Informativity and Identifiability for Identification of Networks of Dynamical Systems

1 Informati vity and Identifiability for Identification of Networks of Dynamical Systems Anders Hansson, Senior Member , IEEE , João V ictor Galvão da Mata, and Martin S. Andersen Abstract —In this paper , we show ho w inf ormativity and iden- tifiability for netw orks of dynamical systems can be in vestigated using Gröbner bases. W e provide a sufficient condition f or informati vity in terms of positiv e definiteness of the spectrum of external signals and full generic rank of the transfer function relating the exter nal signals to the inputs of the predictor . Moreo ver , we show how generic local network identifiability can be in vestigated by computing the dimension of the fiber associated with the closed loop transfer function from external measurable signals to the measured outputs. Index T erms —System Identification, Networks, Dynamical Sys- tems, Informativity , Generic local network identifiability . I . I N T R O D U C T I O N S YSTEM identification for networks of dynamical systems is an acti ve research field with a long history . Some early references are [1]–[3]. Some of the work has been focusing on identifying the full network, e.g. [3]–[5], while other work has been focusing on identifying parts of the network, a so- called sub-network , e.g. [2], [6]–[11]. The latter is an example of system identification under feedback, a topic that also has a long history , see e.g. [12]. Many system identification methods are based on a so- called pr edictor , and the system identification methods using predictors are called prediction err or methods . T wo funda- mental questions related to these methods are what is called informativity and identifiability . The first question is about if the signals used for system identification are such that the predictor can be uniquely determined. The second question is whether the network topology is such that the open loop trans- fer functions can be uniquely determined from the predictor . This paper will focus on these questions under the assumption that a stable predictor exists. W e do not care if this predictor is linear or nonlinear in the open loop transfer functions. W e will in vestigate these questions both for the full network case and for the sub-network case. W e will see that we can handle more general network configurations than what has been possible hitherto in the literature. Specifically , we will consider the partial measurement case , i.e. not all of the node signals in the (sub)-network are in the measured outputs. Moreov er , we will allow the measurement equation to contain transfer functions to be identified, and we will also allow for direct feed-through from the inputs to the measured outputs. In a seamless way , This work was supported by ELLIIT , and by the Novo Nordisk Foundation under grant number NNF20OC0061894. Anders Hansson is with the Department of Electrical Engineering, Linköping Univ ersity (e-mail: anders.g.hansson@liu.se). João V ictor Galvão da Mata and Martin S. Andersen are with the Depart- ment of Applied Mathematics and Computer Science, T echnical University of Denmark (e-mail: jogal@dtu.dk; mskan@dtu.dk). we also allow for some transfer functions to be known and for rational constraints relating transfer functions to one another . Regarding informativity , we focus on sufficient conditions in terms of positive definiteness of the spectrum of external signals and full generic rank of the transfer function relating the external signals to the inputs of the predictor . This has previously been studied in e.g. [2], [13]. W e specifically generalize this work to the partial measurement case, and to some of the transfer functions to be known. W e give both a graph-theoretic and an algebraic geometry characterization of the generic rank. Moreov er , we will deriv e conditions for generic local net- work identifiability in terms of dimensions of affine vari- eties. This is a generalization of the w ork in [3], [14] and complements the work in [15]. Specifically , we are able to consider some of the transfer functions to be known, and constraints on transfer functions. W e are also able to allo w for the measurement equation to contain transfer functions to be identified. I I . M O D E L O F N E T W O R K W e consider a linear dynamic network model defined by P ( q ) w k = Q ( q ) u k z k = R ( q ) w k + S ( q ) u k (1) where the quadruple ( P, Q, R, S ) are transfer function matri- ces in the forward shift operator q . The dimensions of these transfer function matrices are n × n , n × m , p × n , and p × m , respectively . This also defines the dimensions of all the signals, where u k is referred to as the input signal, z k as the output signal, and w k as the node signal. W e assume that P is in vertible for almost all q such that the closed loop system is well-posed . W e may sometimes replace q with the complex variable z . W e assume that some of the elements of the transfer functions we have defined are dependent on some parameter θ , which is going to be identified. If we want to emphasize the parameter dependence we may e.g. write P ( q ; θ ) for transfer functions and y k ( θ ) for signals. W e will often drop parameter dependence and dependence on q . I I I . I N N OV A T I O N F O R M A N D P R E D I C T O R W e di vide u k in a part r k that we know or measure, and another part e k , which we refer to as noise. W e write u k = ( r k , e k ) , where r k has m r components and e k has m e components. W e also partition Q ( q ) and S ( q ) conformably as Q ( q ) =  Q r ( q ) Q e ( q )  , S ( q ) =  S r ( q ) S e ( q )  . 2 W e assume that z k can be partioned as z k = ( y k , r k ) , where all of z k is a signal that we have full information about. Here y k has p y components, and r k has p r components. W e conformally write  R ( q ) S ( q )  =  R ( q ) S r ( q ) S e ( q )  =  R y ( q ) S y r ( q ) S y e ( q ) 0 I 0  for appropriately defined transfer function matrices. From this, we may conclude that the closed loop system from u k to y k may be written y k = G c ( q ) r k + H c ( q ) e k (2) where  G c ( q ) H c ( q )  = R y ( q ) P ( q ) − 1  Q r ( q ) Q e ( q )  +  S r ( q ) S e q  . (3) System identification is very often based on a predictor , and to this end we define the so-called innovation form , i.e. we write ˆ y k = G c ( q ) r k + G o ( q ) ϵ k (4) y k = ˆ y k + ϵ k (5) for some transfer function matrix G o ( q ) , where the innovations ϵ k are zero mean equally distributed and independent random variables. W e remark that obtaining the innov ation form is by no means trivial and often in volv es solving an algebraic Riccati equation, e.g. [16]. Howe ver , we will only need the existence of the inno v ation form in this work, and no explicit knowledge of the transfer function G o ( q ) is needed. From the above equations we realize that we may write ˆ y k = W ( q ) z k (6) where W ( q ) =  W y ( q ) W r ( q )  = ( I + G o ( q )) − 1  G o ( q ) G c ( q )  . This is called the predictor for y k . Here we need to make the assumption that W ( q ) has all poles strictly inside the unit circle. This is not a priori giv en, and has to be verified for each giv en example. It follo ws using simple manipulations of the above equations that G c ( q ) can be recov ered from the predictor as G c ( q ) = ( I − W y ( q )) − 1 W r ( q ) . (7) W e will no w revisit some special cases of the above network formulation that are popular in the literature. Most e xisting work assume that P ( q ) = I − G ( q ) for some transfer function matrix G ( q ) which is assumed to ha ve a zero diagonal. Most existing work also only consider S ( q ) = 0 . Moreover , it is common to hav e R y ( q ) = I implying y k = w k , which is referred to as the full measurement case. The partial measur ement case is when R y ( q ) is a matrix having ro ws equal to standard basis vectors, which means that y k contains components of w k . It is also very common that Q r ( q ) and Q e ( q ) are diagonal matrices. Moreover , it is not uncommon that they are independent of q . This specifically goes for Q r ( q ) . Notice that we do not make any assumptions similar to the special cases above. I V . I N F O R M A T I V I T Y W e will now in vestigate what is called informativity for estimation of the parameter θ . T o this end, the following definition is useful: Definition 1. The signal z k is said to be informative enough if for any two parameters θ 1 and θ 2 such that the closed loop system is stable, it holds that E h ( ˆ y k ( θ 1 ) − ˆ y k ( θ 2 )) ( ˆ y k ( θ 1 ) − ˆ y k ( θ 2 )) T i = 0 (8) implies that W  e iω ; θ 1  = W  e iω ; θ 2  for almost all ω . It holds that ˆ y ( θ 2 ) k − ˆ y ( θ 1 ) k = ∆ W ( q ) z k where ∆ W ( q ) = W ( q ; θ 1 ) − W ( q ; θ 2 ) . From Parse v al’ s formula, it follows that (8) is equiv alent to ∆ W  e iω  Φ z ( ω )∆ W  e − iω  T = 0 where Φ z is the spectrum of z k . Hence, a sufficient con- dition for the above equality to imply that W  e iω ; θ 1  = W  e iω ; θ 2  for almost all ω , is that Φ z ( ω ) is positive definite for almost all ω . W e have that the closed-loop transfer function from u k = ( r k , e k ) to z k is gi ven by Π( q ) = R ( q ) P ( q ) − 1 Q ( q ) + S ( q ) and that the spectrum of z k is positiv e definite if Π  e iω ; θ 0  is full row rank for almost all ω and the spectrum of u k is positiv e definite. This condition depends on the true system described by θ 0 , and hence it is a difficult condition to verify . Let M ( q ) =  P ( q ) Q ( q ) − R ( q ) S ( q )  . (9) Similarly as in [13], we will instead only in vestigate the so- called generic rank defined as rank g Π = max M rank Π . Here we consider Π to just be a function of the free entries of the matrix M . 1 The only property we need regarding these free entries when we in vestigate generic rank is that they are algebraically independent 2 commuting indeterminates ov er the coefficient field. The coefficient field is the field of rational functions of transfer functions. This is a field with the standard definition of addition and multiplication. W e will later on see that the generic rank is equal to the rank for almost all M . Since Π is a rational function in z , it follo ws that rank g Π( z ) = max z rank g Π( z ) for almost all z ∈ C , and hence rank Π( z ) = max z rank g Π( z ) 1 When no ambiguity arises we will often just refer to M even if we mean the free entries of M . 2 Algebraically independent is the formal way of saying that the entries are free. 3 for almost all M and z . Using this result, it is possible to give a suf ficient condition for informativity . Theorem 1. Assume that the spectrum of u k = ( r k , e k ) is positive definite and that the generic rank of Π is full, then z k is informative enough. Remark 1. If we do not want to use all of u k for e xcitation, i.e. if we have some of the components equal to zer o, then we just erase the corr esponding columns in M ( q ) . Similarly , we might not always be inter ested in having uniqueness of all of W ( q ) , i.e. we may only be interested in some of the columns. This is easily accommodated by just r emoving the corr esponding r ows in M ( q ) . What has been presented above is a generalization of the results in [13] for a much more general network configuration. W e will now give conditions on the generic rank of Π in terms of both a graph and in terms of so-called Gröbner basis . A. Graphical Condition Since we are inv estigating generic properties, we drop the dependence on the forw ard shift operator q or the complex variable z . The following equality is easily proven by multi- plying together the matrices:  I 0 RP − 1 I  M  I − P − 1 Q 0 I  =  P 0 0 RP − 1 Q + S  . From well-posedness, it holds that P is in vertible. Hence, the rank of M is the same as the rank of Π plus the dimension of P , which is n . This is a generalization of the formula in (5) in [17] to the direct term case. In that paper, it is shown for real-v alued structured 3 matrices ( A, B , C ) and s ∈ C , how the generic rank of the transfer function C ( sI − A ) − 1 B can be given a graph-theoretic characterization. From algebraic geometry , e.g. [18], it follo ws that the proof of Theorem 1 in [17] still holds true in our setting, i.e. { M | rank Π < rank g Π } is a proper af fine variety of the set of all M . Hence it holds that rank g Π = rank Π for almost all M . Then it is straightforward to see that Theorem 2 of that paper can also be generalized. W e are therefore able to deriv e a graph-theoretic characterization of the generic rank of Π . T o this end, let us define a directed graph G = ( V , E ) with verte x set V and edge set E , where V = U ∪ W ∪ Z with U = {U 1 , . . . , U m } , W = {W 1 , . . . , W n } , Z = {Z 1 , . . . , Z p } . W e can no w associate each row and column in M with elements in V . The first n columns are labeled W 1 to W n , and the next m columns U 1 to U m . W e proceed similarly with the rows. W e then define the edges in E as all the ordered two-tuples ( v i , v j ) with v i , v j ∈ V such that M v j ,v i is not a structural zero. W e do ho wever , not include any self-loop edges, i.e. edges for which v i = v j . If there are vertices v 1 , . . . , v k in V such that 3 Notice that we also have structured matrices, since the transfer function matrices are often sparse. ( v i , v i +1 ) ∈ E for i = 1 , . . . , k − 1 , we say that there is a path in G from v 1 to v k . If v 1 ∈ U and v k ∈ Z we say that there is a path from U to Z . W e say that two paths from U to Z are disjoint if they hav e no vertex in common. W e say that an l -tuple of paths from U to Z is disjoint if each pair of paths in the l -tuple is disjoint. W e are now able to state the follo wing result, which is a generalization of Theorem 2 in [17]. Theorem 2. The maximum number of disjoint paths in V fr om U to Z is equal to the generic rank of Π . Remark 2. The pr oblem of finding the maximum number of disjoint paths can be formulated as a maximum flow pr oblem for an associated flow graph. Remark 3. F or the above pr oof to hold, we need to assume that all non-zer o entries of M are fr ee variables. This assump- tion is not needed for the Gröbner basis condition to follow . B. Gröbner Basis Condition W e will no w in vestig ate the generic rank of Π using Gröbner basis. W e are still considering Π to be a rational function of the free entries of M . W e let D i be the least common multiple of the denominators of the i th row of Π . Multiply each row of Π with this polynomial to obtain the polynomial matrix ˜ Π . Then the generic rank of Π , when well-posedness holds, is the same as the generic rank of ˜ Π when D i  = 0 . Now , define the ideal I k generated by all the minors of ˜ Π of dimension k together with the polynomial 1 − tD , where D is the product of all D i and t is an additional variable. Notice that the equation 1 − tD = 0 always has a solution where D  = 0 and t = 1 /D . 4 From this the following theorem follows. Theorem 3. Given the ideal I k defined above it holds: 1) If the r educed Gröbner basis of I k is one, then rank Π ≥ k . 2) If the reduced Gröbner basis of I k , excluding the basis depending on t , is zer o, then rank Π < k . 3) Otherwise rank g Π ≥ k . Then the Gröbner basis, excluding the basis depending on t , defines the set of entries M for which rank Π < rank g Π . Remark 4. The le xicogr aphic or dering should be used when the Gröbner basis is computed, and the variable t should be the lar gest variable accor ding to that ordering . Remark 5. Notice how known entries of M are seamlessly tr eated since we work over the field of rational functions of transfer functions. Remark 6. In case some of the entries of M ar e constr ained by any rational equation, then one can just add this to the ideal. Specifically , one may consider constraining some entries to be the same. However , then it is easier to just r emove the duplicate variable to start with. Notice that the generalizations in the remarks abov e are trivial to include in the Gröbner basis condition. This is not 4 Notice that det P might have zeros that are not zeros of D , and hence the graph results might be conserv ative as compared to the Gröbner basis results. 4 so easy to do in the graphical condition of the previous subsection, which assumes all non-zero transfer functions to be free variables. V . I D E N T I FI A B I L I T Y As was discussed in Section III the closed loop transfer function can be recov ered from the predictor using (7), which we can uniquely do under the assumption of informativity . The next question that arises is whether we can recover parts of M from G c . From (3), it holds that G c = R y P − 1 Q r + S y r which is a rational expression in the free entries of ( P , Q r , R y , S y r ) , which we will denote by the k -dimensional vector X . Hence, a relev ant question is when a rational equation has a unique solution. Howe ver , in many cases, this is too much to ask for . What is a more relev ant question is whether there is a finite number of isolated solutions for a giv en G c . T o in vestig ate this question, let F be a function defined by the rule F ( X ) = R y P − 1 Q r + S y r . W e let the domain V o of F to be the set of all vectors X of dimension k with elements being rational transfer functions for which the in verse of P e xists. This is a dense open subset of the set for which we do not care if the inv erse exists or not. W e then define the set of admissible G c as V c = F ( V o ) . If we take V c to be the co-domain of F , it follows that F is a surjecti ve function. Unique solution and finite number of isolated solutions parallels the discussion in [19] regarding generic network identifiability and generic local network identifiability , where it is argued that the latter is easier to in vestigate than the former . There it is also conjectured that generic local network identifiability implies generic network identifiability . Definition 2. W e say that we have generic local network identifiability if for almost all G c ∈ V c the following hold. Given a solution X ∈ V o to F ( X ) = G c (10) ther e exists ϵ > 0 such that for any ˜ X satisfying ∥ ˜ X − X ∥ < ϵ and F ( X ) = F ( ˜ X ) it follows that ˜ X = X . Remark 7. F rom this definition it can be shown that if a network is generically locally identifiable, then it can be recov- er ed up to a discrete ambiguity , i.e. the set of X corr esponding to the same G c is discr ete except for a set of G c of measur e zer o, [19]. Similarly as in this r eference we ar e only r eferring to one single frequency z , and then the norm is well-defined as the complex Euclidian norm. In that paper ther e is also a discussion about why it is no limitation to just consider one fr equency . Then we have the following theorem. Theorem 4. It holds that X is generically locally network identifiable for any admissible G c fr om (10) if and only if dim( V o ) = dim( V c ) . Pr oof. For a fixed G c ∈ V c define the fiber F − 1 ( G c ) = { X ∈ V o | F ( X ) = G c } . It holds that generic local network identifiability is equiv alent to that dim  F − 1 ( G c )  = 0 for almost all G c ∈ V c . Since F is surjective, it follows from the fiber dimension theorem in algebraic geometry that dim( V o ) = dim( V c ) + dim  F − 1 ( G c )  for almost all G c ∈ V c . The result is immediate from this. Remark 8. Since V o is a dense subset of a set with dimension k it follows that dim( V o ) = k . W e define the polynomial matrices S and T such that T ⊙ F ( X ) = S where ⊙ is the Hadamar d pr oduct of matrices. W e let D be the least common multiple of the entries of T . Then we define the polynomial matrix T ⊙ G c − S , and the ideal generated by the entries of this matrix and 1 − tD . W e then compute the Gröbner basis for this ideal using the lexicogr aphic or dering, wher e the variables in G c should be the smallest variables accor ding to this or dering. The part of the basis containing only the variables in G c defines the smallest variety that contains V c , c.f. [20, Ch.3.3]. The dimension of this variety is the same as the dimension of V c . Then the algorithm in [21] can be used to compute the dimension. Remark 9. In case some of the entries of X are known, we just e xclude them when defining V o and the variable X . Remark 10. It is possible to put rational constraints on X . In case the constraint is that one of the entries is the same as another entry , then it is easier to just r eplace one variable with the other to r emove the constraint W e will discuss this in an e xample later on in mor e detail. Remark 11. Notice that it is possible to discuss identifiability also when not all of the predictor is estimated uniquely . Then we just consider the closed loop transfer function corr espond- ing to the part of the predictor we estimate uniquely . Then ther e will be fewer equations that determine the open loop transfer functions, but there could still be enough equations to guarantee generic network identifiability . W e will discuss this in more detail for an e xample later on. What has been presented above is a generalization of the results in [3] to the case when Q r  = I . It is also a generalization of the results in [14] to the case when R y  = I . Our result also generalizes the results in [15], where it is assumed that Q r and R y are giv en zero-one matrices. W e also consider a direct term S y r , which other work do not. V I . S U B - N E T W O R K S So far we hav e discussed the identification of a whole network. Often one is only interested in parts of a network. W ithout loss of generality we will assume that the part of the network we are interested in corresponds to the leading entries w A k of w k = ( w A k , w B k ) . W e then partition M as M =   P Q r Q e R y S y r S y e 0 I 0   =     P A P AB Q Ar Q Ae P B A P B Q B r Q B e R y A R y B S y r S y e 0 0 I 0     . 5 W e hav e P A w A k =  − P AB Q Ar   w B k r k  + Q Ae e k y k = R y A w A k +  R y B S y r   w B k r k  + S y e e k . Hence, we can in vestigate generic local network identifiability of network A by making the following replacements in our previous definitions: P A → P  − P AB Q Ar  → Q Ar R y A → R y  R y B S y r  → S y r w A k → w k ( w B k , r k ) → r k . W e then obtain the follo wing rational equation for inv estigat- ing identifiability: R y A P − 1 A  − P AB Q Ar  +  R y B S y r  = G c . If there are columns in  − P AB Q Ar R y B S y r  that are zero, then they may be remo ved from the rational function. Similarly any zero rows in  R y A S y r  may be remov ed. When it comes to informati vity , we realize that the B-part of the netw ork provides excitation of the A-part of the network, and hence the whole network should be considered. Since w B k is part of the exogenous input, we should consider M =       P A P AB Q Ar Q Ae P B A P B Q B r Q B e R y A R y B S y r S y e 0 I 0 0 0 0 I 0       when we in vestigate informativity . Ho wever , rows of the second last block row that correspond to the zero columns in  − P AB R y B  discussed abov e should be remov ed. As seen abov e, the exogenous signal contains also measure- ments of the node signals in the B-part of the network. W e also need to estimate models for parts of the B-network. Because of sparsity this might not be a big problem, since we may not hav e to measure all of w B k . Howe ver , this is in many cases still not desirable. If we instead of measuring the node signals w B k can measure  ˜ w B k ˆ w B k  =  − P AB R y B  w B k , we will not need to estimate models for any parts of the B- network. Then we obtain the equations: P A w A k =  I 0 Q Ar    ˜ w B k ˆ w B k r k   + Q Ae e k y k = R y A w A k +  0 I S y r    ˜ w B k ˆ w B k r k   + S y e e k . The replacement will instead be: P A → P  I 0 Q Ar  → Q r Q Ae → Q e R y A → R y  0 I S y r  → S y r S y e → S y e w A k → w k ( ˜ w B k , ˆ w B k r k ) → r k . W e then obtain the follo wing rational equation for inv estigat- ing identifiability: R y A P − 1 A  I 0 Q Ar  +  0 I S y r  = G c which now contains dynamics of only the A-part of the network. Any zero ro ws in  Ry A S y r  may be removed. If there are zero rows in  − P AB R y B  it follows that there are zero signals in ( ˜ w B k , ˆ w B k ) . The corre- sponding columns in the above rational matrix may then be remov ed. Also an y zero columns in  Q Ar S y r  may be remov ed. Since ( ˜ w B k , ˆ w B k ) is part of the exogenous input, we should consider M =         P A P AB Q Ar Q Ae P B A P B Q B r Q B e R y A R y B S y r S y e 0 − P AB 0 0 0 R y B 0 0 0 0 I 0         when we in vestigate informati vity . Howe ver , any zero rows in the fourth and fifth block rows should be remov ed, since they correspond to zero signals in ( ˜ w B k , ˆ w B k ) . Notice that the criterion for informati vity holds for any pattern of the matrices defining the M -matrix. Based on this, we no w realize that we are able to analyze informativity and generic local network identifiability for sub-networks using the methods we have de veloped for a whole network. It is possible to combine the two abov e methods depending on what can be measured so that only some of the columns of P AB and R y B are estimated. V I I . E X A M P L E In this section, we will inv estigate the theory we hav e dev eloped on some examples. Example 1. W e will look at the very simple example in F igure 1 to illustrate why the conditions we have presented ar e mor e general than the ones pr eviously pr esented in the literatur e. W e assume that v 1 k = H 1 e 1 k , v 2 k = H 2 e 2 k and that r 1 k = Q 1 r k and r 2 k = Q 2 r k . Notice that we have the same single r k entering the loop in two places. W e are inter ested in estimating ( G 1 , Q 1 ) , and use a predictor for pr edicting w 1 k . 6 + G 1 + G 2 + + r 1 w 2 v 1 r 2 w 4 v 2 w 3 w 1 Fig. 1. Block diagram for a simple network. W e can measur e ( w 3 k , r k ) in addition to w 1 k , and ther efor e the pr edictor input is z k = ( w 1 k , w 3 k , r k ) . W e obtain the following equations     w 1 k w 2 k w 3 k w 4 k     =     0 G 1 0 0 0 0 1 0 0 0 0 G 2 1 0 0 0         w 1 k w 2 k w 3 k w 4 k     +     0 H 1 0 Q 1 0 0 0 0 H 2 Q 2 0 0       r k e 1 k e 2 k   z k =   1 0 0 0 0 0 1 0 0 0 0 0       w 1 k w 2 k w 3 k w 4 k     +   0 0 1   r k . (11) It is straightforwar d to see that there are three disjoint paths: ( E 1 , W 1 , Z 1 ) , ( E 2 , W 3 , Z 2 ) and ( R 1 , Z 3 ) , and hence we have informativity , assuming that the spectrum for ( r k , e 1 k , e 2 k ) is positive definite. W e can also verify this by dir ectly looking at the transfer function Π =    G 1 ( Q 1 + G 2 Q 2 ) 1 − G 1 G 2 1 1 − G 1 G 2 G 1 1 − G 1 G 2 G 2 ( G 2 Q 1 + Q 2 ) 1 − G 1 G 2 G 2 1 − G 1 G 2 1 1 − G 1 G 2 1 0 0    which has determinant − 1 / (1 − G 1 G 2 ) . Hence, both the rank and the generic rank are full when we have well-posedness. The closed loop system transfer function fr om ( w 3 k , r k ) to w 1 k is given by  G 1 c G 2 c  =  1 0   1 − G 1 0 1  − 1  0 0 1 Q 1  =  G 1 G 1 Q 1  . The dimension of the affine varieties are dim V c = dim V o = 2 and we conclude that ( G 1 , Q 1 ) = ( G 1 c , G 2 c /G 1 c ) . Hence, we have generic local network identifiability . The r eason we do not have network identifiability is that ther e only exists a unique solution for all G 2 c when G 1 c  = 0 . Under this assumption, ther e is only one solution for each given G c , and hence we do not only have g eneric local network identifiability , but generic network identifiability . Example 2. Let us now consider the case wher e we ar e only inter ested in estimating G 1 , and wher e we do not car e about any of the other transfer functions. W e also assume that Q 1 is known. W e still use the same measur ed signals and the same predictor . The closed system transfer functions are the same. Now , the equations for generic network identifiability ar e linear in G 1 since Q 1 is no longer a variable b ut known. Ther e are two equations, but only one unknown. Because of this, we have more information than what is needed. W e can ther efore, if we so desir e, consider to only estimate parts for the predictor , either the part corresponding to w 3 k or to r k . Hence, we only need informativity for one of these signals. Ther efore it is enough if either the spectrum for ( r k , e 1 k ) or the spectrum for ( e 1 k , e 2 k ) is positive definite. This shows an inter esting interplay between informativity and generic local network identifiability . F or more complicated networks, it is ther efore not so easy to know what signals should be excited to be able to obtain consistent estimates. Example 3. W e consider again the network in F igure 1, but this time we ar e inter ested in estimating both tr ansfer functions ( G 1 , G 2 ) . W e perform two separate experiments wher e we first excite with only r 1 k and measur e w 1 k , and then we excite with only r 2 k and measure w 3 k . It is not difficult to conclude that we have informativity for both experiments, irr espective of what the noises are . It is also straightforwar d to conclude that w 1 k = G 1 c r 1 k , w 3 k = G 2 c r 2 k wher e G 1 c = G 1 1 − G 1 G 2 , G 2 c = G 2 1 − G 1 G 2 . If we r eplace ( G 1 , G 2 ) with ( − 1 /G 2 , − 1 /G 1 ) we obtain the same ( G 1 c , G 2 c ) . Hence, this is an e xample where local identifiability does not imply identifiability . Notice that     w 1 k w e k w 3 k w 4 k     = 1 1 − G 1 G 2     G 1 G 1 G 2 1 G 2 G 1 G 2 G 2 G 1 1      r 1 k r 2 k  and it is not possible to pick out these two closed loop transfer functions by multiplying fr om the left and the right with zer o-one matrices. However , if we duplicate the network and constrain the tr ansfer functions in the two networks to be the same, it is possible to pick out them with zer o-one matrices. This shows that the conjectur e in [19] that local identifiability implies identifiability is not necessarily true if the same transfer functions appear twice in a network. V I I I . N U M E R I C A L E X A M P L E S The abov e examples were possible to analyze without using any of the theory we have dev eloped. In this section, we will show how a Matlab implementation based on Gröbner basis calculations can be used to analyze fairly large examples, that are not so easy to analyze by hand. 7 Example 4. Consider the dynamical network defined by ( P , Q r , R y , S y r ) , wher e P = I − G , S y r = 0 and G =       0 G 12 0 0 0 0 0 G 23 0 0 0 0 0 G 34 0 0 0 0 0 G 45 G 51 0 0 0 0       Q r =       Q 1 1 0 0 0 0 Q 2 1 0 0 0 0 Q 3 1 0 0 0 0 Q 4 1 Q 5 0 0 0 1       R y =   R 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0   . F or this example , we have dim V o = 11 , and using Matlab, we can compute dim V c = 11 in 11 minutes on a laptop computer . Hence, we have generic local network identifiability . In a second we can verify that the r ow rank of Π is full for the case when Q e = I and S y e = 0 , and ther efor e we also have informativity if the spectrum of ( r k , e k ) is positive definite. Example 5. W e consider the same dynamic network as in the pr evious example, b ut now we ar e only inter ested in identifying the transfer functions ( G 12 , Q 2 , Q 2 , R 1 ) . W e let y k = ( w 1 k , w 2 k ) and we measure ˜ w B k = G 23 w 3 k fr om the B- part of the network. W e then have that F ( X ) =  R 1 0 0 1   1 − G 12 0 1  − 1  0 Q 1 1 0 1 0 Q 2 1  with the input ( ˜ w B k , r 1 k , r 2 k , r 3 k ) and the output y k . The M - matrix for investigating informativity is M =     P Q r Q e R y S y r S y e ˜ P 0 0 0 I 0     wher e P and Q r ar e as in the pre vious example , Q e = I , S y r = 0 , S y e = 0 ,and R y =  R 1 0 0 0 0 0 1 0 0 0  , ˜ P =  0 0 G 23 0 0  . Notice that the zer o matrices have one fewer r ow as compared to the pr evious example. In about a second we can verify that we have generic local network identifiability and informativity . W e can also in vestigate if ( r 4 k , r 5 k ) can be equal to zer o. W e just put the corr esponding rows and columns equal to zer o in M , and we obtain that we still have informativity . It is also possible to in vestigate if we in addition can have r 3 k equal to zer o. However , then we have to investigate generic local network identifiability for the case when we remove the last column in F ( X ) . It turns out that we have both informativity and generic local network identifiability also for this case. I X . C O N C L U S I O N S In this paper , we have shown how to in vestig ate informa- tivity and generic local network identifiability for networks of dynamical systems. Our informativity results are characterized in terms of verte x-disjoint paths of a graph associated with the dynamic network. Informativity can also be cast as an alge- braic geometry problem where Gröbner bases are computed. The generic local network identifiability result is phrased as two af fine varieties ha ving the same dimension. 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