ABSTRACT
This section deals with physical dynamical systems in general and control systems in particular, which are mathematically described directly in the form of a time-invariant linear vector Input-Output (IO) differential equation of the classical form (2.1), ∑ k = 0 k = ν A k Y ( k ) ( t ) = ∑ k = 0 k = η D k D ( k ) ( t ) + ∑ k = 0 k = μ B k U ( k ) ( t ) = ∑ k = 0 k = ξ H k I ( k ) ( t ) , ∀ t ∈ T 0 , v ≥ 1 , ξ = max ( η , μ ) , Y ( k ) ( t ) = d k Y ( t ) d t k , 0 ≤ η ≤ ν , 0 ≤ μ ≤ ν , A k ∈ ℜ N × N , D k ∈ ℜ N × d , B k ∈ ℜ N × r , k = 0 , 1 , . . . , v , d e t A ν ≠ 0 , η < v ⇒ D i = O N , d , i = η + 1 , η + 2 , . . , v . μ < ν ⇒ B i = O N , r , i = μ + 1 , μ + 2 , . . . , v . $$ \begin{gathered} \mathop \sum \limits_{{k = 0}}^{{k = \nu }} A_{k} {\text{Y}}^{{(k)}} (t) = \mathop \sum \limits_{{k = 0}}^{{k = \eta }} D_{k} {\text{D}}^{{(k)}} (t) + \mathop \sum \limits_{{k = 0}}^{{k = \mu }} B_{k} {\text{U}}^{{(k)}} (t) = \mathop \sum \limits_{{k = 0}}^{{k = \xi }} H_{k} {\text{I}}^{{(k)}} (t),\,\forall t \in {\mathfrak{T}}_{0} , \hfill \\ v \ge 1,\,\,\,\xi = {\text{~max~}}(\eta ,~\mu ),\,\,{\text{Y}}^{{(k)}} (t) = \frac{{d^{k} {\text{Y}}(t)}}{{dt^{k} }},\,\,0 \le \eta \le \nu ,\,\,\,\,0 \le \mu \le \nu , \hfill \\ A_{k} \in {\Re }^{{N \times N}} ,~D_{k} \in {\Re }^{{N \times d}} ,~B_{k} \in {\Re }^{{N \times r}} ,~k = 0,~1,...,v,\,\,~~detA_{\nu } \ne 0, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\eta < v \Rightarrow ~D_{i} = O_{{N,d}} ,~\,\,i = \eta + 1,~\eta + 2,~..,v. \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mu < \nu \Rightarrow ~B_{i} = O_{{N,r}} ,~i = \mu + 1,~\mu + 2,...,~v. \hfill \\ \end{gathered} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/math2_1.tif"/>
