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The reference Feedforward control concepts through Interactive Tools says, to minimise the effect of the disturbance $d$ on the process output $y$, a compensator $G_{\text{ff}}=P_3/P_1$ shall be used. From a theory point of view, no problem. Even a few simple examples behave as expected.

Block diagram illustrating the feedforward control

Encouraged by that, I decided to implement three-phase inverter control in $dq$ reference frame. Inverter model is represented by LC filter equations: $$ \text{L} \frac{\operatorname{d}}{\operatorname{d}t} \begin{bmatrix} i_\text{d} \\ i_\text{q} \\ \end{bmatrix} = \begin{bmatrix} v_\text{dn} \\ v_\text{qn} \\ \end{bmatrix} - \begin{bmatrix} v_{\text{d load}} \\ v_{\text{q load}} \\ \end{bmatrix} + \text{L} \omega \begin{bmatrix} i_\text{q} \\ -i_\text{d} \\ \end{bmatrix} $$ $$ \text{C} \frac{\operatorname{d}}{\operatorname{d}t} \begin{bmatrix} v_\text{d load} \\ v_\text{q load} \\ \end{bmatrix} = \begin{bmatrix} i_\text{d} \\ i_\text{q} \\ \end{bmatrix} - \begin{bmatrix} i_{\text{d load}} \\ i_{\text{q load}} \\ \end{bmatrix} + \text{C} \omega \begin{bmatrix} v_\text{q load} \\ -v_\text{d load} \\ \end{bmatrix} $$ From there, I derive plant model and control structure in $dq$ reference frame (only $d$ frame is shown): three-phase inverter control strucure and plant model for d reference frame

In this case disturbance $i_{\text{d load}}$ is fed directly to the plant model. i.e. $P_3=1$. Open-loop inner loop equation is: $$ G_\text{co}(s) = \underbrace{K_\text{P} \left( 1 + \frac{1}{s T_\text{I}} \right)} _\text{PI controller} \underbrace{ \left(\frac{1}{s \text{L}}\right) } _{\text{Process}} $$ and closed loop ($P_1$) is $$ G_\text{cc}(s) = \frac{G_\text{co}(s)}{1 + G_\text{co}(s)} = ... = \frac{s T_\text{I}+1}{s^2 \text{L} T_\text{I}/K_\text{P} + s T_\text{I} + 1} $$ Compensator transfer fucntion is $G_{\text{ff}}=1/G_\text{cc}(s)$ and numerator is a higher order than the denominator.

  • I cannot implement this transfer function in MATLAB Simulink. Is there a way forward?
  • I don't see any issues with the method! Is this the right approach? Articles deal little with feed-forward signal or cross-coupling compensation.
  • When I make $P_3$ equal to $P_1$, i.e. $G_{\text{ff}}=1$, the disturbance is eliminated. In other cases, there is a huge overshoot for step disturbance. It eventually stabilises.
  • I also have to compensate the cross-coupling element $\omega\text{C}v_{\text{q load}}$ using the same compensator?
  • In the real system (e.g. DSP), I will implement only the control part in $dq$ frame. Do I need to add compensators to load current and cross-coupling signals? Also, the plant's transfer function will be different and unknown. How to determine compensator there or leave it at 1 and only tune PI controller to get the desired response.
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I cannot implement this transfer function in MATLAB Simulink. Is there a way forward?

That's because you can't implement it at all.
$$ \frac{1}{G_\text{cc}(s)} = \frac{s^2 \text{L} T_\text{I}/K_\text{P} + s T_\text{I} + 1}{s T_\text{I}+1}, \tag 1 $$ which is impossible because it's not proper. But consider two things: first, $P_3(s)$ may save you, and second, if it doesn't, there's an approximation that will.

If $P_3(s)$ is inherently low-pass, then your $G_{ff} = \frac{P_3(s)}{G_\text{cc}(s)}$ will become proper, or at least will have the same order in the numerator and the denominator. Then it can be realized without pain.

If $P_3(s)$ isn't low-pass then you can approximate $G_{ff}$. Consider $P_3(s) = 1$. With a bit of rearrangement, you can get: $$ \frac{1}{G_\text{cc}(s)} = k_1 s + k_0 + \frac{k_{lp}}{s T_I + 1} $$ The low-pass filter is realizable, the proportional gain is realizable, and only the naked differentiator is unrealizable. So let $$ G_{ff}(s) = \frac{k_1 s}{\tau s + 1} + k_0 + \frac{k_{lp}}{s T_I + 1} \simeq \frac{1}{G_\text{cc}(s)} $$

The smaller the $\tau$, the more this approaches the "ideal", but the more sensitive the system becomes to noise, and the bigger the signal is coming out of it. The larger the $\tau$, the easier the system design gets, but the more error you have.

Choosing the right value for $\tau$ involves the cost of control, the anticipated power spectrum of the disturbance, the robustness of the system, what you really need to achieve, and a whole bunch of other parameters.

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1 - What transfer function are you trying to implement? The controller transfer function or the "plant" transfer function?

2 - You are trying to simulate some kind of inverter/active rectifier, right? Inverters are non-linear, you can simulate them with only linear blocks if you want but you will not be able to test your controller with the non-linearities that are present in real-life.

3 - What are P1 and P3 ?

4 - You kind of compensate your cross-coupling terms in the second image the $\omega Li_q$ and $\omega Li_q$ terms. The cross-coupling terms are compensated to improve the response of your controller, but they are not technically needed.

5 - In a real-life implementation you don't have to compensate the cross-coupling terms but compensating the cross-coupling makes your control system settle faster if I recall correctly.

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