The following demo verifies IAX and IAYI are same as DGX DGYI the Doyle Glover X and inv[Y] Riccati equations in the special case of a linear system.
In[24]:= <<SYStems.m
In[25]:= NCE[IAX[x]//.rulinearall]
Out[25]= tp[x] ** XX ** A ** x + tp[x] ** tp[A] ** tp[x] ** XX +
tp[x] ** tp[C1] ** C1 ** x + tp[x] ** XX ** B1 ** tp[B1] ** tp[x] ** XX -
tp[x] ** XX ** B2 ** inv[e1] ** tp[B2] ** tp[x] ** XX -
tp[x] ** XX ** B2 ** inv[e1] ** tp[D12] ** C1 ** x -
tp[x] ** tp[C1] ** D12 ** inv[e1] ** tp[B2] ** tp[x] ** XX -
tp[x] ** tp[C1] ** D12 ** inv[e1] ** tp[D12] ** C1 ** x
In[26]:= Sub[%,x->1]
Out[26]= XX ** A + tp[A] ** XX + tp[C1] ** C1 + XX ** B1 ** tp[B1] ** XX -
XX ** B2 ** inv[e1] ** tp[B2] ** XX -
XX ** B2 ** inv[e1] ** tp[D12] ** C1 -
tp[C1] ** D12 ** inv[e1] ** tp[B2] ** XX -
tp[C1] ** D12 ** inv[e1] ** tp[D12] ** C1
In[27]:= NCE[%-DGX]
Out[27]= 0
In[28]:= NCE[IAYI[x]//.rulinearall]
Out[28]= tp[x] ** inv[YY] ** A ** x + tp[x] ** tp[A] ** tp[x] ** inv[YY] +
tp[x] ** tp[C1] ** C1 ** x - tp[x] ** tp[C2] ** inv[e2] ** C2 ** x +
tp[x] ** inv[YY] ** B1 ** tp[B1] ** tp[x] ** inv[YY] -
tp[x] ** inv[YY] ** B1 ** tp[D21] ** inv[e2] ** C2 ** x -
tp[x] ** tp[C2] ** inv[e2] ** D21 ** tp[B1] ** tp[x] ** inv[YY] -
tp[x] ** inv[YY] ** B1 ** tp[D21] ** inv[e2] ** D21 ** tp[B1] ** tp[x] **
inv[YY]
In[29]:= Sub[%,x->1]
Out[29]= inv[YY] ** A + tp[A] ** inv[YY] + tp[C1] ** C1 -
tp[C2] ** inv[e2] ** C2 + inv[YY] ** B1 ** tp[B1] ** inv[YY] -
inv[YY] ** B1 ** tp[D21] ** inv[e2] ** C2 -
tp[C2] ** inv[e2] ** D21 ** tp[B1] ** inv[YY] -
inv[YY] ** B1 ** tp[D21] ** inv[e2] ** D21 ** tp[B1] ** inv[YY]
In[30]:= NCE[YY**%**YY-DGY]
Out[32]= 0
In[34]:= Quit