Minimality and identifiability of SARX systems Mih´ aly Petreczky and Laurent Bako and Stephane Lecoeuche Univ Lille Nord de France, F-59000 Lille, France, and EMDouai, IA, F-59500 Douai, France {mihaly.petreczky,laurent.bako,stephane.lecoeuche}@mines-douai.fr

Abstract: The paper address the problem of minimality and identifiability of Switched ARX (abbreviated by SARX) models. We propose a notion of identifiability and minimality for SARX models which depends only on the parameters of the model, not on data. We formulate conditions for minimality and identifiability of SARX systems. In particular, we show that SARX systems are generically identifiable. Keywords: switched ARX, minimization, identification, hybrid, identifiability 1. INTRODUCTION The paper deals with the problem of identifiability and minimality of switched ARX systems. Motivation. Switched ARX systems (abbreviated as SARX ) are popular in hybrid systems community, due to their simplicity and modelling power. In particular, most of hybrid system identification algorithms were developed for switched ARX systems. Despite their popularity, identifiability and minimality of switched ARX are not yet completely understood.

minimality and identifiability is much more direct for SARX systems than for linear switched systems. It is worth noting that a SARX system can be minimal or identifiable, even if none of the ARX subsystems is minimal (resp. identifiable). In fact, the relationship between identifiability and minimality of SARX systems and their ARX subsystems is not straightforward.

Minimality and identifiability are essential for analyzing algorithms for identification and adaptive control. Indeed, only identifiable parameterizations have the chance to be identified correctly by a parameter estimation algorithm. For this reason, identifiability is usually one of the conditions for correctness of parameter estimation algorithms. Minimality is closely related to identifiability: parameters which occur only in the non-minimal system component are not identifiable.

Related work. Identification of hybrid systems is an active research topic, see for example, Bako et al. (2009b,a); Vidal (2008); Verdult and Verhaegen (2004); Juloski et al. (2005b); Roll et al. (2004); Ferrari-Trecate et al. (2003); Nakada et al. (2005); Weiland et al. (2006); Paoletti et al. (2007b); Fliess et al. (2008) and the overview Juloski et al. (2005a); Paoletti et al. (2007a) on the topic. Many of the major contributions are formulated only for SARX systems, Vidal (2008); Ferrari-Trecate et al. (2003); Lauer et al. (2009); Paoletti et al. (2010). The relationship between SARX systems and state-space representations was addressed in Paoletti et al. (2010); Weiland et al. (2006), and in this paper we use some of those results.

Contribution of the paper. We propose a definition and conditions for minimality and identifiability of SARX systems. Minimality and identifiability depend only on the system parameters and not on the data generated by the system. The conditions for minimality and identifiability can be checked by numerical algorithms. We also show that minimality is closely related to identifiability. In addition, we show that SARX parameterizations are generically minimal and are generically identifiable.

To the best of our knowledge, the results of the paper are new. However, Vidal (2008) contains persistence of excitation conditions for SARX systems, which is related to identifiability. The main difference between the two concepts is that the former is a property of the data, while the latter is a property of the parameterization. Vidal (2008) also proposes a definition of minimality of SARX which implies our definition. However, the two definitions are not equivalent.

In order to prove these results, we convert SARX systems to a state-space form, using the regressors as a statevariable. We then analyze minimality and identifiability of the resulting state-space representation by applying Petreczky et al. (2010). However, the results of the paper are not trivial consequences of Petreczky et al. (2010). This is due to the rich structure of SARX systems which allows us to derive stronger results than for general linear switched systems. For example, the relationship between

The technical report ? represents an extended version of the current paper. Outline. In §2 we define SARX systems and the corresponding systems theoretic concepts such as minimality and identifiablity. In §3 we present the transformation of SARX systems into state-space form. In §4 we discuss the relationship between minimality of a SARX system and its subsystems. In §5 we provide sufficient conditions

for strong minimality. In §6 we discuss the relationship between minimality and identifiability. In §7 we show that minimality and identifiability are generic properties. 2. DEFINITION OF SARX SYSTEMS Throughout the paper, p will denote the output dimension and m will denote the input dimension. The set Q will denote the set of discrete modes, and without loss of generality, we assume that Q = {1, . . . , D}. Denote by T = N the time-axis of natural numbers. The definition of SARX systems is as follows. Definition 1. (SARX systems). A SARX system S of type (ny , nu ), where 0 < nu ≤ ny are integers, is a collection S = {nq }q∈Q , where nq , q ∈ Q are p × (ny p + nu m) matrices. We will call a SARX system a SISO SARX system if p = m = 1. Notation 1. In the sequel we will use the following decomposition for the matrices nq :   nq = n1q , n2q , . . . , nnq u +ny , where niq ∈ Rp×p , i = 1, . . . , ny , njq ∈ Rp×m , j = ny + 1, . . . , nu + ny . In order to assign semantics to SARX systems defined above, we need to formalize the concept of input-output behavior for switched systems, of which SARX systems form a subclass. To this end, we will need the notions of hybrid input and input-output map. Notation 2. (Hybrid inputs). Denote by U = Q × Rm . We denote by U + the set of all non-empty and finite) sequences of elements of U. A sequence w = (q0 , u0 ) · · · (qt , ut ) ∈ U + , t ≥ 0 (1) describes the scenario, when the discrete mode qi and the continuous input ui are fed to the system at time i, for i = 0, . . . , t. The input-output behaviors of interest are then maps f : U + → Rp . The value f (w) describes the output of the system at time t, generated as a response of the system to the hybrid input w. Now we are ready to define the semantics of SARX systems. Unlike systems in state-space form, SARX systems describe the relationship between the inputs and outputs of switched system directly. Definition 2. (Input-output maps of SARX systems). The SARX S is a realization of the input-output map f , if for all w ∈ U + of the form (1), the outputs yi = f ((q0 , u0 ) · · · (qi , ui )), i = 0, . . . , t satisfy the equation yt = nqt φt (2)

Definition 3. (Dimension). The dimension of a SARX system S of type (ny , nu ) is the number pny + nu m and it is denoted by dim S. Definition 4. (Minimality). A SARX S of is minimal, if there exists no equivalent SARX of dimension less than dim S. In order to be able to speak of identifiability, we need the notion of parameterization of SARX systems. Notation 3. Denote by SARX(ny , nu , m, p, Q) the set of all SARX systems of type (ny , nu ) with input space Rm , output space Rp , and set of discrete modes Q. Definition 5. (Parametrization) Assume that Θ ⊆ Rd is the set of parameters. A SARX parametrization is a map Π : Θ → SARX(ny , nu , m, p, Q) (4) Definition 6. (Identifiability) The parametrization Π is called identifiable, if for θ1 6= θ2 ∈ Θ, the corresponding SARX Π(θ1 ) and Π(θ2 ) are not equivalent. The intuition behind the above definition is that if a parametrization is not identifiable, then there might exists different parameter values which yield the same observed behavior and hence they cannot be distinguished from each other by input-output experiments. 3. CONVERTING SARX SYSTEMS INTO STATE-SPACE FORM In order to present the transformation of a SARX to a state-space system, we have to recall the definition of discrete-time linear switched systems. For a more detailed exposition, see Sun and Ge (2005b); Liberzon (2003); Petreczky et al. (2010). Definition 7. A linear switched system (abbreviated by DTLSS) is a discrete-time system Σ represented by xt+1 = Aqt xt + Bqt ut and x(0) = 0 (5) yt = Cqt xt . Here xt ∈ Rn is the continuous state at time t ∈ T , ut ∈ Rm is the continuous input at time t ∈ T , yt ∈ Rp is the continuous output at time t ∈ T , qt ∈ Q is the discrete mode (state) at time t, Q is the finite set of discrete modes of Σ. For each discrete mode q ∈ Q, the corresponding matrices are of the form Aq ∈ Rn×n , Bq ∈ Rn×m and Cq ∈ Rp×n . The number n is called the dimension of Σ, and it is denoted by dim Σ. Notation 4. We will use (p, m, n, Q, {(Aq , Bq , Cq ) | q ∈ Q}) as a short-hand notation for DTLSSs of the form (5).

and for all j < 0, we set yj = 0 and uj = 0.

For any hybrid input sequence w ∈ U + of the form (1), denote by yΣ (w) the corresponding output yt . We call the map yΣ : U + 3 w 7→ yΣ (w) ∈ Rp the input-output map of Σ. An input-output map f : U + → Rp is said to be realized by the DTLSS Σ if the input-output map of Σ coincides with f . The DTLSS Σ is ˆ of a minimal realization, if for any DTLSS realization Σ ˆ yΣ , dim Σ ≤ dim Σ.

Two SARXs are called equivalent, if they are realizations of the same input-output map.

In order to present the state-space representation of SARX systems, we use the following notation.

where we define the regressor φt ∈ R(ny p+nu m) as h iT T T T T T yt−2 · · · yt−n u · · · u φt = yt−1 , t−1 t−nu y

(3)

Notation 5. We denote by Id the d×d identity matrix and by Od×l the d × l matrix all entries of which are zero. Now we are ready to present the transformation of a SARX system to state-space form. This transformation is based on Weiland et al. (2006). Definition 8. Let S = {nq }q∈Q be a SARX system of type (ny , nu ). The DTLSS ΣS associated with a SARX S is defined as ΣS = (p, m, n, Q, {(Aq , Bq , Cq ) | q ∈ Q}), where n = pny + mnu , and   Aq = Ayq , Auq  1 2   nq , nq , . . . , nnq y −1 nnq y  I(ny −1)p O(ny −1)p×p   Ayq =   Om×p(ny −1) Om×p  O(nu −1)m×p(ny −1) O(nu −1)m×p   ny +1 ny +2  nq , nq , . . . , nnq y +nu −1 nqnu +ny O(ny −1)p×(nu −1)m O(ny −1)p×m   Auq =   Om×m(nu −1) Om×m I(nu −1)m O(nu −1)m×m C q = nq   Opny ×m  Im Bq =  Om(nu −1)×m (6) where we used the decomposition of nq from Notation 1. Lemma 1. (Weiland et al. (2006)). The SARX S is a realization of the input-output map f if and only if the associated DTLSS ΣS is a realization of f . The main idea behind the proof of Lemma 1 is that for all t ∈ T , the state xt of ΣS equals the regressor φt from (3). The lemma above allows us to reduce the problem of identifiability and minimality for SARX systems to that of DTLSSs of the form (6). This will be done in the subsequent sections. Finally, we note that none of the linear subsystems of ΣS is minimal. However, as we shall see later, the system ΣS is generically minimal. Lemma 2. (Non-minimality). Consider the DTLSS ΣS . For every q ∈ Q, the linear system (Cq , Aq , Bq ) is not observable and hence (Cq , Aq , Bq ) it is not minimal. The proof of Lemma 2 can be found in Appendix A. 4. MINIMALITY OF SARX SYSTEMS In this section we will analyze minimality of SARX systems. We start by stating a number of simple properties of minimal SARX systems. After that, we link minimality of SARX systems with the minimality of the associated DTLSSs systems. This allows us to formulate sufficient conditions for minimality of SARX systems. 4.1 Minimality: elementary conditions Below we will state some elementary properties of minimal SARX systems. Lemma 3. If the SISO SARX system S is minimal, then n +n there must exist q, qˆ ∈ Q such that nq u y 6= 0.

The proof of Lemma 3 can be found in Appendix A. Recall that in the classical literature, a SISO ARX is said to be minimal if and only if the numerator and the denominator of its transfer function are co-prime polynomials. Below we show that our definition of minimality is consistent with the traditional one. Lemma 4. A SISO ARX system is minimal according to Definition 4 if and only if the numerators and denominators of its transfer function are co-prime. The proof of Lemma 4 can be found in Appendix A. Lemma 5. If at least one of the ARX subsystems of a SISO SARX system is minimal, the system is minimal. The proof of Lemma 5 can be found in Appendix A. One might have noticed that our definition of minimality is ambiguous, since an input-output map could have two minimal SARX realizations of type (ny , nu ) and (ˆ ny , n ˆu) respectively with (ny , nu ) 6= (ˆ ny , n ˆ u ). According to the lemma below that this is impossible in the SISO case. Lemma 6. Assume that S1 and S2 are two minimal and equivalent SISO SARX systems such that S1 is of type (ny , nu ) and S2 is of type (ˆ ny , n ˆ u ). Then (ny , nu ) = (ˆ ny , n ˆ u ). The proof of Lemma 6 can be found in Appendix A. 4.2 Minimality of SARX and of the associated state-space representations In this section we analyze the minimality of the DTLSSs associated with SARX systems. The motivation for this lies in the following corollary of Lemma 1. Corollary 1. If the associated DTLSS ΣS is a minimal realization, then S is a minimal realization. The proof of Corollary 1 can be found in Appendix A. Corollary 1 allows us to reduce the problem of minimality of SARX to that of DTLSSs, the latter has already been investigated in Petreczky et al. (2010). It prompts us to propose the following definition. Definition 9. A SARX system S is called strongly minimal, if the corresponding DTLSS ΣS is minimal. By Corollary 1, strong minimality implies minimality. We conclude by presenting a number of examples which clarify the relationship between (strong) minimality of SARX systems and minimality of ARX subsystems. Minimality does not imply strong minimality, Example 1 below is a counter-example. Example 1. Consider the SARX system S with discrete modes Q = {1, 2} such that the ARX system associated with mode 1 is yt = −yt−2 + ut−1 and the ARX system associated with mode 2 is yt = −2yt−2 + 2ut−1 . The two ARX systems are distinct, each of them is minimal, yet the associated DTLSS ΣS is not minimal (in fact, it is not observable). The latter can be checked using the conditions of minimality from Petreczky et al. (2010).

Minimality of the ARX subsystems is not necessary for strong minimality (and hence minimality) of the whole system. Example 2. Consider again the SARX system S with two discrete modes Q = {1, 2} such that the ARX system in mode 1 is of the form yt = 8yt−1 − 15yt−2 + ut−1 − 3ut−2 , and the ARX system in mode 2 is of the form yt = yt−1 + 2yt−2 + ut−1 + ut−2 . The transfer function of the ARX in the first mode is z−3 1 z 2 −8z+15 = z−5 and the transfer function of the second z+1 1 ARX is z2 −z−2 = z−2 , hence neither of them is minimal. Yet, the DTLSS ΣS can easily seen to be minimal, using the conditions of Petreczky et al. (2010). Since strong minimality implies minimality of SARX systems, we get that S is minimal.

SISO transfer has no zero-pole cancellation (i.e. its numeration and denominator are coprime) and its denominator is of degree n, then all its minimal realizations are of order n. Due to the presence of switching, the formulation of Theorem 1 is more involved. In addition, Theorem 1 does not imply the classical results, since condition (B) of the theorem is always false if there is only one discrete state. In order to demonstrate the utility of the above theorem, we present the following examples. Example 3. Let’s apply Theorem 1 to the SARX S from Example 2. We obtain that ny = nu = 2, n1 = n +n [8, −15, 1, −3] and n2 = [1, 2, 1, 1]. Hence, n1 y u = −3 6= 0, and χ1 (z) = z 2 − 8z + 15, ψ1,2,1 (z) = z − 7 υ2 (z) = z + 2 φ2,1 (z) = z − 6. It is clear that the roots of χ1 (z) are 5 and 3 and hence υ2 (z) and χ1 (z) are con prime and φ2,1 (z) and χ1 (z) are coprime. Moreover, n2 y −

5. SUFFICIENT CONDITIONS FOR MINIMALITY

(B) of Theorem 1 hold and thus S is (strongly) minimal.

As we have seen in the previous section, strong minimality implies minimality. By Petreczky et al. (2010) strong minimality and hence minimality can be checked algorithmically. Indeed, strong minimality of a SARX system S means minimality of the associated DTLSS ΣS . The latter can be checked by checking if the rank of each of the finite matrices R(ΣS ) and O(ΣS ) defined in (Petreczky et al., 2010, Theorem 2) equals the dimension of ΣS .

6. IDENTIFIABILITY OF SARX SYSTEMS

We can also formulate sufficient conditions for minimality which do not involve computing DTLSSs. Theorem 1. (Sufficient conditions for (strong) minimality). Consider a SISO SARX system S = {nq }q∈Q of type (ny , nu ). For all modes q, qˆ ∈ Q define the polynomials χq (z) = z ny −

ny X

njq1 z ny −j

j=1 ny

υq (z) = φqˆ,q (z) =

X i=1 nu X

niq z ny −i y nj+n ψqˆ,q,nu −j (z), q

j=1

where ψqˆ,q,j (z) is defined recursively for j = 0, 1, 2 . . . , as follows: ψqˆ,q,0 (z) = 1 and ψqˆ,q,j+1 (z) = zψqˆ,q,j (z) + (nq − nqˆ)dj (7) ny +nu where the vectors dj ∈ R are defined as follows: d0 = e1 and if dj = (dj,1 , . . . , dj,ny , 0, . . . , 0)T , dj,1 , . . . , dj,ny ∈ R, then dj+1 = (nq dj , dj,1 , . . . , dj,ny −1 , 0, . . . , 0)T . Then S is strongly minimal, if the following conditions hold (A) there exists discrete modes q0 and q1 such that the polynomials χq0 (z) and φq0 ,q1 (z) are co-prime, and (B) there exists discrete modes q2 and q3 , such that n +n n υq3 (z) and χq2 (z) are co-prime, nq2u y 6= 0 and nq3y 6= n +nu

nq3y

n +n nq2y u

n

nq2y .

n +ny

n n1 y nu +ny n1

n2 u

In this section we study identifiability of SARX systems. We derive our results by reducing identifiability analysis of SARX systems to that of the associated DTLSSs. This is possible due to the following corollary of Lemma 1. Corollary 2. A SARX parametrization Π is identifiable, if and only if the DTLSS parametrization Πsw : Θ 3 θ 7→ ΣΠ(θ) is identifiable 1 . The proof of Corollary 2 can be found in Appendix A. In order to apply the results of Petreczky et al. (2010), we have to restrict attention to strongly minimal SARX systems. To this end, we need the following terminology. Definition 10. (Minimality of parametrizations) The parametrization Π is called minimal (resp. strongly minimal ), if for all θ ∈ Θ, Π(θ) is minimal (resp. strongly minimal). If a SARX parametrization is strongly minimal, then the corresponding DTLSSs parametrization will be minimal 2 . Hence, we can apply the conditions and algorithms described in Petreczky et al. (2010) for analyzing the identifiability if the latter parametrization. By Corollary 2 the identifiability of the latter parametrization is identifiability of the original SARX parametrization. In fact, for the SISO case (i.e. when p = m = 1), we can derive even stronger results, by showing minimality is sufficient for identifiability. To this end, we need the following definition. Definition 11. (Injective parametrizations) An SARX parametrization Π is said to be injective if Π is an injective map. An injective parametrization allows us to exclude the situation where two different parameter values lead to the same SARX system. The ARX parametrization yt = θ2 yt−1 + ut−1 with θ ∈ R is not injective, since any θ and −θ always lead to the same ARX system. 1

The proof of Theorem 1 can be found in Appendix A. Theorem 1 is analogous to the well-known result that if a

= 2 − 15 3 = −3 6= 0. Hence, conditions (A) and

See Petreczky et al. (2010) for the definition of a parametrization and identifiability of DTLSSs. 2 See Petreczky et al. (2010) for the definition.

The following theorem, which is one of the main results of the paper, describes the relationship between strong minimality and identifiability. Theorem 2. Assume that p = m = 1. If a SISO SARX parametrization Π is injective and strongly minimal, then Π is identifiable.

In order to formalize these results, we need the following terminology. Definition 12. (Generiticity). A subset G of Θ ⊂ Rd is generic, if G is non-empty and there exists a non-zero polynomial P (X1 , . . . , Xd ) in d variables such that G = {θ ∈ Θ | P (θ) 6= 0}.

The proof of Theorem 2 can be found in Appendix A. In order to prove Theorem 2, we will need the following result which is interesting on its own right. Theorem 3. Consider two SISO SARX systems S1 = {nq }q∈Q and S2 = {hq }q∈Q of type (ny , nu ) and assume n n +n that for some q ∈ Q, either nq y 6= 0 or nq y u 6= 0. If there 3 exists an isomorphism between the associated DTLSSs ΣS1 and ΣS2 , then this isomorphism is the identity map.

That is, a generic subset of Θ is a non-empty subset whose complement in Θ satisfies a polynomial equation. Definition 13. (Generic identifiability and minimality) The parametrization Π is said to be generically identifiable if there exists a generic subset G of Θ, such that the parametrization Π|G : G 3 θ 7→ Π(θ) is identifiable.

The proof of Theorem 3 can be found in Appendix A. Theorem 3 implies that under some mild conditions that the transformation of two different SARX systems to statespace representations cannot result in isomorphic systems. Recall that strong minimality of a SARX system does not imply minimality of its ARX subsystems. Similarly, identifiability of a SARX parametrization does not imply the identifiability of the corresponding parametrization of ARX subsystems. The example below demonstrates this point. Example 4. Consider the SARX parametrization Π with Θ = R2 , and consider the parametrization Π((θ1 , θ2 )) = {nq (θ1 , θ2 )}q∈Q , where n1 = [(θ1 + θ2 ) −θ1 θ2 , 1 −θ2 ] n2 = [(2 + θ2 ) −2θ2 , 1 −θ2 ] Define the set G = {(θ1 , θ2 ) | θ1 6= 2}. Consider the restriction Π|G of Π to G. Using Theorem 1 one can check that for any (θ1 , θ2 ) ∈ G, the SARX system Π((θ1 , θ2 )) is strongly minimal. Hence, the parametrization Π|G is identifiable by Theorem 2. Identifiability of Π|G can also be checked by considering the switching sequence 112 and input u0 = 1, ut = 0, t > 0 and noticing that then y0 , y1 = 1, y2 = θ1 , y3 = 2θ1 + θ2 θ1 − 2θ2 from which 1) θ2 = (yθ31−2θ −2 . Hence, θ1 and θ2 can be determined from the outputs y2 and y3 . Note however, that for any (θ1 , θ2 ), the ARX subsystems of Π(θ1 , θ2 ) are not identifiable, since their dynamics does not depend on θ2 . 7. MINIMALITY AND IDENTIFIABILITY ARE GENERIC In the previous sections we have established that strong minimality is sufficient for minimality and that it is also sufficient for identifiability. However, we have also demonstrated that for some minimal SARX systems strong minimality does not hold. Hence, one may wonder how typical strong minimality is. Below we will show that strong minimality is a generic property, i.e. it holds for almost all SARX systems, if |Q| > 1. This also means that identifiability is a generic property. In other words, strong minimality occurs very frequently. 3 See Petreczky et al. (2010) for the definition of isomorphism between DTLSSs

Similarly, Π is generically minimal (respectively generically strongly minimal ), if there exists a generic subset G of Θ, such that the parametrization Π|G : G 3 θ 7→ Π(θ) is minimal (respectively strongly minimal). Definition 14. (Polynomial parametrization) Let K = (pny + mnu )|Q|. Then any SARX system of type (ny , nu ) can be identified with a point in RK , by identifying the system with its parameters {nq }q∈Q . Thus, SARX(ny , nu , m, p, Q) can be identified with the space RK . A parametrization Π is said to be polynomial, if Θ is an affine algebraic variety and Π is a polynomial map from Θ to SARX(ny , nu , m, p, Q). Intuitively, if a property is generic for a parametrization, then every member of the parametrization can be approximated with arbitrary accuracy by another member which has the this property. Another interpretation is that if we randomly generate parameters, then the property will hold for the thus obtained random parametrization with probability one. Example 5. Consider the parametrization Π from Example 4. The set G from Example 4 is generic. Hence, since the parametrization Π|G is strongly minimal and identifiable, the parametrization Π is generically strongly minimal, generically minimal, and generically identifiable. Theorem 4. (Generic minimality). If |Q| > 1, Π is a polynomial parametrization and Π contains a strongly minimal SARX system, (i.e. for some θ ∈ Θ, Π(θ) is strongly minimal), then Π is generically minimal. The proof of Theorem 4 can be found in Appendix A. Notice that Theorem 2 implies the following corollary. Corollary 3. Consider the SISO case, i.e. p = m = 1. If a SARX parametrization is injective, it is polynomial, and it is generically strongly minimal, then it is generically identifiable. The proof of Corollary 3 can be found in Appendix A. Corollary 3 and Theorem 4 yield the following. Corollary 4. Assume that p = m = 1. If a SISO SARX parametrization is polynomial and it contains a strongly minimal element, then it is generically identifiable. The proof of Corollary 4 can be found in Appendix A. The trivial SISO SARX parametrization Πtriv is the SARX parametrization defined as follows: Θ = R|Q|(nu +ny )

and Πtriv is the identity map. From Corollary 3-4 we obtain that Corollary 5. The trivial parametrization is generically minimal and in the SISO case it is generically identifiable. The proof of Corollary 5 can be found in Appendix A. 8. CONCLUSIONS In this paper we proposed definitions for minimality and identifiability of SARX systems. We have shown that minimal SARX parametrizations are also identifiable. We have also formulated sufficient and necessary conditions for minimality of SARX systems. Future research is aimed at finding a more complete characterization of minimality and extending the results to MIMO systems and systems with autonomous switching. REFERENCES Bako, L., Merc´ere, G., and Lecoeuche, S. (2009a). Online structured subspace identification with application to switched linear systems. International Journal of Control, 82, 1496–1515. Bako, L., Merc´ere, G., Vidal, R., and Lecoeuche., S. (2009b). Identification of switched linear state space models without minimum dwell time. In IFAC Symposium on System Identification, Saint Malo, France. Ferrari-Trecate, G., Muselli, M., Liberati, D., and Morari, M. (2003). A clustering technique for the identification of piecewise affine systems. Automatica, 39, 205–217. Fliess, M., Join, C., and Perruquetti, W. (2008). Realtime estimation for switched linear systems. In IEEE Conference on Decision and Control, Cancun, Mexico. Gantmacher, F. (2000). The theory of matrices, volume I. AMS Chelsea Publishing. Juloski, A., Heemels, W., Ferrari-Trecate, G., Vidal, R., Paoletto, S., and Niessen, J. (2005a). Comparison of four procedures for the identification of hybrid systems. In Hybrid Systems: Computation and Control, volume 3414 of LNCS, 354–369. Springer-Verlag, Berlin. Juloski, A.L., Weiland, S., and Heemels, W. (2005b). A bayesian approach to identification of hybrid systems. IEEE Transactions on Automatic Control, 50, 1520– 1533. Lauer, F., Vidal, R., and Bloch, G. (2009). A producterror framework on linear hybrid system identification. In 15th IFAC Symposium on System Identification. Liberzon, D. (2003). Switching in Systems and Control. Birkh¨ auser, Boston. Nakada, H., Takaba, K., and Katayama, T. (2005). Identification of piecewise affine systems based on statistical clustering technique. Automatica, 41, 905–913. Paoletti, S., Juloski, A., Ferrari-Trecate, G., and Vidal, R. (2007a). Identification of hybrid systems: A tutorial. European Journal of Control, 13(2-3), 242 – 260. Paoletti, S., Roll, J., Garulli, A., and Vicino, A. (2007b). Input/ouput realization of piecewise affine state space models. In 46th IEEE Conf. on Dec. and Control. Paoletti, S., Roll, J., and Garulli, A.and Vicino, A. (2010). On the input-output representation of piecewise affine state space models. IEEE Transactions on Automatic Control, 55, 60 – 73.

Petreczky, M., Bako, L., and van Schuppen, J. (2010). Identifiability of discrete-time linear switched systems. In Hybrid Systems: Computation and Control, 141–150. ACM. Roll, J., Bemporad, A., and Ljung, L. (2004). Identification of piecewise affine systems via mixed-integer programming. Automatica, 40, 37–50. Sun, Z. and Ge, S.S. (2005a). Switched Linear Systems – Control and Design. Springer. Sun, Z. and Ge, S.S. (2005b). Switched linear systems : control and design. Springer, London. Verdult, V. and Verhaegen, M. (2004). Subspace identification of piecewise linear systems. In Proc.Conf. Decision and Control. Vidal, R. (2008). Recursive identification of switched ARX systems. Automatica, 44(9), 2274 – 2287. Weiland, S., Juloski, A.L., and Vet, B. (2006). On the equivalence of switched affine models and switched ARX models. In 45th IEEE Conf. on Decision and Control. Appendix A. PROOFS Proof. [Proof of Lemma 2] Notice that for each q ∈ Q, Aq contains a zero row, hence rankAq < nu + ny . This means that λ = 0 is an eigenvalue of Aq . By the Hautus-criterion, (CqT, Aq ) is an  observable pair if and only if the matrix Cq , λI − ATq has rank ny + nu for all the eigenvalues of Aq . We will show that for λ = 0 this matrix cannot be of Tfull row  rank. Indeed, for λ = 0 the matrix becomes Cq , −ATq . But Cq equals the first row of Aq multiplied   by −1. Hence, CqT , −ATq will have the same rank as Aq and that is smaller than nu + ny . Proof. [Proof of Lemma 3] Assume the contrary, i.e. that n +n ˆq ∈ nq u y = 0 for all q ∈ Q. Define the vectors n Rny +nu −1 , q ∈ Q   n ˆ q = n1q . . . nnq u +ny −1 Define the regressors φˆt as h iT T T . . . , yt−n uTt−1 . . . uTt−ny −nu −1 , φˆt = yt−1 y where we used the convention that yj = 0 and uj = 0 for j < 0. It then follows that yt = nq φt = n ˆ q φˆt ˆ = ({ˆ for all t ∈ T . Hence, S nq }q∈Q ) realizes the same ˆ is smaller input-output map as S. But the dimension of S than that of S, which contradicts the minimality of S. Proof. [Proof of Lemma 4] The proof follows from the classical linear theory, by observing that two ARX systems realize the same input-output map if and only if they have the same transfer function (modulo zero/pole cancelation). Consider an ARX system yt = nq φt and assume that it is minimal. If its transfer function admits a zeropole cancellation, then the degrees of the numerator and denominator of the transfer function decrease by one. The latter means that the transfer function can be realized by an ARX of type (ny −1, nu −1). The dimension of the latter is ny + nu − 2 and hence smaller than that of the original system, which was supposed to be minimal. Moreover, this new ARX system will realize the same input-output map as the original one.

Conversely, consider an ARX system S whose transfer function does not allow zero/pole cancellation. Let f be the input-output map of S and assume that the ARX ˆ is a minimal realization of f . Then the transfer system S function HSˆ (z) cannot allow a zero/pole cancellation and it must be equal to the transfer function HS (z) of S. Since neither HS (z) nor HSˆ (z) allow zero/pole cancellation, their equality implies the equality of the numerators and denominators respectively, viewed as polynomials. In particular, the corresponding coefficients are the same and hence the parameters of the two ARX systems are the same too. In particular, the dimensions of the two systems will be the same, and hence S is then a minimal realization of its input-output map. Proof. [Proof of Lemma 5] Consider S = ({nq }q∈Q ) and assume that for some qm ∈ Q, the ARX yt = nqm φt is minimal.Assume that S is not minimal and hence there exists a SARX Sm = ({ˆ nq }q∈Q ) such that dim Sm ≤ dim S and Sm realizes the same input-output map as S. It then follows that the dimension of the ARX yt = nqm φt is larger than that of yt = n ˆ qm φt . It also follows that both yt = n ˆ qm φt and yt = nqm φt realizes the same linear input-output map 4 . This contradicts the minimality of yt = nqm φt . Proof. [Proof of Lemma 6] Pick any discrete state q and consider the transfer functions Hi (z),i = 1, 2 of the ARX system in mode q associated with the SARX Si ,i = 1, 2. Since S1 and S2 are equivalent, they produce the same response to any input if the discrete mode is kept to be q. Hence, the ARX systems corresponding to the mode q are also equivalent, i.e. H1 (z) and H2 (z) describe the same input-output behavior. This means that the transfer functions H1 (z) and H2 (z) are equal as rational expressions, after possibly performing zero/pole cancelation. The degrees of the numerators of H1 (z) and H2 (z) are respectively nu and n ˆ u and the degrees of the denominators of H1 (z) and H2 (z) are respectively ny and nˆy . Performing zero/pole cancellation does not change the difference between the degree of the numerator and the degree of the denominator. Hence, we obtain that ny − nu = n ˆy − n ˆ u must hold. But since both S1 and S2 are minimal SARX realizations of the same input-output map, their dimensions must agree and hence ny + nu = n ˆy + n ˆu. It is easy to see that the only solution to the system of equations  ny − nu = n ˆy − n ˆu ny + nu = n ˆy + n ˆu is ny = n ˆ y and nu = n ˆu. Proof. [Proof of Corollary 1] Assume that S is not min0 0 imal. Then there exists an equivalent Sm of type (ny , nu ) 0 0 such that ny + nu < ny + nu . But this implies that 0 0 dim ΣSm = ny +nu < ny +nu = dim ΣS , which contradicts to the minimality of ΣS . Proof. [Proof of Corollary 2] Consider two SARX systems Si = {niq }q∈Q , i = 1, 2 of type (ny , nu ). Notice that the 4

Namely, the map which maps input u0 , . . . , ut to the output yt = f ((qm , u0 ) · · · (qm , ut )), where f is the input-output map of S

associated DTLSSs ΣS1 and ΣS2 realize the same inputoutput map Assume that the parametrization Π is identifiable, but Πsw is not identifiable. Then there exist two parameters θ1 , θ2 ∈ Θ, θ1 6= θ2 , such that Πsw (θ1 ) and Πsw (θ2 ) realize the same input-output map. Since Πsw (θi ) = ΣΠ(θi ) , i = 1, 2, by the remark above it follows that Π(θ1 ) and Π(θ2 ) are equivalent. This contradicts the identifiability of Π. Conversely, assume that Πsw is identifiable, but Π is not identifiable. Then there exists parameters θ1 , θ2 ∈ Θ, θ1 6= θ2 , such that Π(θ1 ) and Π(θ2 ) are equivalent. This means that ΣΠ(θ1 ) = Πsw (θ1 ) and ΣΠ(θ2 ) = Πsw (θ2 ) realize the same input-output map. But this contradicts the identifiability of Πsw . For the proof of Theorem 1, we will need a number of auxiliary results. Below, we consider a S = {nq }q∈Q . We denote by Aq the corresponding matrix of the DTLSS ΣS . We will by ei the ith standard basis vector of Rny +nu . Lemma 7. Let X1 = Span{e1 , . . . , eny }. It then follows that for any q ∈ Q, (1) Aq ej = njq e1 + ej+1 for all j = 1, . . . , ny + nu , j 6= ny n and Aq eny = nq y e1 . (2) The space X1 is Aq invariant and Span{Aiq e1 | i = 0, . . . , ny − 1} = X1 (3) Ajq eny +1 ∈ X1 + eny +j+1 , j = 0, . . . , nu − 1, and Anq u eny +1 ∈ X1 . (4) For any qˆ ∈ Q, ψqˆ,q,j (Aqˆ)e1 = Ajq e1 , where ψqˆ,q,0 (z) = 1 and ψqˆ,q,j+1 (z) = zψqˆ,q,j (z)+(nq −nqˆ)ψqˆ,q,j (Aqˆ)e1 , where where the vectors dj ∈ Rny +nu are defined as follows: d0 = e1 and if dj = (dj,1 , . . . , dj,ny , 0, . . . , 0)T , dj,1 , . . . , dj,ny ∈ R, then dj+1 = (nq dj , dj,1 , . . . , dj,ny −1 , 0, . . . , 0)T Proof. [Proof of Lemma 7] Part 1 follows by a simple computation. Part 2 follows from Part 1 by taking into account that Aq eny ∈ Span{e1 }. Part 3 follows from the n +1 definition of Aq by induction Indeed, Aq eny +1 = nq y e1 + eny +2 ∈ X1 + eny +2 and if Ajq eny +1 ∈ X1 + ej+1 , then n +j+2 Aj+1 eny +1 ∈ X1 +Aq ej+1 ⊆ X1 +nq y e1 +ej+2 ⊆ X1 + q ej+2 . Finally, ψqˆ,q,j (Aqˆ)e1 = Ajq e1 we will prove by induction. For j = 0, the equality is trivial. Notice that Aq ei = niq e1 + ei+1 = Aqˆei + (niq − niqˆ)e1 for all i = 1, . . . , ny − 1, and n n n Aq eny = nq y e1 = Aqˆeny + (nq y − nqˆ y )e1 . Hence, for any Pny Pny x = i=1 xi ei , Aq x = Aqˆx + i=1 xi (niq − niqˆ)e1 = Aq1 x + ((nq − nqˆ)x)e1 .. Hence, if ψqˆ,q,j (Aqˆ)e1 = Ajq e1 holds, then Aj+1 e1 = Aq ψqˆ,q,j (Aqˆ)e1 = q . (A.1) Aqˆψqˆ,q,j (Aqˆ)e1 + +((nq − nqˆ)ψqˆ,q,j (Aqˆ)e1 )e1 = Finally, notice that dj = Ajq ej for all j = 0, . . . , ny . Pny Indeed, i=1 dj,i ei , then Aq dj = Pny d0 i= e1 and Pny if dj = ( i=1 dj,i nq )e1 + i=2 dj,i−1 ei = dj+1 . Hence, by replacing ψqˆ,q,j (Aqˆ)e1 = Ajq e1 by dj in A.1, we obtain that Aj+1 e1 = ψqˆ,q,j+1 (Aqˆ)e1 . q

Hence, by induction we get that last statement of the lemma. n +n Lemma 8. If nq y u 6= 0, then {eTny Ajq | j = 0, . . . , ny + nu − 1} spans R1×(ny +nu ) . n −i eTny Aq y ,

eTi

eTny +j ny

Moreover, = i = 1, . . . , ny , and = eTny χq (Aq )γj,q (Aq ), j = 1, . . . , nu , where χq (z) = z − Pny j ny −j and the polynomial υj (z), j = 1, . . . , nu is j=1 nq z defined recursively as follows: γ1,q (z) = γi,q (z) =

1 n +n nq u y

z nu −1

1

nu −i − nu +ny (z

nq

i−1 X

γj,q (z)nnq y +nu −i+j )

j=1

Proof. [Proof of Lemma 8] In this proof we will view Aq as a linear map xT 7→ xT Aq , defined on the space of row vectors xT ∈ R1×(ny +nu ) . From the structure of Aq it then follows that eTj Aq = eTj−1 , for j = {2, . . . , ny } ∪ {ny + 2, . . . , ny +nu }. Hence, eTny Ajq , j = 0, . . . , ny −1 spans X1 = Pny +nu −1 i T Span{eT1 , . . . , eTny }. Notice eT1 Aq = nq ei + i=1 n +ny T eny +nu .

nq u

Hence

ny +nu

xT =

X

niq eTi = eT1 Aq −

i=ny +1

ny X

niq eTi

(A.2)

i=1

belongs to the linear span of

eTny Ajq ,

j = 0, . . . , ny .

We proceed to prove that xT Ajq , j = 0, . . . , nu − 1 span X2 = Span{eTny +1 , . . . eTny +nu }. From this the first statement of the lemma follows. Notice that eTny +1 Aq = 0 and eTny +j Aq = eTny +j−1 for all j = 2, . . . , nu . Hence, xT Ajq =

nX u −j

nnq y +i+j eTny +i

χAq (z) = z nu (z ny −

Proof. [Proof of Lemma 9] From Lemma 8 it follows that R1×(nu +ny ) is a cyclic subspace with respect to the linear operator Aˆq : xT 7→ xT Aq 5 . By (Gantmacher, 2000, Theorem 4, Chapter VII), it then follows that the minimal polynomial of the linear operator Aˆq equals its characteristic polynomial and it is of degree ny + nu . Note that in the standard basis eT1 , . . . , eTnu +ny , the basis of the linear operator Aˆq is ATq . Hence, the minimal polynomial and characteristic polynomial of ATq coincide. But these polynomials are the same for the matrices Aq and ATq . Moreover, from Lemma 8 it also follows that eTny is the generating element of the cyclic space R1×(ny +nu ) . Hence, by (Gantmacher, 2000, §4.1,Chapter VII), a polynomial ψ(z) is a minimal polynomial of Aˆq , if ψ(Aˆq )eTn = y

eTny ψ(Aq ) = 0 and it has the smallest possible degree. By the discussion above, the degree of the minimal polynomial of Aˆq must be ny + nu . Hence, the minimal polynomial of Aˆq is the unique monic polynomial ψ(z) of degree ny + nu , such that eTny ψ(Aq ) = 0. If we show that eTny χAq (Aq ) = 0, then the statement of the lemma follows. To this end, notice that if y T ∈ X2 = Span{eTny +1 , . . . , eTny +nu }, then y T Anq u = 0. In Pny i T n T addition, eTny Aq y = eT1 Aq = i=1 nq ei + x , where Pnu T i T x = i=ny +1 nq ei ∈ X2 . Hence, by taking into account n −i

the remark above and that eTi = eny Aq y eTny Anq y +nu =

n +nu

From (A.3) and nq y

n −j

ny X

, i = 1, . . . , ny ,

niq eTi Anq u + xT Anq u =

i=1 ny X

i=1

Finally, the statement eTj = eTny Aq y , j = 1, . . . , ny follows from the definition of Aq . The statement that eTny +j = eTny χ(Aq )γj,q (Aq ), j = 1, . . . , nu can be shown as follows. From (A.2) it follows that xT = eTny χq (Aq ). From (A.4) and (A.5) it follows that eTny +j = xT γj,q (Aq ) for all j = 1, . . . , nu . Combining the above statements implies the second statement of the lemma. n +n Lemma 9. Assume that nq u y 6= 0. The characteristic polynomial of Aq coincides with its minimal polynomial and it equals

niq z ny −i )

i=1

(A.3)

6= 0 it then follows that 1 eTny +1 = nu +ny xT Anq u −1 (A.4) nq and if eTny +1 , . . . , eTny +j have already been obtained from the linear combinations of xT Aiq , i = nu − j, . . . , nu − 1, then j X 1 eTny +j+1 = nu +ny (xT Anq u −j−1 − nnq y +nu −j−1+i eny +i ). nq i=1 (A.5) Hence, xT Ajq , j = 0, . . . , nu − 1 span X2 .

ny X

.

niq eTny Anq u +ny −i

i=1

The latter is exactly equivalent to eTny χAq (Aq ) = 0. Proof. [Proof Theorem 1] We will show that if Part (A) holds, then ΣS is reachable, and if Part (B) holds, then ΣS is observable. Proof of Part (A) We will show that if the conditions of (A) hold, then (Aq0 , Anq1u Bq1 ) is a controllable pair. By Sun and Ge (2005a) it then follows that the DTLSS ΣS is reachable. From Lemma 7 it follows that Ajq1 Bq1 = Ajq1 eny +1 = n +j nq1y Aqj−1 e1 + eny +j+1 for j = 1, . . . , nu − 1 and hence 1 Anq1u Bq1 =

nu X

nqn1y +j Anq1u −j e1

j=1

From Lemma 7 it also follows that ψq0 ,q1 ,j (Aq0 ) = Ajq1 and hence the polynomial φq0 ,q1 (z) satisfies Anq1u Bq = φq0 ,q1 (Aq0 )e1 . 5

see (Gantmacher, 2000, §4, Chapter VII) for the definition of cyclic subspaces

From Lemma 7 it follows that φq0 ,q1 (Aq0 )e1 ∈ X1 and X1 is Aq1 invariant, where X = Span{e1 , . . . , eny }. In addition, from the construction of Aq0 it follows that with respect to the basis e1 , . . . , eny , the matrix representation of the restriction of Aq0 to X1 is of the form  1   nq0 . . . nnq0y −1 nnq0y Aˆq0 = . Iny −1 O(ny −1)×1 The above matrix is in companion form and it is known that its characteristic polynomial equals its minimal polynomial and it equals χq0 (z). That is, χq0 (z) is the minimal polynomial of the linear operator Aq0 restricted to X1 . Moreover, from Lemma 7 it follows that Ajq0 e1 , j = 0, . . . , ny − 1 generate the space X1 , i.e. X1 is a cyclic subspace w.r.t. to Aq . Then by (Gantmacher, 2000, §4.1,Chapter VII) χq0 (z) is a minimal polynomial of e1 with respect to Aq0 , i.e. χq0 (Aq0 )e1 = 0 and χq0 (z) has the smallest degree among all the polynomials ψ(z) such that ψ(Aq0 )e1 = 0. Suppose now that χq0 (z) and φq0 ,q1 (z) are coprime, but (Aq0 , Anq1u Bq1 ) = (Aq0 , φq (Aq0 )e1 ) is not a controllable pair. Then the vectors Ajq0 x, j = 0, . . . , ny − 1, x = φq0 ,q1 (Aq0 )e1 are linearly dependent, i.e. there exists a non-zero polynomial κ(z) of degree at most ny − 1 such that κ(Aq0 )x = 0. By substituting x = φq0 ,q1 (Aq0 )e1 , we get κ(Aq0 )φq0 ,q1 (Aq0 )e1 = 0. That is, for the polynomial φ(z) = κ(z)φq0 ,q1 (z), φ(Aq0 )e1 = 0. This implies by Gantmacher (2000) that the minimal polynomial χq0 (z) divides φ(z) = κ(z)φq0 ,q1 (z). Since χq0 (z) and φq0 ,q1 (z) are co-prime, then this is possible only if χq0 (z) divides κ(z). But the degree of κ(z) is strictly smaller than the degree of χq0 (z), hence κ(z) cannot be divisible by χq0 (z). We arrived to a contradiction. That is, we can conclude that (Aq0 , Anq1u Bq1 ) is a controllable pair. Proof of Part (B) We will show that (Cq3 , Aq2 ) is an observable pair. By Sun and Ge (2005a) this is sufficient for observability of ΣS . To this end, using the notation of Lemma 8 define the polynomial nu X ˆ ψ(z) = υq3 (z) + nnq3y +j γj,q2 (z)χq2 (z). j=1

We will argue that if the conditions of Part (B) hold, ˆ ˆ then ψ(z) and z nu χq2 (z) are co-prime. Indeed, if ψ(z) and z nu χq2 (z) are not co-prime, then there exists an irreducible ˆ polynomial q(z) which divides both ψ(z) and z nu χq2 (z). If q(z) is an irreducible polynomial which divides z nu χq2 (z), then it either equals z or it divides χq2 (z). If q(z) = z ˆ ˆ ˆ and it divides ψ(z), then 0 is a root of ψ(z), i.e. ψ(0) = 0. Notice that by induction it follows that for j = 1, . . . , nu − 1, γj,q2 (0) = 0 and γnu ,q2 (0) = nu1+ny . Hence, from nq2

n ˆ ˆ the definition of ψ(z) it follows that ψ(0) = nq3y + n +n n +n nq3u y nq3u y ny ny ˆ n +n χq (0) = nq3 − n +n nq2 . Hence, ψ(0) = 0 implies nq2u

y

nq2u

2

n +ny

nq3u

n

that nq3y =

n +ny

nq2u

y

n

nq2y , which contradicts to the condition

ˆ of (B). If q(z) divides χq2 (z) and it divides ψ(z), the it Pnu nu +i ˆ divides υq3 (z) = ψ(z) − ( i=1 nq3 γi,q2 (z))χq2 (z). But this contradicts to the assumption that ψq3 (z) and χq2 (z) are co-prime. ˆ Hence, by the discussion above, ψ(z) and z nu χq2 (z) are coprime, so if z nu χq2 (z) divides P (z), it then must divide κ(z). But the degree of κ(z) is strictly smaller than that of z nu χq2 (z), hence z nu χq2 (z) cannot divide κ(z). We arrived to a contradiction. Hence, (Cq3 , Aq2 ) must be an observable pair. Proof. [Proof of Theorem 3] Assume that ΣS1 = (p, m, n, Q, {(Aq , Bq , Cq ) | q ∈ Q}) 0

0

0

0

ΣS2 = (p, m, n , Q, {(Aq , Bq , Cq ) | q ∈ Q}). 0

with n = n = n = ny + nu and p = m = 1. Consider an isomorphism M between ΣS1 and ΣS2 . Denote by ei the ith standard unit vector of Rn . Then eT1 , . . . , eTn form the standard basis in R1×n The proof depends on the following series of technical results. Proposition 1. eT1 Aq = eT1 MAq (A.6) Proof. [Proof of Proposition 1] From the construction of 0 0 ΣSi , i = 1, 2 it then follows that Cq = eT1 Aq , Cq = eT1 Aq . From the definition of isomorphism between DTLSSs, it 0 follows that Cq M = Cq , q ∈ Q. Hence, we obtain that 0

ˆ q ). Then from Lemma 8 it follows that Cq3 = eTny ψ(A 2 Assume that (Cq3 , Aq2 ) is not an observable pair. Then Cq3 Ajq2 , j = 0, . . . , ny − 1 are linearly independent. Hence, there exists a polynomial κ(z) of degree less than ny , such that Cq3 κ(Aq2 ) = 0. Hence, we obtain that ˆ q )κ(Aq ) = 0. In other words, the polynomial eTny ψ(A 2 2 ˆ P (z) = ψ(z)κ(z) is an annihilating polynomial with respect to the operator Aˆq : x 7→ xAq 6 of eT . Since 2

2

ny

by Lemma 8 eTny Ajq2 , j = 0, . . . , ny + nu generate the whole space, it then follows that P (z) is the annihilating polynomial of the whole space, i.e. P (Aq2 ) = 0. It then follows that P (z) is divisible by the minimal polynomial of Aˆq2 which coincides with that of Aq2 . From Lemma 9 it follows that the minimal polynomial of Aq2 is z nu χq2 (z). 6

See (Gantmacher, 2000, §1) for the definition

eT1 Aq = eT1 Aq M. 0

But Aq M = MAq by the definition of a DTLSS isomorphism, and hence we obtain the claim of the proposition. Proposition 2. The columns of Aq span the space Span{e1 , . . . , enu +ny } \ {eny +1 }. Proof. [Proof of Proposition 2] Indeed, Aq en = nnq e1 , n Aq eny = nq y e1 , Aq ej = ej+1 + njq e1 for all j ∈ {1, . . . , n − n 1} \ {ny }. Hence, if either nq y 6= 0 or nnq 6= 0, then e1 belongs to the column space of Aq , and hence ej = Aq ej−1 − nqj−1 e1 belongs to the column space of Aq , for j ∈ {2, . . . , n} \ {ny + 1}. Proposition 3. For any i = 1, . . . , nu + ny , if eTi Aq = eTi MAq , then eTi = eTi M. Proof. [Proof of Proposition 3] Indeed, if eTi Aq = eTi MAq , then this implies that (eTi −eTi M)Aq = 0. By Proposition 2

this implies that (eTi − eTi M)ej = 0 for all j ∈ {1, . . . , nu + ny } \ {ny + 1}. Notice that from the construction of ΣS1 , ΣS2 and the definition of a DTLSS morphism it follows 0 that eny +1 = Bq = MBq = Meny +1 . Hence, (eTi − eTi M)eny +1 = 0 and thus (eTi − eTi M)ej = 0, j = 1, . . . , ny + nu . This is just an alternative way of formulating the conclusion of the proposition. Proposition 4. If eTj−1 M = eTj−1 , then eTj Aq = eTj MAq for all j = {2, . . . , ny + nu } \ {ny + 1}. 4] Notice that eTj Aq = eTj−1 , 0 and eTj Aq = eTj−1 , eTj Aq =

Proof. [Proof of Proposition 0 eTj Aq = eTj−1 , j = ny , . . . , 2, 0 eTj−1 , for j = ny + nu , . . . , ny + 2. Hence, by using Aq M = MAq , we derive 0

eTj−1 M = eTj Aq M = eTj MAq

(A.7)

for all j ∈ {2, . . . , ny } ∪ {ny + 2, . . . , ny + nu }. Since eTj−1 M = eTj−1 , and eTj Aq = eTj−1 for all j = {2, . . . , nu + ny } \ {ny + 1}, from (A.7) we obtain the claim of the proposition. The rest of the proof proceeds as follows. We will prove that eTj = MeTj , j = 1, . . . , ny + nu , (A.8) which is just another way of saying that M is the identity matrix. To this end, from (A.7) and Proposition 3 it follows that (A.8) holds for j = 1. Moreover, the ny + 1th row of 0 0 Aq and Aq are both zero, hence, 0 = eTny +1 Aq 0 = eTny +1 Aq 0 and thus 0 = eny +1 Aq M = eTny +1 MAq . From this we get that eTny +1 Aq = eTny +1 MAq and by Proposition 3 this implies that (A.8) holds for j = ny + 1. Notice that if (A.8) holds for j = k ∈ {1, . . . , nu + ny − 1} \ {ny }, then by Proposition 4, eTk+1 Aq = eTk+1 MAq . By Proposition 3, the latter implies that (A.8) holds for j = k + 1. Hence, by induction we get that (A.8) holds for all j. Proof. [Proof of Theorem 2] e will show that the DTLSS parameterization Πsw : Θ 3 θ 7→ ΣΠ(θ) is identifiable. By Corollary 2 this is sufficient for identifiability of Π. Since Π is strongly minimal, the DTLSS parameterization Πsw is minimal, see Petreczky et al. (2010) for definition. In order to show identifiability of Πsw , by (Petreczky et al., 2010, Corollary 1) it is enough to show that the only isomorphism between elements of Πsw is the identity. Consider now two elements Σi = ΣΠ(θi ) , θi ∈ Θ, i = 1, 2 of Πsw . Notice that Π(θ1 ) is minimal, since it is strongly minimal, and thus if Π(θ1 ) = {nq }q∈Q , then by Lemma 3 n +n nq u y 6= 0. But then from Theorem 3 it follows that the only isomorphism between Σ1 and Σ2 is the identity map.

Any n × n minor of O(ΣS ) or of R(ΣS ) can be viewed as a polynomial in the entries of the matrices of ΣS . Since the entries of the matrices of ΣS are linear functions of the entries of the parameters of S, it follows that any n × n minor of O(ΣS ) and R(ΣS ) can be viewed as polynomial in S, where S is identified with an element of RK . Let P1 and P2 be the set of all n × n minors of O(ΣS ) and respectively R(ΣS ), each minor being viewed as a polynomial RK . Define the polynomials X Pobs (X1 , . . . , XK ) = (P (X1 , . . . , XK ))2 P ∈P1

Pcontr (X1 , . . . , XK ) =

X

(P (X1 , . . . , XK ))2

P ∈P2

It is then clear that Pobs (S) 6= 0 if and only if at least one of the n × n minors of O(ΣS ) is not zero, i.e. if and only if rankO(ΣS ) = n. That is, Pobs (S) 6= 0 if and only if ΣS is observable. Similarly, Pcontr (S) 6= 0 if and only if rankR(ΣS ) 6= 0, i.e if and only if ΣS is reachable. Define now Pmin = Pcontr Pobs . Then Pmin (S) 6= 0 if and only if ΣS is both observable and reachable, i.e. if and only if ΣS is minimal. Finally, consider a polynomial parametrization Π such that Π contains a strongly minimal element. The fact that Π is a polynomial parametrization implies that there exists polynomials Πi in variables X1 , . . . , Xd , i = 1, . . . , K such that Π(θ) = (Π1 (θ), . . . , ΠK (θ)) for all θ ∈ Θ. Here we used the identification of a SARX system of type (ny , nu ) with a point in RK . Consider the polynomial Qmin (X1 , . . . , Xd ) = Pmin (Π1 (X1 , . . . , Xd ), . . . , . . . ΠK (X1 , . . . , Xd )). Notice that the set of parameters from Θ which do not yield a minimal SARX system all satisfy the equation Qmin (θ) = 0. From the assumption that Π contains a strongly minimal element it follows that for some θ ∈ Θ, Qmin (θ) = Pmin (Π(θ)) 6= 0. Hence, the set G = {θ ∈ Θ | Qmin (θ) 6= 0} is a non-empty subset of Θ and it is clearly generic. That is, Π is generically strongly minimal, and hence minimal. Proof. [Proof of Corollary 3] If Π is generically strongly minimal, then there exists a generic set G ⊆ Θ such that the parametrization Π|G : G 3 θ 7→ Π(θ) is strongly minimal. Hence, by Theorem 2, Π|G is identifiably. This means that Π is generically strongly identifiable. Proof. [Proof of Corollary 4] If Π contains a strongly minimal element, then by Theorem 4 Π is generically strongly minimal. The rest follows from Corollary 3.

Proof. [Proof of Corollary 5] By Example 2, there exists a strongly minimal SARX system, i.e. SARXtriv contains a strongly minimal element. Moreover, SARXtriv is clearly injective and polynomial. The statement follows now TheProof. [Proof of Theorem 4] Let K = (pny +mnu )|Q|.Then orem 4 and Corollary 4. any SARX system of type (ny , nu ) can be identified with a point in RK , by identifying the system with the collection of its parameters {nq }q∈Q , nq ∈ Rp×(pny +mnu ) . First, we construct a polynomial Pmin (X1 , . . . , XK ), such that Pmin (S) 6= 0 if and only if S is strongly minimal. To this end, consider the DTLSS ΣS and consider the observability and controllability matrix O(ΣS ) and R(ΣS ).