Add inf. about rotational speed for each plot
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@ -153,7 +153,6 @@ To study the dynamics of the system, the differential equations of motions eqref
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\begin{equation}
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\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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with $\bm{G}_d$ a $2 \times 2$ transfer function matrix
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#+name: eq:Gd_m_k_c
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\begin{equation}
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@ -177,7 +176,7 @@ One can verify that without rotation ($\Omega = 0$) the system becomes equivalen
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# Change of variables
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In order to make this study less dependent on the system parameters, the following change of variable is performed:
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- $\omega_0 = \sqrt{\frac{k}{m}}$: Natural frequency of the mass-spring system in $\si{\radian/\s}$
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- $\omega_0 = \sqrt{\frac{k}{m}}$: Undamped natural frequency of the mass-spring system in $\si{\radian/\s}$
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- $\xi = \frac{c}{2 \sqrt{k m}}$: Damping ratio
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The transfer function matrix eqref:eq:Gd_m_k_c becomes equal to
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@ -209,20 +208,17 @@ Supposing small damping ($\xi \ll 1$), two pairs of complex conjugate poles are
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#+name: eq:pole_values
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\begin{subequations}
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\begin{align}
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p_{+} &= - \xi \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \pm j \left( \omega_0 + \Omega \right) \\
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p_{-} &= - \xi \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right) \pm j \left( \omega_0 - \Omega \right)
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p_{+} &= - \xi \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \pm j \omega_0 \left( 1 + \frac{\Omega}{\omega_0} \right) \\
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p_{-} &= - \xi \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right) \pm j \omega_0 \left( 1 - \frac{\Omega}{\omega_0} \right)
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\end{align}
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\end{subequations}
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When the rotation speed in non-null, the resonance frequency is split into two pairs of complex conjugate poles.
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The real part and complex part of these two pairs of complex conjugate poles are represented in Figure ref:fig:campbell_diagram as a function of the rotational speed $\Omega$.
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As the rotation speed increases, $p_{+}$ goes to higher frequencies and $p_{-}$ to lower frequencies.
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# The system goes unstable at some frequency w0
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When the rotational speed $\Omega$ reaches $\omega_0$, the real part $p_{-}$ is positive meaning the system becomes unstable.
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The stiffness of the X-Y stage is too low to hold to rotating payload hence the instability.
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Stiff positioning platforms should be used if high rotational speeds or heavy payloads are used.
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This is graphically represented with the Campbell Diagram in Figure ref:fig:campbell_diagram.
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When the rotational speed $\Omega$ reaches $\omega_0$, the real part $p_{-}$ becomes positive rendering the system unstable.
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Physically, the negative stiffness term induced by centrifugal forces exceeds the spring stiffness.
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Thus, stiff positioning platforms should be used when working at high rotational speeds.
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#+name: fig:campbell_diagram
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#+caption: Campbell Diagram : Evolution of the complex and real parts of the system's poles as a function of the rotational speed $\Omega$
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@ -232,8 +228,8 @@ This is graphically represented with the Campbell Diagram in Figure ref:fig:camp
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# Bode Plots for different ratio W/w0
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Looking at the transfer function matrix $\bm{G}_d$ eqref:eq:Gd_w0_xi_k, one can see it has two distinct terms that can be studied separately:
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- the direct (diagonal) terms
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- the coupling (off-diagonal) terms
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- the direct (diagonal) terms (Figure ref:fig:plant_compare_rotating_speed_direct)
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- the coupling (off-diagonal) terms (Figure ref:fig:plant_compare_rotating_speed_coupling)
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The bode plot of the direct and coupling terms are shown in Figure ref:fig:plant_compare_rotating_speed for several rotational speed $\Omega$.
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@ -248,11 +244,11 @@ When the
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| file:figs/plant_compare_rotating_speed_direct.pdf | file:figs/plant_compare_rotating_speed_coupling.pdf |
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| <<fig:plant_compare_rotating_speed_direct>> Direct Terms $d_u/F_u$, $d_v/F_v$ | <<fig:plant_compare_rotating_speed_coupling>> Coupling Terms $d_v/F_u$, $d_u/F_v$ |
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In the rest of this study, $\Omega < \omega_0$
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In the rest of this study, rotational speeds smaller than the undamped natural frequency of the system are used ($\Omega < \omega_0$).
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* Decentralized Integral Force Feedback
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** System Schematic and Control Architecture
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Force Sensors are added in series with the actuators as shown in Figure [[fig:system_iff]].
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** Force Sensors and Control Architecture
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In order to apply Decentralized Integral Force Feedback to the system, force sensors are added in series of the two actuators (Figure ref:fig:system_iff).
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# Reference to IFF control
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@ -335,7 +331,7 @@ It increases with the rotational speed $\Omega$.
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\end{equation}
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# Problem of zero with a positive real part
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Also, as one zero has a positive real part, the *IFF control is no more unconditionally stable*.
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Also, as one zero has a positive real part, the IFF control is no more unconditionally stable.
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This is due to the fact that the zeros of the plant are the poles of the closed loop system with an infinite gain.
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Thus, for some finite IFF gain, one pole will have a positive real part and the system will be unstable.
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@ -381,7 +377,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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# Explain that now the low frequency loop gain does not reach a gain more than 1 (if g not so high)
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#+name: fig:loop_gain_modified_iff
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#+caption: Bode Plot of the Loop Gain for IFF with and without the HPF
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#+caption: Bode Plot of the Loop Gain for IFF with and without the HPF, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/loop_gain_modified_iff.pdf]]
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@ -392,7 +388,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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\end{equation}
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#+name: fig:root_locus_modified_iff
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#+caption: Root Locus for IFF with and without the HPF
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#+caption: Root Locus for IFF with and without the HPF, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/root_locus_modified_iff.pdf]]
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@ -408,7 +404,7 @@ This means that at low frequency, the system is decoupled (the force sensor remo
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# Trade off
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#+name: fig:root_locus_wi_modified_iff
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#+caption: Root Locus for several HPF cut-off frequencies $\omega_i$
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#+caption: Root Locus for several HPF cut-off frequencies $\omega_i$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/root_locus_wi_modified_iff.pdf]]
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@ -486,7 +482,7 @@ The overall stiffness $k$ stays constant:
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\end{equation}
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#+name: fig:plant_iff_kp
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#+caption: Bode Plot of $f_u/F_u$ without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$
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#+caption: Bode Plot of $f_u/F_u$ without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/plant_iff_kp.pdf]]
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@ -498,7 +494,7 @@ The overall stiffness $k$ stays constant:
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# Show that it is the case on the root locus
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#+name: fig:root_locus_iff_kp
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#+caption: Root Locus for IFF without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$
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#+caption: Root Locus for IFF without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/root_locus_iff_kp.pdf]]
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@ -515,12 +511,11 @@ The overall stiffness $k$ stays constant:
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# Example with kp = 5 m Omega
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#+name: fig:root_locus_iff_kps_opt
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#+caption: Root Locus for IFF when parallel stiffness is used
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#+caption: Root Locus for IFF when parallel stiffness is used, $\Omega = 0.1 \omega_0$
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#+attr_latex: :environment subfigure :width 0.49\linewidth :align c
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| file:figs/root_locus_iff_kps.pdf | file:figs/root_locus_opt_gain_iff_kp.pdf |
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| <<fig:root_locus_iff_kps>> Three values of $k_p$ | <<fig:root_locus_opt_gain_iff_kp>> $k_p = 5 m \Omega^2$, optimal damping is shown |
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* Direct Velocity Feedback :noexport:
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** System Schematic and Control Architecture
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# Basic Idea of DVF
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@ -593,7 +588,7 @@ With:
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** Attainable Damping
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#+name: fig:comp_root_locus
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#+caption: Root Locus for the three proposed decentralized active damping techniques: IFF with HFP, IFF with parallel springs, and relative DVF
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#+caption: Root Locus for the three proposed decentralized active damping techniques: IFF with HFP, IFF with parallel springs, and relative DVF, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/comp_root_locus.pdf]]
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@ -611,7 +606,7 @@ With:
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# The roll-off is -1 instead of -2
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#+name: fig:comp_active_damping
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#+caption: Comparison of the three proposed Active Damping Techniques
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#+caption: Comparison of the two proposed Active Damping Techniques, $\Omega = 0.1 \omega_0$
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#+attr_latex: :environment subfigure :width 0.45\linewidth :align c
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| file:figs/comp_compliance.pdf | file:figs/comp_transmissibility.pdf |
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| <<fig:comp_compliance>> Transmissibility | <<fig:comp_transmissibility>> Compliance |
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