Is Ziegler-Nichols tuning method for PID controllers to second order systems (under damped systems) efficient?
There are actually two methods proposed by Ziegler and Nichols: the step response and the ultimate sensitivity method.
The step response may be interpreted as a method based on the plant approximation by a delayed first order integrator. Usually, it leads to oscillatory responses. If you wish a more reliable tuning appropriate also for underdamped 2nd order systems, see e.g. https://www.researchgate.net/publication/269632329_PD-Controller_Tuning_Based_on_I2Td_and_I2T1_Models.
The integral term and an usually required filtration may effectively be added by a disturbance observer, as e.g. in https://www.researchgate.net/publication/261058017_Modular_PID-controller_design_with_different_filtering_properties
Conference Paper PD-Controller Tuning Based on I2Td and I2T1 Models
Conference Paper Modular PID-controller design with different filtering properties
Ziegler-Nichols is not an efficient method for tuning PID controllers. It is rather a method for getting sure that your controller will not make whole system unstable. Try to google "pid tuning methods".Here is a comparison of tuning methods: http://www.ie.itcr.ac.cr/einteriano/control/clase/Zomorrodi_Shahrokhi_PID_Tunning_Comparison.pdf
I think you should have a look at the book "Modern Control Systems" by R.C. Dorf and R.H. Bishop. There are listed methods of PID-controller tuning based on error measures minimizations. For example you can use ITAE (integral of time-weighted absolute error) criterion for tuning your PID-controller.
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