Relay Auto Tuning in pure first order system. Theoretically how ultimate period possible?

Hi Lalu,

There was a similar question from you earlier [What are the prominent PID tuning method (other than pole placement/optimisation related methods) for pure first order process?] that I happened to read.

Together with this one, there appears to be some fundamental confusion regarding the Ziegler-Nichols tuning rules, which perhaps needs to be clarified !

(Sorry in case I am missing anything !!)

The ZN rules apply to a large class of control problems that are either lag type or oscillatory type. These were the two broad class of chemical and mechanical engineering problems that formed the interest of control systems designers between the two World Wars, due to which considerable interest was generated around them at that time. One culmination of the effort, which has stood the test of time due to its amazing range of applicability, is what we call as the famous ZN rules.

In the fundamental statement of the ZN tuning rules (the descriptions are scattered across the internet - you can easily search them out) there is no assumption on linearity of systems, let alone order of transfer functions ! They may well be totally nonlinear internally, and the lag type or oscillatory type response may simply be dominant dynamics - nothing more !! As a matter of fact, the best part of the ZN rules lies just there - that very little assumptions are made regarding linearity and order of systems, and they still very much work !!

In view of #2, if you do know a transfer function to be of precisely of first order (obviously, in that case it must be truly linear as well !), why use ZN rules at all ?? You can use more refined methods for such cases (like root locus) ! ZN rules are intended for much worse situations - for systems that are actually known to lesser degree of thoroughness...

Finally, to sort out between the lag/delay and ultimate gain/period ! It is important to note that the lag type and oscillatory type systems are two distinct system types (not interchangeable !) with distinct ZN tuning rules ! For lag type, the rules are based on open-loop step input, with consequent measurement of steady gain, lag time, and delay time. For oscillatory type, the rules are based on closed loop step input with consequent measurement of ultimate gain and period. The two should not be confused mutually !!

We may imagine the plight of control engineers in 1930-40's with primitive design methods - struggling to get the better of nonlinear/linear control problems with vague knowledge of internal details, where certain problems would always show sluggish monotonic response and never oscillate, and some others that always oscillated and were difficult to damp out !

Sorry if that was a drag, but I hope I didn't miss much ! Why don't you hunt around the web for some papers on the ZN rules alone, so that a better feel of their applicability can be obtained ! If you are lucky, you may actually locate the original 1942 paper by Ziegler and Nichols - believe me, it is fascinating to go through their thorough experimental results..

Cheers !

-Sanjay

Lalu Seban

Dear Sanjoy,

Thanks for your elaborate answer and time.

I was convinced with the fact, but we can find a number papers in reputed journals using Relay Auto Tuning in first order systems. I also simulated the results and verified them. But the tuning parameters are entirely different from ZN open loop tuning method (using approximated FOPTD model).

Here in Relay Auto Tuning we ultimately find the process oscillating at a uniform period. But how we can take it as ultimate period and corresponding calculate gain as ultimate gain.

Hope this add some clarification to the main question.

Sanjoy Roy

Lalu,

My opinion is that we can not take it to be ultimate period. The oscillations in this case are due to the original first order system together with the relay nonlinearity.

So we are making measurements for the relay + original system, not exclusively the original system !

There is only a sort of "compromise justification" that I can think of, though! (Almost as an afterthought.)

In most PID controllers, the control signal as generated by the PID combo is passed through a pair of saturation limits that avoid what is known as the integral windup or integrator windup. If the saturation limits are decided before application of the ZN rules, and included with the system block, then the tuning is automatically for the saturation + original system!

But even if we accept this argument, there are two riders to it:

The saturation limits are (ideally speaking !) not open to modification once the PID parameters have been set according to ZN.

Saturation limits are equivalent (not exactly the same !) to relay strictly without hysteresis. So all this only justifies involvement of a relay without hysteresis, not one that includes the latter.

-Sanjay

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