Realization theory for time-varying, discrete-time linear systems
Consider a family of difference equations \(x_{k+1}=A_ k x_ k + B_ k u_ k\), \(y_ k = C_ k x_ k\), where \(B_ k : \mathbb{C}^ m \to {\mathfrak H},\) \(A_ k: {\mathfrak H} \to {\mathfrak H}\) and \(C_ k : {\mathfrak H} \to C^ p\) are linear bounded operators (\({\mathfrak H}\) is a Hilbert space). Construct the input-output operator \[ F = \left[ \begin{matrix} 0 & 0 & 0 & \cdots \\ C_1 B_0 & 0 & 0 & \cdots \\ C_2 A_1 B_0 & C_2 B_1 & 0 & \cdots \\ C_3 A_2 A_1 B_0 & C_3 A_2 B_1 & C_3 B_2 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{matrix} \right] \] and the transfer function \(F(\lambda) = U(\lambda)FU(\lambda)^{-1}\), \(|\lambda| = 1\), where \[ U(\lambda) \langle x_0, x_1, x_2, \ldots \rangle = \langle x_0, \lambda^{-1} x_ 1,\lambda^{-2} x_2, \ldots\rangle. \] It turns out that \(F(\lambda)\) admits an extension (again denoted by \(F(\lambda)\)) to a bounded analytic operator function on the unit disc. Realizations of \(F(\lambda)\) are studied, i.e., representations of the form \(F(\lambda) = \lambda C(I-\lambda A)^{-1}B\), where \(A\), \(B\) and \(C\) are Hilber t space linear bounded operators. In particular, a minimal realization for \(F(\lambda)\) is constructed.
scientific article; zbMATH DE number 4066714
scientific article