Now the demo starts by taking the Hamiltonian for the closed loop system sHW and optimizing it in W to get what we denote sHWo. Soon the demo requires serious knowledge of [BHW] so if the reader has problems he is urged to read SYSHinfTAC.m a file which gives a rather complete account of the terse and unmotivated calculations here.
In[4]:= <<SYStems.m
NOTE: SYStems.m loads in the following files:
NCAlgbra.m, NCAliasFunctions.m, SYSDefIA.m,
SYSSpecialize.m, and SYSHinfFormulas.m
In[5]:= Substitute[sHW,dd[z]->0]
Out[5]= -tp[W] ** W + (tp[c[z]] ** tp[D12[x]] + tp[C1[x]]) **
(C1[x] + D12[x] ** c[z]) + ((tp[W] ** tp[B1[x]] + tp[c[z]] ** tp[B2[x]] +
tp[A[x]]) ** GEx[x, z] +
tp[GEx[x, z]] ** (A[x] + B1[x] ** W + B2[x] ** c[z]))/2 +
(tp[GEz[x, z]] ** (b[z] ** (C2[x] + D21[x] ** W) + a[z]) +
((tp[W] ** tp[D21[x]] + tp[C2[x]]) ** tp[b[z]] + tp[a[z]]) ** GEz[x, z])
/2
The critical W, found by taking the gradient of % and setting it to 0,
is
In[6]:= CriticalPoint[%,W]
Out[6]= {W -> tp[B1[x]] ** GEx[x, z]/2 +
tp[D21[x]] ** tp[b[z]] ** GEz[x, z]/2}
In[7]:= Substitute[%%,%]
Out[7]= -(tp[GEx[x, z]] ** B1[x]/2 + tp[GEz[x, z]] ** b[z] ** D21[x]/2) **
(tp[B1[x]] ** GEx[x, z]/2 + tp[D21[x]] ** tp[b[z]] ** GEz[x, z]/2) +
(tp[c[z]] ** tp[D12[x]] + tp[C1[x]]) ** (C1[x] + D12[x] ** c[z]) +
(((tp[GEx[x, z]] ** B1[x]/2 + tp[GEz[x, z]] ** b[z] ** D21[x]/2) **
tp[B1[x]] + tp[c[z]] ** tp[B2[x]] + tp[A[x]]) ** GEx[x, z] +
tp[GEx[x, z]] ** (A[x] + B1[x] **
(tp[B1[x]] ** GEx[x, z]/2 + tp[D21[x]] ** tp[b[z]] ** GEz[x, z]/2) \
+ B2[x] ** c[z]))/2 +
(tp[GEz[x, z]] ** (b[z] ** (C2[x] +
D21[x] ** (tp[B1[x]] ** GEx[x, z]/2 +
tp[D21[x]] ** tp[b[z]] ** GEz[x, z]/2)) + a[z]) +
(((tp[GEx[x, z]] ** B1[x]/2 + tp[GEz[x, z]] ** b[z] ** D21[x]/2) **
tp[D21[x]] + tp[C2[x]]) ** tp[b[z]] + tp[a[z]]) ** GEz[x, z])/2
------Check this against the stored formula for sHWo
In[8]:= NCExpand[%-sHWo]
Out[8]= -tp[c[z]] ** e1[x] ** c[z] +
tp[c[z]] ** tp[D12[x]] ** D12[x] ** c[z] -
tp[GEz[x, z]] ** b[z] ** e2[x] ** tp[b[z]] ** GEz[x, z]/4 +
tp[GEz[x, z]] ** b[z] ** D21[x] ** tp[D21[x]] ** tp[b[z]] ** GEz[x, z]/4
In[9]:= Substitute[%%,rue]
NOTE: rue = {tp[D12[x_]] ** D12[x_] :> e1[x], D21[x_] **
tp[D21[x_]] :> e2[x]}
Out[9]= 0
The output sHWo is also recorded in the file SYSDefIA.m
above. Note that neither sHWo nor sHWoWIA depends
on a or c.