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3.8 Pole/Zero Analysis 331
Estimate the number of
remaining terms.
In[9]:= ComplexityEstimate[sbgp2]
Out[9]= 4
Show complexity of
original system.
In[10]:= ComplexityEstimate[eqs]
Out[10]= 132
The result shows that the algorithm has eliminated the two high-frequency poles and has approximated
the low-frequency pole p near s ⩵ by zero. On the contrary, the pole p has been preserved,
and its numerical value lies in the specified tolerance region. In addition, the complexity of
the characteristic polynomial has been greatly reduced. Therefore, we can expect to obtain an
interpretable formula for p .
Compute the
approximated
characteristic polynomial.
In[11]:= detp2 = Det[GetMatrix[sbgp2]] // Together
Out[11]=
C2 R1 R2 s beta$Q1 C2 R1 RE s beta$Q1 C2 R2 RE s
beta$Q1 C1 C2 R1 R2 RE s 2 R1 R2 Rbe$Q1 RC RE
Solve for the poles. In[12]:= sbgpoles = Solve[detp2 == 0, s]
Extract the pole of interest.
Out[12]=
R1 R2 beta$Q1 R1 RE beta$Q1 R2 RE
s 0, s
beta$Q1 C1 R1 R2 RE
In[13]:= p2 = s /. Last[sbgpoles] // Simplify
1
R1 1
R2 1
beta$Q1
RE
Out[13]=
C1
Next, we try to extract the right-half plane zero z near s ⩵ ⨯ .
Approximate the equations
with respect to z .
In[14]:= sbgz3 = ApproximateDeterminant[eqs, V$5, 2.0*^8, 0.05,
GEPSolver −> GeneralizedEigensystem]
Out[14]= DAEAC, 10 10
Compute the remaining
zeros numerically.
Compute the
approximated polynomial.
In[15]:= ZerosByQZ[sbgz3, V$5]
Out[15]= 0., 0., 2.0000000000000003 10 8
In[16]:= detz3 = Det[GetMatrix[sbgz3]] // Simplify
Out[16]=
beta$Q1 C1 C2 s2 1 Cbc$Q1 RE s
Rbe$Q1 RE
Solve for the zeros. In[17]:= sbgzeros = Solve[detz3 == 0, s]
1
Out[17]= s 0, s 0, s
Cbc$Q1 RE
Extract the zero of interest.
In[18]:= z3 = s /. Last[sbgzeros]
1
Out[18]=
Cbc$Q1 RE