I am sorry to disagree, but a PR controller in alpha-beta frame is not equivalent to a PI controller in synchronous frame. In reality, a PR controller is equivalent to PI controllers in a pair of synchronous frames, one rotating forward and one rotating backwards, with mutual interaction among them. See, e.g., (4) and (5) in Article Frequency Response Analysis of Current Controllers for Selec...

Another way to see that they are not equivalent: a PI controller in synchronous frame has one pole, whereas the PR controller has two poles. Thus, the response is not the same. That being said, to assume that mainly the proportional part sets the bandwidth and the stability margins is often reasonable. But do not be mistaken: a PR controller is not equivalent to a PI controller.

PR controller is similar to PI controller.

PI controller is used for dc (d-q parameter) and PR is used for ac (alpha-beta, abc) kind of system.

The suggested way to tune the PR controller is to replace PR by PI and tune the PI controller.

Further replace the PI by PR with the same parameter gains and it will work.

Please refer to attached paper.

The best way to tune it will be by the genetic algorithim toolbox present within the matlab simulink environment because it helps in tuning the inner loop in accordance with the desired response thus tracking the system achieve its goal

@Zahid Dar: Can you provide any demo model of the same ??

I am sorry to disagree, but a PR controller in alpha-beta frame is not equivalent to a PI controller in synchronous frame. In reality, a PR controller is equivalent to PI controllers in a pair of synchronous frames, one rotating forward and one rotating backwards, with mutual interaction among them. See, e.g., (4) and (5) in Article Frequency Response Analysis of Current Controllers for Selec...

Another way to see that they are not equivalent: a PI controller in synchronous frame has one pole, whereas the PR controller has two poles. Thus, the response is not the same. That being said, to assume that mainly the proportional part sets the bandwidth and the stability margins is often reasonable. But do not be mistaken: a PR controller is not equivalent to a PI controller.

Dear Alejandro:

I do not completely agree with you. Using the complex shift property (also called the frequency shift property) of the Laplace transform, it is possible to show that multiplying by exp(-jwt) a dynamic equation in the time domain is equivalent, in the Laplace domain, to replace the Laplace operator "s" by "s+jwt".

Therefore if you take a PR controller in the alpha-beta frame yo can get some sort of equivalent PI controller in the d-q frame and viceversa. I understand this has been discussed in one book of Prof. Matavelli and has also been analysed in " A Generalized Class of Stationary Frame-Current Controllers for Grid-Connected AC–DC Converters", IEEE Transactions on Power Delivery Vol. 25, Nr. 4, 2010.

For instance if you replace each "s" of a PI controller (in the d-q frame) by (s+jw) you will get a PR controller in the alpha-beta frame. It is not going to be exactly a PR controller, but definitively it is going to be a resonant controller (with some cross-coupling terms). Therefore for each resonant controller in the stationary frame you can have an equivalent d-q controller in the synchronous rotating frame and vice versa.

Regarding the two poles of the PR controller compared to one pole of the PI controller. Probably this is produced because each frequency in the dq frame represents two frequencies in the alpha-beta frame. Therefore a single PI controller in d-q frame represents two resonant controllers in the alpha-beta frame. I have not thought about that, but I believe that this can be also analysed using the shift frequency property of the Laplace transform.

Dear Roberto,

I do agree that the Frequency shift property is a very useful tool to analyze this topic. That is the reason why I referred to an equation using it, i.e., to (4) and (5) of Article Frequency Response Analysis of Current Controllers for Selec...

I do also agree that by applying this property to an integrator 1/s, you obtain some sort of complex resonator 1/(s+j*w1), with 21 being the fundamental frequency. This was already done more than 30 years ago: Article A New Synchronous Current Regulator and an Analysis of Curre...

However, the person who asked the question here was clearly referring to a PR controller of the form Kp+s/(s^2+w1^2), which it can be easily seen by means of the frequency shift property that is equivalent to an integrator rotating backwards (which gives 1/(s+j*w1) in stationary frame) plus an integrator rotating forward (which gives 1/(s-j*w1)). By adding these two terms, you get the well-known s/(s^2+w1^2) form.

Thanks,

Alejandro

Thank you for your comment Alejandro.

If I take a conventional PI in the d-q frame the transfer function is Kp(s+a)/s.

If I apply the frequency shift property to this controller to referr to a stationary frame operating at frequency w1, I will have

Kp(s+a)/s is equivalent to kp [(s-jw1)+a]/(s-jw1).

After some manipulation I will get

Kp(s+a)/a is equivalent to Kp+Ki*s/ (s^2+w1^2)+Ki*jw1/ (s^2+w1^2)

which is a resonant controller with cross coupling terms .

However it is not exactly equal to the PR controller discussed in the original question and you are right about that.

Dear Alejandro and Roberto

Thanks for the insight view regarding PI controller and PR controller.

I agree with your points and would like to add in some comments.

PI controller is equivalent to PR controller in terms of minimizing steady state error in case of rotating dq domain and minimizing steady state oscillation in case of rotating frame in alpha-beta domain.

I think can say that both are not same but may be equivalent.

The suggested way of using PI gains for PR controller is to use initial stage. Definitely, the stability analysis is essential for PR controller and it can be checked using PI gains at starting step.

Thanks.

Ravi

Dear Dr. Tripathi,

A PI controller (Kp+Ki/s) in positive-sequence synchronous frame is not equivalent to a PR controller (Kp+Kr*s/(s^2+w0^2)) in stationary frame, neither in transient nor in steady state.

The combination of PI controller in a positive-sequence synchronous frame plus another one in a negative-sequence synchronous frame is equivalent to a PR controller in stationary frame, both in transient and in steady state.

I am sorry, but I don't understand your last paragraph.

Anyway, thanks for your contribution.

Alejandro

Dear Dr. Alejandro

Yes, I agree with the point that PI controller in combination of positive-sequence and negative sequence synchronous frame is equivalent PR controller.

Thank you for the detailed explanation.

I am sorry, there is writing mistake in the last paragraph of my previous comments.

I wanted to say that, the suggested way of using PI controller gains is to use the controller gains for PR at initial stage. Definitely, the stability analysis is essential for PR controller and it can be checked using PI gains at initial step. And, further can be tuned more appropriately.

Hope that i have made my clear through explanation in this comment.

Thanks for the making understanding and providing in-depth analysis.

Ravi Tripathi

Dear all, you can mathematically show that PI and PR controllers are equivalent if and only if their tuning is properly done and their implementation is performed based on their state-space models. I have published a brief note on arxiv.org (considering also anti-windup for both controller types), please feel free to check out:

https://arxiv.org/pdf/1610.07133.pdf or Working Paper On the equivalence of proportional-integral and proportional...

All the best,

Christoph.

Thank you Christoph.

Mathematically, the commonly used PR controller is not equivalent to PI controller. It can be deduced using the theory presented in the appendix of [1] that the controller in stationary frame shown in the attached figure is equivalent to a PI controller in synchronous frame.

[1] D. N. Zmood; D. G. Holmes. "Stationary frame current regulation of PWM inverters with zero steady-state error," IEEE Transactions on Power Electronics, 2003, 18(3): 814 - 822.

You can find in the attached figure that there exist coupling terms between the alpha axis-controller and the beta-axis controller. The commonly used PR controller in stationary frame does't include such coupling terms. The coupling terms come from the controller itself, not from the plant. If completely decoupled controller is required, the coupling terms generated by the plant model (e.g. the typical R-L model) should also be included.

I have done some test in matlab/simulink some time ago that such a controller performs exactly the same with the PI controller in stationary frame, if the parameters Kp and Ki have the same values.

However, in practice, the performance difference between the commonly used PR controller and PI controller may not be obvious. Therefore, PR controller and PI controller can be considered equivalent from the viewpoint of engineering.

In addition, I would like to recommend the an advanced resonant controller, i.e. the vector-proportional-integral (VPI) controller. Its expression is s(Kps+Ki)/(s^2+w^2). Its numerator could be used to cancel the pole of the R-L plant. VPI controller could eliminate undesired peaks in the bode plot. According to my experience, it performs really better than the common PR controller. More information about the VPI controller could be found in [2] and other literature.

[2] Alejandro G. Yepes; Francisco D. Freijedo; Jesús Doval-Gandoy; Óscar López; Jano Malvar; Pablo Fernandez-Comesaña. " Effects of Discretization Methods on the Performance of ResonantControllers," IEEE Transactions on Power Electronics, 2010, 25(7): 1692 - 1712.

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The coupled terms are for the positive sequence. If the negative sequence is considered. The coupled terms could be canceled.

Therefore, I agree with Alejandro that " The combination of PI controller in a positive-sequence synchronous frame plus another one in a negative-sequence synchronous frame is equivalent to a PR controller in stationary frame, both in transient and in steady state."

Dear all,

the identical but more general result in state space considering anti-windup is shown in my note, Working Paper On the equivalence of proportional-integral and proportional...

Book Non-identifier Based Adaptive Control in Mechatronics

All the best,

Christoph.

Hello,

As Dr. Yepes mentioned, one main difference between PR and PI controllers is the number of poles and zeros. One main advantage of PR is that cross-coupling is avoided when working in alpha-beta frame.

Just to provide a qualitative approach to the original answer, I would mention that, since an open loop pole-zero cancelation to get rid of the plant pole (i.e., IMC) is not a suitable technique for PR , something different (and maybe not so forced) can be tried. E.g., in the following paper

Article Discrete-Time Domain Modeling of Voltage Source Inverters in...

we proposed to place the PR zeros as further as possible from the right half plane (RHP), so you assure that the closed loop poles are in the LHP and are damped: from Evans root locus approach, the closed loop poles converge to the open loop zeros.

Anyway, as already mentioned in this thread, other reasonable approaches can be found in the literature. e.g.,

Article Frequency-domain passivity-based current controller design

provides and discusses a relative wide region for the resonant gains .

Best Regards,

F

I cannot support using a linear compensation network for a non-linear system. I also know that compensation Bode Plot margins leads to inaccurate results with poor correlation to step load response due to >2nd order effects .The best analysis should be used such as state-space and Stability factor from Root Locus / NyQuist plots to get perfect agreement with various step load responses over entire load range.

This test is what I have to validate the fastest , most accurate servo systems I know (rotary VCM in HDD’s). I have many patterns to stimulate any possible resonance in both directions and results can be widely different in settling response, if the servos do not do the following;

Precise sampling for feedback is critical along with controlled input ramps to reduce the chance of instability from simultaneous external disturbances. Non-linearities such as temperature sensitive permeability or PM magnet strength must be sensed and compensated. All other inputs must anticipate non-linear and delayed responses to feed-forward the expect feedback to rely then on more linearized (corrected) gain controls to match expected feedback under dynamic input changes. This requires better models of non-linear behaviour and more sensors and signal sampling for peak, and True RMS Current with computed torque, acceleration vs acceleration input, measured velocity vs velocity demand input and measured position or phase error vs expected profile at the end of demand profile, system frequency and non-linear correction factors. This way no sensors are integrated and stability relies more on precompensated nonlinear (feedforward) drive to create a more linear system. Then less integral gain is needed which always increases system overshoot and setting time.

From me experience, the proportional factor is always small can be even zero due to the ubiquitous noises. Then fix the PR bandwith, and this may be quite difficult in a variable-frequency speed control system. And finally increase the resonant factor little by little otherwise it could easily causes oscillations. If the PR is used in a fixed-frequency system, the adjusting can be much easier. My suggestion is that simulate your system with all algorithms complied in s-function/Simulink and system parameters identical to the real ones. This can help you find the closest values for use in the test rig. I did so and it always worked.