ABSTRACT

The classes of the systems studied in this book are the following discrete-time time-invariant linear dynamical systems: the known but not sufficiently explored IO systems, the very well known and studied but still not completely explored ISO systems, and new, more general, IIO systems. We start with the Input‐Output (IO) systems determined by (2.1): ∑ r = 0 r = ν A r E r Y ( k ) = ∑ r = 0 r - μ B r E r I ( k ) , det A ν ≠ 0 , ∀ k ∈ N 0 , v ≥ 1 , 0 ≤ μ ≤ v , $$ \mathop \sum \limits_{{r = 0}}^{{r = \nu }} A_{r} E^{r} {\text{Y}}(k) = \sum\limits_{{r = 0}}^{{r - \mu }} {B_{r} } E^{r} {\mathbf{I}}(k),\,{\text{~det~}}A_{\nu } \ne 0,\,\forall k \in N_{0} ,v \ge 1,\,0 \le \mu \le v, $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781138039629/a28624dc-7a42-40c0-abd8-1d07ecbd20fe/content/math2_1.tif"/> E r Y ( k ) = Y ( k + r ) , A r ∈ R N × N , B r ∈ R N × M , | r | = 0 , 1 , · · · , V , $$ E^{r} {\mathbf{Y}}(k) = {\mathbf{Y}}(k + r),\,\,A_{r} \in {\mathcal{R}}^{{N \times N}} ,\,\,{\mathbf{B}}_{r} \in {\mathcal{R}}^{{N \times M}} ,|r| = 0,1, \cdot \cdot \cdot \,,V, $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781138039629/a28624dc-7a42-40c0-abd8-1d07ecbd20fe/content/math2_2.tif"/> μ < V ⇒ B i = O N , M , i = μ + 1 , μ + 2 , ⋯ | , V . $$ \mu < V \Rightarrow B_{i} = O_{{N,M}} ,~i = \mu + 1,~\mu + 2,~ \cdots |,~V. $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781138039629/a28624dc-7a42-40c0-abd8-1d07ecbd20fe/content/math2_3.tif"/>