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b/paper/paper.org @@ -494,19 +494,28 @@ Thus, if the added parallel stiffness $k_p$ is higher than the negative stiffnes This is confirmed by the Bode plot in Figure ref:fig:plant_iff_kp. # while recovering the alternating poles and zeros near the imaginary axis. -#+name: fig:plant_iff_kp -#+caption: Bode Plot of $f_u/F_u$ without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$ -#+attr_latex: :scale 1 -[[file:figs/plant_iff_kp.pdf]] # Root Locus plot Figure ref:fig:root_locus_iff_kp shows Root Loci plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator eqref:eq:Kf_pure_int. It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system stay in the (stable) right half-plane, and hence the unconditional stability of IFF is recovered. +#+attr_latex: :options [b]{0.42\linewidth} +#+begin_minipage +#+name: fig:plant_iff_kp +#+caption: Bode Plot of $f_u/F_u$ without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$ +#+attr_latex: :scale 1 :float nil +[[file:figs/plant_iff_kp.pdf]] +#+end_minipage +\hfill +#+attr_latex: :options [b]{0.52\linewidth} +#+begin_minipage #+name: fig:root_locus_iff_kp #+caption: Root Locus for IFF without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$ -#+attr_latex: :scale 1 +#+attr_latex: :scale 1 :float nil [[file:figs/root_locus_iff_kp.pdf]] +#+end_minipage + + ** Optimal Parallel Stiffness # The zero is the poles of the system without the force sensors => w0 = sqrt(kp/m) +/- Omega ?? => seems not true