From b5d460c65fad3e68e434f66f05745fdd8a5ad736 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 17 Dec 2024 11:16:07 +0100 Subject: [PATCH] Update Content - 2024-12-17 --- content/zettels/feedforward_control.md | 19 ++++++++++--------- 1 file changed, 10 insertions(+), 9 deletions(-) diff --git a/content/zettels/feedforward_control.md b/content/zettels/feedforward_control.md index cb4ae58..7de9e6f 100644 --- a/content/zettels/feedforward_control.md +++ b/content/zettels/feedforward_control.md @@ -7,6 +7,8 @@ draft = false Tags : +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 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 (Lambrechts, Boerlage, and Steinbuch 2004)). +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 (Lambrechts, Boerlage, and Steinbuch 2004)). ## Model Based Feedforward Control for Second Order resonance plant {#model-based-feedforward-control-for-second-order-resonance-plant} -See (Schmidt, Schitter, and Rankers 2020) (Section 4.2.1). +See (Schmidt, Schitter, and Rankers 2020) (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
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Boerlage, M., M. Steinbuch, P. Lambrechts, and M. van de Wal. 2003. “Model-Based Feedforward for Motion Systems.” In Proceedings of 2003 IEEE Conference on Control Applications, 2003. CCA 2003. doi:10.1109/cca.2003.1223174.
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Lambrechts, P., M. Boerlage, and M. Steinbuch. 2004. “Trajectory Planning and Feedforward Design for High Performance Motion Systems.” In Proceedings of the 2004 American Control Conference. doi:10.23919/acc.2004.1384042.
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Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. The Design of High Performance Mechatronics - Third Revised Edition. Ios Press.
+
Boerlage, M., M. Steinbuch, P. Lambrechts, and M. van de Wal. 2003. “Model-Based Feedforward for Motion Systems.” In Proceedings of 2003 Ieee Conference on Control Applications, 2003. Cca 2003. https://doi.org/10.1109/cca.2003.1223174.