Abstract
We identify a general, i.e. not necessarily denominator-separable Roesser model from 2D discrete vector-geometric trajectories generated by a controllable, quarter-plane 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 input-output 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 input-output data and the computed state trajectories.
Highlights
Introduction and problem statementDiscrete Roesser state-space models, introduced in Roesser (1975), are of the form σ1 x1 σ2 x2 = A11 A12 A21 A22 x1 x2 +B1 B2 u y = C1 C2 + Du, (1)where xi (k1.k2) ∈ Rni for all (k1, k2) ∈ Z2, Ai j ∈ Rni ×n j i, j = 1, 2; u(k1, k2) ∈ Rm and y(k1, k2) ∈ Rp for all (k1, k2) ∈ Z2; and B := B1 B2 ∈ R(n1+n2)×m, C := C1 C2 ∈ Rp×(n1+n2), D ∈ Rp×m
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
Proof The fact that the state trajectory x is vector-geometric follows in a straightforward way from the second equation in (1) and the fact that u and y) are vector-geometric and associated with the same 2D frequency (λ1, λ2)
Summary
In this paper we solve the following identification problem: we are given a finite set consisting of N polynomial vector-geometric input-output trajectories wi :=. Our approach is essentially an application of the consequences of duality, rather than shift-invariance as in subspace identification: in our procedure, state trajectories are computed by factorizing constant matrices built from the data and its dual, rather than Hankel-type matrices consisting of shifts of the data in the two independent variables. Such aspect makes our method conceptually simple, and it helps to reduce. We define vec as the linear map defined by vec : Rm×n → Rmn vec ai j i=1,...,m, j=1,...,n := a11 . . . a1n . . . am1 . . . amn , and mat as the linear map defined by mat : Rmn → Rm×n and mat a11 . . . a1n . . . am1 . . . amn := ai j i=1,...,m, j=1,...,n
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