Structural and Strongly Structural Input and State Observability of Linear Network Systems

Transcription

1 Strutural an Strongly Strutural Input an State Oservaility of Linear Network Systems Sein Gray Feeria Garin Alain Kiangou To ite this version: Sein Gray Feeria Garin Alain Kiangou Strutural an Strongly Strutural Input an State Oservaility of Linear Network Systems IEEE Transations on Control of Network Systems IEEE (4) pp <101109/TCNS > <hal > HAL I: hal Sumitte on 13 De 2017 HAL is a multi-isiplinary open aess arhive for the eposit an issemination of sientifi researh ouments whether they are pulishe or not The ouments may ome from teahing an researh institutions in Frane or aroa or from puli or private researh enters L arhive ouverte pluriisiplinaire HAL est estinée au épôt et à la iffusion e ouments sientifiques e niveau reherhe puliés ou non émanant es étalissements enseignement et e reherhe français ou étrangers es laoratoires pulis ou privés

2 1 Strutural an Strongly Strutural Input an State Oservaility of Linear Network Systems Sein Gray Feeria Garin an Alain Y Kiangou Astrat This paper stuies linear network systems affete y multiple unknown inputs with the ojetive of reonstruting oth the initial state an the unknown input with one timestep elay We state onitions uner whih oth the whole network state an the unknown input an e reonstrute from output measurements over every winow of length N N eing the imension of the system for all system matries that share a ommon zero/non-zero pattern (uniform N-step strongly strutural input an state oservaility) or at least for almost all system matries that share a ommon zero/non-zero pattern (uniform N-step strutural input an state oservaility) Base on some speifi assumptions on the struture of the interations etween the unknown input an the network states we show that suh a haraterization epens only on strongly strutural (resp strutural) oservaility properties of a suitale susystem Inex Terms Linear Network systems Input an State Oservaility (ISO) Strutural Oservaility Strongly Strutural Oservaility Cyer-Physial Seurity I INTRODUCTION The fiels of appliation of network systems span from ritial infrastruture omains suh as power networks water an gas istriution networks to healthare systems flight ontrol systems among others Given the uiquitous nature of their usage it is of paramount importane to ensure that eah iniviual susystem funtions as esire The notion of oservaility enales one to ahieve this y expening as few resoures as possile However suh systems are quite likely to malfuntion ue to loal attaks y maliious agents moele as external unknown inputs [1 whih oul have signifiant onsequenes as eviene y the failure of wastewater management systems in Marohy Australia in early 2000 [2 multiple power lakouts in Brazil [3 to ite a few Hene in aition to eing ale to oserve the state in the presene of unknown inputs (also known as strong oservaility [4 [5) it is ruial that the unknown input e oserve as well This notion is known as Input an State Oservaility whih hereafter is referre to as ISO In the ontext of Linear Time Varying (LTV) systems a system eing ISO over an interval oes not neessarily mean that the sai system woul e ISO over every suffiiently long interval The onept of uniform δ-step ISO (ie ISO over every time winow of length δ) gets ri of this rawak The notion of ISO is of partiular importane in esigning uniase minimum-variane filters that simultaneously estimate oth state an unknown input [6 [8 It is well-known The authors are with Univ Grenole Alpes CNRS Inria Grenole INP GIPSA-La F Grenole Frane ( seingray@inriafr feeriagarin@inriafr alainkiangou@univ-grenole-alpesfr) that algerai approahes towars haraterizing ISO involve the lassi Kalman-like rank onition or a variant of the Popov-Belevith-Hautus (PBH) test (see eg [9 [8) Both tests require exat knowlege of entries in the matries of interest an are omputationally heavy as the imension of the system grows while the latter is not suitale for LTV systems This leas to the stuy of ISO ase on the struture of the unerlying network (represente y a graph) an the orresponing line of work is known as struture systems We say that a linear system is struture if the system matries have oeffiients that are either a fixe zero or a free parameter (ie the oeffiients may take any value in R) Thus one an apture a family of systems that oey a ertain rule Uner suh a setup if a property hols for almost all hoies of entries in the non-zero positions of the system matries it is alle strutural [10 whereas if it hols for all non-zero hoies of entries in the non-zero positions of the system matries it is alle strongly strutural [11 (s-strutural) For linear time-invariant (LTI) systems strutural ontrollaility or the ual notion of oservaility has een stuie sine [10 while graph-theoreti haraterizations for s-strutural ontrollaility were first provie in [11 The survey paper [12 revises some graph-theoreti haraterizations for oservaility in aition to realling similar onitions for solvaility issues like isturane rejetion input-output eoupling an so on while equivalent haraterizations for s-strutural ontrollaility have een provie in [13 More reently for s- strutural ontrollaility [14 provies neessary an suffiient onitions in terms of uniquely restrite mathing (preise efinition appears in Setion VI) while [15 oes so in terms of zero foring sets Notie that all the results mentione as yet are for LTI systems where oth parameters an the struture remain onstant over the time In ontrast it is natural to assume that the parameters an evolve over the time while the struture remains fixe (LTV with fixe topology) Uner suh a senario neessary an suffiient onitions for strutural oservaility of LTV systems are given in [16 while neessary an suffiient onitions for s-strutural oservaility are availale in [17 However these results are not appliale for LTV systems with unknown inputs To the est of our knowlege for isrete-time linear struture systems a graph-theoreti haraterization for the more general ISO prolem enompassing multiple unknown inputs an aounting for LTV ynamis is missing For ontinuoustime LTI systems [18 gives neessary an suffiient onitions for strutural ISO whih when translate into a isretetime setup yiels strutural ISO with some elay L The

3 present paper eals with the notion of ISO with elay-1 a notion that is essential for running input an state estimation filters For isrete-time LTV systems [19 gives a haraterization of s-strutural ISO ut not of strutural ISO while an earlier work from the authors gives haraterizations of oth strutural an s-strutural ISO for LTI systems affete y a single unknown input [20 As suh the main ontriutions of this paper are threefol; uner suitale assumptions on the struture of the input an output matries first we show equivalene etween ISO of a linear system an oservaility of a suitaly efine susystem Seon we give a haraterization of uniform N-step strutural (see Theorem 1) (resp uniform N-step strongly strutural (see Theorem 2)) input an state oservaility ie the onitions uner whih oth the whole network state an the unknown input an e reonstrute for almost all (resp all) system matries that share a ommon zero/non-zero pattern over every time winow of length N This equivalene enales one to stuy strutural (resp s- strutural) ISO using the graph tehniques given in [21 [14 (resp [14 [15) The organization of this paper is as follows: We state the prolem in Setion II Setion III gives an algerai haraterization of the ISO prolem while Setion IV uner suitale assumptions on the input an output matries shows the equivalene etween ISO an oservaility of an appropriate susystem Setion V isusses strutural ISO while Setion VI stuies the stronger notion of s-strutural ISO Differently from Setion V an VI Setion VII explores strutural an s-strutural ISO without assumptions on input an output matries while Setion VIII gives onluing remarks along with isussing future lines of researh Notations R R an Z enote the set of real numers non-zero real numers an integers respetively e j;n represents the j th vetor of the anonial asis of R N Alternatively assuming that the length is lear from ontext we woul enote the same as just e j [A ij enotes the entry in matrix A that orrespons to its i th ron j th olumn I N enotes an ientity matrix of size N Given two matries A an B let A B an A B enote the entrywise prout an Kroneker prout respetively A = iag(a 1 A 2 A N ) enotes a lok iagonal matrix whose loks along the iagonal are A 1 A 2 A n In ase A 1 = A 2 = = A N we get iag(a 1 A 2 A N ) = I N A A k } k1 enotes a sequene of matries A k k = +1 k 1 X enotes arinality of a set X a enotes the smallest integer greater than or equal to a II PROBLEM STATEMENT Consier a linear network system with N noes represente y a graph G = V E} where V is the vertex set an E = (j i) V V [A G ij = 1}; A G eing the ajaeny matrix of G In this network some states an e iretly measure They efine the set O = j 1 j 2 j M } V M eing the numer of oserve states From an analysis of the network we assume that V an also e partitione into assailale noes an reliale ones We efine y A = i 1 i 2 i R } the set of the R assailale noes that may e attake y P external maliious agents efining the set enote y I the attak eing moele as a unknown input An illustration is given in Fig 1 where three maliious noes namely x y z an attak the network with vertex set V through agents k j an i A setup of this sort an e use as an astration to moel attaks on multiple noes inluing eeption attaks [22 false ata injetion [23 fault iagnosis an etetion [24 input estimation in physiologial systems [25 z k O y j g f I x i e Fig 1: Graph representation of a network system affete y external agents where I O an V are the sets of attakers oserve noes an state noes respetively while the set of assailale noes is A = i j k} The ynamis of the linear network system esrie aove is given y the following equations: V a x k+1 = W k x k + A B u k y k = A C x k (1) with state vetor x k R N unknown input vetor u k R P an output vetor y k R M Furthermore W k R N N A B R N P an A C R M N where W k A B an A C are state matrix at time instant k input matrix an output matrix respetively For the rest of this paper over a given interval [ k 1 we enote the ynamis of a LTV system as while that of a LTI system is iniate y W AB A C } k1 In what follows we assume that for all k elonging to Z W k W or in partiular W k W where matrix set W = Z 1 A G Z 1 R N N } an W = Z 1 A G Z 1 R N N } Both W A B A C } an W A B A C } impose a fixe zero struture The remaining oeffiients (ie not fixe to zero positions) of the matries are referre to as free parameters The free parameters of matries elonging to W an take any values while those elonging to W stritly take non-zero values Notie that this assumption implies that the topology of G remains fixe ut the entries orresponing to the free parameters of the system matries may vary For the partiular ase of LTI systems the sai entries remain onstant We narrow our attention to the ase wherein eah unknown input affets exatly one noe of G an eah noe is at most affete y a single unknown input This leas to the following assumption: Assumption 1 (A1): A B = [ e i1;n e i2;n e ip ;N

4 A T C = [ e j1;n e j2;n e jm ;N In the ontext of network systems it is natural to think of states as loal variales that are in ifferent physial loations whereas unknown inputs oul e isolate entities that are at est ale to attak a single state For instane the topology of a power istriution network an e onsiere as the onnetivity etween the meters installe at the sustation feeers transformers an onsumer mains An attak orrespons to aition or raining of ative power while the state at eah noe an e measure using smart meters As a onsequene of assumption A1 we rule out senarios wherein a linear omination of multiple unknown inputs affets a single noe in G Therefore we have R = P On the other han the unknown inputs are of aritrary nature an for the partiular ase in whih some of the unknown inputs are the same we woul have a single unknown input affeting multiple noes in G an as suh we provie suffiient ut not neessary onitions for this setup as well In this paper we first stuy onitions uner whih it is possile to jointly estimate oth the initial state an the sequene of multiple unknown inputs for an LTV system as well as the partiular ase of an LTI system W A B A C } k1 from measurements of a suset of state verties Thereafter ase only on the struture of the graph G we will haraterize ISO for i) almost all hoies of entries in W (see Setion V) an ii) every hoie of entries in W (see Setion VI) over all suffiiently long time winows III ALGEBRAIC CHARACTERIZATION In this setion we esrie some algerai riteria for oservaility an for ISO reviewing the relevant lassial results together with some new variation of them onerning ISO see in partiular Prop 4 A Definitions The onept of oservaility was first introue y Kalman in his seminal paper [26 We reall it in the following efinition Definition 1: The system W k A C } k1 is oservale on [ k 1 if any initial state x k0 is uniquely etermine y the orresponing measure output sequene y k0 y k0+1 y k1 } It is worthwhile to notie here that Definition 1 expliitly asks that the initial state x k0 e reonstrute assuming that the input is known On the other han the notion of strong oservaility asks that the initial state x k0 e reonstrute even in the presene of an unknown input while that of left invertiility with elay 1 requires that the inputs from u k0 up to u k1 1 e reonstrute from the outputs up to y k1 These two notions namely strong oservaility an left invertiility with elay 1 give rise to the efinition of ISO Definition 2: The system W k A B A C } k1 is ISO with elay 1 on the interval [ k 1 if the initial onition x k0 R N an the unknown inputs sequene u k0 u k0+1 u k1 1} an e uniquely etermine from the measure output sequene y k0 y k0+1 y k1 } A stronger notion of oservaility is that of uniform δ-step oservaility whih requires that a system e oservale over every time winow of length δ [27 Analogously we efine uniform δ-step ISO as follows: Definition 3: The system W k A B A C } k Z is uniformly δ- step ISO if Z W k A B A C } k0+δ is ISO over [ + δ Remark 1: Notie that although uniform δ-step ISO (resp oservaility) is with respet to all intervals of length δ it turns out that it an e rephrase onsiering all intervals of length at least δ For oservaility this is immeiate: if a system is oservale over [ + δ then it is also oservale over [ + η for all η δ For ISO one nees to reonstrut all inputs up to +η 1 an not only those up to +δ 1 If the system is uniformly δ-step oservale it is possile to use δ-step ISO over suessive time winows of length δ to ensure that all the require inputs are inee reonstrute Remark 2: It is well-known that either LTI systems are not oservale or are uniformly N-step oservale in whih ase we woul simply all it as oservale B Oservaility Invertiility Input an State Oservaility Matries Let Θ k0k 1 Γ k0k 1 an Ψ k0k 1 represent the oservaility matrix invertiility matrix an input an state oservaility (ISO) matrix respetively over the interval [ k 1 These are efine as follows: A C A C W k0 A C W k0+1w k0 Θ k0k 1 = A C W k1 1 W k0 0 0 A C A B 0 Γ k0k 1 = A C W k0+1a B A C A B 0 A C W k1 1 W k0+1a B A C A B Ψ k0k 1 = [Θ k0k 1 Γ k0k 1 (2) This leas us to the following lassial Kalman-like haraterization of ISO Proposition 1: The system W k A B A C } k1 is ISO over [ k 1 if an only if Ψ k0k 1 is full olumn rank ie rank(ψ k0k 1 ) = N + (k 1 )P Proof: Let y k0:k 1 an u k0:k 1 1 enote the vetors of onatenate outputs an unknown inputs over [ k 1 respetively Therefore from (1) an (2) the following an e reaily otaine y k0:k 1 = Θ k0k 1 x k0 + Γ k0k 1 u k0:k 1 1 = Ψ k0k 1 [ x k0 u k0:k 1 1

5 Base on Definition 2 it is immeiate that input an state oservaility is equivalent to uniqueness of the aove system of linear equations Prop 1 enales one to exploit the struture of Ψ k0k 1 so as to fin some simple neessary onitions for Ψ k0k 1 to have full olumn rank The following proposition riefly summarizes them Proposition 2: The following onitions are neessary for the system W k A B A C } k1 to e ISO over [ k 1 : i) rank(θ k0k 1 ) = N ii) rank (A C A B ) = P iii) M P iv) N P In ase N neessary: v) M > P vi) k 1 > P then the following onitions are also N M M P In ase P = N then the following onitions are neessary an suffiient: M = N k [ k 1 rank(a C ) = N an k [ k 1 1 rank(a B ) = N Proof: The proof is reporte in the Appenix Notie that Prop 2 fully haraterizes the ISO prolem for the partiular ase of P = N where uner A1 the system is ISO if an only if all noes are oserve (ie O = V) In this paper we restrit our attention to the non-trivial ase of N > P ie not all the noes are assailale Therefore from Prop 2 M > P is a neessary onition for ISO From Prop 2 we know that the following are neessary onitions for ISO: 1) all the assailale noes are oserve ie i 1 i 2 i P } j 1 j 2 j M } an 2) all of the assailale noes are istint ie there oes not exist h k elonging to 1 2 P } suh that i h = i k We also assume that all of the oserve noes are istint ie there oes not exist h k elonging to 1 2 M} suh that j h = j k This ensures that there are no repeate or epenent rows in C Therefore one an relael the noes in G in the following manner: i 1 = j 1 = 1 i 2 = j 2 = 2 i P = j P = P The aforesai relaeling allows us to rewrite A B an A C as follows: Assumption 2 (A2): A B = [ e 1;N e 2;N e P ;N A T C = [ e 1;N e 2;N e P ;N e jp +1 ;N e jm ;N C Alternative Algerai Charaterization Prop 1 haraterizes ISO in terms of rank of Ψ k0k 1 However the elements in Ψ k0k 1 are otaine y taking prouts of the state matries over the interval [ k 1 Consequently the zero/non-zero pattern is not preserve In orer to overome this rawak in this susetion we provie an alternative algerai haraterization for oth oservaility an ISO Theorem 641 in [28 gives an alternative haraterization of ontrollaility The following proposition oes the same for oservaility Proposition 3: The system W k A C } k1 is oservale over [ k 1 if an only if rank(q k0k 1 ) = (k 1 + 1)N where A C A C 0 Q k0k 1 = 0 A C W k0 I N 0 0 W k0+1 I N 0 0 W k1 1 I N with Q k0k 1 R (k1 k0 + 1)M + (k1 k0)n (k1 k0 + 1)N Proof: Notie that the prolem of reonstruting x k0 from y k0:k 1 is equivalent to the prolem of reonstruting x k0 x k0+1 x k1 The relationship etween the states an outputs an e expresse via a system of linear equations as follows From Eq (1) an setting u(k) = 0 P we have: k [ k 1 1 W k x k x k+1 = 0 N an k [ k 1 C k x k = y k [ y This an e rewritten as: Q k0k 1 x k0:k 1 = k0:k 1 0 (k1 1)N Hene the system W k A B A C } k1 is oservale over [ k 1 if an only if the aove system of linear equations has a unique solution It turns out that similar arguments an e mae for ISO as well an will e shown in Prop 4 As a first step we efine the following matrix: [ 0 J k0k 1 = Q B k0k 1 k0k 1 where B k0k 1 = I k1 A B R (k1 k0)n (k1 k0)p Proposition 4: The system W k A B A C } k1 is ISO over [ k 1 if an only if rank (J k0k 1 ) = (k 1 )P + (k 1 + 1)N Proof: The prolem of reonstruting x k0 an u k0:k 1 1 from y k0:k 1 is equivalent to the prolem of reonstruting x k0 x k0+1 x k1 an u k0:k 1 1 From (1) we have: k [ k 1 1 W k x k + A B u k x k+1 = 0 N an k [ k 1 A C x k = y k Hene oth the state equation an output equation at eah time instant an e expresse as a linear omination of x k0 x k0+1 x k1 as well as u k0 u k0+1 u k1 1 in the following manner: [ [ uk0:k J 1 1 y k0k 1 = k0:k 1 x k0:k 1 0 (k1 )N Hene the system W k A B A C } k1 is ISO over [ k 1 if an only if the aove system of linear equations has a unique solution IV ISO AS OBSERVABILITY OF AN APPROPRIATE SUBSYSTEM The ojetive here is to eompose the system into two susystems an show that ISO is equivalent to oservaility of one of the susystems It is ruial to notie here that the ientity of the noes eing

6 assailale remains fixe an aoring to assumption A2 equal to 1 2 P } Consequently the noes laele from i P +1 i N are not assailale This enales us to eompose the state vetor in two loks: ˆx k enoting states that are iretly affete y the unknown inputs an x k for the remaining states; a orresponing partitioning is also one for the output vetor otaining [ˆxk x k = x k y k = [ŷk with ˆx k R P x k R N-P ŷ k R P an ỹ k R M-P Moreover thanks to Assumption A2 the input an output matries an e rewritten as follows: A B = [ IP 0 ỹ k [ IP 0 A C = 0 Ã C Therefore the system W k A B A C } k1 into two susystems as follows: an e eompose ˆxk+1 = Ŵk ˆx k + Λ x k + u k ŷ k = ˆx k (3) xk+1 = W k x k + Ωˆx k ỹ k = ÃC x k (4) where we use the notation [Ŵk W k = Ω k From (3) it is lear that ˆx k is iretly oserve Hene (3) represents a system with known state ut two unknown inputs namely x k an u k while (4) represents a system with unknown state ut known input Hene we have the following proposition Proposition 5: Uner A2 the system W k A B A C } k1 is ISO over [ k 1 if an only if the system k1 Wk ÃC} is oservale over [ k 1 Proof: We efine the matries Q N an Q N as follows: [ [ Ip 0 Q N = QN = 0 Λ k Wk I N P Let Π 1 an Π 2 represent ron olumn permutation matries respetively efine as follows For olumn permutations we put at the eginning the first P olumns of eah ourrene of A C otaining [ R1 0 R J Π 2 = 3 R 2 B k0k 1 R 4 where R 1 = I k1 +1 A C Q N W k0 Q N Q N 0 N P R 2 = 0 N P W k1 1Q N Q N R 3 = I k1 +1 A C QN an W k0 QN Q N 0 R 4 = 0 W Q k1 1 N Q N For row permutations onsier the following steps: we first arrange the (k 1 +1) row loks orresponing to the first P rows of eah ourrene of A C then the (k 1 ) row loks orresponing to the first P rows of eah ourrene of A B an finally the remaining rows so as to otain where Π 1 J Π 2 = I (k1 +1)P 0 0 P 1 I (k1 )P P C P 3 0 W Ŵ k0 I P 0 P 1 = 0 Ŵ k1 1 I P Λ k0 0 P 2 = 0 Λ k1 1 0 Ω k0 0 P 3 = 0 Ω k1 1 0 C = I k1 +1 A C an W k0 I N P 0 0 Wk0+1 I N P 0 W = 0 Wk1 1 I N P Let J = Π1 J Π 2 an ˆ J = I (k1 )P P 2 0 C 0 W J = [ C W Notie that J is lok lower triangular with the loks over the iagonal I (k1 +1)P an J ˆ This implies rank ( J ) = (k 1 + 1)P + rank ( J ˆ ) J ˆ is lok upper triangular with loks over the iagonal I (k1 )P an J Therefore the following hols: rank ( J ) = (k 1 + 1)P + (k 1 )P + rank ( J ) From Prop 4 we know that W k A B A C } k1 is ISO over [ k 1 if an only if rank ( J ) = (k 1 )P +(k 1 +1)N whih in turn is equivalent to rank ( J ) = (k 1 + 1)(N P ) From Prop 3 the latter orrespons to oservaility of k1 Wk ÃC} over [ k 1 The result in Prop 5 an e interprete as follows: when the susystem W k1 k ÃC} is oservale over [ k 1 one of the two unknown inputs in (3) namely x k is known an hene it is possile to ompute u k sine ˆx k is iretly measure For LTI systems alternatively the PBH rank test may also e use to prove Prop 5 The intereste reaer may e inspire y Prop 4 in [20 an proof therein

7 As mentione previously Prop 1 an Prop 4 haraterize ISO in terms of rank onitions of matries namely Ψ k0k 1 an J k0k 1 respetively These algerai tehniques work well provie we have aess to the exat values of all the oeffiients of the aforesai matries Moreover from a omputational stanpoint this tehnique is rather limite sine the omputational omplexity inreases as the size of the network grows Therefore in the sequel we turn our attention to strutural (resp s-strutural) results ie the fous is on fining onitions suh that the system is ISO for almost all hoies of free parameters (resp every hoie of free parameters) of the system matries V STRUCTURAL ISO The main ojetive of this setion is to haraterize ISO for almost all hoies of free parameters A Definition an Impliations We enote y W A B A C } LTV the family of all LTV systems as given in (1) an having the same zero/non-zero pattern as given y W A B an A C while W A B A C } LTI represents the orresponing family of all LTI systems As mentione previously struture systems have fixe zero positions an free parameters Let E enote the numer of ones in A G Uner suh a setup the free parameters an take values in R ( E )(k1 k0) where [ k 1 represents the time winow over whih the system W A B A C } LTV is eing oserve Notie that eah element in R ( E )(k1 k0) yiels a hoie of free parameters Strutural ISO then asks that there e at least one memer in W A B A C } LTV whih is oservale This leas us to the following efinition Definition 4: W A B A C } LTV is struturally ISO on [ k 1 k 1 > if there exists at least one system with W k W; suh that W k A B A C } k1 is ISO Analogously one an efine strutural oservaility an uniform N-step strutural oservaility for LTV systems In partiular efinitions for strutural ISO strutural oservaility an uniform N-step strutural oservaility an also e otaine for LTI systems where the spae of free parameters is R E an the same free parameters are repeate at eah time instant It is well-known that oservaility (an ontrollaility) is a property (see [29 [12) suh that either there is no hoie of parameters that makes it true or it is true for almost all hoies of parameters Almost all hoies of parameters means all hoies of parameter exept those lying in some proper algerai variety of the spae of free parameters This means there are some non-trivial polynomials (one or more ut finitely many) suh that the property is true for all parameters exept those whih are zeros of this system of polynomials The polynomials eing non-trivial (ie not ientially zero) ensures that the variety is proper (ie not the whole spae of free parameters) an therefore has Leesgue measure zero This an e interprete as the property eing true with proaility one if the parameters are thrown at ranom aoring to any ontinuous proaility istriution Furthermore small variations in the parameter values woul not lea to loss of property It turns out that the aove isussion also hols for ISO as shown in the following: Proposition 6: The set of parameters for whih W AB A C } LTV is not ISO is either the whole parameter spae R ( E )(k1 k0) or a proper variety of R ( E )(k1 k0) Proof: The proof is ase on stanar tools an is reporte in Appenix This means that over a given interval [ k 1 if one memer of the family of systems W A B A C } LTV is ISO then almost all memers of W A B A C } LTV are ISO On the other han if W A B A C } LTV is not struturally ISO then none of the memers of W A B A C } LTV is ISO An analogous result to Prop 6 also hols for LTI systems wherein the spae of free parameters R E Definition 4 nees to e seen against this akrop It turns out that strutural ISO for a family of LTI systems implies strutural ISO for the orresponing family of LTV systems an is given y the following remark Remark 3: If the LTI system W A B A C } LTI is struturally ISO then the orresponing LTV system W A B A C } LTV is struturally ISO over all suffiiently long intervals Inee if the system W A B A C } LTI is struturally ISO then there exists W W suh that the triplet (W A B A C ) is ISO Therefore over an interval [ k 1 of length at least N one an set W k = W k [ k 1 otaining a system that is ISO over [ k 1 therey exhiiting a hoie of entries for whih W A B A C } LTV is ISO Consequently from Definition 4 the system W A B A C } LTV is struturally ISO over [ k 1 However the onverse of Remark 3 is open In the rest of this setion we show that uner assumption A2 the onitions given in Remark 3 are equivalent B Uniform N-step strutural ISO for LTV systems From Proposition 5 we an stuy ISO y stuying oservaility of a suitale su-system Here we apply this tehnique to the family of systems W A B A C } LTV efining a suitale family of susystems We efine the set of matries W as W = Q T N W Q N W W} Let W Ã C } LTV represent the family of all LTV systems as given in (4) ut without the known input ˆx k We enote y W Ã C } LTI the ounterpart LTI susystem (ie whose matries have the same zero/nonzero pattern as given y W) As a onsequene of Prop 5 for LTI systems we have the following remark: Remark 4: Uner A2 W AB A C } LTI is uniform N- step strutural ISO if an only if W Ã C } LTI is struturally oservale It turns out that orresponing to Remark 4 onitions for strutural results an also e otaine for LTV systems as shall e eviene in the rest of this susetion An immeiate orollary of Prop 5 is the following Proposition 7: Uner A2 W A B A C } LTV is struturally ISO over [ k 1 if an only if W Ã C } LTV is struturally oservale over [ k 1 The avantage of Prop 7 is that it reaks own the prolem of strutural ISO into an equivalent prolem in strutural oservaility With this in han an rewriting Thm 3 in [30

8 (also see [16) for oservaility we otain equivalene etween strutural oservaility for LTV an LTI systems an is given y the following proposition Proposition 8 (Thm 3 in [30): Uner A2 over any interval [ k 1 of length at least N W Ã C } LTV is struturally oservale if an only if W Ã C } LTI is struturally oservale Prop 7 an Prop 8 together reak own the strutural ISO prolem of LTV systems into a strutural oservaility prolem of a orresponing suitaly efine LTI susystem Thanks to [21 it turns out that the strutural oservaility of an LTI susystem an e etermine y heking ertain graph-theoretial onitions Before proeeing we nee a few onstruts on G Let G e the graph orresponing to W Let S = L 1 L 2 E S } e a ipartite graph assoiate with G with L 1 = Ṽ \ Õ L 2 = Ṽ onstrute in the following manner two verties in L 1 an L 2 that orrespon to the same element v Ṽ are enote as u v an w v respetively an E S = (u i w j ) L 1 L 2 (i j) Ẽ} Similar to [31 ut without the introution of aitional output noes in G we state the following efinitions Definition 5: The graph G with oservation set Õ is sai to e output-onnete if for all v Ṽ there exists a path from v to w for some w Õ Definition 6: A mathing is a set of eges that o not share any ommon verties With Definitions 5 an 6 in han we state the following result rephrase for oservaility Lemma 1 (Thm 1 [14): The system W AC } LTI is struturally oservale if an only if: 1) G is output-onnete; 2) there exists a mathing in S of size N Õ As an asie the aove result previously appeare in [21 an [31 With Lemma 1 in plae we state our first main result Theorem 1: Uner A2 W A B A C } LTV is uniformly N- step struturally ISO if an only if the following onitions are satisfie: 1) G is output-onnete; 2) there exists a mathing in S of size N Õ Proof: From Prop 7 it an e seen that uner A2 the system W AB A C } LTV is struturally ISO over [ k 1 if an only if the susystem W Ã C } LTV is struturally oservale over [ k 1 while from Prop 8 it an e seen that the susystem W Ã C } LTV is struturally oservale over [ k 1 if an only if the orresponing LTI susystem W Ã C } LTI is struturally oservale It is well-known that LTI systems are either oservale over every suffiiently long interval or not oservale at all Thus setting δ = N in Remark 1 an from Prop 7 an Prop 8 it follows that uner A2 the system W AB A C } LTV is struturally ISO over [ k 1 if an only if the susystem W Ã C } LTI is uniform N-step strutural ISO Thereafter from Lemma 1 the proof is omplete Example 1: With referene to the system shown in Figure 1 it an e seen from Figure 2 an Figure 4 that the susystem is output-onnete an its ipartite graph S ontains a mathing of size N Õ an hene the susystem is struturally oservale [14 Therefore from Thm 1 the system given in Õ a g f Fig 2: The susystem G for the system shown in Fig 1 u u u e w w w w e w f w g Fig 3: Bipartite graph S assoiate with G u u u w w w w e w f w g Fig 4: A Mathing M in S Figure 1 is uniformly N-step struturally ISO Item i) or output-onneteness of G an e heke y using a variant of Tarjan s algorithm an has omplexity that is linear in the numer of eges an verties of G (ie O( Ẽ + Ṽ )) [32 On the other han Hoproft-Karp maximum mathing algorithm an e use for heking item ii) an its omplexity is O(( Ẽ + Ṽ ) Ṽ ) [33 VI S-STRUCTURAL ISO The main ojetive of this setion is to haraterize ISO for every hoie of entry in W A Definition S-strutural properties are those that hol for every non-zero hoie of free parameters of the system matries That is s- strutural ISO (resp oservaility) requires that every memer of the family of LTV systems given y W A B A C } LTV e ISO (resp oservale) This leas us to the following efinition Definition 7: Let k 1 Z an k 1 > W A B A C } LTV is s-struturally ISO on [ k 1 if for every system W k A B A C } k1 with W k W is ISO Analogous to Definition 7 one an also efine s-strutural oservaility an uniform N-step s-strutural oservaility for LTV systems In partiular efinitions for s-strutural ISO s-strutural oservaility an uniform N-step s-strutural oservaility an also e otaine for LTI systems It turns out that s-strutural ISO for a family of LTV systems

9 implies s-strutural ISO for the orresponing family of LTI systems an is given y the following remark Remark 5: If the system W A B A C } LTV is s-struturally ISO over an interval [ k 1 then the orresponing LTI system W A B A C } LTI is s-struturally ISO Inee if the system W A B A C } LTI is not s-struturally ISO then from Definition 7 there exists a system W 1 A B A C } with W 1 W suh that W 1 A B A C } is not ISO Over any interval [ k 1 we an set W k = W 1 k [ k 1 suh that the LTV system W k A B A C } k1 is not ISO Consequently from Definition 7 the system W A B A C } LTV is not s- struturally ISO over any interval Notie that for strutural ISO the impliation is in the other iretion (see Remark 3) The onverse of Remark 5 remains open In the rest of this setion we show that uner assumption A2 over suffiiently long intervals the onitions given in Remark 5 are equivalent B Uniform N-step s-strutural ISO for LTV systems The set of matries W is efine analogous to W One an use Prop 5 so as to otain s-strutural ISO results We first fous on LTI systems As another onsequene of Prop 5 we have the following: Remark 6: Uner A2 W A B A C } LTI is uniform N-step s-strutural ISO if an only if W Ã C } LTI is s-struturally oservale It turns out that one an otain similar onitions for LTV systems as well as shall e seen in the rest of this susetion First notie that another immeiate orollary of Prop 5 an e state as follows Proposition 9: Uner A2 W A B A C } LTV is s- struturally ISO over [ k 1 if an only if W ÃC} LTV is s-struturally oservale over [ k 1 Thanks to Prop 9 we an now rephrase s-strutural ISO of a family of LTV systems as an equivalent prolem in s-strutural oservaility of a suitale family of LTV systems Against this akrop it is inee relevant to see if s-strutural ISO of LTV systems is equivalent to s-strutural oservaility of a suitale family of LTI systems The following proposition is immeiate from Corollary IV2 [17 Proposition 10: Uner A2 over any interval [ k 1 of length at least N W ÃC} LTV is s-struturally oservale if an only if W ÃC} LTI is s-struturally oservale Thus from Prop 9 an Prop 10 it an e seen that uner assumption A2 the s-strutural ISO prolem for LTV systems reaks own into an equivalent prolem in s-strutural oservaility for a suitaly efine LTI susystem This equivalene allows us to exploit the literature on s-strutural oservaility as we see in the following Thanks to [14 (also see [15) it turns out that s-strutural oservaility of an LTI system an e assesse y heking some graph-theoretial onitions Here we woul e fousing on the notion of uniquely restrite mathing (also known as onstraine mathing) as in [14 In orer to proee a few onstruts on the graph G are ue E loop E S enote the eges of the form u i w i } if there exists any Notie that E loop orrespons to self-loops in G Let E new enote the set of newly ae self-loops in G ie aing self-loops for those verties i Ṽ that previously i not have one in G Let S = L 1 L 2 E S } where E S = E S E new } enote another ipartite graph on G We reall that a mathing is sai to e uniquely restrite if there is no other mathing involving the same vertex set Equivalent haraterizations of uniquely restrite mathings are isusse in [34 The following result is the same as Thm 5 in [14 ut rewritten for s-strutural oservaility Lemma 2 (Thm 5 [14): The system W ÃC} LTI is s- struturally oservale if an only if: 1) there exists a uniquely restrite mathing M E S of size N Õ ; 2) there exists a uniquely restrite mathing M E S of size N Õ suh that M E loop = With Lemma 2 in plae we present our seon main result in the following theorem Theorem 2: Uner A2 W A B A C } LTV is uniformly N-step s-struturally ISO if an only if the following two onitions are satisfie: 1) there exists a uniquely restrite mathing M E S of size N Õ ; 2) there exists a uniquely restrite mathing M E S of size N Õ suh that M E loop = Proof: From Prop 9 an Prop 10 it an e seen that uner A2 W A B A C } LTV is s-struturally ISO over any interval of length at least N if an only if W ÃC} LTI is s-struturally oservale while from Prop 10 it an also e seen that W ÃC} LTI is s-struturally oservale if an only if W ÃC} LTV is s-struturally oservale over every interval of length at least N Thus Prop 9 an Prop 10 together with setting δ = N in Remark 1 results in the following: uner A2 W A B A C } LTV is s-struturally ISO if an only if the LTI susystem W ÃC} LTI is uniform N- step s-strutural ISO Thereafter from Lemma 2 the proof is omplete The onitions in Theorem 2 an e heke using the algorithm given in [14 with omplexity O( Ṽ 2 ) or with the algorithm introue in [35 whih ahives a linear omplexity O( Ṽ + Ẽ ) y omining sophistiate ata strutures an sparse matrix tehniques Example 1 (ontinue): With respet to the system given in Figure 1 first reall that M = ( w ) (u w ) (u w ) (u w e )} (see Figure 4) is a mathing in S Furthermore there exists no other mathing M E S saturating the same verties as M an hene y efinition M E S is a uniquely restrite mathing The seon onition is heke with respet to the ipartite graph S given in Figure 5 It an e seen that M = ( ) (u w ) (u w ) (u w e )} (see Figure 6) is a mathing in S that satisfies M E loop = Notie that there exists no other mathing ˆM E S saturating the same verties as M Therefore from Thm 2 the system given in Figure 1 is uniformly N-step s-struturally ISO Now onsier another example

10 u u w w e f w g Fig 5: Bipartite graph S in soli re: S S in ashe lue: E new = ( ) (u w ) (u w )} while E loop = (u w )} u u w w e f w g Fig 6: A mathing M in S Example 2: Consier the system given in Figure 7 whose orresponing susystem is given in Figure 8 while a ipartite graph assoiate with the susystem G 1 is given in Figure 9 It is immeiate that the susystem G 1 is output-onnete Furthermore there also exists a mathing of size N Õ on the ipartite graph S 1 Therefore from Thm 1 the system given in Figure 7 is uniformly N-step struturally ISO On the other han from Figures 10(a) an 10() it an e seen that there oes not exist a uniquely restrite mathing over the hoie of vertex sets u u } an w w } The same an e sai with respet to the vertex sets u u } an w w } (see Figures 10() an 10()) u u } an w w w } (see Figures 10(e) an 10(f)) u u } an w w } (see Figures 10(g) an 10(h)) Thus there oes not exist a uniquely restrite mathing of size N Õ on the ipartite graph S 1 an hene from Thm 2 the system given in Figure 7 is not uniformly N-step s-struturally ISO a a I x y z e f g O Fig 7: The system G 1 Õ Fig 8: Susystem G 1 u u w w w Fig 9: Bipartite graph S 1 assoiate with G 1 VII CONCLUSION We have stuie ISO with one time-step elay for linear network systems with a fixe topology Uner the assumptions: (i) eah assailale noe an e attake y at most a single unknown input (ii) eah unknown input affets exatly one noe an (iii) iret measurements of ertain states are availale we provie a haraterization of ISO in terms of oservaility of a suitaly efine susystem Moreover we have stuie uniform N-step ISO for almost all hoies of free parameters in the W as well as for all non-zero hoies of free parameters in W where W stans for the family of systems sharing the same zero/non-zero pattern A future line of investigation woul e to haraterize ISO for more general linear network systems y onsiering time-varying topology an less a priori knowlege on the assailale noes A weaker notion of ISO wherein the fous is on the unique reovery of just a suset of the states for instane only the states that are not affete y an unknown input oul e another iretion of future works Proof of Proposition 2 APPENDIX Item i) requires that the first N olumns of Ψ k0k 1 e linearly inepenent Item ii) requires that the last P olumns of Ψ k0k 1 e linearly inepenent while items iii) an iv) are neessary onitions for item ii) To see the neessity of items v) an vi) notie that in orer for Ψ k0k 1 to e full olumn rank it is neessary that Ψ k0k 1 has at least as many rows as olumns ie M(k 1 + 1) N + (k 1 )P From the aove equation sine (k 1 + 1) > 0 it follows that M P + N P (k 1 +1) If N > P this implies that M > P Then uner M > P item vi) immeiately follows from the aove equation For the partiular ase of M = P = N notie that Ψ k0k 1 is a lok lower triangular matrix with eah of the loks eing square Hene a neessary an suffiient onition for full olumn rank of Ψ k0k 1 is that eah of the iagonal loks have full olumn rank This is equivalent to i) rank(a C A B ) = N k [ + 1 k 1 an ii) rank(a C ) = N Notie that k [ + 1 k 1 rank(a C A B ) = N if an only if: i) rank(a C ) = N an ii) rank(a B ) = N Proof of Proposition 6 From Prop 1 ISO is equivalent to Ψ k0k 1 having rank N + (k 1 )P Hene the system is not ISO if an only if all the square sumatries of size N + (k 1 )P have zero eterminant Notie that the entries of Ψ k0k 1 are polynomials whose variales are the free parameters of W k A B an A C ; the fixe values (zeros) an e interprete as polynomials of egree zero For eah sumatrix of size N + (k 1 )P the eterminant is otaine y multipliations an summations of suh polynomials an hene is itself a polynomial We have foun a finite set of polynomials suh that the system is not ISO if an only if the parameters elong to the zero set of all

11 u w u w u w u w (a) Mathing M 1 on S 1 () Mathing M 2 on S 1 () Mathing M 3 on S 1 () Mathing M 4 on S 1 u w w u w w u w w u w w (e) Mathing M 5 on S 1 (f) Mathing M 6 on S 1 (g) Mathing M 7 on S 1 (h) Mathing M 8 on S 1 Fig 10: All maximum mathings on S 1 these polynomials Either suh polynomials are all trivial (ie onstantly equal to zero) an hene all hoies of parameters result in a system not ISO or at least one of the polynomials is non-trivial an hene the set of parameters for whih the system is not ISO is a proper variety of parameter spae REFERENCES [1 F Pasqualetti F Dorfler an F Bullo Control-theoreti methos for yerphysial seurity: Geometri priniples for optimal ross-layer resilient ontrol systems IEEE Control Systems vol 35 no 1 pp [2 J Slay an M Miller Lessons learne from the Maroohy water reah in International Conferene on Critial Infrastruture Protetion Springer 2007 pp [3 J P Conti The ay the sama stoppe Engineering & Tehnology vol 5 no 4 pp [4 M Hautus Strong 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