Consider an ideal series PID controller with parameters Kc, Ta and Tb (satisfying some closed loop requirements) written as
PID1(s)=Kc(1+Tas)(1+Tbs)/(Tas)=
=Kc(1+1/(Tas))(1+Tbs)
It means that Ta may be interpreted as the integral time constant Ti and Tb as the derivative time constant TD.
In the linear range of control it is obviously equivalent to the controller
PID2(s)=Kc'(1+Tas)(1+Tbs)/(Tbs)=
=Kc'(1+1/(Tbs))(1+Tas)
with Kc'=KcTb/Ta, Ti=Tb, TD=Ta.
Importance of this ambiguity will be shown in constrained control. Do you know about some papers dealing with this topic?
Impact of this question is illustrated in paper
Conference Paper Constrained filtered PID Controller for IPDT plants
Using this ambiguity, ideal input and output shapes of the Integral + Dead-Time-System can be achieved with a serial PID controller with limited output (formerly known as Pre-Act) without the use of any additional anti-windup measures. Just choose the smaller of the two time constants guaranteeing multiple real dominant poles for the integral time constant - the controller is windupless. See Conference Paper Constrained filtered PID Controller for IPDT plants
orArticle Making the PI and PID Controller Tuning Inspired by Ziegler ...
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