ABSTRACT
The linear Higher order Input‐State‐Output (HISO) (dynamical, control) systems have not been studied so far. Their mathematical models contain the α‐th order linear differential state vector equation (5.1) and the linear algebraic output vector equation (5.2), A ( α ) R α ( t ) = D ( μ ) D μ ( t ) + B ( μ ) U μ ( t ) = H ( μ ) I μ ( t ) , ∀ t ∈ T 0 , A ( α ) ∈ ℜ ρ × ( α + 1 ) ρ , R α ∈ ℜ ( α + 1 ) ρ , D ( μ ) ∈ ℜ ρ × ( μ + 1 ) d , B ( μ ) ∈ ℜ ρ × ( μ + 1 ) r , H ( μ ) = [ D ( μ ) ⋮ B ( μ ) ] ∈ ℜ ρ × ( μ + 1 ) ( d + r ) , I μ = [ ( D μ ) T ⋮ ( U μ ) T ] T ∈ ℜ ( μ + 1 ) ( d + r ) , $$ \begin{gathered} A^{{(\alpha )}} {\text{R}}^{\alpha } (t) = D^{{(\mu )}} {\text{D}}^{\mu } (t) + B^{{(\mu )}} {\text{U}}^{\mu } (t) = H^{{(\mu )}} {\text{I}}^{\mu } (t)~,~\forall t \in {\mathfrak{T}}_{0} , \hfill \\ A^{{(\alpha )}} \in {\Re }^{{\rho \times (\alpha + 1)\rho }} ,{\text{R}}^{\alpha } \in {\Re }^{{(\alpha + 1)\rho }} ,D^{{(\mu )}} \in {\Re }^{{\rho \times (\mu + 1)d}} ,B^{{(\mu )}} \in {\Re }^{{\rho \times (\mu + 1)r}} , \hfill \\ H^{{(\mu )}} = [D^{{(\mu )}} \vdots B^{{(\mu )}} ] \in {\Re }^{{\rho \times (\mu + 1)(d + r)}} ,\,{\text{I}}^{\mu } = [({\text{D}}^{\mu } )^{T} \vdots ~({\text{U}}^{\mu } )^{T} ]^{T} \in {\Re }^{{(\mu + 1)(d + r)}} , \hfill \\ \end{gathered} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/math5_1.tif"/> Y ( t ) = R ( α ) R α ( t ) + V D ( t ) + U U ( t ) = R ( α ) R α ( t ) + Q I ( t ) , ∀ t ∈ T 0 , V ∈ ℜ N × d , U ∈ ℜ Nxr , Q = [ V ⋮ U ] ∈ ℜ N × ( d + r ) , R ( α ) = [ R 0 ⋮ R 1 ⋮ ⋯ ⋮ R α - 1 ⋮ O N , ρ ] , R α = O N , ρ . $$ \begin{gathered} {\text{Y}}(t) = R^{{(\alpha )}} {\text{R}}^{\alpha } (t) + V{\text{D}}(t) + U{\text{U}}(t) = R^{{(\alpha )}} {\text{R}}^{\alpha } (t) + Q{\text{I}}(t),\,\,\forall t \in {\mathfrak{T}}_{0} , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,V \in {\Re }^{{N \times d}} ,~U \in {\Re }^{{Nxr}} ,~Q = ~[V \vdots U]~ \in {\Re }^{{N \times (d + r)}} , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,R^{{(\alpha )}} = ~[R_{0} \vdots R_{1} \vdots \cdots \vdots R_{{\alpha - 1}} \vdots O_{{N,\rho }} ]~,~R_{\alpha } = O_{{N,\rho }} . \hfill \\ \end{gathered} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429432149/418cc552-67ff-4e7e-a408-87b73829f6e7/content/math5_2.tif"/>
