Input-output equations and identifiability of linear ODE models

IEEE TRANSACTIONS ON A UTOMA TIC CON TROL 1 Input-out p ut equation s an d ident i fi ability of linear O DE models Alex e y Ovchinniko v , Gleb P ogudin, and P eter Thompson Abstract — Structural identifiability is a property of a differe n- tial model with parame ters that allows for the parameters to be determined from the model equations in the absence of noise. The method of input-output equations is one method for ve rifying structura l identifiability . This method stands out in its impor tance because the additional insights it pro vides can be used to analyze and improv e models. Howeve r , its complete theoretical gr ounds and applicability are s till t o b e established. A subtlety and key for this method to work correc tly i s kno wing whether the coeffic ients of these equations are identifiable. In this paper , to address this, we prove identifiability of the coefficient s of input -output equations for t ypes of diff erential models that often appear i n practice, such as l inear models with one output and linear compartment models in whic h, from eac h compartment, one can reach either a leak or an input. This shows that check ing identifiability via input-output equations for these models is legitimate and, as we pro ve, that the field of identifiable functions is generated by the coefficient s of the input-output equations. Finally , we exploit a conn ection bet ween input- output equations and the transf er funct ion matrix to show that, for a linear compartment model with an input and strongly connected graph, the field of all identifiable functions is generated by the coefficient s of the transfer function matrix even if the initial con- ditions are generic. Index T erms — identifiable functions, i nput-output equations, linear compartment models, structura l parameter i dentifiability I . I N T R O D U C T I O N A. Background Structural global iden tifiability (in what follows, we will say just “identifiability” for simp licity) is a property o f a differential mod el with param e ters th a t allo ws for the param- eters to be un iquely determine d from the m o del equations, noiseless data and sufficiently exciting in p uts (also known as the persistence of excitation, see [1]–[3]) . Performing iden ti- fiablity analysis is an importan t first step in e valuating and, if needed, adjusting the model. There ar e different approach e s This w ork was partially supported by the NSF grants CCF-156394 2, CCF-1564132, CCF-1319632, DMS-1760448, CCF- 1708884, DMS-1853650, DMS-1853482; NSA grant #H98230-18-1-0016; and CUNY grants PSC-CUNY #69827-0047 , #60098-00 48. A. Ovchinnikov is with CUNY Queens College , Depar tment of Mathematics, 65-30 Kissena Blvd, Q ueens, NY 11367, USA and CUNY Graduate Center , Ph.D . Programs in Mathematics and Computer Science, 365 Fifth Ave nue, New Y ork, NY 10016 , USA (e-mail: aovchinnik ov@qc.cuny .edu). G. P ogudin was with Courant I nstitute of Mathematical Sciences, New Y ork University , New Y ork, NY 10012, USA and the Higher School of E conomics, Fa c- ulty of Computer Science , Moscow , 109028, Russia. He is now with LIX, CNRS, ´ Ecole P olytechnique , Institute P olytechnique de P aris, 1 rue Honor ´ e d’Estienne d’Or ves , 91120 , Pa laiseau, F ran ce (e-mail: gleb.pogu din@polytechnique .edu). P . Thompson was with CUNY Graduate Center , Ph.D . Prog ram in Mathe- matics, 365 Fifth Av enue , New Y ork, NY 10016, USA (e-mail: peter thompson- math@gmail.com, peter.thompson@liu.se). to a ssessing identifiability (see [4] –[6] fo r d escriptions of methods). If structura l id entifiability is established, one can assess practical iden tifiability before d oing reliable p arameter identification [7], [8]. There is a relaxed version of id entifiability , namely , lo- cal identifiability . It refers to th e possibility o f determining finitely many feasible p arameter values. The r e are efficient algorithm s [9], [1 0] for ch e c king whether a g iv en function of parameters is locally identifiable. T o the best o f our knowledge, ther e are no complete and efficient algorithms for finding all locally iden tifiable fun ctions of par ameters (see [11, page 7] for a partial algorithm) , a ke y to efficient mo d el reparame tr ization for improving the mo del. 1) How the errors t hat we pre vent occur in existing methods : On e of the approach es, which is widely used, is based o n input- output equations [12] –[23] , and h a s app eared in so ftware packages such as COMBOS, D AISY , and th e ir successors. An existing challeng e is to understand the a prior i applicab ility of the method , as the above software packages make incorrect identifiability co nclusions fo r some models. W e address th is challenge in the pr esent paper . W e will now discuss th is in more detail. Rou ghly spe a k ing, input-o u tput equations ar e “min imal” eq uations that depend only o n the inp u t and outpu t variables and p arameters ( see [2 4] for applications other than ide ntifiability). W e will d escribe a typical algorith m based on th is app roach using the f ollowing linear compartmen t model as a r unning example:      x ′ 1 = − ( a 01 + a 21 ) x 1 + a 12 x 2 + u , x ′ 2 = a 21 x 1 − a 12 x 2 , y = x 2 . (1) In the above system, • x 1 and x 2 are the state variables; • y is the output observed in the experiment; • u is th e inpu t (contr ol) fu nction to be cho sen by the experimenter; • a 01 , a 12 , a 21 are unknown scalar parameters. The question is whether the values of the parameters a 01 , a 12 , a 21 can be d etermined from y and u . A typical algorithm operates as follows : (1) Find input-outp ut eq uations, wr iting them as (differential) polyno mials in the input and ou tput variables. For ( 1 ), a calculation shows that the inp ut-outp ut equation is y ′′ + ( a 01 + a 12 + a 21 ) y ′ + a 01 a 12 y − a 21 u = 0 . (2) (2) Use the following Assumption (A) : 2 IEEE TRANSACTIONS ON A UTOMA TIC CON TROL a function of parameters is identifia ble if a nd only if it can be e xpress ed as a rational function o f the coefficients of the input-o utput equ ations . In ou r example, this amounts to assuming th at a function of parameter s is identifiable if and o nly if it can b e expressed as a r ational fu nction o f a 01 + a 12 + a 21 , a 01 a 12 , and a 21 . One possible ration ale behind this assumption is th e “solvability” condition from [13 , Remar k 3 ] : du e to the “minimality” of th e in p ut-outp ut equation s, one would expect that there exist N and t 1 , . . . , t N ∈ R such that the linear system        y ′′ ( t 1 ) + c 1 y ′ ( t 1 ) + c 2 y ( t 1 ) + c 3 u ( t 1 ) = 0 . . . y ′′ ( t N ) + c 1 y ′ ( t N ) + c 2 y ( t N ) + c 3 u ( t N ) = 0 (3) in c 1 , c 2 , c 3 has a uniqu e solution in terms of y ( t i ) , y ′ ( t i ) , y ′′ ( t i ) , u ( t i ) , 1 6 i 6 N , so the coefficients of ( 2 ) are ide ntifiable. H owever , the assumption is not always satisfied an d , co nsequently , suc h N and t 1 , . . . , t N might not exist at all. This is a re ason, e.g., wh y D AISY may miss the non -identifiability of some o f th e p arameters in those systems. An exam p le is given in Section IV -A.1 ( see also [5, Example 2 .14] and [ 25, Sec tions 5.2 an d 5.3]). (3) Set up a system o f poly nomial equa tio ns in the parame ters setting the coefficients of ( 2 ) eq u al to n ew variables,      a 01 + a 12 + a 21 = c 1 a 01 a 12 = c 2 − a 21 = c 3 , (4) and verify if ( 4 ) as a system in th e a ’ s with coefficients in the field C ( c 1 , c 2 , c 3 ) has a uniqu e solution. Th is can be done, e.g., using Gr ¨ obner b a ses. Altern ativ ely , f or ( 4 ), one can see th at a 21 = − c 3 can be uniquely recovered, but the values of a 01 and a 12 are kn own on ly up to exchange due to th e sym metry o f ( 4 ) with respect to a 01 and a 12 . 2) Impor tance of the I O-equation method: finds all identifiable com- binations and helps with repar ametrization : Even though there are complete algorithms (th at is, no t relying on any assumption like Assumption (A) ab ove) for assessing structural identi- fiability (see, e.g., [2 6 ]), establishin g when the input-ou tput equation method is v alid is important b ecause: • T his method can produ ce all id entifiable fu nctions (also referred to as “true pa r ameters” in [2 4, Remark 2 ]), n ot ju st assess ide ntifiability of specific param e ters. More p r ecisely , [27, Cor ollary 5.8] shows that the field g enerated by the coefficients of the input-o utput equations contains all of the identifiable functions. In example ( 1 ), the field of ide n tifiable functions is gener- ated by the c o efficients of ( 2 ), so it is equ a l to C ( a 01 + a 12 + a 21 , a 01 a 12 , a 21 ) = C ( a 01 + a 12 , a 01 a 12 , a 21 ) . Generators of the field of ide n tifiable function s can b e used to reparame tr ize the model [12], [2 8], [29]. • T his m e thod can be used f or pr oving general th eorems about classes of mo dels [14], [1 5]. • For a large class o f linear compa rtment models, there are efficient method s for co mputing their in put-ou tp ut equations [14], [ 15], [22]. B. The problem As was described above, the approach to assessing iden- tifiability via input-o utput eq u ations h as been used much in the last thr ee decad es and has its own distinctiv e featur e s. Howe ver , it hea vily relies on Assumption (A) , which is n ot always true (see [5, Ex ample 2.14 ] and [25, Section 5 .2]). It ca n be verified by an alg orithm [30 , Section 4.1] and [3 1, Section 3.4 ] but is not verified in a ny imp lementation we h av e seen (inclu ding [17 ], [20]) . The general problem studied in this paper is: to determine classes of ODE models that satisfy Assumption (A) a priori; con sequently , the ap pr oach via input-outp ut equatio ns gives correct result for these m o dels . Discrepancy between d ifferent notions of id entifiability is not unusual given the wid e rang e of experimental setups and mathematical tools inv olved. W e refer the reader to a re c ent revie w [32 ] (see also [3]) presenting a numb er of no tions of id entifiability together with som e kn own (in)equivalences between them. Ou r work clarifies this big pictu re b y g iving explicit an d easy to check (unlike [27]) co nditions for equiv- alence of d ifferent ways to assess iden tifiability . C. Our resul ts The fi rst part of o ur results sh ows th at Assumption (A) is a p riori satisfied for th e f ollowing classes of models o ften appearin g in practice [ 7], [ 2 0], [ 33]–[ 38]: • linear models with one output ( Main Result 1 ); • linear compar tment mode ls such that, f rom ev ery vertex of the graph of the m odel, at least one leak or inp ut is reachable (Main Result 2 ). Checking wheth er the model is of one of these types can b e done just by v isu al inspection. For instance, as we will see in Example 1 , each o f the se theorems is applicable to m odel ( 1 ). Main Result 1 cannot be streng thened to m ore than o ne output if all linear models are allowed, see Section IV -A.1 a n d, for non-lin e ar systems, see [39, Le m ma 5.1]. The second p art is devoted to relaxing the “minimality” con- dition o n th e input-ou tp ut equations. For linear c ompartm e nt models, elegant relations involving only p arameters, inputs, and o utputs wer e pro p osed in [14, T heorem 2] b ased on Cramer’ s rule (see a lso [1 5, Pro p osition 2.3] ). In gener al, using these eq uations instead of the “m in imal” r elations in the algo- rithm above would give incor rect results [15 , Remark 3.11 ]. Howe ver , in Main Result 3 , we sh ow that, for lin ear com- partment m odels with an inp ut an d whose graph is strongly connected , on e can use these equ ations as the input-o utput equations and obtain the full field of identifiable functions. Furthermo re, we app ly Main Result 3 to the transfe r f unc- tion method [40, page 444]. It is kn own that, in case o f multiple outputs, using only the coe fficients of th e transfer function m atrix (as opp o sed to the full outp ut transfo rms) may O VCHINNI K O V , P OGUDIN, AN D TH OMPSON: I NPUT -OUTPUT EQUA TIONS AND I DENTIFIABI LITY OF LIN EAR ODE MODE LS 3 lead to inco rrect identifiability conclu sions [40, Example 10.6] . As a corollary of our results (Corollary 3 ), we show that this is not the ca se fo r su ch linear co mpartmen t models. W e state th e conseq uences of our results for algorithms for computin g identifiable functio n s in Section II-C and illustrate the condition s in our main r esults in Section IV . D . Structu r e of the p aper Basic notions an d notation fro m d ifferential algebra, iden- tifiability , and linear compartment m odels are given in Sec- tion II . The m ain results in a br ief for m a r e stated in Section III and then stated a nd proved in Section V . I n Section IV , we illustrate our main results with examples, e.g., showing how existing ide ntifiability appro aches cou ld fail. The app endix has results we use relatin g th e notions used in the paper fo r lin ear models to the co rrespon ding notions for n onlinear systems. I I . P R E L I M I N A R I E S In this section, we recall the no tation/notio ns f ound in the literature and intr o duce our own n o tation/notio ns to state ou r main results in Section III . All fields ha ve char acteristic zero. A. Id e ntifiabi lity of li near mode ls Fix positi ve integers λ , n , m , and κ for the remain der of the paper . Let µ µ µ = ( µ 1 , . . . , µ λ ) , x = ( x 1 , . . . , x n ) , y = ( y 1 , . . . , y m ) , and u = ( u 1 , . . . , u κ ) . Consider a system of ODEs Σ =      x ′ = f ( x , µ µ µ , u ) , y = g ( x , µ µ µ , u ) , x ( 0 ) = x ∗ , (5) where f = ( f 1 , . . . , f n ) and g = ( g 1 , . . . , g m ) are tuples of polyno mials in x , u over C ( µ µ µ ) of de gr ee at most one . For a rational f unction h ( µ µ µ ) ∈ C ( µ µ µ ) , we will defin e two notions of identifiability : identifiability and IO-identifia bility . The former is me a n ingful fr om the mod eling stand point; th e latter is what th e alg orithm outlined in the introdu ction checks. 1) Identifiability : W e fix notation to giv e rigorous defin itions: Notation 1 ( Auxiliary ana lytic n otation): (a) Let C ∞ ( 0 ) d e note the set o f all functio n s that are co mplex analytic in some n eighbo rhood of t = 0. (b) L e t Ω ⊂ C λ be the complement to the set where at least one of the deno minators o f the coefficients of ( 5 ) in C ( µ µ µ ) vanishes. (c) For every h ∈ C ( µ µ µ ) , we set Ω h : = C n × { ˆ µ µ µ ∈ Ω | h ( ˆ µ µ µ ) we ll- defined } × ( C ∞ ( 0 )) κ . (d) For ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) such that ˆ µ µ µ ∈ Ω , let X ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) and Y ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) den ote the u n ique solution over C ∞ ( 0 ) of the instance of Σ with x ∗ = ˆ x ∗ , µ µ µ = ˆ µ µ µ , and u = ˆ u (see [41, Theorem 2.2.2] ) . (e) For any positive integer s , a sub set U ⊂ C s is called Zariski open if there exists a polyno mial P on C s such that U is the co mplement to the zer o set of P . (f) For a ny positive in teger s , a subset U ⊂ ( C ∞ ( 0 )) s is called Zariski open if there exists a polynomial P in z 1 , . . . , z s and their deri vati ves such that U = { ˆ z z z ∈ ( C ∞ ( 0 )) s | P ( ˆ z z z ) | t = 0 6 = 0 } . (g) For any positi ve in teger s and X = C s or ( C ∞ ( 0 )) s , the set of all n o nempty Zariski ope n subsets of X will be denoted by τ ( X ) . Definition 1 (Identifia b ility , see [5, Definition 2.5]): W e say that h ( µ µ µ ) ∈ C ( µ µ µ ) is identifiable if ∃ Θ ∈ τ ( C n × C λ ) ∃ U ∈ τ (( C ∞ ( 0 )) κ ) ∀ ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) ∈ ( Θ × U ) ∩ Ω h | S h ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) | = 1 , where S h ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) : = { h ( ˜ µ µ µ ) | ∃ ( ˜ x ∗ , ˜ µ µ µ , ˆ u ) ∈ Ω h such that Y ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) = Y ( ˜ x ∗ , ˜ µ µ µ , ˆ u ) } . The field { h ∈ C ( µ µ µ ) | h is iden tifiable } will be called the fi eld of identifiable fu nctions . 2) Input-output i dentifiability : Th e n otion of IO- id entifiability can be defined fo r systems with rational right- hand side ( see Section A f r om the Appendix ). Here we give a specialization of th e gen eral definition to the lin ear case (the equiv alence of Definition 3 and Definitio n 7 restricted to the linear ca se is established in Proposition 1 ). For this, we w ill first recall se veral standard n otions from d ifferential algebr a : Notation 2 ( Differ ential ring s and ideals): (a) A differ ential ring ( R , δ ) is a commutative ring with a deriv ation ′ : R → R , that is, a map such th at, for all a , b ∈ R , ( a + b ) ′ = a ′ + b ′ and ( ab ) ′ = a ′ b + ab ′ . (b) The ring of differ ential polyn omials in the variables x 1 , . . . , x n over a field K is the ring K [ x ( i ) j | i > 0 , 1 6 j 6 n ] with a deriv ation d efined on the ring by ( x ( i ) j ) ′ : = x ( i + 1 ) j . This d ifferential ring is denoted by K { x 1 , . . . , x n } . (c) For a differential polyno m ial P ∈ K { x 1 , . . . , x n } an d 1 6 i 6 n , the order of P with r espect to x i is the order o f the highest deriv ativ e of x i appearin g in P ( − ∞ if x i does not appear in P ). I t is deno ted by ord x i P . (d) An ideal I of a differential ring ( R , δ ) is called a differ en- tial ideal if , fo r all a ∈ I , δ ( a ) ∈ I . F or F ⊂ R , the smallest differential ideal containing set F is d enoted by [ F ] . (e) For Σ as in ( 5 ), let I Σ = [ x ′ − f , y − g ] ⊂ C ( µ µ µ ) { x , y , u } be the d ifferential ideal of Σ . Informally , I Σ is th e ide al of all relations am ong compon ents of a generic solu tion of Σ . Definition 2 (a fu ll set o f in put-ou tp ut equatio ns): For Σ as in ( 5 ), a tuple ( p 1 , . . . , p m ) of differential p o lynomia ls from C ( µ µ µ ) { y , u } is called a fu ll set o f inp ut-outp ut equation s if there is an o rdering of the ou tp ut variables, which we will assume to b e y 1 < y 2 < . . . < y m to simplify no tation, such that (1) p 1 is the linear differential poly nomial in y 1 and u in I Σ of the smallest possible order in y 1 such that the coefficient of the h ig hest deri vati ve of y 1 is one. (2) For every ℓ > 1, p ℓ is the linear differential po lynomial in y 1 , . . . , y ℓ and u in I Σ such that • o rd y j p ℓ < ord y j p j for e very 1 6 j < ℓ ; • th e co efficient of the high e st deriv ati ve of y ℓ in p ℓ is 1 ; • o rd y ℓ p ℓ is the smallest p ossible. Definition 3 (IO-identifia ble function): For a system Σ , consider a full set E of input- output equations. Then the subfield k of C ( µ µ µ ) gener ated by the coefficients of E over C is called the field of input-outp ut identifiable (IO-identifiab le) functions . W e call h ∈ C ( µ µ µ ) IO-identifia ble if h ∈ k . Remark 1: Pro position 1 establishes the eq uiv alence of th is definition to Definition 7 , which is applicab le to a genera l 4 IEEE TRANSACTIONS ON A UTOMA TIC CON TROL rational ODE systems. Prop osition 1 also imp lies that th e field of inp ut-outp u t id entifiable functio ns does no t depend on the choice of a f ull set of in put-ou tput equations. For example s of input- output eq u ations an d IO-identifiable function s, see Section IV . 3) Comparison of identifiability and IO-identifiability : Remark 2 (Meaning of IO-id entifiability) : One can see that the field of IO-identifiab le fu n ctions is exactly what will b e computed by the first two steps of the alg o rithm o utlined in the introdu c tion (see also Algor ithm II.1 ). The g eneral problem as stated in Section I-B c a n be restated as: Determine classes of ODE mod els for which identifiab le ⇐ ⇒ IO-iden tifiable . [27, The o rem 4.2] toge th er with [5 , Example 2. 1 4] (see also Section IV -A.1 and [ 2 5, Sections 5 .2 and 5.3] with non- constant dynamics and o utputs) imply th at: Identifiable ( IO-id entifiable . (6) B. Linear compar t me nt mode ls In this section, we discuss lin ear compartment mode ls [42 ]. Such a model consists of a set o f co mpartme n ts in wh ich material is transferred from some co mpartme n ts to other com- partments. W e also allo w for leak age of material f r om some compartm ents out o f the system, an d for in p ut of mater ial into some compartmen ts from outside the system. W e use the notation of [1 4, Section 2] but ou r construction will be slightly mo r e general (allowing scaling for inpu ts and outputs). Let G b e a simp le directed graph with n vertices V an d edges E . Let In, Ou t, and Leak be subsets o f V . Th e coefficients of material transfer are { a ji | j ← i ∈ E } and { a 0 i | i ∈ Leak } , and there may be some ad ditional parameter s, we will deno te all the parameters b y µ µ µ as befor e. For i = 1 , . . . , n , let x i be th e quantity of material in compartmen t i . If i ∈ In, let b i ( µ µ µ ) u i be the r ate at which the experimenter inp uts ma ter ial into the i -th compartm ent, where b i ∈ C ( µ µ µ ) \ { 0 } . If i ∈ Out, let y i = c i ( µ µ µ ) x i , where c i ∈ C ( µ µ µ ) \ { 0 } . Without loss of generality , we assume Out = { 1 , . . . , m } . Now the system o f equ ations governing the dynam ic s of x 1 , . . . , x n is gi ven by Σ = ( x ′ = A ( G ) x + u , y i = c i ( µ µ µ ) x i , for e very i ∈ Out , (7) where x = ( x 1 , . . . , x n ) T , u is the n × 1 matrix whose i -th entry is b i ( µ µ µ ) u i if i ∈ I n an d 0 otherwise, and A ( G ) is the matrix (genera lizin g the Laplacian of the graph) define d by A ( G ) i j =              − a 0 i − ∑ k : i → k ∈ E a ki , i = j , i ∈ L eak − ∑ k : i → k ∈ E a ki , i = j , i 6∈ Leak a i j , j → i ∈ E 0 , otherwise . (8) In the notation o f ( 19 ), we h av e x = { x 1 , . . . , x n } , y = { y 1 , . . . , y m } , u = { u i | i ∈ In } . Definition 4: A system Σ is ca lled a linear co m p artment model if the r e exists a simple dire cted gr a ph G with edges E and vertices V , subsets In, Out, an d Leak of V , and fu nctions b i , c j ∈ C ( µ µ µ ) \ { 0 } such that Σ has the form of ( 7 ). It was o bserved in [14, Th e o rem 2] that, for a linear compartm ent model, one ca n ob tain r elations am ong in puts, outputs, and p arameters as follows. Let ∂ be the differentiation operator . Let M ji ( G ) be the submatrix of ∂ I − A ( G ) obtained by d eleting the j -th row and i -th column. T hen [14 , Th e o- rem 2] y ields that sy stem ( 7 ) implies tha t fo r e very i ∈ Out, det ( ∂ I − A )( y i ) − 1 c i ( µ µ µ ) ∑ j ∈ In ( − 1 ) i + j det ( M ji )( b j ( µ µ µ ) u j ) = 0 . (9) [15, Theorem 3.8] gives a refined version of ( 9 ) coinciding with ( 9 ) for the cases we co nsider in o u r main r e sults. Definition 5 (Reachability): W e say verte x v is r eachable fr om vertex w or one ca n reach vertex v fr o m verte x w if there exists a dire c ted path from w to v . For example, in the grap h 1 → 2, vertex 2 is reachable fro m vertex 1 . W e say a leak (resp. inpu t) is reach a ble from w if there exists a vertex v in Leak (resp. In) such tha t v is reac h able fr om w . Example 1: Con sid er the gra p h 1 2 u a 12 a 21 a 01 Here G is the grap h given by V = { 1 , 2 } and E = { 1 → 2 , 2 → 1 } . The arr ow leaving compar tment 1 indicates th at Leak = { 1 } , the arrow entering compa rtment 1 indicates that In = { 1 } , and the o ther de c o ration to comp artment 2 indicates that Out = { 2 } . Note that the input and leak arrows , as well as the output decoration , are not considered part of the graph. One can see that the correspondin g system of d ifferential equation s coincides with ( 1 ) and can be written as  x 1 x 2  ′ =  − ( a 01 + a 21 ) a 12 a 21 − a 12   x 1 x 2  +  u 0  , y = x 2 . One can see that this sy stem satisfies the conditio ns of Theorem s 1 , 2 , and 3 . A direct computation shows that the input-o u tput equation ( 2 ) is a special case of ( 9 ). C. Existing al g ori thms used in practice yet to be j ustified In this section, we will present an d justify (r e phrasing our Main Results 1 , 2 , an d 3 ) th e co rrectness o f two versions (Algorithm s II.1 an d II.2 ) of th e algorithm ou tlines in Sec- tion I-A that were no t pr eviously fully ju stified. Algor ith m II.1 is on e of the key compo nents of, e.g. , D AISY [17 ], and Algorithm II.2 summarizes th e appr o ach fr om [15, Defin i- tion 3. 9 ]. Our ju stifications are b ased on the assum ptions stated O VCHINNI K O V , P OGUDIN, AN D TH OMPSON: I NPUT -OUTPUT EQUA TIONS AND I DENTIFIABI LITY OF LIN EAR ODE MODE LS 5 in Corollar ies 1 and 2 . Omitting so m e of the assumption s cou ld lead to incorrect co nclusions, as we show in Sectio n IV -A . Algorithm II.1 Comp uting id entifiable f u nctions Input System Σ as in ( 19 ) Output Generator s of th e field of identifiab le functions of Σ (see Corollary 1 ) (Step 1 ) Comp ute a full set C of in put-ou tput equ ations of Σ . (Step 2 ) Return the co efficients of C considered a s differential polyno mials in y a n d u . Cor ollary 1: Assume that Σ satisfies one of th e following: (1) Σ is as in ( 5 ) and has exactly o ne output; (2) Σ is a linear co mpartmen t model such that one can reach a leak o r an input fro m e very vertex. Then Algorithm II.1 will produce a cor rect result fo r Σ . Pr oof: Algorithm I I.1 will com pute generators of the field of IO-iden tifiable functions. Main Results 1 and 2 imply that, for Σ that we consider, th e field of IO-identifiab le function s coincides with the field of iden tifiab le fun c tions. Algorithm II.2 Comp uting id entifiable f u nctions Input System Σ as in ( 19 ) correspond ing to a linear compart- ment model with gr aph G Output Generator s of th e field of identifiab le functions of Σ (see Corollary 2 ) (Step 1 ) For every i ∈ Out, com p ute an input-ou tput equation p i as in ( 9 ) (or a refined version f rom [ 15, Theorem 3.8]). (Step 2 ) Return the co efficients of { p i | i ∈ Out } co nsidered as differential polyno mials in y and u . Cor ollary 2: In the notation o f Alg orithm II.2 , if graph G is stro ngly con nected and h as at least on e inpu t, then Algorithm II.2 will produce a cor rect result. Pr oof: Follows fro m Main Result 3 . I I I . M A I N R E S U L T S In this section, we will state ou r main results in a conden sed form. For the d e tailed statements, see th e cor respond ing theorems in Section V . In Section II- C , we show ho w our main r esults app ly to justify ing an algorithm com puting all identifiable fu nctions of an ODE model. In Section IV , we present examples (both of applied an d o f p urely mathematical nature) illu strating the importanc e and use of th e co nditions in th e statemen ts of our main results. No te that, while th e first result is restricted to MISO systems, the second and third ar e applicable to MIMO system s as well. Main Result 1 (see Theo r em 1 ): If system Σ as in ( 5 ) has exactly one outpu t, then IO-ide n tifiable functions coinc id e with identifiable f unctions. Main Result 2 (see Theo r em 2 ): If the gra p h of a line a r compartm ent model is such th at one can reach a leak or an input fro m every vertex, then IO-identifiab le function s coincide with identifiable functions. Pr oblem 1 : W ill Main Result 2 remain true if th e cond ition on the graph is rem oved or r elaxed? In other words, Main Results 1 an d 2 provide classes o f models for which the approach via input-outp ut equ a tions outlined in the intr oduction g iv es the correct resu lt. Main Result 3 (see Theo r em 3 ): For a linear compartme n t model with at least one input an d whose gr a ph is stron gly connected , th e field of all id entifiable fu nctions is gen erated by the co efficients of equations ( 9 ). This th eorem combin ed with Lemma 5 yields: Cor ollary 3: For a linear co mpartmen t mo del satisfying the assumptions of Main Result 3 , the field of all identifiable function s is gener ated by the coefficients o f the entries of th e transfer function m atrix (see Section C in Appendix). I V . E X A M P L E S A. How iden tifiabili ty methods could make mistakes In this section, we will consider several examples to illus- trate how the me thods based on IO-equ ations, formula ( 17 ), and tran sfer function s may lead to in correct conc lu sions ab out identifiability . Th is is to make th e reader mor e aw are of the condition s in our main r esults. 1) F ailure to detect non-iden tifiability with multiple outpu ts using IO- equations : W e will d iscuss a simp le example o f a linear system such that the classical metho d of IO- e quations will no t be able to decide on the (non- ) identifiability of 2 o f the 3 parameters. The example will also show that the co ndition of having only one output cannot be rem oved from o ur Main Result 1 . W e begin with an ODE for radioactive decay x ′ = − ax , with a being an unk nown decay r ate. Supp ose now that we hav e an unknown co n stant in flow b , and so x ′ = − ax + b . Co n sider the following outpu t (e.g. , th e ra d iation level): y = cx , in wh ich the unknown paramete r c represents the pro perties of the medium between th e observer an d the rad ioactive species. Suppose now that th e re is a known fixed outflow w (e.g., throug h a h ole o f fixed size), and so the ODE model beco mes      x ′ = − ax + b − w w ′ = 0 y 1 = cx , y 2 = w (10) W e then h av e y ′ 1 = cx ′ = − cax + cb − cw = − ay 1 + cb − cy 2 y ′ 2 = w ′ = 0 , (11) which can be sho wn to b e a full set of input-ou tput equation s. T o check the solv ability condition, consider system ( 3 ):      y ′ 1 ( t 1 ) = − ay 1 ( t 1 ) + cb − cy 2 ( t 1 ) y ′ 1 ( t 2 ) = − ay 1 ( t 2 ) + cb − cy 2 ( t 2 ) y ′ 1 ( t 3 ) = − ay 1 ( t 3 ) + cb − cy 2 ( t 3 ) , (12) which we consider as a linear system in a , cb , and c . The matrix of the system is A : =   − y 1 ( t 1 ) 1 − y 2 ( t 1 ) − y 1 ( t 2 ) 1 − y 2 ( t 2 ) − y 1 ( t 3 ) 1 − y 2 ( t 3 )   Since y 2 is a con stan t, the second and th ir d co lumns of the matrix are p ropor tional. T h erefor e , system ( 12 ) has in fin itely 6 IEEE TRANSACTIONS ON A UTOMA TIC CON TROL many so lutions for the corre sponding c o efficients of the inp ut- output equ ations, cb and b . Hence, the matrix is rank -deficient and the solvability condition is not satisfied. Therefore, pro- ceeding further with try ing to check the identifiab ility of b an d c based just on ( 12 ) co uld (a n d will for th is example, as we will see) cause an incorrect conclusion as the validity of this method is cu r rently guar anteed und er the solvability condition . For example, the software D AISY (which is based on input- output equations) ap plied to this mo del con cludes that all of a , b , and c are globally id entifiable. Ho wever , neither b , nor c is e ven lo cally identifiable. This can b e seen, e.g., by noticin g that the following is an output-p reserving transformation of system ( 10 ) f or all non-zero k : x → kx , c → c k , b → kb + w − kw . Therefo re, Assumption (A) is n o t satisfied. F or an analogous example witho ut constant states, see [5, Remark 2.15] . 2) The tra nsfer function method and ge neric initial c onditions (see also [40, Example 10.6]) : Consider the linear com partment model u 1 3 2 a 31 a 32 in which an input function u is applied to comp artment 1, the quantity in compartme n t 1 is measured, an d material flo ws from comp artment 1 to compartment 3 and fr om compar tment 2 to compartment 3. T h e c orrespon ding system is   x 1 x 2 x 3   ′ =   − a 31 0 0 0 − a 32 0 a 31 a 32 0     x 1 x 2 x 3   +   u 0 0   y 1 = x 1 . (13) Since the system satisfies the hypothesis of Theorem 1 , we can find the field of identifiable function s, C ( a 31 ) , by looking at the input-ou tput equation: y ′ 1 + a 31 y 1 − u = 0 . The transfer functio n (see Section C in Appen dix) for th e system is 1 s + a 31 , so the transfer function metho d gives the same correct result although the initial cond itions are not zero but generic and the assumptions of Theorem 3 are no t satisfied. Howe ver , using transfer fu nction will lead to err o neous r esults for this mo d el if we move the outpu t to com partment 3 (that is, replace y 1 = x 1 with y 1 = x 3 ). In this case, the tran sfer function is a 31 s ( s + a 31 ) , in dicating that a 32 is not identifiable. Howe ver , it actually is identifiable. This can be sho wn again by using Algorithm 1 (as the assump tion of Theo rem 1 is still satisfied), that is, considering th e fo llowing input-o utput equation : y ′′′ 1 + ( a 31 + a 32 ) y ′′ 1 + a 31 a 32 y ′ 1 − a 31 a 32 u − a 31 u ′ . Thus, the hyp otheses of Corollary 3 cann o t be omitted. Note that Alg orithm I I.2 will give a c orrect re su lt for this case even though the assumptions of Th eorem 3 are no t satisfied. B. P ositiv e examples for app lying our the or y Below we give examples from the literature satisfying at least one of the sufficient conditions f rom o ur main results. 1) Kinetics of lead i n humans and our results for one output. : T he following system of equatio ns is used in [42 , Section 4A] to model the kinetics of lead in the human body:          x ′ 1 = k 1 x 1 + k 2 x 2 + k 3 x 3 + k 4 x ′ 2 = k 5 x 1 + k 6 x 2 x ′ 3 = k 7 x 1 − k 3 x 3 y 1 = x 1 A full set of input-ou tput eq u ations is uniq ue in this case an d consists of a sing le d ifferential polyn omial: y ′′′ 1 − ( k 1 + k 3 + k 6 ) y ′′ 1 + ( − k 1 k 3 + k 1 k 6 − k 2 k 5 − k 3 k 6 − k 3 k 7 ) y ′ 1 + ( k 1 k 3 k 6 − k 2 k 3 k 5 + k 3 k 6 k 7 ) y 1 + k 3 k 4 k 6 . By Cor o llary 1 (con dition (1) ), the field of id entifiable fu nc- tions is generated by k 1 + k 3 + k 6 , − k 1 k 3 + k 1 k 6 − k 2 k 5 − k 3 k 6 − k 3 k 7 , k 3 ( k 1 k 6 − k 2 k 5 + k 6 k 7 ) , k 3 k 4 k 6 . In other words, these parameter combinations are identifiable, and m o reover any other iden tifiable comb ination of pa r ameters can be written as a ration al co mbination o f these. 2) Hepatobiliary kinetics of bro mosulfoph thalein : Th e following linear compartmen t model is taken from [43 , Section 6.3]:                    x ′ 1 = − k 31 x 1 + k 13 x 3 + u x ′ 2 = − k 42 x 2 + k 24 x 4 x ′ 3 = k 31 x 1 − ( k 03 + k 13 + k 43 ) x 3 x ′ 4 = k 42 x 2 + k 43 x 3 − ( k 04 + k 24 ) x 4 y 1 = x 1 , y 2 = x 2 . A fu ll set of IO-eq uations is too large to display here but th eir coefficients are k 13 , k 31 , k 04 k 42 , k 24 k 43 , k 03 + k 43 , k 04 + k 24 + k 42 . (14) Hence, by Corollary 1 , th e field of identifiab le fun ctions is generated by ( 14 ). This refine s the analysis p erforme d in [44, Example 3], where it was shown that ( 1 4 ) generate the field of IO-identifiab le functio ns ( as we h av e seen, fo r some examples, there are I O-identifiable functions th at are n ot identifiable). 3) Cyclic model : The following model can be obtained f r om [35, Mod el M] by ad ding extra leak s and an o utput for a b etter illustration of the com putation in co nnection to o u r results:          x ′ 1 = a 13 x 3 − a 21 x 1 − a 01 x 1 x ′ 2 = a 21 x 1 − a 32 x 2 − a 02 x 2 + u x ′ 3 = a 32 x 2 − a 13 x 3 y 1 = x 1 , y 2 = x 2 Using the co efficients of a full set o f IO-equatio n s or of th e transfer fun c tion matrix, we obtain by Co r ollaries 1 an d 3 th at the field of iden tifiable functions is gen e rated b y : a 21 , ( a 01 + a 21 ) a 13 , a 01 + a 13 , a 13 a 32 , a 02 + a 32 . O VCHINNI K O V , P OGUDIN, AN D TH OMPSON: I NPUT -OUTPUT EQUA TIONS AND I DENTIFIABI LITY OF LIN EAR ODE MODE LS 7 V . P R O O F S A. “Id entifiabi lity ⇐ ⇒ IO-identifiab i lity ” for l inear systems with one o utput (proof of The o rem 1 ) In this sectio n, we p rove one of the main results, Theor em 1 , which shows that, fo r a linear system with one outp ut, IO- identifiability an d identifiability are equiv alent. W e begin with showing a preliminar y result. Lemma 1: Let K be a field. Conside r • th e differential p olynom ial ring K { y , u } with deriv ation ∂ satisfying ∂ ( K ) = 0, • P ∈ K { y , u } of the for m P = D P ( y ) + U P , wh ere D P ∈ K [ ∂ ] is a linear differential op erator over K with lea ding co efficient 1 and U P ∈ K { u } . Let W be the Wronskian of all the mono mials of P except for the one o f the highest o rder with r espect to y . The n W / ∈ [ P ] . Pr oof: Since the coefficients of P and W ar e in K , the membersh ip W ∈ [ P ] would be the same con sid e red over K or its algebraic closure. Hence, replacing K with its algebraic closure if necessary , we assume tha t K is alg ebraically closed. Consider a le xicog raphic mon omial ordering in duced b y an orderin g of the variables su c h th at y ( i + 1 ) > y ( i ) for e very i > 0 an d y is greater than any deriv ati ve of u . Since for all r P , P ′ , . . . , P ( r ) is a Gr ¨ obner basis for [ P ] ∩ K [ y , y ′ , . . . , y ( r ) , u , u ′ , . . . , u ( r ) ] , it follows from [45, Lemma 1.5] that P , P ′ , . . . form a Gr ¨ obner basis of [ P ] with r espect to this ordering as define d by [4 5, Definition 1.4]. Since the leading terms of a Gr ¨ obner basis are linear, [ P ] is a prime idea l. Th us, we can introduce L : = Fra c ( K { y , u } / [ P ]) . Denote the field of co n stants o f L by C ( L ) a nd the imag e s of y and u in L by ¯ y and ¯ u , respectively . Since none of der iv ati ves of u appea r in th e leading terms o f the Gr ¨ ob ner b a sis, ¯ u and their deriv ati ves are algeb raically indep endent over K . Assume that the statement of the lemma is not tr ue. D u e to [46, The o rem 3.7, p. 21], this implies that the images in L of the monomia ls of P except fo r the o ne of the highest order in y are lin early dependen t over C ( L ) . Therefo re, there exists a nonzero p olynom ial Q = D Q ( y ) + U Q , where D Q ∈ C ( L )[ ∂ ] is monic and U Q ∈ C ( L ) { u } , such that Q ( y , u ) = 0 and ord D Q < ord D P . Let D 0 be the gcd of D P and D Q with the leading coefficient 1. Then ord D 0 < ord D P . If F is an alg ebraically closed field, p ∈ F [ X ] , and p is divisible by a q ∈ E [ X ] with the leading coefficient 1, where E is an extension of F , then q ∈ F [ X ] . Hence , as D 0 divides D P and K is alg ebraically closed, D 0 ∈ K [ ∂ ] a nd ther e is D 1 ∈ K [ ∂ ] such that D P = D 1 D 0 . Ther e also ar e A , B ∈ C ( L )[ ∂ ] such that D 0 = AD P + BD Q . Consider R : = A ( P ) + B ( Q ) = D 0 ( y ) + U R , where U R = A ( U P ) + B ( U Q ) . Th e n R ( y , u ) = 0 . Since P − D 1 ( R ) ∈ C ( L ) { u } vanishes on u and u is differentially independen t over C ( L ) , it follows that P = D 1 ( R ) . Considering a basis o f C ( L ) over K , we can write U R = U 0 + e 1 U 1 + . . . + e N U N , where U 0 , . . . , U N ∈ K { u } and 1 , e 1 , e 2 , . . . , e N ∈ C ( L ) are lin- early indep endent over K . Since D 1 ( U R ) = U P and D 1 ∈ K [ ∂ ] , U 1 , . . . , U N ∈ ker D 1 , where we consider D 1 as a f unction from C ( L ) { y , u } to C ( L ) { y , u } . Th ere are two cases: • D 1 is not divisible by ∂ . Then ker D 1 = { 0 } . Hence, U 1 = . . . = U N = 0 . • D 1 is divisible by ∂ . Then ker D 1 = C ( L ) . T hus, U 1 , . . . , U N ∈ K . Howe ver , since U P = D 1 ( U R ) , U P does not contain a term in K . Hence, U Q does not co ntain a term in C ( L ) and, consequen tly , U R does not contain a ter m in C ( L ) . T hus, U 1 = . . . = U N = 0 . In both cases, we have sho wn that U R ∈ K { u } . Thus, R ∈ K { y , u } and R ∈ [ P ] . But this is impossible becau se P , P ′ , P ′′ , . . . is a Gr ¨ o bner basis of [ P ] with respect to the mon omial order ing introdu c ed in the beginnin g of the p r oof, and ord D 0 < o rd D P , so R is n ot reducible with re spect to this ba sis. Theor em 1 (Ma in Result 1 ): For every Σ as in ( 5 ) with m = 1 (that is, sing le o utput), for all h ∈ C ( µ µ µ ) , h is iden tifiable ⇐ ⇒ h is I O-identifiable. Pr oof: [27 , T heorem 4 . 2] implies tha t iden tifiable func- tions are always IO-iden tifiable, so it rem ains to show the reverse inclusion. Con sider a full set of input-o utput equa- tions for Σ . Since m = 1, it will con sist of a single linea r differential polyn omial p ∈ C ( µ µ µ ) { y , u } . Then , Lem ma 1 an d [27, Lemma 4 .6] imply tha t its coefficients are identifiable, so the re verse inclusion ho lds as well. B. Sufficient con dition for “iden t ifiabili ty ⇐ ⇒ IO-id entifiab i lity ” for line a r compar tmen t model s (proof of The o rem 2 ) For the notation , see Section II-B . Lemma 2: Let F = Frac ( C ( µ µ µ ) { x , y , u } / I Σ ) . The field of constants o f F lies in the subfield of F generated by C , µ µ µ and x . Pr oof: Observe that F as a field is generated b y µ µ µ , x , and all the deri vati ves of u , and a ll these elements ar e algebraically indep endent. Assume th at there exists ℓ > 0 and h ∈ C ( µ µ µ , x , u , . . . , u ( ℓ ) ) such tha t h ′ = 0 and, witho ut loss of generality , ∂ ∂ u ( ℓ ) κ h 6 = 0. Then we have h ′ = ℓ ∑ i = 0 κ ∑ r = 1 u ( i + 1 ) r ∂ ∂ u ( i ) r h + n ∑ j = 1 x ′ j ∂ ∂ x j h = u ( ℓ + 1 ) κ ∂ ∂ u ( ℓ ) κ h + a , a ∈ C ( µ µ µ , x , u , . . . , u ( ℓ ) , u ( ℓ + 1 ) 1 , . . . , u ( ℓ + 1 ) κ − 1 ) . Now h ′ = 0 yields a co ntradiction since u ( ℓ + 1 ) κ is tran scen dental over C ( µ µ µ , x , u , . . . , u ( ℓ ) , u ( ℓ + 1 ) 1 , . . . , u ( ℓ + 1 ) κ − 1 ) and ∂ ∂ u ( ℓ ) κ h 6 = 0. Lemma 3: Con sider a g r aph G such that, fr om every vertex, at least o ne leak can be reached . Th en the eigenvalues of A ( G ) are distinct and a lg ebraically independe n t ov er Q . Pr oof: Let H be a direc ted span ning f o rest of G con- structed b y a breadth- fir st search (depth-first search would work as well) with the set Leak as the source such that, from ev ery vertex, there is a path to some element of Leak. Relabeling vertices if necessary , A ( H ) is upper triangu lar with algebraically indepen dent diagonal entries. It is well known 8 IEEE TRANSACTIONS ON A UTOMA TIC CON TROL that a br eadth-first searc h on a g r aph will constru ct a spanning forest containin g all vertices r eachable from the so urce set (cf. [47, Section 22.2]). W e illustrate our procedure with an example. Let G be the graph shown below , with Leak = { 1 , 6 } : 1 2 3 4 5 6 The steps of a breadth-first search with sou rce set { 1 , 6 } are the first three upper left, u pper right, and lower left graphs shown b elow . The fou r th lo wer right gr aph is a relabeling of the third as described ab ove. 1 6 1 2 3 5 6 1 2 3 4 5 6 1 2 4 6 5 3 T aking H to be the four th gra p h, we have A ( H ) =         − a 01 a 12 − a 12 − a 03 a 34 a 35 − a 34 − a 35 a 56 − a 56         . Since the diag onal entries are a lg ebraically indepen dent ov er Q and alg ebraic over the field extension of Q gener a te d by the co efficients of the characteristic po ly nomial of A ( H ) , it follows that the coefficients of th e char acteristic po lynomia l of A ( H ) are algebraically in depend ent over Q . For all i , j , if th e co efficients of the char a c teristic p o ly- nomial of A ( G ) | a i , j = 0 are algebraically independen t, then the coefficients of the chara c teristic po ly nomial o f A ( G ) are al- gebraically in depend ent. Since A ( H ) can be obtaine d fro m A ( G ) by setting equal to 0 those a i , j such that H h as no edge from j to i , it follows that the coefficients of the ch aracteristic polyno mial o f A ( G ) are n on-zero and algebraically ind epen- dent. Since these n coefficients belo ng to the field extension of Q genera te d by n eig en values, the eig en values must be algebraically independ ent as well. Theor em 2 (Main Result 2 ) : L et Σ b e a linear compa r tment model with gr aph G such th at, from every vertex o f G , at least one leak o r inpu t is reacha b le. Then the fields of identifiab le and IO-identifiable functions c oincide. Pr oof: Let K : = Frac ( C ( µ µ µ ) { x , y , u } / I Σ ) . W e will show that Σ d oes not have a rational first in tegral, that is C ( K ) = C ( µ µ µ ) . Then the theo r em will follow fr o m [27, Theorem 4.7]. Consider a mo del Σ ∗ with a gr aph G ∗ obtained fro m G by replacing every inpu t with a leak (if there was a vertex with an inpu t an d a leak , we simply r emove the inpu t). The theore m will follow from the following two claims. Claim: I f Σ has a rational fi rst in te gral, then Σ ∗ also do es. Consider a first integral of Σ , that is, an elemen t of C ( K ) \ C ( µ µ µ ) . Lemm a 2 implies that th ere exists R ∈ C ( µ µ µ , x ) \ C such that c is th e im age of R in K . Since C [ µ µ µ , x ] { u } ∩ I Σ = 0 due to [5, Lemma 3 .1] and the image of R in K is a con stan t, the Lie deri vati ve of R with respect to Σ , L Σ ( R ) : = n ∑ i = 1 ∂ R ∂ x i f i , where f 1 , . . . , f n are as in ( 19 ), is zer o. If there exists i ∈ In such that x i appears in R , then L Σ ( R ) will be o f the form L Σ ( R ) = ∂ R ∂ x i b i ( µ µ µ ) u i + ( something not in volving u i ) 6 = 0 . Thus, R does not inv olve a ny x i with i ∈ I n. Then, due to the con struction of G ∗ , L Σ ∗ ( R ) = L Σ ( R ) = 0, so Σ ∗ also has a rational fir st integral. Claim: Σ ∗ does not h ave rational first inte grals. Lemma 3 implies th at the eige n values of A ( G ∗ ) are algebraically inde- penden t. Then [48 , Theor e m 10.1.2, p. 118] implies that Σ ∗ does not h av e rational first in tegrals. C. Using more convenient IO-equa tions (proo f of The orem 3 ) For the notation , see Section II-B . Lemma 4: Let K b e a field. For all a , b , c ∈ K [ x ] such that gcd ( a , b ) = 1, th e r e exists at most one p air ( p , q ) of elements of K [ x ] such that a p + b q = c and deg p < deg b . Pr oof: Suppose ( p , q ) and ( p 1 , q 1 ) are distinct pairs satisfying the two pro perties above. I t fo llows that a ( p − p 1 ) + b ( q − q 1 ) = 0 . (15) Since ( p , q ) 6 = ( p 1 , q 1 ) , ( 15 ) im plies that p 6 = p 1 . Since deg ( p − p 1 ) < deg b , ( 15 ) im plies gcd ( a , b ) 6 = 1, contradicting o ur hypothesis. Cor ollary 4: Let K be a field con taining C and a , b , c ∈ K [ x ] with g cd ( a , b ) = 1. If th ere is a pair of polynomials ( p , q ) with a p + b q = c and deg p < deg b , then the co efficients of p and q belon g to the field extension of C generated b y the coe fficients of a , b , and c . Pr oof: Suppo se som e co efficient of p o r q do es not belong to the field generated by the co efficients of a , b , and c . By [49 , Theorem 9.29 , p. 11 7], there is a field automorp hism σ of K that fixes the field extension of C gene r ated b y th e coefficients of a , b , a n d c and m oves this coefficient. W e extend σ to K [ x ] by σ ( x ) = x . Applying σ to both sides of a p + bq = c gives us a σ ( p ) + b σ ( q ) = c . Using K for K in Lemma 4 , we arrive at a con tradiction. Theor em 3 (Ma in Result 3 ): Let Σ b e a linear compa r tment model with a graph G . Le t A = A ( G ) and M ji be the su bmatrix of ∂ I − A o b tained b y deletin g th e j -th row and the i -th column of ∂ I − A . Recall that (see ( 9 )), f or every solution of Σ , we have for e very i ∈ Out, det ( ∂ I − A )( y i ) = 1 c i ( µ µ µ ) ∑ j ∈ In ( − 1 ) i + j det ( M ji )( b j ( µ µ µ ) u j ) . If G is strongly connected and has at least one input, then the coefficients of these d ifferential polyno mials with respect to y ’ s and u ’ s g enerate the field of ide ntifiable fu nctions o f Σ . O VCHINNI K O V , P OGUDIN, AN D TH OMPSON: I NPUT -OUTPUT EQUA TIONS AND I DENTIFIABI LITY OF LIN EAR ODE MODE LS 9 Pr oof: W ithout loss of generality , assume Out = { 1 , . . . , m } . W e set, for i = 1 , . . . , m , h i : = det ( ∂ I − A )( y i ) − 1 c i ( µ µ µ ) ∑ j ∈ In ( − 1 ) i + j det ( M ji ) b j ( µ µ µ ) u j . (16) Let also D = det ( ∂ I − A ) and, for i = 1 , . . . , m , let Q i be th e 1 × n ma trix of o p erators d efined b y ( ( Q i ) j = ( − 1 ) i + j c i ( µ µ µ ) det ( M ji ) b j ( µ µ µ ) , j ∈ In , ( Q i ) j = 0 , j / ∈ In . (17) Observe that, f or i = 1 , . . . , m h i = D ( y i ) − Q i · u , where u is the n × 1 matrix defin ed by u j = u j if j ∈ In and u j = 0 otherwise. First we show that th e coefficients of h 1 , . . . , h m are IO- identifiable. Fix i . Consider an ordering of the outputs such that y i is the sm a llest one. Let p 1 , . . . , p m be a full set o f in p ut- output equation s with respect to th is orderin g (see Definition 2 ) which exists du e to Proposition 1 . Then p 1 is of the f o rm E ( y i ) + B · u , where E is a linear differential operator and B is a 1 × n matrix of linear differential o perators, both with coef ficients in C ( µ µ µ ) . Since h i ∈ I Σ and h i in volves only y i and u , the second p a r t of Proposition 1 implies that h i ∈ [ p 1 ] , so the r e exists a differential operator D 0 ∈ C ( µ µ µ )[ ∂ ] such th at h i = D 0 p 1 . Sinc e G is strong ly connected and has an in put, b y [15, Proposition 3 .19], gcd ( D ∪ { ( Q i ) j | ( Q i ) j 6 = 0 } ) = 1 . Thus D 0 has o rder zero, so h i and p 1 are pro portion al. Therefo re, the coefficients of ( 16 ) are IO-identifiable. Next, we show that th e field ge n erated b y the c o efficients of h 1 , . . . , h m contains the field o f IO-identifiable fun c tions. Fix an ordering on the o utputs y m > . . . > y 1 . W e will show that the full set p 1 , . . . , p m of input-o utput eq uations with respect to this ordering satisfies: ord y 1 p 1 = n , ord y i p i = 0 for ev ery 2 6 i 6 m . (18) The fact that or d y 1 p 1 = n is im plied by the previous paragr aph. From ( 5 ), we see th at th e transcenden ce degree of C ( µ µ µ ) { x , y , u } / I Σ over C ( µ µ µ ) { u } is eq ual to n , so the tran scendence d egree of C ( µ µ µ ) { y , u } / ( I Σ ∩ C ( µ µ µ ) { y , u } ) over C ( µ µ µ ) { u } is less than or equ al to n . From th e for m of p 1 , we have that y 1 , y ′ 1 , . . . , y ( n − 1 ) 1 are algeb raically in - depend ent over C ( µ µ µ ) { u } , so fo r i = 2 , . . . , m , the elements y i , y 1 , y ′ 1 , . . . , y ( n − 1 ) 1 must be alg ebraically d epende nt over C ( µ µ µ ) { u } . Hence, th e eq uation for y i has order 0 in y i . Thus, p 1 = D ( y 1 ) − Q 1 · u and, for e very , 2 6 i 6 m , w e can write p i = y i + D i ( y 1 ) + P i · u , where P i is a 1 × n matr ix of linea r differential o perators and the order of o perator D i is at m ost n − 1. W e show that the coefficients of p 1 , . . . , p m can be written in terms of the coefficients of h 1 , . . . , h m . Sinc e h 1 equals D ( y 1 ) − Q 1 · u , this is true for the coefficients o f D an d Q 1 . It r emains to show this for the coefficients of D 2 , . . . , D m and P 2 , . . . , P m . Note that for all i an d for j 6∈ In we have ( P i ) j = 0 , so we need only ad dress the coefficients of ( P i ) j for j ∈ In. Fix i > 1 and let g = y i + D i ( y 1 ) + P i ( u ) . W e hav e that D ( g ) − D i ( h 1 ) = D ( y i ) + ( DP i + D i Q 1 )( u ) ∈ I Σ . It follows that D ( y i ) + ( DP i + D i Q 1 )( u ) = h i , so , f or all j , D ( P i ) j + D i ( Q 1 ) j = − ( Q i ) j . By the hypoth esis of the theorem , In 6 = ∅ . Fix j ∈ In. W e apply [15, Propo sition 3.1 9] to the model obtained from Σ b y deleting all the inputs except for j and obtain, using D 6 = 1, that gcd ( D , ( Q 1 ) j ) = 1 f or ev ery j ∈ I n. By Coro llary 4 , the coefficients of ( P i ) j and D i belong to the field extension of C generated by the coefficients o f D , ( Q 1 ) j , and ( Q i ) j . W e showed that the field extension of C gener a te d b y the coef- ficients of h 1 , . . . , h m is the field of IO-identifiab le fu nctions. By Theorem 2 , this is the field of iden tifiable fu nctions. A P P E N D I X A. Gen eral defini tion of iden t ifiabili ty In th is section , we will generalize the notions fr o m Sec- tion II- A to ODE systems with rational rig h t-hand side. Fix positive in teger s λ , n , m , an d κ for th e remainder of the ap- pendix. Let µ µ µ = ( µ 1 , . . . , µ λ ) , x = ( x 1 , . . . , x n ) , y = ( y 1 , . . . , y m ) , and u = ( u 1 , . . . , u κ ) . Consider a system o f ODEs Σ =            x ′ = f ( x , µ µ µ , u ) Q ( x , µ µ µ , u ) , y = g ( x , µ µ µ , u ) Q ( x , µ µ µ , u ) , x ( 0 ) = x ∗ , (19) where f = ( f 1 , . . . , f n ) and g = ( g 1 , . . . , g m ) are tuples o f elements of C [ µ µ µ , x , u ] and Q ∈ C [ µ µ µ , x , u ] \{ 0 } . Notation 3 ( Ideal I Σ ): (a) For an id eal I and element a in a ring R , we denote I : a ∞ = { r ∈ R | ∃ ℓ : a ℓ r ∈ I } . This set is also an idea l in R . (b) Given Σ as in ( 19 ), we defin e the differential ideal of Σ : I Σ = [ Q x ′ − f , Q y − g ] : Q ∞ ⊂ C ( µ µ µ ) { x , y , u } . For the case of a linear system as in ( 5 ), this ideal coincides with th e one from No tation 2 . Notation 4 ( Auxiliary ana lytic n otation): (a) For every given h ∈ C ( x ∗ , µ µ µ ) , let Ω = { ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) ∈ C n × C λ × ( C ∞ ( 0 )) κ | Q ( ˆ x ∗ , ˆ µ µ µ , ˆ u ( 0 )) 6 = 0 } Ω h = Ω ∩ ( { ( ˆ x ∗ , ˆ µ µ µ ) ∈ C n + λ | h ( ˆ x ∗ , ˆ µ µ µ ) well-defined } × ( C ∞ ( 0 )) κ ) . (b) For ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) ∈ Ω , let X ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) and Y ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) denote the unique solution over C ∞ ( 0 ) of the instance of Σ with x ∗ = ˆ x ∗ , µ µ µ = ˆ µ µ µ , and u = ˆ u (see [41, Th eorem 2.2.2]). 10 IEEE TRANSACTIONS ON A UTOMA TIC CON TROL Definition 6 (Identifia b ility , see [5, Defin ition 2 . 5]): W e say that h ( x ∗ , µ µ µ ) ∈ C ( x ∗ , µ µ µ ) is identifiable if ∃ Θ ∈ τ ( C n × C λ ) ∃ U ∈ τ (( C ∞ ( 0 )) κ ) ∀ ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) ∈ ( Θ × U ) ∩ Ω h | S h ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) | = 1 , where S h ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) : = { h ( ˜ x ∗ , ˜ µ µ µ ) | ( ˜ x ∗ , ˜ µ µ µ , ˆ u ) ∈ Ω h Y ( ˆ x ∗ , ˆ µ µ µ , ˆ u ) = Y ( ˜ x ∗ , ˜ µ µ µ , ˆ u ) } . In this paper , we are interested in com paring identifiability and IO-iden tifiab ility (Definition 7 ), and the latter is defined for fun ctions in µ µ µ , no t in µ µ µ and x ∗ . Thu s, just for the purpose of comparison, we will restrict our selves to the field { h ∈ C ( µ µ µ ) | h is identifiable } , which we will ca ll the fie ld of identifiab le functio n s . Definition 7 (IO-identifia bility): The smallest field k such that C ⊂ k ⊂ C ( µ µ µ ) and I Σ ∩ C ( µ µ µ ) { y , u } is generated (as an ideal or as a differential ideal) by I Σ ∩ k { y , u } is called the field of IO-identifia b le functions . W e call h ∈ C ( µ µ µ ) IO-identifia ble if h ∈ k . B. Specializat i on to t he linea r case Pr oposition 1 : For ev ery system Σ o f the form ( 5 ): (1) f or every ordering of ou tput variables, there exists a unique fu ll set o f inpu t-outpu t eq u ations with respect to this ordering ; (2) if p 1 , . . . , p m is th e full set of input-outpu t equatio ns with respect to y 1 < . . . < y m , then the derivati ves of p 1 , . . . , p m form a Gr ¨ obn e r basis of I Σ ∩ C ( µ µ µ ) { y , u } with respect to any lexicog raphic mono mial ord ering such that • a ny deriv ativ e of any of y ’ s is greater any derivati ve of any o f u ’ s; • y ( j 1 ) i 1 > y ( j 2 ) i 2 iff i 1 > i 2 or i 1 = i 2 and j 1 > j 2 . An analo gous statement holds fo r any ord ering of ou tputs. (3) Defin itions 7 an d 3 defin e the same field. In particular, the field defined in Defin ition 3 d oes no t de p end on the choice of a f ull set of in put-ou tput equations. Pr oof: W e fix an or dering y 1 < . . . < y m of ou tputs. Assume th at th e re are full sets o f in put-ou tput eq uations p 1 , . . . , p m and q 1 , . . . , q m with respect to this ord e ring. L et ℓ be the smallest integer such that p ℓ 6 = q ℓ . By the defin itio n, ord y ℓ p ℓ = ord y ℓ q ℓ . Then ord y i ( p ℓ − q ℓ ) < ord y i p i for e very i 6 ℓ ; this con tradicts the d efinition of a f u ll set of in put-ou tput equations. T o finish the pr oof of p art (1) of th e p roposition , we will sho w th e existence of a full set of inp ut-outp ut equatio ns. Let J : = I Σ ∩ C ( µ µ µ ) { y , u } . Con sid e r the set of d ifferential polyno mials S : = { x ′ − f , x ′′ − f ′ , . . . , y − g , y ′ − g ′ , . . . } . By the definition of I Σ , S generates I Σ . Since these generato rs are linear, I Σ has a lin ear Gr ¨ obn er basis (see [45, Definition 1.4]) with respect to any mono mial o rdering . Since J is an elimination ideal of I Σ , it also has a linear Gr ¨ obner basis with respect to a ny mo nomial orde ring. Con sider any lexicograp hic monom ial orde r ing on C ( µ µ µ ) { x , y , u } such that • a ny d e riv ati ve o f a ny of y 1 , . . . , y m is greater than any deriv ative of any of x 1 , . . . , x n ; • any d e r iv ati ve of any of x 1 , . . . , x n is greater than a ny deriv ative of any of u 1 , . . . , u κ ; • for a = x , y , a ( j 1 ) i 1 > a ( j 2 ) i 2 iff i 1 > i 2 or i 1 = i 2 and j 1 > j 2 . Observe that S is a Gr ¨ o bner basis o f I Σ with respect to any such mo nomial ord e r ing. T h erefor e , u an d their der iv ati ves ar e algebraically ind e penden t mod ulo I Σ , and the tran scendence degree of C ( µ µ µ ) { x , y , u } over C ( µ µ µ ) { u } modulo I Σ is finite. Consider the restriction o f the ord ering d escribed above to C ( µ µ µ ) { y , u } . Consider the reduced Gr ¨ ob ner basis B o f J with respect to this orderin g. As we have sho wn, it is linear . Since the tr anscenden ce degree o f C ( µ µ µ ) { y , u } over C ( µ µ µ ) { u } modu lo J is finite, for every 1 6 i 6 m , there is a deriv ative of y i among the leading term s of B . Moreover, by differentiating the correspo n ding elemen t of B , we see that all higher deriv ativ es of y i will appear as leading term s of B . For each 1 6 i 6 m , w e set p i to be the element in B with the leading term b eing y ( j ) i such that j is the smallest po ssible. Then the fact that p 1 , . . . , p m are a part of th e r e d uced Gr ¨ obner basis implies that they form a full set o f in put-ou tput equation s with respect to the ord ering y 1 < y 2 < . . . < y m . This finish e s the proof o f part (1) o f the proposition. T o prove pa r t (2 ) o f th e p roposition , o bserve th at the deriv a- ti ves o f p 1 , . . . , p m form a Gr ¨ obn e r basis of [ p 1 , . . . , p m ] with respect to the described ordering. Thus, it remains to show that [ p 1 , . . . , p m ] = J . Assume that there is q ∈ J \ [ p 1 , . . . , p m ] . By red ucing it with respect to appr opriate d eriv ativ es of p 1 , . . . , p m , we can assume that ord y i q < ord y i p i for e very 1 6 i 6 m . But th is would imply that p 1 , . . . , p m is not a full set of inp ut-outp ut eq uations, proving part (2) of the prop osition. T o prove part (3) of the propo sition, note that, since a full set of input-ou tp ut equ a tions is a part o f a reduced Gr ¨ obner basis of J , its coefficients are in th e field of defin ition of J . On the other hand, since the set of all deriv ativ es o f p 1 , . . . , p m forms a Gr ¨ obn er basis of J and the coefficients of these d eriv ativ es are the same as th e co efficients of p 1 , . . . , p m , th e co efficients of p 1 , . . . , p m generate the field of d efinition o f J . C. Input-ou tput equation s based on th e Cramer’s rule and the transfer function matr ix Recall [40, p age 444] th at the tr ansfer fu nction matrix o f a linear system ( x ′ = A ( µ µ µ ) x + B ( µ µ µ ) u , y = C ( µ µ µ ) x . is defined by H ( µ µ µ , s ) : = C ( µ µ µ )( sI − A ( µ µ µ )) − 1 B ( µ µ µ ) , (20) where s is a new algebr a ic variable and I is the iden tity m atrix. This m atrix relate s the Lap lac e transfo r ms of y and u u n der the assumption that the initial co n ditions ar e zer o [40 , page 75]. The formulas ( 9 ) and ( 20 ) look similar and in fact a r e re la ted . W e gi ve a connection we ar e interested in as Lemma 5 below , and refer fo r further co nnection to a n u pcomin g paper [50]. For a ration al function f ∈ C ( µ µ µ )( s ) in s , by the coefficients of f , we will und erstand the union of the coefficients of the nu merator and den ominator in th e reduced form if th e denomin ator is taken to b e m o nic. O VCHINNI K O V , P OGUDIN, AN D TH OMPSON: I NPUT -OUTPUT EQUA TIONS AND I DENTIFIABI LITY OF LIN EAR ODE MODE LS 11 Lemma 5: Con sider a linear compa r tment model with at least one in put and who se graph is strong ly connecte d . Then the following sets generate the same subfield in C ( µ µ µ ) • c o efficients of the input-outpu t equations ( 9 ); • c o efficients of the entries of the tran sfer function matrix. Pr oof: Since each o f the equ a tions ( 9 ) inv olves only one ou tp ut and eac h row in th e matrix ( 20 ) corre sponds to an output, pr oving the lemma for th e sing le-outpu t case will yield the g eneral case by taking the u nion of the respective generato r s. W e write ( 9 ) as p ( ∂ ) y = q 1 ( ∂ ) u 1 + . . . + q r ( ∂ ) u r for nonzero p ( s ) , q 1 ( s ) , . . . , q r ( s ) ∈ C ( µ µ µ )[ s ] such that p ( s ) is monic. L e t F 1 be the field generated by the co efficients of p , q 1 , . . . , q r . Since the g raph is stro ngly c o nnected and h as an inp ut, [15, Proposition 3.19 ] implies that gcd ( p , q 1 , . . . , q r ) = 1. A direct computatio n shows that H ( µ µ µ ) defined b y ( 20 ) is equal to H ( µ µ µ ) : = ( h 1 ( s ) , h 2 ( s ) , . . . , h r ( s )) =  q 1 ( s ) p ( s ) , q 2 ( s ) p ( s ) , . . . , q r ( s ) p ( s )  . Let F 2 be the field generated by the coefficients o f h 1 , . . . , h r . For all in tegers n 1 , . . . , n r , th e coefficients of n 1 h 1 + . . . + n r h r belong to F 2 . Since q 1 , . . . , q r , p are copr ime, there exist integers n 1 , . . . , n r such that n 1 q 1 + . . . + n r q r is coprime with p , so p is the denominator of n 1 h 1 + . . . + n r h r . Henc e , the coefficients of p belo ng to F 2 . Let g i = gcd ( p , q i ) and p i = p / g i . By d efinition, th e coefficients of p i are in F 2 , so the coefficients of g i are in F 2 . By definition, the coefficients of q i / g i are in F 2 , so the co efficients o f q i are in F 2 . Th us, F 1 ⊂ F 2 . T o prove F 2 ⊂ F 1 , note that th e coefficients of the remainder and quotient of tw o po ly nomials belong to the field gener ated by the coefficients of the poly nomials. Since the n umerato r and denomin ator of h i are equ a l to q i / gcd ( p , q i ) and p / g cd ( p , q i ) , respectively , we have F 2 ⊂ F 1 , so F 1 = F 2 . A C K N OW L E D G M E N T S W e are gratefu l to the CCiS at CUNY Queens College for the computation al resources and to Julio Banga, Joseph DiStefano, Marisa Eisenberg, Nikki Meshkat, Maria Pia Sac- comani, Ann e Shiu, Seth Sulliv ant, Alejandr o V illa verde, and to referees f or useful discussions and sugg estions. R E F E R E N C E S [1] L. Ljung and T . Glad, “On global ide ntifiabili ty for arbitrary model parametri zations, ” Automatica , vol. 30, no. 2, pp. 265 –276, 1994. [Online]. 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