Continuity of dynamical systems: A system theoretic approach
The paper studies the convergence of time-invariant linear discrete-time systems from a ``behavioural'' point of view [see \textit{J. C. Willems}, Automatica 22, 675-694 (1986; Zbl 0628.62088); and ibid. 23, 87-115 (1987; Zbl 0628.62089)]. The systems are described by autoregressive linear equations and are identified with the solution space \(B\subset W^{{\mathbb{Z}}}\) of these equations, where \(W={\mathbb{R}}^ q\) is the corresponding signal alphabet. A family of systems \(\Sigma_{\epsilon}\), \(\epsilon >0\) is said to converge to a system \(\Sigma_ 0\) if the associated closed, linear shift-invariant subspaces \(B_{\epsilon}\) of \(W^{{\mathbb{Z}}}\) (provided with the pointwise topology) converge to the subspace \(B_ 0\). Under the assumption that the system lags are uniformly bounded it is shown that \(\Sigma_{\epsilon}\to \Sigma_ 0\) as \(\epsilon\) \(\to 0\) if and only if the system \(\Sigma_{\epsilon}\) can be described by autoregressive equations whose coefficients converge to the coefficients of a suitable system equation for \(\Sigma_ 0\).
time-invariant linear discrete-time systems
scientific article; zbMATH DE number 4074888