FRF measurement functionality

I would like to see an FRF measurement functionality implemented in the VESC. This functionality gives you the amplitude and phase response of for example the current loop.

Required "ingredients"

1. Noise injection.

2. Trace functionality.

3. Averaging.

4. FRF calculation.

5. Bode plot functionality.

1. Noise injection.

A noise generator needs to be implemented, which can then be connected to various parts where you want to inject the noise. The first points to add the noise would be:

Vd and Vq, such that the current loops can be measured. Later other injection points can be added.

PRBS is a good and easy way to generate noise. It has a flat frequency spectrum and all frequency components are in there. It also has an optimal crest factor of 1. Here is code I am using that makes a repeating sequence of 2047 samples:

unsigned int NdownsamplePRBS = 1;
unsigned int downsamplePRBS  = 1;
uint16_t PRBSstate = 0xACE1u; /* Any nonzero start state will work. */
uint16_t noisebit;       /* Must be 16bit to allow bit<<15 later in the code */
float PRBSnoise;

// noise generation using PRBS, see https://en.wikipedia.org/wiki/Pseudorandom_binary_sequence
downsamplePRBS++;
if ( downsamplePRBS > NdownsamplePRBS) {
  downsamplePRBS = 1;
  noisebit  = ((PRBSstate >> 5) ^ (PRBSstate >> 7) ) & 1;  // taps: 11 9; feedback polynomial: x^11 + x^9 + 1
  PRBSstate =  (PRBSstate >> 1) | (noisebit  << 15);
}

PRBSnoise = 2 * (noisebit - 0.5)

The PRBSnoise variable can now be used as source for the noise.

The downsampling is not really needed and could be left out. This would be used to lower the frequency content of the noise signal, but since the VESC cannot make large traces at the full frequency this would cause aliasing to appear. Still this could be kind useful, but one has to be careful when interpreting the results. An interesting thing is that Siemens S120 drives actually implement it like that, with the aliasing problem in there.

To inject the noise is simple:

float Vq_distgain; //These you need to be able to set from the SW
float Vd_distgain; //These you need to be able to set from the SW

Vq += PRBSnoise * Vq_distgain;
Vd += PRBSnoise * Vd_distgain;

In principle the prbs generator can always run. To enable the noise just make any of the distgain variables non-zero.

2. Trace functionality.

Samples need to be taken at the full sample frequency of the VESC.

Trace of the following signals is needed (when injecting in d the q can be skipped and visa-versa, but maybe it is simpler to allways trace both):

• PRBSnoise
• Vd and Vq, directly after the noise is added.
• Id and Iq

The trace needs to be started after starting the noise and letting the system settle for at least 1 second (or settable parameter amount of time)

3. Averaging.

To get a good quality measurement, averaging is required. Typical is 10 to 20 averages. Much more doesn't help much, as averaging improves the SNR with sqrt( N_averages ).

Since the PRBS is a repeating sequence of 2047 samples, this exact same length should be captured, but it does not really matter what starting point you take. To get averaging in the FRF, simply sum in the time domain. Dividing by the number of averages is not required, as this will drop out when calculating the FRF.

Example:

sample 1 to 2047         simply go into the buffer values 1 to 2047

sample 2048 to 4095  add the values in the buffer at location 1 to 2047

and so on.

4. FRF calculation.

This can be performed on the host pc.

Get the data from the buffer, which should be exactly 2047 samples long.

Simply take the FFT of all the signals, whereby the signals need to be exactly 2047 samples long. You should see that the FFT of the noise has a flat magnitude.

FFT will give the response for both negative and positive frequencies. In this case the negative frequencies can just be ignored. The frequency vector of the FFT will have a spacing of Fs / 2047 (Where Fs is the sample frequency).

To get the electrical plant of the system: Plant_q = FRF_iq / FRF_Vq. (or d if you injected noise in the d axis) .  

If the current loop was turned on during the measurement, you can also get back the sensitivity as follows: Sq = FRF_Vq / (PRBSnoise * Vq_distgain)

From the sensitivity you can get back the Open Loop response, which is typically used by control engineers to tune controllers and check margins: OLq = (1/Sq)  -1

5. Bode plot functionality.

Typically the FRF data is represented in a Bode plot, with a logarithmic frequency scale in Hz, the magnitude in dB and the phase in degrees.

When plotting Plant_d and Plant_q you should clearly see the R L behavior  (1 / (R+L*s)), i.e. a flat 1/R line which then transitions into a -1 slope of the L*s. The graph is the inverse of the impedence.

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What do you think, is this something you could implement?