Discrete Roesser state models from 2D frequency data
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Abstract
We identify a general, i.e. not necessarily denominatorseparable Roesser model from 2D discrete vectorgeometric trajectories generated by a controllable, quarterplane causal system. Our procedure consists of two steps: the first one is the computation of state trajectories from the factorization of constant matrices directly constructed from inputoutput data. The second step is the computation of the state, output, and input matrices of a Roesser model as solutions of a system of linear equations involving the given inputoutput data and the computed state trajectories.
Keywords
2D systems Roesser models Bilinear difference formsMathematics Subject Classification
93A30 93B15 93B20 93B30 93C201 Introduction and problem statement
 1.
\(m\le n\)
 2.
\(\deg (d_n(z_1)\ge \deg (n_i(z_1))\), \(i=0,\ldots ,m1\)
 3.
\(\deg (d_n(z_1))\ge \deg (d_i(z_1))\), \(i=0,\ldots ,n\).
We want to find matrices A, B, C, D such that (1) holds for the data (2) and some associated state trajectories \(\widehat{x}_i:=\begin{bmatrix} \widehat{x}_{i,1}\\ \widehat{x}_{i,2}\end{bmatrix}\), \(i=1,\ldots ,N\). Such quadruple (A, B, C, D) will be called an unfalsified Roesser model for the data (2).
Roesser model system identification has been considered previously, see Cheng et al. (2017), Farah et al. (2014), Ramos (1994), Ramos et al. (2011), Ramos and dos Santos (2011), and Ramos and Mercère (2016, 2017a, b), and it has been applied in modelling the spatial dynamics of deformable mirrors (see Voorsluys 2015), heat exchangers (see Farah et al. 2016), batch processes controlled by iterative learning control (see Wei et al. 2015), and in image processing (see Ramos and Mercère 2017b). Our approach to compute an unfalsified model differs fundamentally from previous work. It is based on an idea pursued in the 2D continuoustime case in Rapisarda and Antoulas (2016), and derived from the 1D Loewner framework, see Antoulas and Rapisarda (2015) and Rapisarda and Antoulas (2015) [and also Rapisarda and Schaft (2013) and Rapisarda and Trentelman (2011) for analogous approaches to 1D identification based on the factorization of “energy” matrices]. Namely, we use the data (2) to compute state trajectories corresponding to it, and subsequently we compute a state representation for the data and such state trajectories by solving linear equations in the unknown matrices A, B, C, D. From a methodological point of view our twostage procedure is thus analogous to 2D subspace identification algorithms: first compute state trajectories compatible with the data, then fit a statespace model to the inputoutput and the computed state trajectories. However, our approach is essentially an application of the consequences of duality, rather than shiftinvariance as in subspace identification: in our procedure, state trajectories are computed by factorizing constant matrices built from the data and its dual, rather than Hankeltype matrices consisting of shifts of the data in the two independent variables. Such aspect makes our method conceptually simple, and it helps to reduce the amount of bookkeeping necessary for calculations. Moreover, approaching the problem from a frequencydomain and a duality point of view allows us to avoid imposing restrictive assumptions on the datagenerating system, such as the separabilityinthedenominator property required by earlier work on 2D subspace identification such as Cheng et al. (2017), Ramos (1994), Ramos et al. (2011) and Ramos and dos Santos (2011). We note that the recent publication Ramos and Mercère (2017b), provides a subspace algorithm for the identification of general, i.e. not necessarily separableinthedenominator, Roesser models.
The paper is structured as follows. In Sect. 2 we gather the necessary background material; this section contains several original results in the theory of 2D bilinear and quadratic difference forms, a tool extensively used in our approach. In Sect. 3 we illustrate some original results on duality of Roesser models, including a “pairing” result crucial for our identification procedure. In Sect. 6 we illustrate our method for the identifying Roesser models. Section 7 contains some concluding remarks.
Notation We denote by \(\mathbb {R}^{m\times n}\) (respectively \(\mathbb {C}^{m\times n}\)) the set of all \(m\times n\) matrices with entries in \(\mathbb {R}\) (respectively \(\mathbb {C}\)). \(\mathbb {C}^{\bullet \times n}\) denotes the set of matrices with n columns and an unspecified (finite) number of rows. Given \(A\in \mathbb {C}^{m\times n}\), we denote by \(A^*\) its conjugate transpose. If A has full column rank, we denote by \(A^\dagger \) a leftinverse of A. If A, B are matrices with the same number of columns (or linear maps acting on the same space), \({{\mathrm{\mathrm col}}}(A,B)\) is the matrix (map) obtained stacking A on top of B.
\(\mathbb {C}[z_1^{1},z_1,z_2^{1},z_2]\) is the ring of bivariate Laurent polynomials in the indeterminates \(z_1\), \(z_2\) with complex coefficients, and \(\mathbb {C}^{m\times n}[z_1^{1},z_1,z_2^{1},z_2]\) that of \(m\times n\) bivariate Laurent polynomial matrices. The ring of \(m\times n\) Laurent polynomial matrices with real coefficients in the indeterminates \(\zeta _1\), \(\zeta _2\), \(\eta _1\), \(\eta _2\) is denoted by \(\mathbb {R}^{m\times n}[\zeta _1^{1},\zeta _1,\zeta _2^{1},\zeta _2,\eta _1^{1},\eta _1,\eta _2^{1},\eta _2]\).
We denote by \(\left( \mathbb {C}^\mathtt{w}\right) ^{\mathbb {Z}^2}\) the set \(\left\{ w:\mathbb {Z}^2\rightarrow \mathbb {C}^\mathtt{w}\right\} \) consisting of all sequences of \(\mathbb {Z}^2\) taking their values in \(\mathbb {C}^\mathtt{w}\), and by \(\ell _2(\mathbb {Z}^2,\mathbb {C}^\mathtt{w})\) the set of squaresummable sequences in \(\left( \mathbb {C}^\mathtt{w}\right) ^{\mathbb {Z}^2}\). The notation \((\cdot ,\cdot )\) appended to a symbol (e.g. u) is used to denote a trajectory \(u:\mathbb {Z}^2\rightarrow \mathbb {C}^\mathtt{{u}}\). With slight abuse of notation, given \(\lambda \in \mathbb {C}\) we denote by \(\exp _{\lambda }\) the geometric series whose value at \(k\in \mathbb {Z}\) is \(\exp _{\lambda }(k):=\lambda ^k\).
2 Background material
2.1 Controllable 2D behaviors
In this paper we follow the behavioral and algebraic terminology of Rocha (1990) and Rocha and Willems (1991) [see also Rocha and Zerz (2005) for refinements of such behavioral concepts and for the corresponding algebraic characterizations]. A \(\mathtt{g}\times \mathtt{w}\) Laurent polynomial matrix R in the indeterminates \(z_1,z_2\) is called left prime if the existence of some \(D, R^\prime \in \mathbb {R}^{\bullet \times \bullet }[z_1^{1},z_1,z_2^{1},z_2]\) for which equality \(R=DR^\prime \) holds implies that D is unimodular. The property of rightprimeness is defined analogously. The following is a characterization of controllable behaviors [see p. 414 of Rocha and Willems (1991) for the definition].
Proposition 1
 1.
\(\mathfrak {B}\in \mathcal{L}^{\mathtt{w}}_2\) is controllable;
 2.
\(\mathfrak {B}\) is the closure of \({\mathfrak {B}\cap \ell _2\left( \mathbb {Z}^2,\mathbb {R}^\mathtt{w}\right) }\) in the topology of pointwise convergence;
 3.
\(\exists \) \(\mathtt{g}\in \mathbb {N}\), \(R\in \mathbb {R}^{\mathtt{g}\times \mathtt{w}}[z_1^{1},z_1,z_2^{1},z_2]\) left prime such that \(\mathfrak {B}=\ker ~R(\sigma _1,\sigma _2)\);
 4.\(\exists \) \(\mathtt{l}\in \mathbb {N}\), \(M\in \mathbb {R}^{\mathtt{w}\times \mathtt{l}}[z_1^{1},z_1,z_2^{1},z_2]\) right prime such that$$\begin{aligned} \mathfrak {B}=\left\{ w \text{ s.t. } \text{ there } \text{ exists } \ell \in \left( \mathbb {C}^\mathtt{w}\right) ^{\mathbb {Z}^2} \text{ s.t. } w=M\left( \sigma _1^{1},\sigma _1,\sigma _2^{1},\sigma _2\right) \ell \right\} \end{aligned}$$(4)
Proof
See Theorem 1 p. 415 of Rocha and Willems (1991). \(\square \)
It follows from Proposition 2 of Rocha and Willems (1991) that a controllable behavior is uniquely identified by its transfer function [see p. 415 of Rocha and Willems (1991) for the definition]. Controllability is crucial for the definition of the dual of a \(\mathcal {L}_2^{\mathtt{w}}\)behavior, as defined in the next section.
2.2 Dual discrete 2D behaviors
Proposition 2
Proof
The second equality follows from \(RM=0\), \(J^2=I_\mathtt{w}\), and standard results in behavioral system theory (see Rocha 1990).
\(\square \)
Corollary 1
Assume that \(\mathfrak {B}\in \mathcal {L}^\mathtt{w}_2\) is controllable, and let \(J=J^\top \) be such that \(J^2=I_\mathtt{w}\). Denote the transfer function of \(\mathfrak {B}\) by \(H(z_1,z_2)\) and the transfer function of \(\mathfrak {B}^{\perp _J}\) by \(H^\prime (z_1,z_2)\). Then \(H^\prime (z_1,z_2)=H(z_1^{1},z_2^{1})^\top \).
We conclude this section showing how to compute trajectories of the dual system from those of the primal one with the mirroring technique [see also Kaneko and Rapisarda (2003), Kaneko and Rapisarda (2007) and Rapisarda and Willems (1997) for the use of such idea in the 1D case]. Given the importance of dual trajectories in our identification procedure, such technique is crucial to our approach.
Proposition 3
Let \(\mathfrak {B}\in \mathcal {L}_2^\mathtt{w}\) be controllable, and \(J=J^\top \in \mathbb {R}^{\mathtt{w}\times \mathtt{w}}\) be such that \(J^2=I_\mathtt{w}\). Let \(\overline{w}\in \mathbb {C}^\mathtt{w}\), and denote by \(w(\cdot ,\cdot )\in \mathfrak {B}\) a trajectory whose value at \((k_1,k_2)\) is \(\overline{w} {\lambda _1}^{k_1} {\lambda _2}^{k_2}\). Assume that \(\overline{v}\in \mathbb {C}^\mathtt{w}\) satisfies \(\overline{v}^*\overline{w}=0\). Then the trajectory \(v(\cdot ,\cdot )\) whose value at \((k_1,k_2)\) is \(J\overline{v} \left( \frac{1}{\lambda _1^{*}}\right) ^{k_1} \left( \frac{1}{\lambda _2^{*}}\right) ^{k_2}\) belongs to \(\mathfrak {B}^{\perp _J}\).
Proof
Example 1
2.3 2D bilinear and quadratic difference forms
Bilinear and quadratic difference forms (BdF and QdF in the following) have been used in the analysis of stability of 2D discrete systems in Kojima et al. (2007, 2011), NappAvelli et al. (2011a, b) and Rapisarda and Rocha (2012). We briefly review them here, and introduce some novel results.
Proposition 4
 1.
\(\partial \left( \varPhi (\zeta ,\eta )\right) =0_{\mathtt{w}_1\times \mathtt{w}_2}\);
 2.
There exists a vBdFs \({{\mathrm{\mathrm col}}}(\varPsi _1,\varPsi _2)\) such that \(\varPhi (\zeta ,\eta )=\nabla \varPsi (\zeta ,\eta )\);
 3.
\(\sum _{k_1=\infty }^{+\infty } \sum _{k_2=\infty }^{+\infty } Q_\varPhi (w)(k_1,k_2)=0\) for all \(w\in \ell _2(\mathbb {Z}^2,\mathbb {R}^\mathtt{w})\).
Proof
See Proposition 1 p. 1524 of Kojima and Kaneko (2014). \(\square \)
The following result states that there are nonzero vBdFs whose divergence is zero; this is wellknown in the continuous case, even considering nonconstant fields.
Proposition 5
 1.
\(\nabla ~{{\mathrm{\mathrm col}}}(L_{\varPsi _1},L_{\varPsi _2})=0\);
 2.
\((\zeta _1\eta _11) \varPsi _1(\zeta ,\eta )+(\zeta _2\eta _21) \varPsi _2(\zeta ,\eta )=0\);
 3.There exists \(\varPsi \in \mathbb {R}^{\mathtt{w}\times \mathtt{w}}[\zeta _1^{1},\zeta _1,\zeta _2^{1},\zeta _2,\eta _1^{1},\eta _1,\eta _2^{1},\eta _2]\) such that$$\begin{aligned} \varPsi _1(\zeta ,\eta )= & {} (\zeta _2\eta _21)\varPsi (\zeta ,\eta )\\ \varPsi _2(\zeta ,\eta )= & {} (\zeta _1\eta _11)\varPsi (\zeta ,\eta )\; . \end{aligned}$$
Proof
The equivalence of statements (1) and (2) follows from the relation (9) between the discrete divergence and its fourvariable polynomial representation. The implication \((3) \Longrightarrow (2)\) follows from straightforward verification.
To prove the implication \((2) \Longrightarrow (3)\) observe first that if \((\zeta _1\eta _11) \varPsi _1(\zeta ,\eta )=(\zeta _2\eta _21) \varPsi _2(\zeta ,\eta )\), then \(\varPsi _1(\zeta ,\eta )\) is divisible by \((\zeta _2\eta _21)\), and \(\varPsi _2(\zeta ,\eta )\) by \((\zeta _1\eta _11)\). Consequently, there exist \(\varPsi _j^\prime (\zeta ,\eta )\), \(j=1,2\) such that \(\varPsi _j(\zeta ,\eta )=(\zeta _i\eta _i1)\varPsi _j^\prime (\zeta ,\eta )\), \(i,j=1,2\), \(i\ne j\). Statement (3) follows readily from such equality. \(\square \)
3 Dual Roesser representations and pairing
To the best of the author’s knowledge and despite the popularity of Roesser models, what the dual system is of one admitting such a representation, and whether such dual system admits a Roesser representation itself, have not been investigated before. In this section we prove some statements related to such issues, which moreover are crucial for our system identification approach. Foremost among them is a “pairing” result, showing how bilinear forms on the external variables of the primal and the dual system are related to bilinear forms on the state variables of the two systems.
In the following result we show that the transfer function of the dual system (10) is related to that of (1) by a straightforward transformation.
Proposition 6
Proof
\(\square \)
Proposition 7
Proof
We can reformulate the result of Proposition 7 saying that the bilinear form induced by \(\,J\) on the external trajectories of the primal and the dual system is the divergence of the field induced by the inner products on the first and second state components of the primal and the dual system. Such relation is of paramount importance for our system identification procedure.
4 The data matrix and the 2D Stein matrix equation
The following result [that uses the definition of observability on p. 6 of Roesser (1975)] implies that if the external trajectories are vectorgeometric, also the corresponding state trajectories of the primal system are vectorgeometric.
Proposition 8
Let \({{\mathrm{\mathrm col}}}(x,u,y)\) satisfy (1). Assume that \(u=\overline{u}~ {\exp _{\lambda _1} \exp _{\lambda _2}}\) and \(y=\overline{y}~ {\exp _{\lambda _1} \exp _{\lambda _2}}\) for some \( \overline{u}\in \mathbb {C}^{\mathtt{m}}\) and \( \overline{y}\in \mathbb {C}^{\mathtt{p}}\), respectively, and some \(\lambda _i\in \mathbb {C}\), \(i=1,2\). Assume that the representation (1) is observable. Then there exists a unique \(\overline{x}\in \mathbb {C}^{\mathtt{n}_1+\mathtt{n}_2}\) such that the state trajectory corresponding to u and y is \(x=\overline{x}~ {\exp _{\lambda _1} \exp _{\lambda _2}}\).
Proof
The fact that the state trajectory x is vectorgeometric follows in a straightforward way from the second equation in (1) and the fact that u and y) are vectorgeometric and associated with the same 2D frequency \((\lambda _1,\lambda _2)\). The uniqueness of the state trajectory, and consequently of the state direction \(\overline{x}\), follows from the observability of (1). \(\square \)
A result analogous to Proposition 8 also holds true for the dual system represented by (10) and equivalently by (11); we will not state it explicitly here.
Proposition 9
Proof
The claim follows in a straightforward way by applying the pairing equation (12) to the vectorgeometric data \(w_i^\prime \), \(w_j\) and the associated state trajectories \(x_i^\prime \), \(x_j\). \(\square \)
The solutions of the linear matrix equation (18) can be parametrized in a straightforward way, as we now show. We first show how to compute one pair of solutions, and then consider the homogeneous matrix equation associated with (18).
Proposition 10
 1.
The pair \((\mathcal {S}_1^\prime ,\mathcal {S}_2^\prime )\) is a solution of (20);
 2.The following equality holds:\(k,j=1,\ldots ,N\);$$\begin{aligned} \mathcal {S}_1^\prime (k,j)\left( \lambda ^j_1 \mu _1^{k*}1 \right) +\mathcal {S}_2^\prime (k,j)\left( \lambda ^j_2 \mu _2^{k*} 1\right) =0\; , \end{aligned}$$
 3.The following equality holds:\(k,j=1,\ldots ,N\).$$\begin{aligned} \mathcal {S}_1^\prime (k,j)=\mathcal {S}_2^\prime (k,j)\frac{\lambda ^j_2 \mu _2^{k*} 1 }{\lambda ^j_1 \mu _1^{k*}1 }\; , \end{aligned}$$(21)
On the basis of the result of Proposition 10, we characterize all solutions to (18) as follows.
Corollary 2
5 Computation of state trajectories
 Input:

Primal and dual data as in (14)
 Output:

An unfalsified Roesser model for the data.
 Step 1:

Construct the matrix \(\mathcal {D}\) defined by (15) from the data (14).
 Step 2:

Compute a pair \((\mathcal {S}_1,\mathcal {S}_2)\) of solutions to the matrix equation (23).
 Step 3:
 Perform a rankrevealing factorization of \(\mathcal {S}_k=F_{k}^{\prime \top } F_k\), i.e.$$\begin{aligned} {{\mathrm{\mathrm rank}}}(\mathcal {S}_k)={{\mathrm{\mathrm rank}}}(F_k)={{\mathrm{\mathrm rank}}}(F^\prime _k)\; , \; k=1,2\; . \end{aligned}$$
 Step 4:
 Defineand solve for \(A_{ij}\), \(B_{i}\), \(C_i\), \(i,j=1,2\) and D in$$\begin{aligned} Y:= & {} \begin{bmatrix} \overline{y}_1&\ldots&\overline{y}_N\end{bmatrix}\in \mathbb {C}^{\mathtt{p}\times N}\nonumber \\ U:= & {} \begin{bmatrix} \overline{u}_1&\ldots&\overline{u}_N\end{bmatrix}\in \mathbb {C}^{\mathtt{m}\times N}\; , \end{aligned}$$(24)$$\begin{aligned} \begin{bmatrix} F_1 \varLambda _1\\F_2\varLambda _2\\Y \end{bmatrix}=\begin{bmatrix} A_{11}&A_{12}&B_1\\ A_{21}&A_{22}&B_2\\ C_1&C_2&D \end{bmatrix} \begin{bmatrix} F_1\\ F_2\\ U \end{bmatrix}\; . \end{aligned}$$(25)
 Step 5:

Return A, B, C, D.
 Identifiability:

Recall from Proposition 5 that \(\overline{\mathcal {S}}_k\), \(k=1,2\) defined in (16) are not the only solutions of 2D matrix Stein equation (23): if \(\mathcal {S}_k^\prime \), \(k=1,2\) satisfy the homogeneous 2D Stein matrix equation (20), then also \(\overline{\mathcal {S}}_k+\mathcal {S}_k^\prime \), \(k=1,2\) are solutions of (23). The converse also holds: since the matrix equation (23) is linear, it follows that every solution pair to it can be written as \(\overline{\mathcal {S}}_k+\mathcal {S}_k^\prime \) for some pair \((\mathcal {S}_1^\prime ,\mathcal {S}_2^\prime )\) solving (20) and \(\overline{\mathcal {S}}_k\) defined in (16). Such nonunicity of the solutions to (23) is an unavoidable consequence of the noninvertibility of the divergence operator, or equivalently the existence of nonzero solution pairs to (20). Consequently, identifiability is not a wellposed question in the identification approach sketched in Algorithm 1.
 Complexity:

Another issue arising from the nonunicity of the solutions to (23) is the fact that Algorithm 1 may compute models having larger state dimension than that of the generating system. Note that the sum of the ranks of \(\mathcal {S}_k\), \(k=1,2\) coming from a generic solution pair computed in Step 2 will generically be \(N+N\), and consequently presumably higher than the minimal state dimension of a Roesser model of the datagenerating system. Thus generically the Roesser model computed in Step 4 would be highdimensional and impractical for use e.g. in simulation, control, and so forth. In Sect. 5.1 we illustrate a procedure using rankminimization to compute a minimal complexity model; see also Remark 1 where an alternative approach using Gröbner basis computation is sketched.
 Computation of A, B, C, D:

Finally, sufficient conditions must be established guaranteeing that solutions A, B, C, D exist to the system of linear equations (25).
The rest of this section is devoted to modifying Algorithm 1 to address the complexity issue; the solvability of the system of linear equations (25) is considered in Sect. 6. We will show that with small modifications our data matrixbased approach to Roesser model identification offers the opportunity to address in a conceptually simple way the problem of deriving a minimalcomplexity unfalsified Roesser model.
5.1 Computation of minimal complexity state trajectories
Proposition 11
Proof
We can now refine Algorithm 1 as follows.
 Input:

Primal and dual data as in (14)
 Output:

A minimal complexity unfalsified Roesser model for the data.
 Step 1:

Construct the matrix \(\mathcal {D}\) defined by (15) from the data (14).
 Step 2:

Define \(\mathfrak {C}\) as in Proposition 11, and solve the optimization problem (26).
 Step 3:
 Perform a rankrevealing factorization of \(\mathcal {S}_k=F_{k}^{\prime \top } F_k\), i.e.$$\begin{aligned} {{{\mathrm{\mathrm rank}}}(\mathcal {S}_k)={{\mathrm{\mathrm rank}}}(F_k)={{\mathrm{\mathrm rank}}}(F^\prime _k)\; , \; k=1,2\; .} \end{aligned}$$
 Step 4:
 Defineand solve for \(A_{ij}\), \(B_{i}\), \(C_i\), \(i,j=1,2\) and D in$$\begin{aligned} {Y}&{:=}&{\begin{bmatrix} \overline{y}_1&\ldots&\overline{y}_N\end{bmatrix}\in \mathbb {C}^{\mathtt{p}\times N}}\nonumber \\ {U}&{:=}&{\begin{bmatrix} \overline{u}_1&\ldots&\overline{u}_N\end{bmatrix}\in \mathbb {C}^{\mathtt{m}\times N}}\; , \end{aligned}$$(27)$$\begin{aligned} {\begin{bmatrix} F_1 \varLambda _1\\F_2\varLambda _2\\Y \end{bmatrix}=\begin{bmatrix} A_{11}&A_{12}&B_1\\ A_{21}&A_{22}&B_2\\ C_1&C_2&D \end{bmatrix} \begin{bmatrix} F_1\\ F_2\\ U \end{bmatrix}}\; . \end{aligned}$$(28)
 Step 5:

Return A, B, C, D.
Remark 1
In Rapisarda (2017) a Gröbner basis approach to solve the rank minimization problem (26) is illustrated. Such approach uses a parametrization similar to that of Proposition 10 in order to transform the problem of finding fixedrank matrices solving the 2D Sylvester equation (the continuous counterpart of the Stein equation) into a polynomial algebraic problem. In order to compute a minimal complexity Roesser model for the data, beginning with \(c=1\) we check whether there exist solution pairs \((\mathcal {S}_1,\mathcal {S}_2)\) to (23) such that \({{\mathrm{\mathrm rank}}}(\mathcal {S}_k)=\mathtt{n}_k\), \(k=1,2\) and \(\mathtt{n}_1+\mathtt{n}_2=c\). If no such solution pair exists, we increment c by 1 and repeat the check. Note that working under the assumption \(\lambda _k^i \mu _k^j\ne 1\), \(i,j=1,\ldots ,N\), \(k=1,2\), equation (23) is solved by \((\overline{\mathcal {S}}_1,\overline{\mathcal {S}}_2)\), where \(\overline{\mathcal {S}}_i\), \(i=1,2\), are the unique solution to (19) when \(\mathcal {Q}=\frac{1}{2}\mathcal {D}\), \(M=M_i\), \(\varLambda =\varLambda _i\), \(i=1,2\). Consequently, such search ends after at most N steps.
The largest part of the computational effort of such approach is due to the complexity of Gröbner basis calculations, which becomes especially heavy for problems involving more than ten data trajectories. However, such approach has the advantage that a parametrization of all solutions to (23) with a given total rank is obtained; a numerical approach based on rankminimization algorithms only produces one solution among many. Consequently, such parametrization opens up the possibility of exploring the space of unfalsified models of given complexity, with potential application to 2D datadriven model order reduction [see Rapisarda and Schaft (2013) and Rapisarda and Trentelman (2011) for the 1D case]. Such procedure also shares with the one sketched in Algorithm 2 a conceptually appealing simplicity that avoids some difficulties inherent in other approaches based on shiftinvariance. \(\square \)
6 Identification of Roesser models
Proposition 12
Let \(\mathfrak {B},\mathfrak {B}^{\perp }\in \mathcal {L}^\mathtt{w}_2\) be controllable. Let data (14) be given and define \(\mathcal {D}\) by (15), \(\varLambda _i,M_i\), \(i=1,2\) by (16), U, Y by (24), and \(U^\prime \), \(Y^\prime \) by (29).
 1.
\({{\mathrm{\mathrm im}}}~M_1^*F^{\prime *}_1\cap {{\mathrm{\mathrm im}}}~M_2^*F^{\prime *}_2=\{0\}\);
 2.
\({{\mathrm{\mathrm im}}}~\begin{bmatrix} M_1^*F^{\prime *}_1&M_2^*F^{\prime *}_2\end{bmatrix}\cap {{\mathrm{\mathrm im}}}~ U^{\prime *}=\{0\}\),
 3.
\({{\mathrm{\mathrm im}}}~Y^{\prime *}\cap {{\mathrm{\mathrm im}}}~U^{\prime *}=\{0\}\).
Proof
Remark 2
The sufficient conditions stated in Proposition 12 fall short of being completely satisfactory, since they involve the matrices arising from the factorizations of \(\mathcal {S}_k\) rather than the matrices \(\mathcal {S}_k\), \(k=1,2\), themselves, or in the best case, the inputoutput data itself. We also do not make any claim about the conservativeness of such sufficient conditions. The issue of deriving tighter conditions expressed only in terms of the inputoutput data is a pressing issue for further research. \(\square \)
7 Conclusions
We have presented a novel approach to the identification of unfalsified Roesser discrete models from vectorgeometric data, based on the idea of first computing state trajectories compatible with the given inputoutput trajectories, and secondly using such trajectories together with the data in order to compute the A, B, C, D matrices of the Roesser equations. Our procedure is based on new results concerning duality of such models (Sect. 3) and on a parametrization of the solutions to the 2D Stein matrix equation (Sect. 4). Such results lead to an algorithm for the computation of state trajectories (Sect. 5), which can be refined in a straightforward way to one for the computation of minimal complexity ones (Sect. 5.1). The 2D Stein equation is exploited once again in order to find an unfalsified model for the inputoutput data and the computed state matrices (see Sect. 6).
In several preceding publications concerned with linear timeinvariant systems, the author and his collaborators put forward an “energy”based approach to identification. Given the abundance of powerful methods to solve system identification problems for such class of systems, it can be argued that such results amounted to a relatively minor contribution. The author hopes that the application of such ideas to 2D systems as in Rapisarda and Antoulas (2016) and in the present paper, may shift the balance of judgment more in his favour. The ideas underlying the approach presented here and in the germane publication Rapisarda and Antoulas (2016) can be applied to a wider class of systems, and to more general classes of data than vectorgeometric or exponential ones. Their potential lies in the generality of the idea of duality, which we believe can be used to overcome the difficulties (e.g. of bookkeeping) inherent in applying shiftinvariance techniques to multidimensional systems, or to bypass them altogether for system classes where such property is not satisfied (e.g. 1D timevarying and nonlinear systems, for which promising results are being obtained as we write).
Limiting ourselves to the class of multidimensional systems considered in this paper, three areas of research are currently being investigated. Firstly, we need to generalize our approach to the case of data other than vectorgeometric through the use of compactsupport trajectories and infinite series involving their shifts [as in the 1D case, see Rapisarda and Trentelman (2011)]. Secondly, we want to identify other classes of 2D systems than the Roesser one, amenable to be identified with duality ideas. The main issue to be addressed is to determine classes of systems admit a “pairing relation” with their dual, which can be expressed as the divergence of a field involving the state trajectories, and if possible algebraically characterize such property. Moreover, it is important to ascertain whether such divergence is amenable to a computationally straightforward treatment; for example, in Rem. 8 p. 2751 of Rapisarda and Antoulas (2016) it has been shown that (continuoustime) FornasiniMarchesini models have been shown to admit a pairing relation, but one which does not seem to be conducive to a direct exploitation to derive from it state trajectories. Finally, we plan to investigate whether duality relations can be used to compute minimal Roesser model from nonminimal ones, and in the problem of statespace realisation from transfer functions.
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