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I have been solving following problem. I have two open loop state estimators used for estimation of the unmeasurable states of a given linear dynamic system. The first estimator provides estimate $\hat{\psi}_{r_{{\alpha\beta}_i}}$ which is accurate at low frequencies and inaccurate at high frequencies. The second estimator provides estimate $\hat{\psi}_{r_{{\alpha\beta}_v}}$ which is accurate at high frequencies and inaccurate at low frequencies. The goal is to combine both the estimates $\hat{\psi}_{r_{{\alpha\beta}_i}}$ and $\hat{\psi}_{r_{{\alpha\beta}_v}}$ into the resulting estimate $\hat{\psi}_{r_{\alpha\beta}}$ which will be accurate at the same time in the low frequencies and the high frequencies.

I have found in some arcticle following approach

enter image description here

I was curious how does this structure work. So I have used the Laplace transform and derived following formula

$$\hat\Psi_{r_{\alpha\beta}}(s) = \frac{s\cdot K_p + K_i}{s^2 + K_p\cdot s + K_i}\cdot\hat\Psi_{r_{{\alpha\beta}_{i}}}(s) + \frac{s^2}{s^2 + K_p\cdot s + K_i }\cdot\hat\Psi_{r_{{\alpha\beta}_{v}}}(s)$$

Via comparison with the standard second order system

$$\frac{\omega^2_n}{s^2 + 2\zeta\omega_n\cdot s + \omega_n^2}$$

I have found that following formulas hold: $K_p = 2\zeta\omega_n$, $K_i = \omega^2_n$. I have arbitrarily chosen $\zeta = \frac{1}{\sqrt{2}}$ and $\omega_n=2\pi\cdot 10$ and created the Bode plot

enter image description here

From the frequency response it is apparent that the above mentioned control loop actually has the desired properties i.e. the resulting estimate $\hat{\psi}_{r_{\alpha\beta}}$ consists of the $\hat{\psi}_{r_{{\alpha\beta}_i}}$ estimate at the low frequencies and the $\hat{\psi}_{r_{{\alpha\beta}_v}}$ estimate at the high frequencies.

My question is:

  1. Why is it necessary to use the control loop consisting from the PI controller ($K_p + \frac{K_i}{s}$) cascaded with the integrator ($\frac{1}{s}$)? Why isn't sufficient just to apply the individual filters on the estimates and sum the filtered signals?
  2. Let's say the control loop consisting from the PI controller in cascade with the integrator is inevitable. For practical implementation I need to find some reasonable values for the upper and lower limit of the PI controller and also for the integrator. How can I find these limits?
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  • $\begingroup$ "I have found in some article". Could you please edit your question with a citation for the article? If possible, include a link to a version that's not behind a paywall. If not, provide a link (or a good old citation to the article on paper). $\endgroup$
    – TimWescott
    Commented Dec 15, 2022 at 22:00

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Why is it necessary to use the control loop consisting from the PI controller ($K_p+\frac{K_{i}}{s}$) cascaded with the integrator ($\frac 1 s$)? Why isn't sufficient just to apply the individual filters on the estimates and sum the filtered signals?

You could do that. If you did you'd need a set of filters with two pairs of nominally identical modes instead of just two modes, and there would be a startup transient where the pairs of modes were, in fact, not identical.

Let's say the control loop consisting from the PI controller in cascade with the integrator is inevitable. For practical implementation I need to find some reasonable values for the upper and lower limit of the PI controller and also for the integrator. How can I find these limits?

Regardless of how you implement it, you need to find the right filter settings. You would do this by looking at the dynamics of $\hat{\psi}_{r_{{\alpha\beta}_i}}$ and $\hat{\psi}_{r_{{\alpha\beta}_v}}$, so that you can choose the optimal crossover frequency. This is probably the frequency where $\hat{\psi}_{r_{{\alpha\beta}_i}}$ and $\hat{\psi}_{r_{{\alpha\beta}_v}}$ are close to being equally good, but the exact best frequency (or even whether a second-order system is best to do the crossover) depends on the dynamics of $\hat{\psi}_{r_{{\alpha\beta}_i}}$ and $\hat{\psi}_{r_{{\alpha\beta}_v}}$.

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  • $\begingroup$ Thank you for your answer. Let's say I would like to implement the control loop in software. I see that the crossover frequency has to be set based on the dynamics of the individual estimates. But what I still don't understand is how to set the limits of the PI controller output and the integrator output. Without appropriate limits I am not able to practically implement neither the PI controller nor the integrator. Or do you think that it is possible? $\endgroup$
    – Steve
    Commented Dec 16, 2022 at 8:01
  • $\begingroup$ It is a feedback loop, but it's not really a "control loop" -- it's just a two-input, one-output, IIR filter. If by "limit the integrators" you mean integrator anti-windup, you don't have an actuator to saturate, so you shouldn't need to do that. As long as the system is BIBO stable and the inputs are, indeed, bounded -- just make sure that the worst-case input amplitudes can't make the states exceed their reasonable ranges, and you'll be fine. $\endgroup$
    – TimWescott
    Commented Dec 16, 2022 at 15:49

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