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content/zettels/feedforward_control.md
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Depending on the physical system to be controlled, several feedforward controllers can be used: 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. 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} ## 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> <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): 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} \\] \\[ 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. - 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**: **2nd order setpoint generation**:
If we compute the fourier transform of the generated acceleration, we get the following signal (-20db/dec): If we compute the fourier transform of the generated acceleration, we get the following signal (-20db/dec).
![](/ox-hugo/feedforward_2nd_order_fourier.png)
{{< 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}}}\\). 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. 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. 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" >}} {{< figure src="/ox-hugo/feedforward_schematic_snap.png" >}}
## References
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body"> <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_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>
<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> </div>