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:tangle matlab/detail_control_2_decoupling.m :END: @@ -1814,7 +1817,18 @@ Future work will aim at developing a complementary filter synthesis method that ** Introduction :ignore: -- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Decoupling Strategies][Decoupling Strategies]] +When dealing with MIMO systems, a typical strategy is to: +- First decouple the plant dynamics (discussed in this section) +- Apply SISO control for the decoupled plant (discussed in section ref:sec:detail_control_optimization) + +Another strategy would be to apply a multivariable control synthesis to the coupled system. +Strangely, while H-infinity synthesis is a mature technology, it use for the control of Stewart platform is not yet demonstrated. +From [[cite:&thayer02_six_axis_vibrat_isolat_system]]: +#+begin_quote +Experimental closed-loop control results using the hexapod have shown that controllers designed using a decentralized single-strut design work well when compared to full multivariable methodologies. +#+end_quote + +- [ ] Review of [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Decoupling Strategies][Decoupling Strategies]] for stewart platforms - [ ] Add some citations about different methods #+name: tab:detail_control_decoupling_review @@ -1836,25 +1850,12 @@ Future work will aim at developing a complementary filter synthesis method that | Piezoelectric | Force, Position | Vibration isolation, Model-Based, *Modal control*: 6x PI controllers | [[cite:&yang19_dynam_model_decoup_contr_flexib]] | |---------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------| | PZT | Geophone (struts) | *H-Infinity* and mu-synthesis | [[cite:&lei08_multi_objec_robus_activ_vibrat]] | -| Voice Coil | Force sensors (strus) + accelerometer (cartesian) | Decentralized Force Feedback + Centralized H2 control based on accelerometers | [[cite:&xie17_model_contr_hybrid_passiv_activ]] | +| Voice Coil | Force sensors (struts) + accelerometer (cartesian) | Decentralized Force Feedback + Centralized H2 control based on accelerometers | [[cite:&xie17_model_contr_hybrid_passiv_activ]] | | Voice Coil | Accelerometers | MIMO H-Infinity, active damping | [[cite:&jiao18_dynam_model_exper_analy_stewar]] | -When dealing with MIMO systems, a typical strategy is to: -- first decouple the plant dynamics -- apply SISO control for the decoupled plant - -Another strategy would be to apply a multivariable control synthesis to the coupled system. -From [[cite:&thayer02_six_axis_vibrat_isolat_system]]: -#+begin_quote -Experimental closed-loopcontrol results using the hexapod have shown that controllers designed using a decentralized single-strut design work well when compared to full multivariable methodologies. -#+end_quote - Assumptions: - parallel manipulators -Review of decoupling strategies for Stewart platforms: -- [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Decoupling Strategies][Decoupling Strategies]] - The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators. It is structured as follow: @@ -1884,81 +1885,335 @@ It is structured as follow: #+begin_src matlab :noweb yes <> + +%% Initialize Frequency Vector +freqs = logspace(0, 3, 1000); #+end_src ** Test Model <> -Let's consider a parallel manipulator with several collocated actuator/sensors pairs. +- a test model will be used to compare all the decoupling strategies +- to render the analysis simpler, the system of Figure ref:fig:detail_control_model_test_decoupling_detail is used +- It has 3DoF, and has 3 parallels struts whose model is shown in Figure ref:fig:detail_control_strut_model +- It is quite similar to the Stewart platform (parallel architecture, as many struts as DoF) -System in Figure ref:fig:detail_control_model_test_decoupling will serve as an example. +- [ ] Write the equation of motion + - Write the equation of motion at the center of mass + - mass + Inertia, stiffness, damping, actuator forces (mapped using Jacobian) + - Write it in matrix form +- [X] Show the used parameters (table ?) -We will note: -- $b_i$: location of the joints on the top platform -- $\hat{s}_i$: unit vector corresponding to the struts direction -- $k_i$: stiffness of the struts -- $\tau_i$: actuator forces -- $O_M$: center of mass of the solid body -- $\mathcal{L}_i$: relative displacement of the struts +\begin{equation} + \bm{M}_{\{M\}} = \begin{bmatrix} + m & 0 & 0 \\ + 0 & m & 0 \\ + 0 & 0 & I + \end{bmatrix} +\end{equation} -#+name: fig:detail_control_model_test_decoupling -#+caption: Model use to compare decoupling techniques +\begin{equation} + \bm{\mathcal{K}} = \begin{bmatrix} + k & 0 & 0 \\ + 0 & k & 0 \\ + 0 & 0 & k + \end{bmatrix}, \quad \bm{\mathcal{C}} = \begin{bmatrix} + c & 0 & 0 \\ + 0 & c & 0 \\ + 0 & 0 & c + \end{bmatrix} +\end{equation} + +\begin{equation} + \bm{J}_{\{M\}} = \begin{bmatrix} + 1 & 0 & h_a \\ + 0 & 1 & -l_a \\ + 0 & 1 & l_a \\ + \end{bmatrix} +\end{equation} + +Recall the Jacobian relationships: +Let's link forces and torques applied at the CoM + +\begin{equation} +M_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}} + \bm{J}_{\{M\}}^t \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}} + \bm{J}_{\{M\}}^t \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}} = \bm{\mathcal{F}}_{\{M\}} +\end{equation} + +with $\bm{\mathcal{X}}_{\{M\}}$ the two translation and one rotation expressed with respect to the center of mass. +$\bm{\mathcal{F}}_{\{M\}}$ forces and torque applied at the center of mass. + +\begin{equation} + \bm{\mathcal{X}}_{\{M\}} = \begin{bmatrix} + x \\ + y \\ + R_z + \end{bmatrix}, \quad \bm{\mathcal{F}}_{\{M\}} = \begin{bmatrix} + F_x \\ + F_y \\ + M_z + \end{bmatrix} +\end{equation} + + +#+name: fig:detail_control_model_test_decoupling_detail +#+caption: 3DoF model used to study decoupling strategies +#+attr_latex: :options [htbp] +#+begin_figure +#+attr_latex: :caption \subcaption{\label{fig:detail_control_model_test_decoupling}Geometrical parameters} +#+attr_latex: :options {0.58\textwidth} +#+begin_subfigure +#+attr_latex: :scale 1 [[file:figs/detail_control_model_test_decoupling.png]] +#+end_subfigure +#+attr_latex: :caption \subcaption{\label{fig:detail_control_strut_model}Strut model} +#+attr_latex: :options {0.38\textwidth} +#+begin_subfigure +#+attr_latex: :scale 1 +[[file:figs/detail_control_strut_model.png]] +#+end_subfigure +#+end_figure + +$\tau_i$ are the actuator forces +$\mathcal{L}_i$ are the relative displacement of the struts + +Two frames are defined: +- $\{M\}$ with origin $O_M$ at the Center of mass of the solid body +- $\{K\}$ with origin $O_K$ at the Center of mass of the parallel manipulator + +#+name: tab:detail_control_decoupling_test_model_params +#+caption: Model parameters +#+attr_latex: :environment tabularx :width 0.9\linewidth :align cXc +#+attr_latex: :center t :booktabs t :font \scriptsize +| *Parameter* | *Description* | *Value* | +|-------------+---------------------------------------------------+-------------------| +| $b_i$ | Location of the joints on the top platform | $b_1 = $ | +| $\hat{s}_i$ | Unit vector corresponding to the struts direction | | +| $l_a$ | | $0.5\,m$ | +| $h_a$ | | $0.2\,m$ | +| $k$ | Actuator stiffness | $10\,N/\mu m$ | +| $c$ | Actuator damping | $200\,Ns/m$ | +| $m$ | Payload mass | $40\,\text{kg}$ | +| $I$ | Payload rotational inertia | $5\,\text{kg}m^2$ | #+begin_src matlab -%% System parameters -l = 1.0; % Length of the mass [m] -h = 2*1.7; % Height of the mass [m] +%% Analytical Formula for the Jacobian Matrix +% Create symbolic variables for all parameters +syms l h la ha m I real +syms c1 c2 c3 real +syms k1 k2 k3 real -la = l/2; % Position of Act. [m] -ha = h/2; % Position of Act. [m] +% Unit vectors of the actuators (symbolic) +s1 = [1; 0]; % Actuator 1 direction (horizontal) +s2 = [0; 1]; % Actuator 2 direction (vertical) +s3 = [0; 1]; % Actuator 3 direction (vertical) -m = 400; % Mass [kg] -I = 115; % Inertia [kg m^2] +% Location of the joints with respect to the center of mass (symbolic) +Mb1 = [-l/2; -ha]; % Joint 1 position vector +Mb2 = [-la; -h/2]; % Joint 2 position vector +Mb3 = [ la; -h/2]; % Joint 3 position vector -%% Actuator Damping [N/(m/s)] -c1 = 2e1; -c2 = 2e1; -c3 = 2e1; +% Jacobian matrix (Center of Mass) +J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1); + s2', Mb2(1)*s2(2) - Mb2(2)*s2(1); + s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)]; -%% Actuator Stiffness [N/m] -k1 = 15e3; -k2 = 15e3; -k3 = 15e3; +% Display the symbolic Jacobian matrix +disp('Symbolic Jacobian Matrix (J_CoM):'); +disp(J_CoM); -%% Unit vectors of the actuators +% Jacobian at the Center of stiffness {K} +Mb1 = [-l/2; 0]; +Mb2 = [-la; -h/2+ha]; +Mb3 = [ la; -h/2+ha]; + +J_CoK = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1); + s2', Mb2(1)*s2(2) - Mb2(2)*s2(1); + s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)]; + +% Display the symbolic Jacobian matrix +disp('Symbolic Jacobian Matrix (J_CoK):'); +disp(J_CoK); +#+end_src + +#+begin_src matlab +%% Analytical Formula for the Modal Decoupling +syms l h la ha m I k c s real +syms omega1 omega2 omega3 real % Natural frequencies + +% Unit vectors of the actuators +s1 = [1; 0]; % Actuator 1 direction (horizontal) +s2 = [0; 1]; % Actuator 2 direction (vertical) +s3 = [0; 1]; % Actuator 3 direction (vertical) + +% Location of the joints with respect to the center of mass (symbolic) +Mb1 = [-l/2; -ha]; % Joint 1 position vector +Mb2 = [-la; -h/2]; % Joint 2 position vector +Mb3 = [la; -h/2]; % Joint 3 position vector + +% Calculate the Jacobian matrix (Center of Mass) symbolically +J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1); + s2', Mb2(1)*s2(2) - Mb2(2)*s2(1); + s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)]; + +disp('Symbolic Jacobian Matrix (J_CoM):'); +disp(J_CoM); + +% Define system matrices +M = diag([m, m, I]); +K_struts = diag([k, k, k]); +C_struts = diag([c, c, c]); + +% Transform stiffness and damping to Cartesian space +K = J_CoM' * K_struts * J_CoM; +C = J_CoM' * C_struts * J_CoM; + +disp('Mass Matrix (M):'); +disp(M); + +disp('Stiffness Matrix (K):'); +disp(K); + +disp('Damping Matrix (C):'); +disp(C); + +% Define the plant in the frame of the struts +% G_L = J_CoM * inv(M*s^2 + C*s + K) * J_CoM' +D_cart = M*s^2 + C*s + K; % Denominator in Cartesian space +disp('Dynamic Matrix in Cartesian Space (M*s^2 + C*s + K):'); +disp(D_cart); + +% Modal Decomposition +% Calculate the eigenvalues and eigenvectors of M\K +% For a symbolic approach, we'll use the general form of eigenvectors +% [V,D] = eig(M\K) + +% Instead of direct symbolic eigendecomposition (which is complex), +% we'll use known properties of modal analysis for analytical expressions + +% First, calculate M\K (inverse mass matrix times stiffness matrix) +MK = simplify(M\K); +disp('M\K Matrix:'); +disp(MK); + +% For a mechanical system with 3 DOF, we expect 3 eigenmodes +% Let's define symbolic eigenvectors in a general form +% According to vibration theory, the eigenvectors should be orthogonal with respect to M + +% Define symbolic eigenvectors +V = sym('v', [3, 3]); + +% Define the symbolic eigenvalues (squared natural frequencies) +D = diag([omega1^2, omega2^2, omega3^2]); + +% The eigenvectors should satisfy the equation (M\K)*V = V*D +% This is equivalent to K*V = M*V*D +% We can derive this symbolically, but it's complex for 3D systems + +% For an analytical approach, we can use physics to guide us +% For this system, we expect modes corresponding to: +% 1. Horizontal translation +% 2. Vertical translation +% 3. Rotation + +% Calculate modal mass matrix (mu = V'*M*V) +mu = simplify(V' * M * V); +disp('Modal Mass Matrix (mu):'); +disp(mu); + +% Modal output matrix +Cm = simplify(J_CoM * V); +disp('Modal Output Matrix (Cm):'); +disp(Cm); + +% Modal input matrix +Bm = simplify(inv(mu) * V' * J_CoM'); +disp('Modal Input Matrix (Bm):'); +disp(Bm); + +% Plant in the modal space +% For a fully decoupled system, Gm should be diagonal +Gm = simplify(inv(Cm) * J_CoM * inv(D_cart) * J_CoM' * inv(Bm')); +disp('Plant in Modal Space (Gm):'); +disp(Gm); +#+end_src + +** Decentralized Plant / Control in the frame of the struts + +#+begin_src latex :file detail_control_decoupling_control_struts.pdf +\begin{tikzpicture} + \node[block] (Gl) {$\bm{G}_{\{\mathcal{L}\}}$}; + + % Connections and labels + \draw[<-] (Gl.west) -- ++(-0.8, 0) node[above right]{$\bm{\tau}$}; + \draw[->] (Gl.east) -- ++( 0.8, 0) node[above left]{$\bm{\mathcal{L}}$}; +\end{tikzpicture} +#+end_src + +#+RESULTS: +[[file:figs/detail_control_jacobian_decoupling_arch.png]] + +\begin{equation} + \frac{\bm{\mathcal{L}}}{\bm{\mathcal{\tau}}}(s) = \bm{G}_{\mathcal{L}}(s) = \left( \bm{J}_{\{M\}}^{-t} M_{\{M\}} \bm{J}_{\{M\}}^{-1} s^2 + \bm{\mathcal{C}} s + \bm{\mathcal{K}} \right)^{-1} +\end{equation} + +At low frequency the plant converges to a diagonal constant matrix whose diagonal elements are linked to the actuator stiffnesses. + +\begin{equation} + \bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}} +\end{equation} + +#+begin_src matlab +%% Compute Equation of motion +l = 1; h=2; +la = 0.5; % Horizontal position of actuators [m] +ha = 0.2; % Vertical of actuators [m] + +m = 40; % Payload mass [kg] +I = 5; % Payload rotational inertia [kg m^2] + +% Actuator Damping [N/(m/s)] +c1 = 2e2; +c2 = 2e2; +c3 = 2e2; + +% Actuator Stiffness [N/m] +k1 = 1e6; +k2 = 1e6; +k3 = 1e6; + +% Unit vectors of the actuators s1 = [1;0]; s2 = [0;1]; s3 = [0;1]; -%% Location of the joints +% Stiffnesss and Damping matrices of the struts +Kr = diag([k1,k2,k3]); +Cr = diag([c1,c2,c3]); + +% Location of the joints with respect to the center of mass Mb1 = [-l/2;-ha]; Mb2 = [-la; -h/2]; Mb3 = [ la; -h/2]; -%% Jacobian matrix -J = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1); - s2', Mb2(1)*s2(2)-Mb2(2)*s2(1); - s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)]; +% Jacobian matrix (Center of Mass) +J_CoM = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1); + s2', Mb2(1)*s2(2)-Mb2(2)*s2(1); + s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)]; -%% Stiffnesss and Damping matrices of the struts -Kr = diag([k1,k2,k3]); -Cr = diag([c1,c2,c3]); - -%% Mass Matrix in frame {M} +% Mass Matrix in frame {M} M = diag([m,m,I]); -%% Stiffness Matrix in frame {M} -K = J'*Kr*J; +% Stiffness Matrix in frame {M} +K = J_CoM'*Kr*J_CoM; -%% Damping Matrix in frame {M} -C = J'*Cr*J; +% Damping Matrix in frame {M} +C = J_CoM'*Cr*J_CoM; -%% Plant in frame {M} -G = J*inv(M*s^2 + C*s + K)*J'; +% Plant in the frame of the struts +G_L = J_CoM*inv(M*s^2 + C*s + K)*J_CoM'; #+end_src -The magnitude of the coupled plant $G$ is shown in Figure ref:fig:detail_control_coupled_plant_bode. +The magnitude of the coupled plant $\bm{G}_{\mathcal{L}}$ is shown in Figure ref:fig:detail_control_coupled_plant_bode. #+begin_src matlab :exports none figure; @@ -1967,11 +2222,12 @@ tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None'); for out_i = 1:3 for in_i = 1:3 nexttile; - plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), 'k-', ... + plot(freqs, abs(squeeze(freqresp(G_L(out_i,in_i), freqs, 'Hz'))), 'k-', ... 'DisplayName', sprintf('$\\mathcal{L}_%i/\\tau_%i$', out_i, in_i)); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlim([1e-1, 2e1]); ylim([1e-6, 1e-2]); - legend('location', 'northeast', 'FontSize', 8); + xlim([freqs(1), freqs(end)]); ylim([1e-8, 1e-4]); + leg = legend('location', 'northeast', 'FontSize', 8); + leg.ItemTokenSize(1) = 18; if in_i == 1 ylabel('Mag. [m/N]') @@ -1997,9 +2253,9 @@ exportFig('figs/detail_control_coupled_plant_bode.pdf', 'width', 'full', 'height #+RESULTS: [[file:figs/detail_control_coupled_plant_bode.png]] -** Decentralized Plant / Control in the frame of the struts ** Jacobian Decoupling <> +**** Jacobian Matrix The Jacobian matrix can be used to: - Convert joints velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$: @@ -2015,7 +2271,7 @@ By wisely choosing frame $\{O\}$, we can obtain nice decoupling for plant: The obtained plan corresponds to forces/torques applied on origin of frame $\{O\}$ to the translation/rotation of the payload expressed in frame $\{O\}$. -#+begin_src latex :file detail_control_jacobian_decoupling_arch.pdf +#+begin_src latex :file detail_control_decoupling_control_jacobian.pdf \begin{tikzpicture} \node[block] (G) {$\bm{G}$}; \node[block, left=0.6 of G] (Jt) {$J_{\{O\}}^{-T}$}; @@ -2047,8 +2303,202 @@ The inputs and outputs of the decoupled plant $\bm{G}_{\{O\}}$ have physical mea It is then easy to include a reference tracking input that specify the wanted motion of the payload in the frame $\{O\}$. +Decoupling properties depends on the chosen frame $\{O\}$. + +**** Center Of Mass + +#+begin_src latex :file detail_control_decoupling_control_jacobian_CoM.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.6 of G] (Jt) {$J_{\{M\}}^{-T}$}; + \node[block, right=0.6 of G] (Ja) {$J_{\{M\}}^{-1}$}; + + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$}; + + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{M\}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +\begin{equation} + J_{\{M\}} = \begin{bmatrix} + 1 & 0 & h_a \\ + 0 & 1 & -l_a \\ + 0 & 1 & l_a \\ + \end{bmatrix}, \quad J_{\{M\}}^{-1} = \begin{bmatrix} + 1 & \frac{h_a}{2 l_a} & \frac{-h_a}{2 l_a} \\ + 0 & \frac{1}{2} & \frac{1}{2} \\ + 0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a} \\ + \end{bmatrix} +\end{equation} + +Analytical formula of the plant: + +\begin{equation} + \frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^t \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^t \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1} +\end{equation} + +At high frequency, converges towards the inverse of the mass matrix: + +\begin{equation} + \bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix} +1/m & 0 & 0 \\ +0 & 1/m & 0 \\ +0 & 0 & 1/I +\end{bmatrix} +\end{equation} + + + +Plant is well decoupled above the suspension mode with the highest frequency. +Such strategy is usually applied on systems with low frequency suspension modes, such that the plant corresponds to decoupled mass lines. + +- [ ] Reference to some papers about vibration isolation or ASML? + +#+begin_src matlab +%% Jacobian Decoupling - Center of Mass +G_CoM = pinv(J_CoM)*G_L*pinv(J_CoM'); +G_CoM.InputName = {'Fx', 'Fy', 'Mz'}; +G_CoM.OutputName = {'Dx', 'Dy', 'Rz'}; +#+end_src + +#+begin_src matlab :exports none +figure; +hold on; +for i_in = 1:3 + for i_out = [i_in+1:3] + plot(freqs, abs(squeeze(freqresp(G_CoM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +plot(freqs, abs(squeeze(freqresp(G_CoM(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... + 'DisplayName', '$G_{CoM}(i,j)\ i \neq j$'); +set(gca,'ColorOrderIndex',1) +for i_in_out = 1:3 + plot(freqs, abs(squeeze(freqresp(G_CoM(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{CoM}(%d,%d)$', i_in_out, i_in_out)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Magnitude'); +ylim([1e-10, 1e-3]); +leg = legend('location', 'northeast', 'FontSize', 8); +leg.ItemTokenSize(1) = 18; +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/detail_control_jacobian_plant_CoM.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:detail_control_jacobian_plant_CoM +#+caption: Plant decoupled using the Jacobian matrices $G_x(s)$ +#+RESULTS: +[[file:figs/detail_control_jacobian_plant_CoM.png]] + +**** Center Of Stiffness + +#+begin_src latex :file detail_control_decoupling_control_jacobian_CoK.pdf +\begin{tikzpicture} + \node[block] (G) {$\bm{G}$}; + \node[block, left=0.6 of G] (Jt) {$J_{\{K\}}^{-T}$}; + \node[block, right=0.6 of G] (Ja) {$J_{\{K\}}^{-1}$}; + + % Connections and labels + \draw[<-] (Jt.west) -- ++(-1.8, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$}; + \draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{\mathcal{L}}$}; + \draw[->] (Ja.east) -- ++( 1.8, 0) node[above left]{$\bm{\mathcal{X}}_{\{K\}}$}; + + \begin{scope}[on background layer] + \node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=16pt] (Gx) {}; + \node[below right] at (Gx.north west) {$\bm{G}_{\{K\}}$}; + \end{scope} +\end{tikzpicture} +#+end_src + +\begin{equation} + J_{\{K\}} = \begin{bmatrix} + 1 & 0 & 0 \\ + 0 & 1 & -l_a \\ + 0 & 1 & l_a + \end{bmatrix}, \quad J_{\{K\}}^{-1} = \begin{bmatrix} + 1 & 0 & 0 \\ + 0 & \frac{1}{2} & \frac{1}{2} \\ + 0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a} + \end{bmatrix} +\end{equation} + +Frame $\{K\}$ is chosen such that $\bm{J}_{\{K\}}^t \bm{\mathcal{K}} \bm{J}_{\{K\}}$ is diagonal. +Typically, it can me made based on physical reasoning as is the case here. + +\begin{equation} + \frac{\bm{\mathcal{X}}_{\{F\}}}{\bm{\mathcal{F}}_{\{F\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^t \bm{J}_{\{M\}}^{-T} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^t \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^t \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1} +\end{equation} + +Plant is well decoupled below the suspension mode with the lowest frequency. +This is usually suited for systems which high stiffness. + +\begin{equation} + \bm{G}_{\{K\}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{J}_{\{K\}}^{-1} \bm{\mathcal{K}}^{-1} \bm{J}_{\{K\}}^{-t} +\end{equation} + +#+begin_src matlab +%% Jacobian Decoupling - Center of Mass +% Location of the joints with respect to the center of stiffness +Mb1 = [-l/2; 0]; +Mb2 = [-la; -h/2+ha]; +Mb3 = [ la; -h/2+ha]; + +% Jacobian matrix (Center of Stiffness) +J_CoK = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1); + s2', Mb2(1)*s2(2)-Mb2(2)*s2(1); + s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)]; + +G_CoK = pinv(J_CoK)*G_L*pinv(J_CoK'); +G_CoK.InputName = {'Fx', 'Fy', 'Mz'}; +G_CoK.OutputName = {'Dx', 'Dy', 'Rz'}; +#+end_src + +#+begin_src matlab :exports none +figure; +hold on; +for i_in = 1:3 + for i_out = [i_in+1:3] + plot(freqs, abs(squeeze(freqresp(G_CoK(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +plot(freqs, abs(squeeze(freqresp(G_CoK(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... + 'DisplayName', '$G_{CoK}(i,j)\ i \neq j$'); +set(gca,'ColorOrderIndex',1) +for i_in_out = 1:3 + plot(freqs, abs(squeeze(freqresp(G_CoK(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{CoK}(%d,%d)$', i_in_out, i_in_out)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Magnitude'); +ylim([1e-10, 1e-3]); +leg = legend('location', 'northeast', 'FontSize', 8); +leg.ItemTokenSize(1) = 18; +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/detail_control_jacobian_plant_CoK.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:detail_control_jacobian_plant_CoK +#+caption: Plant decoupled using the Jacobian matrices $G_x(s)$ +#+RESULTS: +[[file:figs/detail_control_jacobian_plant_CoK.png]] + ** Modal Decoupling <> +**** Theory :ignore: Let's consider a system with the following equations of motion: \begin{equation} @@ -2165,34 +2615,133 @@ The obtained system on the diagonal are second order resonant systems which can Using this decoupling strategy, it is possible to control each mode individually. +**** Example + +For the system in Figure ref:fig:detail_control_model_test_decoupling, we have: +\begin{align} +\bm{x} &= \begin{bmatrix} x \\ y \\ R_z \end{bmatrix} \\ +\bm{y} &= \mathcal{L} = J \bm{x}; \quad C_{ox} = J; \quad C_{ov} = 0 \\ +M &= \begin{bmatrix} +m & 0 & 0 \\ +0 & m & 0 \\ +0 & 0 & I +\end{bmatrix}; \quad K = J' \begin{bmatrix} +k & 0 & 0 \\ +0 & k & 0 \\ +0 & 0 & k +\end{bmatrix} J; \quad C = J' \begin{bmatrix} +c & 0 & 0 \\ +0 & c & 0 \\ +0 & 0 & c +\end{bmatrix} J +\end{align} + +In order to apply the architecture shown in Figure ref:fig:modal_decoupling_architecture, we need to compute $C_{ox}$, $C_{ov}$, $\Phi$, $\mu$ and $J$. + +- [ ] Is it possible to obtained the analytical formulas for decoupling matrices? + +#+begin_src matlab +%% Modal Decoupling +% Modal Decomposition +[V,D] = eig(M\K); + +% Modal Mass Matrix +mu = V'*M*V; + +% Modal output matrix +Cm = J_CoM*V; + +% Modal input matrix +Bm = inv(mu)*V'*J_CoM'; + +% Plant in the modal space +Gm = inv(Cm)*G_L*inv(Bm); +#+end_src + +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(Bm, {}, {}, ' %.4f '); +#+end_src + +#+name: tab:modal_decoupling_Bm +#+caption: $B_m$ matrix +#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| -0.0004 | -0.0007 | 0.0007 | +| -0.0151 | 0.0041 | -0.0041 | +| 0.0 | 0.0025 | 0.0025 | + +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(Cm, {}, {}, ' %.1f '); +#+end_src + +#+name: tab:modal_decoupling_Cm +#+caption: $C_m$ matrix +#+attr_latex: :environment tabularx :width 0.2\linewidth :align ccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| -0.1 | -1.8 | 0.0 | +| -0.2 | 0.5 | 1.0 | +| 0.2 | -0.5 | 1.0 | + +#+begin_src matlab :exports none +figure; +hold on; +for i_in = 1:3 + for i_out = [i_in+1:3] + plot(freqs, abs(squeeze(freqresp(Gm(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... + 'HandleVisibility', 'off'); + end +end +plot(freqs, abs(squeeze(freqresp(Gm(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... + 'DisplayName', '$G_m(i,j)\ i \neq j$'); +set(gca,'ColorOrderIndex',1) +for i_in_out = 1:3 + plot(freqs, abs(squeeze(freqresp(Gm(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_m(%d,%d)$', i_in_out, i_in_out)); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Magnitude'); +ylim([1e-7, 1e0]); +leg = legend('location', 'northeast', 'FontSize', 8); +leg.ItemTokenSize(1) = 18; +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/detail_control_modal_plant.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:detail_control_modal_plant +#+caption: Modal plant $G_m(s)$ +#+RESULTS: +[[file:figs/detail_control_modal_plant.png]] + ** SVD Decoupling <> +**** Singular Value Decomposition + +- Introduction to SVD [[cite:&brunton22_data]] +- Applied to parallel manipulator? + +**** Control Architecture + +- [ ] Have notation for the measured FRF +- [ ] And for the real approximation + Procedure: - Identify the dynamics of the system from inputs to outputs (can be obtained experimentally) + Frequency Response Function, which is a complex matrix obtained for several frequency points. - Choose a frequency where we want to decouple the system (usually, the crossover frequency is a good choice) -#+begin_src matlab :eval no -%% Decoupling frequency [rad/s] -wc = 2*pi*10; - -%% System's response at the decoupling frequency -H1 = evalfr(G, j*wc); -#+end_src - Compute a real approximation of the system's response at that frequency -#+begin_src matlab :eval no -%% Real approximation of G(j.wc) -D = pinv(real(H1'*H1)); -H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); -#+end_src -- Perform a Singular Value Decomposition of the real approximation -#+begin_src matlab :eval no -[U,S,V] = svd(H1); -#+end_src + As /real/ V and U matrices need to be obtained, a real approximation of the complex measured response needs to be computed. + - [ ] Find reference to do so. +- Perform a Singular Value Decomposition of the real approximation. + Unitary U and V matrices are then obtained such that: + V-t Greal U-1 is a diagonal matrix + - Use the singular input and output matrices to decouple the system as shown in Figure ref:fig:detail_control_decoupling_svd \[ G_{svd}(s) = U^{-1} G(s) V^{-T} \] -#+begin_src matlab :eval no -Gsvd = inv(U)*G*inv(V'); -#+end_src #+begin_src latex :file detail_control_decoupling_svd.pdf \begin{tikzpicture} @@ -2232,194 +2781,31 @@ The inputs and outputs are ordered from higher gain to lower gain at the chosen - [ ] Do we loose any physical meaning of the obtained inputs and outputs? - [ ] Can we take advantage of the fact that U and V are unitary? -** Comparison -<> -**** Jacobian Decoupling -Decoupling properties depends on the chosen frame $\{O\}$. +**** Example -Let's take the CoM as the decoupling frame. +- [ ] Analytical formulas in this case? +- [ ] Do we have something special when applying SVD to a collocated MIMO system? +- *Verify why such a good decoupling is obtained!* +# - When applying SVD on a non-collocated MIMO system, we obtained a decoupled plant looking like the one in Figure ref:fig:detail_control_gravimeter_svd_plant #+begin_src matlab -Gx = pinv(J)*G*pinv(J'); -Gx.InputName = {'Fx', 'Fy', 'Mz'}; -Gx.OutputName = {'Dx', 'Dy', 'Rz'}; -#+end_src +%% SVD Decoupling -#+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); -figure; +wc = 2*pi*200; % Decoupling frequency [rad/s] +% System's response at the decoupling frequency +H1 = evalfr(G_L, j*wc); -% Magnitude -hold on; -for i_in = 1:3 - for i_out = [i_in+1:3] - plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'DisplayName', '$G_{x}(i,j)\ i \neq j$'); -set(gca,'ColorOrderIndex',1) -for i_in_out = 1:3 - plot(freqs, abs(squeeze(freqresp(Gx(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{x}(%d,%d)$', i_in_out, i_in_out)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Magnitude'); -ylim([1e-7, 1e-1]); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/detail_control_jacobian_plant.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:detail_control_jacobian_plant -#+caption: Plant decoupled using the Jacobian matrices $G_x(s)$ -#+RESULTS: -[[file:figs/detail_control_jacobian_plant.png]] - -**** Modal Decoupling -For the system in Figure ref:fig:detail_control_model_test_decoupling, we have: -\begin{align} -\bm{x} &= \begin{bmatrix} x \\ y \\ R_z \end{bmatrix} \\ -\bm{y} &= \mathcal{L} = J \bm{x}; \quad C_{ox} = J; \quad C_{ov} = 0 \\ -M &= \begin{bmatrix} -m & 0 & 0 \\ -0 & m & 0 \\ -0 & 0 & I -\end{bmatrix}; \quad K = J' \begin{bmatrix} -k & 0 & 0 \\ -0 & k & 0 \\ -0 & 0 & k -\end{bmatrix} J; \quad C = J' \begin{bmatrix} -c & 0 & 0 \\ -0 & c & 0 \\ -0 & 0 & c -\end{bmatrix} J -\end{align} - -In order to apply the architecture shown in Figure ref:fig:modal_decoupling_architecture, we need to compute $C_{ox}$, $C_{ov}$, $\Phi$, $\mu$ and $J$. - -#+begin_src matlab -%% Modal Decomposition -[V,D] = eig(M\K); - -%% Modal Mass Matrix -mu = V'*M*V; - -%% Modal output matrix -Cm = J*V; - -%% Modal input matrix -Bm = inv(mu)*V'*J'; -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(Bm, {}, {}, ' %.4f '); -#+end_src - -#+name: tab:modal_decoupling_Bm -#+caption: $B_m$ matrix -#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc -#+attr_latex: :center t :booktabs t :float t -#+RESULTS: -| -0.0004 | -0.0007 | 0.0007 | -| -0.0151 | 0.0041 | -0.0041 | -| 0.0 | 0.0025 | 0.0025 | - -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(Cm, {}, {}, ' %.1f '); -#+end_src - -#+name: tab:modal_decoupling_Cm -#+caption: $C_m$ matrix -#+attr_latex: :environment tabularx :width 0.2\linewidth :align ccc -#+attr_latex: :center t :booktabs t :float t -#+RESULTS: -| -0.1 | -1.8 | 0.0 | -| -0.2 | 0.5 | 1.0 | -| 0.2 | -0.5 | 1.0 | - -And the plant in the modal space is defined below and its magnitude is shown in Figure ref:fig:detail_control_modal_plant. -#+begin_src matlab -Gm = inv(Cm)*G*inv(Bm); -#+end_src - -#+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); -figure; - -% Magnitude -hold on; -for i_in = 1:3 - for i_out = [i_in+1:3] - plot(freqs, abs(squeeze(freqresp(Gm(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'HandleVisibility', 'off'); - end -end -plot(freqs, abs(squeeze(freqresp(Gm(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ... - 'DisplayName', '$G_m(i,j)\ i \neq j$'); -set(gca,'ColorOrderIndex',1) -for i_in_out = 1:3 - plot(freqs, abs(squeeze(freqresp(Gm(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_m(%d,%d)$', i_in_out, i_in_out)); -end -hold off; -set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -xlabel('Frequency [Hz]'); ylabel('Magnitude'); -ylim([1e-6, 1e2]); -legend('location', 'northeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace -exportFig('figs/detail_control_modal_plant.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:detail_control_modal_plant -#+caption: Modal plant $G_m(s)$ -#+RESULTS: -[[file:figs/detail_control_modal_plant.png]] - -Let's now close one loop at a time and see how the transmissibility changes. - -**** SVD Decoupling -#+begin_src matlab -%% Decoupling frequency [rad/s] -wc = 2*pi*10; - -%% System's response at the decoupling frequency -H1 = evalfr(G, j*wc); - -%% Real approximation of G(j.wc) +% Real approximation of G(j.wc) D = pinv(real(H1'*H1)); H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2)))); [U,S,V] = svd(H1); -Gsvd = inv(U)*G*inv(V'); +Gsvd = inv(U)*G_L*inv(V'); #+end_src -#+begin_src matlab :exports results :results value table replace :tangle no -data2orgtable(H1, {}, {}, ' %.2g '); -#+end_src - -#+caption: Real approximate of $G$ at the decoupling frequency $\omega_c$ -#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc -#+attr_latex: :center t :booktabs t :float t -#+RESULTS: -| -8e-06 | 2.1e-06 | -2.1e-06 | -| 2.1e-06 | -1.3e-06 | -2.5e-08 | -| -2.1e-06 | -2.5e-08 | -1.3e-06 | - -- [ ] Do we have something special when applying SVD to a collocated MIMO system? -- *Verify why such a good decoupling is obtained!* -# - When applying SVD on a non-collocated MIMO system, we obtained a decoupled plant looking like the one in Figure ref:fig:detail_control_gravimeter_svd_plant - #+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); figure; - -% Magnitude hold on; for i_in = 1:3 for i_out = [i_in+1:3] @@ -2436,8 +2822,9 @@ end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('Magnitude'); -% ylim([1e-8, 1e-2]); -legend('location', 'northeast'); +ylim([1e-10, 2e-4]); +leg = legend('location', 'northeast', 'FontSize', 8); +leg.ItemTokenSize(1) = 18; #+end_src #+begin_src matlab :tangle no :exports results :results file replace @@ -2449,6 +2836,55 @@ exportFig('figs/detail_control_svd_plant.pdf', 'width', 'wide', 'height', 'norma #+RESULTS: [[file:figs/detail_control_svd_plant.png]] +#+begin_src matlab :exports results :results value table replace :tangle no +data2orgtable(H1, {}, {}, ' %.2g '); +#+end_src + +#+caption: Real approximate of $G$ at the decoupling frequency $\omega_c$ +#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| -8e-06 | 2.1e-06 | -2.1e-06 | +| 2.1e-06 | -1.3e-06 | -2.5e-08 | +| -2.1e-06 | -2.5e-08 | -1.3e-06 | + +** Comparison +<> + +#+name: tab:detail_control_decoupling_strategies_comp +#+caption: Comparison of decoupling strategies +#+attr_latex: :environment tabularx :width \linewidth :align lXXX +#+attr_latex: :center t :booktabs t :font \scriptsize +| | *Jacobian* | *Modal* | *SVD* | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Philosophy* | Topology Driven | Physics Driven | Data Driven | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Requirements* | Known geometry | Known equations of motion | Identified FRF | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Decoupling Matrices* | Decoupling using $J$ obtained from geometry | Decoupling using $\Phi$ obtained from modal decomposition | Decoupling using $U$ and $V$ obtained from SVD | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Decoupled Plant* | $\bm{G}_{\{O\}} = J_{\{O\}}^{-1} \bm{G} J_{\{O\}}^{-T}$ | $\bm{G}_m = C_m^{-1} \bm{G} B_m^{-1}$ | $\bm{G}_{svd}(s) = U^{-1} \bm{G}(s) V^{-T}$ | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Implemented Controller* | $\bm{K}_{\{O\}} = J_{\{O\}}^{-T} \bm{K}_{d}(s) J_{\{O\}}^{-1}$ | $\bm{K}_m = B_m^{-1} \bm{K}_{d}(s) C_m^{-1}$ | $\bm{K}_{svd}(s) = V^{-T} \bm{K}_{d}(s) U^{-1}$ | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Physical Interpretation* | Forces/Torques to Displacement/Rotation in chosen frame | Inputs to excite individual modes | Directions of max to min controllability/observability | +| | | Output to sense individual modes | | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Decoupling Properties* | Decoupling at low or high frequency depending on the chosen frame | Good decoupling at all frequencies | Good decoupling near the chosen frequency | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Pros* | Physical inputs / outputs | Target specific modes | Good Decoupling near the crossover | +| | Good decoupling at High frequency (diagonal mass matrix if Jacobian taken at the CoM) | 2nd order diagonal plant | Very General | +| | Good decoupling at Low frequency (if Jacobian taken at specific point) | | | +| | Easy integration of meaningful reference inputs | | | +| | | | | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Cons* | Coupling between force/rotation may be high at low frequency (non diagonal terms in K) | Need analytical equations | Loose the physical meaning of inputs /outputs | +| | Limited to parallel mechanisms (?) | | Decoupling depends on the real approximation validity | +| | If good decoupling at all frequencies => requires specific mechanical architecture | | Diagonal plants may not be easy to control | +|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| +| *Applicability* | Parallel Mechanisms | Systems whose dynamics that can be expressed with M and K matrices | Very general | +| | Only small motion for the Jacobian matrix to stay constant | | Need FRF data (either experimentally or analytically) | + ** TODO Robustness of the decoupling strategies? :noexport: *** Introduction :ignore: What happens if we add an additional resonance in the system (Figure ref:fig:model_test_decoupling_spurious_res). @@ -2543,7 +2979,6 @@ Gxr.OutputName = {'Dx', 'Dy', 'Rz'}; The obtained plant is shown in Figure ref:fig:detail_control_jacobian_plant_spurious and is not much different than for the plant without the spurious resonance. #+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); figure; % Magnitude @@ -2585,7 +3020,6 @@ Gmr = inv(Cm)*Gr*inv(Bm); The obtained decoupled plant is shown in Figure ref:fig:detail_control_modal_plant_spurious. Compare to the decoupled plant in Figure ref:fig:detail_control_modal_plant, the added resonance induces some coupling, especially around the frequency of the added spurious resonance. #+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); figure; % Magnitude @@ -2640,7 +3074,6 @@ Gsvdr = inv(U)*Gr*inv(V'); The obtained plant is shown in Figure ref:fig:detail_control_svd_plant_spurious. #+begin_src matlab :exports none -freqs = logspace(-1, 2, 1000); figure; % Magnitude @@ -2689,39 +3122,6 @@ Conclusion about NASS: - Prefer to use Jacobian decoupling as we get more physical interpretation - Also, it is possible to take into account different specifications in the different DoF -#+name: tab:detail_control_decoupling_strategies_comp -#+caption: Comparison of decoupling strategies -#+attr_latex: :environment tabularx :width \linewidth :align lXXX -#+attr_latex: :center t :booktabs t :font \scriptsize -| | *Jacobian* | *Modal* | *SVD* | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Philosophy* | Topology Driven | Physics Driven | Data Driven | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Requirements* | Known geometry | Known equations of motion | Identified FRF | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Decoupling Matrices* | Decoupling using $J$ obtained from geometry | Decoupling using $\Phi$ obtained from modal decomposition | Decoupling using $U$ and $V$ obtained from SVD | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Decoupled Plant* | $\bm{G}_{\{O\}} = J_{\{O\}}^{-1} \bm{G} J_{\{O\}}^{-T}$ | $\bm{G}_m = C_m^{-1} \bm{G} B_m^{-1}$ | $\bm{G}_{svd}(s) = U^{-1} \bm{G}(s) V^{-T}$ | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Implemented Controller* | $\bm{K}_{\{O\}} = J_{\{O\}}^{-T} \bm{K}_{d}(s) J_{\{O\}}^{-1}$ | $\bm{K}_m = B_m^{-1} \bm{K}_{d}(s) C_m^{-1}$ | $\bm{K}_{svd}(s) = V^{-T} \bm{K}_{d}(s) U^{-1}$ | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Physical Interpretation* | Forces/Torques to Displacement/Rotation in chosen frame | Inputs to excite individual modes | Directions of max to min controllability/observability | -| | | Output to sense individual modes | | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Decoupling Properties* | Decoupling at low or high frequency depending on the chosen frame | Good decoupling at all frequencies | Good decoupling near the chosen frequency | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Pros* | Physical inputs / outputs | Target specific modes | Good Decoupling near the crossover | -| | Good decoupling at High frequency (diagonal mass matrix if Jacobian taken at the CoM) | 2nd order diagonal plant | Very General | -| | Good decoupling at Low frequency (if Jacobian taken at specific point) | | | -| | Easy integration of meaningful reference inputs | | | -| | | | | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Cons* | Coupling between force/rotation may be high at low frequency (non diagonal terms in K) | Need analytical equations | Loose the physical meaning of inputs /outputs | -| | Limited to parallel mechanisms (?) | | Decoupling depends on the real approximation validity | -| | If good decoupling at all frequencies => requires specific mechanical architecture | | Diagonal plants may not be easy to control | -|---------------------------+----------------------------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------| -| *Applicability* | Parallel Mechanisms | Systems whose dynamics that can be expressed with M and K matrices | Very general | -| | Only small motion for the Jacobian matrix to stay constant | | Need FRF data (either experimentally or analytically) | * Closed-Loop Shaping using Complementary Filters :PROPERTIES: @@ -2762,7 +3162,8 @@ It will be experimentally demonstrated with the NASS. *SISO control design methods* - frequency domain techniques - manual loop-shaping - key idea: modification of the controller such that the open-loop is made according to specifications [[cite:&oomen18_advan_motion_contr_precis_mechat]]. -This works well because the open loop transfer function is linearly dependent of the controller. + This works well because the open loop transfer function is linearly dependent of the controller. + Different techniques for open loop shaping [[cite:&lurie02_system_archit_trades_using_bode]] However, the specifications are given in terms of the final system performance, i.e. as closed-loop specifications. @@ -3935,7 +4336,4 @@ addpath('./src/'); % Path for functions #+BEGIN_SRC matlab %% Colors for the figures colors = colororder; - -%% Initialize Frequency Vector -freqs = logspace(-1, 3, 1000); #+END_SRC diff --git a/nass-control.pdf b/nass-control.pdf index 3018bed..8d6c1de 100644 Binary files a/nass-control.pdf and b/nass-control.pdf differ diff --git a/nass-control.tex b/nass-control.tex index 41224be..a13875d 100644 --- a/nass-control.tex +++ b/nass-control.tex @@ -1,4 +1,4 @@ -% Created 2025-04-04 Fri 11:38 +% Created 2025-04-04 Fri 19:20 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} @@ -32,7 +32,6 @@ Several considerations: \item Section \ref{sec:detail_control_optimization}: How to design the controller \end{itemize} \chapter{Multiple Sensor Control} -\label{sec:orgf67d083} \label{sec:detail_control_multiple_sensor} \textbf{Look at what was done in the introduction \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/nass-introduction.org}{Stewart platforms: Control architecture}} @@ -135,7 +134,6 @@ In this section, we wish to study if sensor fusion can be an option for multi-se \item optimize the dynamical uncertainty \end{itemize} \section{Sensor fusion - Introduction} -\label{sec:org119298d} Measuring a physical quantity using sensors is always subject to several limitations. First, the accuracy of the measurement is affected by several noise sources, such as electrical noise of the conditioning electronics being used. @@ -183,13 +181,11 @@ Although many design methods of complementary filters have been proposed in the Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters \cite{dehaeze19_compl_filter_shapin_using_synth}. Based on that, this work introduces a new way to design complementary filters using the \(\mathcal{H}_\infty\) synthesis which allows to shape the complementary filters' magnitude in an easy and intuitive way. \section{Sensor Fusion and Complementary Filters Requirements} -\label{sec:org8c6ec99} \label{ssec:detail_control_sensor_fusion_requirements} Complementary filtering provides a framework for fusing signals from different sensors. As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements. These requirements are discussed in this section. -\paragraph{Sensor Fusion Architecture} -\label{sec:org1e5ae2a} +\subsubsection{Sensor Fusion Architecture} A general sensor fusion architecture using complementary filters is shown in Figure \ref{fig:detail_control_sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\). The two sensors output signals \(\hat{x}_1\) and \(\hat{x}_2\) are estimates of \(x\). @@ -210,8 +206,7 @@ That is, unity magnitude and zero phase at all frequencies. \begin{equation}\label{eq:detail_control_comp_filter} H_1(s) + H_2(s) = 1 \end{equation} -\paragraph{Sensor Models and Sensor Normalization} -\label{sec:org429b48b} +\subsubsection{Sensor Models and Sensor Normalization} In order to study such sensor fusion architecture, a model for the sensors is required. Such model is shown in Figure \ref{fig:detail_control_sensor_model} and consists of a linear time invariant (LTI) system \(G_i(s)\) representing the sensor dynamics and an input \(n_i\) representing the sensor noise. @@ -253,8 +248,7 @@ The super sensor output \(\hat{x}\) is therefore described by \eqref{eq:detail_c \includegraphics[scale=1]{figs/detail_control_fusion_super_sensor.png} \caption{\label{fig:detail_control_fusion_super_sensor}Sensor fusion architecture with two normalized sensors.} \end{figure} -\paragraph{Noise Sensor Filtering} -\label{sec:org8410a77} +\subsubsection{Noise Sensor Filtering} First, suppose that all the sensors are perfectly normalized \eqref{eq:detail_control_perfect_dynamics}. The effect of a non-perfect normalization will be discussed afterwards. @@ -290,8 +284,7 @@ This is the simplest form of sensor fusion with complementary filters. However, the two sensors have usually high noise levels over distinct frequency regions. In such case, to lower the noise of the super sensor, the norm \(|H_1(j\omega)|\) has to be small when \(\Phi_{n_1}(\omega)\) is larger than \(\Phi_{n_2}(\omega)\) and the norm \(|H_2(j\omega)|\) has to be small when \(\Phi_{n_2}(\omega)\) is larger than \(\Phi_{n_1}(\omega)\). Hence, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor. -\paragraph{Sensor Fusion Robustness} -\label{sec:orgda97814} +\subsubsection{Sensor Fusion Robustness} In practical systems the sensor normalization is not perfect and condition \eqref{eq:detail_control_perfect_dynamics} is not verified. @@ -351,15 +344,13 @@ For instance, the phase \(\Delta\phi(\omega)\) added by the super sensor dynamic As it is generally desired to limit the maximum phase added by the super sensor, \(H_1(s)\) and \(H_2(s)\) should be designed such that \(\Delta \phi\) is bounded to acceptable values. Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain. \section{Complementary Filters Shaping} -\label{sec:orgc4a4ceb} \label{ssec:detail_control_hinf_method} As shown in Section \ref{ssec:detail_control_sensor_fusion_requirements}, the noise and robustness of the super sensor are a function of the complementary filters' norm. Therefore, a synthesis method of complementary filters that allows to shape their norm would be of great use. In this section, such synthesis is proposed by writing the synthesis objective as a standard \(\mathcal{H}_\infty\) optimization problem. As weighting functions are used to represent the wanted complementary filters' shape during the synthesis, their proper design is discussed. Finally, the synthesis method is validated on an simple example. -\paragraph{Synthesis Objective} -\label{sec:orgec6a952} +\subsubsection{Synthesis Objective} The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:detail_control_comp_filter}. This is equivalent as to finding proper and stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:detail_control_hinf_cond_complementarity}, \eqref{eq:detail_control_hinf_cond_h1} and \eqref{eq:detail_control_hinf_cond_h2} are satisfied. @@ -372,8 +363,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\) & |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:detail_control_hinf_cond_h2} \end{align} \end{subequations} -\paragraph{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis} -\label{sec:orgc16ac7f} +\subsubsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis} The synthesis objective can be easily expressed as a standard \(\mathcal{H}_\infty\) optimization problem and therefore solved using convenient tools readily available. Consider the generalized plant \(P(s)\) shown in Figure \ref{fig:detail_control_h_infinity_robust_fusion_plant} and mathematically described by \eqref{eq:detail_control_generalized_plant}. @@ -418,8 +408,7 @@ Therefore, applying the \(\mathcal{H}_\infty\) synthesis to the standard plant \ Note that there is only an implication between the \(\mathcal{H}_\infty\) norm condition \eqref{eq:detail_control_hinf_problem} and the initial synthesis objectives \eqref{eq:detail_control_hinf_cond_h1} and \eqref{eq:detail_control_hinf_cond_h2} and not an equivalence. Hence, the optimization may be a little bit conservative with respect to the set of filters on which it is performed, see \cite[,Chap. 2.8.3]{skogestad07_multiv_feedb_contr}. -\paragraph{Weighting Functions Design} -\label{sec:org81bbe63} +\subsubsection{Weighting Functions Design} Weighting functions are used during the synthesis to specify the maximum allowed complementary filters' norm. The proper design of these weighting functions is of primary importance for the success of the presented \(\mathcal{H}_\infty\) synthesis of complementary filters. @@ -450,8 +439,7 @@ The typical magnitude of a weighting function generated using \eqref{eq:detail_c }\right)^n \end{equation} \end{minipage} -\paragraph{Validation of the proposed synthesis method} -\label{sec:org4c438a2} +\subsubsection{Validation of the proposed synthesis method} The proposed methodology for the design of complementary filters is now applied on a simple example. Let's suppose two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that: @@ -498,7 +486,6 @@ The \(\mathcal{H}_\infty\) norm is here found to be close to one which indicates This is confirmed by the bode plots of the obtained complementary filters in Figure \ref{fig:detail_control_hinf_filters_results}. This simple example illustrates the fact that the proposed methodology for complementary filters shaping is easy to use and effective. \section{Synthesis of a set of three complementary filters} -\label{sec:org6c33bf8} \label{sec:detail_control_hinf_three_comp_filters} Some applications may require to merge more than two sensors \cite{stoten01_fusion_kinet_data_using_compos_filter,fonseca15_compl}. @@ -602,7 +589,6 @@ Filter \(H_1(s)\) is defined using \eqref{eq:detail_control_h1_compl_h2_h3} thus Figure \ref{fig:detail_control_three_complementary_filters_results} displays the three synthesized complementary filters (solid lines) which confirms that the synthesis is successful. \section*{Conclusion} -\label{sec:org0d8cdf5} A new method for designing complementary filters using the \(\mathcal{H}_\infty\) synthesis has been proposed. It allows to shape the magnitude of the filters by the use of weighting functions during the synthesis. This is very valuable in practice as the characteristics of the super sensor are linked to the complementary filters' magnitude. @@ -614,15 +600,28 @@ Future work will aim at developing a complementary filter synthesis method that \begin{itemize} \item Talk about the possibility to use H2 to minimize the RMS value of the super sensor noise? (or maybe make a section about that?) +There is a draft paper about that. \item For the NASS, it was shown that the HAC-IFF strategy works fine and is easy to understand and tune \item It would be very interesting to see how sensor fusion (probably between the force sensor and the external metrology) compares in term of performance and robustness \end{itemize} \chapter{Decoupling} -\label{sec:orgbbaf347} \label{sec:detail_control_decoupling} +When dealing with MIMO systems, a typical strategy is to: \begin{itemize} -\item[{$\square$}] \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies} +\item First decouple the plant dynamics (discussed in this section) +\item Apply SISO control for the decoupled plant (discussed in section \ref{sec:detail_control_optimization}) +\end{itemize} + +Another strategy would be to apply a multivariable control synthesis to the coupled system. +Strangely, while H-infinity synthesis is a mature technology, it use for the control of Stewart platform is not yet demonstrated. +From \cite{thayer02_six_axis_vibrat_isolat_system}: +\begin{quote} +Experimental closed-loop control results using the hexapod have shown that controllers designed using a decentralized single-strut design work well when compared to full multivariable methodologies. +\end{quote} + +\begin{itemize} +\item[{$\square$}] Review of \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies} for stewart platforms \item[{$\square$}] Add some citations about different methods \end{itemize} @@ -647,34 +646,17 @@ Voice Coil & Geophone + Eddy Current (Struts, collocated) & Decentralized (Sky H Piezoelectric & Force, Position & Vibration isolation, Model-Based, \textbf{Modal control}: 6x PI controllers & \cite{yang19_dynam_model_decoup_contr_flexib}\\ \midrule PZT & Geophone (struts) & \textbf{H-Infinity} and mu-synthesis & \cite{lei08_multi_objec_robus_activ_vibrat}\\ -Voice Coil & Force sensors (strus) + accelerometer (cartesian) & Decentralized Force Feedback + Centralized H2 control based on accelerometers & \cite{xie17_model_contr_hybrid_passiv_activ}\\ +Voice Coil & Force sensors (struts) + accelerometer (cartesian) & Decentralized Force Feedback + Centralized H2 control based on accelerometers & \cite{xie17_model_contr_hybrid_passiv_activ}\\ Voice Coil & Accelerometers & MIMO H-Infinity, active damping & \cite{jiao18_dynam_model_exper_analy_stewar}\\ \bottomrule \end{tabularx} \end{table} -When dealing with MIMO systems, a typical strategy is to: -\begin{itemize} -\item first decouple the plant dynamics -\item apply SISO control for the decoupled plant -\end{itemize} - -Another strategy would be to apply a multivariable control synthesis to the coupled system. -From \cite{thayer02_six_axis_vibrat_isolat_system}: -\begin{quote} -Experimental closed-loopcontrol results using the hexapod have shown that controllers designed using a decentralized single-strut design work well when compared to full multivariable methodologies. -\end{quote} - Assumptions: \begin{itemize} \item parallel manipulators \end{itemize} -Review of decoupling strategies for Stewart platforms: -\begin{itemize} -\item \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}{Decoupling Strategies} -\end{itemize} - The goal of this section is to compare the use of several methods for the decoupling of parallel manipulators. It is structured as follow: @@ -687,7 +669,6 @@ It is structured as follow: \item Conclusions are drawn on the three decoupling methods \end{itemize} \section{Test Model} -\label{sec:orgf9db9a6} \label{ssec:detail_control_decoupling_comp_model} Let's consider a parallel manipulator with several collocated actuator/sensors pairs. @@ -705,9 +686,19 @@ We will note: \end{itemize} \begin{figure}[htbp] -\centering -\includegraphics[scale=1]{figs/detail_control_model_test_decoupling.png} -\caption{\label{fig:detail_control_model_test_decoupling}Model use to compare decoupling techniques} +\begin{subfigure}{0.58\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/detail_control_model_test_decoupling.png} +\end{center} +\subcaption{\label{fig:detail_control_model_test_decoupling}Geometrical parameters} +\end{subfigure} +\begin{subfigure}{0.38\textwidth} +\begin{center} +\includegraphics[scale=1,scale=1]{figs/detail_control_strut_model.png} +\end{center} +\subcaption{\label{fig:detail_control_strut_model}Strut model} +\end{subfigure} +\caption{\label{fig:detail_control_model_test_decoupling_detail}3DoF model used to study decoupling strategies} \end{figure} The magnitude of the coupled plant \(G\) is shown in Figure \ref{fig:detail_control_coupled_plant_bode}. @@ -718,9 +709,7 @@ The magnitude of the coupled plant \(G\) is shown in Figure \ref{fig:detail_cont \caption{\label{fig:detail_control_coupled_plant_bode}Magnitude of the coupled plant.} \end{figure} \section{Decentralized Plant / Control in the frame of the struts} -\label{sec:orgf446497} \section{Jacobian Decoupling} -\label{sec:orgfcc4afb} \label{ssec:detail_control_comp_jacobian} The Jacobian matrix can be used to: @@ -754,8 +743,10 @@ The inputs and outputs of the decoupled plant \(\bm{G}_{\{O\}}\) have physical m \end{itemize} It is then easy to include a reference tracking input that specify the wanted motion of the payload in the frame \(\{O\}\). +\subsubsection{Center Of Mass} + +\subsubsection{Center Of Stiffness} \section{Modal Decoupling} -\label{sec:org896946d} \label{ssec:detail_control_comp_modal} Let's consider a system with the following equations of motion: @@ -861,7 +852,6 @@ The obtained system on the diagonal are second order resonant systems which can Using this decoupling strategy, it is possible to control each mode individually. \section{SVD Decoupling} -\label{sec:org40b06bc} \label{ssec:detail_control_comp_svd} Procedure: @@ -894,10 +884,8 @@ The inputs and outputs are ordered from higher gain to lower gain at the chosen \item[{$\square$}] Can we take advantage of the fact that U and V are unitary? \end{itemize} \section{Comparison} -\label{sec:org766237f} \label{ssec:detail_control_decoupling_comp} -\paragraph{Jacobian Decoupling} -\label{sec:orgfc21963} +\subsubsection{Jacobian Decoupling} Decoupling properties depends on the chosen frame \(\{O\}\). Let's take the CoM as the decoupling frame. @@ -907,8 +895,7 @@ Let's take the CoM as the decoupling frame. \includegraphics[scale=1]{figs/detail_control_jacobian_plant.png} \caption{\label{fig:detail_control_jacobian_plant}Plant decoupled using the Jacobian matrices \(G_x(s)\)} \end{figure} -\paragraph{Modal Decoupling} -\label{sec:org124e143} +\subsubsection{Modal Decoupling} For the system in Figure \ref{fig:detail_control_model_test_decoupling}, we have: \begin{align} \bm{x} &= \begin{bmatrix} x \\ y \\ R_z \end{bmatrix} \\ @@ -962,8 +949,7 @@ And the plant in the modal space is defined below and its magnitude is shown in \end{figure} Let's now close one loop at a time and see how the transmissibility changes. -\paragraph{SVD Decoupling} -\label{sec:org3f3f14e} +\subsubsection{SVD Decoupling} \begin{table}[htbp] \caption{\label{}Real approximate of \(G\) at the decoupling frequency \(\omega_c\)} \centering @@ -986,7 +972,6 @@ Let's now close one loop at a time and see how the transmissibility changes. \caption{\label{fig:detail_control_svd_plant}Svd plant \(G_m(s)\)} \end{figure} \section*{Conclusion} -\label{sec:org79124be} The three proposed methods clearly have a lot in common as they all tend to make system more decoupled by pre and/or post multiplying by a constant matrix However, the three methods also differs by a number of points which are summarized in Table \ref{tab:detail_control_decoupling_strategies_comp}. @@ -1041,7 +1026,6 @@ Conclusion about NASS: \end{tabularx} \end{table} \chapter{Closed-Loop Shaping using Complementary Filters} -\label{sec:org5dd9a93} \label{sec:detail_control_optimization} Performance of a feedback control is dictated by closed-loop transfer functions. @@ -1077,8 +1061,9 @@ It will be experimentally demonstrated with the NASS. \begin{itemize} \item frequency domain techniques \item manual loop-shaping - key idea: modification of the controller such that the open-loop is made according to specifications \cite{oomen18_advan_motion_contr_precis_mechat}. -\end{itemize} This works well because the open loop transfer function is linearly dependent of the controller. +Different techniques for open loop shaping \cite{lurie02_system_archit_trades_using_bode} +\end{itemize} However, the specifications are given in terms of the final system performance, i.e. as closed-loop specifications. @@ -1102,10 +1087,8 @@ In this paper, we propose a new controller synthesis method \item direct translation of requirements such as disturbance rejection and robustness to plant uncertainty \end{itemize} \section{Control Architecture} -\label{sec:org34a8e50} \label{ssec:detail_control_control_arch} -\paragraph{Virtual Sensor Fusion} -\label{sec:org95786c4} +\subsubsection{Virtual Sensor Fusion} Let's consider the control architecture represented in Figure \ref{fig:detail_control_sf_arch} where \(G^\prime\) is the physical plant to control, \(G\) is a model of the plant, \(k\) is a gain, \(H_L\) and \(H_H\) are complementary filters (\(H_L + H_H = 1\) in the complex sense). The signals are the reference signal \(r\), the output perturbation \(d_y\), the measurement noise \(n\) and the control input \(u\). @@ -1143,8 +1126,7 @@ y &= \frac{1}{1+G^{\prime} K H_L} dy &&+ \frac{G^{\prime} K}{1+G^{\prime} K u &= \frac{-K H_L}{1+G^{\prime} K H_L} dy &&+ \frac{K}{1+G^{\prime} K H_L} r &&- \frac{K H_L}{1+G^{\prime} K H_L} n \end{alignat} with \(K = \frac{k}{1 + H_H G k}\) -\paragraph{Asymptotic behavior} -\label{sec:org96ac3b2} +\subsubsection{Asymptotic behavior} We now want to study the asymptotic system obtained when using very high value of \(k\) \begin{equation} \lim_{k\to\infty} K = \lim_{k\to\infty} \frac{k}{1+H_H G k} = \left( H_H G \right)^{-1} @@ -1177,7 +1159,6 @@ We obtain a sensitivity transfer function equals to the high pass filter \(S = \ Assuming that we have a good model of the plant, we have then that the closed-loop behavior of the system converges to the designed complementary filters. \section{Translating the performance requirements into the shapes of the complementary filters} -\label{sec:org5196a79} \label{ssec:detail_control_trans_perf} The required performance specifications in a feedback system can usually be translated into requirements on the upper bounds of \(\abs{S(j\w)}\) and \(|T(j\omega)|\) \cite{bibel92_guidel_h}. The process of designing a controller \(K(s)\) in order to obtain the desired shapes of \(\abs{S(j\w)}\) and \(\abs{T(j\w)}\) is called loop shaping. @@ -1185,8 +1166,7 @@ The process of designing a controller \(K(s)\) in order to obtain the desired sh The equations \eqref{eq:detail_control_cl_system_y} and \eqref{eq:detail_control_cl_system_u} describing the dynamics of the studied feedback architecture are not written in terms of \(K\) but in terms of the complementary filters \(H_L\) and \(H_H\). In this section, we then translate the typical specifications into the desired shapes of the complementary filters \(H_L\) and \(H_H\).\\ -\paragraph{Nominal Stability (NS)} -\label{sec:org6c13475} +\subsubsection{Nominal Stability (NS)} The closed-loop system is stable if all its elements are stable (\(K\), \(G^\prime\) and \(H_L\)) and if the sensitivity function (\(S = \frac{1}{1 + G^\prime K H_L}\)) is stable. For the nominal system (\(G^\prime = G\)), we have \(S = H_H\). @@ -1194,8 +1174,7 @@ For the nominal system (\(G^\prime = G\)), we have \(S = H_H\). Nominal stability is then guaranteed if \(H_L\), \(H_H\) and \(G\) are stable and if \(G\) and \(H_H\) are minimum phase (to have \(K\) stable). Thus we must design stable and minimum phase complementary filters.\\ -\paragraph{Nominal Performance (NP)} -\label{sec:org4d66e50} +\subsubsection{Nominal Performance (NP)} Typical performance specifications can usually be translated into upper bounds on \(|S(j\omega)|\) and \(|T(j\omega)|\). Two performance weights \(w_H\) and \(w_L\) are defined in such a way that performance specifications are satisfied if @@ -1218,15 +1197,13 @@ The translation of typical performance requirements on the shapes of the complem \end{itemize} We may have other requirements in terms of stability margins, maximum or minimum closed-loop bandwidth.\\ -\paragraph{Closed-Loop Bandwidth} -\label{sec:org3d13ca8} +\subsubsection{Closed-Loop Bandwidth} The closed-loop bandwidth \(\w_B\) can be defined as the frequency where \(\abs{S(j\w)}\) first crosses \(\frac{1}{\sqrt{2}}\) from below. If one wants the closed-loop bandwidth to be at least \(\w_B^*\) (e.g. to stabilize an unstable pole), one can required that \(|S(j\omega)| \le \frac{1}{\sqrt{2}}\) below \(\omega_B^*\) by designing \(w_H\) such that \(|w_H(j\omega)| \ge \sqrt{2}\) for \(\omega \le \omega_B^*\). Similarly, if one wants the closed-loop bandwidth to be less than \(\w_B^*\), one can approximately require that the magnitude of \(T\) is less than \(\frac{1}{\sqrt{2}}\) at frequencies above \(\w_B^*\) by designing \(w_L\) such that \(|w_L(j\omega)| \ge \sqrt{2}\) for \(\omega \ge \omega_B^*\).\\ -\paragraph{Classical stability margins} -\label{sec:org7217eab} +\subsubsection{Classical stability margins} Gain margin (GM) and phase margin (PM) are usual specifications on controlled system. Minimum GM and PM can be guaranteed by limiting the maximum magnitude of the sensibility function \(M_S = \max_{\omega} |S(j\omega)|\): \begin{equation} @@ -1240,8 +1217,7 @@ For the nominal system \(M_S = \max_\omega |S| = \max_\omega |H_H|\), so one can |H_H(j\omega)| \le 2 \quad \forall\omega \end{equation} and thus obtain acceptable stability margins.\\ -\paragraph{Response time to change of reference signal} -\label{sec:orge54cf0a} +\subsubsection{Response time to change of reference signal} For the nominal system, the model is accurate and the transfer function from reference signal \(r\) to output \(y\) is \(1\) \eqref{eq:detail_control_cl_performance_y} and does not depends of the complementary filters. However, one can add a pre-filter as shown in Figure \ref{fig:detail_control_sf_arch_class_prefilter}. @@ -1258,8 +1234,7 @@ Typically, \(K_r\) is a low pass filter of the form K_r(s) = \frac{1}{1 + \tau s} \end{equation} with \(\tau\) corresponding to the desired response time.\\ -\paragraph{Input usage} -\label{sec:org2711dd4} +\subsubsection{Input usage} Input usage due to disturbances \(d_y\) and measurement noise \(n\) is determined by \(\big|\frac{u}{d_y}\big| = \big|\frac{u}{n}\big| = \big|G^{-1}H_L\big|\). Thus it can be limited by setting an upper bound on \(|H_L|\). @@ -1267,8 +1242,7 @@ Thus it can be limited by setting an upper bound on \(|H_L|\). Input usage due to reference signal \(r\) is determined by \(\big|\frac{u}{r}\big| = \big|G^{-1} K_r\big|\) when using a pre-filter (Figure \ref{fig:detail_control_sf_arch_class_prefilter}) and \(\big|\frac{u}{r}\big| = \big|G^{-1}\big|\) otherwise. Proper choice of \(|K_r|\) is then useful to limit input usage due to change of reference signal.\\ -\paragraph{Robust Stability (RS)} -\label{sec:orge7938b5} +\subsubsection{Robust Stability (RS)} Robustness stability represents the ability of the control system to remain stable even though there are differences between the actual system \(G^\prime\) and the model \(G\) that was used to design the controller. These differences can have various origins such as unmodelled dynamics or non-linearities. @@ -1307,8 +1281,7 @@ Robust stability is then guaranteed by having the low pass filter \(H_L\) satisf \end{equation} To ensure robust stability condition \eqref{eq:detail_control_nominal_perf_hl} can be used if \(w_L\) is designed in such a way that \(|w_L| \ge |w_I| (2 + |w_I|)\).\\ -\paragraph{Robust Performance (RP)} -\label{sec:orge9f7a36} +\subsubsection{Robust Performance (RP)} Robust performance is a property for a controlled system to have its performance guaranteed even though the dynamics of the plant is changing within specified bounds. For robust performance, we then require to have the performance condition valid for all possible plants in the defined uncertainty set: @@ -1345,13 +1318,10 @@ Robust performance is then guaranteed if \eqref{eq:detail_control_robust_perf_a} One should be aware than when looking for a robust performance condition, only the worst case is evaluated and using the robust stability condition may lead to conservative control. \section{Analytical formulas for complementary filters?} -\label{sec:org6512345} \label{ssec:detail_control_analytical_complementary_filters} \section{Numerical Example} -\label{sec:org16118e8} \label{ssec:detail_control_simulations} -\paragraph{Procedure} -\label{sec:org56ef466} +\subsubsection{Procedure} In order to apply this control technique, we propose the following procedure: \begin{enumerate} @@ -1365,8 +1335,7 @@ If one does not want to use the \(\mathcal{H}_\infty\) synthesis, one can use pr \item Design a pre-filter \(K_r\) if requirements on input usage or response to reference change are not met \item Control implementation: Filter the measurement with \(H_L\), implement the controller \(K\) and the pre-filter \(K_r\) as shown on Figure \ref{fig:detail_control_sf_arch_class_prefilter} \end{enumerate} -\paragraph{Plant} -\label{sec:orgdce9917} +\subsubsection{Plant} Let's consider the problem of controlling an active vibration isolation system that consist of a mass \(m\) to be isolated, a piezoelectric actuator and a geophone. We represent this system by a mass-spring-damper system as shown Figure \ref{fig:detail_control_mech_sys_alone} where \(m\) typically represents the mass of the payload to be isolated, \(k\) and \(c\) represent respectively the stiffness and damping of the mount. @@ -1392,8 +1361,7 @@ Its bode plot is shown on Figure \ref{fig:detail_control_bode_plot_mech_sys}. \includegraphics[scale=1]{figs/detail_control_bode_plot_mech_sys.png} \caption{\label{fig:detail_control_bode_plot_mech_sys}Bode plot of the transfer function \(G(s)\) from \(F\) to \(x\)} \end{figure} -\paragraph{Requirements} -\label{sec:org450d439} +\subsubsection{Requirements} The control objective is to isolate the displacement \(x\) of the mass from the ground motion \(w\). The disturbance rejection should be at least \(10\) at \(\SI{2}{\hertz}\) and with a slope of \(-2\) below \(\SI{2}{\hertz}\) until a rejection of \(10^4\). @@ -1430,8 +1398,7 @@ All the requirements on \(H_L\) and \(H_H\) are represented on Figure \ref{fig:d \end{subfigure} \caption{\label{fig:detail_control_spec_S_T_obtained_filters}Caption with reference to sub figure (\subref{fig:detail_control_spec_S_T}) (\subref{fig:detail_control_hinf_filters_result_weights})} \end{figure} -\paragraph{Design of the filters} -\label{sec:org571449a} +\subsubsection{Design of the filters} \textbf{Or maybe use analytical formulas as proposed here: \href{file:///home/thomas/Cloud/research/papers/dehaeze20\_virtu\_senso\_fusio/matlab/index.org}{Complementary filters using analytical formula}} @@ -1451,8 +1418,7 @@ After the \(\hinf\text{-synthesis}\), we obtain \(H_L\) and \(H_H\), and we plot H_H &= \frac{0.9936 (s+111.1) (s^2 + 0.3988s + 0.08464)}{(s+57.99) (s^2 + 65.77s + 2981)} \end{align} \end{subequations} -\paragraph{Controller analysis} -\label{sec:org418c661} +\subsubsection{Controller analysis} The controller is \(K = \left( H_H G \right)^{-1}\). A low pass filter is added to \(K\) so that it is proper and implementable. @@ -1481,8 +1447,7 @@ It is implemented as shown on Figure \ref{fig:detail_control_mech_sys_alone_ctrl \end{subfigure} \caption{\label{fig:detail_control_bode_Kfb_loop_gain}Caption with reference to sub figure (\subref{fig:detail_control_bode_Kfb}) (\subref{fig:detail_control_bode_plot_loop_gain_robustness})} \end{figure} -\paragraph{Robustness analysis} -\label{sec:orge5d85d1} +\subsubsection{Robustness analysis} The robust stability can be access on the nyquist plot (Figure \ref{fig:detail_control_nyquist_robustness}). The robust performance is shown on Figure \ref{fig:detail_control_robust_perf}. @@ -1503,12 +1468,10 @@ The robust performance is shown on Figure \ref{fig:detail_control_robust_perf}. \caption{\label{fig:fig_label}Caption with reference to sub figure (\subref{fig:detail_control_nyquist_robustness}) (\subref{fig:detail_control_robust_perf})} \end{figure} \section{Experimental Validation?} -\label{sec:org23a68bd} \label{ssec:detail_control_exp_validation} \href{file:///home/thomas/Cloud/research/papers/dehaeze20\_virtu\_senso\_fusio/matlab/index.org}{Experimental Validation} \section*{Conclusion} -\label{sec:orge845c83} \begin{itemize} \item[{$\square$}] Discuss how useful it is as the bandwidth can be changed in real time with analytical formulas of second order complementary filters. Maybe make a section about that. @@ -1522,7 +1485,6 @@ Maybe give analytical formulas of second order complementary filters in the digi \end{itemize} \end{itemize} \chapter*{Conclusion} -\label{sec:org08e0307} \label{sec:detail_control_conclusion} \printbibliography[heading=bibintoc,title={Bibliography}] \end{document}