To specialize to the linear case, just apply
∙ rulinearsys to make the systems linear
∙ rulinearEB to make the energy function quadratic
∙ ruGE1 then ruGEXY to make the energy function solve the Hinf problem (max entropy soln.)
∙ rulinearall contains all of the rules above and is what we usually use.
(See Glossary)
A special class of IA systems are those satisfying
(See Glossary)
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 |