ABSTRACT
A slightly more general class than the ISO systems (3.1), (3.2) is the family of the Extended Input‐State‐Output systems (EISO systems) described in terms of the total coordinates by d X ( t ) dt = A X ( t ) + D ( μ ) D μ ( t ) + B ( μ ) U μ ( t ) = A X ( t ) + P ( μ ) I μ ( t ) , ∀ t ∈ T 0 , A ∈ ℜ nxn , D ( μ ) ∈ ℜ n × ( μ + 1 ) d , B ( μ ) ∈ ℜ n × ( μ + 1 ) r , P ( μ ) ∈ ℜ n × ( μ + 1 ) M , $$ \begin{aligned} \frac{{d{\text{X}}(t)}}{{dt}} = A{\text{X}}(t) + D^{{(\mu )}} {\text{D}}^{\mu } (t) + B^{{(\mu )}} {\text{U}}^{\mu } (t) = A{\text{X}}(t) + P^{{(\mu )}} {\text{I}}^{\mu } (t),\,\,\forall t \in {\mathfrak{T}}_{0} , \\ A \in {\Re }^{{nxn}} ,\,D^{{(\mu )}} \, \in {\Re }^{{n \times (\mu + 1)d}} ,B^{{(\mu )}} \, \in {\Re }^{{n \times (\mu + 1)r}} ,P^{{(\mu )}} \, \in {\Re }^{{n \times (\mu + 1)M}} ,\,\, \\ \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/math4_1.tif"/> Y ( t ) = C X ( t ) + V D ( t ) + U U ( t ) = C X ( t ) + Q I ( t ) , ∀ t ∈ T 0 . C ∈ ℜ N × n , V ∈ ℜ N × d , U ∈ ℜ N × r , Q ∈ ℜ N × M . $$ \begin{aligned} \,\,\,{\text{Y}}(t) = C{\text{X}}(t) + V{\text{D}}(t) + U{\text{U}}(t) = C{\text{X}}(t) + Q{\text{I}}(t)~,~\forall t \in {\mathfrak{T}}_{0} . \\ C \in {\Re }^{{N \times n}} ,~V \in {\Re }^{{N \times d}} ,~U \in {\Re }^{{N \times r}} ,~Q \in {\Re }^{{N \times M}} . \\ \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/math4_2.tif"/>
