I have a previous knowledge of the system, so I've the lower and upper bound of the frequency response test.
I'm performing the response frequency test, as follow:
The input to the system is given one frequency per time, and after X cycles at the same frequency,the frequency is increased by a factor, F.
The X cycles are chosen appropriately to let the transient of the system vanish, and the increment F is chosen to have enough frequency to recreate the bode diagram.
To get the gain and the phase of the system for a given frequency, I've chosen only the portion that the system is in the steady state, and after I applied the least square method to the input and measurement (output of the system).
My doubts regarding about the implementation in the real world, because I cannot sampled the output and write in the input at the same time. Then, the input u(k) and output(k) are not synchronize.
I suppose the phase of the system in high frequency may be unreliable due to this problem. But, what is the correct sequence to minimize the undesirable effect on the test, without increasing the sampling frequency?
Do the calculation of the sine - Write at the Input of the system - Read the output
Read the output - do the calculation of the sine - Write at the Input
Second question.
I should compensate the zero-order-hold, during the phase estimation? I mean I should increase the phase by a factor ws*T/2, where ws is the sampling frequency?
Thank you in advanced.
The non-synchronicity of input and output isn't a big issue, having a common reference time and both sapling times are adequate for the input frequency you investigate.
Alternatively, if you are working with a digital system, instead of generating the frequency response, you could try to estimate the location of the poles and zeros of the plant (connected to the ZOH) directly, by simply inputting a step (or some other waveform) and seeing how your plant responds! When I tried this I was very pleased (and a bit surprised!) by the results I got for a simple system I was working on.
See slide 18 in ..
https://www.researchgate.net/publication/282134990_Slides_from_my_conference_presentation_%28Part_II_-_Filter_Application%29
Data Slides from my conference presentation (Part II - Filter Application)
I'm working with a digital system. The frequency response method is kind of double check on the model that was identified by another method. I mean, first I use PRBS (Pseudo Random Binary Sequence) at input of the system and read the output of the system, and after I do the identification.
The frequency response is just to check the first identification.
... OK. Have a look at this then ...
https://www.researchgate.net/publication/282211166_Built-In_Stepped-Sine_Measurements_for_Digital_Control_Systems
You may also find this one interesting (his keynote from the same conference) ...
https://www.researchgate.net/publication/282211207_Trying_to_Keep_it_Real_25_Years_of_Trying_to_Get_the_Stuff_I_Learned_in_Grad_School_to_Work_on_Mechatronic_Systems
Conference Paper Built-In Stepped-Sine Measurements for Digital Control Systems
Conference Paper Trying to Keep it Real: 25 Years of Trying to Get the Stuff ...
Not clear to me why you speak of single sine waves to accumulate a Bode diagram, and then speak of pseudorandom sequence. The first is very tedious and old-hat. The second is much better as it can contain a large bandwidth in a single signal, and the bode digram can be estimated from cross-spectral analysis or cross-correlation (impulse response) - I presume you think your system is linear?
When you do use sine waves, how did you determine the time needed to reach steady state? That should require a time interval at least 5 times as long as the longest time constant estimated for your system. If there is any noise, a single measure from one test at a fixed time will be very unreliable. Multiple measurements from several peaks, or searching for phase shift and gain over multiple cycles by regression would be more robust - the latter is easily seen using Lissajous plots
Thank you, for your answer Henrietta L.
I assumed that system is linear, and the model is validity in the neighborhood of the operation point. I calculated the time to reach the steady-state, as you mentioned. The estimate of the DC gain and phase were obtained, using the least squares with the data related to steady-state (the number of cycles on the same frequency was chosen appropriately to mitigate the degradation due to the noise).
The PRBS signal was used just to do a double check on the identification.
Dear Adolfo,
I would use a cross-spectral analysis for the estimation of the plant frequency response when dealing with linear system and wideband PRBS excitation signal. For example the Welch’s method of averaged periodograms proved to be a numerically robust tool, see
Welch, P.D. "The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms".
The corresponding routine is available in Matlab Signal Processing Toolbox or you can implement it yourself easily using the Fast Fourier Transform.
As an alternative, you can use a swept sine wave (chirp) input signal with variable frequency, amplitude and DC component. This allows you to gain more control over the spectral power density of the excitation signal over the tested frequency range which may be beneficial to improve signal to noise ratio, avoid excitation of some types of nonlinearities, limit the range of the plant output and perform a closed-loop identification at the cost of longer experiment execution time. The plant frequency response can be estimated online or offline by means of an observer, see the provided links for more details.
Conference Paper Kalman filter based observer design for real-time frequency ...
Data Process Control 2015 presentation - Kalman filter based obse...
Conference Paper Robust frequency identification of oscillatory electromechan...
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