FOPDT Model Characterization
FOPDT Model Characterization
FOPDT Model Characterization
It is necessary, to be able to tune a controller in a feedback loop, to have information about the
dynamic behavior of the process we want to control. This behavior is obtained through the
mathematical equations that describe the physical laws involved in the system. These equations
stablish a cause-effect relationship, where we can distinguish a dependent (process response) and
an independent variable (manipulation) as you can see in Figure 1. The order and type of
resulting equations will depend en the physical laws between the dependent and the independent
variable.
When the equations are hard to obtain in an analytical way, we can make experimental testing to
get graphical responses than then we can compare them with known responses, excited by the
same manipulation function, to determine the type and order of the system. From these
graphical methods, we can distinguish the Process Step Testing. This method consists of applying
a step function to a process and get a graphic of the transient response. Then, the transient
response can be approximated to known simple mathematical models as the First Order Plus Dead
Time (FOPDT) or the Second Order Plus Dead Time (SOPDT) as shown in Figure 2.
The FOPDT is a frequently used model to characterize the transient response of a process (open
loop). The transfer function in Laplace form of a FOPDT is:
C( s) K −θs
=Gp ( s )= e
M ( s) τs+1
(Eq 1)
Where C(s) is the transient response or Process variable and M(s) is the Step Function. In a control
loop, M(s) corresponds to the output of the controller (%CO).
K = The process steady-state gain, which defines the sensitivity of the process. This is,
how much the output, C(s), changes per unit change in the input, M(s).
τ = The time constant of the process. It determines the reaction velocity of the process:
With a bigger τ, the process reacts slower and vice versa.
θ = The dead time, is the delay from then a controller output, M(s), signal is used until
when the measured process variable, C(s), first begins to respond.
Eq 3
c ( t 1 ) =0.632 ∆ c (t)
Eq 4
τ
t 1= +θ
3
Eq 5
t 2=τ +θ
Eq 6
K ωn2
G ( s )= 2 2
e−θs
s + 2ζ ωn s+ ωn
Eq 7
4
t s=
ζ ωn
Eq 8
% Mp: overshoot, corresponds to the fraction (or percent) of the final steady-state
change by which the first peak exceeds this change.
A
%Mp= × 100
B
Eq 9
ess: steady-state error, which correspond to the deviation between the process variable
and the setpoint in a steady-state.
tr: rise time, corresponds to the time it takes for the response to first reach its final
steady-state value.
t r=
π− ( 180π ° ) ⋅ β ; β=cos −1
(ζ )
ωd
Eq 11
tp: peak time, corresponds to the time it takes for the response to reach the max peak
Eq 12
ζ=
√ |ln(m p)|
2
|ln(m p )| + π 2
Eq 13
The evaluation of the performance using the indicators showed above also allow us to obtain the
parameters of a SOPDT underdamping model.
There is a method to characterize second order systems no matter if they are underdamping,
critical underdamping or overdamping processes. This method consists of obtaining the natural
oscillation frequency and the damping factor from the plot shown in Figure 5.
Figure 7 (Cecil Smith, fig. 6.7, pg. 144) allow us to calculate wn and To know these parameters
you need to look the times where the final response is about 20%(t2) and 60%(t6). We obtain the
ratio t2/t6 and obtain, according to the curve, And then from t6*Wn we calculate wn.
Bibliography
Smith, C. A., & Corripio, A. B. (2006). Principles and practice of automatic process control.
Hoboken, NJ: Wiley.