# Can gain and phase margins be obtained from step responses?

We need to comply to specific gain and phase margins requirements whereas the controller has been tuned on-line without knowing the plant transfer function (in order to save time). The response does not exhibit any overshoot, but we need to make sure we have 14dB and 30° of gain and phase margins. Can step responses be used for this?

For information, the controller is a PI type and the plant is a piezo actuator which twists a ring.

"Can step responses be used for this?"

The answer is yes as long as the PHASE margin is concerned - provided you have a linear second-order system.

Relevant books and articles contain a formula which relates the phase margin to the pole Q of the system. Because - on the other hand - the pole Q is related to the step respose overshoot, there is a curve which connects overshoot (in %) and the phase margin. For example, for a phase margin of 30deg the corresponding overshoot is app. 38%.
(see, for example, S. Franco: "Design with operational amplifiers and analog integrated circuits", McGraw-Hill, 2nd ed., page 354).

As far as I know, a similar relationship does not exist for the gain margin.

• Interesting, thanks. I know first order systems do not have overshoots, but is it possible that such a system (c.f. the last sentence of my post) is a third order such that I can't use those figures? Jan 16 '15 at 17:23
• A third-order system has one single real pole and a conjugate-complex pole pair (if stability is an issue). Also in this case, I would assume, are the characteristics of the step response a reliable information about stability properties (comparable to 2nd-order systems).
– LvW
Jan 17 '15 at 9:16
• So regardless of the actual order of the system, if I assume it is a second order will the phase margin I get from the step response's overshoot be close to the actual system's phase margin? Jan 21 '15 at 15:19
• ...close to...? Hard to answer. For a 3rd order system I would say: Yes. However, for a higher order sysytem I would be very cautious.
– LvW
Jan 21 '15 at 20:07
• Thanks for your help, it's not the first time you answer my questions related to control. However I can't be certain I meet the requirements this way then. For this time I identified the plant from the step response and used bode analysis - I don't think that's optimal, but at least I'm reasonably sure I meet requirements this way. Jan 21 '15 at 20:43

Unless there are some big wiggles in the gain curve 14dB sounds like plenty. How do you determine the frequency to measure these parameters?
As far as measuring these numbers in the step response... I'm not sure.
If you want to see that the HF response is 14 dB down, then (I think) that information is contained in the short time behavior of the step. (Dang I'm not sure that is right.) And it will be hard to pick out the details. I'd rather just crank up the gain and see where the system oscillates. (But maybe you can't do that?)

• I can crank up the gain and see where the system oscillates, I was just wondering if it could be determined from a single step response - you say "contained", but how? Also I'm not sure I get what you mean by "HF" response for a step, I'm tempted to understand it as a "high frequency" square wave (high frequency here is ~200Hz). The frequency at which GM and PM are measured is a very good point - do all systems have GM and PM decreasing with frequency? Jan 16 '15 at 17:17

When you do a step response, if there are no overshoots, generally you can state that the phase margin is good. However, you are talking about a "plant" and there will be non-linearities and these may cause worsening instability issues that are load or demand defendant.

There is no excuse for NOT doing the job properly!! Lives could be at risk!

If you want to read what the clever people at TI say, read this paper entitled "Simplifying Stability Checks" - it applies mainly to op-amps. Note that on page 3 there is a section entitled: -

"Comparison of Load Step Responses for Varying Phase Margin".

This underlies what I said at the top of this answer. It nicely demonstrates how you get more oscillations as you approach a low phase margin.

• I am asking this question because I'm doing my job properly. If it's impossible to tell without modelling, I'll do it, though the system is quite complex so we would like to stick with the very widespread solution of on-line tuning without transfer function extraction. You're talking about phase margin, but what about gain margin? Jan 15 '15 at 14:02

One method I've found is to identify the system using the Strejc method (which works for any order, sorry the link is in French), then fine tune the model to fit the real system, and finally plot the bode diagram to get the gain and phase margins.

I'm open to other ideas, but that's the best I found.

• The link you provided appears to be broken Aug 7 '15 at 4:35