open and dense (x), g(x), h(x) are analytc unctons Example 4 As an llustraton o the structure algorthm, let us revst the system dened n Example 1 Step 1 o the algorthm ỹ1 ẏ1 yelds ẏ = ŷ = ẏ2 = u Usng F 1 (x) = x 3 /x 1, we get 1 x1 0 x 3 0 ŷ 1 = ẏ 2 = (x 3 /x 1 )ẏ 1 In Step 2, ater derentatng ŷ 1 = ẏ 2, we get the ollowng set o equatons: ỹ 1 = ẏ 1 = x 1 u 1 ỹ 2 = ÿ 2 = x 3ÿ 1 ẏ 1 ẏ 2 + ẏ 1 u 2 x 1 B2 = x1 0 0 ẏ 1 /x 1 So, B2 has ran 2, the number o nputs Hence, α = 2; M α = {x R 3 x 1 0}; Y s = {z R 2 z 1 = 0}, and Y s = {y : [t 0, T) R 2 ẏ(t) Y s or almost all t [t 0, t 0 + δ) [t 0, T), where δ > 0 s arbtrary} Remar 5 Unle n the SISO case, we need some derentablty assumptons on the nput sgnals to characterze the nput space or MIMO systems In the structure algorthm, Step 1 gves ỹ 1 that has already u on the rght-hand sde and the α-th step o the algorthm nvolves {ỹ (j) 1 α 1, j α} Thus ỹ (α 1) must be absolutely contnuous so that ỹ (α) exsts almost everywhere For the nput space, t means that u (α 1) must be Lebesgue measurable and locally essentally bounded These constrants characterze the nput space U or MIMO case and Y s the correspondng set o outputs From the structure algorthm, we deduce that the system s nvertble on Y α = Y \ Y s Based on the structure algorthm, we now study the condtons or unctonal reproducblty o multvarable nonlnear systems Usng the notaton derved n the structure algorthm, denote by Z the vector h(x) Z x, ỹ 1,, ỹ (α 1) ĥ 1 (x, ỹ 1 ) α 1 := ĥ α 1 x, ỹ 1,, ỹ (α 1) and let y ŷ 1 ŷ :=, ŷ (α 1) α 1 yˆ d := y d ŷ d1 ŷ (α 1) dα 1 α 1 (20) So Z s bascally a concatenaton o the expressons appearng at each step o Sngh s structure algorthm whch get derentated and ŷ s the concatenaton o the correspondng expressons on the let-hand sde so that Z x, ỹ 1,, ỹ (α 1) α 1 ŷ = 0 The ollowng result s along the same lne as Lemma 2 and has appeared n Sngh (1982, Theorem 1) The proo s gven n the Appendx and s developed derently than Sngh (1982)

9 1970 A Tanwan, D Lberzon / Automatca 46 (2010) Lemma 4 I the system gven by (1), wth x(t 0 ) =, has a relatve order α <, then there exsts a control nput u such that Γ O (u) = y d ( ) and only ŷ d (t 0 ) = Z, ỹ d1 (t 0 ),, ỹ () d (t 0 ) = 0, 1,, α 1 (21) Another verson o ths result n terms o jet spaces s gven by Responde (1990) Smlarly to the SISO case, the dea s that the porton o the output whch s not drectly aected by u s determned ntally by the value o state varables; and the nput u, or whch Γ O (u) = y d ( ), s gven by (19) wth y replaced by y d n that ormula Example 5 Consder the system gven n Example 1 The vector ŷ s the porton o the output that gets derentated and thereore, ŷ = y1 y 2 ẏ 2 ŷ d = yd1 y d2 ẏ d2 The vector Z(x, y d1, y 2, ẏ d1 ) s gven by x 1 Z(x, y d1, y 2, ẏ d1 ) = x 2 ẏ d1 (x 3 /x 1 ) Usng Lemma 4 and calculatons n Example 4, we have ŷ d (t 0 ) = Z, y d1 (t 0 ), y 2 (t 0 ), ẏ d1 (t 0 ) then the control whch produces y d as an output, on a small nterval, s gven by u 1 = ẏd 1 x 1 u 2 = x 1ÿ d2 x 3 ÿ d1 + ẏ d1 ẏ d2 ẏ d1 I y d ( ) Y α or all tmes and the correspondng state trajectory x( ) M α, then the system can produce y d ( ) as an output over an arbtrary tme nterval Lemma 4 gves the ollowng condton or the vercaton o swtch-sngular pars Lemma 5 For MIMO swtched systems, (, y) s a swtch-sngular par o two subsystems Γ p, Γ q and only y Y p Y q and y ŷ 1 = ŷ α κ 1 (α κ 1) h κ ( ) ĥ 1 κ (, ỹ 1 ) ĥ α κ 1 κ (, ỹ 1,, ỹ (α κ 1) α κ 1 ) or κ = p, q, and α κ denotes the relatve order o subsystem Γ κ (22) The procedure or constructng the nverse rom ths pont onwards s exactly the same as dscussed earler or the SISO case wth u gven by (19) nstead o (5) Remar 6 The results n ths secton can also be extended to nclude the case when there are state jumps at swtchng tmes Denote by ψ p,q : M M the reset map when swtchng rom subsystem p to subsystem q, p, q P Thus ar, we have consdered the case o dentty reset maps only, where ψ p,q (x) = x p, q P, x M For nondentty reset maps, Denton 3 s moded to (, y) s a swtch-sngular par o the two subsystems Γ p, Γ q there exst u 1, u 2 and ε > 0 such that Γ O p, (u 1[t0,t 0 +ε)) = Γq,ψ O p,q ( ) (u 2[t 0,t 0 +ε)) = y [t0,t 0 +ε) or Γp,ψ O q,p ( ) (u 1[t 0,t 0 +ε)) = Γ O q, (u 2[t0,t 0 +ε)) = y [t0,t 0 +ε) Essentally, ths means that the output s ndstngushable between the two subsystems, tang nto account the eect o state jumps In case o SISO systems, nstead o Eq (9), we chec or swtch-sngular pars usng y y (r κ 1) or y y (r κ 1) (t 0 ) = (t 0 ) = h κ (ψ p,κ ( )) L r κ 1 κ, (23) h κ (ψ p,κ ( )) h κ (ψ q,κ ( )) (24) h κ (ψ q,κ ( )) L r κ 1 κ where κ = p, q P and ψ p,p ( ) = ψ q,q ( ) =, p, q P Eq (22) would also be moded smlarly when dealng wth MIMO systems The statement o Theorem 1, n ether case, remans unchanged Another generalzaton s to nclude swtchng mechansms, such as swtchng suraces Denote by S p,q the swtchng surace or subsystem p, where the swtched system jumps to subsystem q Then we only need to chec or the swtch-sngularty o S p,q and S q,p nstead o M or the two subsystems Γ p, Γ q 5 Output generaton In the prevous secton, we consdered the queston o let nvertblty where the objectve was to recover (σ, u) unquely or all y n some output set Y α In ths secton, we address a derent problem whch concerns wth ndng (σ, u) (that may not be unque) such that H x0 (σ, u) = y d or a gven uncton y d and a state For the nvertblty problem, we ound condtons on the subsystems and the output set Y so that the map H x0 s njectve Here, we are gven one partcular (, y d ) and wsh to nd ts premage under the map H x0 For the swtched system (3), denote by H 1 (y d ) the premage o a uncton y d, H 1 (y d ) := {(σ, u) : H x0 (σ, u) = y d } (25) I y d s not n the mage set o H x0 then by conventon H 1 = When H 1 (y d ) s a sngleton, the system s nvertble at We want to nd condtons and an algorthm to generate H 1 (y d ) when H 1 (y d ) s a nte set We requre the ndvdual subsystems to be nvertble at because ths s not the case, then the set H 1 (y d ) may be nnte When a square nonswtched nonlnear system s not nvertble, the matrx B 1 α n (19) s not dened and the expresson or u s moded to: u(t) = B Ď α [Ỹ α Ã α ] + K(x, Ỹ α 1 )v (26) where K s a matrx whose columns orm a bass or the null space o Bα and B Ď α := B T α ( Bα B T α ) 1 s a rght pseudo-nverse o Bα I an output s generated by some nput u obtaned rom (26) wth some ntal state, then due to arbtrary choce o v, there always exst nntely many derent nputs that generate the same output wth the same ntal state Hence to avod nnte loop reasonng, we wll assume that the ndvdual subsystems Γ p are nvertble at or all p P However, we do not assume that the swtched system s nvertble as the subsystems may have swtch-sngular pars We wll only consder the unctons y d ( ) over nte tme ntervals so that there s only a nte number o swtches under consderaton We now present a swtchng nverson algorthm or swtched systems smlar to the one gven by Vu and Lberzon (2008) The algorthm taes the parameters M, y d F pc (dened over

10 A Tanwan, D Lberzon / Automatca 46 (2010) a nte nterval) and returns the set H 1 (y d ) It uses the ndexmatchng map 7 Σ 1 : M F pc 2 P dened as Σ 1 (, y d ) := {p such that y d Y α p and y d satses (21)}, obtaned va the structure algorthm o Γ p The ndex-matchng map returns the ndexes o the subsystems that are capable o generatng y d startng rom I the returned set s empty, no subsystem s able to generate that y d startng rom Note that the ndex-matchng map Σ 1 s dened or every par (, y d ) and always returns a set, whereas the ndex nverson uncton Σ 1 n (16) s dened only or (, y d ) whch are not swtch-sngular pars and returns an element o P In the algorthm, Γ 1,O p, (y d ) denotes the output o the nverse subsystem Γ 1 p ; the symbol reads assgned as, and := reads dened as The concatenaton o an element η and a set S s η S := {η ζ, ζ S} By conventon, η =, η Fnally, the concatenaton o two sets S and T s S T := {η ζ, η S, ζ T} begn H 1 (y d[t0,t)) Let P := {p P : y d[t0,t 0 +ε) Y α p and M α p, ε > 0} Let t := mn{t [t 0, T) : y d[t,t+ε) / Y α p or some p P, ε > 0} otherwse t = T Let P := Σ 1 (, y d[t0,t 0 +ε)) P then Let A := oreach p P do Let x := Γ 1 p, (y d[t0,t )) x M α p and y d[t 0,t ) Y α p then Let u := Γ 1,O p, (y d[t0,t )) T := {t (t 0, t ) : (x(t), y d (t)) s a swtch-sngular par o Γ p, Γ q or some q p} T s a nte set then oreach τ T do let ξ := Γ p (u)(τ) A A {(σ [t0,τ), u [t0,τ)) H 1 ξ (y d[τ,t) )} else T = and t < T then let ξ := Γ p (u)(t ) A A {(σ [t0,t ), u) H 1 ξ (y d[t,t))} else T = and t = T then A A {(σ [t0,t), u)} else A := else A := else A := return H 1 (y d ) := A end The return set A s always nte and, nonempty, t contans the pars o swtchng sgnals and nputs that generate the gven y d startng rom I the return s an empty set, t means that there s no swtchng sgnal and nput that generate y d, or there s an nnte number o possble swtchng tmes Also by our concatenaton notaton, at any nstant o tme, the return o the procedure s an empty set, then that branch o the search wll be empty because η = Based on the semgroup property or the trajectores o dynamcal systems, the algorthm determnes the swtchng sgnal and the nput on a subnterval [t 0, t) o [t 0, T) and then concatenates these sgnals wth the correspondng premage on 7 The set 2 P denotes the set o all subsets o the set P the rest o the nterval [t, T) I t s the rst swtchng tme ater t 0, then we can nd H 1 (y d[t0,t)) by snglng out whch subsystems are capable o generatng y d[t0,t) usng the ndex-matchng map The obvous canddate or rst swtchng tme, denoted by t n the algorthm, s the tme at whch the output loses smoothness Note that n the SISO case, t s the tme at whch one o the rst r 1 dervatves o the output lose contnuty (see Secton 41) But, t s entrely possble that we have a swtchng at some tme nstant τ and the output s stll smooth (see Example 6) In ths case, (x(τ), y [τ,τ+ε) ) orms a swtch-sngular whch, n SISO case, can be checed by usng (9), or or the systems wth reset maps, usng (23) or (24) The algorthm eeps trac o all the swtchsngular pars encountered along the trajectory o the moton and uses a swtch at a later tme to recover a hdden swtch earler (eg a swtch at whch the output s smooth) Ths maes the swtchng nverson algorthm a recursve procedure callng tsel wth derent parameters wthn the man algorthm (eg the uncton H 1 (y d ) uses the returns o H 1 ξ (y d[t,t))) The ollowng example should help understand ths algorthm Example 6 Consder a SISO swtched system wth two modes x1 x ẋ = u, M = R Γ 1 : 2 x 2 1 y = x 2 0 e x 2 ẋ = + Γ 2 : x 1 e x 2 u, M = R 2 y = x 1 We wsh to reconstruct the swtchng sgnal σ ( ) and the nput u( ) whch wll generate the ollowng output: cos t t [0, t y d (t) = ) 2 cos t t [t, T) where t = π and T = 45, wth the gven ntal state = (0, 1) T In ths example, (, y [t0,t 0 +ε)) orm a swtch-sngular par, or some c R, = and y(t 0 ) = c c c We now use the above swtchng nverson algorthm to nd (σ, u) such that Γ O,σ (u) = y d We have P = {1, 2} and P := Σ 1 (, y d[0,t )) = {1} by usng the ndex-matchng map wth gven and y d (0) = 1 The nverse o Γ 1 on [0, t ) s z1 z Γ 1 ż = ẏ 1 : 0 1 d, M α = 1 R2 u(t) = z 2 + ẏ d wth z(0) =, whch then gves 0 z(t) = =: x(t) cos t t [0, t ) (27) u(t) = cos t sn t We want to nd T := {t t : (x(t), y d[t,t )) s a swtch-sngular par o Γ 1, Γ 2 }, whch s equvalent to solvng cos t = x 1 (t) = 0, t (0, t ) Ths equaton has a soluton t = π/2 =: t 1 < t, and hence T = {t 1 }, a nte set We have ξ = x(t 1 ) = (0, 0) T and we repeat the procedure or the ntal state ξ and the output y d[t1,t) wth P := Σ 1 (ξ, y d[t1,t )) = {1, 2} We analyze these two cases: Case 1 p = 1 Ths mples t 1 s not a swtchng tme, e, σ (t) = 1 or t [t 0, t ) and u(t), x(t) are gven by (27) or 0 t < t, whch gves ξ = x(t ) = (0, 1) T At t, Γ 2 must be actve, but then y(t ) = x 1 (t ) = 0 2 = y d (t ), thus the ndex-matchng map returns an empty set, Σ 1 (ξ, y d[t,t)) =

11 1972 A Tanwan, D Lberzon / Automatca 46 (2010) Case 2 p = 2, whch means that t 1 s a swtchng nstant So we wor wth the nverse system o Γ 2, 0 1 Γ 1 ż = + ẏ 2 : z 1 1 d, M α = 1 R2 u(t) = e z2ẏ d wth ntal state z(t 1 ) = ξ, whch gves cos t z(t) = =: x(t) cos t + sn t 1 t t 1 u(t) = e (1 cos t sn t) sn t We nd T = {t 1 < t t : (x(t), y d[t,t )) s a swtch-sngular par o Γ 1, Γ 2 }, whch s equvalent to solvng or π cos t = cos t + sn t 1, 2 = t 1 < t t = π It s easy to see that ths equaton has no soluton and thus there exst no swtch-sngular pars n the nterval (t 1, t ) So, we let ξ = x(t ) = ( 1, 2) T and repeat the procedure wth ξ and y d[t,t), whch gves the unque soluton σ [t,t) = 1 and u [t,t) = 2(cos t + sn t) Thus, the swtchng nverson algorthm returns (σ, u), where (1, cos t sn t), 0 t < t 1 (σ, u) = (2, e (1 cos t sn t) sn t), (1, 2(cos t + sn t)), t 1 t < t t t T In ths example, two swtches are requred to generate the gven output One o the swtchng nstants s t as the output loses smoothness at that nstant The other swtchng nstant s t 1 where the output does not lose smoothness Wthout the concept o swtch-sngular pars, one may try all the our possble combnatons wth t as the only swtchng nstant and arrve at the alse concluson that there s no swtchng sgnal and nput that generate y d (t); but nstead the use o the swtchng nverson algorthm allows us to construct the nput and swtchng sgnal 6 Conclusons In ths paper, we addressed the nvertblty problem o swtched nonlnear systems The concepts ntroduced by Vu and Lberzon (2008) or lnear systems were extended to nonlnear systems We gave a necessary and sucent condton or a swtched system to be nvertble, accordng to whch the ndvdual subsystems should be nvertble and there should be no swtchsngular pars We developed ormulae or checng (, y) s a swtch-sngular par o two subsystems and then gave an algorthm that nds swtchng sgnals and nputs, possbly nonunque, whch generate a gven output wth a gven ntal state For uture wor, one nterestng problem s to develop condtons or checng the exstence o swtch-sngular pars whch are more constructve as t s n general not easble to very (22) or every output and state Another research drecton s to approach the problem geometrcally and nvestgate characterzatons equvalent to nonexstence o swtch-sngular pars to obtan geometrc crtera or let nvertblty o swtched systems Acnowledgement We would le to than Lnh Vu or the nsghtul dscussons related to the problem o nvertblty Appendx Proo o Lemma 4 (Necessty) Supposng ε > 0 and nput u dened over the nterval [t 0, t 0 + ε), such that Γ O (u(t)) = y d (t), t [t 0, t 0 + ε), then y d (t 0 ) = y(t 0 ) = h( ) ŷ d1 (t 0 ) = ŷ 1 (t 0 ) = ĥ 1 (x, ỹ 1 ) = ĥ 1 (x, ỹ d1 ) ŷ (α 1) dα 1 (t 0) = ŷ (α 1) α 1 (t 0) = ĥ α 1, ỹ 1,, ỹ (α 1) 1,, ỹ (α 1) α 1 = ĥ α 1, ỹ d1,, ỹ (α 1),, ỹ (α 1) d 1 dα 1 and hence Eq (21) s satsed Sucency: I we nject y d (t) nto the nverse system, then the control nput produced by ths nverse system s gven by (19) wth ỹ replaced by ỹ d, and substtutng t n the α-th step o the structure algorthm ỹ 1 = ã 1 (x) + b1 (x)u, ỹ (α) α we get = ã α (x, {ỹ (j) 1 α 1, j α}) + bα (x, {ỹ (j) 1 α 1, j α 1})u, ỹ 1 (t) = ỹ d1 (t), t [t 0, t 0 + ε) (28) Here t 0 + ε characterzes the tme nstant at whch the trajectory o the nverse system hts the sngular pont n the state space As the system s strongly nvertble at, t s guaranteed that ε > 0 Usng hypothess (21), we have h( ) = y d (t 0 ), and ntegratng (28) on both sdes over the nterval [t 0, t 0 + ε) to get ỹ 1 (t) = ỹ d1 (t), t [t 0, t 0 + ε) (29) Usng the ntal condtons characterzed by (21), the desred result can now be derved by nducton Suppose Eqs (28) and (29) are true or ndex, that s ỹ () (t) = ỹ() d (t) ỹ (t) = ỹ d (t) t [t 0, t 0 + ε), t [t 0, t 0 + ε) Snce ỹ (+1) +1 = ã +1 (x, {ỹ (j) 1, j + 1}) + b +1 (x, {ỹ (j) 1, j })u, substtutng u rom (19) as generated by the nverse system, wth ỹ replaced by ỹ d, gves ỹ (+1) +1 (t) = ỹ(+1) d +1 (t) t [t 0, t 0 + ε) Agan usng hypothess (21) and ntegratng both sdes, we get ỹ +1 (t) = ỹ d+1 (t) t [t 0, t 0 + ε) As y(t) = (ỹ 1 (t),, ỹ α (t)), we get Γ O (u(t)) = y d (t), t [t 0, t 0 + ε) Reerences Babaal, M, & Pappas, G J (2005) Observablty o swtched lnear systems n contnuous tme In LNCS: Vol 3414 Hybrd systems: computaton and control (pp ) Berln: Sprnger Brocett, R W, & Mesarovc, M D (1965) The reproducblty o multvarable systems Journal o Mathematcal Analyss and Applcatons, 11, Chab, S, Boutat, D, Banal, A, & Kratz, F (2007) Invertblty o swtched nonlnear systems Applcaton to mssle aults reconstructon In Proc IEEE 46th con on decson and control (pp ) D Benedetto, M D, Grzzle, J W, & Moog, C H (1989) Ran nvarants o nonlnear systems SIAM Journal on Control and Optmzaton, 27(3),

12 A Tanwan, D Lberzon / Automatca 46 (2010) Hrschorn, R M (1979a) Invertblty o multvarable nonlnear control systems IEEE Transactons on Automatc Control, 24(6), Hrschorn, R M (1979b) Invertblty o nonlnear control systems SIAM Journal on Control and Optmzaton, 17(2), Isdor, A (1995) Nonlnear control systems (3rd ed) Berln: Sprnger Isdor, A, & Moog, C H (1988) On the nonlnear equvalent o the noton o transmsson zeros In C I Byrnes, & A Kurzhans (Eds), Lecture notes n control and normaton scences: Vol 105 Modelng and adaptve control (pp ) Berln: Sprnger-Verlag Lberzon, D (2003) Swtchng n systems and control Boston: Brhäuser Mlleroux, G, & Daaouz, J (2007) Invertblty and latness o swtched lnear dscrete-tme systems In A B A Bcch, & G Buttazzo (Eds), LNCS: Vol 4416 Hybrd systems: computaton and control (pp ) Berln: Sprnger-Verlag Njmejer, H (1982) Invertblty o ane nonlnear control systems: a geometrc approach Systems and Control Letters, 2(3), Njmejer, H, & Schumacher, J M (1985) On the nherent ntegraton structure o nonlnear systems IMA Journal o Mathematcal Control & Inormaton, 2(2), Njmejer, H, & van der Schat, A J (1990) Nonlnear dynamcal control systems New Yor: Sprnger-Verlag Responde, W (1990) Rght and let nvertblty o nonlnear control systems In H Sussmann (Ed), Nonlnear controllablty and optmal control (pp ) New Yor: Marcel Deer San, M K, & Massey, J L (1969) Invertblty o lnear tme-nvarant dynamcal systems IEEE Transactons on Automatc Control, 14, Slverman, L M (1969) Inverson o multvarable lnear systems IEEE Transactons on Automatc Control, 14(3), Sngh, S (1981) A moded algorthm or nvertblty n nonlnear systems IEEE Transactons on Automatc Control, 26(2), Sngh, S N (1982) Generalzed unctonal reproducblty condton or nonlnear systems IEEE Transactons on Automatc Control, 27(4), Sontag, E D (1998) Texts n appled mathematcs: Vol 6 Mathematcal control theory: determnstc nte dmensonal systems (2nd ed) Sprnger Sundaram, S, & Hadjcosts, C N (2006) Desgnng stable nverters and state observers or swtched lnear systems wth unnown nputs In Proc IEEE 45th con on decson and control Vdal, R, Chuso, A, & Soatto, S (2002) Observablty and dentablty o jump lnear systems In Proc IEEE 41st con on decson and control Vol 4 (pp ) Vdal, R, Chuso, A, Soatto, S, & Sastry, S (2003) Observablty o lnear hybrd systems In LNCS: Vol 2623 Hybrd systems: computaton and control (pp ) Berln: Sprnger Vu, L, & Lberzon, D (2008) Invertblty o swtched lnear systems Automatca, 44(4), Aneel Tanwan receved hs undergraduate degree n Mechatroncs engneerng rom Natonal Unversty o Scences and Technology, Pastan n 2005 He completed hs DEA n Automatque-Productque rom Insttut Natonal Polytechnque de Grenoble, France whle worng at Laboratore Automatque de Grenoble He dd hs Masters n Electrcal engneerng at Unversty o Illnos, Urbana- Champagn, USA where he s currently enrolled as a PhD canddate n the Department o Electrcal and Computer Engneerng Hs research nterests nclude swtched and hybrd dynamcal systems, nonlnear control, systems theory and related applcatons Danel Lberzon was born n the ormer Sovet Unon on Aprl 22, 1973 He was a student n the Department o Mechancs and Mathematcs at Moscow State Unversty rom 1989 to 1993 and receved the PhD degree n mathematcs rom Brandes Unversty n 1998 (under the supervson o Pro Roger W Brocett o Harvard Unversty) Followng a postdoctoral poston n the Department o Electrcal Engneerng at Yale Unversty rom 1998 to 2000, he joned the Unversty o Illnos at Urbana-Champagn, where he s now an assocate proessor n the Electrcal and Computer Engneerng Department and a research assocate proessor n the Coordnated Scence Laboratory Dr Lberzon s research nterests nclude swtched and hybrd systems, nonlnear control theory, control wth lmted normaton, and uncertan and stochastc systems He s the author o the boo Swtchng n Systems and Control (Brhauser, 2003) and the author or coauthor o over thrty journal papers Dr Lberzon receved the IFAC Young Author Prze and the NSF CAREER Award, both n 2002, and was elected a senor member o IEEE n 2004 He receved the Donald P Ecman Award rom the Amercan Automatc Control Councl n 2007, and the Xerox Award or Faculty Research rom the UIUC College o Engneerng also n 2007 He delvered a plenary lecture at the 2008 Amercan Control Conerence Snce 2007, he has served as Assocate Edtor or the IEEE Transactons on Automatc Control