For my feeling, this is a very good and important question.

Quote: "..if maximum overshoot is given 12%, then how much gain margin and phase margin fulfill this requirement ? "

Yes - I agree with you that the desired correlation between time domain (overshoot, settling time) and frequency domain (stability margin), in most - if not in all - books and papers is given for 2nd order systems only.

Perhaps anybody has more information?

I always have this problem with higher order system.. co-relation between time and frequency domain parameters. I didnt find any literature on it. Second order systems are good but not perfect always have gain margin issues while designing controller. in-fact i tried these controller real time and destroyed an actuators and amplifier..

According to my experience, many higher order systems can be treated - as far as stability and step response is concerned - as second order systems, as long as the loop gain goes through the 0 dB line with a slope of app. -20 db/dec.

But this gives onl a rough picture of the real system.

The issue is exactly that, while calculating loop gains a reduced order system might not give exact value and may induce instability..... these estimates are sometime not suitable in real time. Such as a system shown in the figure attached. The second order model fit is not appropriate. The damping ratio here is not exactly estimated and thus the gain margin for this is not the actual one. Now we can tune the PID controller if it can be tuned online however if we want to implement the controller on hardware such as FPAA it is very difficult to tune.

I think in this example, the main problem is the first small peak after 10^4. The value of the gain at the peak is close to the gain at the low frequency. Furthermore, the small peak is very close to the natural frequency of the system. This makes the system goes through 0dB line with a very steep slope (as Lutz said we always pursue a slope of -20dB/dec). The small resonant peak is often come from a structural vibration and a band-stop filter is often used to press the vibration(in an passive way). I have no idea about your system. You can refer to some literatures about the vibration control if the small peaks are raised from the structural vibration. In electronic systems the passive filter is often realised using a twin -T circuit.

This issue has been addressed in publications about "Quantitative Feedback Theory". If you can Botanik Horowitz' "Quantitative Feedback design theory" or Yaniv's "Quantitative Feedback design of linear and nonlinear Control Systems" you'll find some interesting answers.

Personally I like to work with "SISOtool" from Matlab's Control Systems toolbox, which allows to design in the frequency domain while closed-loop behaviour is calculated and plotted in the time-domain instantaneously.

Blind model reduction of the original high order model to a second order plus time delay model is dangerous as mentioned by the above several researchers. A pre-processing step should be used to get a proper model reduction. It should be noted that the original high order model has zeroes while the second order plus time delay model has usually no zero. This means that blind model reduction tries to fit the poles (that is, damping factor and time constant) of the second order model to the dynamics of the zeroes of the original high order model, resulting in totally unacceptable poles (equivalently, damping factor and time constant) of the reduced model if the original model has a strong zero dynamics (as shown in the Bode plot of Sachin Wadikhaye). To overcome the problem, the zeroes of the original model should be replaced by an equivalent time delay (this is called pre-processsing step). After the pre-processing step, subsequent model reduction would provide a reasonable second order plus time delay model.

Please, refer to the following papers and textbook for details.

Process Identification and PID Control, Su Whan Sung/Jitae Lee/In-Beum Lee, John Wiley & Sons, April, 2009

S. Skogestad, Simple analytic rules for model reduction and PID controller tun-ing, J. Process Control 13 (2003) 291–309.

Su Whan Sung and In-Beum Lee, Limitations and Countermeasures of PID Controllers, Ind. Eng. Chem. Res., Vol. 35, 2596-2610, 1996

Moving along with either the time domain or frequency domain specifications alone, cannot lead to a proper controller design. The system needs to be analyzed in both frequency domain and time domain. I think sisotool in MATLAB helps a lot in this case.

i'm surprised how you can analyse a bode diagram just based on only the magnitude diagram and ignore the phase diagram! To my knowledge, your question is straight forward. to approximate the higher order by a 2nd order, you have to meet some conditions that ensures the ignoring pole is not effective in neighbor frequencies of the next pole. for that i suggest to use some techniques to shift the poles of the system by controllers.

but the main question about how to switch from time to freq domain or vice versa. Basically, the characteristics of the controller as you said is cleared in time domain since what appears in the time domain response of a system has root in frequency-wise behavior of the system. for example if a step response of a system has high overshoot it means that the high frequencies in that controller have higher gain(overshoot in frequency domain). and this overshoot in frequency domain has a reverse relationship with stability margins. and again if we want to relate this issue to damping we can easily discuss it in this way that a damped response means slower response and this means slower transients which this cross one thing in mind that higher frequencies are cut off or damped( again we come to frequency domain and see the evidence in bode diagram by noticing a damping in frequency response)

i hope this hepls you

Some high order systems are fairly easy to approximate to second-order plus deadtime (FOPDT) model; but for some deadtime systems which possess multiple zeros, or/and unstable pole/s, approximating such systems to low order models can be very tedious and quite inaccurate. Nevertheless, for a control design purpose, one may afford to have fairly crude approximation (low order model) of a given high-order system so long that the model captures the dominant dynamics, especially those poles or/and zeros which are closed to the origin in the complex plane. In this case, it is often useful to combine both frequency domain (GM, PM, Ms, etc) and time domain (overshoot, settling time, etc) criteria to achieve desired tradeoffs between robustness and performance. I think it very difficult to develop  general relationships between the time domain criteria and frequency domain criteria for complex high-order systems. One might find it is quite attractive to write a programme of the design procedure using Matlab rather than simply relying on the standard available tools, e.g., Maltab SISO design tool. For sharing, one may adopt the multi-scale control scheme as an alternative way to systematically design a PID controller based on complex high-order models.