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

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
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
In[]:=
Solve0.0955==Exp-ζ
ω
n
+
-
2
ω
n
+
2
ζ
2
ω
n
100/.ζ->1,
ω
n

Solve
:Inverse functions are being used by Solve​, so some solutions may not be found; use Reduce for complete solution information.
Out[]=
{{
ω
n
234.863}}
In[]:=
234.863/(2*Pi)
Out[]=
37.3796
In[]:=
Solve0.422==Exp-ζ
ω
n
+
-
2
ω
n
+
2
ζ
2
ω
n
100/.ζ->1,
ω
n

Solve
:Inverse functions are being used by Solve​, so some solutions may not be found; use Reduce for complete solution information.
Out[]=
{{
ω
n
86.275}}
In[]:=
86.275/(2*Pi)
Out[]=
13.7311
In[]:=
NSolveRe[Log[0.8321+I0.3825]]==-ζ
ω
n
*0.01,Im[Log[0.8321+I0.3825]]==
ω
n
1-
2
ζ
*.01,{ζ,
ω
n
}
Out[]=
{{ζ0.200002,
ω
n
43.976}}
In[]:=
NSolveRe[Log[0.8321-I0.3825]]==-ζ
ω
n
*0.01,Im[Log[0.8321-I0.3825]]==
ω
n
1-
2
ζ
*.01,{ζ,
ω
n
}
Out[]=
{{ζ-0.200002,
ω
n
-43.976}}
In[]:=
NSolve0.8321+I0.3825==Exp-ζ
ω
n
-
ω
n
2
ζ
-1
*.01&&Element[{ζ,
ω
n
},Reals],{ζ,
ω
n
}
NSolve
:NSolve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.
Out[]=
ζ
-8.79531
-
1.
-1933.89+54145.3

1
-394784.
2

1
if

1
∈&&

1
≤0
,
ω
n

-
1.
-
1.
-1933.89+54145.3

1
-394784.
2

1
if

1
∈&&

1
≤0
,ζ
-574466.+1.67542×
7
10

1
-1.22158×
8
10
2

1
(-86.175+1256.64

1
)(166653.-7.09617×
6
10

1
+1.02062×
8
10
2

1
-4.961×
8
10
3

1
)
if

1
∈&&

1
≥1.
,
ω
n

8.79531
-574466.+1.67542×
7
10

1
-1.22158×
8
10
2

1
(-86.175+1256.64

1
)(166653.-7.09617×
6
10

1
+1.02062×
8
10
2

1
-4.961×
8
10
3

1
)
+0.5
7116.7-216581.

1
+1.57914×
6
10
2

1
+
309.43(-574466.+1.67542×
7
10

1
-1.22158×
8
10
2

1
)
(-86.175+1256.64

1
)(166653.-7.09617×
6
10

1
+1.02062×
8
10
2

1
-4.961×
8
10
3

1
)
if

1
∈&&

1
≥1.
