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tdehaeze committed 2026-10-12 11:16:59 +02:00
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@@ -1,4 +1,11 @@
#+TITLE: A brief and practical introduction to $\mathcal{H}_\infty$ Control #+TITLE: A brief and practical introduction to $\mathcal{H}_\infty$ Control
#+hugo_section: research
#+hugo_bundle: 2020-h-infinity-lecture
#+export_file_name: _index
#+hugo_custom_front_matter: :year 2020 :pubtype "lecture"
#+hugo_custom_front_matter: :pdf "https://git.tdehaeze.xyz/tdehaeze/lecture-h-infinity/raw/branch/master/index.pdf"
#+hugo_custom_front_matter: :video "https://www.youtube.com/watch?v=KdJsm_X8uks"
#+hugo_custom_front_matter: :code "https://git.tdehaeze.xyz/tdehaeze/lecture-h-infinity"
:DRAWER: :DRAWER:
#+STARTUP: overview #+STARTUP: overview
@@ -6,13 +13,6 @@
#+EMAIL: dehaeze.thomas@gmail.com #+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas #+AUTHOR: Dehaeze Thomas
#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ../index.html
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+BIND: org-latex-image-default-option "scale=1" #+BIND: org-latex-image-default-option "scale=1"
#+BIND: org-latex-image-default-width "" #+BIND: org-latex-image-default-width ""
@@ -47,11 +47,7 @@
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END: :END:
#+begin_export html yt:KdJsm_X8uks
<hr>
<p>This document is also available as a <a href="./index.pdf">pdf</a>.</p>
<hr>
#+end_export
* LaTeX Config :noexport: * LaTeX Config :noexport:
#+begin_src emacs-lisp :tangle no #+begin_src emacs-lisp :tangle no
@@ -226,7 +222,10 @@ The general structure of this document is as follows:
Such technique is presented in Section [[sec:closed-loop-shaping]] Such technique is presented in Section [[sec:closed-loop-shaping]]
- Finally, complete examples of the use of $\mathcal{H}_\infty$ Control for practical problems are provided in Section [[sec:h_infinity_mixed_sensitivity]] - Finally, complete examples of the use of $\mathcal{H}_\infty$ Control for practical problems are provided in Section [[sec:h_infinity_mixed_sensitivity]]
* Matlab Init :noexport:ignore: * Introduction to Model Based Control
<<sec:model_based_control>>
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>> <<matlab-dir>>
#+end_src #+end_src
@@ -239,9 +238,6 @@ The general structure of this document is as follows:
addpath('matlab') addpath('matlab')
#+end_src #+end_src
* Introduction to Model Based Control
<<sec:model_based_control>>
** Model Based Control - Methodology ** Model Based Control - Methodology
<<sec:model_based_control_methodology>> <<sec:model_based_control_methodology>>
@@ -617,6 +613,9 @@ The Bode plots of $G(s)$ and $G_d(s)$ are shown in Figures [[fig:bode_plot_examp
[[file:figs/bode_plot_example_Gd.png]] [[file:figs/bode_plot_example_Gd.png]]
* Classical Open Loop Shaping * Classical Open Loop Shaping
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/h_infinity_1_open_loop_shaping.m
:END:
<<sec:open_loop_shaping>> <<sec:open_loop_shaping>>
** Introduction :ignore: ** Introduction :ignore:
@@ -627,6 +626,28 @@ Such Loop Shaping is usually performed manually with tools coming from the class
However, the $\mathcal{H}_\infty$ synthesis can be used to automate the Loop Shaping process. However, the $\mathcal{H}_\infty$ synthesis can be used to automate the Loop Shaping process.
This is presented in Section [[sec:h_infinity_open_loop_shaping]] and applied on the same example in Section [[sec:h_infinity_open_loop_shaping_example]]. This is presented in Section [[sec:h_infinity_open_loop_shaping]] and applied on the same example in Section [[sec:h_infinity_open_loop_shaping_example]].
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
Let's define the system parameters on Matlab.
#+begin_src matlab +n
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
#+end_src
And now the system dynamics $G(s)$ and $G_d(s)$.
#+begin_src matlab +n -r
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
#+end_src
** Introduction to Loop Shaping ** Introduction to Loop Shaping
<<sec:open_loop_shaping_introduction>> <<sec:open_loop_shaping_introduction>>
@@ -636,8 +657,9 @@ This is presented in Section [[sec:h_infinity_open_loop_shaping]] and applied on
#+begin_definition #+begin_definition
The *Loop Gain* (or Loop transfer function) $L(s)$ usually refers to as the product of the controller and the plant (see Figure [[fig:open_loop_shaping]]): The *Loop Gain* (or Loop transfer function) $L(s)$ usually refers to as the product of the controller and the plant (see Figure [[fig:open_loop_shaping]]):
#+name: eq:loop_gain
\begin{equation} \begin{equation}
L(s) = G(s) \cdot K(s) \label{eq:loop_gain} L(s) = G(s) \cdot K(s)
\end{equation} \end{equation}
Its name comes from the fact that this is actually the "gain around the loop". Its name comes from the fact that this is actually the "gain around the loop".
@@ -679,7 +701,7 @@ It is widely used and generally successful as many characteristics of the closed
- *Robust stability*: $L$ small (neglected dynamics) - *Robust stability*: $L$ small (neglected dynamics)
The shaping of the Loop Gain is done manually by combining several leads, lags, notches... The shaping of the Loop Gain is done manually by combining several leads, lags, notches...
This process is very much simplified by the fact that the loop gain $L(s)$ depends *linearly* on $K(s)$ eqref:eq:loop_gain. This process is very much simplified by the fact that the loop gain $L(s)$ depends *linearly* on $K(s)$ [[eq:loop_gain]].
A typical wanted Loop Shape is shown in Figure [[fig:open_loop_shaping_shape]]. A typical wanted Loop Shape is shown in Figure [[fig:open_loop_shaping_shape]].
Another interesting Loop shape called "Bode Step" is described in cite:lurie02_system_archit_trades_using_bode. Another interesting Loop shape called "Bode Step" is described in cite:lurie02_system_archit_trades_using_bode.
@@ -771,7 +793,7 @@ Let's say we came up with the following controller.
The bode plot of the Loop Gain is shown in Figure [[fig:loop_gain_manual_afm]] and we can verify that we have the wanted stability margins using the =margin= command: The bode plot of the Loop Gain is shown in Figure [[fig:loop_gain_manual_afm]] and we can verify that we have the wanted stability margins using the =margin= command:
#+begin_src matlab #+begin_src matlab
[Gm, Pm, ~, Wc] = margin(G*K) [Gm, Pm, ~, Wc] = margin(G*K);
#+end_src #+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
@@ -966,7 +988,7 @@ Let's briefly analyze the obtained controller which bode plot is shown in Figure
Let's finally compare the obtained stability margins of the $\mathcal{H}_\infty$ controller and of the manually developed controller in Table [[tab:open_loop_shaping_compare]]. Let's finally compare the obtained stability margins of the $\mathcal{H}_\infty$ controller and of the manually developed controller in Table [[tab:open_loop_shaping_compare]].
#+begin_src matlab :exports none #+begin_src matlab :exports none
[Gm_2, Pm_2, ~, Wc_2] = margin(G*K) [Gm_2, Pm_2, ~, Wc_2] = margin(G*K);
#+end_src #+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
@@ -1275,6 +1297,9 @@ The procedure to convert a typical control architecture as the one shown in Figu
#+end_exercice #+end_exercice
* Modern Interpretation of Control Specifications * Modern Interpretation of Control Specifications
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/h_infinity_2_control_specifications.m
:END:
<<sec:modern_interpretation_specification>> <<sec:modern_interpretation_specification>>
** Introduction :ignore: ** Introduction :ignore:
@@ -1300,6 +1325,28 @@ The robustness (stability margins) of the system can also be linked to the shape
Links between typical control specifications and shapes of the closed-loop transfer functions are summarized in Section [[sec:other_requirements]]. Links between typical control specifications and shapes of the closed-loop transfer functions are summarized in Section [[sec:other_requirements]].
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
Let's define the system parameters on Matlab.
#+begin_src matlab +n
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
#+end_src
And now the system dynamics $G(s)$ and $G_d(s)$.
#+begin_src matlab +n -r
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
#+end_src
** Closed Loop Transfer Functions and the Gang of Four ** Closed Loop Transfer Functions and the Gang of Four
<<sec:closed_loop_tf>> <<sec:closed_loop_tf>>
@@ -1428,22 +1475,22 @@ The comparison of the sensitivity functions shapes and their effect on the step
wc = 2*pi*1; L_w = 8; wi = 2*pi*0.02; wc = 2*pi*1; L_w = 8; wi = 2*pi*0.02;
Kr = 1/((s + wi)^2) * ... % Double Integrator Kr = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead (1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
Kr = Kr/abs(evalfr(Kr*G, j*wc)); Kr = Kr/abs(evalfr(Kr*G, 1j*wc));
wc = 2*pi*2; L_w = 8; wi = 2*pi*0.045; wc = 2*pi*2; L_w = 8; wi = 2*pi*0.045;
K1 = 1/((s + wi)^2) * ... % Double Integrator K1 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead (1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K1 = K1/abs(evalfr(K1*G, j*wc)); K1 = K1/abs(evalfr(K1*G, 1j*wc));
wc = 2*pi*1; L_w = 2; wi = 2*pi*0.03; wc = 2*pi*1; L_w = 2; wi = 2*pi*0.03;
K2 = 1/((s + wi)^2) * ... % Double Integrator K2 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead (1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K2 = K2/abs(evalfr(K2*G, j*wc)); K2 = K2/abs(evalfr(K2*G, 1j*wc));
wc = 2*pi*1; L_w = 8; wi = 2*pi*0.2; wc = 2*pi*1; L_w = 8; wi = 2*pi*0.2;
K3 = 1/((s + wi)^2) * ... % Double Integrator K3 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead (1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K3 = K3/abs(evalfr(K3*G, j*wc)); K3 = K3/abs(evalfr(K3*G, 1j*wc));
Sr = 1/(1 + Kr*G); Sr = 1/(1 + Kr*G);
@@ -1468,7 +1515,7 @@ The comparison of the sensitivity functions shapes and their effect on the step
plot(freqs, abs(squeeze(freqresp(Sr, freqs, 'Hz'))), 'k-', 'DisplayName', '$K_r(s)$'); plot(freqs, abs(squeeze(freqresp(Sr, freqs, 'Hz'))), 'k-', 'DisplayName', '$K_r(s)$');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frquency [Hz]'); ylabel('Sensitivity Magnitude'); xlabel('Frequency [Hz]'); ylabel('Sensitivity Magnitude');
hold off; hold off;
legend('location', 'southeast'); legend('location', 'southeast');
ylim([1e-3, 3]); ylim([1e-3, 3]);
@@ -1644,7 +1691,7 @@ At a frequency little bit above 100Hz, the phase of the loop gain reaches -180 d
It is confirmed by checking the stability of the closed loop system: It is confirmed by checking the stability of the closed loop system:
#+begin_src matlab :results value replace #+begin_src matlab :results value replace
isstable(feedback(Gr,K)) isstable(feedback(Gr,Kt))
#+end_src #+end_src
#+RESULTS: #+RESULTS:
@@ -1835,6 +1882,9 @@ And we now understand why setting an upper bound on the magnitude of $S$ is gene
| Robustness | Phase/Gain margins | Module margin: $\Vert S\Vert_\infty$ small | | Robustness | Phase/Gain margins | Module margin: $\Vert S\Vert_\infty$ small |
* $\mathcal{H}_\infty$ Shaping of closed-loop transfer functions * $\mathcal{H}_\infty$ Shaping of closed-loop transfer functions
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/h_infinity_3_shaping_cl_tf.m
:END:
<<sec:closed-loop-shaping>> <<sec:closed-loop-shaping>>
** Introduction :ignore: ** Introduction :ignore:
@@ -1855,6 +1905,28 @@ Multiple closed-loop transfer functions can be shaped at the same time.
Such synthesis is usually called *Mixed-sensitivity Loop Shaping* and is one of the most powerful tool of the robust control theory. Such synthesis is usually called *Mixed-sensitivity Loop Shaping* and is one of the most powerful tool of the robust control theory.
Some insight on the use and limitations of such techniques are given in Section [[sec:shaping_multiple_tf]]. Some insight on the use and limitations of such techniques are given in Section [[sec:shaping_multiple_tf]].
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
Let's define the system parameters on Matlab.
#+begin_src matlab +n
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
#+end_src
And now the system dynamics $G(s)$ and $G_d(s)$.
#+begin_src matlab +n -r
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
#+end_src
** How to Shape closed-loop transfer function? Using Weighting Functions! ** How to Shape closed-loop transfer function? Using Weighting Functions!
<<sec:weighting_functions>> <<sec:weighting_functions>>
@@ -1904,14 +1976,15 @@ Now, the closed-loop transfer function from $w$ to $z$ is equal to $W_s(s)S(s)$
#+end_important #+end_important
Let's now show how this is equivalent as *shaping* the sensitivity function: Let's now show how this is equivalent as *shaping* the sensitivity function:
#+name: eq:sensitivity_shaping
\begin{align} \begin{align}
& \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\ & \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\
\Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega\nonumber \\ \Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega\nonumber \\
\Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega \label{eq:sensitivity_shaping} \Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega
\end{align} \end{align}
#+begin_important #+begin_important
As shown in Equation eqref:eq:sensitivity_shaping, the objective of the $\mathcal{H}_\infty$ synthesis applied on the /weighted/ plant is to make the norm sensitivity function smaller than the inverse of the norm of the weighting function, and that at all frequencies. As shown in Equation [[eq:sensitivity_shaping]], the objective of the $\mathcal{H}_\infty$ synthesis applied on the /weighted/ plant is to make the norm sensitivity function smaller than the inverse of the norm of the weighting function, and that at all frequencies.
Therefore, the choice of the weighting function $W_s(s)$ is very important: its inverse magnitude will define the wanted *upper bound* of the sensitivity function magnitude over all frequencies. Therefore, the choice of the weighting function $W_s(s)$ is very important: its inverse magnitude will define the wanted *upper bound* of the sensitivity function magnitude over all frequencies.
#+end_important #+end_important
@@ -2021,7 +2094,7 @@ The Matlab code below produces a weighting function with the following character
plot(freqs, abs(squeeze(freqresp(Ws, freqs, 'Hz'))), 'k-'); plot(freqs, abs(squeeze(freqresp(Ws, freqs, 'Hz'))), 'k-');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frquency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off; hold off;
#+end_src #+end_src
@@ -2041,12 +2114,13 @@ Quite often, higher orders weights are required.
In such case, the following formula can be used: In such case, the following formula can be used:
#+name: eq:weight_formula_advanced
\begin{equation} \begin{equation}
W(s) = \left( \frac{ W(s) = \left( \frac{
\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
}{ }{
\left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}} \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
}\right)^n \label{eq:weight_formula_advanced} }\right)^n
\end{equation} \end{equation}
The parameters permit to specify: The parameters permit to specify:
@@ -2055,7 +2129,7 @@ The parameters permit to specify:
- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$ - the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
- the absolute slope between high and low frequency: $n$ - the absolute slope between high and low frequency: $n$
A Matlab function implementing Equation eqref:eq:weight_formula_advanced is shown below: A Matlab function implementing Equation [[eq:weight_formula_advanced]] is shown below:
#+begin_src matlab :tangle matlab/generateWeight.m :comments none :eval no #+begin_src matlab :tangle matlab/generateWeight.m :comments none :eval no
function [W] = generateWeight(args) function [W] = generateWeight(args)
@@ -2103,7 +2177,7 @@ The obtained shapes are shown in Figure [[fig:high_order_weight]].
'DisplayName', '$n = 3$'); 'DisplayName', '$n = 3$');
hold off; hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frquency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'northeast'); legend('location', 'northeast');
#+end_src #+end_src
@@ -2112,7 +2186,7 @@ The obtained shapes are shown in Figure [[fig:high_order_weight]].
#+end_src #+end_src
#+name: fig:high_order_weight #+name: fig:high_order_weight
#+caption: Higher order weights using Equation eqref:eq:weight_formula_advanced #+caption: Higher order weights using Equation [[eq:weight_formula_advanced]]
#+attr_latex: :float nil #+attr_latex: :float nil
#+RESULTS: #+RESULTS:
[[file:figs/high_order_weight.png]] [[file:figs/high_order_weight.png]]
@@ -2772,6 +2846,9 @@ These regions are summarized in Figure [[fig:h-infinity-4-blocs-constrains]].
#+end_warning #+end_warning
* Mixed-Sensitivity $\mathcal{H}_\infty$ Control - Example * Mixed-Sensitivity $\mathcal{H}_\infty$ Control - Example
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/h_infinity_4_mixed_sensitivity_example.m
:END:
<<sec:h_infinity_mixed_sensitivity>> <<sec:h_infinity_mixed_sensitivity>>
** Introduction :ignore: ** Introduction :ignore:
@@ -2785,6 +2862,15 @@ The important step of interpreting the specifications as wanted shape of closed-
Finally, the shaping of closed-loop transfer functions is performed in Sections [[sec:ex_shaping_S]], [[sec:ex_shaping_GS]] and [[sec:ex_shaping_T]]. Finally, the shaping of closed-loop transfer functions is performed in Sections [[sec:ex_shaping_S]], [[sec:ex_shaping_GS]] and [[sec:ex_shaping_T]].
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
** Control Problem ** Control Problem
<<sec:ex_control_problem>> <<sec:ex_control_problem>>
@@ -3002,7 +3088,7 @@ The Generalized plant of Figure [[fig:ex_general_plant_sim]] is defined on Matla
#+end_src #+end_src
Time domain simulations will be performed by first computing the closed-loop system using the =lft= command and then using the =lsim= command on the closed-loop system: Time domain simulations will be performed by first computing the closed-loop system using the =lft= command and then using the =lsim= command on the closed-loop system:
#+begin_src matlab +n :tangle no :exec :no #+begin_src matlab +n :tangle no :eval no
% Compute the closed-Loop System, K is the controller % Compute the closed-Loop System, K is the controller
P_CL = lft(Psim, K); P_CL = lft(Psim, K);
@@ -3049,7 +3135,7 @@ In such case, we want to shape $S$, $GS$ and $T$.
To do so, we use to generalized plant shown in Figure [[fig:ex_general_plant]] for the synthesis where the three closed-loop tranfert functions from $w$ to $[z_1\,,z_2\,,z_3]$ are respectively $S$, $GS$ and $T$. To do so, we use to generalized plant shown in Figure [[fig:ex_general_plant]] for the synthesis where the three closed-loop tranfert functions from $w$ to $[z_1\,,z_2\,,z_3]$ are respectively $S$, $GS$ and $T$.
This generalized plant is defined on Matlab as follows: This generalized plant is defined on Matlab as follows:
#+begin_src matlab +n :tangle no :exec no #+begin_src matlab +n
P = [1 -1 P = [1 -1
G -G G -G
0 1 0 1
@@ -3207,7 +3293,7 @@ The $\mathcal{H}_\infty$ synthesis is performed and the obtained closed-loop tra
freqs = logspace(-1, 3, 1000); freqs = logspace(-1, 3, 1000);
figure; figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
ax1 = nexttile; ax1 = nexttile;
hold on; hold on;
@@ -3234,6 +3320,17 @@ The $\mathcal{H}_\infty$ synthesis is performed and the obtained closed-loop tra
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('T'); title('T');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('K');
linkaxes([ax1, ax2, ax3, ax4], 'x')
xlim([freqs(1), freqs(end)]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
@@ -3262,7 +3359,7 @@ We can see:
#+begin_src matlab :exports none #+begin_src matlab :exports none
figure; figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
nexttile; nexttile;
hold on; hold on;
@@ -3271,16 +3368,10 @@ We can see:
hold off; hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]'); xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast'); legend('location', 'southeast');
nexttile;
hold on;
plot(t, u1);
hold off;
xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/ex_time_domain_1.pdf', 'width', 'full', 'height', 'tall'); exportFig('figs/ex_time_domain_1.pdf', 'width', 'wide', 'height', 'tall');
#+end_src #+end_src
#+name: fig:ex_time_domain_1 #+name: fig:ex_time_domain_1
@@ -3317,7 +3408,7 @@ The time domain signals are shown in Figure [[fig:ex_time_domain_1b]] and it is
freqs = logspace(-1, 3, 1000); freqs = logspace(-1, 3, 1000);
figure; figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
ax1 = nexttile; ax1 = nexttile;
hold on; hold on;
@@ -3347,6 +3438,18 @@ The time domain signals are shown in Figure [[fig:ex_time_domain_1b]] and it is
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('T'); title('T');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('K');
linkaxes([ax1, ax2, ax3, ax4], 'x')
xlim([freqs(1), freqs(end)]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
@@ -3368,9 +3471,7 @@ The time domain signals are shown in Figure [[fig:ex_time_domain_1b]] and it is
#+begin_src matlab :exports none #+begin_src matlab :exports none
figure; figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
hold on; hold on;
plot(t, 1e6*y1, 'DisplayName', 'Step 1'); plot(t, 1e6*y1, 'DisplayName', 'Step 1');
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b'); plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
@@ -3378,17 +3479,10 @@ The time domain signals are shown in Figure [[fig:ex_time_domain_1b]] and it is
hold off; hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]'); xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast'); legend('location', 'southeast');
nexttile;
hold on;
plot(t, u1);
plot(t, u1b);
hold off;
xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/ex_time_domain_1b.pdf', 'width', 'full', 'height', 'tall'); exportFig('figs/ex_time_domain_1b.pdf', 'width', 'wide', 'height', 'tall');
#+end_src #+end_src
#+name: fig:ex_time_domain_1b #+name: fig:ex_time_domain_1b
@@ -3425,7 +3519,7 @@ The $\mathcal{H}_\infty$ Synthesis is performed and the obtained closed-loop tra
freqs = logspace(-1, 3, 1000); freqs = logspace(-1, 3, 1000);
figure; figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
ax1 = nexttile; ax1 = nexttile;
hold on; hold on;
@@ -3459,6 +3553,19 @@ The $\mathcal{H}_\infty$ Synthesis is performed and the obtained closed-loop tra
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('T'); title('T');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('K');
linkaxes([ax1, ax2, ax3, ax4], 'x')
xlim([freqs(1), freqs(end)]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
@@ -3485,7 +3592,6 @@ If is shown that indeed, the disturbance rejection performance are much better a
figure; figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
hold on; hold on;
plot(t, 1e6*y1, 'DisplayName', 'Step 1'); plot(t, 1e6*y1, 'DisplayName', 'Step 1');
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b'); plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
@@ -3494,14 +3600,6 @@ If is shown that indeed, the disturbance rejection performance are much better a
hold off; hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]'); xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast'); legend('location', 'southeast');
nexttile;
hold on;
plot(t, u1);
plot(t, u1b);
plot(t, u2);
hold off;
xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
@@ -3566,7 +3664,7 @@ The obtained closed-loop transfer functions are shown in Figure [[fig:ex_results
freqs = logspace(-1, 3, 1000); freqs = logspace(-1, 3, 1000);
figure; figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
ax1 = nexttile; ax1 = nexttile;
hold on; hold on;
@@ -3604,6 +3702,20 @@ The obtained closed-loop transfer functions are shown in Figure [[fig:ex_results
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude'); xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('T'); title('T');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K3, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('K');
linkaxes([ax1, ax2, ax3, ax4], 'x')
xlim([freqs(1), freqs(end)]);
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
@@ -3630,9 +3742,7 @@ This can be seen when zooming on the output signal in Figure [[fig:ex_time_domai
#+begin_src matlab :exports none #+begin_src matlab :exports none
figure; figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None'); tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
hold on; hold on;
plot(t, 1e6*y1, 'DisplayName', 'Step 1'); plot(t, 1e6*y1, 'DisplayName', 'Step 1');
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b'); plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
@@ -3642,19 +3752,10 @@ This can be seen when zooming on the output signal in Figure [[fig:ex_time_domai
hold off; hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]'); xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast'); legend('location', 'southeast');
nexttile;
hold on;
plot(t, u1);
plot(t, u1b);
plot(t, u2);
plot(t, u3);
hold off;
xlabel('Time [s]'); ylabel('Control Input $u$ [N]');
#+end_src #+end_src
#+begin_src matlab :tangle no :exports results :results file replace #+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/ex_time_domain_3.pdf', 'width', 'full', 'height', 'tall'); exportFig('figs/ex_time_domain_3.pdf', 'width', 'wide', 'height', 'tall');
#+end_src #+end_src
#+name: fig:ex_time_domain_3 #+name: fig:ex_time_domain_3
@@ -3708,10 +3809,12 @@ If you want to know more about the "$\mathcal{H}_\infty$ and robust control worl
For a complete treatment of multivariable robust control, I would highly recommend this book cite:skogestad07_multiv_feedb_contr. For a complete treatment of multivariable robust control, I would highly recommend this book cite:skogestad07_multiv_feedb_contr.
If you want to nice reference book in French, look at cite:duc99_comman_h. If you want to nice reference book in French, look at cite:duc99_comman_h.
You can also look at the very good lectures below. Here is a series of tech talks by Brian Douglas:
[[yt:?listType=playlist&list=PLn8PRpmsu08qFLMfgTEzR8DxOPE7fBiin][Robust Control - Brian Douglas]] yt:A7wHSr6GRnc
[[yt:?listType=playlist&list=PLsjPUqcL7ZIFHCObUU_9xPUImZ203gB4o][Control Bootcamp - Steve Brunton]] Here is a Control Bootcamp by Steve Brunton:
yt:Pi7l8mMjYVE
#+latex: \printbibliography #+latex: \printbibliography
matlab/h_infinity_1_open_loop_shaping.m
+204
View File
@@ -0,0 +1,204 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% Let's define the system parameters on Matlab.
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
% And now the system dynamics $G(s)$ and $G_d(s)$.
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
% #+HTML: <details><summary>Hint</summary>
% #+LATEX: \tcbsubtitle{Hint}
% You can follow this procedure:
% 1. In order to have good disturbance rejection at low frequency, add a simple or double *integrator*
% 2. In terms of the loop gain, the *bandwidth* can be defined at the frequency $\omega_c$ where $|l(j\omega_c)|$ first crosses 1 from above.
% Therefore, adjust the *gain* such that $L(j\omega)$ crosses 1 at around 10Hz
% 3. The roll-off at high frequency for noise attenuation should already be good enough.
% If not, add a *low pass filter*
% 4. Add a *Lead* centered around the crossover frequency (10 Hz) and tune it such that sufficient phase margin is added.
% Verify that the gain margin is good enough.
% #+HTML: </details>
% #+end_exercice
% Let's say we came up with the following controller.
K = 14e8 * ... % Gain
1/(s^2) * ... % Double Integrator
1/(1 + s/2/pi/40) * ... % Low Pass Filter
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
% The bode plot of the Loop Gain is shown in Figure [[fig:loop_gain_manual_afm]] and we can verify that we have the wanted stability margins using the =margin= command:
[Gm, Pm, ~, Wc] = margin(G*K);
% #+RESULTS:
% | Requirements | Manual Method |
% |-----------------------------+---------------|
% | Gain Margin $> 3$ [dB] | 3.1 |
% | Phase Margin $> 30$ [deg] | 35.4 |
% | Crossover $\approx 10$ [Hz] | 10.1 |
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ylim([1e-4, 1e1])
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*K, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-360, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
% <<sec:h_infinity_open_loop_shaping_example>>
% To apply the $\mathcal{H}_\infty$ Loop Shaping Synthesis, the wanted shape of the loop gain should be determined from the specifications.
% This is summarized in Table [[tab:open_loop_shaping_specifications]].
% Such shape corresponds to the typical wanted Loop gain Shape shown in Figure [[fig:open_loop_shaping_shape]].
% #+name: tab:open_loop_shaping_specifications
% #+caption: Wanted Loop Shape corresponding to each specification
% #+attr_latex: :environment tabularx :width \linewidth :align lXX
% | | *Specification* | *Corresponding Loop Shape* |
% |--------------+----------------------------------------+------------------------------------------|
% | *Dist. Rej.* | Highest possible rejection below 1Hz | Slope of -40dB/dec at low frequency |
% | *Pos. Speed* | Bandwidth of approximately 10Hz | $L$ crosses 1 at 10Hz |
% | *Noise Att.* | Roll-off of -40dB/decade past 30Hz | Roll-off of -40dB/decade past 30Hz |
% | *Robustness* | $\Delta G > 3dB$, $\Delta \phi > 30^o$ | Slope of -20dB/decade near the crossover |
% Then, a (stable, minimum phase) transfer function $L_w(s)$ should be created that has the same gain as the wanted shape of the Loop gain.
% For this example, a double integrator and a lead centered on 10Hz are used.
% Then the gain is adjusted such that the $|L_w(j2 \pi 10)| = 1$.
% Using Matlab, we have:
Lw = 2.3e3 * ...
1/(s^2) * ... % Double Integrator
(1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead
% The $\mathcal{H}_\infty$ open loop shaping synthesis is then performed using the =loopsyn= command:
[K, ~, GAM] = loopsyn(G, Lw);
% The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]] and matches the specified one by a factor $\gamma \approx 2$.
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz'))), 'k--', 'DisplayName', '$L_w(s)$');
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz')))*GAM, 'k-.', 'DisplayName', '$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$');
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz')))/GAM, 'k-.', 'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast');
ylim([1e-4, 1e2]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G*K, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-360, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:open_loop_shaping_hinf_L
% #+caption: Obtained Open Loop Gain $L(s) = G(s) K(s)$ and comparison with the wanted Loop gain $L_w$
% #+RESULTS:
% [[file:figs/open_loop_shaping_hinf_L.png]]
% #+begin_important
% When using the $\mathcal{H}_\infty$ Synthesis, it is usually recommended to analyze the obtained controller.
% This is usually done by breaking down the controller into simple elements such as low pass filters, high pass filters, notches, leads, etc.
% #+end_important
% Let's briefly analyze the obtained controller which bode plot is shown in Figure [[fig:open_loop_shaping_hinf_K]]:
% - two integrators are used at low frequency to have the wanted low frequency high gain
% - a lead is added centered with the crossover frequency to increase the phase margin
% - a notch is added at the resonance of the plant to increase the gain margin (this is very typical of $\mathcal{H}_\infty$ controllers, and can be an issue, more info on that latter)
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
plot(freqs, abs(squeeze(freqresp(K, freqs, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
plot(freqs, 180/pi*angle(squeeze(freqresp(K, freqs, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-180, 90]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:open_loop_shaping_hinf_K
% #+caption: Obtained controller $K$ using the open-loop $\mathcal{H}_\infty$ shaping
% #+RESULTS:
% [[file:figs/open_loop_shaping_hinf_K.png]]
% Let's finally compare the obtained stability margins of the $\mathcal{H}_\infty$ controller and of the manually developed controller in Table [[tab:open_loop_shaping_compare]].
[Gm_2, Pm_2, ~, Wc_2] = margin(G*K);
matlab/h_infinity_2_control_specifications.m
+308
View File
@@ -0,0 +1,308 @@
%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% Let's define the system parameters on Matlab.
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
% And now the system dynamics $G(s)$ and $G_d(s)$.
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
% The Sensitivity Function
% <<sec:sensitivity_transfer_functions>>
% The sensitivity function is indisputably the most important closed-loop transfer function of a feedback system.
% In this section, we will see how the shape of the sensitivity function will impact the performances of the closed-loop system.
% Suppose we have developed a "/reference/" controller $K_r(s)$ and made three small changes to obtained three controllers $K_1(s)$, $K_2(s)$ and $K_3(s)$.
% The obtained sensitivity functions for these four controllers are shown in Figure [[fig:sensitivity_shape_effect]] and the corresponding step responses are shown in Figure [[fig:sensitivity_shape_effect_step]].
% The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table [[tab:compare_sensitivity_shapes]].
% #+name: tab:compare_sensitivity_shapes
% #+caption: Comparison of the sensitivity function shape and the corresponding step response for the three controller variations
% #+attr_latex: :environment tabularx :width 0.9\linewidth :align lXX
% | *Controller* | *Sensitivity Function Shape* | *Change of the Step Response* |
% |--------------+----------------------------------------------------+----------------------------------|
% | $K_1(s)$ | Larger bandwidth $\omega_b$ | Faster rise time |
% | $K_2(s)$ | Larger peak value $\Vert S\Vert_\infty$ | Large overshoot and oscillations |
% | $K_3(s)$ | Larger low frequency gain $\vert S(j\cdot 0)\vert$ | Larger static error |
wc = 2*pi*1; L_w = 8; wi = 2*pi*0.02;
Kr = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
Kr = Kr/abs(evalfr(Kr*G, 1j*wc));
wc = 2*pi*2; L_w = 8; wi = 2*pi*0.045;
K1 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K1 = K1/abs(evalfr(K1*G, 1j*wc));
wc = 2*pi*1; L_w = 2; wi = 2*pi*0.03;
K2 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K2 = K2/abs(evalfr(K2*G, 1j*wc));
wc = 2*pi*1; L_w = 8; wi = 2*pi*0.2;
K3 = 1/((s + wi)^2) * ... % Double Integrator
(1 + s/(wc/sqrt(L_w)))/(1 + s/(wc*sqrt(L_w))); % Lead
K3 = K3/abs(evalfr(K3*G, 1j*wc));
Sr = 1/(1 + Kr*G);
S1 = 1/(1 + K1*G);
S2 = 1/(1 + K2*G);
S3 = 1/(1 + K3*G);
Tr = Kr*G/(1 + Kr*G);
T1 = K1*G/(1 + K1*G);
T2 = K2*G/(1 + K2*G);
T3 = K3*G/(1 + K3*G);
freqs = logspace(-2, 2, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(S1, freqs, 'Hz'))), 'DisplayName', '$K_1(s)$');
plot(freqs, abs(squeeze(freqresp(S2, freqs, 'Hz'))), 'DisplayName', '$K_2(s)$');
plot(freqs, abs(squeeze(freqresp(S3, freqs, 'Hz'))), 'DisplayName', '$K_3(s)$');
plot(freqs, abs(squeeze(freqresp(Sr, freqs, 'Hz'))), 'k-', 'DisplayName', '$K_r(s)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Sensitivity Magnitude');
hold off;
legend('location', 'southeast');
ylim([1e-3, 3]);
% #+name: fig:sensitivity_shape_effect
% #+caption: Sensitivity function magnitude $|S(j\omega)|$ corresponding to the reference controller $K_r(s)$ and the three modified controllers $K_i(s)$
% #+RESULTS:
% [[file:figs/sensitivity_shape_effect.png]]
t = linspace(0, 5, 1000);
figure;
hold on;
plot(t, step(T1, t), 'DisplayName', '$K_1(s)$')
plot(t, step(T2, t), 'DisplayName', '$K_2(s)$')
plot(t, step(T3, t), 'DisplayName', '$K_3(s)$')
plot(t, step(Tr, t), 'k-', 'DisplayName', '$K_r(s)$')
hold off
xlabel('Time [s]'); ylabel('Step Response');
legend('location', 'northeast');
ylim([0, 1.7])
% Robustness: Module Margin
% <<sec:module_margin>>
% Let's start this section by an example demonstrating why the phase and gain margins might not be good indicators of robustness.
% Will follow a discussion about the module margin, a robustness indicator that can be linked to the $\mathcal{H}_\infty$ norm of $S$ and that will prove to be very useful.
% #+begin_exampl
% Consider the following plant $G_t(s)$:
w0 = 2*pi*100;
xi = 0.1;
k = 1e7;
Gt = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1);
% Let's say we have designed a controller $K_t(s)$ that gives the loop gain shown in Figure [[fig:phase_gain_margin_model_plant]].
Kt = 1.2e6*(s + w0)/s;
% The following characteristics can be determined from the Loop gain in Figure [[fig:phase_gain_margin_model_plant]]:
% - Control bandwidth of $\approx 10\, \text{Hz}$
% - Infinite gain margin (the phase of the loop-gain never reaches $-180^o$)
% - More than $90^o$ of phase margin
% This clearly indicate very good robustness of the closed-loop system!
% Or does it? Let's find out.
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gt*Kt, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gt*Kt, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-200, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:phase_gain_margin_model_plant
% #+caption: Bode plot of the obtained Loop Gain $L(s)$
% #+attr_latex: :float nil
% #+RESULTS:
% [[file:figs/phase_gain_margin_model_plant.png]]
% Now let's suppose the controller is implemented in practice, and the "real" plant $G_r(s)$ as a slightly lower damping factor than the one estimated for the model:
xi = 0.03;
Gr = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1);
% The obtained "real" loop gain is shown in Figure [[fig:phase_gain_margin_real_plant]].
% At a frequency little bit above 100Hz, the phase of the loop gain reaches -180 degrees while its magnitude is more than one which indicates instability.
% It is confirmed by checking the stability of the closed loop system:
isstable(feedback(Gr,Kt))
% #+RESULTS:
% : 0
freqs = logspace(0, 3, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gt*Kt, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
plot(freqs, abs(squeeze(freqresp(Gr*Kt, freqs, 'Hz'))), 'DisplayName', '$L_r(s)$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
legend('location', 'northeast');
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gt*Kt, freqs, 'Hz')))));
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gr*Kt, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-200, 0]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
% #+name: fig:phase_gain_margin_real_plant
% #+caption: Bode plots of $L(s)$ (loop gain corresponding the nominal plant) and $L_r(s)$ (loop gain corresponding to the real plant)
% #+attr_latex: :float nil
% #+RESULTS:
% [[file:figs/phase_gain_margin_real_plant.png]]
% Therefore, even a small change of the plant parameter renders the system unstable even though both the gain margin and the phase margin for the nominal plant are excellent.
% This is due to the fact that the gain and phase margin are robustness indicators corresponding a *pure* change or gain or a *pure* change of phase but not a combination of both.
% #+end_exampl
% Let's now determine a new robustness indicator based on the Nyquist Stability Criteria.
% #+begin_definition
% - Nyquist Stability Criteria (for stable systems) ::
% If the open-loop transfer function $L(s)$ is stable, then the closed-loop system will be unstable for any encirclement of the point $-1$ on the Nyquist plot.
% - Nyquist Plot ::
% The Nyquist plot shows the evolution of $L(j\omega)$ in the complex plane from $\omega = 0 \to \infty$.
% #+end_definition
% #+begin_seealso
% For more information about the /general/ Nyquist Stability Criteria, you may want to look at [[https://www.youtube.com/watch?v=sof3meN96MA][this]] video.
% #+end_seealso
% From the Nyquist stability criteria, it is clear that we want $L(j\omega)$ to be as far as possible from the $-1$ point (called the /unstable point/) in the complex plane.
% This minimum distance is called the *module margin*.
% #+begin_definition
% - Module Margin ::
% The Module Margin $\Delta M$ is defined as the *minimum distance* between the point $-1$ and the loop gain $L(j\omega)$ in the complex plane.
% #+end_definition
% #+begin_exampl
% A typical Nyquist plot is shown in Figure [[fig:module_margin_example]].
% The gain, phase and module margins are graphically shown to have an idea of what they represent.
% Example Plant
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
G = 1/(m*s^2 + c*s + k); % Plant
% Example Controller
K = 14e8 * ... % Gain
1/(s^2) * ... % Double Integrator
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
L = G*K;
L_resp = squeeze(freqresp(L, freqs, 'Hz'));
% Module Margin
Dm = min(abs(1 + L_resp));
% Phase Gain Margin
[Gm, Pm, Wcg, Wcp] = margin(L);
freqs = logspace(0, 3, 1000);
figure;
hold on;
% Gain Margin
plot([-1, -1/Gm], [0, 0], '-', 'DisplayName', sprintf('$\\Delta G = %.1f$', Gm))
% Phase Margin
theta = -pi:0.01:-pi+Pm*pi/180;
plot(cos(theta), sin(theta), '-', 'DisplayName', sprintf('$\\Delta \\phi = %.1f^o$', Pm));
% Module Margin
theta = 0 : 0.01 : 2*pi;
plot(Dm*cos(theta)-1, Dm*sin(theta), '-', 'DisplayName', sprintf('$\\Delta M = %.1f$', Dm));
% Nyquist Plot
plot(real(L_resp), imag(L_resp), 'k-', 'DisplayName', '$L(j\omega)$')
plot(-1, 0, 'k*', 'HandleVisibility', 'off');
hold off;
xlabel('Real Axis'); ylabel('Imaginary Axis');
xlim([-1.5, 0.5]); ylim([-1, 1]);
axis equal;
legend('location', 'southeast');
matlab/h_infinity_3_shaping_cl_tf.m
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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% Let's define the system parameters on Matlab.
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
% And now the system dynamics $G(s)$ and $G_d(s)$.
G = 1/(m*s^2 + c*s + k); % Plant
Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
% with:
% - =dcgain=: low frequency gain
% - =[freq,mag]=: frequency =freq= at which the gain is =mag=
% - =hfgain=: high frequency gain
% #+begin_exampl
% The Matlab code below produces a weighting function with the following characteristics (Figure [[fig:first_order_weight]]):
% - Low frequency gain of 100
% - Gain of 1 at 10Hz
% - High frequency gain of 0.5
Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
freqs = logspace(-2, 2, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Ws, freqs, 'Hz'))), 'k-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
% Let's use this function to generate three weights with the same high and low frequency gains, but but different slopes.
W1 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 1);
W2 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
W3 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 3);
% The obtained shapes are shown in Figure [[fig:high_order_weight]].
freqs = logspace(-2, 2, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), ...
'DisplayName', '$n = 1$');
plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), ...
'DisplayName', '$n = 2$');
plot(freqs, abs(squeeze(freqresp(W3, freqs, 'Hz'))), ...
'DisplayName', '$n = 3$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'northeast');
% #+HTML: </details>
% #+end_exercice
% Let's say we came up with the following weighting function:
Ws = generateWeight('G0', 1e3, ...
'G1', 1/2, ...
'Gc', sqrt(2), 'wc', 2*pi*10, ...
'n', 2);
% The weighting function is then added to the generalized plant.
P = [1 -G;
1 -G];
Pw = blkdiag(Ws, 1)*P;
% And the $\mathcal{H}_\infty$ synthesis is performed on the /weighted/ generalized plant.
K = hinfsyn(Pw, 1, 1, 'Display', 'on');
% #+RESULTS:
% #+begin_example
% Test bounds: 0.5 <= gamma <= 0.51
% gamma X>=0 Y>=0 rho(XY)<1 p/f
% 5.05e-01 0.0e+00 0.0e+00 3.000e-16 p
% Limiting gains...
% 5.05e-01 0.0e+00 0.0e+00 3.461e-16 p
% 5.05e-01 -3.5e+01 # -4.9e-14 1.732e-26 f
% Best performance (actual): 0.503
% #+end_example
% $\gamma \approx 0.5$ means that the $\mathcal{H}_\infty$ synthesis generated a controller $K(s)$ that stabilizes the closed-loop system, and such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer function from $w$ to $z$ is less than $\gamma$:
% \begin{align*}
% & \| W_s(s) S(s) \|_\infty \approx 0.5 \\
% & \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
% \end{align*}
% This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:results_sensitivity_hinf]].
% #+begin_important
% Obtaining $\gamma < 1$ means that the $\mathcal{H}_\infty$ synthesis found a controller such that the specified closed-loop transfer functions are bellow the specified upper bounds.
% Yet, obtaining a $\gamma$ slightly above one does not necessary means the synthesis is unsuccessful.
% It just means that at some frequency, one of the closed-loop transfer functions is above the specified upper bound by a factor $\gamma$.
% #+end_important
figure;
hold on;
plot(freqs, 1./abs(squeeze(freqresp(Ws, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W_s|^{-1}$');
plot(freqs, abs(squeeze(freqresp(1/(1 + K*G), freqs, 'Hz'))), 'k-', 'DisplayName', '$|S|$');
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast', 'FontSize', 8);
matlab/h_infinity_4_mixed_sensitivity_example.m
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%% Clear Workspace and Close figures
clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% #+name: fig:classical_feedback_test_system_bis
% #+caption: Block diagram corresponding to the example system
% #+RESULTS:
% [[file:figs/ex_test_system_feedback.png]]
% The two transfer functions present in the system are derived and defined below:
k = 1e6; % Stiffness [N/m]
c = 4e2; % Damping [N/(m/s)]
m = 10; % Mass [kg]
% Control Plant
G = 1/(m*s^2 + c*s + k);
% Disturbance dynamics
Gd = (c*s + k)/(m*s^2 + c*s + k);
% We also define the inputs signals that will be used for time domain simulations.
% They are graphically shown in Figure [[fig:ex_inputs_signals]].
% Time Vector
t = 0:1e-4:0.5;
% Reference Input
r = zeros(size(t));
r(t>0.05 & t<=0.25) = 1e-4*(t(t>0.05 & t<=0.25)-0.05);
r(t>0.25) = 2e-5;
% Measurement Noise
Fs = 1e3; % Sampling Frequency [Hz]
Ts = 1/Fs; % Sampling Time [s]
n = sqrt(Fs/2)*randn(1, length(t)); % Signal with an ASD equal to one
n = lsim(1e-6*(s + 2*pi*1e2)^2/(s + 2*pi*1e3)^2/(1+s/2/pi/500), n, t)'; % Shaped noise
% Disturbance
d = zeros(size(t));
d(t>0.3) = 5e-6;
figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
plot(t, 1e6*r, 'k-')
xlabel('Time [s]');
ylabel('Reference Signal $[\mu m]$');
nexttile;
plot(t, 1e6*n, 'k-')
xlabel('Time [s]');
ylabel('Measurement Noise $[\mu m]$');
nexttile;
plot(t, 1e6*d, 'k-')
xlabel('Time [s]');
ylabel('Disturbance Motion $[\mu m]$');
% #+name: fig:ex_general_plant_sim
% #+caption: Generalized plant that will be used for simulations
% #+RESULTS:
% [[file:figs/ex_general_plant_sim.png]]
% The Generalized plant of Figure [[fig:ex_general_plant_sim]] is defined on Matlab as follows:
Psim = [0 0 Gd G
0 0 0 1
1 -1 -Gd -G];
Psim.InputName = {'r', 'n', 'd', 'u'};
Psim.OutputName = {'y', 'u', 'e'};
% Modern Interpretation of control specifications
% <<sec:ex_specification_interpretation>>
% #+begin_exercice
% 1. Translate the control specifications into wanted shape of closed-loop transfer functions
% 2. Conclude and the closed-loop transfer functions to be shaped
% 3. Chose a general configuration architecture that allows to shape these transfer function
% 4. Using Matlab, define the generalized plant
% #+HTML: <details><summary>Hint</summary>
% #+LATEX: \tcbsubtitle{Hint}
% 1. Make use of Table [[tab:usual_shaping_gang_four]]
% 2. Make use of Table [[tab:usual_shaping_gang_four]]
% 3. See Section [[sec:shaping_multiple_tf]]
% 4. See Section [[sec:shaping_multiple_tf]]
% #+HTML: </details>
% #+end_exercice
% After converting the control specifications into wanted shape of closed-loop transfer functions, we might come up with the Table [[tab:ex_specification_shapes]].
% In such case, we want to shape $S$, $GS$ and $T$.
% #+name: tab:ex_specification_shapes
% #+caption: Control Specifications and associated wanted shape of the closed-loop transfer functions
% #+attr_latex: :environment tabularx :width 0.7\linewidth :align llX
% | <l> | <c> | <l> |
% | *Specification* | *TF* | *Wanted Shape* |
% |--------------------------+-----------+------------------------------------|
% | Follow Step Reference | $S$ | +40dB of slope at low frequency |
% | Reject Disturbances | $S$, $GS$ | Small gain |
% | Reject measurement noise | $T$ | Small high frequency (>100Hz) gain |
% | Robust System | $S$ | Small $\Vert S \Vert_\infty$ |
% To do so, we use to generalized plant shown in Figure [[fig:ex_general_plant]] for the synthesis where the three closed-loop tranfert functions from $w$ to $[z_1\,,z_2\,,z_3]$ are respectively $S$, $GS$ and $T$.
% This generalized plant is defined on Matlab as follows:
P = [1 -1
G -G
0 1
G -G];
% Step 1 - Shaping of $S$
% <<sec:ex_shaping_S>>
% Let's first shape the Sensitivity function as it is usually the most important of the /Gang of four/ closed-loop transfer functions.
% To do so, we have to create a weight $W_1(s)$ that defines the wanted upper bound on $|S(j\omega)|$:
% - small low frequency gain: $|S(j\cdot 0)| = 10^{-3}$
% - minimum crossover frequency of $\approx 10Hz$: $|S(j2\pi 10)| < \frac{1}{\sqrt{2}}$
% - small maximum peak magnitude for robustness properties: $\|S\|_\infty < 2$
% The weighting function is design using the =generateWeigh= function and its inverse shape can be seen in Figure
W1 = generateWeight('G0', 1e3, ...
'G1', 1/2, ...
'Gc', sqrt(2), 'wc', 2*pi*10, ...
'n', 1);
% To not constrain $GS$ and $T$ for the shaping of $S$, $W_2$ and $W_3$ are first taken as very small gains:
W2 = tf(1e-8);
W3 = tf(1e-8);
% The $\mathcal{H}_\infty$ synthesis is performed and the obtained closed-loop transfer functions $S$, $GS$, and $T$ and compared with the upper bounds set by the weighting functions in Figure [[fig:ex_results_1]].
Pw = blkdiag(W1, W2, W3, 1)*P;
K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
% #+RESULTS:
% #+begin_example
% Test bounds: 0.5 <= gamma <= 0.51
% gamma X>=0 Y>=0 rho(XY)<1 p/f
% 5.05e-01 0.0e+00 0.0e+00 5.511e-14 p
% Limiting gains...
% 5.05e-01 0.0e+00 0.0e+00 1.867e-14 p
% Best performance (actual): 0.502
% #+end_example
% Closed-loop tranfer functions are computed
Z1 = lft(P, K1);
freqs = logspace(-1, 3, 1000);
figure;
tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1');
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W|^{-1}$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('S');
legend('location', 'southeast');
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('GS');
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('T');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('K');
linkaxes([ax1, ax2, ax3, ax4], 'x')
xlim([freqs(1), freqs(end)]);
% #+name: fig:ex_results_1
% #+caption: Obtained Shape Closed-Loop transfer functions (dashed black lines indicate inverse magnitude of the weighting functions)
% #+RESULTS:
% [[file:figs/ex_results_1.png]]
% Time domain simulation is then performed and the obtained output displacement and control inputs are shown in Figure [[fig:ex_time_domain_1]].
% We can see:
% - we are not able to follow the ramp input. This have to be solved by modifying the weighting function $W_1(s)$
% - we have poor rejection of disturbances. This we be solve by shaping $GS$ in Section [[sec:ex_shaping_GS]]
% - we have quite large effect of the measurement noise. This will be solved by shaping $T$ in Section [[sec:ex_shaping_T]]
% Closed-loop simulation
Z1sim = lft(Psim, K1);
z = lsim(Z1sim, [r;n;d], t);
y1 = z(:,1);
u1 = z(:,2);
figure;
tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
nexttile;
hold on;
plot(t, 1e6*y1, 'DisplayName', 'Step 1');
plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast');
% #+name: fig:ex_time_domain_1
% #+caption: Time domain simulation results
% #+RESULTS:
% [[file:figs/ex_time_domain_1.png]]
% Remember that in order to follow ramp inputs, the sensitivity function should have a slope of +40dB/dec at low frequency (Table [[tab:usual_shaping_gang_four]]).
% To do so, let's modify $W_1$ to impose a slope of +40dB/dec at low frequency.
% This can simple be done by using a second order weight:
W1 = generateWeight('G0', 1e3, ...
'G1', 1/2, ...
'Gc', sqrt(2), 'wc', 2*pi*15, ...
'n', 2);
% The $\mathcal{H}_\infty$ synthesis is performed using the new weights and the obtained closed-loop shaped are shown in figure [[fig:ex_results_1b]].
% The time domain signals are shown in Figure [[fig:ex_time_domain_1b]] and it is confirmed that the ramps are now follows without static errors.
Pw = blkdiag(W1, W2, W3, 1)*P;
K1b = hinfsyn(Pw, 1, 1, 'Display', 'on');
% Compute the closed-loop transfer function
Z1b = lft(P, K1b);
freqs = logspace(-1, 3, 1000);
figure;
tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1');
plot(freqs, abs(squeeze(freqresp(Z1b(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1b');
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W^{-1}|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('S');
legend('location', 'southeast');
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z1b(2,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('GS');
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z1b(3,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('T');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('K');
linkaxes([ax1, ax2, ax3, ax4], 'x')
xlim([freqs(1), freqs(end)]);
% #+name: fig:ex_results_1b
% #+caption: Obtained Shape Closed-Loop transfer functions
% #+RESULTS:
% [[file:figs/ex_results_1b.png]]
% Closed-loop simulations
Z1bsim = lft(Psim, K1b);
z = lsim(Z1bsim, [r;n;d], t);
y1b = z(:,1);
u1b = z(:,2);
figure;
tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
hold on;
plot(t, 1e6*y1, 'DisplayName', 'Step 1');
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast');
% Step 2 - Shaping of $GS$
% <<sec:ex_shaping_GS>>
% Looking at Figure [[fig:ex_results_2]], it is clear that the rejection of disturbances is not satisfactory.
% This can also be seen by the large peak of $GS$ in Figure [[fig:ex_results_1b]].
% This poor rejection of disturbances is actually due to the fact that the obtain controller has at notch at the resonance frequency of the plant.
% To overcome this issue, we can simply increase the magnitude of $W_2$ to limit the peak magnitude of $GS$
% Let's take $W_2$ as a simple constant gain:
W2 = tf(4e5);
% The $\mathcal{H}_\infty$ Synthesis is performed and the obtained closed-loop transfer functions are shown in Figure [[fig:ex_results_2]].
Pw = blkdiag(W1, W2, W3, 1)*P;
K2 = hinfsyn(Pw, 1, 1, 'Display', 'on');
Z2 = lft(P, K2);
freqs = logspace(-1, 3, 1000);
figure;
tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1');
plot(freqs, abs(squeeze(freqresp(Z1b(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1b');
plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 2');
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W^{-1}|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('S');
legend('location', 'southeast');
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z1b(2,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('GS');
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z1b(3,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('T');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K2, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('K');
linkaxes([ax1, ax2, ax3, ax4], 'x')
xlim([freqs(1), freqs(end)]);
% #+name: fig:ex_results_2
% #+caption: Obtained Shape Closed-Loop transfer functions
% #+RESULTS:
% [[file:figs/ex_results_2.png]]
% Time domain simulation results are shown in Figure [[fig:ex_time_domain_2]].
% If is shown that indeed, the disturbance rejection performance are much better and only very small oscillation is obtained.
Z2sim = lft(Psim, K2);
z = lsim(Z2sim, [r;n;d], t);
y2 = z(:,1);
u2 = z(:,2);
figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
hold on;
plot(t, 1e6*y1, 'DisplayName', 'Step 1');
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
plot(t, 1e6*y2, 'DisplayName', 'Step 2');
plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast');
% Step 3 - Shaping of $T$
% <<sec:ex_shaping_T>>
% Finally, we want to limit the effect of the noise on the displacement output.
% To do so, $T$ is shaped such that its high frequency gain is reduced.
% This is done by increasing the high frequency gain of the weighting function $W_3$ until the $\mathcal{H}_\infty$ synthesis gives $\gamma \approx 1$.
% The final weighting function $W_3$ is defined as follows:
W3 = generateWeight('G0', 1e-1, ...
'G1', 1e4, ...
'Gc', 1, 'wc', 2*pi*70, ...
'n', 2);
% The $\mathcal{H}_\infty$ synthesis is performed and $\gamma$ is closed to one.
Pw = blkdiag(W1, W2, W3, 1)*P;
K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
% #+RESULTS:
% #+begin_example
% Test bounds: 0.622 <= gamma <= 1.62
% gamma X>=0 Y>=0 rho(XY)<1 p/f
% 1.01e+00 -1.3e+07 # -6.4e-18 3.459e-04 f
% 1.28e+00 0.0e+00 0.0e+00 5.369e-02 p
% 1.13e+00 3.8e-15 0.0e+00 1.695e-01 p
% 1.07e+00 -5.0e-08 -1.2e-19 1.622e+00 # f
% 1.10e+00 0.0e+00 0.0e+00 3.147e-01 p
% 1.08e+00 0.0e+00 0.0e+00 5.327e-01 p
% 1.08e+00 3.4e-15 0.0e+00 8.052e-01 p
% Best performance (actual): 1.08
% #+end_example
% The obtained closed-loop transfer functions are shown in Figure [[fig:ex_results_3]] and we can obverse that:
% - The high frequency gain of $T$ is indeed reduced
% - This comes as the expense of a large magnitude both $GS$ and $S$.
% This means we will probably have slightly less good disturbance rejection and tracking performances.
Z3 = lft(P, K3);
freqs = logspace(-1, 3, 1000);
figure;
tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1');
plot(freqs, abs(squeeze(freqresp(Z1b(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 1b');
plot(freqs, abs(squeeze(freqresp(Z2(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 2');
plot(freqs, abs(squeeze(freqresp(Z3(1,1), freqs, 'Hz'))), 'DisplayName', 'Step 3');
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W^{-1}|$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('S');
legend('location', 'southeast');
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(2,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z1b(2,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z2(2,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z3(2,1), freqs, 'Hz'))));
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('GS');
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(Z1(3,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z1b(3,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z2(3,1), freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Z3(3,1), freqs, 'Hz'))));
plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), 'k--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('T');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(K1, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K1b, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(K3, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
title('K');
linkaxes([ax1, ax2, ax3, ax4], 'x')
xlim([freqs(1), freqs(end)]);
% #+name: fig:ex_results_3
% #+caption: Obtained Shape Closed-Loop transfer functions
% #+RESULTS:
% [[file:figs/ex_results_3.png]]
% The time domain simulation signals are shown in Figure [[fig:ex_time_domain_3]].
% We can indeed see a slightly less good disturbance rejection.
% However, the vibrations induced by the sensor noise is well reduced.
% This can be seen when zooming on the output signal in Figure [[fig:ex_time_domaim_3_zoom]].
Z3sim = lft(Psim, K3);
z = lsim(Z3sim, [r;n;d], t);
y3 = z(:,1);
u3 = z(:,2);
figure;
tiledlayout(1, 1, 'TileSpacing', 'None', 'Padding', 'None');
hold on;
plot(t, 1e6*y1, 'DisplayName', 'Step 1');
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
plot(t, 1e6*y2, 'DisplayName', 'Step 2');
plot(t, 1e6*y3, 'DisplayName', 'Step 3');
plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast');
% #+name: fig:ex_time_domain_3
% #+caption: Time domain simulation results
% #+RESULTS:
% [[file:figs/ex_time_domain_3.png]]
figure;
hold on;
plot(t, 1e6*y1, 'DisplayName', 'Step 1');
plot(t, 1e6*y1b, 'DisplayName', 'Step 1b');
plot(t, 1e6*y2, 'DisplayName', 'Step 2');
plot(t, 1e6*y3, 'DisplayName', 'Step 3');
plot(t, 1e6*r, 'k--', 'DisplayName', '$r$');
hold off;
xlabel('Time [s]'); ylabel('Output $y$ [$\mu m$]');
legend('location', 'southeast');
xlim([0.4, 0.5])
preamble.tex
+10 -10
View File
@@ -50,17 +50,17 @@
\tcbuselibrary{minted} \tcbuselibrary{minted}
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\newtcolorbox{hint}{ enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint} \newtcolorbox{hint}[1][]{ enhanced,breakable,colback=my-pale-grey,colframe=my-grey,fonttitle=\bfseries,title=Hint,#1}
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\newtcolorbox{my-quote}[1]{% \newtcolorbox{my-quote}[1]{%
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