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Review, Section 4

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Thomas Dehaeze 2020-07-08 11:29:49 +02:00
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@ -307,15 +307,15 @@ This can be seen in the Root Locus plot (Figure ref:fig:root_locus_pure_iff) whe
Physically, this can be explain like so: at low frequency, the loop gain is very large due to the pure integrators in $K_F$.
The control system is thus canceling the spring forces which makes the suspended platform no able to hold the payload against centrifugal forces, hence the instability.
In order to apply Decentralized IFF on rotating positioning stages, two solutions are proposed to deal with this instability problem.
In order to apply Decentralized IFF on rotating platforms, two solutions are proposed to deal with this instability problem.
The first one consists of slightly modifying the control law (Section ref:sec:iff_hpf) while the second one consists of adding springs in parallel with the force sensors (Section ref:sec:iff_kp).
* Integral Force Feedback with High Pass Filter
<<sec:iff_hpf>>
** Modification of the Control Low :ignore:
As was explained in the previous section, the instability when using IFF with pure integrators comes from high controller gain at low frequency.
As was explained in the previous section, the instability comes in part from the high gain at low frequency caused by the pure integrators.
In order to limit the low frequency controller gain, an high pass filter (HPF) can be added to the controller
In order to limit this low frequency controller gain, an high pass filter (HPF) can be added to the controller
#+name: eq:IFF_LHF
\begin{equation}
\bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}, \quad K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
@ -329,7 +329,7 @@ This is however not the case in this study as it will become clear in the next s
#+latex: \par
** Feedback Analysis :ignore:
The loop gains for the decentralized controllers $K_F(s)$ with and without the added HPF are shown in Figure ref:fig:loop_gain_modified_iff.
The loop gains, $K_F(s)$ times the direct dynamics $f_u/F_u$, with and without the added HPF are shown in Figure ref:fig:loop_gain_modified_iff.
The effect of the added HPF limits the low frequency gain as expected.
The Root Loci for the decentralized IFF with and without the HPF are displayed in Figure ref:fig:root_locus_modified_iff.
@ -359,34 +359,34 @@ It is interesting to note that $g_{\text{max}}$ also corresponds to the gain whe
#+latex: \par
** Optimal Control Parameters :ignore:
Two parameters can be tuned for the controller eqref:eq:IFF_LHF: the gain $g$ and the pole's location $\omega_i$.
Two parameters can be tuned for the modified controller eqref:eq:IFF_LHF: the gain $g$ and the pole's location $\omega_i$.
The optimal values of $\omega_i$ and $g$ are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
In order to visualize how $\omega_i$ does affect the attainable damping, the Root Loci for several $\omega_i$ are displayed in Figure ref:fig:root_locus_wi_modified_iff.
It is shown that even though small $\omega_i$ seem to allow more damping to be added to the system resonances, the control gain $g$ may be limited to small values due to Eq. eqref:eq:gmax_iff_hpf.
It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes, the control gain $g$ may be limited to small values due to Eq. eqref:eq:gmax_iff_hpf.
#+name: fig:root_locus_wi_modified_iff
#+caption: Root Locus for several HPF cut-off frequencies $\omega_i$, $\Omega = 0.1 \omega_0$
#+attr_latex: :scale 1
[[file:figs/root_locus_wi_modified_iff.pdf]]
In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of the ratio $\omega_i/\omega_0$.
The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also display and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure ref:fig:mod_iff_damping_wi).
Three regions can be observed:
- $\frac{\omega_i}{\omega_0} < 0.02$: the added damping is limited by the maximum allowed control gain $g_{\text{max}}$
- $0.02 < \frac{\omega_i}{\omega_0} < 0.2$: good amount of damping can be added for $g \approx 2$
- $0.2 < \frac{\omega_i}{\omega_0}$: the added damping becomes small due to the shape of the Root Locus (Figure ref:fig:root_locus_wi_modified_iff)
In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of the $\omega_i/\omega_0$.
The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure ref:fig:mod_iff_damping_wi).
#+name: fig:mod_iff_damping_wi
#+caption: Attainable damping ratio $\xi_\text{cl}$ as a function of the ratio $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown
#+attr_latex: :scale 1
[[file:figs/mod_iff_damping_wi.pdf]]
Three regions can be observed:
- $\omega_i/\omega_0 < 0.02$: the added damping is limited by the maximum allowed control gain $g_{\text{max}}$
- $0.02 < \omega_i/\omega_0 < 0.2$: the attainable damping ratio is maximized and is reached for $g \approx 2$
- $0.2 < \omega_i/\omega_0$: the added damping decreases as the $\omega_i/\omega_0$ increases
* Integral Force Feedback with Parallel Springs
<<sec:iff_kp>>
** Stiffness in Parallel with the Force Sensor :ignore:
As was explained in section ref:sec:iff_pure_int, the instability when using decentralized IFF for rotating positioning platforms is due to Gyroscopic effects and, more precisely, due to the negative stiffness induced by centrifugal forces.
As was explained in section ref:sec:iff_pure_int, the instability when using decentralized IFF for rotating platforms is due to Gyroscopic effects and, more precisely, due to the negative stiffness induced by centrifugal forces.
In this section additional springs in parallel with the force sensors are added to counteract this negative stiffness.
Such springs are schematically shown in Figure ref:fig:system_parallel_springs where $k_a$ is the stiffness of the actuator and $k_p$ the stiffness in parallel with the actuator and force sensor.
@ -510,7 +510,7 @@ For any $k_p > m \Omega^2$, the control gain $g$ can be tuned such that the maxi
* Comparison and Discussion
<<sec:comparison>>
** Introduction :ignore:
Two modifications to the decentralized IFF for rotating positioning stages have been proposed.
Two modifications to the decentralized IFF for rotating platforms have been proposed.
The first modification concerns the controller and consists of adding an high pass filter to $K_F$ eqref:eq:IFF_LHF.
The system was shown to be stable for gains up to $g_\text{max}$ eqref:eq:gmax_iff_hpf.