An Interconnection-Based Interpretation of the Loewner Matrices*

We introduce a novel interpretation of the Loewner and shifted Loewner matrices constructed from tangentatial interpolation data. This interpretation requires the development of new objects: the left- and right-Loewner matrices. Once the interpretation is introduced, a method for the on-line estimation of the Loewner and shifted Loewner matrices is presented.


I. INTRODUCTION
The generalized realization problem [1] for linear systems involves the construction of a minimal state-space model of an underlying system consistent with data obtained by sampling the transfer matrix at points of the complex plane [1]. The problem is closely related to the rational interpolation problem, which seeks to construct a rational matrix function consistent with given interpolation data [1], [2], [3]. A particular general case of the rational interpolation problem involves using tangential data, that is matrix data sampled in specific directions. This has been used for the model reduction of multi-variable systems in [4] and generalized state-space (descriptor systems) in [1]. A relaxed version of the rational interpolation problem is the so-called problem of rational interpolation without a specific area of analyticity. This has been widely studied and solution approaches include the generating system approach, which produces a matrix function that parameterizes the set of all interpolants [3], [5], [6], and the method of using the Hankel matrix to directly construct state-space models [7], [8], which has been used in the first treatments of the partial realization problem [9], [10].
Finally, another important tool to solve the generalized realization problem is the Loewner matrix, also known as the divided-difference matrix [1]. This matrix is closely related to the Hankel matrix [11], [12], and has been first used to solve rational interpolation problems in [13]. The Loewner matrix can be factored into generalized controllability and observability matrices, which can then be used to construct state-space models [14]. A recent result shows that the Loewner framework of [1] can be interpreted as a special case of a two-sided moment-matching procedure for * a.astolfi@imperial.ac.uk descriptor systems [15], following the development of timedomain definitions of moments in [16]. In the literature there are multiple methods for constructing state-space realizations from the knowledge of the Loewner matrices, which vary depending on the rank and size of such matrices [1]. While interesting, the actual construction of the realization using the Loewner matrices is not the objective of this paper: the interested reader should refer to Sections 4 and 5 of [1] or to [3].
One of our goals is to provide an intuitive and efficient method for determining the Loewner matrix by sampling the input and output signals of a linear system. The entries of the matrix are collected by probing the system and filtering its output with two auxiliary systems. This provides a system theoretic interpretation (interconnection based) of the Loewner matrix as the input and output "gains" of a transformed system. Such an interpretation, which only relies on an interconnection perspective, has the potential to allow defining the "Loewner matrix" for more general classes of systems, and hence to extend the tangential interpolation methods of [1] and [3] to (classes of) nonlinear systems and time-varying systems.
The structure of this paper is as follows. In Section 2 notation and terminology are presented, along with preliminary results and a discussion of the problem. In Section 3 a few new objects related to the Loewner and shifted Loewner matrices are introduced. In Section 4 a conceptual experimental setup yielding an interconnection-based interpretation of the Loewner and shifted Loewner matrices is introduced. Finally, in Section 5 we present results on estimating the entries of the Loewner and shifted Loewner matrices and provide bounds on the estimation error as a function of time.

II. NOTATION, PRELIMINARY RESULTS, AND PROBLEM SETUP
In what follows we use standard notation. R denotes the set of real numbers, Z the set of integers, C the set of complex numbers, C n the set of complex vectors having n rows, and C n×m the set of complex matrices having n rows and m columns. C − denotes the elements of C having negative real part. σ(A) denotes the spectrum of a square matrix A. e i denotes a vector with all entries equal to 0 except for the i-th entry which is equal to 1. δ(t) denotes the Dirac delta function.
The paper [1] has provided methods for constructing generalized state-space representations described by equations of the form 1 with state x(t) ∈ C n , input u(t) ∈ C m , output y(t) ∈ C p , and matrices E, A, B, C, and D of appropriate dimensions, consistent with a set of so-called tangential interpolation data, or matrix data sampled in specific directions. In what follows we specialize the definitions and results of [1] considering the special case in which E = I and D = 0, therefore we consider systems of the forṁ The following assumptions hold throughout the paper. Assumption 1: The triple of matrices (A, B, C) is a minimal realization of the system (1)-(2), i.e. the system (1)-(2) is reachable and observable.
To pose any interpolation, or realization, problem, and, in particular, to define the Loewner and shifted Loewner matrices one has to introduce tangential data (see [1]). Tangential data are composed of right and left interpolation data. The right interpolation data are described by the set (3) These can be defined, equivalently, by the matrices The left interpolation data are described by the set or, equivalently, by the matrices The next assumption is required to guarantee the existence and uniqueness of matrices introduced in what follows. Assumption 2: The matrices A, Λ, and M have no common eigenvalues, that is Assumption 3: The matrix R is full rank and the matrix L is full rank. The objective of the generalized realization problem is to determine a state-space realization of the form (1)-(2) such that the associated transfer matrix, namely the rational matrix function H(s) = C(sI − A) −1 B, satisfies the interpolation conditions H(λ i )r i = w i , i = 1, . . . , ρ, and j H(µ j ) = v j , j = 1, . . . , v. The Loewner matrix, L, and the shifted Loewner matrix, σL, are defined in terms of the tangential data (3) and (4) as (see [14]) respectively. Note that, since the rational transfer matrix H(s) generates the data, the shifted Loewner matrix is the Loewner matrix associated to the transfer matrix sH(s). Furthermore, the Loewner matrix is the unique (by Assumption 2) solution of the Sylvester equation and the shifted Loewner matrix is the unique (again by Assumption 2) solution of the Sylvester equation where The matrices Y and X are referred to as the tangential generalized observability and tangential generalized controllability matrices, respectively. These are the solutions of the Sylvester equations respectively.
For the purposes of this paper it is sufficient to know that in order to construct the realization we must obtain the matrices 2 W , V , L, and σL, however, because of Assumption 3 and because σL − LΛ = V R and σL − M L = LW , we need only be concerned with determining L and σL. As a result, the primary focus of the paper is to develop a system theoretic interpretation of the Loewner matrices, L and σL, with the additional goal of developing a framework allowing the experimental determination of the Loewner matrices for the system (1)-(2) when A, B, and C are unknown.

III. THE LEFT-AND RIGHT-LOEWNER MATRICES
Before presenting the main results of the paper we must define a few additional objects. These objects are all welldefined due to Assumption 2. First, define the left-Loewner matrix, L , and the right-Loewner matrix, L r , as the unique solutions of the Sylvester equations In a similar fashion, we define the left-shifted-Loewner matrix, σL , and the right-shifted-Loewner matrix, σL r , as the unique solutions of the Sylvester equations These definitions, along with (5)- (6), carry the implication that and that σL = σL + σL r .

IV. THE CONCEPTUAL EXPERIMENTAL SETUP
In this section we develop a conceptual experimental setup for the construction of a sampled Loewner matrix corresponding to tangential interpolation data of the form (3) and (4). To this end we use the tangential data in matrix form to construct the two systems 3 with states ζ r (t) ∈ C ρ and ζ (t) ∈ C v , inputs ∆(t) ∈ C ρ and χ(t) ∈ C p , and outputs v(t) ∈ C m and η(t) ∈ C v . Consider now the interconnection equations v = u and χ = y, see Figure 1. The state-space realization of the resulting interconnected system is therefore that is a system with state ζ r x ζ , input ∆, and output η. Note that all information on the Loewner matrices and the generalized controllability and observability matrices are (somehow) encoded in the system (21)-(22), but it is not obvious how to retrieve such information, i.e. the matrices L , L r , X, and Y do not appear explicitly in the realization (21)-(22).
To expose such matrices one has to select a specific set of coordinates, which allows rewriting the interconnected system (21)-(22) as a parallel interconnection (note that (21)-(22) has a natural series interconnection form).

Theorem 1: Consider the system (21)-(22). The coordinates transformation
is such that the system in the new coordinates is described by the equations Proof: The proof is obtained via a direct calculation. Differentiating with respect to time z c = x − Xζ r yieldṡ By Assumption 2, X is the unique solution of (8), hencė z c = Az c − X∆.
Consider now z = ζ + Y x + L ζ r . Again, a direct differentiation yieldṡ which, by (9) and (10), yieldṡ Finally, which, utilizing (7) and (14), yields hence the claim. Theorem 1 lends itself to a simple interpretation: the left-Loewner, right-Loewner, tangential generalized observability, and tangential generalized controllability matrices are the input and output "gains" of three systems interconnected in parallel, as illustrated in Figure 2, such that their overall input-output behaviour coincides with that of the system (21)-(22). To obtain an analagous interpretation for the shifted Loewner matrices one has to perform time differentiation of the output of each of the systems (17)-(20). This yields the systemsζ v(t) = Rζ r (t) = RΛζ r (t) + R∆(t), with states ζ r (t) ∈ C ρ and ζ (t) ∈ C v , inputs ∆(t) ∈ C ρ and χ(t) ∈ C p , and outputs v(t) ∈ C m and η(t) ∈ C v . Note that, similarly to (17)-(20), these are constructed only using interpolation data (in matrix form).

V. ONLINE ESTIMATION OF THE LOEWNER MATRICES
In light of Theorem 1 it is now straightforward to devise a procedure which, under some additional assumptions, allows determining the entries of the Loewner matrix associated with the system (1)-(2) and the systems (17)-(20) (strictly speaking the Loewner matrix is associated to (1)-(2) and the interpolation data, however with some abuse of terminology we can associate the Loewner matrix to (1)-(2) and (17)-(20)).
Selecting the i-th row of η(t) yields η i (t) = e µit e λj t L ij L r ij + e i Y e At Xe j .
Consider now two samples of η i (t) at times t 0 and t 1 and note that one could arrange the available samples as Since, by Assumpion 2 λ j = µ i and by hypothesis t 1 − t 0 = j 2πk λj −µi , for all k ∈ Z, the matrix of exponentials can be inverted, yielding which proves the claim.
Taking any p-norm of both sides, it is easy to see that there exist γ 1 (δ) > 0 and γ 2 (δ) > 0 such that which lends itself to the following convergence result, the proof of which is straightforward, hence omitted.
Proposition 2: Consider the interconnected system (21)-(22). Let t ∈ R and let δ ∈ R be such that δ = j 2πk λj −µi , k ∈ Z. Let the estimate of L ij be (see (29) Similarly to Theorem 1, Theorem 2 can be exploited to devise a procedure which allows determining the entries of the shifted Loewner matrix associated with the system (1)- (2) and the systems (17)-(20).
To prove case (iii), note that if µ i = 0 then σL ij = 0, hence η i (t) = λ j e λj t σL ij + e i Y Ae At AXe j and σL ij = (λ j e λj t ) −1 (η i (t) − e i Y Ae At AXe j ).
For the case (i), consider two samples of η i (t) at times t 0 and t 1 and note that one could arrange the available samples as η i (t 0 ) − e i Y Ae At0 AXe j η i (t 1 ) − e i Y Ae At1 AXe j = µ i e µit0 λ j e λj t0 µ i e µit1 λ j e λj t1 σL ij σL r ij .
Since, by Assumption 2, λ j = µ i and, by hypothesis, t 1 − t 0 = j 2πk λj −µi , for all k ∈ Z, the matrix of exponentials can be inverted, yielding σL ij = λ j e λj t1 − µ i e µit1 µ i e µit0 − λ j e λj t0 µ i λ j (e µit0+λj t1 − e µit1+λj t0 ) × η i (t 0 ) − e i Y Ae At0 AXe j η i (t 1 ) − e i Y Ae At1 AXe j , which proves the result. Similarly to Proposition 2, estimates of the entries of the shifted Loewner matrix can be defined for each of the cases (i), (ii), and (ii) discussed in Proposition 3.

VI. CONCLUSION
We have presented a new interpretation of the Loewner matrices. To this end we have defined new objects: the leftand right-Loewner matrices, the introduction of which is instrumental for the novel interpretation of the Loewner matrices constructed from tangential interpolation data. Using this interpretation we provide a method for online estimation of the Loewner matrices. Due to its systems theoretic nature, this interpretation has the potential to extend the definition of "Loewner matrices" to nonlinear systems and time-varying systems, and hence to extend previously studied tangential interpolation methods to more general classes of systems.