Update Matlab tangled files
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@@ -44,10 +44,10 @@ The matlab code is accessible on [[https://zenodo.org/record/3894343][Zonodo]] a
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To run the Matlab code, go in the =matlab= directory and run the following Matlab files corresponding to each section.
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#+caption: Paper's sections and corresponding Matlab files
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| Sections | Matlab File |
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|------------------------------------+----------------------------|
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| Sections | Matlab File |
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|------------------------------------+--------------------------|
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| Section [[sec:system_description]] | =s1_system_description.m= |
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| Section [[sec:iff_pure_int]] | =s2_iff_pure_int.m= |
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| Section [[sec:iff_pure_int]] | =s2_iff_pure_int.m= |
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| Section [[sec:iff_pseudo_int]] | =s3_iff_hpf.m= |
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| Section [[sec:iff_parallel_stiffness]] | =s4_iff_kp.m= |
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| Section [[sec:comparison]] | =s5_act_damp_comparison.m= |
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@@ -60,16 +60,19 @@ To run the Matlab code, go in the =matlab= directory and run the following Matla
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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addpath('./matlab/');
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addpath('./src/');
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#+begin_src matlab :tangle no :noweb yes
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<<m-init-path>>
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#+end_src
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#+begin_src matlab :eval no :noweb yes
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<<m-init-path-tangle>>
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#+end_src
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** System description
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@@ -440,16 +443,19 @@ Two identical controllers $K_F$ are used to feedback each of the sensed force to
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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addpath('./matlab/');
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addpath('./src/');
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#+begin_src matlab :tangle no :noweb yes
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<<m-init-path>>
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#+end_src
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#+begin_src matlab :eval no :noweb yes
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<<m-init-path-tangle>>
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#+end_src
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** Plant Parameters
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@@ -764,16 +770,19 @@ It is shown that for non-null rotating speed, one pole is bound to the right-hal
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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addpath('./matlab/');
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addpath('./src/');
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#+begin_src matlab :tangle no :noweb yes
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<<m-init-path>>
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#+end_src
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#+begin_src matlab :eval no :noweb yes
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<<m-init-path-tangle>>
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#+end_src
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** Plant Parameters
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@@ -1160,16 +1169,19 @@ To find the optimum, the gain that maximize the simultaneous damping of the mode
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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addpath('./matlab/');
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addpath('./src/');
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#+begin_src matlab :tangle no :noweb yes
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<<m-init-path>>
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#+end_src
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#+begin_src matlab :eval no :noweb yes
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<<m-init-path-tangle>>
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#+end_src
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** Schematic
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@@ -1761,16 +1773,19 @@ These two methods are now compared in terms of added damping, closed-loop compli
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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addpath('./matlab/');
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addpath('./src/');
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#+begin_src matlab :tangle no :noweb yes
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<<m-init-path>>
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#+end_src
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#+begin_src matlab :eval no :noweb yes
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<<m-init-path-tangle>>
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#+end_src
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** Plant Parameters
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@@ -2143,10 +2158,23 @@ Critical Damping corresponds to to $\xi = 1$, and thus:
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#+latex: \printbibliography
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bibliography:ref.bib
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* Helping Functions :noexport:
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** Initialize Path
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#+NAME: m-init-path
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#+BEGIN_SRC matlab
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addpath('./matlab/'); % Path for scripts
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addpath('./matlab/src/'); % Path for functions
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#+END_SRC
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#+NAME: m-init-path-tangle
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#+BEGIN_SRC matlab
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addpath('./src/'); % Path for functions
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#+END_SRC
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* Functions :noexport:
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** Sort Poles for the Root Locus
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:PROPERTIES:
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:header-args:matlab+: :tangle src/rootLocusPolesSorted.m
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:header-args:matlab+: :tangle matlab/src/rootLocusPolesSorted.m
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:header-args:matlab+: :comments none :mkdirp yes :eval no
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:END:
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<<sec:rootLocusPolesSorted>>
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@@ -2291,7 +2319,7 @@ This Matlab function is accessible [[file:src/rootLocusPolesSorted.m][here]].
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#+end_src
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** =computeSimultaneousDamping=
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#+begin_src matlab :tangle src/computeSimultaneousDamping.m
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#+begin_src matlab :tangle matlab/src/computeSimultaneousDamping.m
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function [xi_min] = computeSimultaneousDamping(g, G, K)
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[w, xi] = damp(minreal(feedback(G, g*K)));
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xi_min = 1/min(xi);
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@@ -4,15 +4,17 @@ clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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addpath('./src/'); % Path for functions
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% Numerical Values
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% Let's define initial values for the model.
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k = 1; % Actuator Stiffness [N/m]
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c = 0.05; % Actuator Damping [N/(m/s)]
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m = 1; % Payload mass [kg]
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k = 1; % Actuator Stiffness [N/m]
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c = 0.05; % Actuator Damping [N/(m/s)]
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m = 1; % Payload mass [kg]
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xi = c/(2*sqrt(k*m));
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w0 = sqrt(k/m); % [rad/s]
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xi = c/(2*sqrt(k*m));
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w0 = sqrt(k/m); % [rad/s]
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% Campbell Diagram
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% The Campbell Diagram displays the evolution of the real and imaginary parts of the system as a function of the rotating speed.
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@@ -20,19 +22,19 @@ w0 = sqrt(k/m); % [rad/s]
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% It is shown in Figures [[fig:campbell_diagram_real]] and [[fig:campbell_diagram_imag]], and one can see that the system becomes unstable for $\Omega > \omega_0$ (the real part of one of the poles becomes positive).
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Ws = linspace(0, 2, 51); % Vector of considered rotation speed [rad/s]
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Ws = linspace(0, 2, 51); % Vector of considered rotation speed [rad/s]
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p_ws = zeros(4, length(Ws));
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p_ws = zeros(4, length(Ws));
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for W_i = 1:length(Ws)
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W = Ws(W_i);
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for W_i = 1:length(Ws)
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W = Ws(W_i);
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pole_G = pole(1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2));
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[~, i_sort] = sort(imag(pole_G));
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p_ws(:, W_i) = pole_G(i_sort);
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end
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pole_G = pole(1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2));
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[~, i_sort] = sort(imag(pole_G));
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p_ws(:, W_i) = pole_G(i_sort);
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end
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clear pole_G;
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clear pole_G;
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% Simscape Model
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% In order to validate all the equations of motion, a Simscape model of the same system has been developed.
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@@ -40,152 +42,152 @@ clear pole_G;
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% The rotating speed for the Simscape Model is defined.
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W = 0.1; % Rotation Speed [rad/s]
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W = 0.1; % Rotation Speed [rad/s]
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Kiff = tf(zeros(2));
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Kiff = tf(zeros(2));
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kp = 0; % Parallel Stiffness [N/m]
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cp = 0; % Parallel Damping [N/(m/s)]
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kp = 0; % Parallel Stiffness [N/m]
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cp = 0; % Parallel Damping [N/(m/s)]
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open('rotating_frame.slx');
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open('rotating_frame.slx');
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% The transfer function from $[F_u, F_v]$ to $[d_u, d_v]$ is identified from the Simscape model.
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%% Name of the Simulink File
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mdl = 'rotating_frame';
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%% Name of the Simulink File
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mdl = 'rotating_frame';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/G'], 2, 'openoutput'); io_i = io_i + 1;
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/G'], 2, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io, 0);
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G = linearize(mdl, io, 0);
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%% Input/Output definition
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G.InputName = {'Fu', 'Fv'};
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G.OutputName = {'du', 'dv'};
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%% Input/Output definition
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G.InputName = {'Fu', 'Fv'};
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G.OutputName = {'du', 'dv'};
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% The same transfer function from $[F_u, F_v]$ to $[d_u, d_v]$ is written down from the analytical model.
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Gth = (1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
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[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
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-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
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Gth = (1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
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[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
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-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)];
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% Both transfer functions are compared in Figure [[fig:plant_simscape_analytical]] and are found to perfectly match.
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freqs = logspace(-1, 1, 1000);
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freqs = logspace(-1, 1, 1000);
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figure;
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tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
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figure;
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tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
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% Magnitude
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ax1 = nexttile([2, 1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G(1,1), freqs))), '-')
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plot(freqs, abs(squeeze(freqresp(Gth(1,1), freqs))), '--')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
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title('$d_u/F_u$, $d_v/F_v$');
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% Magnitude
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ax1 = nexttile([2, 1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G(1,1), freqs))), '-')
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plot(freqs, abs(squeeze(freqresp(Gth(1,1), freqs))), '--')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
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title('$d_u/F_u$, $d_v/F_v$');
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ax2 = nexttile([2, 1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G(1,2), freqs))), '-')
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plot(freqs, abs(squeeze(freqresp(Gth(1,2), freqs))), '--')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
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title('$d_u/F_v$, $d_v/F_u$');
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ax2 = nexttile([2, 1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G(1,2), freqs))), '-')
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plot(freqs, abs(squeeze(freqresp(Gth(1,2), freqs))), '--')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
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title('$d_u/F_v$, $d_v/F_u$');
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ax3 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(1,1), freqs))), '-')
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gth(1,1), freqs))), '--')
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
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yticks(-180:90:180);
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ylim([-180 180]);
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hold off;
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ax3 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(1,1), freqs))), '-')
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gth(1,1), freqs))), '--')
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
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yticks(-180:90:180);
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ylim([-180 180]);
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hold off;
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ax4 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(1,2), freqs))), '-', ...
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'DisplayName', 'Simscape')
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gth(1,2), freqs))), '--', ...
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'DisplayName', 'Analytical')
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
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yticks(-180:90:180);
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ylim([-180 180]);
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hold off;
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legend('location', 'southwest', 'FontSize', 8);
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ax4 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G(1,2), freqs))), '-', ...
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'DisplayName', 'Simscape')
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gth(1,2), freqs))), '--', ...
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'DisplayName', 'Analytical')
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
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yticks(-180:90:180);
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ylim([-180 180]);
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hold off;
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legend('location', 'southwest', 'FontSize', 8);
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linkaxes([ax1,ax2,ax3,ax4],'x');
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xlim([freqs(1), freqs(end)]);
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linkaxes([ax1,ax2],'y');
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linkaxes([ax1,ax2,ax3,ax4],'x');
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xlim([freqs(1), freqs(end)]);
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linkaxes([ax1,ax2],'y');
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% Effect of the rotation speed
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% The transfer functions from $[F_u, F_v]$ to $[d_u, d_v]$ are identified for the following rotating speeds.
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Ws = [0, 0.2, 0.7, 1.1]*w0; % Rotating Speeds [rad/s]
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Ws = [0, 0.2, 0.7, 1.1]*w0; % Rotating Speeds [rad/s]
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Gs = {zeros(2, 2, length(Ws))};
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Gs = {zeros(2, 2, length(Ws))};
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for W_i = 1:length(Ws)
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W = Ws(W_i);
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for W_i = 1:length(Ws)
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W = Ws(W_i);
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Gs(:, :, W_i) = {(1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
|
||||
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)]};
|
||||
end
|
||||
Gs(:, :, W_i) = {(1/k)/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2), 2*W*s/(w0^2) ; ...
|
||||
-2*W*s/(w0^2), (s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)]};
|
||||
end
|
||||
|
||||
|
||||
|
||||
% They are compared in Figures [[fig:plant_compare_rotating_speed_direct]] and [[fig:plant_compare_rotating_speed_coupling]].
|
||||
|
||||
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gs{W_i}(1,1), freqs))), ...
|
||||
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Ws(W_i)/w0))
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
|
||||
leg = legend('location', 'southwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 6;
|
||||
ylim([1e-4, 1e2]);
|
||||
title('$d_u/F_u$, $d_v/F_v$');
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gs{W_i}(1,1), freqs))), ...
|
||||
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Ws(W_i)/w0))
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
|
||||
leg = legend('location', 'southwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 6;
|
||||
ylim([1e-4, 1e2]);
|
||||
title('$d_u/F_u$, $d_v/F_v$');
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{W_i}(1,1), freqs))))
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{W_i}(1,1), freqs))))
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
|
||||
|
||||
@@ -195,32 +197,32 @@ xlim([freqs(1), freqs(end)]);
|
||||
% [[file:figs/plant_compare_rotating_speed_direct.png]]
|
||||
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gs{W_i}(2,1), freqs))))
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
|
||||
ylim([1e-4, 1e2]);
|
||||
title('$d_u/F_v$, $d_v/F_u$');
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gs{W_i}(2,1), freqs))))
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
|
||||
ylim([1e-4, 1e2]);
|
||||
title('$d_u/F_v$, $d_v/F_u$');
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{W_i}(2,1), freqs))))
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{W_i}(2,1), freqs))))
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
+128
-126
@@ -4,160 +4,162 @@ clear; close all; clc;
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
addpath('./src/'); % Path for functions
|
||||
|
||||
% Plant Parameters
|
||||
% Let's define initial values for the model.
|
||||
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
|
||||
kp = 0; % [N/m]
|
||||
cp = 0; % [N/(m/s)]
|
||||
kp = 0; % [N/m]
|
||||
cp = 0; % [N/(m/s)]
|
||||
|
||||
% Comparison of the Analytical Model and the Simscape Model
|
||||
% The rotation speed is set to $\Omega = 0.1 \omega_0$.
|
||||
|
||||
W = 0.1*w0; % [rad/s]
|
||||
W = 0.1*w0; % [rad/s]
|
||||
|
||||
Kiff = tf(zeros(2));
|
||||
Kiff = tf(zeros(2));
|
||||
|
||||
open('rotating_frame.slx');
|
||||
open('rotating_frame.slx');
|
||||
|
||||
|
||||
|
||||
% And the transfer function from $[F_u, F_v]$ to $[f_u, f_v]$ is identified using the Simscape model.
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
Giff = linearize(mdl, io, 0);
|
||||
Giff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'fu', 'fv'};
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'fu', 'fv'};
|
||||
|
||||
|
||||
|
||||
% The same transfer function from $[F_u, F_v]$ to $[f_u, f_v]$ is written down from the analytical model.
|
||||
|
||||
Giff_th = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
Giff_th = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
|
||||
|
||||
|
||||
% The two are compared in Figure [[fig:plant_iff_comp_simscape_analytical]] and found to perfectly match.
|
||||
|
||||
freqs = logspace(-1, 1, 1000);
|
||||
freqs = logspace(-1, 1, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
figure;
|
||||
tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs))), '-')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_th(1,1), freqs))), '--')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
title('$f_u/F_u$, $f_v/F_v$');
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs))), '-')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_th(1,1), freqs))), '--')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
title('$f_u/F_u$, $f_v/F_v$');
|
||||
|
||||
ax2 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,2), freqs))), '-')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_th(1,2), freqs))), '--')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
title('$f_u/F_v$, $f_v/F_u$');
|
||||
ax2 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,2), freqs))), '-')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_th(1,2), freqs))), '--')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
title('$f_u/F_v$, $f_v/F_u$');
|
||||
|
||||
ax3 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs))), '-')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,1), freqs))), '--')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
ax3 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs))), '-')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,1), freqs))), '--')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
|
||||
ax4 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,2), freqs))), '-', ...
|
||||
'DisplayName', 'Simscape')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,2), freqs))), '--', ...
|
||||
'DisplayName', 'Analytical')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
legend('location', 'northeast', 'FontSize', 8);
|
||||
ax4 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,2), freqs))), '-', ...
|
||||
'DisplayName', 'Simscape')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,2), freqs))), '--', ...
|
||||
'DisplayName', 'Analytical')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
legend('location', 'northeast', 'FontSize', 8);
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'y');
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'y');
|
||||
|
||||
% Effect of the rotation speed
|
||||
% The transfer functions from $[F_u, F_v]$ to $[f_u, f_v]$ are identified for the following rotating speeds.
|
||||
|
||||
Ws = [0, 0.2, 0.7]*w0; % Rotating Speeds [rad/s]
|
||||
Ws = [0, 0.2, 0.7]*w0; % Rotating Speeds [rad/s]
|
||||
|
||||
Gsiff = {zeros(2, 2, length(Ws))};
|
||||
Gsiff = {zeros(2, 2, length(Ws))};
|
||||
|
||||
for W_i = 1:length(Ws)
|
||||
W = Ws(W_i);
|
||||
for W_i = 1:length(Ws)
|
||||
W = Ws(W_i);
|
||||
|
||||
Gsiff(:, :, W_i) = {1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]};
|
||||
end
|
||||
Gsiff(:, :, W_i) = {1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2]};
|
||||
end
|
||||
|
||||
|
||||
|
||||
% The obtained transfer functions are shown in Figure [[fig:plant_iff_compare_rotating_speed]].
|
||||
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gsiff{W_i}(1,1), freqs))), ...
|
||||
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Ws(W_i)/w0))
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
leg = legend('location', 'southeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 6;
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, abs(squeeze(freqresp(Gsiff{W_i}(1,1), freqs))), ...
|
||||
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Ws(W_i)/w0))
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
leg = legend('location', 'southeast', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 6;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsiff{W_i}(1,1), freqs))))
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsiff{W_i}(1,1), freqs))))
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
% Decentralized Integral Force Feedback
|
||||
% The decentralized IFF controller consists of pure integrators:
|
||||
@@ -172,29 +174,29 @@ xlim([freqs(1), freqs(end)]);
|
||||
% It is shown that for non-null rotating speed, one pole is bound to the right-half plane, and thus the closed loop system is unstable.
|
||||
|
||||
|
||||
figure;
|
||||
figure;
|
||||
|
||||
gains = logspace(-2, 4, 100);
|
||||
gains = logspace(-2, 4, 100);
|
||||
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
set(gca,'ColorOrderIndex',W_i);
|
||||
plot(real(pole(Gsiff{W_i})), imag(pole(Gsiff{W_i})), 'x', ...
|
||||
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Ws(W_i)/w0));
|
||||
set(gca,'ColorOrderIndex',W_i);
|
||||
plot(real(tzero(Gsiff{W_i})), imag(tzero(Gsiff{W_i})), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
set(gca,'ColorOrderIndex',W_i);
|
||||
cl_poles = pole(feedback(Gsiff{W_i}, g/s*eye(2)));
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-2, 0.5]); ylim([0, 2.5]);
|
||||
hold on;
|
||||
for W_i = 1:length(Ws)
|
||||
set(gca,'ColorOrderIndex',W_i);
|
||||
plot(real(pole(Gsiff{W_i})), imag(pole(Gsiff{W_i})), 'x', ...
|
||||
'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Ws(W_i)/w0));
|
||||
set(gca,'ColorOrderIndex',W_i);
|
||||
plot(real(tzero(Gsiff{W_i})), imag(tzero(Gsiff{W_i})), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
set(gca,'ColorOrderIndex',W_i);
|
||||
cl_poles = pole(feedback(Gsiff{W_i}, g/s*eye(2)));
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-2, 0.5]); ylim([0, 2.5]);
|
||||
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
+128
-126
@@ -4,18 +4,20 @@ clear; close all; clc;
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
addpath('./src/'); % Path for functions
|
||||
|
||||
% Plant Parameters
|
||||
% Let's define initial values for the model.
|
||||
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
|
||||
kp = 0; % [N/m]
|
||||
cp = 0; % [N/(m/s)]
|
||||
kp = 0; % [N/m]
|
||||
cp = 0; % [N/(m/s)]
|
||||
|
||||
% Modified Integral Force Feedback Controller
|
||||
% Let's modify the initial Integral Force Feedback Controller ; instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:
|
||||
@@ -29,93 +31,93 @@ cp = 0; % [N/(m/s)]
|
||||
|
||||
% Let's arbitrary choose the following control parameters:
|
||||
|
||||
g = 2;
|
||||
wi = 0.1*w0;
|
||||
g = 2;
|
||||
wi = 0.1*w0;
|
||||
|
||||
Kiff = (g/(wi+s))*eye(2);
|
||||
Kiff = (g/(wi+s))*eye(2);
|
||||
|
||||
|
||||
|
||||
% And the following rotating speed.
|
||||
|
||||
W = 0.1*w0;
|
||||
W = 0.1*w0;
|
||||
|
||||
Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
|
||||
|
||||
|
||||
% The obtained Loop Gain is shown in Figure [[fig:loop_gain_modified_iff]].
|
||||
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1)*(g/s), freqs))))
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1)*Kiff(1,1), freqs))))
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Loop Gain');
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1)*(g/s), freqs))))
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1)*Kiff(1,1), freqs))))
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Loop Gain');
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1)*(g/s), freqs))), ...
|
||||
'DisplayName', 'IFF')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1)*Kiff(1,1), freqs))), ...
|
||||
'DisplayName', 'IFF + HPF')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
leg = legend('location', 'southwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 6;
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1)*(g/s), freqs))), ...
|
||||
'DisplayName', 'IFF')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1)*Kiff(1,1), freqs))), ...
|
||||
'DisplayName', 'IFF + HPF')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
leg = legend('location', 'southwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 6;
|
||||
|
||||
hold off;
|
||||
hold off;
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
% Root Locus
|
||||
% As shown in the Root Locus plot (Figure [[fig:root_locus_modified_iff]]), for some value of the gain, the system remains stable.
|
||||
|
||||
|
||||
figure;
|
||||
figure;
|
||||
|
||||
gains = logspace(-2, 4, 100);
|
||||
gains = logspace(-2, 4, 100);
|
||||
|
||||
hold on;
|
||||
% Pure Integrator
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', 'DisplayName', 'IFF');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', 'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
clpoles = pole(feedback(Giff, (g/s)*eye(2)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off');
|
||||
end
|
||||
% Modified IFF
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', 'DisplayName', 'IFF + HPF');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', 'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
clpoles = pole(feedback(Giff, (g/(wi+s))*eye(2)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-2, 0.5]); ylim([-1.25, 1.25]);
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
hold on;
|
||||
% Pure Integrator
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', 'DisplayName', 'IFF');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', 'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
clpoles = pole(feedback(Giff, (g/s)*eye(2)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off');
|
||||
end
|
||||
% Modified IFF
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', 'DisplayName', 'IFF + HPF');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', 'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
clpoles = pole(feedback(Giff, (g/(wi+s))*eye(2)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-2, 0.5]); ylim([-1.25, 1.25]);
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
|
||||
|
||||
|
||||
@@ -125,40 +127,40 @@ xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
% [[file:figs/root_locus_modified_iff.png]]
|
||||
|
||||
|
||||
xlim([-0.2, 0.1]); ylim([-0.15, 0.15]);
|
||||
legend('hide');
|
||||
xlim([-0.2, 0.1]); ylim([-0.15, 0.15]);
|
||||
legend('hide');
|
||||
|
||||
% What is the optimal $\omega_i$ and $g$?
|
||||
% In order to visualize the effect of $\omega_i$ on the attainable damping, the Root Locus is displayed in Figure [[fig:root_locus_wi_modified_iff]] for the following $\omega_i$:
|
||||
|
||||
wis = [0.01, 0.1, 0.5, 1]*w0; % [rad/s]
|
||||
wis = [0.01, 0.1, 0.5, 1]*w0; % [rad/s]
|
||||
|
||||
figure;
|
||||
figure;
|
||||
|
||||
gains = logspace(-2, 4, 100);
|
||||
gains = logspace(-2, 4, 100);
|
||||
|
||||
hold on;
|
||||
for wi_i = 1:length(wis)
|
||||
set(gca,'ColorOrderIndex',wi_i);
|
||||
wi = wis(wi_i);
|
||||
L(wi_i) = plot(nan, nan, '.', 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wi./w0));
|
||||
for g = gains
|
||||
clpoles = pole(feedback(Giff, (g/(wi+s))*eye(2)));
|
||||
set(gca,'ColorOrderIndex',wi_i);
|
||||
plot(real(clpoles), imag(clpoles), '.');
|
||||
end
|
||||
end
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'kx');
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko');
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-2.3, 0.1]); ylim([-1.2, 1.2]);
|
||||
xticks([-2:1:2]); yticks([-2:1:2]);
|
||||
leg = legend(L, 'location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
hold on;
|
||||
for wi_i = 1:length(wis)
|
||||
set(gca,'ColorOrderIndex',wi_i);
|
||||
wi = wis(wi_i);
|
||||
L(wi_i) = plot(nan, nan, '.', 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wi./w0));
|
||||
for g = gains
|
||||
clpoles = pole(feedback(Giff, (g/(wi+s))*eye(2)));
|
||||
set(gca,'ColorOrderIndex',wi_i);
|
||||
plot(real(clpoles), imag(clpoles), '.');
|
||||
end
|
||||
end
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'kx');
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'ko');
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-2.3, 0.1]); ylim([-1.2, 1.2]);
|
||||
xticks([-2:1:2]); yticks([-2:1:2]);
|
||||
leg = legend(L, 'location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
|
||||
clear L
|
||||
clear L
|
||||
|
||||
|
||||
|
||||
@@ -168,9 +170,9 @@ clear L
|
||||
% [[file:figs/root_locus_wi_modified_iff.png]]
|
||||
|
||||
|
||||
xlim([-0.2, 0.1]); ylim([-0.15, 0.15]);
|
||||
xticks([-0.2:0.1:0.1]); yticks([-0.2:0.1:0.2]);
|
||||
legend('hide');
|
||||
xlim([-0.2, 0.1]); ylim([-0.15, 0.15]);
|
||||
xticks([-0.2:0.1:0.1]); yticks([-0.2:0.1:0.2]);
|
||||
legend('hide');
|
||||
|
||||
|
||||
|
||||
@@ -192,37 +194,37 @@ legend('hide');
|
||||
% To find the optimum, the gain that maximize the simultaneous damping of the mode is identified for a wide range of $\omega_i$ (Figure [[fig:mod_iff_damping_wi]]).
|
||||
|
||||
|
||||
wis = logspace(-2, 1, 100)*w0; % [rad/s]
|
||||
wis = logspace(-2, 1, 100)*w0; % [rad/s]
|
||||
|
||||
opt_xi = zeros(1, length(wis)); % Optimal simultaneous damping
|
||||
opt_gain = zeros(1, length(wis)); % Corresponding optimal gain
|
||||
opt_xi = zeros(1, length(wis)); % Optimal simultaneous damping
|
||||
opt_gain = zeros(1, length(wis)); % Corresponding optimal gain
|
||||
|
||||
for wi_i = 1:length(wis)
|
||||
wi = wis(wi_i);
|
||||
Kiff = 1/(s + wi)*eye(2);
|
||||
for wi_i = 1:length(wis)
|
||||
wi = wis(wi_i);
|
||||
Kiff = 1/(s + wi)*eye(2);
|
||||
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
|
||||
|
||||
[g_opt, xi_opt] = fminsearch(fun, 0.5*wi*((w0/W)^2 - 1));
|
||||
opt_xi(wi_i) = 1/xi_opt;
|
||||
opt_gain(wi_i) = g_opt;
|
||||
end
|
||||
[g_opt, xi_opt] = fminsearch(fun, 0.5*wi*((w0/W)^2 - 1));
|
||||
opt_xi(wi_i) = 1/xi_opt;
|
||||
opt_gain(wi_i) = g_opt;
|
||||
end
|
||||
|
||||
figure;
|
||||
yyaxis left
|
||||
plot(wis, opt_xi, '-', 'DisplayName', '$\xi_{cl}$');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,1]);
|
||||
ylabel('Damping Ratio $\xi$');
|
||||
figure;
|
||||
yyaxis left
|
||||
plot(wis, opt_xi, '-', 'DisplayName', '$\xi_{cl}$');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,1]);
|
||||
ylabel('Damping Ratio $\xi$');
|
||||
|
||||
yyaxis right
|
||||
hold on;
|
||||
plot(wis, opt_gain, '-', 'DisplayName', '$g_{opt}$');
|
||||
plot(wis, wis*((w0/W)^2 - 1), '--', 'DisplayName', '$g_{max}$');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,10]);
|
||||
ylabel('Controller gain $g$');
|
||||
yyaxis right
|
||||
hold on;
|
||||
plot(wis, opt_gain, '-', 'DisplayName', '$g_{opt}$');
|
||||
plot(wis, wis*((w0/W)^2 - 1), '--', 'DisplayName', '$g_{max}$');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,10]);
|
||||
ylabel('Controller gain $g$');
|
||||
|
||||
xlabel('$\omega_i/\omega_0$');
|
||||
set(gca, 'XScale', 'log');
|
||||
legend('location', 'northeast', 'FontSize', 8);
|
||||
xlabel('$\omega_i/\omega_0$');
|
||||
set(gca, 'XScale', 'log');
|
||||
legend('location', 'northeast', 'FontSize', 8);
|
||||
+236
-234
@@ -4,108 +4,110 @@ clear; close all; clc;
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
addpath('./src/'); % Path for functions
|
||||
|
||||
% Plant Parameters
|
||||
% Let's define initial values for the model.
|
||||
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
|
||||
kp = 0; % [N/m]
|
||||
cp = 0; % [N/(m/s)]
|
||||
kp = 0; % [N/m]
|
||||
cp = 0; % [N/(m/s)]
|
||||
|
||||
% Comparison of the Analytical Model and the Simscape Model
|
||||
% The same transfer function from $[F_u, F_v]$ to $[f_u, f_v]$ is written down from the analytical model.
|
||||
|
||||
W = 0.1*w0; % [rad/s]
|
||||
W = 0.1*w0; % [rad/s]
|
||||
|
||||
kp = 1.5*m*W^2;
|
||||
cp = 0;
|
||||
kp = 1.5*m*W^2;
|
||||
cp = 0;
|
||||
|
||||
Kiff = tf(zeros(2));
|
||||
Kiff = tf(zeros(2));
|
||||
|
||||
open('rotating_frame.slx');
|
||||
open('rotating_frame.slx');
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/K'], 1, 'openinput'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/G'], 1, 'openoutput'); io_i = io_i + 1;
|
||||
|
||||
Giff = linearize(mdl, io, 0);
|
||||
Giff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'fu', 'fv'};
|
||||
%% Input/Output definition
|
||||
Giff.InputName = {'Fu', 'Fv'};
|
||||
Giff.OutputName = {'fu', 'fv'};
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_th = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
Giff_th.InputName = {'Fu', 'Fv'};
|
||||
Giff_th.OutputName = {'fu', 'fv'};
|
||||
Giff_th = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
Giff_th.InputName = {'Fu', 'Fv'};
|
||||
Giff_th.OutputName = {'fu', 'fv'};
|
||||
|
||||
freqs = logspace(-1, 1, 1000);
|
||||
freqs = logspace(-1, 1, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
figure;
|
||||
tiledlayout(3, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs))), '-')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_th(1,1), freqs))), '--')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
title('$f_u/F_u$, $f_v/F_v$');
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs))), '-')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_th(1,1), freqs))), '--')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
title('$f_u/F_u$, $f_v/F_v$');
|
||||
|
||||
ax2 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,2), freqs))), '-')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_th(1,2), freqs))), '--')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
title('$f_u/F_v$, $f_v/F_u$');
|
||||
ax2 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,2), freqs))), '-')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_th(1,2), freqs))), '--')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
title('$f_u/F_v$, $f_v/F_u$');
|
||||
|
||||
ax3 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs))), '-', ...
|
||||
'DisplayName', 'Simscape')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,1), freqs))), '--', ...
|
||||
'DisplayName', 'Analytical')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
legend('location', 'southwest', 'FontSize', 8);
|
||||
ax3 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs))), '-', ...
|
||||
'DisplayName', 'Simscape')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,1), freqs))), '--', ...
|
||||
'DisplayName', 'Analytical')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
legend('location', 'southwest', 'FontSize', 8);
|
||||
|
||||
ax4 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,2), freqs))), '-')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,2), freqs))), '--')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
ax4 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,2), freqs))), '-')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_th(1,2), freqs))), '--')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'y');
|
||||
linkaxes([ax1,ax2,ax3,ax4],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'y');
|
||||
|
||||
% Effect of the parallel stiffness on the IFF plant
|
||||
% The rotation speed is set to $\Omega = 0.1 \omega_0$.
|
||||
|
||||
W = 0.1*w0; % [rad/s]
|
||||
W = 0.1*w0; % [rad/s]
|
||||
|
||||
|
||||
|
||||
@@ -119,69 +121,69 @@ W = 0.1*w0; % [rad/s]
|
||||
% One can see that for $k_p > m \Omega^2$, the systems shows alternating complex conjugate poles and zeros.
|
||||
|
||||
|
||||
kp = 0;
|
||||
kp = 0;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
|
||||
kp = 0.5*m*W^2;
|
||||
k = 1 - kp;
|
||||
kp = 0.5*m*W^2;
|
||||
k = 1 - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_s = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
Giff_s = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
|
||||
kp = 1.5*m*W^2;
|
||||
k = 1 - kp;
|
||||
kp = 1.5*m*W^2;
|
||||
k = 1 - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_l = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
Giff_l = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs))), 'k-', ...
|
||||
'DisplayName', '$k_p = 0$')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_s(1,1), freqs))), 'k--', ...
|
||||
'DisplayName', '$k_p < m\Omega^2$')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_l(1,1), freqs))), 'k:', ...
|
||||
'DisplayName', '$k_p > m\Omega^2$')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
ylim([1e-4, 2e1]);
|
||||
legend('location', 'southeast', 'FontSize', 8);
|
||||
% Magnitude
|
||||
ax1 = nexttile([2, 1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs))), 'k-', ...
|
||||
'DisplayName', '$k_p = 0$')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_s(1,1), freqs))), 'k--', ...
|
||||
'DisplayName', '$k_p < m\Omega^2$')
|
||||
plot(freqs, abs(squeeze(freqresp(Giff_l(1,1), freqs))), 'k:', ...
|
||||
'DisplayName', '$k_p > m\Omega^2$')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
|
||||
ylim([1e-4, 2e1]);
|
||||
legend('location', 'southeast', 'FontSize', 8);
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs))), 'k-')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_s(1,1), freqs))), 'k--')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_l(1,1), freqs))), 'k:')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(1,1), freqs))), 'k-')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_s(1,1), freqs))), 'k--')
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Giff_l(1,1), freqs))), 'k:')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
hold off;
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
|
||||
% IFF when adding a spring in parallel
|
||||
% In Figure [[fig:root_locus_iff_kp]] is displayed the Root Locus in the three considered cases with
|
||||
@@ -200,55 +202,55 @@ xlim([freqs(1), freqs(end)]);
|
||||
% \end{equation}
|
||||
|
||||
|
||||
gains = logspace(-2, 2, 100);
|
||||
gains = logspace(-2, 2, 100);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', ...
|
||||
'DisplayName', '$k_p = 0$');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
cl_poles = pole(feedback(Giff, (g/s)*eye(2)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', ...
|
||||
'DisplayName', '$k_p = 0$');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
cl_poles = pole(feedback(Giff, (g/s)*eye(2)));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(pole(Giff_s)), imag(pole(Giff_s)), 'x', ...
|
||||
'DisplayName', '$k_p < m\Omega^2$');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(tzero(Giff_s)), imag(tzero(Giff_s)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
cl_poles = pole(feedback(Giff_s, (g/s)*eye(2)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(pole(Giff_s)), imag(pole(Giff_s)), 'x', ...
|
||||
'DisplayName', '$k_p < m\Omega^2$');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(tzero(Giff_s)), imag(tzero(Giff_s)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
cl_poles = pole(feedback(Giff_s, (g/s)*eye(2)));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
set(gca,'ColorOrderIndex',3);
|
||||
plot(real(pole(Giff_l)), imag(pole(Giff_l)), 'x', ...
|
||||
'DisplayName', '$k_p > m\Omega^2$');
|
||||
set(gca,'ColorOrderIndex',3);
|
||||
plot(real(tzero(Giff_l)), imag(tzero(Giff_l)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
set(gca,'ColorOrderIndex',3);
|
||||
cl_poles = pole(feedback(Giff_l, (g/s)*eye(2)));
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-1, 0.2]); ylim([0, 1.2]);
|
||||
set(gca,'ColorOrderIndex',3);
|
||||
plot(real(pole(Giff_l)), imag(pole(Giff_l)), 'x', ...
|
||||
'DisplayName', '$k_p > m\Omega^2$');
|
||||
set(gca,'ColorOrderIndex',3);
|
||||
plot(real(tzero(Giff_l)), imag(tzero(Giff_l)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
set(gca,'ColorOrderIndex',3);
|
||||
cl_poles = pole(feedback(Giff_l, (g/s)*eye(2)));
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-1, 0.2]); ylim([0, 1.2]);
|
||||
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
|
||||
|
||||
|
||||
@@ -258,97 +260,97 @@ leg.ItemTokenSize(1) = 8;
|
||||
% [[file:figs/root_locus_iff_kp.png]]
|
||||
|
||||
|
||||
xlim([-0.04, 0.06]); ylim([0, 0.1]);
|
||||
legend('hide');
|
||||
xlim([-0.04, 0.06]); ylim([0, 0.1]);
|
||||
legend('hide');
|
||||
|
||||
% Effect of $k_p$ on the attainable damping
|
||||
% However, having large values of $k_p$ may decrease the attainable damping.
|
||||
|
||||
% To study the second point, Root Locus plots for the following values of $k_p$ are shown in Figure [[fig:root_locus_iff_kps]].
|
||||
|
||||
kps = [2, 20, 40]*m*W^2;
|
||||
kps = [2, 20, 40]*m*W^2;
|
||||
|
||||
|
||||
|
||||
% It is shown that large values of $k_p$ decreases the attainable damping.
|
||||
|
||||
figure;
|
||||
figure;
|
||||
|
||||
gains = logspace(-2, 4, 500);
|
||||
gains = logspace(-2, 4, 500);
|
||||
|
||||
hold on;
|
||||
for kp_i = 1:length(kps)
|
||||
kp = kps(kp_i);
|
||||
k = 1 - kp;
|
||||
hold on;
|
||||
for kp_i = 1:length(kps)
|
||||
kp = kps(kp_i);
|
||||
k = 1 - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
|
||||
set(gca,'ColorOrderIndex',kp_i);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', ...
|
||||
'DisplayName', sprintf('$k_p = %.1f m \\Omega^2$', kp/(m*W^2)));
|
||||
set(gca,'ColorOrderIndex',kp_i);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
Kiffa = (g/s)*eye(2);
|
||||
cl_poles = pole(feedback(Giff, Kiffa));
|
||||
set(gca,'ColorOrderIndex',kp_i);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-1.2, 0.2]); ylim([0, 1.4]);
|
||||
set(gca,'ColorOrderIndex',kp_i);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', ...
|
||||
'DisplayName', sprintf('$k_p = %.1f m \\Omega^2$', kp/(m*W^2)));
|
||||
set(gca,'ColorOrderIndex',kp_i);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
Kiffa = (g/s)*eye(2);
|
||||
cl_poles = pole(feedback(Giff, Kiffa));
|
||||
set(gca,'ColorOrderIndex',kp_i);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-1.2, 0.2]); ylim([0, 1.4]);
|
||||
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
|
||||
alphas = logspace(-2, 0, 100);
|
||||
alphas = logspace(-2, 0, 100);
|
||||
|
||||
opt_xi = zeros(1, length(alphas)); % Optimal simultaneous damping
|
||||
opt_gain = zeros(1, length(alphas)); % Corresponding optimal gain
|
||||
opt_xi = zeros(1, length(alphas)); % Optimal simultaneous damping
|
||||
opt_gain = zeros(1, length(alphas)); % Corresponding optimal gain
|
||||
|
||||
Kiff = 1/s*eye(2);
|
||||
Kiff = 1/s*eye(2);
|
||||
|
||||
for alpha_i = 1:length(alphas)
|
||||
kp = alphas(alpha_i);
|
||||
k = 1 - alphas(alpha_i);
|
||||
for alpha_i = 1:length(alphas)
|
||||
kp = alphas(alpha_i);
|
||||
k = 1 - alphas(alpha_i);
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
Giff = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2];
|
||||
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, Kiff);
|
||||
|
||||
[g_opt, xi_opt] = fminsearch(fun, 2);
|
||||
opt_xi(alpha_i) = 1/xi_opt;
|
||||
opt_gain(alpha_i) = g_opt;
|
||||
end
|
||||
[g_opt, xi_opt] = fminsearch(fun, 2);
|
||||
opt_xi(alpha_i) = 1/xi_opt;
|
||||
opt_gain(alpha_i) = g_opt;
|
||||
end
|
||||
|
||||
figure;
|
||||
yyaxis left
|
||||
plot(alphas, opt_xi, '-', 'DisplayName', '$\xi_{cl}$');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,1]);
|
||||
ylabel('Damping Ratio $\xi$');
|
||||
figure;
|
||||
yyaxis left
|
||||
plot(alphas, opt_xi, '-', 'DisplayName', '$\xi_{cl}$');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,1]);
|
||||
ylabel('Damping Ratio $\xi$');
|
||||
|
||||
yyaxis right
|
||||
hold on;
|
||||
plot(alphas, opt_gain, '-', 'DisplayName', '$g_{opt}$');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,2.5]);
|
||||
ylabel('Controller gain $g$');
|
||||
yyaxis right
|
||||
hold on;
|
||||
plot(alphas, opt_gain, '-', 'DisplayName', '$g_{opt}$');
|
||||
set(gca, 'YScale', 'lin');
|
||||
ylim([0,2.5]);
|
||||
ylabel('Controller gain $g$');
|
||||
|
||||
xlabel('$\alpha$');
|
||||
set(gca, 'XScale', 'log');
|
||||
legend('location', 'northeast', 'FontSize', 8);
|
||||
xlabel('$\alpha$');
|
||||
set(gca, 'XScale', 'log');
|
||||
legend('location', 'northeast', 'FontSize', 8);
|
||||
@@ -4,103 +4,105 @@ clear; close all; clc;
|
||||
%% Intialize Laplace variable
|
||||
s = zpk('s');
|
||||
|
||||
addpath('./src/'); % Path for functions
|
||||
|
||||
% Plant Parameters
|
||||
% Let's define initial values for the model.
|
||||
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
k = 1; % Actuator Stiffness [N/m]
|
||||
c = 0.05; % Actuator Damping [N/(m/s)]
|
||||
m = 1; % Payload mass [kg]
|
||||
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
xi = c/(2*sqrt(k*m));
|
||||
w0 = sqrt(k/m); % [rad/s]
|
||||
|
||||
kp = 0; % [N/m]
|
||||
cp = 0; % [N/(m/s)]
|
||||
kp = 0; % [N/m]
|
||||
cp = 0; % [N/(m/s)]
|
||||
|
||||
|
||||
|
||||
% The rotating speed is set to $\Omega = 0.1 \omega_0$.
|
||||
|
||||
W = 0.1*w0;
|
||||
W = 0.1*w0;
|
||||
|
||||
% Root Locus
|
||||
% IFF with High Pass Filter
|
||||
|
||||
wi = 0.1*w0; % [rad/s]
|
||||
wi = 0.1*w0; % [rad/s]
|
||||
|
||||
Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
Giff = 1/(((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))^2 + (2*W*s/(w0^2))^2) * ...
|
||||
[(s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2)) + (2*W*s/(w0^2))^2, - (2*xi*s/w0 + 1)*2*W*s/(w0^2) ; ...
|
||||
(2*xi*s/w0 + 1)*2*W*s/(w0^2), (s^2/w0^2 - W^2/w0^2)*((s^2)/(w0^2) + 2*xi*s/w0 + 1 - (W^2)/(w0^2))+ (2*W*s/(w0^2))^2];
|
||||
|
||||
|
||||
|
||||
% IFF With parallel Stiffness
|
||||
|
||||
kp = 5*m*W^2;
|
||||
k = k - kp;
|
||||
kp = 5*m*W^2;
|
||||
k = k - kp;
|
||||
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
w0p = sqrt((k + kp)/m);
|
||||
xip = c/(2*sqrt((k+kp)*m));
|
||||
|
||||
Giff_kp = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
Giff_kp = 1/( (s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2)^2 + (2*(s/w0p)*(W/w0p))^2 ) * [ ...
|
||||
(s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2, -(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p));
|
||||
(2*xip*s/w0p + k/(k + kp))*(2*(s/w0p)*(W/w0p)), (s^2/w0p^2 + kp/(k + kp) - W^2/w0p^2)*(s^2/w0p^2 + 2*xip*s/w0p + 1 - W^2/w0p^2) + (2*(s/w0p)*(W/w0p))^2 ];
|
||||
|
||||
k = k + kp;
|
||||
k = k + kp;
|
||||
|
||||
figure;
|
||||
figure;
|
||||
|
||||
gains = logspace(-2, 2, 100);
|
||||
gains = logspace(-2, 2, 100);
|
||||
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', ...
|
||||
'DisplayName', 'IFF + LFP');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
Kiff = (g/(wi + s))*eye(2);
|
||||
cl_poles = pole(feedback(Giff, Kiff));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(pole(Giff)), imag(pole(Giff)), 'x', ...
|
||||
'DisplayName', 'IFF + LFP');
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
Kiff = (g/(wi + s))*eye(2);
|
||||
cl_poles = pole(feedback(Giff, Kiff));
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(pole(Giff_kp)), imag(pole(Giff_kp)), 'x', ...
|
||||
'DisplayName', 'IFF + $k_p$');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(tzero(Giff_kp)), imag(tzero(Giff_kp)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
Kiffa = (g/s)*eye(2);
|
||||
cl_poles = pole(feedback(Giff_kp, Kiffa));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-1.2, 0.05]); ylim([0, 1.25]);
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(pole(Giff_kp)), imag(pole(Giff_kp)), 'x', ...
|
||||
'DisplayName', 'IFF + $k_p$');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(tzero(Giff_kp)), imag(tzero(Giff_kp)), 'o', ...
|
||||
'HandleVisibility', 'off');
|
||||
for g = gains
|
||||
Kiffa = (g/s)*eye(2);
|
||||
cl_poles = pole(feedback(Giff_kp, Kiffa));
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(real(cl_poles), imag(cl_poles), '.', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
axis square;
|
||||
xlim([-1.2, 0.05]); ylim([0, 1.25]);
|
||||
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
xlabel('Real Part'); ylabel('Imaginary Part');
|
||||
leg = legend('location', 'northwest', 'FontSize', 8);
|
||||
leg.ItemTokenSize(1) = 8;
|
||||
|
||||
% Controllers - Optimal Gains
|
||||
% In order to compare to three considered Active Damping techniques, gains that yield maximum damping of all the modes are computed for each case.
|
||||
|
||||
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, (1/(wi+s))*eye(2));
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff, (1/(wi+s))*eye(2));
|
||||
|
||||
[opt_gain_iff, opt_xi_iff] = fminsearch(fun, 0.5*(w0^2/W^2 - 1)*wi);
|
||||
opt_xi_iff = 1/opt_xi_iff;
|
||||
[opt_gain_iff, opt_xi_iff] = fminsearch(fun, 0.5*(w0^2/W^2 - 1)*wi);
|
||||
opt_xi_iff = 1/opt_xi_iff;
|
||||
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff_kp, 1/s*eye(2));
|
||||
fun = @(g)computeSimultaneousDamping(g, Giff_kp, 1/s*eye(2));
|
||||
|
||||
[opt_gain_kp, opt_xi_kp] = fminsearch(fun, 2);
|
||||
opt_xi_kp = 1/opt_xi_kp;
|
||||
[opt_gain_kp, opt_xi_kp] = fminsearch(fun, 2);
|
||||
opt_xi_kp = 1/opt_xi_kp;
|
||||
|
||||
% Passive Damping - Critical Damping
|
||||
% \begin{equation}
|
||||
@@ -113,116 +115,116 @@ opt_xi_kp = 1/opt_xi_kp;
|
||||
% \end{equation}
|
||||
|
||||
|
||||
c_opt = 2*sqrt(k*m);
|
||||
c_opt = 2*sqrt(k*m);
|
||||
|
||||
% Transmissibility And Compliance
|
||||
% <<sec:comp_transmissibilty>>
|
||||
|
||||
|
||||
open('rotating_frame.slx');
|
||||
open('rotating_frame.slx');
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
%% Name of the Simulink File
|
||||
mdl = 'rotating_frame';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/dw'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/fd'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Meas'], 1, 'output'); io_i = io_i + 1;
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/dw'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/fd'], 1, 'input'); io_i = io_i + 1;
|
||||
io(io_i) = linio([mdl, '/Meas'], 1, 'output'); io_i = io_i + 1;
|
||||
|
||||
% Open Loop :ignore:
|
||||
|
||||
Kiff = tf(zeros(2));
|
||||
Kiff = tf(zeros(2));
|
||||
|
||||
kp = 0;
|
||||
cp = 0;
|
||||
kp = 0;
|
||||
cp = 0;
|
||||
|
||||
G_ol = linearize(mdl, io, 0);
|
||||
G_ol = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_ol.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_ol.OutputName = {'Dx', 'Dy'};
|
||||
%% Input/Output definition
|
||||
G_ol.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_ol.OutputName = {'Dx', 'Dy'};
|
||||
|
||||
% Passive Damping
|
||||
|
||||
kp = 0;
|
||||
cp = 0;
|
||||
kp = 0;
|
||||
cp = 0;
|
||||
|
||||
c_old = c;
|
||||
c = c_opt;
|
||||
c_old = c;
|
||||
c = c_opt;
|
||||
|
||||
Kiff = tf(zeros(2));
|
||||
Kiff = tf(zeros(2));
|
||||
|
||||
G_pas = linearize(mdl, io, 0);
|
||||
G_pas = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_pas.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_pas.OutputName = {'Dx', 'Dy'};
|
||||
%% Input/Output definition
|
||||
G_pas.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_pas.OutputName = {'Dx', 'Dy'};
|
||||
|
||||
c = c_old;
|
||||
c = c_old;
|
||||
|
||||
% Pseudo Integrator IFF :ignore:
|
||||
|
||||
kp = 0;
|
||||
cp = 0;
|
||||
kp = 0;
|
||||
cp = 0;
|
||||
|
||||
Kiff = opt_gain_iff/(wi + s)*tf(eye(2));
|
||||
Kiff = opt_gain_iff/(wi + s)*tf(eye(2));
|
||||
|
||||
G_iff = linearize(mdl, io, 0);
|
||||
G_iff = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_iff.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_iff.OutputName = {'Dx', 'Dy'};
|
||||
%% Input/Output definition
|
||||
G_iff.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_iff.OutputName = {'Dx', 'Dy'};
|
||||
|
||||
% IFF With parallel Stiffness :ignore:
|
||||
|
||||
kp = 5*m*W^2;
|
||||
cp = 0.01;
|
||||
kp = 5*m*W^2;
|
||||
cp = 0.01;
|
||||
|
||||
Kiff = opt_gain_kp/s*tf(eye(2));
|
||||
Kiff = opt_gain_kp/s*tf(eye(2));
|
||||
|
||||
G_kp = linearize(mdl, io, 0);
|
||||
G_kp = linearize(mdl, io, 0);
|
||||
|
||||
%% Input/Output definition
|
||||
G_kp.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_kp.OutputName = {'Dx', 'Dy'};
|
||||
%% Input/Output definition
|
||||
G_kp.InputName = {'Dwx', 'Dwy', 'Fdx', 'Fdy'};
|
||||
G_kp.OutputName = {'Dx', 'Dy'};
|
||||
|
||||
% Transmissibility :ignore:
|
||||
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff({'Dx'}, {'Dwx'}), freqs))), ...
|
||||
'DisplayName', 'IFF + HPF')
|
||||
plot(freqs, abs(squeeze(freqresp(G_kp( {'Dx'}, {'Dwx'}), freqs))), ...
|
||||
'DisplayName', 'IFF + $k_p$')
|
||||
plot(freqs, abs(squeeze(freqresp(G_pas({'Dx'}, {'Dwx'}), freqs))), ...
|
||||
'DisplayName', 'Passive')
|
||||
plot(freqs, abs(squeeze(freqresp(G_ol( {'Dx'}, {'Dwx'}), freqs))), 'k-', ...
|
||||
'DisplayName', 'Open-Loop')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylim([1e-2, 3e1]);
|
||||
xlabel('Frequency [rad/s]'); ylabel('Transmissibility [m/m]');
|
||||
legend('location', 'southwest', 'FontSize', 8);
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff({'Dx'}, {'Dwx'}), freqs))), ...
|
||||
'DisplayName', 'IFF + HPF')
|
||||
plot(freqs, abs(squeeze(freqresp(G_kp( {'Dx'}, {'Dwx'}), freqs))), ...
|
||||
'DisplayName', 'IFF + $k_p$')
|
||||
plot(freqs, abs(squeeze(freqresp(G_pas({'Dx'}, {'Dwx'}), freqs))), ...
|
||||
'DisplayName', 'Passive')
|
||||
plot(freqs, abs(squeeze(freqresp(G_ol( {'Dx'}, {'Dwx'}), freqs))), 'k-', ...
|
||||
'DisplayName', 'Open-Loop')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylim([1e-2, 3e1]);
|
||||
xlabel('Frequency [rad/s]'); ylabel('Transmissibility [m/m]');
|
||||
legend('location', 'southwest', 'FontSize', 8);
|
||||
|
||||
% Compliance :ignore:
|
||||
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
freqs = logspace(-2, 1, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff({'Dx'}, {'Fdx'}), freqs))), ...
|
||||
'DisplayName', 'IFF + HPF')
|
||||
plot(freqs, abs(squeeze(freqresp(G_kp( {'Dx'}, {'Fdx'}), freqs))), ...
|
||||
'DisplayName', 'IFF + $k_p$')
|
||||
plot(freqs, abs(squeeze(freqresp(G_pas({'Dx'}, {'Fdx'}), freqs))), ...
|
||||
'DisplayName', 'Passive')
|
||||
plot(freqs, abs(squeeze(freqresp(G_ol( {'Dx'}, {'Fdx'}), freqs))), 'k-', ...
|
||||
'DisplayName', 'Open-Loop')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylim([1e-2, 3e1]);
|
||||
xlabel('Frequency [rad/s]'); ylabel('Compliance [m/N]');
|
||||
legend('location', 'southwest', 'FontSize', 8);
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G_iff({'Dx'}, {'Fdx'}), freqs))), ...
|
||||
'DisplayName', 'IFF + HPF')
|
||||
plot(freqs, abs(squeeze(freqresp(G_kp( {'Dx'}, {'Fdx'}), freqs))), ...
|
||||
'DisplayName', 'IFF + $k_p$')
|
||||
plot(freqs, abs(squeeze(freqresp(G_pas({'Dx'}, {'Fdx'}), freqs))), ...
|
||||
'DisplayName', 'Passive')
|
||||
plot(freqs, abs(squeeze(freqresp(G_ol( {'Dx'}, {'Fdx'}), freqs))), 'k-', ...
|
||||
'DisplayName', 'Open-Loop')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylim([1e-2, 3e1]);
|
||||
xlabel('Frequency [rad/s]'); ylabel('Compliance [m/N]');
|
||||
legend('location', 'southwest', 'FontSize', 8);
|
||||
@@ -0,0 +1,10 @@
|
||||
% =computeSimultaneousDamping=
|
||||
|
||||
function [xi_min] = computeSimultaneousDamping(g, G, K)
|
||||
[w, xi] = damp(minreal(feedback(G, g*K)));
|
||||
xi_min = 1/min(xi);
|
||||
|
||||
if xi_min < 0
|
||||
xi_min = 1e8;
|
||||
end
|
||||
end
|
||||
@@ -0,0 +1,93 @@
|
||||
function [poles] = rootLocusPolesSorted(G, K, gains, args)
|
||||
% rootLocusPolesSorted -
|
||||
%
|
||||
% Syntax: [poles] = rootLocusPolesSorted(G, K, gains, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - G, K, gains, args -
|
||||
%
|
||||
% Outputs:
|
||||
% - poles -
|
||||
|
||||
arguments
|
||||
G
|
||||
K
|
||||
gains
|
||||
args.minreal double {mustBeNumericOrLogical} = false
|
||||
args.p_half double {mustBeNumericOrLogical} = false
|
||||
args.d_max double {mustBeNumeric} = -1
|
||||
end
|
||||
|
||||
if args.minreal
|
||||
p1 = pole(minreal(feedback(G, gains(1)*K)));
|
||||
[~, i_uniq] = uniquetol([real(p1), imag(p1)], 1e-10, 'ByRows', true);
|
||||
p1 = p1(i_uniq);
|
||||
|
||||
poles = zeros(length(p1), length(gains));
|
||||
poles(:, 1) = p1;
|
||||
else
|
||||
p1 = pole(feedback(G, gains(1)*K));
|
||||
[~, i_uniq] = uniquetol([real(p1), imag(p1)], 1e-10, 'ByRows', true);
|
||||
p1 = p1(i_uniq);
|
||||
|
||||
poles = zeros(length(p1), length(gains));
|
||||
poles(:, 1) = p1;
|
||||
end
|
||||
|
||||
if args.minreal
|
||||
p2 = pole(minreal(feedback(G, gains(2)*K)));
|
||||
[~, i_uniq] = uniquetol([real(p2), imag(p2)], 1e-10, 'ByRows', true);
|
||||
p2 = p2(i_uniq);
|
||||
poles(:, 2) = p2;
|
||||
else
|
||||
p2 = pole(feedback(G, gains(2)*K));
|
||||
[~, i_uniq] = uniquetol([real(p2), imag(p2)], 1e-10, 'ByRows', true);
|
||||
p2 = p2(i_uniq);
|
||||
poles(:, 2) = p2;
|
||||
end
|
||||
|
||||
for g_i = 3:length(gains)
|
||||
% Estimated value of the poles
|
||||
poles_est = poles(:, g_i-1) + (poles(:, g_i-1) - poles(:, g_i-2))*(gains(g_i) - gains(g_i-1))/(gains(g_i-1) - gains(g_i - 2));
|
||||
|
||||
% New values for the poles
|
||||
poles_gi = pole(feedback(G, gains(g_i)*K));
|
||||
[~, i_uniq] = uniquetol([real(poles_gi), imag(poles_gi)], 1e-10, 'ByRows', true);
|
||||
poles_gi = poles_gi(i_uniq);
|
||||
|
||||
% Array of distances between all the poles
|
||||
poles_dist = sqrt((poles_est-poles_gi.').*conj(poles_est-poles_gi.'));
|
||||
|
||||
% Get indices corresponding to distances from lowest to highest
|
||||
[~, c] = sort(min(poles_dist));
|
||||
|
||||
as = 1:length(poles_gi);
|
||||
|
||||
% for each column of poles_dist corresponding to the i'th pole
|
||||
% with closest previous poles
|
||||
for p_i = c
|
||||
% Get the indice a_i of the previous pole that is the closest
|
||||
% to pole c(p_i)
|
||||
[~, a_i] = min(poles_dist(:, p_i));
|
||||
|
||||
poles(as(a_i), g_i) = poles_gi(p_i);
|
||||
|
||||
% Remove old poles that are already matched
|
||||
% poles_gi(as(a_i), :) = [];
|
||||
poles_dist(a_i, :) = [];
|
||||
as(a_i) = [];
|
||||
end
|
||||
end
|
||||
|
||||
if args.d_max > 0
|
||||
poles = poles(max(abs(poles(:, 2:end) - poles(:, 1:end-1))') > args.d_max, :);
|
||||
end
|
||||
|
||||
if args.p_half
|
||||
poles = poles(1:round(end/2), :);
|
||||
end
|
||||
|
||||
[~, s_p] = sort(imag(poles(:,1)), 'descend');
|
||||
poles = poles(s_p, :);
|
||||
|
||||
poles = poles.';
|
||||
@@ -1,10 +0,0 @@
|
||||
% =computeSimultaneousDamping=
|
||||
|
||||
function [xi_min] = computeSimultaneousDamping(g, G, K)
|
||||
[w, xi] = damp(minreal(feedback(G, g*K)));
|
||||
xi_min = 1/min(xi);
|
||||
|
||||
if xi_min < 0
|
||||
xi_min = 1e8;
|
||||
end
|
||||
end
|
||||
@@ -1,93 +0,0 @@
|
||||
function [poles] = rootLocusPolesSorted(G, K, gains, args)
|
||||
% rootLocusPolesSorted -
|
||||
%
|
||||
% Syntax: [poles] = rootLocusPolesSorted(G, K, gains, args)
|
||||
%
|
||||
% Inputs:
|
||||
% - G, K, gains, args -
|
||||
%
|
||||
% Outputs:
|
||||
% - poles -
|
||||
|
||||
arguments
|
||||
G
|
||||
K
|
||||
gains
|
||||
args.minreal double {mustBeNumericOrLogical} = false
|
||||
args.p_half double {mustBeNumericOrLogical} = false
|
||||
args.d_max double {mustBeNumeric} = -1
|
||||
end
|
||||
|
||||
if args.minreal
|
||||
p1 = pole(minreal(feedback(G, gains(1)*K)));
|
||||
[~, i_uniq] = uniquetol([real(p1), imag(p1)], 1e-10, 'ByRows', true);
|
||||
p1 = p1(i_uniq);
|
||||
|
||||
poles = zeros(length(p1), length(gains));
|
||||
poles(:, 1) = p1;
|
||||
else
|
||||
p1 = pole(feedback(G, gains(1)*K));
|
||||
[~, i_uniq] = uniquetol([real(p1), imag(p1)], 1e-10, 'ByRows', true);
|
||||
p1 = p1(i_uniq);
|
||||
|
||||
poles = zeros(length(p1), length(gains));
|
||||
poles(:, 1) = p1;
|
||||
end
|
||||
|
||||
if args.minreal
|
||||
p2 = pole(minreal(feedback(G, gains(2)*K)));
|
||||
[~, i_uniq] = uniquetol([real(p2), imag(p2)], 1e-10, 'ByRows', true);
|
||||
p2 = p2(i_uniq);
|
||||
poles(:, 2) = p2;
|
||||
else
|
||||
p2 = pole(feedback(G, gains(2)*K));
|
||||
[~, i_uniq] = uniquetol([real(p2), imag(p2)], 1e-10, 'ByRows', true);
|
||||
p2 = p2(i_uniq);
|
||||
poles(:, 2) = p2;
|
||||
end
|
||||
|
||||
for g_i = 3:length(gains)
|
||||
% Estimated value of the poles
|
||||
poles_est = poles(:, g_i-1) + (poles(:, g_i-1) - poles(:, g_i-2))*(gains(g_i) - gains(g_i-1))/(gains(g_i-1) - gains(g_i - 2));
|
||||
|
||||
% New values for the poles
|
||||
poles_gi = pole(feedback(G, gains(g_i)*K));
|
||||
[~, i_uniq] = uniquetol([real(poles_gi), imag(poles_gi)], 1e-10, 'ByRows', true);
|
||||
poles_gi = poles_gi(i_uniq);
|
||||
|
||||
% Array of distances between all the poles
|
||||
poles_dist = sqrt((poles_est-poles_gi.').*conj(poles_est-poles_gi.'));
|
||||
|
||||
% Get indices corresponding to distances from lowest to highest
|
||||
[~, c] = sort(min(poles_dist));
|
||||
|
||||
as = 1:length(poles_gi);
|
||||
|
||||
% for each column of poles_dist corresponding to the i'th pole
|
||||
% with closest previous poles
|
||||
for p_i = c
|
||||
% Get the indice a_i of the previous pole that is the closest
|
||||
% to pole c(p_i)
|
||||
[~, a_i] = min(poles_dist(:, p_i));
|
||||
|
||||
poles(as(a_i), g_i) = poles_gi(p_i);
|
||||
|
||||
% Remove old poles that are already matched
|
||||
% poles_gi(as(a_i), :) = [];
|
||||
poles_dist(a_i, :) = [];
|
||||
as(a_i) = [];
|
||||
end
|
||||
end
|
||||
|
||||
if args.d_max > 0
|
||||
poles = poles(max(abs(poles(:, 2:end) - poles(:, 1:end-1))') > args.d_max, :);
|
||||
end
|
||||
|
||||
if args.p_half
|
||||
poles = poles(1:round(end/2), :);
|
||||
end
|
||||
|
||||
[~, s_p] = sort(imag(poles(:,1)), 'descend');
|
||||
poles = poles(s_p, :);
|
||||
|
||||
poles = poles.';
|
||||
Reference in new issue
Block a user