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Fixing the value of wn for a second order system

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garimella

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Fixing value of wn for a second order system

I have to design a second order system that has to respond for a sawtooth input with frequency of 1kHz. With this input, how to go about fixing the value of natural frequency of the system?
 

Re: Fixing value of wn for a second order system

Depends on the intended waveform quality, of course. You realize that ideal sawtooth has infinite bandwidth?

Presume you are designing a low-pass transfer function. Choosing an appropriate Q is important as well. Guess you'll end up with Bessel or even lower Q characteristic and e.g. 20 times the fundamental frequency. Why not try out in a signal processing tool or circuit simulator?
 

Re: Fixing value of wn for a second order system

I want to avoid trial method to arrive at the solution. Wanted to know if there is a method to make the process easier. But it is important that this system should track the input sawtooth with less than 2% error. Now how to fix the value of wn and proceed forward
 

natural frequency - should we care?

A second order system is characterized by wn and zeta. if bandwidth of the system equals wn , my system is stable with sufficient zeta, should we really care how much wn value should be. In other words, what role wn, plays in designing control system?
 

Re: Fixing value of wn for a second order system

You should definitely care. I think you don't yet imagine how the sawtooth signal is distorted when passing the control system. As a first step, you may want to specify your 2% error criterion clearly. Please look at the examples with 10 and 50 kHz Bessel low-pass (corresponds to F0 of 12.7 respectively 63.5 kHz and Q = 0.58).

sawtooth.PNG

You can of course improve the control system tracking quality by limiting the set point slew rate, changing the sawtooth wave into an asymmetric triangle.

tri.PNG

Regarding your previous question, once you have defined the waveform and error criterion unequivocally, you can optimize the controller parameters by numerical methods.
 

Re: Fixing value of wn for a second order system

I'm not sure if I understand the question correctly. The closed loop transfer function of a control system is set up by the combination of "plant" and controller (or "compensator"). The former is usually given and can't be modified at will. Or you need to change the hardware.

The previously posted response waveforms are under the assumption that the "plant" is fast enough to implement the respective transfer functions. The examples show how F0 (or wn) influences the control system output.
 

Re: Fixing value of wn for a second order system

Should I understand that cut off frequency and wn are identical? In that case wn does influence the output. I am trying to uncover the concept of how someone can approach wn and cause instability in the system. If wn is 100hz, does it mean ,that I should not excite the input beyond 100Hz? I am unable to relate wn with respect to filter
 

Re: Fixing value of wn for a second order system

I commented the relation of wn and low pass cut-off frequency previously.

Unfortunately you didn't yet describe your hypothetical control system completely. I assumed a low-pass closed loop transfer function in post #2 in lack of a specification.

Instead of discussing the relation of technical terms, define the control system topology and the transfer function you are looking at.
 

Re: Fixing value of wn for a second order system

I am looking for a 2nd order low pass filter. Wanted to understand the TF in relation to control system terminology like zeta, wn and bandwidth.
 

Re: Fixing value of wn for a second order system

Q = 1/(2 zeta), I think. Second order Bessel in post #5 corresponds to zeta = 0.87. Relation of 3 dB cut-off frequency and F0 was already given (factor 1.27 for 2nd order Bessel).
 

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