In[]:=
H[s_]=s^2+2*ζ*
ω
n
*s+
ω
n
^2
Out[]=
2
s
+2sζ
ω
n
+
2
ω
n
In[]:=
Roots[H[s]==0,s]
Out[]=
s-ζ
ω
n
-
-
2
ω
n
+
2
ζ
2
ω
n
||s-ζ
ω
n
+
-
2
ω
n
+
2
ζ
2
ω
n
Continuous Time to Discrete Time
In[]:=
z==Exp[s.T]
Out[]=
z
s.T

LInes of constant ζ
In[]:=
ShowTableParametricPlotReImExp-ζ
ω
n
+
-
2
ω
n
+
2
ζ
2
ω
n
,{
ω
n
,0,Pi},PlotRange->All,{ζ,0,1,0.1}
Out[]=
-1.0
-0.5
0.5
1.0
0.2
0.4
0.6
0.8
1.0
Lines of constant
ω
n
In[]:=
ShowTableParametricPlotReImExp-ζ
ω
n
+
-
2
ω
n
+
2
ζ
2
ω
n
,{ζ,0,1},PlotRange->All,{
ω
n
,0,Pi,0.1*Pi}
Out[]=
-1.0
-0.5
0.5
1.0
0.2
0.4
0.6
0.8
1.0
Solving
ζ,
ω
n
of continuous time system, starting from poles in z-domain, here real poles
In[]:=
NSolve0.422==Exp-ζ
ω
n
+
ω
n
2
ζ
-1
*.01,0.0955==Exp-ζ
ω
n
-
ω
n
2
ζ
-1
*.01,{ζ,
ω
n
}
NSolve
:Inverse functions are being used by NSolve​, so some solutions may not be found; use Reduce for complete solution information.
Out[]=
{{ζ1.12801,
ω
n
142.347}}
In[]:=
s1=-ζ
ω
n
-
-
2
ω
n
+
2
ζ
2
ω
n
/.{ζ1.1280072197296975`,
ω
n
142.34744868083004`}
Out[]=
-234.863
In[]:=
Log[0.0955]*100
Out[]=
-234.863
In[]:=
-234.863/(2*Pi)
Out[]=
-37.3796
In[]:=
s2=-ζ
ω
n
+
-
2
ω
n
+
2
ζ
2
ω
n
/.{ζ1.1280072197296975`,
ω
n
142.34744868083004`}
Out[]=
-86.275
In[]:=
Log[0.422]*100
Out[]=
-86.275
In[]:=
-86.275/(2*Pi)
Out[]=
-13.7311
Solving
ζ,
ω
n
of continuous time system, starting from poles in z-domain, here complex conjugate poles
In[]:=
NSolve0.832066760726599+I0.382547088403759==Exp-ζ
ω
n
+
ω
n
2
ζ
-1
*.01,0.832066760726599-I0.382547088403759==Exp-ζ
ω
n
-
ω
n
2
ζ
-1
*.01,{ζ,
ω
n
}
NSolve
:Inverse functions are being used by NSolve​, so some solutions may not be found; use Reduce for complete solution information.
Out[]=
{{ζ0.2,
ω
n
43.9823}}
In[]:=
s1=-ζ
ω
n
-
-
2
ω
n
+
2
ζ
2
ω
n
/.{ζ0.19999999999999937`,
ω
n
43.982297150257104`}
Out[]=
-8.79646-43.0937
In[]:=
Exp[s1*0.01]
Out[]=
0.832067-0.382547
In[]:=
s2=-ζ
ω
n
+
-
2
ω
n
+
2
ζ
2
ω
n
/.{ζ0.19999999999999937`,
ω
n
43.982297150257104`}
Out[]=
-8.79646+43.0937
In[]:=
Exp[s2*0.01]
Out[]=
0.832067+0.382547