MagnetoShield: A first principles model

[gergelytakacs]

A first principles model of the MagnetoShield shall be developed. This could be used as a starting point, but remember, that we have an additional attraction between the two permanent magnets! Also, There is a parallel capacitor added there to stabilize the voltage signal, but that could be possibly neglected.

[gergelytakacs]

Some parameters:

• m = 0.76 [g] (weight of the magnet, measured)
• R = 198.3 [Ohm] (Resistance of the electromagnet, measured)
• L = 0.239 [H] (Inductance of the electromagnet, measured)
• r = 8 [mm] (Core diameter of the electromagnet, measured and from datasheet)
• y0 = 15 [mm] (Distance from electromagnet for linearization, suggested).

[gergelytakacs]

Amateur hour: essential questions Q: Can an input to output (u(k)->y(k)) measurement be used in closed loop identification (CLI)? Does the presence of the controller affect the results? How and why?

TLDR; Yes.

A: Yes it can be, it is called direct closed loop identification [01,02,03,04,05] and it is commonly used. It neglects the effect of the controller [01],05], however, may affect the results under certain conditions, since the output is correlated with the input [05] and disturbances. Some identification methods assume this noise is zero, thus shall not be used for closed loop cases [04]. Closed loop identification may be even beneficial, since the point of operation and input spectrum is the same as intended use [02]. About 30 years ago it was thought that closed loop identification should be avoided, however now it is actually preferred [02,05] at least for some situations.

The possible pitfalls of CLI are [02]:

• The input signal contains effects of past output (b/c feedback) and past disturbances. Disturbance is thus correlated with inputs [07].
• Lack of causality.
• Too simple controller used.

The solution to make direct closed loop identification usable is:

• Use adequately complex controllers. Even if the input is in itself persistently exciting, the experiment may not be informative if the controller is too simple [02].
• Use a persistently exciting reference or other probe signal [02].
• The model must contain the true model [02,03]
• Noise model must be good if it is not white noise but having a high signal to noise ratio or minimizing the feedback contribution to the spectrum will likely minimize the effect [03,04]. Noise model may be also estimated, this is the suggested approach of Matlab [07].

A note on identification methods and target model types:

• A straightforward spectral analysis will give erroneous results [02]. Other non-parametric methods such as impulse-response and bode are to be avoided as well [03,05]
• If impulse response is identified by correlation methods, the result will be biased [02, 05].
• The instrument variable (IV) will also give incorrect results [04]
• Subspace methods will typically not give a consistent estimate when applied to closed-loop [02,05].
• Prediction error methods will give consistent estimates, given the data is informative and the model set contains the true system [02,06,06].
• Some methods may contain an expectation that there should be no algebraic loops, i.e. there is a delay somewhere in the loop [01].

The alternative to direct closed loop identification is :

• Indirect identification, when the controller information is used to reconstruct the model [01,05],
• Joint i/o identification, when models between reference and input,output are created, then the plant model is reconstructed [01,05].

Matlab:

• According to [07][07] the paper 05 and book [06][06] prove that using PEM and estimating a noise model will identify a plant even if it was operated closed-loop. This is the official The Mathworks stance. An example is

  model = pem(data, NX, 'dist', 'estimate')
  model2 = idss(drss(NX)) % or some other template you created using IDSS command
  model2.DisturbanceMOdel = 'estimate'
  model2 = pem(data, model2, 'focus', 'prediction')

Notes:

• As long as the controller is perfectly linear, time-invariant and noise-free; there is nothing to gain from the alternate methods [05].
• Some claim that CLI cannot be used for unstable systems [01], yet some say that PEM can be applied as long as the predictor is stable [05]. ARX, ARMAX satisfy this automatically.

[gergelytakacs]

@mgulan The usual model, with simplified electrical response is: After linearisation leading to:

[gergelytakacs]

@mgulan If you wish to try grey box (or any other type) of identification, here is a *.mat file of closed loop behavior for identification purposes. MagnetoShield.zip

The columns go: u(k) - input [V], w(k) - injected noise [V], y(k) - position [mm], sampled at Ts=0.004 s.

[gergelytakacs]

The linearized state-space models are third-order. A third-order black box model yields fairly sensible results: Given one uses detrended data ("delta") formulation and that is

 Options = ssestOptions;                              
 Options.Display = 'on';                              
 Options.WeightingFilter = 'prediction';              
 Options.InitialState = 'estimate';                   
 Options.N4Horizon = [15 39 39];                                                                          
 ss27 = ssest(Magnetic Levitationd, 3, Options)  

[gergelytakacs]

Here is the usual nonlinear and linearized model for magnetic levitation: Note, the model structure is the same as here.

[gergelytakacs]

Amateur hour: essential questions Q: What is the capacitance of the solenoid? How does the extra capacitor parallel to the solenoid affect the dynamics? A The terminal of the capacitor is in parallel with the inductor, which is a long wire. We can assume that the capacitor is shorted, therefore bypassed - thus can be essentially ignored in the analysis.

[gergelytakacs]

Accounting for the permanent magnet shall yield the same linearized model:

[gergelytakacs]

Also, according to [Barie and Chiasson] and their references, inductance is distance dependent, so we get a slightly more complex model as follows: From the viewpoint of the linearized model, this adds another term to the last equation, however, I could not make this one converge either.

[gergelytakacs]

@mgulan As of 3rd December the 3rd order black-box model seems to be decent, however, I had no luck with linearized and nonlinear first principle models yet. Neither of the latter two converge to anything useful. If you are up for the challenge, yo may try to make them work - the current "skeletons" for the identification are here. I'd probably try the lienarized grey-box at first, the nonlinear is even worse.

[gergelytakacs]

@mgulan A sensible black box TF model (1st order!) added to the collection. Maybe it would be fun to use the data (or even the hardware) in the Identification classes:

model =

  From input "Solenoid Voltage" to output "Position":
   0.02382
  ---------
  s + 46.93

Continuous-time identified transfer function.

Parameterization:
   Number of poles: 1   Number of zeros: 0
   Number of free coefficients: 2
   Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                                         
Estimated using TFEST on time domain data "Magnetic Levitation".
Fit to estimation data: 74.69% (data prefiltered)               
FPE: 7.842e-09, MSE: 7.665e-09  

[gergelytakacs]

@mgulan @erik1392 Amateur hour: Q: If you have an unstable model, testing its time-domain response to input data will necessarily cause an unstable response? A: ? My guess is yes,

[gergelytakacs]

@mgulan Successful grey-box linearized TF model:

  From input "Solenoid Voltage" to output "Position":
    -1.215
  ----------
  s^2 - 2218

Continuous-time identified transfer function.

Parameterization:
   Number of poles: 2   Number of zeros: 0
   Number of free coefficients: 2
   Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                                
Estimated using TFEST on frequency domain data "dataf".
Fit to estimation data: 88.95%                         
FPE: 1.465e-09, MSE: 1.46e-09 

This conforms to the "classical" model, where the mechanics of the ball are augmented with an U=Ki*I approach

[gergelytakacs]

@mgulan Success with linearized grey-box SS:

  A = 
           x1      x2      x3
   x1       0       1       0
   x2    2145       0  -217.6
   x3       0       0  -261.2

  B = 
       Solenoid Vol
   x1             0
   x2             0
   x3          1.48

  C = 
             x1  x2  x3
   Position   1   0   0

  D = 
             Solenoid Vol
   Position             0

  K = 
       Position
   x1         0
   x2         0
   x3         0

Parameterization:
   STRUCTURED form (some fixed coefficients in  A, B, C).
   Feedthrough: none
   Disturbance component: none
   Number of free coefficients: 4
   Use "idssdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                                
Estimated using SSEST on frequency domain data "dataf".
Fit to estimation data: 82.96%                         
FPE: 4.206e-09, MSE: 4.182e-09     

And this actually puts some trust in the model:

---Parameter Comparison---
L (measured): 0.239 [H],   L (model): 0.76 H
R (measured): 198.3 [Ohm], R (model): 199 [Ohm]

But since inductance is related to the object, possibly also to its permanent magnetic properites, this difference is not actually that significant.

So, all in all: (Will re-visit nonlinear ID as well).

[gergelytakacs]

One thing: when R2 will be ready, we will have a new measured state: coil current. This can make system identification more interesting, let's hope it will make things a bit smoother.

[gergelytakacs]

Amateur hour: Q: Can MATLAB's nlgreyest() handle unstable models?

I'll just ask.

[mgulan]

Hard to find out. Ver. 2018b of Scientific Toolbox provides only few useless information on p. 17-129/130.

[gergelytakacs]

Read the manual, but I am no smarter. :) (Scientific Toolbox, really?:)

On Fri, 7 Dec 2018, 16:34 mgulan <notifications@github.com wrote:

Hard to find out. Ver. 2018b of Scientific Toolbox provides only few useless information on p. 17-129/130.

— You are receiving this because you were assigned. Reply to this email directly, view it on GitHub https://github.com/gergelytakacs/AutomationShield/issues/75#issuecomment-445269151, or mute the thread https://github.com/notifications/unsubscribe-auth/Aj6tcM_Y1uTBQwReVaN-iW419uwHOeC6ks5u2opwgaJpZM4YvCmp .

[gergelytakacs]

After a quick test none of the first principles models work, have to find out the reason... Haven't tested a lot yet, but it doesn't work "out of the box" as of now.

[gergelytakacs]

There is something strange with the TF sys ID...

• K is by definition positive
• The data is OK: the more voltage (current) the closer it gets to the magnet, that is, the less distance.
• This logic should be mirrored by the transfer function. From a reference position (linearization point) the ball should "fall" to negative values, not run to positive (towards the magnet). This means, that the gain, and thus b0 is negative.
• Yet, the system identification toolbox consistently tells, that the gain is positive.

Could this problem be caused by:

• the effect of the feedback,
• linearization,
• the fact that it is identified from the frequency function?

The thing is, that if I put "-model" to the compare, it is still the same. So the ferquency peaks don't have an orientation in this sense.

Don't know where my logic fails.

[Update] That parameter consistently turns out to be positive, not negative. It could be the orientation of the measurement or somehting...

[gergelytakacs]

The first principles linearized models are quite cool actually, but there is still no directly identified (or compared) nonlinear model. I will make a new issue out of this,