Experimental testing: Is a system really first-order?
Suppose we run an experiment to acquire a step response of a system, and the result looks like
Fig. 1. We suspect the system may be first-order, we estimate the time constant from the 63.2%
mark, and we use this time constant to mathematically model the system and predict its behavior.
ym
t
Fig. 1: Experimental step response (from System Dynamics by Doebelin).
In the following discussion, we develop a method for determining how close such a system is to
being truly first-order, and if it is, determining a best estimate of its time constant using all the
experimental data, not just the one data point at the 63.2% mark.
Step response
Suppose we apply a step input u to a system
and we measure the output ym and time t so
that we have a set of n+1 points for ym as
follows:
Taking the loge of both sides yields

y 
t
ln 1 − m  = − .
Gu0 
τ
14243
(5)
Z (t )
Time t
t(0)
t(1)
Output ym ym (0) ym (1)
…
…
t(n)
ym (n)
The model of a first-order system is
τ y& + y = Gu(t ),
(1)
where τ is the time constant and G is the static
gain. The solution to this ODE (with zero ICs,
and a step input of magnitude uo ) is given by
t

− 
τ 

y (t ) = Gu0 1 − e 
(2)


If the experimental system is first order, then
the experimental data ought to satisfy
t

− 
y m (t ) = Gu0  1 − e τ 
(3)


Here’s how we check. We manipulate (3) to
obtain
t
−
y
1− m = e τ
(4)
Gu0
and we designate the left-hand side Z, called
the log of the incomplete-response,
 y (t) 
Z ( t ) = ln 1 − m  .
(6)
Gu0 

We compute Z(t) from the experimental data
and we compare it to (-t/τ) from the right-hand
side of (5). Note that the ideal Z is a linear
function of t with a slope of −1/τ.
Procedure: Given experimental step response
data ym , estimate the time constant (τest ) from
the 63.2% point. Extract from the experimental
data the first 4τest values of t and ym . Use (6) to
compute the log incomplete response Z(t), plot
Z(t), and perform a linear curve fit. Figure 2
shows a representative plot. The closer a linear
curve fits the data, the closer the system is to
being first order with a time constant τ equal
to the negative inverse of the slope of the fitted
curve. Conversely, if the plot of Z is clearly
not a straight line, then the system is not first
order and (1) is not a good model.
Z(t)
t
Fig. 2: Curve -fit to a plot of Z(t) (from System Dynamics by Doebelin).
IC response
A similar approach can be used when we
have experimental data from an initialcondition response (or an impulse response).
Suppose we give a system an initial
condition y0 and measure the output ym and
time t so that we have a set of n+1 points for
ym as follows:
Time t
t(0)
t(1)
Output ym ym (0) ym (1)
…
…
t(n)
ym (n)
The model of a first-order system is
τ y& + y = 0,
(7)
where τ is the time constant and the IC is
given by y(0) = y0 . The response is given by
y (t ) = y 0 e
−
t
τ
(8)
If the experimental system is first order, then
the experimental data ought to satisfy
y m ( t ) = y0 e
−
t
τ
(9)
Following a procedure like the one for the
step response, we get an expression for Z
given by
 y (t ) 
Z ( t ) = ln  m  .
(10)
 y0 
Again, we compute Z(t) from the
experimental data and we compare it to the
ideal
Z(t) = (-t/τ).
Again, the ideal Z is a linear function of t
with a slope of −1/τ.
Procedure: Given experimental ICresponse data ym , estimate the time constant
(τest ) from the 63.2% point. Extract from the
experimental data the first 4τest values of t
and ym . Use (10) to compute the log
incomplete-response Z(t), plot Z(t), and
perform a linear curve fit. Figure 2 shows a
representative plot. The closer a linear curve
fits the data, the closer the system is to being
first order with a time constant τ equal to the
negative inverse of the slope of the fitted
curve. Conversely, if the plot of Z is clearly
not a straight line, then the system is not first
order and (7) is not a good model.
In-class exercise
For the step-response data given below,
compute and plot the log incomplete
response Z(t). Plot a linear curve-fit and
determine the time constant. Can we
conclude that the system is first-order?
Time (s)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
ym (in)
0.00
0.97
1.79
2.35
2.83
3.15
3.37
3.53
3.73
3.82
3.91
3.98
4.03
4.06
4.06
4.09
4.13
4.14
4.13