143 lines
6.8 KiB
Markdown
143 lines
6.8 KiB
Markdown
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title = "Feedforward Control"
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author = ["Dehaeze Thomas"]
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draft = false
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Depending on the physical system to be controlled, several feedforward controllers can be used:
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- <sec:rigid_body_feedforward>
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- <sec:fourth_order_feedforward>
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- <sec:model_based_feedforward>
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## Rigid Body Feedforward {#rigid-body-feedforward}
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<span class="org-target" id="org-target--sec-rigid-body-feedforward"></span>
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Second order trajectory planning: the acceleration and velocity can be bound to wanted values.
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Such trajectory is shown in Figure [1](#figure--fig:feedforward-second-order-trajectory).
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<a id="figure--fig:feedforward-second-order-trajectory"></a>
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{{< figure src="/ox-hugo/feedforward_second_order_trajectory.png" caption="<span class=\"figure-number\">Figure 1: </span>Second order trajectory" >}}
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Here, it is supposed that the driven system is a simple mass \\(m\\) with a damper \\(c\\).
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In that case, the feedforward force should be:
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\begin{equation}
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F\_{ff} = m a + c v
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\end{equation}
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## Fourth Order Feedforward {#fourth-order-feedforward}
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<span class="org-target" id="org-target--sec-fourth-order-feedforward"></span>
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The main advantage of "fourth order feedforward" is that it takes into account the flexibility in the system (one resonance between the actuation point and the measurement point, see Figure [2](#figure--fig:feedforward-double-mass-system)).
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This can lead to better results than second order trajectory planning as demonstrated [here](https://www.20sim.com/control-engineering/snap-feedforward/).
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<a id="figure--fig:feedforward-double-mass-system"></a>
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{{< figure src="/ox-hugo/feedforward_double_mass_system.png" caption="<span class=\"figure-number\">Figure 2: </span>Double mass system" >}}
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The equations of motion are:
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\begin{align}
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m\_1 \ddot{x}\_1 &= -c\_1 \dot{x}\_1 - k(x\_1 - x\_2) - c (\dot{x}\_1 - \dot{x}\_2) + F \\\\
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m\_2 \ddot{x}\_2 &= k(x\_1 - x\_2) + c (\dot{x}\_1 - \dot{x}\_2)
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\end{align}
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From the equation of motion, two transfer functions are computed:
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\begin{align}
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\frac{x\_2}{F}(s) &= \frac{c s + k}{(m\_1 s^2 + c\_1 s)(m\_2 s^2 + c s + k) + m\_2 s^2 (cs + k)} \\\\
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\frac{x\_1}{F}(s) &= \frac{m\_2 s^2 + c s + k}{(m\_1 s^2 + c\_1 s)(m\_2 s^2 + c s + k) + m\_2 s^2 (cs + k)}
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\end{align}
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Depending on whether \\(x\_1\\) or \\(x\_2\\) is to be positioned, two feedforward controllers can be used.
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If \\(x\_2\\) is to be positioned, the ideal feedforward force \\(F\_{f2}\\) is:
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\begin{equation}
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F\_{f2} = \frac{q\_1 s^4 + q\_2 s^3 + q\_3 s^2 + q\_4 s}{k\_{12} s + c} \cdot x\_2
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\end{equation}
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with:
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\begin{align}
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q\_1 &= m\_1 m\_2 \\\\
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q\_2 &= (m\_1 + m\_2) k\_{12} + m\_1 k\_2 + m\_2 k\_1 \\\\
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q\_3 &= (m\_1 + m\_2)c + k\_1 k\_2 + (k\_1 + k\_2) k\_{12} \\\\
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q\_4 &= (k\_1 + k\_2) c
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\end{align}
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This means that if a fourth-order trajectory for \\(x\_2\\) is used, the feedforward architecture shown in Figure [3](#figure--fig:feedforward-fourth-order-feedforward-architecture) can be used:
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\begin{equation}
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F\_{f2} = \frac{1}{k\_12 s + c} (q\_1 d + q\_2 j + q\_3 q + q\_4 v)
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\end{equation}
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<a id="figure--fig:feedforward-fourth-order-feedforward-architecture"></a>
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{{< figure src="/ox-hugo/feedforward_fourth_order_feedforward_architecture.png" caption="<span class=\"figure-number\">Figure 3: </span>Fourth order feedforward implementation" >}}
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Similarly, if \\(x\_1\\) is to be positioned, the perfect feedforward force \\(F\_{f1}\\) is:
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\begin{equation}
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F\_{f1} = \frac{1}{m\_2 s^2 + c s + k} \cdot (q\_1 s + q\_2 j + q\_3 a + q\_4 v)
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\end{equation}
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with:
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\begin{align}
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q\_1 &= m\_1 m\_2 \\\\
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q\_2 &= (m\_1 + m\_2) c + m\_2 c\_1 \\\\
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q\_3 &= (m\_1 + m\_2) k + c\_1 c \\\\
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q\_4 &= c\_1 k
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\end{align}
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and \\(s\\) the snap, \\(j\\) the jerk, \\(a\\) the acceleration and \\(v\\) the velocity.
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The same architecture shown in Figure [3](#figure--fig:feedforward-fourth-order-feedforward-architecture) can be used.
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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>)).
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## Model Based Feedforward Control for Second Order resonance plant {#model-based-feedforward-control-for-second-order-resonance-plant}
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<span class="org-target" id="org-target--sec-model-based-feedforward"></span>
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See (<a href="#citeproc_bib_item_2">Schmidt, Schitter, and Rankers 2020</a>) (Section 4.2.1).
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Suppose we have a second order plant (could typically be a piezoelectric stage):
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\\[ G(s) = \frac{C\_f \omega\_0^2}{s^2 + 2\xi \omega\_0 s + \omega\_0^2} \\]
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<a id="figure--fig:feedforward-second-order-plant"></a>
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{{< figure src="/ox-hugo/feedforward_second_order_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Bode plot of a second order system with fitted model" >}}
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The idea is to design a feedforward controller that corresponds to the plant inverse:
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\\[ C\_{ff}(s) = \frac{s^2 + 2\xi \omega\_0 s + \omega\_0^2}{C\_f \omega\_0^2} \\]
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This controller has a pair of zeros, corresponding to an anti-resonance at the eigenfrequency of the first eigenmode of the system, with equal damping.
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The controller needs to be modified in such a way that it becomes realisable.
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In this case it is decided to create a resulting overall transfer function of the controller and the plant that acts like a well damped mass-spring system with the same natural frequency as the plant and an additional reduction of the excitation of higher frequency eigenmodes.
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In order to realise this controller first two poles have to be added, placed at the same frequency as the resonance but with a higher damping ratio.
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Typically a damping ratio between aperiodic and critical (\\(0.7 < \xi < 1\\)) is applied to avoid oscillations.
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For \\(\xi = 1\\) this results in the following transfer function:
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\\[ C\_{ff}(s) = \frac{s^2 + 2\xi \omega\_0 s + \omega\_0^2}{s^2 + 2 \cdot 1 \cdot \omega\_0 s + \omega\_0^2}\\]
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<a id="figure--fig:feedforward-compensated-system"></a>
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{{< figure src="/ox-hugo/feedforward_compensated_system.png" caption="<span class=\"figure-number\">Figure 5: </span>Bode plot of the feedforward controlled system" >}}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<div class="csl-entry"><a id="citeproc_bib_item_1"></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>, nil. doi:<a href="https://doi.org/10.23919/acc.2004.1384042">10.23919/acc.2004.1384042</a>.</div>
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<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. <i>The Design of High Performance Mechatronics - Third Revised Edition</i>. Ios Press.</div>
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</div>
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