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<title>IEEE</title>
<id>http://www.zotero.org/styles/ieee</id>
<link href="http://www.zotero.org/styles/ieee" rel="self"/>
<link href="https://ieeeauthorcenter.ieee.org/wp-content/uploads/IEEE-Reference-Guide.pdf" rel="documentation"/>
<link href="https://journals.ieeeauthorcenter.ieee.org/your-role-in-article-production/ieee-editorial-style-manual/" rel="documentation"/>
<author>
<name>Michael Berkowitz</name>
<email>mberkowi@gmu.edu</email>
</author>
<contributor>
<name>Julian Onions</name>
<email>julian.onions@gmail.com</email>
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<contributor>
<name>Rintze Zelle</name>
<uri>http://twitter.com/rintzezelle</uri>
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<name>Stephen Frank</name>
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<name>Giuseppe Silano</name>
<email>g.silano89@gmail.com</email>
<uri>http://giuseppesilano.net</uri>
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<contributor>
<name>Patrick O'Brien</name>
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<contributor>
<name>Brenton M. Wiernik</name>
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<summary>IEEE style as per the 2018 guidelines, V 11.12.2018.</summary>
<updated>2020-06-15T03:21:46+00:00</updated>
<rights license="http://creativecommons.org/licenses/by-sa/3.0/">This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License</rights>
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<term name="chapter" form="short">ch.</term>
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<date-part name="day" form="numeric-leading-zeros" suffix=", "/>
<date-part name="year"/>
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<number variable="issue" form="numeric"/>
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<macro name="title">
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<else>
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<text variable="publisher"/>
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</group>
</else>
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<macro name="event">
<choose>
<if type="paper-conference speech" match="any">
<choose>
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<text variable="container-title" font-style="italic"/>
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<if type="webpage post post-weblog" match="any">
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@ -14,8 +14,6 @@
#+HTML_MATHJAX: align: center tagside: right font: TeX
#+CSL_STYLE: ieee.csl
#+PROPERTY: header-args:matlab :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
@ -80,21 +78,22 @@ The general structure of this document is as follows:
** Introduction :ignore:
# - Section [[sec:model_based_control_methodology]]
# - Section [[sec:comp_classical_modern_robust_control]]
# - Section [[sec:example_system]]
- Section [[sec:model_based_control_methodology]]
- Section [[sec:comp_classical_modern_robust_control]]
- Section [[sec:example_system]]
** Model Based Control - Methodology
<<sec:model_based_control_methodology>>
The typical methodology for *Model Based Control* techniques is schematically shown in Figure [[fig:control-procedure]].
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure [[fig:control-procedure]].
It consists of three steps:
1. *Identification or modeling*: a mathematical model $G(s)$ representing the plant dynamics is obtained
1. *Identification or modeling*: $\Longrightarrow$ mathematical model
2. *Translate the specifications into mathematical criteria*:
- _Specifications_: Response Time, Noise Rejection, Maximum input amplitude, Robustness, ...
- _Mathematical Criteria_: Cost Function, Shape of transfer function, Phase/Gain margin, Roll-Off, ...
3. *Synthesis*: research of a controller $K(s)$ that satisfies the specifications for the model of the system
- _Mathematical Criteria_: Cost Function, Shape of TF
# - Cost Function, Needed Bandwidth, Roll-off, ...
# - $\Longrightarrow$ We will use the $\mathcal{H}_\infty$ Norm
3. *Synthesis*: research of $K$ that satisfies the specifications for the model of the system
#+begin_src latex :file control-procedure.pdf
\begin{tikzpicture}
@ -126,44 +125,20 @@ It consists of three steps:
#+RESULTS:
[[file:figs/control-procedure.png]]
In this document, we will suppose a model of the plant is available (step 1 already performed), and we will focus on steps 2 and 3.
In this document, we will mainly focus on steps 2 and 3.
In Section [[sec:open_loop_shaping]], steps 2 and 3 will be described for a control techniques called *classical (open-)loop shaping*.
Then, steps 2 and 3 for the *$\mathcal{H}_\infty$ Loop Shaping* of closed-loop transfer functions will be discussed in Sections [[sec:modern_interpretation_specification]], [[sec:closed-loop-shaping]] and [[sec:h_infinity_mixed_sensitivity]].
Step 2 will be discussed in Section [[sec:modern_interpretation_specification]].
There are two main methods for the controller synthesis (step 3):
- /open/ loop shaping discussed in Section [[sec:open_loop_shaping]]
- /closed/ loop shaping discussed in Sections [[sec:modern_interpretation_specification]] and [[sec:h_infinity_mixed_sensitivity]]
** From Classical Control to Robust Control
<<sec:comp_classical_modern_robust_control>>
Many different model based control techniques have been developed since the birth of /classical control theory/ in the '30s.
*Classical control* methods were developed starting from 1930 based on tools such as the Laplace and Fourier transforms.
It was then natural to study systems in the frequency domain using tools such as the Bode and Nyquist plots.
Controllers were manually tuned to optimize criteria such as control bandwidth, gain and phase margins.
The '60s saw the development of control techniques based on a state-space.
Linear algebra and matrices were used instead of the frequency domain tool of the class control theory.
This allows multi-inputs multi-outputs systems to be easily treated.
Kalman introduced the well known /Kalman estimator/ as well the notion of optimality by minimizing quadratic cost functions.
This set of developments is loosely termed *Modern Control* theory.
By the 1980's, modern control theory was shown to have some robustness issues and to lack the intuitive tools that the classical control methods were offering.
This led to a new control theory called *Robust control* that blends the best features of classical and modern techniques.
This robust control theory is the subject of this document.
The three presented control methods are compared in Table [[tab:comparison_control_methods]].
Note that in parallel, there have been numerous other developments, including non-linear control, adaptive control, machine-learning control just to name a few.
#+name: tab:comparison_control_methods
#+caption: Table summurazing the main differences between classical, modern and robust control
| <l> | <c> | <c> | <c> |
| | *Classical Control* | *Modern Control* | *Robust Control* |
| <l> | <c> | <c> | <c> |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Date* | 1930- | 1960- | 1980- |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
@ -172,7 +147,6 @@ Note that in parallel, there have been numerous other developments, including no
| | Bode Plots | | Open/Closed Loop Shaping |
| | Phase and Gain margins | | Weighting Functions |
| | | | Disk margin |
| | | | Singular Value Decomposition |
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *Control Architectures* | Proportional, Integral, Derivative | Full State Feedback, LQR | General Control Configuration |
| | Leads, Lags | Kalman Filters, LQG | Generalized Plant |
@ -185,7 +159,6 @@ Note that in parallel, there have been numerous other developments, including no
|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
| *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 |
| | No clear way to limit input usage | | |
#+begin_src latex :file robustness_performance.pdf
\begin{tikzpicture}
@ -256,23 +229,17 @@ Note that in parallel, there have been numerous other developments, including no
\end{tikzpicture}
#+end_src
# #+name: fig:robustness_performance
# #+caption: Comparison of the performance and robustness of classical control methods, modern control methods and robust control methods. The required information on the plant to succesfuly apply each of the control methods are indicated by the colors.
# #+RESULTS:
# [[file:figs/robustness_performance.png]]
#+name: fig:robustness_performance
#+caption: Comparison of the performance and robustness of classical control methods, modern control methods and robust control methods. The required information on the plant to succesfuly apply each of the control methods are indicated by the colors.
#+RESULTS:
[[file:figs/robustness_performance.png]]
** Example System
<<sec:example_system>>
Throughout this document, multiple examples and practical application of presented control strategies will be provided.
Most of them will be applied on a physical system presented in this section.
This system is shown in Figure [[fig:mech_sys_1dof_inertial_contr]].
It could represent an active suspension stage supporting a payload.
The /inertial/ motion of the payload is measured using an inertial sensor and this is feedback to a force actuator.
Such system could be used to actively isolate the payload (disturbance rejection problem) or to make it follow a trajectory (tracking problem).
The notations used on Figure [[fig:mech_sys_1dof_inertial_contr]] are listed and described in Table [[tab:example_notations]].
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}
@ -352,7 +319,7 @@ You can follow this generic procedure:
#+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 architecture as shown in Figure [[fig:open_loop_shaping]].
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}
@ -374,7 +341,7 @@ Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure
#+end_src
#+name: fig:classical_feedback_test_system
#+caption: Block diagram corresponding to the example system of Figure [[fig:mech_sys_1dof_inertial_contr]]
#+caption: Block diagram corresponding to the example system
#+RESULTS:
[[file:figs/classical_feedback_test_system.png]]
@ -386,14 +353,12 @@ Let's define the system parameters on Matlab.
m = 10; % Mass [kg]
#+end_src
And now the system dynamics $G(s)$ and $G_d(s)$.
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
The Bode plots of $G(s)$ and $G_d(s)$ are shown in Figures [[fig:bode_plot_example_afm]] and [[fig:bode_plot_example_Gd]].
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
@ -457,27 +422,24 @@ The Bode plots of $G(s)$ and $G_d(s)$ are shown in Figures [[fig:bode_plot_examp
** Introduction :ignore:
After an introduction to classical Loop Shaping in Section [[sec:open_loop_shaping_introduction]], a practical example is given in Section [[sec:loop_shaping_example]].
Such Loop Shaping is usually performed manually with tools coming from the classical control theory.
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]].
- Section [[sec:open_loop_shaping_introduction]]
- Section [[sec:loop_shaping_example]]
- Section [[sec:h_infinity_open_loop_shaping]]
- Section [[sec:h_infinity_open_loop_shaping_example]]
** Introduction to Loop Shaping
<<sec:open_loop_shaping_introduction>>
#+begin_definition
*Loop Shaping* refers to a control design procedure that involves explicitly shaping the magnitude of the *Loop Transfer Function* $L(s)$.
*Loop Shaping* refers to a design procedure that involves explicitly shaping the magnitude of the *Loop Transfer Function* $L(s)$.
#+end_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* $L(s)$ usually refers to as the product of the controller and the plant ("Gain around the loop", see Figure [[fig:open_loop_shaping]]):
\begin{equation}
L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
\end{equation}
Its name comes from the fact that this is actually the "gain around the loop".
#+begin_src latex :file open_loop_shaping.pdf
\begin{tikzpicture}
\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
@ -504,19 +466,17 @@ Its name comes from the fact that this is actually the "gain around the loop".
[[file:figs/open_loop_shaping.png]]
#+end_definition
This synthesis method is one of main way controllers are design in the classical control theory.
It is widely used and generally successful as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$ such as:
- *Good Tracking*: $L$ large
This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$ such as:
- *Performance*: $L$ large
- *Good disturbance rejection*: $L$ large
- *Attenuation of measurement noise on plant output*: $L$ small
- *Small magnitude of input signal*: $L$ small
- *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 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.
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.
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 (an example is shown in Figure [[fig:open_loop_shaping_shape]]).
#+begin_src latex :file open_loop_shaping_shape.pdf
\begin{tikzpicture}
@ -537,12 +497,12 @@ Another interesting Loop shape called "Bode Step" is described in cite:lurie02_s
\path[shift={(0,1.8)}, fill=red!50!white] (0.5, 1.25) -- (2, 0.5) -| coordinate[near start](lfshaping) cycle;
\path[shift={(0,2.2)}, fill=red!50!white] (6, -0.5) -- (7.5, -1.25) |- coordinate[near end](hfshaping) cycle;
\draw[<-] (lfshaping) -- ++(0, -0.8) node[below, align=center]{{\scriptsize Ref. tracking}\\{\scriptsize Dist. rejection}};
\draw[<-] (hfshaping) -- ++(0, 0.8) node[above, align=center]{{\scriptsize Noise attenuation}};
\draw[<-] (lfshaping) -- ++(0, -0.8) node[below, align=center]{Reference\\Tracking};
\draw[<-] (hfshaping) -- ++(0, 0.8) node[above, align=center]{Noise\\Rejection};
% Crossover frequency
\node[below] (wc) at (4,2){$\omega_c$};
\draw[<-] (wc.south) -- ++(0, -0.4) node[below, align=center]{{\scriptsize Bandwidth}};
\draw[<-] (wc.south) -- ++(0, -0.4) node[below, align=center]{Bandwidth};
% Phase
\draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4, -1.25)[out=0, in=-180] to
@ -550,7 +510,7 @@ Another interesting Loop shape called "Bode Step" is described in cite:lurie02_s
-1.25)[out=0, in=-180] to (6, -2) -- (7.5, -2);
% Phase Margin
\draw[->, dashed] (4, -2) -- (4, -1.25) node[above]{{\scriptsize Phase Margin}};
\draw[->, dashed] (4, -2) -- (4, -1.25) node[above]{Phase Margin};
\draw[dashed] (0, -2) node[left]{$-\pi$} -- (7.5, -2);
\end{tikzpicture}
#+end_src
@ -560,42 +520,27 @@ Another interesting Loop shape called "Bode Step" is described in cite:lurie02_s
#+RESULTS:
[[file:figs/open_loop_shaping_shape.png]]
The shaping of *closed-loop* transfer functions is obviously not as simple as they don't depend linearly on $K(s)$.
But this is were the $\mathcal{H}_\infty$ Synthesis will be useful!
More details on that in Sections [[sec:modern_interpretation_specification]] and [[sec:closed-loop-shaping]].
** Example of Manual Open Loop Shaping
** Example of Open Loop Shaping
<<sec:loop_shaping_example>>
#+begin_exampl
Let's take our example system described in Section [[sec:example_system]] and design a controller using the Open-Loop shaping synthesis approach.
The specifications are:
1. *Disturbance rejection*: Highest possible rejection below 1Hz
2. *Positioning speed*: Bandwidth of approximately 10Hz
3. *Noise attenuation*: Roll-off of -40dB/decade past 30Hz
4. *Robustness*: Gain margin > 3dB and Phase margin > 30 deg
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 fulfills the specifications.
Using =SISOTOOL=, design a controller that fulfill the specifications.
#+begin_src matlab :eval no :tangle no
sisotool(G)
#+end_src
#+HTML: <details><summary>Hint</summary>
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.
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
@ -603,22 +548,6 @@ Let's say we came up with the following controller.
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
#+end_src
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
[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]'}, {'Requirements', 'Manual Method'}, ' %.1f ');
#+end_src
#+RESULTS:
| Requirements | Manual Method |
|-----------------------------+---------------|
| Gain Margin $> 3$ [dB] | 3.1 |
| Phase Margin $> 30$ [deg] | 35.4 |
| Crossover $\approx 10$ [Hz] | 10.1 |
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
@ -655,62 +584,106 @@ The bode plot of the Loop Gain is shown in Figure [[fig:loop_gain_manual_afm]] a
#+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]'}, {'Requirements', 'Manual Method'}, ' %.1f ');
#+end_src
#+RESULTS:
| Requirements | 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
<<sec:h_infinity_open_loop_shaping>>
The synthesis of controllers based on the Loop Shaping method can be automated using the $\mathcal{H}_\infty$ Synthesis.
The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ Synthesis.
Using Matlab, it can be easily performed using the =loopsyn= command:
Even though we will not go into details, we will provide one example.
Using Matlab, the $\mathcal{H}_\infty$ Loop Shaping Synthesis can be performed using the =loopsyn= command:
#+begin_src matlab :eval no :tangle no
K = loopsyn(G, Lw);
K = loopsyn(G, Gd);
#+end_src
where:
- =G= is the (LTI) plant
- =Lw= is the wanted loop shape
- =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
Therefore, by just providing the wanted loop shape and the plant model, the $\mathcal{H}_\infty$ Loop Shaping synthesis generates a /stabilizing/ controller such that the obtained loop gain $L(s)$ matches the specified one with an accuracy $\gamma$.
Even though we will not go into details and explain how such synthesis is working, an example is provided in the next section.
** 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]].
Let's reuse the previous plant.
Such shape corresponds to the typical wanted Loop gain Shape shown in Figure [[fig:open_loop_shaping_shape]].
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
#+name: tab:open_loop_shaping_specifications
#+caption: Wanted Loop Shape corresponding to each specification
| | Specification | Corresponding Loop Shape |
|-------------------------+---------------------------------------------+-----------------------------------------------------------------|
| *Disturbance Rejection* | Highest possible rejection below 1Hz | Slope of -40dB/decade at low frequency to have a high loop gain |
| *Positioning Speed* | Bandwidth of approximately 10Hz | $L$ crosses 1 at 10Hz: $\vert L_w(j2 \pi 10)\vert = 1$ |
| *Noise Attenuation* | Roll-off of -40dB/decade past 30Hz | Roll-off of -40dB/decade past 30Hz |
| *Robustness* | Gain margin > 3dB and Phase margin > 30 deg | 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:
#+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 $\mathcal{H}_\infty$ open loop shaping synthesis is then performed using the =loopsyn= command:
The $\mathcal{H}_\infty$ optimal open loop shaping synthesis is performed using the =loopsyn= command:
#+begin_src matlab
[K, ~, GAM] = loopsyn(G, Lw);
#+end_src
The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]] and matches the specified one by a factor $\gamma \approx 2$.
#+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 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)
#+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]] and matches the specified one by a factor $\gamma$.
#+begin_src matlab :exports none
freqs = logspace(0, 3, 1000);
@ -752,51 +725,7 @@ The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]] and m
#+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)
#+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]]
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 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)