I need to know the possible range of the three parameters of the PID control. How can I do that? Is it possible to calculate using the Ziegler-Nichols Method?
It depends on the response of the systems. If open loop response or close loop response fit to the Ziegler-Nichols Method required response, yes, you can.
Dear Elaine,
You can use the relay auto-tuning method which is a modification of the Ziegler-Nichols method. This method is safer because it doesn't require you to run the system in open loop.
You may refer these papers for details
1) K.J. Astrom, T. Hagglund, “Automatic tuning of PID”,Instrument Society of America,
1988.
2) Cheng-Ching Yu, “Autotuning of PID Controllers-A Relay Feedback Approach”,
Springer 2nd Edition, 2006
There are many PID tuning relations and rules for different types of systems. But you need a model of the system, e.g., first-order plus deadtime (FOPDT) model in order to use the formulas. There are a few methods which require no system model at all. Ziegler-Nichols is one such method where minimum knowledge of the system is required.
You can develop your own PID controller tuning relations provided you have some basic knowledge on feedback structure. Fr example, if your system is given by a FOPDT model,
P(s) = Kp*pexp(-D*s)/(Tp*s+1) ; Kp = system gain, D = deadtime and Tp = time constant
Using first-order Taylor series, you can approximate this system as
P(s) = Kp*(1-D*s)/(Tp*s+1);
Let say you use PI controller
Gc(s) = Kc(1 + 1/Ti*s); Ti = reset time and Kc = controller gain
The closed-loop characteristic equation is given as
1 + Gc(s)*P(s) = 0
which can be simplified to a polynomial equation as follows
Ti*(Tp - K*D)*s^2 + (Ti + K*(Ti-D))*s + K = 0
From the necessary criterion of Routh stability, all coefficients of the closed-loop characteristic polynomial must be positive. Thus, from the coefficient of s^2, we get an upper limit on the loop gain K = Kc*Kp
K < Tp/D
From the coefficient of s and s^0, we have lower limits
K > -Ti/(Ti-D), where Ti > D
K > 0
So, the range of the Kc should be between thee upper limit and maximum lower limit:
0 < Kc < Tp/(Kp*D)
while the reset time should take a range like
Ti > D
Note, if you choose Ti < D, then you will have two upper limits on the loop gain and one lower limit. The stabilizing range of K must be between the minimum upper limit and lower limit.
So, it's not very difficult to devise your won tuning formulas. All you need is some basic knowledge on feedback control.
This explanation is from the engineering point of view. For a mathematician, the procedures can be more mathematically tedious, but the end result shall be the same - stability as the first requirement and then to achieve certain performance as a second.
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