Affine Dependence of Network Controllability/Observability on Its Subsystem Parameters and Connections

1 Afﬁne Dependence of Network Observabili ty/Contr ollability on Its Subsystem Parameters and Connections T ong Zhou † and Y uyu Zhou Abstract —This p ap er in vestiga tes observability/controllability of a networ ked dynamic system (NDS) in which system ma- trices of its subsystems are expressed through linear fractional transfo rmations (LFT). Some rela tions ha ve been obtained be- tween this NDS and descr iptor systems about their obser v- ability/controllability . A necessary and sufﬁcient condition is established with the associated matrices dependin g afﬁ nely on subsystem p arameters/connections. An attractive property of this condit ion is that all the required calculations are per - fo rmed independ en tly on each ind ividual subsystem. Except well- posedness, not any other conditions are asked for sub system parameters/connections. This is in sharp contrast to recent results on structural obser va bility/controllability which is p ro ven to be NP hard. Some characteristics ar e established for a subsystem which are helpful in constructing an obser vable/contr ollable NDS. It has been made clear that su b systems with an input matrix of full column rank are helpful in constructin g an observable NDS, while sub systems with an outp ut matrix of full row rank are h elpful in constructin g a controllable NDS. These results are extended to an NDS with descriptor f orm subsystems. As a byproduct, th e full normal rank condition of p re vious works on n etwork observ ability/controllability has been completely remo v ed. On the other hand, satisfaction of this condi tion is shown to be appreciativ e in buildin g an observa ble/controllability NDS. Index T erms —controllability , descriptor system, Kronecker canonical for m, large scale system, LFT , networked d ynamic system, observ abil ity . I . I N T RO D U C T I O N In system de sig ns, it is essential to at ﬁrst build a p lan t capable of reachin g goo d perf ormance s. When a networked dynamic system (NDS) is to be designed, this p roblem is related to both selecting sub system para meters and design ing subsystem conn ections [2, 13, 17, 22]. T o achieve this objec- ti ve, some explicit relations seem necessary b etween system achiev ab le perfo r mances and its parameters/con nections. On the other hand, observability/controllability is essential fo r a system to p roperly work, noting that they are closely related to various imp ortant system properties. Examples include f au lt detection, o p timal con trol, pole placemen t, state estimation , etc. It is widely believed that an unco ntrollable/u nobservable plant can h ardly be anticip ated to have satisfactory regula- tion/estimation per forman ces [12, 17, 18, 22]. Observability an d con trollability are now well developed concepts in system analysis and synthesis, and v ariou s criteria This work was supported in part by the NNSFC under Grant 61733008 and 61573209. This work has been submitted to the IEEE for possible public ation. Copyri ght may be transferred without notice, after which this ver s ion may no longer be accessible . T ong Zhou † (correspond ing author) and Y uyu Zhou are with the Departmen t of Automatio n, Ts inghua Univ ersity , Beijing , 100084, P . R. China (email: tzhou@mail.ts inghua.edu.cn , zhouyy18@mail s.tsinghua.ed u.cn ). have been established , such as th e PBH test, co ntrollabil- ity/observability matrix, etc. [ 12, 18, 22]. I n ad dition, th ese concepts have been stud ied from various distinctive aspects. For example, it is n ow extensi vely known that f or various typ es of system s, con trollability and observability are generic system proper ties. Th at is, r ather th a n n umerical values o f sy stem matrices, it is the state connections, as well as the connections from an input to the system states (the con nections f rom the system states to an ou tput), that determ ine the contro llability (observability) of a system . Motivated by these obser vations, structural controllability /observability is d ev e loped and studied by many r esearchers [7, 14, 22]. [5] and [20] reveal that the minima l inp ut/outpu t number guaranteeing the structural controllab ility/observability of an NDS is determin ed by its subsystem dynamics. The prob lem is proved to b e NP-h ard in [15] of searching the sparest o utput/inpu t m atrix with the associated NDS observable/contro llable. [16] suggests som e metrics for qu antitativ ely analy zing hardn ess in co ntrolling a system . It has been m ade clear in [20] that to constru ct an observable/controllab le NDS, each sub system sh ould b e observable/contro llab le. Another o bservation there is that th e minimal numbe r for subsystem outputs/inp uts is equal to the maximal geometric multiplicity of its state transition matrix. And so on . While various results have b e en ob tained for NDS observ - ability/contr o llability , many importa n t issues remain u nsolved. Inﬂuences of subsystem con n ections/dyn amics etc., o n the observability/contro llability of the whole system, are some examples [3, 4 , 22]. In a d dition, techn ology developments, stringent performance requir e m ents, etc. ar e rapidly producing complicated systems with an increasing n umber of subsystems that have d istinguished working mechan ics [ 2, 13, 19]. For these systems, nu m erical stability and computatio nal co sts are essential issues with g reat challeng es [15, 17, 22]. When sy stem matrices o f each subsystem can be expressed as an linear fr actional transform ation ( LFT) of its ﬁrst p rinciple parameters (FPP), NDS structur al contr ollability has been recently studied in [23]. It h as been sho wn th a t con trollability for these systems is a generic prop e r ty and their structu ral controllab ility veriﬁcation is in general NP hard. This is quite discourag ing, no ting that LFT is a very effectiv e expression in system analysis and synthesis, and various systems have the proper ty that althoug h elements of its system matrices are not algebraically indepe ndent of each o ther, they can be written as an LFT o f its algeb raically independ ent FPPs [18, 22]. T o settle the issue of NDS o bservability/controllab ility veriﬁcations under the situatio n that system matrices of its subsystems are expressed b y LFTs, in this pap er , rather than their structural counterp arts, ob servability and con trollability are directly studied. Surprising ly , it has been observed that 2 this veriﬁcatio n problem can be converted to the veriﬁcation of the co ntrollability/o bservability of a par ticular descripto r system. By m eans of the Kr onecker cano nical fo rm (KCF) of a matrix pencil, a ra n k based co ndition is derived with the associated matrix dependin g af ﬁne ly on both subsystem parameters an d conne ctions. T his con d ition keeps the attractive proper ties of the veriﬁcatio n procedur e rep orted in [19, 22] that in o btaining the a ssoc iate d matrices, the in volved numerical calculations are perform ed in depend ently on each subsystem, which make its veriﬁcation scalable fo r a large scale NDS. In der iving these results, except a well-posedn ess co ndition, not any oth er req u irements are asked for a subsy stem FPP or a subsystem con nection. This explicit r e lation with sub sys- tem par ameters/conn ections seem usefu l in design ing system topolog ies and selecting subsystem parameters. A by p rodu c t of th is inv e stigation is that the fu ll n ormal ran k conditio n asked in [19, 22] has been com p letely r emoved. On the othe r han d, it has been mad e clear that satisfaction of th is rank conditio n by a subsystem is app reciative in re ducing difﬁculties of constructing a c o ntrollable/o bservable NDS. On th e basis of th is co ndition, it is shown that a sub system with an input matrix of full column ran k (FCR) is helpf u l in con stru cting an observable NDS that rece i ves signals from other subsystem, wh ile a sub system with an output matrix of full row rank ( FRR) is helpful in co nstructing a c ontrollab le NDS tha t sends signals to other subsystems. Some ran k con- ditions ha ve also been deriv ed fo r a subsystem with which a n observable/contro llab le NDS can b e constructed more easily , and these cond itio ns can b e indep endently veriﬁed for eac h subsystem. These results are expected to be help ful in study ing optimal sensor/actu a to r placeme nts. Ex tensions to an NDS with descriptor form subsystems have also been dealt with. Similar results h av e been obtained. The ab ove results are in sharp con tr ast to those about structural con trollability/ob servability , and ma ke co ntrollabil- ity/observability veriﬁcation in princip le feasible f or a large scale NDS with LFT p a rameterized system matrices for ea ch subsystem, no ting that in [2 3], structural con trollability veriﬁ- cation h as been proven to be NP hard f or these systems, and an effective veriﬁcation alg orithm is developed o nly fo r the case in whic h the m atrix has a diag onal p arametrization that is constructed fr om all subsystem FPPs and the SCM, which appears to be a gr e a t restriction o n the applicability of the obtained results to a practical pr oblem. The rem a ining of this paper is structur ed as follows. The next section gives an NDS model and some prelim inary re- sults. NDS ob servability/controllability is d ealt w ith in Sectio n III. An applica tio n of th ese resu lts to sensor/actu ator place- ments are discussed in Section IV , while Sectio n V inv estig a te s how to exten d them to an NDS when the d ynamics of some subsystems are d escribed by a descriptor f orm. A n umerical example is provid ed in Section VI to illustrate the ob tained theoretical results. Finally , con clusions are given in Section VII which reveals some fu rther issues. Four ap pendices a r e attached to p rovide proofs of several technic a l results. The following symbols and n otation are ad o pted. R n and C represent r espectively th e n dimensional real Euclidean space and the set of com plex numb ers. de t ( · ) stands for the determin ant of a squa re matrix, while · ⊥ the matrix whose co lumns form a base of the nu ll space of a matrix. diag { X i | L i =1 } d enotes a bloc k diagon al matrix with its i -th diagona l block being X i , while col { X i | L i =1 } the matrix/vector stacked by X i | L i =1 with its i -th ro w block matrix /vector being X i . When A is an m × n dimensio n al matrix and k / K an element/subset of { 1 , 2 , · · · , n } , A (1 : k ) represen ts the m atrix constructed from its ﬁrst k column s, while A ( K ) th e matrix constructed f rom its column s in d exed by th e set K . 0 m × n and 0 m stand resp e ctiv ely for the m × n dim e nsional zero matrix and the m d imensional z ero colum n vector . Th eir subscripts are often omitted when the omission does not cause confusion s. Superscripts T an d H are ado pted to repr esent the transpose and the conjugate transpo se of a vector/matrix respectively . If a t some values of its variable(s), the v alu e of a matrix valued func tion is of FRR/FCR, it is said to b e of f ull normal row/column rank (FNRR/FNCR). Results of Section II I in this paper have been pr e sen ted without proo f in th e 58th IE EE Conferen ce on Decision and Control [21]. In addition to p roviding p roofs of th e se results, this paper also investigates NDS actuator/sensor placements, as well as N D S controllability/ob servability veriﬁcations when subsystem dyn a mics are d e scr ibed through a descriptor form. I I . N D S M O D E L A N D P R E L I M I N A RY R E S U LT S In real world prob lems, NDS sub systems u sually do not have identical input-outp ut relations. In [1 9, 22], an appr o ach is sug gested to d escribe the dynamics of a general linear time in variant (L TI) NDS. In this paper, to clarify relations between subsystem FPPs and its system matrices, th e fo llowing model is adopted to describe th e d ynamics of th e i -th subsystem Σ i of an NDS Σ constituted f rom N subsystems. This mod el is also utilized in [23] for con tin uous time NDSs.   δ ( x ( t, i )) z ( t, i ) y ( t, i )   =         A [0] xx ( i ) A [0] xv ( i ) B [0] x ( i ) A [0] zx ( i ) A [0] zv ( i ) B [0] z ( i ) C [0] x ( i ) C [0] v ( i ) D [0] ( i )    +   H 1 ( i ) H 2 ( i ) H 3 ( i )   × P ( i )[ I − G ( i ) P ( i )] − 1  F 1 ( i ) F 2 ( i ) F 3 ( i )     x ( t, i ) v ( t, i ) u ( t, i )   (1) Here, δ ( · ) denotes either th e deriv ative of a fu nction with respect to time or a f orward time shift ope ration. In oth er words, the above model can be either co n tinuou s tim e or discrete time. Mo reover , t stands f or the tem poral variable, x ( t, i ) the state vector of this subsystem, y ( t, i ) and u ( t, i ) its external outpu t a n d in p ut vectors respectively , z ( t, i ) and v ( t, i ) its inter nal o utput and input vectors resp e ctiv ely , re p- resenting signals sent to o th er subsystems and sign a ls g o tten from other subsystems. All subsystem parameter s are giv en in the matrix P ( i ) , which may be a con centration o r a rea c tio n ratio in bio logical/chem ic a l processes, a resistor, an ind uctor or a capacitor in electrical/electron ic systems, a mass, a spring or a damper in mechanical systems, etc. They are usually called FPPs (ﬁrst principle p arameter), and can b e chosen/tuned in system designs. The matr ices G ( i ) , H j ( i ) | 3 j =1 and F j ( i ) | 3 j =1 are intro duced to ind icate how th e system matr ices of this subsystem is changed by its FPPs. Th ese matrices, as well as 3 the matr ice s A [0] ∗ # ( i ) , B [0] ∗ ( i ) , C [0] ∗ ( i ) and D [0] ( i ) , in which ∗ , # = x , u , v , y or z , ar e often used to represent chemical, biological, p hysical or e lec trical prin ciples governing subsys- tem dynamics, such as Netwon’ s mechanics, the Kirchhoff ’ s current law , etc., which imp lies tha t they are u sually prescribed and can hardly be chosen or tu ned in designin g a system. In this mo del, the matrix P ( i ) with i ∈ { 1 , 2 , · · · , N } is in principle con stituted fr om ﬁxed zero elemen ts and FPPs of Subsystem Σ i . In some applications, a simple FPP f unction may be more con venient, such as the square root o f a FP P , the multiplication of so me FPPs, etc. T h ese transformatio n s do n ot change con clusions in this pap er , pr ovided th at they for m a bijective glo bal transformatio n. T o a void awkward statements, they ar e called pseudo FPPs (PFPP) in this pape r, and are often assumed to b e independ ent o f each o ther algebraically . Compared with the subsystem model utilized in [1 9, 22], it is believed th a t the above mode l is closer to actual inpu t-outpu t relations o f a dynamic p lant, due to that each system matrix, that is, A ∗ # ( i ) , B ∗ ( i ) , C ∗ ( i ) a n d D ( i ) , in which ∗ , # = x , u , v , y or z , is represen ted as a matrix valued function of the param eter matrix P ( i ) , which reﬂects the well known fact that in an actual system, elements in its system m atrices are usually no t algebraica lly ind ependen t of each othe r, an d some of them can ev en not be tun ed in system designs. Notin g th at an L FT is cap able of representing any rationa l functions [18], it is believed that a large class of systems can be described by the a f oremen tio ned mo del. A more d etailed discussion can be found in [23] o n en gineering m o tiv ations o f the aforemen - tioned mo del. T o hav e a concise presentation, the d epende nce of a system matrix of th e subsy stem Σ i on its parameter matrix P ( i ) is usually no t explicitly expressed, e x cept when this omission may cause some signiﬁcan t confusions. When effects fr om part of the subsystem FPPs o n the perfor mances of the who le NDS are to be in vestigated , the aforemen tioned model can also be applied. This can b e don e simply throu gh prescribin g the oth e r FPPs to some speciﬁc numerical values. Denote vectors col { z ( t, i ) | N i =1 } and col { v ( t, i ) | N i =1 } respec- ti vely by z ( t ) an d v ( t ) . I n this pap er , it is assumed as in [19, 22] that NDS subsystem interactio ns are described by v ( t ) = Φ z ( t ) (2) The matrix Φ is called subsystem conn ection matrix (SCM), which describes inﬂuences among dif f erent NDS subsystems. If each subsystem is r egarded as a node and each non zero element of its SCM as an edge, a graph can be constructed for an NDS, which is usually called the structure or topo logy of the co rrespond ing NDS. Throu g hout this pap er , the following a ssum ptions are adopted . • The vectors u ( t, i ) , v ( t, i ) , x ( t, i ) , y ( t, i ) and z ( t, i ) respectively ha ve a d imension o f m u i , m v i , m x i , m y i and m z i . • Every NDS subsystem Σ i with i ∈ { 1 , 2 , · · · , N } is well- posed. • The NDS Σ itself is well-po sed. The ﬁrst assumption is introd uced to clarify vector size, while well- p osedness of a system mea n s that its states respo nd solely to each pair of the ir in itial values and externa l inpu ts, that is nece ssary for a system to p roper ly work [12, 17, 18, 22]. This means that all these three assumptions sho uld be met by a practical system. In o ther words, the assumptions adopted here are not quite restrictive. By means of the above symb ols, integers M x i , M v i , M x and M v are d e ﬁn ed r e spectiv e ly as M x = P N k =1 m x k , M v = P N k =1 m v k , an d M x i = M v i = 0 when i = 1 , M x i = P i − 1 k =1 m x k , M v i = P i − 1 k =1 m v k when 2 ≤ i ≤ N . Then the d imension of the SCM Φ is M v × M z . When this matrix is p artitioned consistently with the dim ensions of the vecto rs z ( t, i ) | N i =1 and v ( t, i ) | N i =1 , its i - th row j -th column block, den oted by Φ ij , has a dimension of m v i × m z j . This sub matrix reﬂec ts direct ef f ects of Subsystem Σ j on Subsystem Σ i , i, j = 1 , 2 , · · · , N . Brieﬂy , in system analysis and syn thesis, a system is called controllab le if th ere exists an extern al input vecto r that ca n maneuver its sate vector fr om any prescr ib ed initial value to any prescribed ﬁnal value, and it is called obser vable if the value o f its initial state vector can b e rec overed from the time h istor y of its extern al in p ut/outpu t vectors. It is well known that contro llability and ob ser vability are dual p roperties of a sy stem, which means that observability of a system is equiv alen t to the con trollability of its transpose, and v ice versa [12, 18]. These characteristics keep v alid fo r an NDS [19, 2 2]. T o d ev e lo p a compu tationally feasible c o ndition for NDS observability/contro llability veriﬁcation s, the following well known results of matrix analyses are introduc e d [9, 10]. Lemma 1: Divide a m atrix A as A =  A T 1 A T 2  T , an d assume that A 1 is not of FCR. T h en the matrix A is of FCR, if and o n ly if the matr ix A 2 A ⊥ 1 is. When th e matr ix A 1 is o f FCR, A ⊥ 1 = 0 . In this case, fo r an arbitrary matrix A 2 with a compa tib le dimen sion, the matrix A =  A T 1 A T 2  T is of FCR obvio usly . Based on these con clusions, the following results ar e de- riv e d . They are g reatly helpful in exp loiting blo ck diago nal structures of the matrices in NDS observability/controllability veriﬁcations. Lemma 2: Assume that A i | 3 i =1 and B i | 3 i =1 are some ma- trices having compatible dimensions, and the matrix A 2 is of FCR. Th en the matrix  diag { A 1 , A 2 , A 3 } [ B 1 B 2 B 3 ]  is o f FCR, if and only if the matrix  diag { A 1 , A 3 } [ B 1 B 3 ]  is. Proof: Note that     A 1 0 0 0 A 2 0 0 0 A 3 B 1 B 2 B 3     =     0 I 0 0 I 0 0 0 0 0 I 0 0 0 0 I         A 2 0 0 0 A 1 0 0 0 A 3 B 2 B 1 B 3     ×   0 I 0 I 0 0 0 0 I   (3) in which the ide n tity matrices and th e zero matrices in general have different dimensions. Obvio usly , a necessary and sufﬁcient condition f o r the matrix  diag { A 1 , A 2 , A 3 } [ B 1 B 2 B 3 ]  4 being o f FCR is that the matrix  diag { A 2 , A 1 , A 3 } [ B 2 B 1 B 3 ]  has this prop erty . When the matrix A 2 is of FCR, direct alg ebraic operation s show that [ A 2 0 0 ] ⊥ = col { [0 0] , [ I 0] , [0 I ] } . As   0 A 1 0 0 0 A 3 B 2 B 1 B 3     0 0 I 0 0 I   =   A 1 0 0 A 3 B 1 B 3   (4) the proof can be co mpleted thr o ugh a direct ap plication of Lemma 1. ✸ From this lemma, the f o llowing co nclusions can be directly obtained. As the proof is quite straigh tforward, it is omitted . Corollary 1: Assume that A [ j ] i | i =3 ,j = m i =1 ,j =1 and B [ j ] i | i =3 ,j = m i =1 ,j =1 are some matr ices having compatible dimensions, an d the matrix h A [1] 2 A [2] 2 · · · A [ m ] 2 i is of FCR. Th en the matrix   diag n A [1] 1 , A [1] 2 , A [1] 3 o · · · diag n A [ m ] 1 , A [ m ] 2 , A [ m ] 3 o h B [1] 1 B [1] 2 B [1] 3 i · · · h B [ m ] 1 B [ m ] 2 B [ m ] 3 i   is o f FCR, if and only if the following m atrix has this pro perty   diag n A [1] 1 , A [1] 3 o · · · diag n A [ m ] 1 , A [ m ] 3 o h B [1] 1 B [1] 3 i · · · h B [ m ] 1 B [ m ] 3 i   In order to d erive a co mputation ally scalab le necessary and suf ﬁcient condition for NDS observability/controllab ility veriﬁcations, the following results on m atrix pencils ar e in- troduced , which are g iv en in many published work s in cluding [1, 11]. Deﬁnition 1: L et G and H be two arb itrary m × n di- mensional r eal matr ices. A matrix valued polyn omial Ψ( λ ) = λG + H is called a m a trix pencil. • This ma tr ix p encil is called regular, whenever m = n and det (Ψ( λ )) 6≡ 0 . • If both the matrices G an d H are inv e r tible, th en this matrix penc il is called strictly regular . • If there exist two n onsingu lar r e al m a trices U and V , such that Ψ( λ ) = U ¯ Ψ( λ ) V are satisﬁed by two matrix pencils Ψ ( λ ) an d ¯ Ψ( λ ) , then th ese tw o matrix pencils are said to b e strictly equivalent. Throu g hout this pap er , for an ar bitrary positiv e integer m , the symb ol H m ( λ ) stands f or an m × m dime nsional strictly regular matrix pencil, wh ile the symbols K m ( λ ) , N m ( λ ) , L m ( λ ) and J m ( λ ) respectively for matr ix pen cils having th e following deﬁnitions, K m ( λ ) = λI m +  0 I m − 1 0 0  , N m ( λ ) = λ  0 I m − 1 0 0  + I m (5) L m ( λ ) =  K m ( λ )  0 1   , J m ( λ ) =  K T m ( λ ) [0 1 ]  (6) These matrix p encils are often used in constructing the Kro- necker canon ical fo rm (KCF) of a general matrix pe n cil. Obviously , the d imensions of the matrix pencils K m ( λ ) and N m ( λ ) are m × m , while the matrix pencils L m ( λ ) and J m ( λ ) respectively have a dime nsion of m × ( m + 1 ) and ( m + 1) × m . Moreover , when m = 0 , L m ( λ ) is a 0 × 1 zero m atrix whose existence means ad ding a zero colum n vector in a KCF with out increasing its rows, while J m ( λ ) is a 1 × 0 zero matrix whose existence m e ans adding a zero ro w vecto r in a KCF without increasing its colu mns. On the o th er hand, J m ( λ ) = L T m ( λ ) . For a c lea r presentation, howe ver, it a ppears better to introdu ce these two matrix pen c ils simultaneou sly . In other words, th e capital letters H , K , N , J an d L are used in this paper to indicate the type of the associated matrix pencil, while the subscript m its dim ensions. When a matrix pen cil is b lock diagonal with the diagonal blocks having th e form H ∗ ( λ ) , K ∗ ( λ ) , N ∗ ( λ ) , L ∗ ( λ ) and J ∗ ( λ ) , it is called KCF . It is now extensively known that any matrix pencil is strictly equ ivalent to a KCF [1, 9, 11], which can be stated as follows. Lemma 3 : For any matrix pencil Ψ( λ ) , there are some unique nonn egati ve integers ξ H , ζ K , ζ L , ζ N , ζ J , ξ L ( j ) | ζ L j =1 and ξ J ( j ) | ζ J j =1 , as well as some unique positive integers ξ K ( j ) | ζ K j =1 and ξ N ( j ) | ζ N j =1 , such that Ψ( λ ) is strictly equ iv alent to the block diag onal matrix pencil ¯ Ψ( λ ) deﬁn ed as ¯ Ψ( λ ) = diag n H ξ H ( λ ) , K ξ K ( j ) ( λ ) | ζ K j =1 , L ξ L ( j ) ( λ ) | ζ L j =1 , N ξ N ( j ) ( λ ) | ζ N j =1 , J ξ J ( j ) ( λ ) | ζ J j =1 o (7) The f o llowing lemma explicitly characterizes the null spaces of th e ma tr ix pencils H ∗ ( λ ) , K ∗ ( λ ) , N ∗ ( λ ) , L ∗ ( λ ) and J ∗ ( λ ) . This c h aracterization is helpf u l in clarify ing subsystems with which an observable/controllable NDS ca n b e mo re ea sily constructed . Its proof is deferr ed to Appendix A. Lemma 4: Let m be an arbitrar y p ositiv e integer . Then the matrix pe ncils deﬁned respe c ti vely in Equatio ns (5) and (6 ) have the following nu ll spaces. • H m ( λ ) is no t of full rank (FR) only at m isolated complex values of the variable λ . All these values are not equal to zero. • N m ( λ ) is always of FR. • J m ( λ ) is a lways of FCR. • K m ( λ ) is sing ular only at λ = 0 , and K ⊥ m (0) = col { 1 , 0 m − 1 } . • L m ( λ ) is not of FCR at e very co mplex λ , and L ⊥ m ( λ ) = col n 1 , ( − λ ) j   m j =1 o . I I I . O B S E RV A B I L I T Y A N D C O N T RO L L A B I L I T Y O F A N N D S Note that parallel, cascade and feedb ack con nections o f LFTs can still b e expressed as an LFT [1 8]. O n th e other hand, [19] h as already made it clear that th e system matrices of the whole NDS can be represented as an L FT of its SCM, provided that all the subsystems are c o nnected by their internal inputs/outp uts. These make it possible to r ewrite the NDS Σ in a form wh ich is com pletely the same as that of [19], in which all the (pseudo ) FPPs of each subsystem, as well as the subsystem connection ma tr ix, a r e included in a single matrix. This has also been performed in [23]. As the associated ex- pressions are important in study in g actuator/sensor pla c ements in Section V , and the d eriv ation s are not very leng thy , both of them ar e inclu d ed in this section to m a ke the pr esentation mo r e easily und erstandable. 5 More sp eciﬁcally , for every subsystem Σ i with i ∈ { 1 , 2 , · · · , N } , the follo win g two auxiliary inter nal o utput and input vectors z [ a ] ( t, i ) and v [ a ] ( t, i ) are introd uced, z [ a ] ( t, i ) = [ F 1 ( i ) F 2 ( i ) F 3 ( i )]   x ( t, i ) v ( t, i ) u ( t, i )   + G ( i ) v [ a ] ( t, i ) (8) v [ a ] ( t, i ) = P ( i ) z [ a ] ( t, i ) (9) Deﬁne vectors ¯ z ( t, i ) and ¯ v ( t, i ) respectively as ¯ z ( t, i ) = col { z ( t, i ) , z [ a ] ( t, i ) } and ¯ v ( t, i ) = c o l { v ( t, i ) , v [ a ] ( t, i ) } , an d assume that their dimen sions are respecti vely m ¯ z i and m ¯ v i . Straightfor ward alg ebraic manipula tio ns sh ow tha t when this subsystem is well-posed, i.e, when the matrix I − G ( i ) P ( i ) is regular , its input-outpu t relations can be rewritten as (9) and the next equa tio n   δ ( x ( t, i )) ¯ z ( t, i ) y ( t, i )   =   A xx ( i ) A xv ( i ) B x ( i ) A zx ( i ) A zv ( i ) B z ( i ) C x ( i ) C v ( i ) D ( i )     x ( t, i ) ¯ v ( t, i ) u ( t, i )   (10) in which D ( i ) = D [0] ( i ) and A xx ( i ) = A [0] xx ( i ) , A xv ( i ) = h A [0] xv ( i ) H 1 ( i ) i A zx ( i )=  A [0] zx ( i ) F 1 ( i )  , A zv ( i ) =  A [0] zv ( i ) H 2 ( i ) F 2 ( i ) G ( i )  B ( i ) x = B [0] x ( i ) , B z ( i ) =  B [0] z ( i ) F 3 ( i )  C x ( i ) = C [0] x ( i ) , C v ( i ) = h C [0] v ( i ) H 3 ( i ) i Denote vectors col  ¯ z ( t, i ) | N i =1  and col  ¯ v ( t, i ) | N i =1  respec- ti vely b y ¯ z ( t ) and ¯ v ( t ) . Moreover , co nstruct a matrix ¯ Φ as ¯ Φ =          Φ 11 Φ 12 · · · Φ 1 N P (1) 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . Φ N 1 Φ N 2 · · · Φ N N 0 0 · · · P ( N )          (11) Then Equ ations (2) and (9) can be comp actly r ewritten as ¯ v ( t ) = ¯ Φ ¯ z ( t ) (12) Here, Φ ij with i, j ∈ { 1 , 2 , · · · , N } stands for the ( i, j ) -th submatrix of the SCM Φ when it is divided com patibly with the d imensions of the system inter nal output and input vectors. T o emphasize similar ities in system analyses and syntheses between th e matrices ¯ Φ and Φ , as well as to distinguish their engine e ring signiﬁcance, etc., the matrix ¯ Φ is called the augmen te d SCM in the remaining of this paper . Equation s (1 0) and ( 12) give an eq uiv alent descriptio n for the input-ou tput relation s of the NDS Σ , which has completely the same fo rm as that for the NDS in vestigated in [19]. This equiv alen t fo rm is beneﬁted from th e inv ar iance pro perties of LFTs, and makes results of [1 9] straigh tforwardly app licable to the NDS Σ , which are gi ven in the next lemma. Lemma 5: Assum e that the NDS Σ , and each of its sub - systems Σ i | N i =1 , are well-posed. Then this NDS is observable if and only if for an arbitrar y co mplex scalar λ , the following matrix penc il M ( λ ) is of FCR, M ( λ ) =   λI M x − A xx − A xv − C x − C v − ¯ Φ A zx I M z − ¯ Φ A zv   (13) Here, A ∗ # = di ag  A ∗ # ( i ) | N i =1  , C ∗ = di ag  C ∗ ( i ) | N i =1  , in which ∗ , # = x , v , or z . Assume that th ere ar e M su b systems in th e NDS Σ , denote their in dices by k ( j ) | M j =1 , with their system matrix [ C x ( k ( j )) C v ( k ( j ))] not being of FCR. For a clear pre- sentation, assum e without any loss of gener ality th at 1 ≤ k (1) < k (2 ) < · · · < k ( M ) ≤ N . The n from m a trix theo ries [9, 10], we have that [ C x ( k ( j )) C v ( k ( j ))] ⊥ is of FCR for each j = 1 , 2 , · · · , M . Let N cx ( k ( j )) and N cv ( k ( j )) den ote respectively the ﬁrst m x k ( j ) rows an d the rem aining m ¯ v k ( j ) rows of th is m atrix. Using these notations, the following con- clusions are der iv ed , which establish som e relations between the observability of the NDS in vestigated in th is paper and that of a descriptor system. Their p roof is deferred to App endix B. Theorem 1: Deﬁne matrices N cx and N cv respectively as N cx =    0 diag (  N cx ( k ( j )) 0      M j =1 )    (14) N cv =    0 diag (  N cv ( k ( j )) 0      M j =1 )    (15) in which the z e ro matrices in gene r al hav e different dimen- sions. Then the ND S Σ is observable if and only if for every complex scalar λ , the following matrix pe ncil Ψ ( λ ) is of FC R, Ψ( λ ) = λ  N cx 0  +  − A xx N cx − A xv N cv N cv − ¯ Φ ( A zx N cx + A zv N cv )  (16) Remark 1: When the matrix N cx is squa r e and the ma- trix col  N cx , N cv − ¯ Φ ( A zx N cx + A zv N cv )  is of FCR, as well as that det ( λN cx − [ A xx N cx + A xv N cv ]) 6≡ 0 , th e condition that the afor e mention matrix pe n cil Ψ( λ ) is of FCR at every comp lex λ , is necessary and sufﬁcient for the observability of the following descriptor system, N cx δ ( x ( t )) = [ A xx N cx + A xv N cv ] x ( t ) y ( t ) =  N cv − ¯ Φ ( A zx N cx + A zv N cv )  x ( t ) Results on the o bservability of descrip tor systems appear directly applicable to that of the NDS Σ , while the former has be en extensively studied an d various conclu sions h av e been established [6, 8]. A direct app lication of the se results, howe ver, can not efﬁciently use the b lock d iagonal stru cture of the a ssoc ia ted matrices, an d usually introduces some un- necessary computation al costs that is not quite attracti ve f or large scale NDS an a lysis and synthesis. In addition, there a r e in genera l not any guarante e s that the matrix pencil λN cx − [ A xx N cx + A xv N cv ] is regular . As a matter of fact, th is matrix pencil may som etimes e ven n ot be square. Furthermor e, the matrix col  N cx , N cv − ¯ Φ ( A zx N cx + A zv N cv )  depend s 6 on th e aug mented SCM ¯ Φ , and can no t be guaranteed to be always of FCR. Remark 2: The results of Theo rem 1 essentially mean that when there is a sub system in the NDS Σ with its external output matr ix, that is, [ C x ( i ) C v ( i )] , being o f FCR, then the conclusion s of system o bservability are not chang ed by removing the c o lumn block s associated with this subsystem from the ma tr ix pencil M ( λ ) . This can also be under stood from Corollar y 1. This pr operty is quite inter esting in NDS designs, as it means tha t if a subsystem holds this property , then it will not affect th e observability of the wh ole NDS, n o matter how it is conn ected to other subsystems. T o d erive a compu tationally attractive c o ndition fo r the observability of the NDS of E quations (1) and (2), the KCF for a matrix pencil, which is giv e n in the p revious s e ction, are helpful. It can be claim ed fr om L emma 3 th at for each i ∈ { k (1) , k (2) , · · · , k ( M ) } , the r e are regular r eal m atrices U ( i ) and V ( i ) , and a matrix p encil Ξ( λ, i ) , such that λN cx ( i ) − [ A xx ( i ) N cx ( i ) + A xv ( i ) N cv ( i )] = U ( i )Ξ( λ, i ) V ( i ) (17) in which Ξ( λ, i ) = di ag n H ξ H i ( λ ) , K ξ K i ( j ) ( λ ) | ζ K i j =1 , L ξ L i ( j ) ( λ ) | ζ L i j =1 , N ξ N i ( j ) ( λ ) | ζ N i j =1 , J ξ J i ( j ) ( λ ) | ζ J i j =1 o (18) Here, ξ H i , ζ K i , ζ N i , ζ L i , ζ J i , ξ L i ( j ) | ζ L i j =1 and ξ J i ( j ) | ζ J i j =1 are so m e non n egati ve integers, ξ K i ( j ) | ζ K i j =1 and ξ N i ( j ) | ζ N i j =1 are some po siti ve integers. All th ese numb ers are un iquely determined by the system matrices A xx ( i ) , A xv ( i ) , N cx ( i ) and N cv ( i ) o f th e i -th subsystem Σ i . In the decompo sition of Equation (17), the calcu lations ar e perfor med for each subsystem in dividually . On the other h and, there are extensive studies o n expr e ssing a matrix p encil with the KCF and various computatio nally attractive a lg orithms have already bee n established [1, 9]. It can therefo re be declared that computatio ns in volved in the afo remention ed decomp o sition are in general possible, while the total com- putational com p lexity increases linearly with the increment of the subsystem n umber N . Deﬁne matr ix pencils ¯ Ξ( λ ) an d ¯ Ξ( λ, i ) with i ∈ { k (1) , k (2) , · · · , k ( M ) } respectiv ely as ¯ Ξ( λ, i ) = diag n H ξ H i ( λ ) , K ξ K i ( j ) ( λ ) | ζ K i j =1 , L ξ L i ( j ) ( λ ) | ζ L i j =1 o ¯ Ξ( λ ) = diag  ¯ Ξ( λ, k ( i )) | M i =1  Based on the above obser vations, as well as the expressions in Equatio n (18), the following condition is obtained for NDS observability . Its d eriv ation s are d e ferred to Appen d ix C. Theorem 2: For e very i ∈ { k (1) , k (2) , · · · , k ( M ) } , d enote ξ H i + P ζ K i j =1 ξ K i ( j )+ P ζ L i j =1 ξ L i ( j ) b y s ( i ) . Moreover , represen t the m atrix con structed from the ﬁrst s ( i ) colu m ns of th e in verse o f the matrix V ( i ) by V − 1 i (1 : s ( i )) . Denote the following matrix pencil " ¯ Ξ( λ )  N cv − ¯ Φ( A zx N cx + A zv N cv )  diag n V − 1 k ( i ) (1 : s ( k ( i ))) | M i =1 o # by ¯ Ψ( λ ) . Then the NDS Σ is observable, if and only if th e matrix penc il ¯ Ψ( λ ) is of FCR at each λ ∈ C . From the pro of of Theore m 2, it is clear that if ξ H i = ζ K i = ζ L i = 0 in each Ξ( λ, i ) w ith i ∈ { k (1) , k (2) , · · · , k ( M ) } , which is essentially a condition req uired for each subsystem individually , then the NDS Σ is always ob servable, no matter how the sub systems are con nected and the par ameters o f a subsystem are selected . For each i ∈ { k (1) , k (2) , · · · , k ( M ) } , let Λ ( i ) denote th e set con sisting of the complex λ at which ¯ Ξ( λ, i ) is not of FC R. From Lemma 4, this set is the whole complex plane if ζ L i 6 = 0 . On the other hand , if ζ L i = 0 , then th is set is simply formed by zero and all the comp lex values that lea d to a singular H ξ H i ( λ ) . In add ition, ¯ Ξ ⊥ ( λ, i ) is also b lock diago nal with each of its blocks being completely deter mined by H ⊥ ξ H i ( λ ) , K ⊥ ξ K i ( j ) ( λ ) a nd L ⊥ ξ L i ( j ) ( λ ) . Mor eover , all of them ca n be easily obtained using the results of Lem m a 4. Furthermo re, let Λ den ote the set consisting of the v alu es of the comp lex variable λ at which ¯ Ξ( λ ) is not of FCR, an d M ( λ 0 ) the set of all the subsystem indices with which th e matrix pencil ¯ Ξ( λ, i ) is not of FCR at λ 0 . Fro m the deﬁnitions of ¯ Ξ( λ ) an d ¯ Ξ( λ, i ) | N i =1 , it is obvious that Λ = M [ i =1 Λ ( k ( i )) , ¯ Ξ ⊥ ( λ 0 ) =    0 diag (  ¯ Ξ ⊥ ( λ 0 , i ) 0      i ∈M ( λ 0 ) )    (19) The following results can be im mediately established from Theorem 2 and Le mma 1. Theorem 3 : For a prescribed complex λ 0 , deﬁn e matrices X ( λ 0 ) and Y ( λ 0 ) resp e c ti vely as X ( λ 0 ) = di ag n N cv ( k ( i )) V − 1 k ( i ) (1 : s ( k ( i ))) | M i =1 o ¯ Ξ ⊥ ( λ 0 ) (20) Y ( λ 0 ) = di ag { [ A zx ( k ( i )) N cx ( k ( i )) + A zv ( k ( i )) N cv ( k ( i ))] × V − 1 k ( i ) (1 : s ( k ( i ))) | M i = 1 o ¯ Ξ ⊥ ( λ 0 ) (21) Then the NDS Σ is observable, if and o nly if for e ach λ 0 ∈ Λ , the matrix X ( λ 0 ) − ¯ Φ Y ( λ 0 ) (22) is of FCR. The proo f is omitted due to its obviousness. The above theo rem makes it clear th a t in the matrix pencil ¯ Ξ( λ ) , the e x istence of a matrix p e ncil with the f orm of L ∗ ( λ ) may greatly increase difﬁculties for the satisfaction of the observability requirement by the NDS Σ , as it makes the set Λ equal to the wh ole complex p lane an d requires that the matrix given by Eq u ation (22) is of FCR a t each co mplex λ 0 . The latter lea d s in g eneral to inﬁnitely many con straints on the augmen ted SCM ¯ Φ . It is interesting to see possibilities to av oid o ccurren c e o f this type of matr ix pencils in sub system construction s for an NDS. This will be discussed brieﬂy in the following Section IV . Remark 3: In b oth the deﬁnition of the matr ix X ( λ 0 ) and the deﬁnition of the matrix Y ( λ 0 ) , a ll the in volved matr ices have a con sistent block diagona l structu re. T his means that these two matrices are also block d iagonal, and the compu - tational costs for obtain ing them increase only linearly with 7 the incr ement of the sub sy stem number N . This is a qu ite attractive pro perty in dealing with a large scale N D S which consists of numerou s subsystem s. Remark 4: No te that in the augmented SCM ¯ Φ , which is deﬁned by Eq u ation (11), both subsystem PFPPs a n d con - nection pa rameters are included. Th is implies that the above theorem reﬂects inﬂuences o f these two kinds of parameter s o n NDS o bservability . On the oth er h and, this augmented SCM ¯ Φ clearly ha s a sparse structur e. This means that r esults about sparse com putations, which have been extensively and we ll studied in ﬁelds like numer ical analysis, can be applied to th e veriﬁcation of the con dition in The o rem 3. It is interesting to see possibilities o f developing more num erically efﬁcient methods for th is con dition veriﬁcation, u sing the par ticular sparse structure of the au gmented SCM ¯ Φ an d the consistent block diag onal structure of the matr ices X ( λ 0 ) and Y ( λ 0 ) . Remark 5: While the matrices X ( λ 0 ) and Y ( λ 0 ) of Equa- tion ( 2 2) are calculated in a signiﬁcantly different way from those o f the p revious works repor ted in [19, 22], they have completely th e same f orm. This implies that the associated NDS observability con ditions share the same comp utational advantages in large scale NDS analyses and syntheses. On the other han d , in the deriv a tions of Theorem 3, except the well- posedness assumptions, th ere ar e no t any o ther req u irements on a subsystem of the NDS Σ . That is, the FNCR cond ition on each subsystem, which is requir ed in [1 9, 22] to ge t the associated transmission zeros of each sub sy stem, is completely removed. It is worthwhile to men tion that there ar e various actual systems that d o not ha ve this FNCR p roperty . Obvious examples in clude an NDS with a subsystem that does not have an external output [2 2]. Removal o f this condition is interesting fr om not only a math e matical viewpoint, but also an app lication viewpoint. Remark 6: Com pared with [23], the results of th e ab ove theorem are in a pu re algebr aic form. In system a n alysis and synthesis, they a r e not as illustrative as the results of [23] which are given in a graph ic form. It is interesting to see wheth er or no t a graphic form can be obtain ed f rom Theorem 3 on the ob servability of an NDS. On the other hand, [23] proves that structural obser vability veriﬁcation is in g eneral NP hard f or the NDS Σ , and computation ally feasible results ar e derived on ly fo r the case in wh ich the augmen te d SCM ¯ Φ is diag onal. Th is requirem ent can not be easily satisﬁed by a practical system and signiﬁcantly restricts their applicab ility . In the d e r iv ations of Theorem 3, howe ver, excep t well- p osedness o f each sub system and the whole system, which is also asked in [23] and is necessary for a system to p roperly work, there are not any o ther con straints on either a sub system o r the whole system of the NDS. Th is is a sign iﬁcant advantage of th e results in this paper over those of [ 23], and makes controllability/o bservability veriﬁcation for a lar ge scale NDS much more com putationally feasible than their structu ral counterpar ts. Remark 7: Recall that con tr ollability of an L TI system is equal to observability of its dual system, and this is also true for an NDS [19, 22]. This means that the above results can be dire ctly applied to contro llab ility an alysis for the NDS Σ . As a matter of fact, u sing the d uality b etween controllab ility and observability of a system, as we ll as the eq uiv alence representatio n of th e NDS Σ , which is given b y Equations (1 0) and (12 ) , it can be declared that the NDS Σ is con trollable, if and o n ly if the following matrix pencil is of FRR for each complex λ ,  λI M x − A xx B x − A xv ¯ Φ − A zx B z I M z − A zv ¯ Φ  in wh ich B x = diag  B x ( i ) | N i =1  , B z = diag  B z ( i ) | N i =1  , an d all the oth er matrices hav e th e same deﬁnitions a s those of Lemma 5. Using the KCF of a matrix pencil, as well as a ba sis fo r the left null spa c e of the m atrix col { B x , B z } , similar alg ebraic manipulatio ns give a necessary a n d sufﬁcient condition for NDS con trollability with a similar for m as that of Th eorem 3. I V . C O N D I T I O N S F O R S E N S O R / A C T U A T O R P L A C E M E N T S Sections III makes it cle a r that f o r any 1 ≤ i ≤ N , the existence of a matrix p encil L ∗ ( λ ) / J ∗ ( λ ) in the matrix p encil ¯ Ξ( λ, i ) , which is co mpletely an d independen tly deter mined b y the system matrices of the subsystem Σ i , may make th e ob- servability/controllability condition difﬁcult to be satisﬁed by an NDS. An interesting issue is therefore that in constructing subsystems of an NDS, whethe r it is possible to av oid the existence of this type o f m atrix pencils. I n th is sectio n, we in vestigate how to avoid this o c c urrence under the assumption that C v ( i ) = 0 for every i = 1 , 2 , · · · , N . Note that in the matr ix pen cil ¯ Ξ( λ, i ) , b oth the matrice s C x ( i ) and C v ( i ) are in volved which respectiv ely associate the external outpu t vector y ( t, i ) of the sub system Σ i to its state vector and internal inpu t vector . On th e other han d, sensors are usually used for state measuremen t in a system, an d an essential req uirement for sensor placements is that the associ- ated system is o bservable. These mean that th e aforementio ned issue is clo sely related to NDS sen sor plac ements wh ich is also an important topic in system design s. In ad dition, y ( t, i ) m ay also be used to e valuate the p erform ances of a system in its designs. E xamples inclu de to ask some states of a subsy stem to track an ob jectiv e signal, etc. In e ither of these situations, the external outp ut vector y ( t, i ) usually include only some states of a subsystem [18, 22]. Th ese observations mean that it does not intr oduce very sev er e r estrictions in actual app lications throug h assumin g that C [0] v ( i ) = 0 and H 3 ( i ) = 0 for each i = 1 , 2 , · · · , N in the NDS Σ . Under this hypo thesis, it is obviou s from its deﬁnitio n th at C v ( i ) = 0 . He n ce, the assumption ado p ted in this section is r e a sonable and not quite restrictiv e. T o settle the above issue, th e following results are h elpful which are stan dard in the ana ly sis of a matr ix pencil [9, 11]. Lemma 6: De ﬁne the ran k of an m × n dim ensional matrix pencil Ψ( λ ) as the maxim um dim ension of its subm atrix whose deter m inant is not con stantly equal to z e ro, and de note it by rank { Ψ( λ ) } . Assume th at in the KCF o f th e matrix pencil Ψ ( λ ) , there ar e p ma tr ix pen cils in the form of L ∗ ( λ ) and q m a trix pencils in th e f orm of J ∗ ( λ ) . Then p = n − rank { Ψ( λ ) } , q = m − rank { Ψ( λ ) } (23) 8 An im m ediate result of Lemm a 6 is that there does n ot exist a L ∗ ( λ ) / J ∗ ( λ ) in the K CF of a m atrix pencil, if and only if it is o f FNCR/FNRR. Remark 8: From the deﬁnition of the matrix pencil ¯ Ξ( λ, i ) , it is obvio us that th e no nexistence of a matrix pencil w ith the form of L ∗ ( λ ) is eq ual to that in th e KCF Ξ( λ, i ) . On the other hand, Lem mas 1 and 6, together with E q uation (17), imply th at in th e KCF Ξ( λ, i ) , ζ L i = 0 if and only if th e following matrix pencil is of FNCR,  λI m x i − A xx ( i ) − A xv ( i ) − C x ( i ) − C v ( i )  It can therefore be declared fro m Lemma 2 and the consistent block d iagonal structure o f the ma tr ices A xx , A xv , C x and C v that, there does not exist a subsy stem Σ i with 1 ≤ i ≤ N in th e NDS Σ , su c h that the re is a matr ix pen cil in the form of L ∗ ( λ ) in its associated matrix pencil ¯ Ξ( λ, i ) , if and only if the following matrix pencil is o f FNCR,  λI M x − A xx − A xv − C x − C v  (24) Remark 9: When λ is no t equal to an eigenv alue of the matrix A xx , it is straig h tforward to prove that [ λI M x − A xx − A xv ] ⊥ = col  ( λI M x − A xx ) − 1 A xv , I  It c a n there fore be claimed from Lem ma 1 that the matrix pencil g i ven by Equatio n (24) is of FNCR, if an d on ly if the transfer func tio n matrix (TFM) C v + C x ( λI M x − A xx ) − 1 A xv is. Note that this TFM is exactly the TFM G [1] ( λ ) deﬁned in [19, 22]. The above discussions mean that while the results of [19, 22] are valid only when the T FM G [1] ( λ ) is of FNCR, which appear s to be a se vere restriction on the applicability of the obtained results, its satisf a ction by the subsystems of an NDS may greatly red uce difﬁculties in constructin g an observable NDS. Similar con clusions can also be r eached on the co r respond ing FNRR assumption ad o pted in [19, 22] for controllab ility veriﬁcation o f an NDS. In order to get condition s f or the non- existence of a matrix pencil in ¯ Ξ( λ, i ) that has the fo rm of L ∗ ( λ ) , we at ﬁr st in vestigate inﬂuences of the ran k of th e m atrix [ C x C v ] on the observability of the NDS Σ . Standard algebraic man ipulations show that removing a row in this matrix that depend linearly on the other rows does not affect o bservability of the NDS. This conclusion is obvious from an application view of point. T o be mo re speciﬁc, a line a r dep endence of the r ows of th e matrix [ C x C v ] means that some of th e NDS external outp u ts can be expressed as a linear combination of its othe r externa l outputs. Hence, these external outputs do not contain any ne w informa tio n about the NDS states, and their elim ination doe s not have a ny in ﬂu ences on the ob ser vability of th e NDS. These observations, tog e ther with Lemma 5, further m ean that in the investigation of the observability o f the NDS Σ , the assump tion will not introduce any lose of gen erality that the matrix [ C x C v ] is of FRR. From the co mpatible b lock diagona l s tructures of th e matr ices C v and C x , this hyp othesis is obviously e q uiv alent to that the ma tr ix [ C x ( i ) C v ( i )] is of FRR for each i = 1 , 2 , · · · , N . Under these assumption s, the following con clusions are obtained fro m Lemm a 6 on the nu mber of matrix pen cils L ∗ ( λ ) in the matr ix pencil Ξ( λ, i ) d eﬁned b y Equation s (17) and (18). Theorem 4: Ass u me that C v ( i ) = 0 . Deﬁne a matrix pencil Θ( λ, i ) as Θ( λ, i ) = n I − A xv ( i )  A T xv ( i ) A xv ( i )  − 1 A T xv ( i ) o × [ λI − A xx ( i )] C ⊥ x ( i ) (25) Then there does not exist a matrix penc il in the K CF Ξ( λ, i ) that has the form L ∗ ( λ ) , if a n d only if • the matrix pen c il Θ( λ, i ) is of FNCR; • the matrix A xv ( i ) is o f FCR. A proo f of this theorem is provided in Appendix D. Note that when a matrix is o f FCR, its pr oduct with an arbitrary nonze r o vector o f a co mpatible dimension is certainly not equal to zero. On the other hand, recall that in Sub system Σ i , the matrix A xv ( i ) is actually a n inpu t matrix co nnecting its in ternal inputs, that is, signals sen t from other subsystems, to its state vector . The re quiremen t that th is matrix is of FCR m e a ns that in order to co n struct an o b servable NDS, it is appreciative in sub system selections to guarantee that any nonzer o signals received from o ther sub systems h av e some inﬂuences o n a subsystem state. In o ther word s, inﬂu ences on the states of a subsystem f rom a signal sen t by an other subsystem, are not allowed to be killed by any other signal( s) sent by any oth e r subsystem(s). Generally speaking, the matrix A T xv ( i ) A xv ( i ) may not be in vertible, which lead s to som e difﬁculties in the d eﬁnition of th e ma tr ix pencil Θ( λ, i ) . Ho wev e r, this matrix is certainly positive deﬁnite and therefore has an in verse, provided that the matrix A xv ( i ) is o f FCR. This means that when th e second condition in the above theorem is satisﬁed, the matrix pencil Θ( λ, i ) is well deﬁned. When a problem of sen sor placemen ts is under investigation, it is a general situation that one sensor measures only on e state of p lant. Th is means that in each row of the matrix C x ( i ) , there is only one non zero elem ent. On the other hand, previous discussions r ev e al that the matrix C x ( i ) can b e assume d to be of FRR without any lo ss of ge nerality . These ob servations mean that the assum ption th a t each element of the external output vector y ( t, i ) is related only to one of the states o f the subsystem Σ i , as well as the assumption that all the elements of the extern al ou tp ut vector y ( t, i ) are different from each other, generally do not introduce any restric tio ns in a n in vestigation about sensor pla c ement. In the sub system Σ i , deno te by { k ( j, i ) | j ∈ { 1 , 2 , · · · , m y i } , k ( j, i ) ∈ { 1 , 2 , · · · , m x i }} the set co nsisting of its states that are measured by a sensor . The ab ove discussions mean th at it can be assumed that k ( j 1 , i ) 6 = k ( j 2 , i ) whenever j 1 6 = j 2 without any loss of generality . It can also be d ir ectly shown that the fo llowing two assumptions do no t introduce any restrictions on senso r placement studies also, • 1 ≤ k (1 , i ) < k (2 , i ) < · · · < k ( m y i , i ) ≤ m x i ; 9 • the j -th eleme nt of the external output vector y ( t, i ) is equal to the k ( j, i ) -th e lement of the state vector x ( t, i ) of the sub system Σ i , j = 1 , 2 , · · · , m y i . The details ar e omitted due to their stra ightforwardn ess. On th e oth er hand , the rationality of these assumptions can be easily understoo d f r om an app lication view of points. In particular, any position rea rrangem ents of the NDS external outputs do not chan ge the total information contained in these outputs about its states. Hence, it is not ou t o f imaginations that these assumptions do n ot intro duce any restriction s on observability analysis for the NDS Σ . Let e [ y ] j,i with 1 ≤ j ≤ m y i denote the j -th canonical basis vector o f the Eu clidean space R m y i , an d O ( j, i ) the m y i × [ k ( j, i ) − k ( j − 1 , i ) − 1] dimension al zero matrix in which j = 1 , 2 , · · · , m y i + 1 with k (0 , i ) and k ( m y i + 1 , i ) being respectiv ely deﬁned as k (0 , i ) = 0 and k ( m y i + 1 , i ) = m x i . T h e above discussions show tha t in a sensor p lacement problem , it can be gen erally assumed, without any loss of generality , that C x ( i ) = h O (1 , i ) e [ y ] 1 ,i O (2 , i ) e [ y ] 2 ,i · · · O ( m y i , i ) e [ y ] m y i ,i O ( m y i + 1 , i ) i (26) Under this assumption, the following results can be directly obtained from T heorem 4. Corollary 2: Assume that C v ( i ) = 0 and C x ( i ) satisﬁes Equation (26). Den o te the set { 1 , 2 , · · · , m x i } \ { k (1 , i ) , k (2 , i ) , · · · , k ( m y i , i ) } by S ( i ) , an d the following matrix pencil n I − A xv ( i )  A T xv ( i ) A xv ( i )  − 1 A T xv ( i ) o [ λI − A xx ( i )] I m x i ( S ( i )) by ¯ Θ( λ, i ) . Then th e re does no t e x ist a matrix pencil in the KCF Ξ( λ, i ) that has the form L ∗ ( λ ) , if and o nly if • the matrix A xv ( i ) is o f FCR. • the matrix p encil ¯ Θ( λ, i ) is o f FNCR. Proof: When Equation (26) is satisﬁed by the matrix C x ( i ) , direct matrix man ipulations show that C ⊥ x ( i ) = diag (  I k ( j,i ) − k ( j − 1 ,i ) − 1 0      m y i j =1 , I m x i − k ( m y i ,i ) ) Obviously , th is matrix can b e equiv alen tly expr essed as C ⊥ x ( i ) = h e [ x ] j,i : j ∈ S ( i ) i in wh ich e [ x ] j,i with 1 ≤ j ≤ m x i denote th e j -th cano nical basis vector of the Eu clidean space R m x i . The results can then be obtained by applying this expr ession for th e matrix C ⊥ x ( i ) to the deﬁnition of th e matrix pencil Θ( λ, i ) given in Theorem 4. This comp letes the proof. ✸ From the d eﬁnitions of the matrices A xx ( i ) an d A xv ( i ) , which a r e given im mediately after Equation (10), it is clear that these two matrices depend neither on a PFPP of a subsystem, nor o n a no nzero entry in the SCM. In other words, these two matrices depend on ly on the p rinciples of mechanics, electricity , chemistry , biology , etc . , that govern th e dynamic s of th e subsystem Σ i . Th erefore, the results of Theo rem 4 a nd Corollary 2 ap pear very helpf ul in determin in g th e type of subsystems with which an NDS can be co nstructed that is possible to ach iev e goo d per f ormanc e s mor e ea sily . On the other hand, alth ough the 2n d condition of Corollary 2 is comb inatorial, it dep ends only on a sin- gle subsy stem. As the dimension of the state vector in a subsystem is usually not very large, this condi- tion does not lead to a heavy co mputation al burden in general. In addition, u sing the KCF of th e ma trix pen - cil n I − A xv ( i )  A T xv ( i ) A xv ( i )  − 1 A T xv ( i ) o [ λI − A xx ( i )] , som e more explicit cond itions on sensor positions can be obtained. The details are omitted due to spac e consideration s. On th e ba sis of the du ality between controllab ility and observability of the NDS Σ , similar results can be obtained for actuato r placements. Details are o m itted due to th eir straightfor ward n ess. V . E X T E N S I O N S T O A N N D S W I T H D E S C R I P T O R S U B S Y S T E M S For various tradition a l contro l plants, ra th er than a state space model, it is more con venient to descr ibe its dynamics by a descr iptor form. T yp ical examples inclu de con strained mechanical systems, electrical power systems, etc. [6, 8, 1 1]. An inter esting issue is therefore about prop erties o f an N D S with the dy namics of its subsystems being d e scr ibed by descriptor form s. In this section, comp lete observability and co ntrollability are in vestigated f or an NDS, assuming that the dynamics of its subsystems are described by a descriptor form with its system matrices b e ing a n LFT of some PFPPs, under the hypoth esis that each subsystem and the whole NDS a r e well- posed. Results of th e previous sections are extended . More precisely , assume that in an NDS Σ [ d ] composin g of N sub systems, the dynam ics of its i -th subsystem, den o ted by Σ [ d ] i with i ∈ { 1 , 2 , · · · , N } , is described by the following descriptor form ,   E [0] ( i ) δ ( x ( t, i )) z ( t, i ) y ( t, i )   =         A [0] xx ( i ) A [0] xv ( i ) B [0] x ( i ) A [0] zx ( i ) A [0] zv ( i ) B [0] z ( i ) C [0] x ( i ) C [0] v ( i ) D [0] ( i )    +   H 1 ( i ) H 2 ( i ) H 3 ( i )   P ( i )[ I − G ( i ) P ( i )] − 1  F 1 ( i ) F 2 ( i ) F 3 ( i )       x ( t, i ) v ( t, i ) u ( t, i )   (27) in w h ich E [0] ( i ) is a known square and real matr ix that may not be in vertible. In actu al applications, this m atrix is usually utilized to reﬂecting co nstraints on system states, etc. All the othe r matrices and vectors have the same meanings as those of Equation (1). Moreover, relatio ns among the inter nal subsystem inputs and outputs are stil l assumed to be describ ed by Equatio n (2). T o in vestigate controllability and observability o f this NDS, some related concep ts and results about a descriptor system are at ﬁr st introduce d , wh ich are now well known [ 6, 8, 11]. An L TI p lant is called a descriptor system when its input- output relation s are describ e d by the fo llowing tw o equations, E δ ( x ( t )) = Ax ( t ) + B u ( t ) , y ( t ) = C x ( t ) + D u ( t ) (28 ) 10 Here, A , B , C , D and E a re some constant real matrices with compatible dimen sio ns. If th ere exists a λ ∈ C such that det ( λE − A ) 6 = 0 , then it is said to be regular . A descriptor system is said to be completely observable, if its initial states can be uniquely determined by its inputs and outputs over the whole time in te r val. Regularity is particular a n d impor tant to a descrip tor system. When stimulated by a co nsistent input, a regular descrip to r system has a n unique output. Lemma 7: Assume that the descriptor system de scr ibed by Equation (2 8) is regular . It is co mpletely obser vable, if and only if th e next two conditions are simultaneo usly satisﬁed, • the matrix pen cil  λE − A C  is of FCR at e very λ ∈ C ; • the matrix  E C  is of FCR. Throu g h com pletely th e sam e augmen tations as th ose of Section III for the intern al inpu t an d ou tput vector s of the subsystem Σ i with 1 ≤ i ≤ N , the inp ut-outp u t relations of the NDS Σ [ d ] can be equ ivalently re wr itten as Equ a tio n (12) and the f ollowing e quation with i ∈ { 1 , 2 , · · · , N } ,   E [0] ( i ) δ ( x ( t, i )) ¯ z ( t, i ) y ( t, i )   =   A xx ( i ) A xv ( i ) B x ( i ) A zx ( i ) A zv ( i ) B z ( i ) C x ( i ) C v ( i ) D ( i )     x ( t, i ) ¯ v ( t, i ) u ( t, i )   (29) Here, all the matrices have th e same deﬁn itions as those of Equation s ( 10) and (1 2). Deﬁne a matrix D u as D u = diag  D [0] ( i ) | N i =1  . Us- ing this equivalent representatio n of the NDS Σ [ d ] , it can be straigh tforwardly shown th rough eliminatin g the inter nal input vector ¯ v ( t ) and the internal outpu t vecto r ¯ z ( t ) that, its dynamics can also be described by the descriptor system of Equation (28) with E = di ag  E [0] ( i ) | N i =1  and  A B C D  =  A xx B x C x D u  +  A xv C v  ¯ Φ  I − A zv ¯ Φ  − 1 [ A zx B z ] (30) in w h ich the ma trices A # ∗ , B ∗ and C ∗ with ∗ , # = x , v , or z , have th e same d eﬁnitions as those of L emma 5. From the deﬁnitio n of regu la r ity an d the above equ ation, a condition can be established for the NDS Σ [ d ] being regular . Theorem 5: Den ote a set consisting of M x + 1 arb itrary but different co mplex numbers by ¯ Λ . Th e n the ND S Σ [ d ] is regular if and o nly if there is a λ 0 ∈ ¯ Λ such that the f ollowing matrix is of FCR,  λ 0 E − A xx − A xv − ¯ Φ A zx I M z − ¯ Φ A zv  (31) Proof: For brevity , deno te th e matrix pen cil of E quation (31) by Π( λ ) . When each sub system of the NDS Σ [ d ] and the whole system are well-po sed, it can be directly proven through matrix manipulatio ns that the matrix I M v − ¯ Φ A zv is of FR [19, 2 2]. Hence det { Π( λ ) } = det  I M z − ¯ Φ A zv  × det  λE − h A xx + A xv  I M z − ¯ Φ A zv  − 1 ¯ Φ A zx i = det  I M z − ¯ Φ A zv  × det ( λE − A ) (32) That is, at every λ , the nonsingular ities of th e matrix pe ncils Π( λ ) and λE − A ar e eq uiv alent to each other . Assume now th at there is a λ 0 ∈ ¯ Λ such that the matrix Π( λ 0 ) is of FCR. Then the above argum ents mean that at least at this speciﬁc λ 0 , the in verse o f the matr ix pencil λE − A exists. The regularity of th e NDS Σ [ d ] follows dir ectly from deﬁnitions. On th e contrary , assum e that the matrix pe ncil Π( λ ) is not in vertib le for e very λ 0 ∈ ¯ Λ . Then the nonsingular ity equiv alen ce between the matrices Π( λ 0 ) a n d λ 0 E − A means that det ( λ 0 E − A ) ≡ 0 when ev e r λ 0 ∈ ¯ Λ . Note that det ( λE − A ) is a poly nomial having a degree n ot gr eater than M x . Hence, if det ( λE − A ) 6≡ 0 , then it has at most M x roots. Recall that the set ¯ Λ is f o rmed by M x + 1 distinguished com plex numbers. It can then be claimed that if det (Π( λ 0 )) = 0 for ev er y λ 0 ∈ ¯ Λ , th en det ( λE − A ) ≡ 0 . That is, the NDS Σ [ d ] is not regular . Th is co mpletes the proof . ✸ For a p a rticular λ 0 , the nonsingularity of th e matrix Π( λ 0 ) can b e veriﬁed through a matrix comp letely in the same form of Equa tion (2 2). As a ma tter of fact, a direc t a p plication of Lemma 1 with M 1 = [ λ 0 E − A xx − A xv ] , M 2 =  − ¯ Φ A zx I M z − ¯ Φ A zv  leads to this co nclusion. On th e oth er hand , n ote that the matrix E and the matrix A xx , a s well as the matrix A xv , have a consistent block diag o nal structure. T his mean s that in getting [ λ 0 E − A xx − A xv ] ⊥ , the associated computation s can be do ne on each individual subsystem indepen d ently . On the b asis of Lem ma 7 and Equatio n (30), the f ollowing results are gotten for NDS complete observability , which are very similar to those of Lemm a 5. The latter is established in [19, 22] f or an NDS with its subsystems being describe d b y a state-space mo d el. Theorem 6: Assume th at the NDS is regular . Deﬁne respec- ti vely a matrix M [ d ] ∞ and a matr ix pencil M [ d ] ( λ ) as M [ d ] ( λ ) =   λE − A xx − A xv − C x − C v − ¯ Φ A zx I M z − ¯ Φ A zv   (33) M [ d ] ∞ =   E 0 − C x − C v − ¯ Φ A zx I M v − ¯ Φ A zv   (34) Then the NDS Σ [ d ] is completely o bservable, if and o nly if the next two condition s a r e simultaneo usly satis ﬁed, • the matrix M [ d ] ∞ is of FCR; • at each λ ∈ C , the m atrix pencil M [ d ] ( λ ) is of FCR. Proof: For br evity , deﬁne a matrix pencil ¯ M [ d ] ( λ ) and a matrix ¯ M [ d ] ∞ respectively as ¯ M [ d ] ( λ ) =  λE −  A xx + A xv ( I M v − ¯ Φ A zv ) − 1 ¯ Φ A zx  C x + C v ( I M v − ¯ Φ A zv ) − 1 ¯ Φ A zx  ¯ M [ d ] ∞ =  E C x + C v ( I M v − ¯ Φ A zv ) − 1 ¯ Φ A zx  Then according to Lemma 7 and Equa tio n (30), th e NDS Σ [ d ] is co mpletely observable, if and only if ¯ M [ d ] ∞ is of FCR and ¯ M [ d ] ( λ ) is o f FCR at an a r bitrary λ ∈ C . 11 When th e NDS Σ [ d ] is well- posed, we hav e that the matrix I M v − ¯ Φ A zv is in vertib le. Th erefore  − ¯ Φ A zx I M v − ¯ Φ A zv  ⊥ = col  I , ¯ Φ A zx ( I M v − ¯ Φ A zv ) − 1  (35) The proof can now be co mpleted thro ugh an application of Lemma 1. ✸ Note that in the ab ove theorem, except the replacem e nt of the identity matrix I M x in the matrix pencil M ( λ ) o f Le m ma 5 by the matrix E in the m a tr ix p encil M [ d ] ( λ ) , th e se two matrix pe ncils are com pletely the same. On the oth er hand, note that the results of Lemma 3 are valid f or every matrix pencil with th e form λG + H . It is obviou s that thro ugh similar arguments as those of Section s III and IV , similar conclusions can be obtained for the veriﬁcation of th e second condition of Theor em 6 , and therefore the comp lete ob servability of an NDS whose subsystem s bein g modeled by a descr iptor form, as well as for its sensor plac ements. Concernin g veriﬁcation o f the condition associated with the matrix M [ d ] ∞ , let M 1 and M 2 represent respectively th e ma- trix  E 0 − C x − C v  and the matrix  − ¯ Φ A zx I M v − ¯ Φ A zv  . Then a direct application of L emma 1 leads to an equivalent condition in the form o f Eq uation ( 2 2). On ce aga in, excep t the augmen te d SCM ¯ Φ , the other tw o matrices in this condition can b e obtained from indep endent computatio n s on each individual subsystem. In addition , based o n the du ality between com p lete con - trollability and co mplete observability o f descriptor systems, similar results can b e obtained f or the complete co ntrollability of the NDS Σ [ d ] , as well a s fo r its actuator placeme n ts. V I . A N A RT I FI C I A L E X A M P L E T o illustrate applica bility of the ob tained r esults in system analyses and syntheses, an artiﬁcial NDS is constructed and analyzed in this section, which has N subsystems and each of th em are con stituted from two opera tio nal ampliﬁers, two capacitors and several resistors. Figure 1 gives a schematic illustration of its i -th subsystem with 1 ≤ i ≤ N . The following assumptions are adopted for th is system in which i ∈ { 1 , 2 , · · · , N } . • For each subsystem, the voltage of its righ t ca p acitor is measured. • The i -th su bsystem is directly affected by m ( i ) subsys- tems with th eir indices being ρ 1 ( i ) , ρ 2 ( i ) , · · · , ρ m ( i ) ( i ) . • The internal output of the i -th subsystem, that is, z ( t, i ) , directly a ffects n ( i ) sub systems with their indices bein g ξ 1 ( i ) , ξ 2 ( i ) , · · · , ξ n ( i ) ( i ) . • The i -th subsystem dir ectly affects the ξ j ( i ) -th su bsystem as its η j ( i ) -th inter nal input, 1 ≤ j ≤ n ( i ) . Deﬁne ¯ R i for each i = 1 , 2 , · · · , N as ¯ R i =   n ( i ) X j =1 1 R ξ j ( i ) , η j ( i )   − 1 Moreover , deno te R i C i and R i / ¯ R i respectively by T i and k i . I n addition, let x 1 ( t, i ) and x 2 ( t, i ) represent r espectively the voltages of the left an d righ t capacitors in the i -th subsystem, and d eﬁne its state vector x ( t, i ) as x ( t, i ) = col { x 1 ( t, i ) , x 2 ( t, i ) } . Using these symbols, th e fo llowing model can be straightfo rwardly established fro m circu it prin- ciples for th e dynamics of this subsystem, ˙ x ( t, i ) = 1 (5 + 3 k i ) T i  − 3 − 2 k i 1 1 − 2 − 3 k i  x ( t, i ) + 1 (5 + 3 k i ) T i  2 + k i 1   v ( t, i ) − R ∗ i R i, 0 u ( t, i )  z ( t, i ) = 1 5 + 3 k i  [1 3 ] x ( t, i ) +  v ( t, i ) − R ∗ i R i, 0 u ( t, i )   y ( t, i ) = [0 1] x ( t, i ) In addition, subsystem conn ections are gi ven by the following equation v ( t, i ) = m ( i ) X j =1 R ∗ i R i,j z ( t, ρ j ( i )) (36) This implies that for each 1 ≤ j ≤ m ( i ) and each 1 ≤ i ≤ N , the i -th row ρ j ( i ) -column ele m ent of th e SCM Φ is equal to R ∗ i /R i,j , while all the other elements ar e equal to zero. Clearly , each element in the above system matrices, as well as th e SCM, is a rational func tion of the physical parameter s, which can be further expressed by an LFT . As this expr e ssion does not af fect conclusions of th is example, the details are not included. In add ition, this NDS is well-posed if and o nly if det I − diag ( 1 5 + 3 k i     N i =1 ) Φ ! 6 = 0 From th is model, it can be d irectly proved that for e a c h i = 1 , 2 , · · · , N , N cx ( i ) =  1 0 0 0  , N cv ( i ) = [0 1 ] Using th ese matrices, direct alg ebraic man ipulations show th at  λN cx − A xx N cx − A xv N cv N cv − Φ ( A zx N cx + A zv N cv )  =       diag    " λ + 3+2 k i (5+3 k i ) T i − 2+ k i (5+3 k i ) T i − 1 (5+3 k i ) T i − 1 (5+3 k i ) T i #      N i =1    diag n [0 1 ] | N i =1 o − Φ diag  1 (5+3 k i ) T i [1 1 ]    N i =1        On the oth er hand, for an ar bitrary i ∈ { 1 , 2 , · · · , N } , " λ + 3+2 k i (5+3 k i ) T i − 2+ k i (5+3 k i ) T i − 1 (5+3 k i ) T i − 1 (5+3 k i ) T i # = U ( i )  λ + T − 1 i 0 0 1  V ( i ) in which U ( i ) = " 1 − 2+ k i (5+3 k i ) T i 0 − 1 (5+3 k i ) T i # , V ( i ) =  1 0 1 1  Note that from their deﬁnitions, both T i and k i are positiv e number s, which means that the matrices U ( i ) and V ( i ) are always in vertible. Moreover , N cv ( i ) V − 1 i (1 : 1) = − 1 ( A zx ( i ) N cx ( i ) + A zv ( i ) N cv ( i )) V − 1 i (1 : 1) = 0 12         i R i R i R i R i C i C ( , ) z t i ( , ) u t i , 1 i R # i R # i R * i R * i R * i R ,0 i R  , ( ) i m i R ( , ) v t i - 1 ( , ( )) z t i r ( ) ( , ( )) m i z t i r Fig. 1. The i -th Subsystem of t he NDS Let Λ d enote the set co nstituted fro m − 1 / T i | N i =1 with different values. For each λ 0 ∈ Λ , let M ( λ 0 ) denote the set of all the indices of sub systems with T i = − 1 /λ 0 . Then ¯ Ξ ⊥ ( λ 0 ) = col      0 diag (  1 0      j ∈M ( λ 0 ) )      in which the zero vector s usually ha ve dif f erent dime n sions. Substitute these re sults into the d eﬁnitions of the matrices X ( λ 0 ) and X ( λ 0 ) of T heorem 3,we immed iately hav e tha t X ( λ 0 ) = − ¯ Ξ ⊥ ( λ 0 ) , Y ( λ 0 ) = 0 Hence X ( λ 0 ) − Φ Y ( λ 0 ) ≡ − ¯ Ξ ⊥ ( λ 0 ) , and is therefore always of FCR, no matter what v alu e λ 0 takes from the set Λ and what value the SCM Φ takes. It can the refore be declared that when this artiﬁcial NDS is well-posed, it is always observable, no m a tter how its subsystems are co nnected. When the NDS is well-posed and th e voltage o f the left capacitor is measured for each sub system, similar argumen ts show that the system is ob servable, if and only if th e matrix I + Φ is in vertible. Using the results of the p revious sections, th is system can be pr oved to be also con trollable for arb itr ary subsystem connectio ns, provided that it is well-posed. It is worthwhile to em phasize that in large scale NDS analysis and synth esis, com putational co m plexity is a very importan t issue. Note that the results obtain ed in this pa- per, such as Theor em 3, hav e com pletely the same form as Theo rem 2 of [19]. It can be claimed that they share the same compu tational pro perties. That is, their comp utation costs increase linearly with the subsystem number N , while those using a lumped m odel increa se at least in the order of N 3 . Mor eover , computations with th e se results ar e num erically more stable. Analysis d etails, as well as some numerical simulation comp arisons, can be foun d in [19]. V I I . C O N C L U D I N G R E M A R K S This paper studies observability/contro llability o f networked dynamic systems, in which the system matrices of each subsystem are expr essed throu gh a linear fra ctional tran s- formation o f its ( pseudo) ﬁrst prin ciple para meters. Some explicit connectio ns hav e been obtained betwe e n an NDS and a descripto r system in their ob servability/controllability . By means of the Kronecker can onical form of matrix pencils, a rank based necessary an d sufﬁcient co ndition is go tten with the associated matrix depen ding af ﬁnely on sub system parame- ters/connectio ns. While th e correspon d ing diag onal blocks are computed in a sign iﬁcantly different way , this matrix has a form that completely agrees with those of [19, 2 2]. In addi- tion, in its deriv ation s, the full normal rank condition asked there is no lon ger requ ired. On the other h a n d, this matrix keeps the attractive prop erty that in getting the associated matrices, all the numerical calculations a r e performe d on each individual subsystem independently . T his makes the con dition veriﬁcation scalable fo r a large scale NDS. More over, except well-posedn ess of the wh ole sy stem and its subsystems, ther e are no t any oth er restriction s o n either a subsystem or the subsystem conn ections. On the basis of th is condition , characteristics of a subsystem are clariﬁed with which an observable/controllab le NDS can be more easily con structed. It has been made clear that satis- faction o f the full normal column /row rank c o ndition ado pted in [19, 22] may gr e atly r educe difﬁculties in constructing an observable/contro llab le NDS. In addition, subsystem s with an input matrix of full column ran k are helpful in constructing a n observable NDS that r eceives sig n als fr om o th er subsy stems, while subsystems with an ou tput matrix of full row r ank are helpful in co nstructing a contr ollable NDS that send s sign als to other subsystems. Extensions to an NDS whose subsystems are mo d eled by a descr iptor form have also been a ttac ked. It is ob served that similar results can also be established. Further efforts in clude ﬁnding eng ineering signiﬁcant ex- planations for the obtained results, a s well as exten d ing the obtained re sults to structur a l co ntrollability/o bservability of a networked dynamic system. A P P E N D I X A P R O O F O F L E M M A 4 In this proof, the n ull space is der iv ed for th e associated 5 kinds of matrix p encils individually . A. The Matrix P en cil H m ( λ ) Note that the matrix penc il H m ( λ ) is strictly regular . According to th e deﬁnition of a strictly regular matrix pencil, there ar e two real and nonsing u lar m × m dimensiona l matrices X and Y , such that H m ( λ ) = λX + Y . Hence, H m ( λ ) α = X  λI + X − 1 Y  α det [ H m ( λ )] = det [ X ] × det  λI + X − 1 Y  in wh ich α is an a r bitrary vector with a consistent dimension. It can therefore be dec lar ed that det [ H m ( λ )] = 0 if and only if det  λI + X − 1 Y  = 0 . On the other ha n d, note that det  X − 1 Y  = det [ Y ] det [ X ] 13 which is different fro m zero by the non singularity of the associated two matrices. W e therefore have that the m atrix X − 1 Y is always nonsingu lar . Moreover , H m ( λ ) α = 0 if and only if  λI + X − 1 Y  α = 0 . That is, the matrix pen cil H m ( λ ) is r ank deﬁcien t only a t eigenvalues o f the matrix X − 1 Y which are iso la ted and different from zero. B. The Matrix P en cils N m ( λ ) and J m ( λ ) From the deﬁnitions of these two matr ix p encils, direct algebraic man ipulations show that det [ N m ( λ )] ≡ 1 On the o ther hand, if there is a vecto r α = [ α 1 α 2 · · · α m ] T , such that J m ( λ ) α = 0 is satisﬁed at a p articular value of the complex variable λ , say , λ 0 . Then λ 0 α 1 = 0 α i + λ 0 α i +1 = 0 , i = 1 , 2 , · · · , m − 1 α m = 0 which lead to α 1 = α 2 = · · · = α m = 0 . T h at is, α = 0 . Hence, th e matrix J m ( λ ) is of FCR, no m atter what value is taken by the co mplex v ar iable λ . C. The Matrix P en cil K m ( λ ) Assume that ther e is a vector α = [ α 1 α 2 · · · α m ] T , such that K m ( λ ) α = 0 is satisﬁed at a particular value of the com plex v ariab le λ . Denote it by λ 0 . Then λ 0 α i + α i +1 = 0 , i = 1 , 2 , · · · , m − 1 (a.1) λ 0 α m = 0 (a.2) If λ 0 6 = 0 , the n the last equation means that α m = 0 , which further leads to that α m − 1 = α m − 2 = · · · = α 1 = 0 . Hen ce, the matrix pen cil K m ( λ ) is always of FCR when ev er λ 6 = 0 . On th e other h and, if λ 0 = 0 , then Equ a tion (a.1) implies that α m = α m − 1 = · · · = α 2 = 0 , while α 1 can b e an arbitrary complex number, which is the result need ed to prove. D. The Matrix P enc il L m ( λ ) Assume that the r e is a vector α = [ α 1 α 2 · · · α m ] T , suc h that L m ( λ ) α = 0 is satisﬁed with λ = λ 0 , a p articular value of the co mplex v ar iable λ . Then λ 0 α i + α i +1 = 0 , i = 1 , 2 , · · · , m (a.3) If λ 0 6 = 0 , then Equation (a.3) means that α i +1 = − λ 0 α i , i = 1 , 2 , · · · , m . On the othe r h and, if λ 0 = 0 , this equa tio n implies that α m = α m − 1 = · · · = α 2 = 0 , while α 1 can take any complex value. These prove the results. ✸ A P P E N D I X B P R O O F O F T H E O R E M 1 For brevity , deﬁne sets K and N respectively as K = { k (1) , k (2) , · · · , k ( M ) } , N = { 1 , 2 , · · · , N } Let α x ( i ) b e an arbitrary m x i dimensiona l real vector, wh ile α v ( i ) be an arbitrar y m v i dimensiona l rea l vector , i = 1 , 2 , · · · , N . Denote col  α x ( i ) | N i =1 , α v ( i ) | N i =1  by α . From the bloc k diag o nal structure o f the ma tr ices C x and C v , it is immediate that [ C x C v ] α =      C x (1) α x (1) + C v (1) α v (1) C x (2) α x (2) + C v (2) α v (2) . . . C x ( N ) α x ( N ) + C v ( N ) α v ( N )      (a.4) It can ther efore be declared th a t [ C x C v ] α = 0 if and only if C x ( i ) α x ( i ) + C v ( i ) α v ( i ) = 0 , i = 1 , 2 , · · · , N (a.5) From the adopted assumptio n, we have that when i ∈ K , the matrix [ C x ( i ) C v ( i )] is not of FCR, otherwise it is of FCR. Hence, Equ ation (a.5) imp lies that f or each j ∈ { 1 , 2 , · · · , M } , there is a vector ξ ( j ) , such that  α x ( k ( j )) α v ( k ( j ))  =  N cx ( k ( j )) N cv ( k ( j ))  ξ ( j ) (a.6) Moreover , α x ( i ) = 0 and α v ( i ) = 0 for e ach i ∈ N \ K . Denote the vector col  ξ ( j ) | M j =1  by ξ . These results mean that α =                0 col (  N cx ( k ( j )) ξ ( j ) 0      M j =1 )       0 col (  N cv ( k ( j )) ξ ( j ) 0      M j =1 )                =  N cx N cv  ξ (a.7) On the oth er hand, a repetitive a p plication of Lemm a 2 and Corollary 1 shows that the matrix col { N cx , N cv } is of FCR. It can ther efore be declared th at [ C x C v ] ⊥ = col { N cx , N cv } (a.8) Note that rear ranging rows o f a matrix does not affect its rank. The p roof can now be completed using Lem m a 1. ✸ A P P E N D I X C P R O O F O F T H E O R E M 2 Deﬁne a matrix p encil Ξ( λ ) as Ξ( λ ) = " 0 diag n col { Ξ( λ, k ( i )) , 0 }| M i =1 o # From the compatible block diagonal structures of the matrices N cx , A xx and A xv , it is o bvio us that λN cx − [ A xx N cx + A xv N cv ] = diag n I , diag { U ( k ( i )) , I }| M i =1 o × Ξ( λ ) diag n V ( k ( i )) | M i =1 o (a.9) In these expressions, the zero matr ic e s and the identity ma- trices may have d ifferent d imensions. As their actual values are not very important in the f ollowing deriv ation s, they are omitted for simp licity . 14 T o simplify the a ssoc iate d exp r essions, den ote the block diagonal m a trices diag n I , diag { U ( k ( i )) , I }| M i =1 o and diag  V ( k ( i )) | M i =1  respectively by U an d V . Th en both U and V are in vertible. On the othe r h and, it is o bvious from its deﬁnition that the matrix pencil Ξ( λ ) is block diagon a l, and consists only o f strictly regular m a trix penc ils, th e matr ix pencils in the forms of K ∗ ( λ ) , N ∗ ( λ ) , L ∗ ( λ ) and J ∗ ( λ ) . On the basis o f Equ ations (16) and (a.9), we h av e th a t Ψ( λ ) =  U Ξ( λ ) V N cv − ¯ Φ ( A zx N cx + A zv N cv )  =  U 0 0 I  ˜ Ψ( λ ) V (a.10) in which ˜ Ψ( λ ) =  Ξ( λ ) N cv V − 1 − ¯ Φ  A zx N cx V − 1 + A zv N cv V − 1   As bo th th e matrices U an d V are inv e rtible, it is obviou s from Equation (a.10) that the matrix pencil Ψ( λ ) is of FCR at ev er y complex λ , if and only if the matrix pencil ˜ Ψ( λ ) holds this prop erty . Note that deleting zero rows of a matrix d oes not chang e its rank . Moreover , the matrix pen cils N ∗ ( λ ) and J ∗ ( λ ) are always of FCR. On th e b asis of Lemma 2, this characteristic of N ∗ ( λ ) an d J ∗ ( λ ) in th e m atrix p e n cil Ξ( λ ) , as well as its block diag onal structure, it can be claimed that the matr ix pencil ˜ Ψ( λ ) is alw a ys of FC R, if and only if th e matrix pencil ¯ Ψ( λ ) meets this require m ent. This completes the pro of. ✸ A P P E N D I X D P R O O F O F T H E O R E M 4 For brevity , denote λN cx ( i ) − [ A xx ( i ) N cx ( i )+ A xv ( i ) N cv ( i )] by Ω( λ, i ) . From the assumption that C v ( i ) = 0 , straigh tforward matrix operation s show that [ C x ( i ) C v ( i )] ⊥ = diag  C ⊥ x ( i ) , I  (a.11) From this relatio n, as well a s the deﬁnitions of the m atrices N cx ( i ) an d N cv ( i ) , we have that N cx ( i ) =  C ⊥ x ( i ) 0  , N cv ( i ) = [0 I ] (a.12) Substitute these two equalities into the deﬁnition o f the ma trix pencil Ω( λ, i ) , we have that Ω( λ, i ) =  ( λI − A xx ( i )) C ⊥ x ( i ) A xv ( i )  (a.13) Assume that A xv ( i ) is not o f FCR. T hen there exists a nonzer o vector α v , de n ote it by α v 0 such that A xv ( i ) α v 0 = 0 . Construct a vector α 0 as α 0 = col { 0 , α v 0 } in which the zero vector has a su itable dimension. Obviou sly , this vector is not a zero vector . On the othe r hand, from Equation (a.13), direct algebraic man ip ulations sho w that Ω( λ, i ) α 0 = 0 for an arbitrary complex λ . This means that the matrix pencil Ω( λ, i ) is n ot of FNCR. Hence, to guar antee that this m atrix pencil is of FNCR, it is necessary that the matrix A xv ( i ) is o f FCR. Assume now th at the matrix pencil Ω( λ, i ) is not of FNCR. Then for e a ch λ 0 ∈ C , the matrix Ω( λ 0 , i ) is not of FCR. Let α be a no nzero comp lex vector satisfy in g Ω( λ 0 , i ) α = 0 . Partition it as α = col { α x , α v } in a c o nsistent way as the partition of th e m atrix pencil Ω( λ, i ) . Then Ω( λ 0 , i ) α = ( λ 0 I − A xx ( i )) C ⊥ x ( i ) α x + A xv ( i ) α v = 0 (a.14) When the matrix A xv ( i ) is o f FCR, th e matrix A T xv ( i ) A xv ( i ) is invertible. M oreover , Equation (a.14) im plies that α v = −  A T xv ( i ) A xv ( i )  − 1 A T xv ( i )( λ 0 I − A xx ( i )) C ⊥ x ( i ) α x (a.15) α x 6 = 0 is hence guaran teed by α 6 = 0 . Recall that the m atrix pencil Θ( λ, i ) is deﬁned as Θ( λ, i ) = n I − A xv ( i )  A T xv ( i ) A xv ( i )  − 1 A T xv ( i ) o ( λI − A xx ( i )) C ⊥ x ( i ) . Substitute the above eq uality back into Equa- tion (a.14), we further have th at Θ( λ 0 , i ) α x = 0 (a.16) Hence, α x 6 = 0 and Eq uation (a.16) im ply that the ma tr ix Θ( λ 0 , i ) is n ot of FCR. No te that λ 0 is an arbitrary co mplex number . Fr om the deﬁnition of FNCR, it can be declared that the matrix pen cil Θ( λ, i ) is not of FNCR. On th e contrar y , assume that th e m a tr ix pencil Θ( λ, i ) is not of FNCR. Then fo r an arbitr a ry λ 0 ∈ C , there is at least one vector α x 0 that is n onzero and satisﬁ es Θ( λ 0 , i ) α x 0 = 0 . Construct a vector α as α =  α x 0 −  A T xv ( i ) A xv ( i )  − 1 A T xv ( i )( λI − A xx ( i )) C ⊥ x ( i ) α x 0  Clearly α 6 = 0 . On th e other ha nd, dir ect matrix ope rations lead to Ω( λ 0 , i ) α = Θ( λ 0 , i ) α x 0 = 0 (a.17) which m e ans that the matrix Ω( λ 0 , i ) is also not of FCR. As λ 0 is an arb itrary com plex nu mber, this implies that Ω( λ, i ) is always of rank deﬁcien t, an d hence is n o t of FNCR. This comp letes the proof . ✸ R E F E R E N C E S [1] T . Beelen and P . V an Door e n, ”An improved algorithm for th e co mputation of Kr o necker’ s cano nical form of a sing ular p encil”, Line ar Algebra and Applica tions , V o l. 105, pp.9 ∼ 65, 1988. [2] J. Barreiro-Go mez, G. Ob a n do and N. Quijano, ”Dis- tributed populatio n dyn amics: optimiza tio n and control applications”, I EEE T ransactions on Sy stems, Man , and Cybnetics: Systems , V ol.4 7 , No.2, pp.30 4 ∼ 31 4, 2017. [3] A. Chapman , M.N. Abd o lyyou seﬁ and M. Mesbahi, ”Controllability an d observability of network-of net- works via cartesian p roduc ts”, IEE E T ransactions on Automatic C o ntr ol , V ol.59 , No.1 0, p p.266 8 ∼ 267 9, 20 1 4. [4] J.F . Carvalho, S. Pequito , A . P . Aguiar, S. Kar and K. H. Jo h ansson, ”Composability a n d controllability of structural linear time-inv ariant systems: distributed veri- ﬁcation”, Automatica , V o l.78, pp.12 3 ∼ 134 , 2017. [5] N. J. Cowan, E. J. Chastain, D. A. V ilhena, J. S. Freu den- berg and C. T . Bergstrom, ”Nodal dyna m ics, no t degree distributions, determine the stru ctural controllability of 15 complex networks”, P los One , V o l.7, No .6, e38 398, 2012. [6] L. Dai, Singular Contr o l Systems , Lecture Notes in Con- trol an d Inf ormation Sciences, V ol.11 8, Sprin g er , Berlin, 1989. [7] J. M. Dion, C. Commault and J. V an D e r W ou de, ”Generic pro perties a n d contro l of linear structured sys- tems: a survey”, Automatica , V ol.3 9, pp. 1 125 ∼ 1144, 2003. [8] G. R. Duan, Analysis an d Design of Descriptor Lin - ear Systems , Advances in Mechanics an d Math ematics, V o l. 23, Spring er-V erlag,New Y or k, 2010. [9] F . R. Gantmacher, The Theory of Matrices , Chelsea, New Y or k, 1959. [10] R. A. Ho r n and C. R. John son, T o pics in Matrix Ana lysis , Cambridge University Press, Cam bridge, 199 1 . [11] S. I wata and M. T akamatsu, ”On th e Kro necker canonical form of singular mixed matr ix p encils”, SIAM Journal on Contr ol and Optimizatio n , V ol.5 5, No.3 , pp.2 134 ∼ 2 150, 2017. [12] T . Kailath, A. H. Say ed and B. Hassibi, Linea r Esti- mation , Prentice Hall, Upper Saddle Ri ver, New Jersey , 2000. [13] M. D. I lic, L . Xie, U. A. Khan an d J. M. F . Moura, ”Modeling of future cyber-physical energy systems for distributed sen sin g and control”, IEEE T ransactions on Systems, Man, and Cybnetics-P art A: S ystems and Hu- mans , V ol.40, No . 4, pp.825 ∼ 838, 20 10. [14] C. T . L in, ”Structu ral contro llability”, IEEE T ransactions on A utomatic Contr ol , V ol.19 , No.3 , pp .201 ∼ 208, 1974. [15] A. Olshevsky , ”Minimal co n trollability problem s”, IEEE T ransaction s on Contr o l of Network Systems , V o l. 1, No.3, pp.24 9 ∼ 258 , 2014. [16] F . Pasqualetti, S. Zam pieri, and F . Bullo, ”Controlla- bility m etrics, limitations and algo rithms for complex networks”, IEE E T ransactio ns on Con tr ol o f Network Systems , V ol.1, No. 1, pp.40 ∼ 5 2, 20 14. [17] J. H. van Schup pen, O. Boutin, P . L. Kempker , J. K om enda, T . Masopust, N. P ambakian and A . C. M. Ran, ”Control of distributed systems: tutorial and overview”, Eur ope an Journal of Contr ol , V o l.17, No.5-6, pp.57 9 ∼ 602 , 2011. [18] K. M. Zhou , J. C. Doyle and K. Glover , Robust and Optimal Contr ol , Prentice Hall, Up per Sad dle River , New Jersey , 19 96. [19] T . Zhou , ”On the controllab ility and o bservability of networked dynam ic systems”, A u tomatica , V ol.5 2, pp.63 ∼ 75, 2 015. [20] T . Z hou, ”Min im al inpu ts o utputs for subsystems in a networked system”, Automatica , V ol.94, pp. 161 ∼ 1 69, 2018. [21] T . Zhou, ”T o pology an d subsystem param e ter based veriﬁcation for the co ntrollability/o bservability of a networked dy namic system”, Pr oc. the 58th I EEE Confer en ce o n De cision and Contr o l , Nice, Franc e , pp.35 75 ∼ 35 80, 2019. [22] T . Zhou , K. Y . Y o u and T . Li, Estimation an d Contr o l of Lar ge Sca le Networked Systems , E lsevier , Sao Paulo,, 2018. [23] Y . Zhan g and T . Zhou, ”Structural contr ollability of a networked dynamic system with LFT par ametrized subsystems”, IEEE T ransactions on A uto matic Contr o l , V o l. 64, No.12, pp.4 920 ∼ 4 935, 201 9.

Affine Dependence of Network Controllability/Observability on Its Subsystem Parameters and Connections

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment