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content/zettels/feedforward_control.md
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Below, the "References" heading will be auto-inserted.
Depending on the physical system to be controlled, several feedforward controllers can be used:
-
@ -108,14 +110,14 @@ and \\(s\\) the snap, \\(j\\) the jerk, \\(a\\) the acceleration and \\(v\\) the
The same architecture shown in Figure <fig:feedforward_fourth_order_feedforward_architecture> can be used.
In order to implement a fourth order trajectory, look at [this](https://www.mathworks.com/matlabcentral/fileexchange/16352-advanced-setpoints-for-motion-systems) nice implementation in Simulink of fourth-order trajectory planning (see also (<a href="#citeproc_bib_item_2">Lambrechts, Boerlage, and Steinbuch 2004</a>)).
In order to implement a fourth order trajectory, look at [this](https://www.mathworks.com/matlabcentral/fileexchange/16352-advanced-setpoints-for-motion-systems) nice implementation in Simulink of fourth-order trajectory planning (see also (<a href="#citeproc_bib_item_1">Lambrechts, Boerlage, and Steinbuch 2004</a>)).
## Model Based Feedforward Control for Second Order resonance plant {#model-based-feedforward-control-for-second-order-resonance-plant}
<span class="org-target" id="org-target--sec-model-based-feedforward"></span>
See (<a href="#citeproc_bib_item_3">Schmidt, Schitter, and Rankers 2020</a>) (Section 4.2.1).
See (<a href="#citeproc_bib_item_2">Schmidt, Schitter, and Rankers 2020</a>) (Section 4.2.1).
Suppose we have a second order plant (could typically be a piezoelectric stage):
\\[ G(s) = \frac{C\_f \omega\_0^2}{s^2 + 2\xi \omega\_0 s + \omega\_0^2} \\]
@ -164,8 +166,10 @@ It therefore depends on:
- For 4th order, derivative of jerk is generated over time, and then integrated 4 times to give: jerk, acceleration, velocity and position.
**2nd order setpoint generation**:
If we compute the fourier transform of the generated acceleration, we get the following signal (-20db/dec):
![](/ox-hugo/feedforward_2nd_order_fourier.png)
If we compute the fourier transform of the generated acceleration, we get the following signal (-20db/dec).
{{< figure src="/ox-hugo/feedforward_2nd_order_fourier.png" >}}
Notches are at \\(f\_1\\), \\(2f\_1\\), \\(3f\_1\\), ... with \\(f\_1 = \frac{a\_{\text{max}}}{v\_{\text{max}}}\\).
It is therefore possible to choose the velocity and acceleration such that \\(f\_1\\) (or one of its integral multiple) matches the resonance frequency of the system.
Therefore, the acceleration time constant can be chosen at the inverse of the plant resonance.
@ -225,11 +229,8 @@ This can be solved by using **snap feedforward**
{{< figure src="/ox-hugo/feedforward_schematic_snap.png" >}}
## Bibliography {#bibliography}
## References
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Boerlage, M., M. Steinbuch, P. Lambrechts, and M. van de Wal. 2003. “Model-Based Feedforward for Motion Systems.” In <i>Proceedings of 2003 IEEE Conference on Control Applications, 2003. CCA 2003.</i> doi:<a href="https://doi.org/10.1109/cca.2003.1223174">10.1109/cca.2003.1223174</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Lambrechts, P., M. Boerlage, and M. Steinbuch. 2004. “Trajectory Planning and Feedforward Design for High Performance Motion Systems.” In <i>Proceedings of the 2004 American Control Conference</i>. doi:<a href="https://doi.org/10.23919/acc.2004.1384042">10.23919/acc.2004.1384042</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_3"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Boerlage, M., M. Steinbuch, P. Lambrechts, and M. van de Wal. 2003. “Model-Based Feedforward for Motion Systems.” In <i>Proceedings of 2003 Ieee Conference on Control Applications, 2003. Cca 2003.</i> <a href="https://doi.org/10.1109/cca.2003.1223174">https://doi.org/10.1109/cca.2003.1223174</a>.</div>
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