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@ -38,7 +38,6 @@
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
* TODO Introduction :ignore:
* 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>>
@ -49,7 +48,6 @@
#+end_src
* Introduction to the Control Methodology - Model Based Control
** Control Methodology
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure [[fig:control-procedure]].
It consists of three steps:
@ -93,490 +91,52 @@ It consists of three steps:
In this document, we will mainly focus on steps 2 and 3.
** Some Background: From Classical Control to Robust Control
#+name: tab:comparison_control_methods
#+caption: Table summurazing the main differences between classical, modern and robust control
| | *Classical Control* | *Modern Control* | *Robust Control* |
| <l> | <c> | <c> | <c> |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Date* | 1930- | 1960- | 1980- |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Tools* | Transfer Functions | State Space formulation | Systems and Signals Norms ($\mathcal{H}_\infty$, $\mathcal{H}_2$ Norms) |
| | Nyquist Plots | Riccati Equations | Closed Loop Transfer Functions |
| | Bode Plots | | Open/Closed Loop Shaping |
| | Phase and Gain margins | | Weighting Functions |
| | | | Disk margin |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Control Architectures* | Proportional, Integral, Derivative | Full State Feedback | General Control Configuration |
| | Leads, Lags | LQR, LQG | |
| | | Kalman Filters | |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Advantages* | Study Stability | Automatic Synthesis | Automatic Synthesis |
| | Simple | MIMO | MIMO |
| | Natural | Optimization Problem | Optimization Problem |
| | | | Guaranteed Robustness |
| | | | Easy specification of performances |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Disadvantages* | Manual Method | No Guaranteed Robustness | Required knowledge of specific tools |
| | Only SISO | Difficult Rejection of Perturbations | Need a reasonably good model of the system |
** Example System
Let's consider the model shown in Figure [[fig:mech_sys_1dof_inertial_contr]].
It could represent a suspension system with a payload to position or isolate using an force actuator and an inertial sensor.
The notations used are listed in Table [[tab:example_notations]].
#+begin_src latex :file mech_sys_1dof_inertial_contr.pdf
\begin{tikzpicture}
% Parameters
\def\massw{3}
\def\massh{1}
\def\spaceh{1.8}
% Ground
\draw[] (-0.5*\massw, 0) -- (0.5*\massw, 0);
% Mass
\draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5](m){$m$};
% Spring, Damper, and Actuator
\draw[spring] (-0.3*\massw, 0) -- (-0.3*\massw, \spaceh) node[midway, left=0.1]{$k$};
\draw[damper] ( 0, 0) -- ( 0, \spaceh) node[midway, left=0.3]{$c$};
\draw[actuator] ( 0.3*\massw, 0) -- (0.3*\massw, \spaceh) node[midway](F){};
% Displacements
\draw[dashed] (0.5*\massw, 0) -- ++(0.5, 0);
\draw[->] (0.6*\massw, 0) -- ++(0, 0.5) node[below right]{$d$};
% Inertial Sensor
\node[inertialsensor] (inertials) at (0.5*\massw, \spaceh+\massh){};
\node[addb={+}{-}{}{}{}, right=0.8 of inertials] (subf) {};
\node[block, below=0.4 of subf] (K){$K(s)$};
\draw[->] (inertials.east) node[above right]{$y$} -- (subf.west);
\draw[->] (subf.south) -- (K.north) node[above right]{$\epsilon$};
\draw[<-] (subf.north) -- ++(0, 0.6) node[below right]{$r$};
\draw[->] (K.south) |- (F.east) node[above right]{$u$};
\end{tikzpicture}
#+end_src
#+name: fig:mech_sys_1dof_inertial_contr
#+caption: Test System consisting of a payload with a mass $m$ on top of an active system with a stiffness $k$, damping $c$ and an actuator. A feedback controller $K(s)$ is added to position / isolate the payload.
#+RESULTS:
[[file:figs/mech_sys_1dof_inertial_contr.png]]
#+name: tab:example_notations
#+caption: Example system variables
| *Notation* | *Description* | *Value* | *Unit* |
|--------------------+----------------------------------------------------------------+----------------+-----------|
| $m$ | Payload's mass to position / isolate | $10$ | [kg] |
| $k$ | Stiffness of the suspension system | $10^6$ | [N/m] |
| $c$ | Damping coefficient of the suspension system | $400$ | [N/(m/s)] |
| $y$ | Payload absolute displacement (measured by an inertial sensor) | | [m] |
| $d$ | Ground displacement, it acts as a disturbance | | [m] |
| $u$ | Actuator force | | [N] |
| $r$ | Wanted position of the mass (the reference) | | [m] |
| $\epsilon = r - y$ | Position error | | [m] |
| $K$ | Feedback controller | to be designed | [N/m] |
#+begin_exercice
Derive the following open-loop transfer functions:
\begin{align}
G(s) &= \frac{y}{u} \\
G_d(s) &= \frac{y}{d}
\end{align}
#+HTML: <details><summary>Hint</summary>
You can follow this generic procedure:
1. List all applied forces ot the mass: Actuator force, Stiffness force (Hooke's law), ...
2. Apply the Newton's Second Law on the payload
\[ m \ddot{y} = \Sigma F \]
3. Transform the differential equations into the Laplace domain:
\[ \frac{d\ \cdot}{dt} \Leftrightarrow \cdot \times s \]
4. Write $y(s)$ as a function of $u(s)$ and $w(s)$
#+HTML: </details>
#+HTML: <details><summary>Results</summary>
\begin{align}
G(s) &= \frac{1}{m s^2 + cs + k} \\
G_d(s) &= \frac{cs + k}{m s^2 + cs + k}
\end{align}
#+HTML: </details>
#+end_exercice
Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure [[fig:mech_sys_1dof_inertial_contr]] into a classical feedback form as shown in Figure [[fig:open_loop_shaping]].
#+begin_src latex :file classical_feedback_test_system.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
\node[block, right=0.8 of addfb] (K){$K(s)$};
\node[block, right=0.8 of K] (G){$G(s)$};
\node[addb={+}{}{}{}{}, right=0.8 of G] (addd){};
\node[block, above=0.5 of addd] (Gd){$G_d(s)$};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- (addd.west);
\draw[<-] (Gd.north) -- ++(0, 0.8) node[below right]{$d$};
\draw[->] (Gd.south) -- (addd.north);
\draw[->] (addd.east) -- ++(1.2, 0);
\draw[->] ($(addd.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -1.0) -| (addfb.south);
\end{tikzpicture}
#+end_src
#+name: fig:classical_feedback_test_system
#+caption: Block diagram corresponding to the example system
#+RESULTS:
[[file:figs/classical_feedback_test_system.png]]
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 $G_d(s)$ (their bode plots are shown in Figures [[fig:bode_plot_example_afm]] and [[fig:bode_plot_example_Gd]]).
#+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
#+begin_src matlab :exports none
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, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
hold off;
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
yticks(-360:90:360); ylim([-270, 90]);
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/bode_plot_example_afm.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:bode_plot_example_afm
#+caption: Bode plot of the plant $G(s)$
#+RESULTS:
[[file:figs/bode_plot_example_afm.png]]
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
hold off;
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/bode_plot_example_Gd.pdf', 'width', 'wide', 'height', 'small');
#+end_src
#+name: fig:bode_plot_example_Gd
#+caption: Magnitude of the disturbance transfer function $G_d(s)$
#+RESULTS:
[[file:figs/bode_plot_example_Gd.png]]
* Classical Open Loop Shaping
** Introduction to Open Loop Shaping
#+begin_definition
The *Loop Gain* $L(s)$ usually refers to as the product of the controller and the plant (Figure [[fig:open_loop_shaping]]):
\begin{equation}
L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
\end{equation}
#+begin_src latex :file open_loop_shaping.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
\node[block, right=0.8 of addsub] (K) {$K(s)$};
\node[below] at (K.south) {Controller};
\node[block, right=0.8 of K] (G) {$G(s)$};
\node[below] at (G.south) {Plant};
\draw[<-] (addsub.west) -- ++(-0.8, 0) node[above right]{$r$};
\draw[->] (addsub) -- (K.west) node[above left]{$\epsilon$};
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
\draw[->] (G.east) -- ++(0.8, 0) node[above left]{$y$};
\draw[] ($(G.east) + (0.5, 0)$) -- ++(0, -1.4);
\draw[->] ($(G.east) + (0.5, -1.4)$) -| (addsub.south);
\draw [decoration={brace, raise=5pt}, decorate] (K.north west) -- node[above=6pt]{$L(s)$} (G.north east);
\end{tikzpicture}
#+end_src
#+name: fig:open_loop_shaping
#+caption: Classical Feedback Architecture
[[file:figs/open_loop_shaping.png]]
#+end_definition
#+begin_definition
*Open Loop Shaping* refers to a control design technique where the controller $K(s)$ is designed such that the *Open Loop Gain* $L(s)$ has desirable shape.
#+end_definition
This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$:
- *Performance*: $L$ large
- *Good disturbance rejection*: $L$ large
- *Limitation of measurement noise on plant output*: $L$ small
- *Small magnitude of input signal*: $K$ and $L$ small
- *Nominal stability*: $L$ small (RHP zeros and time delays)
- *Robust stability*: $L$ small (neglected dynamics)
The Open Loop shape is usually done manually has the loop gain $L(s)$ depends linearly on $K(s)$ eqref:eq:loop_gain.
$K(s)$ then consists of a combination of leads, lags, notches, etc. such that $L(s)$ has the wanted shape.
** Example of Open Loop Shaping
#+begin_exampl
Let's take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:
- *Performance*: Bandwidth of approximately 10Hz
- *Noise Attenuation*: Roll-off of -40dB/decade past 30Hz
- *Robustness*: Gain margin > 3dB and Phase margin > 30 deg
#+end_exampl
#+begin_exercice
Using =SISOTOOL=, design a controller that fulfill the specifications.
#+begin_src matlab :eval no
sisotool(G)
#+end_src
#+end_exercice
In order to have the wanted Roll-off, two integrators are used, a lead is also added to have sufficient phase margin.
The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure [[fig:loop_gain_manual_afm]].
#+begin_src matlab
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
#+end_src
#+begin_src matlab :exports none
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)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/loop_gain_manual_afm.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name: fig:loop_gain_manual_afm
#+caption: Bode Plot of the obtained Loop Gain $L(s) = G(s) K(s)$
#+RESULTS:
[[file:figs/loop_gain_manual_afm.png]]
And we can verify that we have the wanted stability margins:
#+begin_src matlab
[Gm, Pm, ~, Wc] = margin(G*K)
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Manual Method'}, ' %.1f ');
#+end_src
#+RESULTS:
| | Manual Method |
|-----------------------------+---------------|
| Gain Margin $> 3$ [dB] | 3.1 |
| Phase Margin $> 30$ [deg] | 35.4 |
| Crossover $\approx 10$ [Hz] | 10.1 |
** $\mathcal{H}_\infty$ Loop Shaping Synthesis
The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ Synthesis.
Even though we will not go into details, we will provide one example.
Using Matlab, the $\mathcal{H}_\infty$ synthesis of a controller based on the wanted open loop shape can be performed using the =loopsyn= command:
#+begin_src matlab :eval no
K = loopsyn(G, Gd);
#+end_src
where:
- =G= is the (LTI) plant
- =Gd= is the wanted loop shape
- =K= is the synthesize controller
#+begin_seealso
Matlab documentation of =loopsyn= ([[https://www.mathworks.com/help/robust/ref/loopsyn.html][link]]).
#+end_seealso
** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
Let's reuse the previous plant.
Translate the specification into the wanted shape of the open loop gain.
- *Performance*: Bandwidth of approximately 10Hz: $|L_w(j2 \pi 10)| = 1$
- *Noise Attenuation*: Roll-off of -40dB/decade past 30Hz
- *Robustness*: Gain margin > 3dB and Phase margin > 30 deg
#+begin_src matlab
Lw = 2.3e3 * ...
1/(s^2) * ... % Double Integrator
(1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead
#+end_src
The $\mathcal{H}_\infty$ optimal open loop shaping is performed using the =loopsyn= command:
#+begin_src matlab
[K, ~, GAM] = loopsyn(G, Lw);
#+end_src
The Bode plot of the obtained controller is shown in Figure [[fig:open_loop_shaping_hinf_K]].
#+begin_important
It is always important to analyze the controller after the synthesis is performed.
In the end, a synthesize controller is just a combination of low pass filters, high pass filters, notches, leads, etc.
#+end_important
Let's briefly analyze this controller:
- 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)
#+begin_src matlab :exports none
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)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/open_loop_shaping_hinf_K.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+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]]
The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]].
#+begin_src matlab :exports none
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)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/open_loop_shaping_hinf_L.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+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]]
Let's now 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
[Gm_2, Pm_2, ~, Wc_2] = margin(G*K)
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([Gm, Gm_2; Pm, Pm_2; Wc/2/pi, Wc_2/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Specifications', 'Manual Method', '$\mathcal{H}_\infty$ Method'}, ' %.1f ');
#+end_src
#+name: tab:open_loop_shaping_compare
#+caption: Comparison of the characteristics obtained with the two methods
#+RESULTS:
| Specifications | Manual Method | $\mathcal{H}_\infty$ Method |
|-----------------------------+---------------+-----------------------------|
| Gain Margin $> 3$ [dB] | 3.1 | 31.7 |
| Phase Margin $> 30$ [deg] | 35.4 | 54.7 |
| Crossover $\approx 10$ [Hz] | 10.1 | 9.9 |
* First Step in the $\mathcal{H}_\infty$ world
** The $\mathcal{H}_\infty$ Norm
* Some Background: From Classical Control to Robust Control
Classical Control (1930)
- Tools:
- TF (input-output)
- Nyquist, Bode, Black, \ldots
- P-PI-PID, Phase lead-lag, \ldots
- Advantages:
- Stability
- Performances
- Robustness
- Disadvantages:
- Manual Method
- Only SISO
Modern Control (1960)
- Tools:
- State Space
- Optimal Command
- LQR, LQG
- Advantages:
- Automatic Synthesis
- MIMO
- Optimisation problem
- Disadvantages:
- Robustness
- Rejection of Perturbations
Robust Control (1980)
- Tools:
- Disk Margin
- Systems and Signals norms ($\mathcal{H}_\infty$ and $\mathcal{H}_2$ norms)
- Closed Loop Transfer Functions
- Loop Shaping
- Advantages:
- Stability
- Performances
- Robustness
- Automatic Synthesis
- MIMO
- Optimization Problem
- Disadvantages:
- Requires the knowledge of specific tools
- Need a reasonably good model of the system
* The $\mathcal{H}_\infty$ Norm
#+begin_definition
The $\mathcal{H}_\infty$ norm is defined as the peak of the maximum singular value of the frequency response
@ -591,24 +151,31 @@ Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ con
#+end_definition
#+begin_exampl
Let's define a plant dynamics:
#+begin_src matlab
w0 = 2*pi; k = 1e6; xi = 0.04;
G = 1/k/(s^2/w0^2 + 2*xi*s/w0 + 1);
#+end_src
And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function:
#+begin_src matlab :results value replace
hinfnorm(G)
#+end_src
#+RESULTS:
: 7.9216e-06
: 1.0013e-05
The magnitude $|G(j\omega)|$ of the plant $G(s)$ as a function of frequency is shown in Figure [[fig:hinfinity_norm_siso_bode]].
The maximum value of the magnitude over all frequencies does correspond to the $\mathcal{H}_\infty$ norm of $G(s)$ as Equation eqref:eq:hinf_norm_siso implies.
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
freqs = logspace(-1, 1, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-');
plot([20, 100], [hinfnorm(G) hinfnorm(G)], 'k--');
text(100, hinfnorm(G), '$\quad \|G\|_\infty$')
plot([0.5, 2], [hinfnorm(G) hinfnorm(G)], 'k--');
text(2, hinfnorm(G), '$\quad \|G\|_\infty$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude $|G(j\omega)|$');
@ -625,7 +192,7 @@ The maximum value of the magnitude over all frequencies does correspond to the $
[[file:figs/hinfinity_norm_siso_bode.png]]
#+end_exampl
** $\mathcal{H}_\infty$ Synthesis
* $\mathcal{H}_\infty$ Synthesis
*Optimization problem*:
$\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the $\mathcal{H}_\infty$ norms of defined transfer functions are minimized.
@ -642,7 +209,7 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
- Fixed-Structure $\mathcal{H}_\infty$ Synthesis
- Signal Based $\mathcal{H}_\infty$ Synthesis
** The Generalized Plant
* The Generalized Plant
#+begin_src latex :file general_plant.pdf
\begin{tikzpicture}
\node[block={2.0cm}{2.0cm}] (P) {$P$};
@ -680,7 +247,45 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
\begin{bmatrix} z \\ v \end{bmatrix} = P \begin{bmatrix} w \\ u \end{bmatrix} = \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w \\ u \end{bmatrix}
\end{equation}
** From a Classical Feedback Architecture to a Generalized Plant
* Problem Formulation
#+begin_important
The $\mathcal{H}_\infty$ Synthesis objective is to find all stabilizing controllers $K$ which minimize
\begin{equation}
\| F_l(P, K) \|_\infty = \max_{\omega} \overline{\sigma} \big( F_l(P, K)(j\omega) \big)
\end{equation}
#+end_important
#+begin_src latex :file general_control_names.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {$P$};
\node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$};
% Input and outputs coordinates
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
% Connections and labels
\draw[<-] (inputw) node[above left, align=right]{(weighted)\\exogenous inputs\\$w$} -- ++(-1.5, 0);
\draw[<-] (inputu) -- ++(-0.8, 0) |- node[left, near start, align=right]{control signals\\$u$} (K.west);
\draw[->] (outputz) node[above right, align=left]{(weighted)\\exogenous outputs\\$z$} -- ++(1.5, 0);
\draw[->] (outputv) -- ++(0.8, 0) |- node[right, near start, align=left]{sensed output\\$v$} (K.east);
\end{tikzpicture}
#+end_src
#+name: fig:general_control_names
#+caption: General Control Configuration
#+RESULTS:
[[file:figs/general_control_names.png]]
* Classical feedback control and closed loop transfer functions
#+begin_src latex :file classical_feedback.pdf
\begin{tikzpicture}
@ -716,6 +321,7 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
| $d$ | Input Disturbance |
| $\epsilon$ | Tracking Error |
* From a Classical Feedback Architecture to a Generalized Plant
The procedure is:
1. define signals of the generalized plant
2. Remove $K$ and rearrange the inputs and outputs
@ -759,12 +365,12 @@ The procedure is:
\begin{scope}[on background layer]
\node[fit={(G.south-|start.west) ($(z1.north west)+(-0.4, 0)$)}, inner sep=6pt, draw, dashed, fill=black!20!white] (P) {};
\node[below] at (P.north) {Generalized Plant $P(s)$};
\node[below right] at (P.north west) {Generalized Plant $P(s)$};
\end{scope}
\end{tikzpicture}
#+end_src
#+begin_exercice
#+begin_exampl
Let's find the Generalized plant of corresponding to the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]]
#+name: fig:classical_feedback_tracking
@ -790,46 +396,7 @@ The procedure is:
0 1;
1 -G]
#+end_src
#+end_exercice
** The General Synthesis Problem Formulation
#+begin_important
The $\mathcal{H}_\infty$ Synthesis objective is to find all stabilizing controllers $K$ which minimize
\begin{equation}
\| F_l(P, K) \|_\infty = \max_{\omega} \overline{\sigma} \big( F_l(P, K)(j\omega) \big)
\end{equation}
#+end_important
#+begin_src latex :file general_control_names.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {$P$};
\node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$};
% Input and outputs coordinates
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
% Connections and labels
\draw[<-] (inputw) node[above left, align=right]{(weighted)\\exogenous inputs\\$w$} -- ++(-1.5, 0);
\draw[<-] (inputu) -- ++(-0.8, 0) |- node[left, near start, align=right]{control signals\\$u$} (K.west);
\draw[->] (outputz) node[above right, align=left]{(weighted)\\exogenous outputs\\$z$} -- ++(1.5, 0);
\draw[->] (outputv) -- ++(0.8, 0) |- node[right, near start, align=left]{sensed output\\$v$} (K.east);
\end{tikzpicture}
#+end_src
#+name: fig:general_control_names
#+caption: General Control Configuration
#+RESULTS:
[[file:figs/general_control_names.png]]
#+end_exampl
* Modern Interpretation of the Control Specifications
** Introduction