ABSTRACT

The general description, in terms of the total vector coordinates, of time-invariant continuous-time linear Input-Internal and Output state systems , for short IIO systems, without a delay, has the following general form: A ( α ) R α ( t ) = D ( μ ) D μ ( t ) + B ( μ ) U μ ( t ) = H ( μ ) I μ ( t ) , ∀ t ∈ T 0 , $$ A^{{(\alpha )}} {\text{R}}^{\alpha } (t) = D^{{(\mu )}} {\text{D}}^{\mu } (t) + B^{{(\mu )}} {\text{U}}^{\mu } (t) = H^{{(\mu )}} {\text{I}}^{\mu } (t),\,\forall t \in {\mathfrak{T}}_{0} , $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/math6_1.tif"/> E ( ν ) Y ν ( t ) = R y ( α - 1 ) R α - 1 ( t ) + V ( μ ) D μ ( t ) + U ( μ ) U μ ( t ) = = R y ( α - 1 ) R α - 1 ( t ) + Q ( μ ) I μ ( t ) , ∀ t ∈ T 0 , $$ E^{{(\nu )}} {\text{Y}}^{\nu } (t) = \left\{ \begin{gathered} R_{y}^{{(\alpha - 1)}} {\text{R}}^{{\alpha - 1}} (t) + V^{{(\mu )}} {\text{D}}^{\mu } (t) + U^{{(\mu )}} {\text{U}}^{\mu } (t) = \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, = R_{y}^{{(\alpha - 1)}} {\text{R}}^{{\alpha - 1}} (t) + Q^{{(\mu )}} {\text{I}}^{\mu } (t) \hfill \\ \end{gathered} \right\},\forall t \in {\mathfrak{T}}_{0} , $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/math6_2.tif"/>