Antenna

Let k = K[t, K1, K2, Te, Kp, Kc] be the commutative polynomial ring in t, K1, K2, Te, Kp, Kc and let us define the Ore algebra A = k⟨∂,δ⟩, where ∂ is differential operator defined by ∂f(t) = f'(t) and δ is time-delay operator defined by δf(t) = f(t-1).

A=OreAlgebra[Der[t],S[-1][t]];

We consider the matrix R ∈ (A^(6×9)) given by

R0={{Der[t],-K1,0,0,0,0,0,0,0},
{0,(K2/Te)+Der[t],0,0,0,0,-((Kp S[-1][t])/Te),-((Kc S[-1][t])/Te),-((Kc S[-1][t])/Te)},
{0,0,Der[t],-K1,0,0,0,0,0},
{0,0,0,(K2/Te)+Der[t],0,0,-((Kc S[-1][t])/Te),-((Kp S[-1][t])/Te),-((Kc S[-1][t])/Te)},
{0,0,0,0,Der[t],-K1,0,0,0},
{0,0,0,0,0,(K2/Te)+Der[t],-((Kc S[-1][t])/Te),-((Kc S[-1][t])/Te),-((Kp S[-1][t])/Te)}};

MatrixForm[R=ToOrePolynomial[R0,A]]

and the left A-module, finitely presented by R, defined by M = (A^(1×9))/(A^(1×6))R.

Let us compute the adjoint of R:

MatrixForm[Radj=Involution[R,A]]

Let us check whether or not the left A-module M is torsion-free, i.e. whether or not the system admits some autonomous elements.

{L0, L1, L2}=Ext1[Radj,A];

MatrixForm@L0

MatrixForm@L1

L2

MatrixForm[L2]

Since the first matrix is identity, M is torsion-free. Let us now compute the obstructions for M to be projective:

P=ObstructionToProjectiveness[R,A]

Let us deduce the obstructions in δ for M to be a projective left A-module.

OreIntersection[Flatten@P,{S[-1][t]},A]

Hence we obtain that B ⊗ M is projective over the ring B = A⟨σ⟩ / ⟨δ σ -1, σδ - 1⟩, where σ is defined by σf(t)=f(t+1) (i.e. the two sided inverse of δ).

AnT=AnnihilatorTorsionModule[R,A]

Let us compute the paramterization of the system

MatrixForm[param=Parametrization[R,ξ,A]]

We note that this parametrization is a minimal one:

MatrixForm/@MinimalParametrizations[R,A]

Let us check whether or not L admits a left inverse:

LeftInverse[L[[1]],A]

We obtain that L is not an injective parametrization over A. Let now define the following Ore algebra

B=OreAlgebra[Der[t]]

and if we substitute S[-1][t] by δ, i.e.

MatrixForm[L1=L[[1]]/.S[-1][t]->δ]

MatrixForm[L2=ChangeOreAlgebra[L1,B]]

then we obtain that the matrix L2 admits the following left inverse:

MatrixForm[T=Factor[iOrePolynomialToNormal[LeftInverse[L2,B]]]]

Therefore if we use the advance operator σ = (δ^(-1)) then flat outputs of the system are defined by ξ = T((Subscript[x, 1]),…,(Subscript[x, 6]),(Subscript[u, 1]),(Subscript[u, 2]),(Subscript[u, 3])).