​Consider an underdamped second-order process with a transfer function (the ratio of output to input of a system), given by
G(s)=
K
g
2
τ
2
s
+2ζτs+1
, where
K
g
is the steady-state gain,
τ
is the time constant, and
ζ
is the damping coefficient (with
0<ζ<1
). This process is subjected to a step input
U(t)=
M
s
. Here, we show plots of the response
y(t)=
-1
ℒ
(G(s)U(s))
and finds its performance characteristics: overshoot
a/b
, decay ratio
c/a
, time to rise
t
r
, time to first peak
t
p
, and period of oscillations
P
. These performance characteristics are reported in a Table at the end of the notebook.
The overshoot is equal to
a/b=exp-πζ
1-
2
ζ

.The decay ratio is equal to
c/a=
2
(a/b)
=exp-2πζ
1-
2
ζ

.The period of oscillation is
P=2πτ
1-
2
ζ
.The time to first peak is given by
t
p
=πτ
1-
2
ζ
.The rise time is the solution
t
r
of
y(
t
r
)=M
K
g
and
0<
t
r
<
t
p
.
For illustration purposes, we adopt the following values : Kg = 5; τ = 1; ζ = 0.2; M = 5;
In[]:=
Kg=5;τ=1;ζ=0.2;M=5;
Letusconstruct:y(t)=
-1
ℒ
(G(s)U(s))
In[]:=
InvTau=1/τ;ω=Sqrt[InvTau^2-(InvTauζ)^2];
In[]:=
yanal[t_]:=KgM+(-2KgMInvTauζ/ω+KgMInvTauζ/ω)Exp[-ζInvTaut]Sin[ωt]-KgMExp[-ζInvTaut]Cos[ωt];
The time to first peak is given by
t
p
=πτ
1-
2
ζ
.
In[]:=
tp=Piτ/Sqrt[1-ζ^2];
The period of oscillation is
P=2πτ
1-
2
ζ
.
In[]:=
P=2Pi/ω;
In[]:=
sol1=FindRoot[yanal[t]MKg,{t,0,tp}]//Chop;sol2=FindMaximum[yanal[t],{t,tp,tp+0.2}];
In[]:=
sol3=Quiet@FindMaximum[yanal[t],{t,tp+P}];
The overshoot is equal to
a/b=exp-πζ
1-
2
ζ

.The decay ratio is equal to
c/a=
2
(a/b)
=exp-2πζ
1-
2
ζ

.
In[]:=
Overshoot=Exp[-Piζ/Sqrt[1-ζ^2]];​​DecayRatio=Exp[-2Piζ/Sqrt[1-ζ^2]];
In[]:=
p1=Plot[yanal[t],{t,0,3P},FrameTrue,GridLinesAutomatic,AspectRatio1.5,FrameLabel{Style["time",16],Style["output",16]},PlotRangeAll,PlotStyle{Red,Thick},ImageSize400{1,1}];
In[]:=
Show[p1,Graphics[{{Dashed,Thickness[0.01],Blue,Line[{{0,MKg},{3P,MKg}}]},Darker@Green,Arrowheads[{-.05,.05}],Arrow[{{tp,0},{tp,MKg}}],Red,Arrowheads[{-.05,.05}],Arrow[{{tp,MKg},{tp,sol2[[1]]}}],Blue,Arrowheads[{-.05,.05}],Arrow[{{tp,sol2[[1]]1.05},{t/.sol3[[2]],sol2[[1]]1.05}}],Style[Text[Style["P",Italic],{((t/.sol3[[2]])+tp)/2,sol2[[1]]1.1},{0,0}],Blue,14,BackgroundWhite],Style[Text[Style["b",Italic],{tp1.2,MKg/2},{0,0}],Darker@Green,14,BackgroundWhite],Style[Text[Style["a",Italic],{tp1.2,MKg/2+sol2[[1]]/2},{0,0}],Red,14,BackgroundWhite],Style[Text[Style["c",Italic],{(t/.sol3[[2]])1.1,MKg/2+sol3[[1]]/2},{0,0}],Black,14,BackgroundWhite],Black,Arrowheads[{-.03,.03}],Arrow[{{(t/.sol3[[2]]),MKg},{(t/.sol3[[2]]),sol3[[1]]}}]}]]​​
Out[]=
In[]:=
Text@Style[Grid[{{"property","value"},​​{​​Row[{"overshoot ",Style["a",Italic],"/",Style["b",Italic]}],​​NumberForm[Overshoot,{5,4}]​​},​​{​​Row[{"decay ratio ",Style["c",Italic],"/",Style["a",Italic]}],​​NumberForm[DecayRatio,{5,4}]​​},​​{​​Row[{"period ",Style["P",Italic]}],​​NumberForm[P,{5,4}]},​​{​​Row[{"rise time ",
Style["t",Italic]
"r"
}],​​NumberForm[t/.sol1,{5,4}]​​},​​{​​Row[{"time to first peak ",
Style["t",Italic]
"p"
}],​​NumberForm[tp,{5,4}]​​}​​},FrameAll,Background{None,{Lighter[Blend[{Blue,Red}],.8],{Lighter[Blend[{Blue,Darker@Green}],.8],Lighter[Green,.9]}}},ItemSize{9.3,3.3},FrameDarker[Gray,.6],ItemStyle14,Spacings{Automatic,.8}],16]
Out[]=
property
value
overshoot a/b
0.5266
decay ratio c/a
0.2773
period P
6.4127
rise time
t
r
1.8087
time to first peak
t
p
3.2064

References

[1] D. E. Seborg, T. F. Edgar, D. A. Mellichamp, and F. J. Doyle III, Process Dynamics and Control, 4th ed., Hoboken, NJ: John Wiley & Sons, Inc., 2017.

CITE THIS NOTEBOOK

Performance characteristics for step response of an underdamped process​
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, March 21, 2024
​https://community.wolfram.com/groups/-/m/t/3145151