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@ -16,6 +16,7 @@
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#+LATEX_HEADER: \usepackage{siunitx}
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#+LATEX_HEADER: \usepackage{tikz}
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#+LATEX_HEADER: \usetikzlibrary{shapes.misc,arrows,arrows.meta}
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:END:
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* Build :noexport:
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@ -36,17 +37,12 @@
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* Title Page :ignore:
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#+begin_export latex
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\title{Active Damping of Rotating Platforms using Integral Force Feedback}
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\title{Active damping of rotating platforms using integral force feedback}
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\author{Thomas Dehaeze$^{1,3}$ and Christophe Collette$^{1,2}$}
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\address{$^1$ Precision Mechatronics Laboratory, University of Liege, Belgium}
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\address{$^2$ BEAMS Department, Free University of Brussels, Belgium}
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\address{$^3$ European Synchrotron Radiation Facility, Grenoble, France}
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\ead{dehaeze.thomas@gmail.com}
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\vspace{10pt}
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\begin{indented}
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\item[]November 2020
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\end{indented}
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#+end_export
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#+begin_export latex
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@ -62,7 +58,9 @@
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\vspace{2pc}
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\noindent{\it Keywords}: Active Damping, IFF
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\ioptwocol
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\submitto{\SMS}
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\maketitle
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% \ioptwocol
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#+end_export
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* Introduction
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@ -92,7 +90,7 @@ Figure ref:fig:system represents the model schematically which is the simplest i
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#+name: fig:system
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#+caption: Schematic of the studied System
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.9
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[[file:figs/fig01.pdf]]
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The rotating stage is supposed to be ideal, meaning it induces a perfect rotation $\theta(t) = \Omega t$ where $\Omega$ is the rotational speed in $\si{\radian\per\second}$.
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@ -122,7 +120,7 @@ The equation of motion corresponding to the constant rotation in the $(\vec{i}_x
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\end{aligned}
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\end{equation}
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Substituting equations eqref:eq:energy_functions_lagrange into eqref:eq:lagrangian_equations for both generalized coordinates gives two coupled differential equations
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Substituting eqref:eq:energy_functions_lagrange into eqref:eq:lagrangian_equations for both generalized coordinates gives two coupled differential equations
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#+name: eq:eom_coupled
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\begin{subequations}
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\begin{align}
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@ -147,20 +145,20 @@ One can verify that without rotation ($\Omega = 0$) the system becomes equivalen
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#+latex: \par
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** Transfer Functions in the Laplace domain :ignore:
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To study the dynamics of the system, the differential equations of motions eqref:eq:eom_coupled are transformed in the Laplace domain and the $2 \times 2$ transfer function matrix $\bm{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ is obtained.
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To study the dynamics of the system, the differential equations of motions eqref:eq:eom_coupled are transformed in the Laplace domain and the $2 \times 2$ transfer function matrix $\mathbf{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ is obtained.
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#+name: eq:Gd_mimo_tf
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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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\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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\begin{subequations}
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\begin{align}
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\bm{G}_{d}(1,1) &= {\frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}} \\
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&= \bm{G}_{d}(2,2) \nonumber \\
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\bm{G}_{d}(1,2) &= {\frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}} \\
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&= -\bm{G}_{d}(1,2) \nonumber
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\mathbf{G}_{d}(1,1) &= {\frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}} \\
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&= \mathbf{G}_{d}(2,2) \nonumber \\
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\mathbf{G}_{d}(1,2) &= {\frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}} \\
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&= -\mathbf{G}_{d}(1,2) \nonumber
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\end{align}
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\end{subequations}
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@ -169,12 +167,12 @@ To simplify the analysis, the undamped natural frequency $\omega_0$ and the damp
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\omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}}
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\end{equation}
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The transfer function matrix $\bm{G}_d$ becomes equal to
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The transfer function matrix $\mathbf{G}_d$ becomes equal to
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#+name: eq:Gd_w0_xi_k
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\begin{subequations}
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\begin{align}
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\bm{G}_{d}(1,1) &= {\scriptstyle \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} } \\
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\bm{G}_{d}(1,2) &= {\scriptstyle \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
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\mathbf{G}_{d}(1,1) &= {\scriptstyle \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} } \\
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\mathbf{G}_{d}(1,2) &= {\scriptstyle \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
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\end{align}
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\end{subequations}
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@ -184,7 +182,7 @@ Even though no system with such parameters will be encountered in practice, conc
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#+latex: \par
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** System Dynamics and Campbell Diagram :ignore:
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The poles of $\bm{G}_d$ are the complex solutions $p$ of eqref:eq:poles.
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The poles of $\mathbf{G}_d$ are the complex solutions $p$ of eqref:eq:poles.
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#+name: eq:poles
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\begin{equation}
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@ -214,13 +212,13 @@ In the rest of this study, rotational speeds smaller than the undamped natural f
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| file:figs/fig02a.pdf | file:figs/fig02b.pdf |
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| <<fig:campbell_diagram_real>> Real Part | <<fig:campbell_diagram_imag>> Imaginary Part |
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Looking at the transfer function matrix $\bm{G}_d$ in Eq. eqref:eq:Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and the two off-diagonal (coupling) terms are opposite.
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Looking at the transfer function matrix $\mathbf{G}_d$ in eqref:eq:Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and the two off-diagonal (coupling) terms are opposite.
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The bode plot of these two terms are shown in Figure ref:fig:plant_compare_rotating_speed for several rotational speeds $\Omega$.
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These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as $\Omega$ increases.
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For $\Omega > \omega_0$, the low frequency pair of complex conjugate poles $p_{-}$ becomes unstable.
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#+name: fig:plant_compare_rotating_speed
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#+caption: Bode Plots for $\bm{G}_d$ for several rotational speed $\Omega$
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#+caption: Bode Plots for $\mathbf{G}_d$ for several rotational speed $\Omega$
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#+attr_latex: :environment subfigure :width 0.48\linewidth :align c
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| file:figs/fig03a.pdf | file:figs/fig03b.pdf |
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| <<fig:plant_compare_rotating_speed_direct>> Direct Terms | <<fig:plant_compare_rotating_speed_coupling>> Coupling Terms |
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@ -234,7 +232,7 @@ The control diagram is schematically shown in Figure ref:fig:control_diagram_iff
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#+name: fig:system_iff
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#+caption: System with added Force Sensor in series with the actuators
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.9
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[[file:figs/fig04.pdf]]
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#+name: fig:control_diagram_iff
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@ -256,18 +254,18 @@ The forces $\begin{bmatrix}f_u & f_v\end{bmatrix}$ measured by the two force sen
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Inserting eqref:eq:Gd_w0_xi_k into eqref:eq:measured_force yields
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#+name: eq:Gf_mimo_tf
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\begin{equation}
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\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \bm{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \mathbf{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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\begin{subequations}
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\label{eq:Gf}
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\begin{align}
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\bm{G}_{f}(1,1) &= {\scriptstyle \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} } \\
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\bm{G}_{f}(1,2) &= {\scriptstyle \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
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\mathbf{G}_{f}(1,1) &= {\scriptstyle \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} } \\
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\mathbf{G}_{f}(1,2) &= {\scriptstyle \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
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\end{align}
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\end{subequations}
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The zeros of the diagonal terms of $\bm{G}_f$ are equal to (neglecting the damping for simplicity)
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The zeros of the diagonal terms of $\mathbf{G}_f$ are equal to (neglecting the damping for simplicity)
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\begin{subequations}
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\begin{align}
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z_c &= \pm j \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} + \frac{\Omega^2}{{\omega_0}^2} + \frac{1}{2} } \label{eq:iff_zero_cc} \\
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@ -280,10 +278,10 @@ The frequency of the pair of complex conjugate zeros $z_c$ eqref:eq:iff_zero_cc
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For non-null rotational speeds, two real zeros $z_r$ eqref:eq:iff_zero_real appear in the diagonal terms inducing a non-minimum phase behavior.
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This can be seen in the Bode plot of the diagonal terms (Figure ref:fig:plant_iff_compare_rotating_speed) where the low frequency gain is no longer zero while the phase stays at $\SI{180}{\degree}$.
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The low frequency gain of $\bm{G}_f$ increases with the rotational speed $\Omega$
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The low frequency gain of $\mathbf{G}_f$ increases with the rotational speed $\Omega$
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#+name: eq:low_freq_gain_iff_plan
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\begin{equation}
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\lim_{\omega \to 0} \left| \bm{G}_f (j\omega) \right| = \begin{bmatrix}
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\lim_{\omega \to 0} \left| \mathbf{G}_f (j\omega) \right| = \begin{bmatrix}
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\frac{\Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\
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0 & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2}
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\end{bmatrix}
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@ -293,7 +291,7 @@ This can be explained as follows: a constant force $F_u$ induces a small displac
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#+name: fig:plant_iff_compare_rotating_speed
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#+caption: Bode plot of the dynamics from a force actuator to its collocated force sensor ($f_u/F_u$, $f_v/F_v$) for several rotational speeds $\Omega$
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.95
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[[file:figs/fig06.pdf]]
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#+latex: \par
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@ -304,7 +302,7 @@ The two IFF controllers $K_F$ consist of a pure integrator
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#+name: eq:Kf_pure_int
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\begin{equation}
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\begin{aligned}
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\bm{K}_F(s) &= \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix} \\
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\mathbf{K}_F(s) &= \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix} \\
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K_F(s) &= g \cdot \frac{1}{s}
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\end{aligned}
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\end{equation}
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@ -360,7 +358,7 @@ It is interesting to note that $g_{\text{max}}$ also corresponds to the gain whe
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#+name: fig:loop_gain_modified_iff
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#+caption: Modification of the loop gain with the added HFP, $g = 2$, $\omega_i = 0.1 \omega_0$ and $\Omega = 0.1 \omega_0$
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.95
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[[file:figs/fig08.pdf]]
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#+name: fig:root_locus_modified_iff
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@ -379,7 +377,7 @@ It is shown that even though small $\omega_i$ seem to allow more damping to be a
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#+name: fig:root_locus_wi_modified_iff
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#+caption: Root Locus for several HPF cut-off frequencies $\omega_i$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.95
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[[file:figs/fig10.pdf]]
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In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$.
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@ -387,7 +385,7 @@ The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also disp
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#+name: fig:mod_iff_damping_wi
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#+caption: Attainable damping ratio $\xi_\text{cl}$ as a function of $\omega_i/\omega_0$. Corresponding control gain $g_\text{opt}$ and $g_\text{max}$ are also shown
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.95
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[[file:figs/fig11.pdf]]
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Three regions can be observed:
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@ -407,13 +405,13 @@ An example of such system is shown in Figure ref:fig:cedrat_xy25xs.
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#+name: fig:system_parallel_springs
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#+caption: Studied system with additional springs in parallel with the actuators and force sensors
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.9
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[[file:figs/fig12.pdf]]
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#+name: fig:cedrat_xy25xs
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#+caption: XY Piezoelectric Stage (XY25XS from Cedrat Technology)
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#+attr_latex: :width 0.8\linewidth
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#+attr_latex: :scale 0.17
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[[file:figs/fig13.pdf]]
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#+latex: \par
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@ -436,20 +434,20 @@ The equations of motion are derived and transformed in the Laplace domain
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#+name: eq:Gk_mimo_tf
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\begin{equation}
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\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
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\bm{G}_k
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\mathbf{G}_k
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\begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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#+name: eq:Gk
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\begin{subequations}
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\begin{align}
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& \bm{G}_{k}(1,1) = \dots \nonumber \\
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& \mathbf{G}_{k}(1,1) = \dots \nonumber \\
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& {\scriptstyle \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2} } \\
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& \bm{G}_{k}(1,2) = {\scriptscriptstyle \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
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& \mathbf{G}_{k}(1,2) = {\scriptscriptstyle \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
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\end{align}
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\end{subequations}
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Comparing $\bm{G}_k$ eqref:eq:Gk with $\bm{G}_f$ eqref:eq:Gf shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed.
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Comparing $\mathbf{G}_k$ eqref:eq:Gk with $\mathbf{G}_f$ eqref:eq:Gf shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed.
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The two real zeros $z_r$ eqref:eq:iff_zero_real that were inducing non-minimum phase behavior are transformed into complex conjugate zeros if the following condition hold
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#+name: eq:kp_cond_cc_zeros
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\begin{equation}
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@ -464,12 +462,12 @@ It is shown that if the added stiffness is higher than the maximum negative stif
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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 $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.95
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[[file:figs/fig14.pdf]]
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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 $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :width \linewidth
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#+attr_latex: :scale 0.95
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[[file:figs/fig15.pdf]]
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#+latex: \par
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@ -492,7 +490,7 @@ This is confirmed in Figure ref:fig:opt_damp_alpha where the attainable closed-l
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|
||||
#+name: fig:opt_damp_alpha
|
||||
#+caption: Optimal Damping Ratio $\xi_\text{opt}$ and the corresponding optimal gain $g_\text{opt}$ as a function of $\alpha$
|
||||
#+attr_latex: :width \linewidth
|
||||
#+attr_latex: :scale 0.95
|
||||
[[file:figs/fig17.pdf]]
|
||||
|
||||
* Comparison and Discussion
|
||||
@ -579,7 +577,7 @@ The Matlab code that was used for this study is available under a MIT License an
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
|
||||
This research benefited from a FRIA grant from the French Community of Belgium.
|
||||
This research benefited from a FRIA grant (grant number: FC 31597) from the French Community of Belgium.
|
||||
|
||||
* References
|
||||
:PROPERTIES:
|
||||
|
||||
Binary file not shown.
@ -1,4 +1,4 @@
|
||||
% Created 2020-11-02 lun. 14:46
|
||||
% Created 2020-11-02 lun. 15:33
|
||||
% Intended LaTeX compiler: pdflatex
|
||||
\documentclass[10pt]{iopart}
|
||||
|
||||
@ -13,7 +13,6 @@
|
||||
\usepackage{siunitx}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{shapes.misc,arrows,arrows.meta}
|
||||
\date{}
|
||||
\hypersetup{
|
||||
pdfauthor={Thomas Dehaeze},
|
||||
pdftitle={},
|
||||
@ -23,18 +22,13 @@
|
||||
pdflang={English}}
|
||||
\begin{document}
|
||||
|
||||
\title{Active Damping of Rotating Platforms using Integral Force Feedback}
|
||||
\title{Active damping of rotating platforms using integral force feedback}
|
||||
\author{Thomas Dehaeze$^{1,3}$ and Christophe Collette$^{1,2}$}
|
||||
\address{$^1$ Precision Mechatronics Laboratory, University of Liege, Belgium}
|
||||
\address{$^2$ BEAMS Department, Free University of Brussels, Belgium}
|
||||
\address{$^3$ European Synchrotron Radiation Facility, Grenoble, France}
|
||||
\ead{dehaeze.thomas@gmail.com}
|
||||
|
||||
\vspace{10pt}
|
||||
\begin{indented}
|
||||
\item[]November 2020
|
||||
\end{indented}
|
||||
|
||||
\begin{abstract}
|
||||
This paper investigates the use of Integral Force Feedback (IFF) for the active damping of rotating mechanical systems.
|
||||
Guaranteed stability, typical benefit of IFF, is lost as soon as the system is rotating due to gyroscopic effects.
|
||||
@ -47,10 +41,12 @@ The results reveal that, despite their different implementations, both modified
|
||||
\vspace{2pc}
|
||||
\noindent{\it Keywords}: Active Damping, IFF
|
||||
|
||||
\ioptwocol
|
||||
\submitto{\SMS}
|
||||
\maketitle
|
||||
% \ioptwocol
|
||||
|
||||
\section{Introduction}
|
||||
\label{sec:org45302f3}
|
||||
\label{sec:orgdf2365d}
|
||||
\label{sec:introduction}
|
||||
There is an increasing need to reduce the undesirable vibration of many sensitive equipment.
|
||||
A common method to reduce vibration is to mount the sensitive equipment on a suspended platform which attenuates the vibrations above the frequency of the suspension modes.
|
||||
@ -70,14 +66,14 @@ Section \ref{sec:iff_kp} proposes to add springs in parallel with the force sens
|
||||
Section \ref{sec:comparison} compares both proposed modifications to the classical IFF in terms of damping authority and closed-loop system behavior.
|
||||
|
||||
\section{Dynamics of Rotating Platforms}
|
||||
\label{sec:org2c62ca5}
|
||||
\label{sec:orgb49fb34}
|
||||
\label{sec:dynamics}
|
||||
In order to study how the rotation does affect the use of IFF, a model of a suspended platform on top of a rotating stage is used.
|
||||
Figure \ref{fig:system} represents the model schematically which is the simplest in which gyroscopic forces can be studied.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig01.pdf}
|
||||
\includegraphics[scale=0.9]{figs/fig01.pdf}
|
||||
\caption{\label{fig:system}Schematic of the studied System}
|
||||
\end{figure}
|
||||
|
||||
@ -106,7 +102,7 @@ The equation of motion corresponding to the constant rotation in the \((\vec{i}_
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
Substituting equations \eqref{eq:energy_functions_lagrange} into \eqref{eq:lagrangian_equations} for both generalized coordinates gives two coupled differential equations
|
||||
Substituting \eqref{eq:energy_functions_lagrange} into \eqref{eq:lagrangian_equations} for both generalized coordinates gives two coupled differential equations
|
||||
\begin{subequations}
|
||||
\label{eq:eom_coupled}
|
||||
\begin{align}
|
||||
@ -131,20 +127,20 @@ One can verify that without rotation (\(\Omega = 0\)) the system becomes equival
|
||||
\end{subequations}
|
||||
|
||||
\par
|
||||
To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are transformed in the Laplace domain and the \(2 \times 2\) transfer function matrix \(\bm{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) is obtained.
|
||||
To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are transformed in the Laplace domain and the \(2 \times 2\) transfer function matrix \(\mathbf{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) is obtained.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:Gd_mimo_tf}
|
||||
\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\bm{G}_{d}(1,1) &= {\frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}} \\
|
||||
&= \bm{G}_{d}(2,2) \nonumber \\
|
||||
\bm{G}_{d}(1,2) &= {\frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}} \\
|
||||
&= -\bm{G}_{d}(1,2) \nonumber
|
||||
\mathbf{G}_{d}(1,1) &= {\frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}} \\
|
||||
&= \mathbf{G}_{d}(2,2) \nonumber \\
|
||||
\mathbf{G}_{d}(1,2) &= {\frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}} \\
|
||||
&= -\mathbf{G}_{d}(1,2) \nonumber
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
@ -153,12 +149,12 @@ To simplify the analysis, the undamped natural frequency \(\omega_0\) and the da
|
||||
\omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}}
|
||||
\end{equation}
|
||||
|
||||
The transfer function matrix \(\bm{G}_d\) becomes equal to
|
||||
The transfer function matrix \(\mathbf{G}_d\) becomes equal to
|
||||
\begin{subequations}
|
||||
\label{eq:Gd_w0_xi_k}
|
||||
\begin{align}
|
||||
\bm{G}_{d}(1,1) &= {\scriptstyle \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} } \\
|
||||
\bm{G}_{d}(1,2) &= {\scriptstyle \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
|
||||
\mathbf{G}_{d}(1,1) &= {\scriptstyle \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} } \\
|
||||
\mathbf{G}_{d}(1,2) &= {\scriptstyle \frac{\frac{1}{k} \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
@ -166,7 +162,7 @@ For all further numerical analysis in this study, we consider \(\omega_0 = \SI{1
|
||||
Even though no system with such parameters will be encountered in practice, conclusions can be drawn relative to these parameters such that they can be generalized to any other set of parameters.
|
||||
|
||||
\par
|
||||
The poles of \(\bm{G}_d\) are the complex solutions \(p\) of \eqref{eq:poles}.
|
||||
The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of \eqref{eq:poles}.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:poles}
|
||||
@ -205,7 +201,7 @@ In the rest of this study, rotational speeds smaller than the undamped natural f
|
||||
\centering
|
||||
\end{figure}
|
||||
|
||||
Looking at the transfer function matrix \(\bm{G}_d\) in Eq. \eqref{eq:Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and the two off-diagonal (coupling) terms are opposite.
|
||||
Looking at the transfer function matrix \(\mathbf{G}_d\) in \eqref{eq:Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and the two off-diagonal (coupling) terms are opposite.
|
||||
The bode plot of these two terms are shown in Figure \ref{fig:plant_compare_rotating_speed} for several rotational speeds \(\Omega\).
|
||||
These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as \(\Omega\) increases.
|
||||
For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p_{-}\) becomes unstable.
|
||||
@ -221,12 +217,12 @@ For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p
|
||||
\caption{\label{fig:plant_compare_rotating_speed_coupling} Coupling Terms}
|
||||
\end{subfigure}
|
||||
\hfill
|
||||
\caption{\label{fig:plant_compare_rotating_speed}Bode Plots for \(\bm{G}_d\) for several rotational speed \(\Omega\)}
|
||||
\caption{\label{fig:plant_compare_rotating_speed}Bode Plots for \(\mathbf{G}_d\) for several rotational speed \(\Omega\)}
|
||||
\centering
|
||||
\end{figure}
|
||||
|
||||
\section{Decentralized Integral Force Feedback}
|
||||
\label{sec:orgaa40c6a}
|
||||
\label{sec:org7b1adf9}
|
||||
\label{sec:iff}
|
||||
In order to apply IFF to the system, force sensors are added in series with the two actuators (Figure \ref{fig:system_iff}).
|
||||
As this study focuses on decentralized control, two identical controllers \(K_F\) are used to feedback each of the sensed force to its associated actuator and no attempt is made to counteract the interactions in the system.
|
||||
@ -234,7 +230,7 @@ The control diagram is schematically shown in Figure \ref{fig:control_diagram_if
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig04.pdf}
|
||||
\includegraphics[scale=0.9]{figs/fig04.pdf}
|
||||
\caption{\label{fig:system_iff}System with added Force Sensor in series with the actuators}
|
||||
\end{figure}
|
||||
|
||||
@ -256,18 +252,18 @@ The forces \(\begin{bmatrix}f_u & f_v\end{bmatrix}\) measured by the two force s
|
||||
Inserting \eqref{eq:Gd_w0_xi_k} into \eqref{eq:measured_force} yields
|
||||
\begin{equation}
|
||||
\label{eq:Gf_mimo_tf}
|
||||
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \bm{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \mathbf{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{subequations}
|
||||
\label{eq:Gf}
|
||||
\begin{align}
|
||||
\bm{G}_{f}(1,1) &= {\scriptstyle \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} } \\
|
||||
\bm{G}_{f}(1,2) &= {\scriptstyle \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
|
||||
\mathbf{G}_{f}(1,1) &= {\scriptstyle \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} } \\
|
||||
\mathbf{G}_{f}(1,2) &= {\scriptstyle \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
The zeros of the diagonal terms of \(\bm{G}_f\) are equal to (neglecting the damping for simplicity)
|
||||
The zeros of the diagonal terms of \(\mathbf{G}_f\) are equal to (neglecting the damping for simplicity)
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
z_c &= \pm j \omega_0 \sqrt{\frac{1}{2} \sqrt{8 \frac{\Omega^2}{{\omega_0}^2} + 1} + \frac{\Omega^2}{{\omega_0}^2} + \frac{1}{2} } \label{eq:iff_zero_cc} \\
|
||||
@ -280,10 +276,10 @@ The frequency of the pair of complex conjugate zeros \(z_c\) \eqref{eq:iff_zero_
|
||||
For non-null rotational speeds, two real zeros \(z_r\) \eqref{eq:iff_zero_real} appear in the diagonal terms inducing a non-minimum phase behavior.
|
||||
This can be seen in the Bode plot of the diagonal terms (Figure \ref{fig:plant_iff_compare_rotating_speed}) where the low frequency gain is no longer zero while the phase stays at \(\SI{180}{\degree}\).
|
||||
|
||||
The low frequency gain of \(\bm{G}_f\) increases with the rotational speed \(\Omega\)
|
||||
The low frequency gain of \(\mathbf{G}_f\) increases with the rotational speed \(\Omega\)
|
||||
\begin{equation}
|
||||
\label{eq:low_freq_gain_iff_plan}
|
||||
\lim_{\omega \to 0} \left| \bm{G}_f (j\omega) \right| = \begin{bmatrix}
|
||||
\lim_{\omega \to 0} \left| \mathbf{G}_f (j\omega) \right| = \begin{bmatrix}
|
||||
\frac{\Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\
|
||||
0 & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2}
|
||||
\end{bmatrix}
|
||||
@ -293,7 +289,7 @@ This can be explained as follows: a constant force \(F_u\) induces a small displ
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig06.pdf}
|
||||
\includegraphics[scale=0.95]{figs/fig06.pdf}
|
||||
\caption{\label{fig:plant_iff_compare_rotating_speed}Bode plot of the dynamics from a force actuator to its collocated force sensor (\(f_u/F_u\), \(f_v/F_v\)) for several rotational speeds \(\Omega\)}
|
||||
\end{figure}
|
||||
|
||||
@ -303,7 +299,7 @@ The two IFF controllers \(K_F\) consist of a pure integrator
|
||||
\begin{equation}
|
||||
\label{eq:Kf_pure_int}
|
||||
\begin{aligned}
|
||||
\bm{K}_F(s) &= \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix} \\
|
||||
\mathbf{K}_F(s) &= \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix} \\
|
||||
K_F(s) &= g \cdot \frac{1}{s}
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
@ -329,7 +325,7 @@ In order to apply decentralized IFF on rotating platforms, two solutions are pro
|
||||
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}).
|
||||
|
||||
\section{Integral Force Feedback with High Pass Filter}
|
||||
\label{sec:orgcc80b31}
|
||||
\label{sec:orgb29360e}
|
||||
\label{sec:iff_hpf}
|
||||
As was explained in the previous section, the instability comes in part from the high gain at low frequency caused by the pure integrators.
|
||||
|
||||
@ -358,7 +354,7 @@ It is interesting to note that \(g_{\text{max}}\) also corresponds to the gain w
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig08.pdf}
|
||||
\includegraphics[scale=0.95]{figs/fig08.pdf}
|
||||
\caption{\label{fig:loop_gain_modified_iff}Modification of the loop gain with the added HFP, \(g = 2\), \(\omega_i = 0.1 \omega_0\) and \(\Omega = 0.1 \omega_0\)}
|
||||
\end{figure}
|
||||
|
||||
@ -377,7 +373,7 @@ It is shown that even though small \(\omega_i\) seem to allow more damping to be
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig10.pdf}
|
||||
\includegraphics[scale=0.95]{figs/fig10.pdf}
|
||||
\caption{\label{fig:root_locus_wi_modified_iff}Root Locus for several HPF cut-off frequencies \(\omega_i\), \(\Omega = 0.1 \omega_0\)}
|
||||
\end{figure}
|
||||
|
||||
@ -386,7 +382,7 @@ The gain \(g_{\text{opt}}\) at which this maximum damping is obtained is also di
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig11.pdf}
|
||||
\includegraphics[scale=0.95]{figs/fig11.pdf}
|
||||
\caption{\label{fig:mod_iff_damping_wi}Attainable damping ratio \(\xi_\text{cl}\) as a function of \(\omega_i/\omega_0\). Corresponding control gain \(g_\text{opt}\) and \(g_\text{max}\) are also shown}
|
||||
\end{figure}
|
||||
|
||||
@ -398,7 +394,7 @@ Three regions can be observed:
|
||||
\end{itemize}
|
||||
|
||||
\section{Integral Force Feedback with Parallel Springs}
|
||||
\label{sec:org7af993d}
|
||||
\label{sec:orgaa66b9e}
|
||||
\label{sec:iff_kp}
|
||||
In this section additional springs in parallel with the force sensors are added to counteract the negative stiffness induced by the rotation.
|
||||
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.
|
||||
@ -409,14 +405,14 @@ An example of such system is shown in Figure \ref{fig:cedrat_xy25xs}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig12.pdf}
|
||||
\includegraphics[scale=0.9]{figs/fig12.pdf}
|
||||
\caption{\label{fig:system_parallel_springs}Studied system with additional springs in parallel with the actuators and force sensors}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.8\linewidth]{figs/fig13.pdf}
|
||||
\includegraphics[scale=0.17]{figs/fig13.pdf}
|
||||
\caption{\label{fig:cedrat_xy25xs}XY Piezoelectric Stage (XY25XS from Cedrat Technology)}
|
||||
\end{figure}
|
||||
|
||||
@ -438,20 +434,20 @@ The equations of motion are derived and transformed in the Laplace domain
|
||||
\begin{equation}
|
||||
\label{eq:Gk_mimo_tf}
|
||||
\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
|
||||
\bm{G}_k
|
||||
\mathbf{G}_k
|
||||
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{subequations}
|
||||
\label{eq:Gk}
|
||||
\begin{align}
|
||||
& \bm{G}_{k}(1,1) = \dots \nonumber \\
|
||||
& \mathbf{G}_{k}(1,1) = \dots \nonumber \\
|
||||
& {\scriptstyle \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2} } \\
|
||||
& \bm{G}_{k}(1,2) = {\scriptscriptstyle \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
|
||||
& \mathbf{G}_{k}(1,2) = {\scriptscriptstyle \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} }
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
Comparing \(\bm{G}_k\) \eqref{eq:Gk} with \(\bm{G}_f\) \eqref{eq:Gf} shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed.
|
||||
Comparing \(\mathbf{G}_k\) \eqref{eq:Gk} with \(\mathbf{G}_f\) \eqref{eq:Gf} shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed.
|
||||
The two real zeros \(z_r\) \eqref{eq:iff_zero_real} that were inducing non-minimum phase behavior are transformed into complex conjugate zeros if the following condition hold
|
||||
\begin{equation}
|
||||
\label{eq:kp_cond_cc_zeros}
|
||||
@ -466,13 +462,13 @@ It is shown that if the added stiffness is higher than the maximum negative stif
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig14.pdf}
|
||||
\includegraphics[scale=0.95]{figs/fig14.pdf}
|
||||
\caption{\label{fig:plant_iff_kp}Bode plot of \(f_u/F_u\) without parallel spring, with parallel springs with stiffness \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\), \(\Omega = 0.1 \omega_0\)}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig15.pdf}
|
||||
\includegraphics[scale=0.95]{figs/fig15.pdf}
|
||||
\caption{\label{fig:root_locus_iff_kp}Root Locus for IFF without parallel spring, with parallel springs with stiffness \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\), \(\Omega = 0.1 \omega_0\)}
|
||||
\end{figure}
|
||||
|
||||
@ -495,12 +491,12 @@ This is confirmed in Figure \ref{fig:opt_damp_alpha} where the attainable closed
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{figs/fig17.pdf}
|
||||
\includegraphics[scale=0.95]{figs/fig17.pdf}
|
||||
\caption{\label{fig:opt_damp_alpha}Optimal Damping Ratio \(\xi_\text{opt}\) and the corresponding optimal gain \(g_\text{opt}\) as a function of \(\alpha\)}
|
||||
\end{figure}
|
||||
|
||||
\section{Comparison and Discussion}
|
||||
\label{sec:orgc3da302}
|
||||
\label{sec:orgd845e3a}
|
||||
\label{sec:comparison}
|
||||
Two modifications to adapt the IFF control strategy to rotating platforms have been proposed in Sections \ref{sec:iff_hpf} and \ref{sec:iff_kp}.
|
||||
These two methods are now compared in terms of added damping, closed-loop compliance and transmissibility.
|
||||
@ -558,7 +554,7 @@ On can see in Figure \ref{fig:comp_transmissibility} that the problem of the deg
|
||||
The addition of the HPF or the use of the parallel stiffness permit to limit the degradation of the compliance as compared with classical IFF (Figure \ref{fig:comp_compliance}).
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:org3e5d606}
|
||||
\label{sec:org68f7028}
|
||||
\label{sec:conclusion}
|
||||
|
||||
Due to gyroscopic effects, decentralized IFF with pure integrators was shown to be unstable when applied to rotating platforms.
|
||||
@ -579,11 +575,11 @@ Future work will focus on the experimental validation of the proposed active dam
|
||||
The Matlab code that was used for this study is available under a MIT License and archived in Zenodo \cite{dehaeze20_activ_dampin_rotat_posit_platf}.
|
||||
|
||||
\section*{Acknowledgments}
|
||||
\label{sec:org2fdf2e5}
|
||||
This research benefited from a FRIA grant from the French Community of Belgium.
|
||||
\label{sec:org49166ff}
|
||||
This research benefited from a FRIA grant (grant number: FC 31597) from the French Community of Belgium.
|
||||
|
||||
\section*{References}
|
||||
\label{sec:org684d4d0}
|
||||
\label{sec:org8c97a5c}
|
||||
\bibliographystyle{iopart-num}
|
||||
\bibliography{ref.bib}
|
||||
\end{document}
|
||||
|
||||
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Reference in New Issue
Block a user