Re-worked Section 3

index.org

@@ -819,23 +819,25 @@ Let's finally compare the obtained stability margins of the $\mathcal{H}_\infty$
 <<sec:h_infinity_introduction>>
 
 ** Introduction                                                      :ignore:
+In this section, the $\mathcal{H}_\infty$ Synthesis method, which is based on the optimization of the $\mathcal{H}_\infty$ norm of transfer functions, is introduced.
 
-- Section [[sec:h_infinity_norm]]
+After the $\mathcal{H}_\infty$ norm is defined in Section [[sec:h_infinity_norm]], the $\mathcal{H}_\infty$ synthesis procedure is described in Section [[sec:h_infinity_synthesis]] .
-- Section [[sec:h_infinity_synthesis]]
+
-- Section [[sec:generalized_plant]]
+The generalized plant, a very useful tool to describe a control problem, is presented in Section [[sec:generalized_plant]].
-- Section [[sec:h_infinity_general_synthesis]]
+The $\mathcal{H}_\infty$ is then applied to this generalized plant in Section [[sec:h_infinity_general_synthesis]].
-- Section [[sec:generalized_plant_derivation]]
+
+Finally, an example showing how to convert a typical feedback control architecture into a generalized plant is given in Section [[sec:generalized_plant_derivation]].
 
 ** The $\mathcal{H}_\infty$ Norm
 <<sec:h_infinity_norm>>
 
 #+begin_definition
- The $\mathcal{H}_\infty$ norm is defined as the peak of the maximum singular value of the frequency response
+ The $\mathcal{H}_\infty$ norm of a multi-input multi-output system $G(s)$ is defined as the peak of the maximum singular value of its frequency response
  \begin{equation}
    \|G(s)\|_\infty = \max_\omega \bar{\sigma}\big( G(j\omega) \big)
  \end{equation}
 
- For a SISO system $G(s)$, it is simply the peak value of $|G(j\omega)|$ as a function of frequency:
+ For a single-input single-output system $G(s)$, it is simply the peak value of $|G(j\omega)|$ as a function of frequency:
  \begin{equation}
    \|G(s)\|_\infty = \max_{\omega} |G(j\omega)| \label{eq:hinf_norm_siso}
  \end{equation}
@@ -850,7 +852,7 @@ Let's compute the $\mathcal{H}_\infty$ norm of our test plant $G(s)$ using the =
 #+RESULTS:
 : 7.9216e-06
 
-We can see that the $\mathcal{H}_\infty$ norm of $G(s)$ does corresponds to the peak value of $|G(j\omega)|$ as a function of frequency as shown in Figure [[fig:hinfinity_norm_siso_bode]].
+We can see in Figure [[fig:hinfinity_norm_siso_bode]] that indeed, the $\mathcal{H}_\infty$ norm of $G(s)$ does corresponds to the peak value of $|G(j\omega)|$.
 
 #+begin_src matlab :exports none
   freqs = logspace(0, 3, 1000);
@@ -879,18 +881,19 @@ We can see that the $\mathcal{H}_\infty$ norm of $G(s)$ does corresponds to the
 <<sec:h_infinity_synthesis>>
 
 #+begin_definition
-  $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimization, Riccati equation) to find a controller that stabilize the system and that *minimizes* the $\mathcal{H}_\infty$ norms of defined transfer functions.
+  The $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimization, Riccati equation) to find a controller that stabilizes the system and that *minimizes* the $\mathcal{H}_\infty$ norms of defined transfer functions.
 #+end_definition
 
-Why optimizing the $\mathcal{H}_\infty$ norm of transfer functions is a pertinent choice will become clear when we will translate the typical control specifications into the $\mathcal{H}_\infty$ norm of transfer functions.
+Why optimizing the $\mathcal{H}_\infty$ norm of transfer functions is a pertinent choice will become clear when we will translate the typical control specifications into the $\mathcal{H}_\infty$ norm of transfer functions in Section [[sec:modern_interpretation_specification]].
 
+#+begin_important
+Then applying the $\mathcal{H}_\infty$ synthesis to a plant, the engineer work usually consists of the following steps:
+1. Write the problem as standard $\mathcal{H}_\infty$ problem using the generalized plant (described in the next section)
+2. Translate the specifications as $\mathcal{H}_\infty$ norms of transfer functions (Section [[sec:modern_interpretation_specification]])
+3. Make the synthesis and analyze the obtained controller
 
-Then applying the $\mathcal{H}_\infty$ synthesis to a plant, the engineer work usually consists of the following steps
+As the $\mathcal{H}_\infty$ synthesis usually gives very high order controllers, an additional step that reduces the controller order is sometimes required for practical implementation.
-1. Write the problem as standard $\mathcal{H}_\infty$ problem
+#+end_important
-2. Translate the specifications as $\mathcal{H}_\infty$ norms of transfer functions
-3. Make the synthesis and analyze the obtain controller
-4. Reduce the order of the controller for implementation
-
 
 Note that there are many ways to use the $\mathcal{H}_\infty$ Synthesis:
 - Traditional $\mathcal{H}_\infty$ Synthesis (=hinfsyn= [[https://www.mathworks.com/help/robust/ref/hinfsyn.html][doc]])
@@ -904,19 +907,17 @@ Note that there are many ways to use the $\mathcal{H}_\infty$ Synthesis:
 
 The first step when applying the $\mathcal{H}_\infty$ synthesis is usually to write the problem as a standard $\mathcal{H}_\infty$ problem.
 This consist of deriving the *Generalized Plant* for the current problem.
-It makes things much easier for the following steps.
 
 The generalized plant, usually noted $P(s)$, is shown in Figure [[fig:general_plant]].
-It has two inputs and two outputs (both could contains many signals).
+It has two /sets/ of inputs $[w,\,u]$ and two /sets/ of outputs $[z\,v]$ such that:
-The meaning of the inputs and outputs are summarized in Table [[tab:notation_general]].
-
-Note that this generalized plant is as its name implies, quite /general/.
-It can indeed represent feedback as well as feedforward control architectures.
-
 \begin{equation}
-  \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}
+  \begin{bmatrix} z \\ v \end{bmatrix} = P \begin{bmatrix} w \\ u \end{bmatrix}
 \end{equation}
 
+The meaning of these inputs and outputs are summarized in Table [[tab:notation_general]].
+
+A practical example about how to derive the generalized plant for a classical control problem is given in Section [[sec:generalized_plant_derivation]].
+
 #+begin_src latex :file general_plant.pdf
   \begin{tikzpicture}
     \node[block={2.0cm}{2.0cm}] (P) {$P$};
@@ -937,12 +938,12 @@ It can indeed represent feedback as well as feedforward control architectures.
   \end{tikzpicture}
 #+end_src
 
+#+begin_important
 #+name: fig:general_plant
 #+caption: Inputs and Outputs of the generalized Plant
 #+RESULTS:
 [[file:figs/general_plant.png]]
 
-#+begin_important
 #+name: tab:notation_general
 #+caption: Notations for the general configuration
 | Notation | Meaning                                         |
@@ -961,10 +962,11 @@ Once the generalized plant is obtained, the $\mathcal{H}_\infty$ synthesis probl
 
 #+begin_important
   - $\mathcal{H}_\infty$ Synthesis applied on the generalized plant ::
-  Find a stabilizing controller $K$ that, using the sensed output $v$, generates a control signal $u$ such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer function from $w$ to $z$ is minimized.
+  Find a stabilizing controller $K$ that, using the sensed outputs $v$, generates control signals $u$ such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer function from $w$ to $z$ is minimized.
 
   After $K$ is found, the system is /robustified/ by adjusting the response around the unity gain frequency to increase stability margins.
-#+end_important
+
+  The obtained controller $K$ and the generalized plant are connected as shown in Figure [[fig:general_control_names]].
 
 #+begin_src latex :file general_control_names.pdf
   \begin{tikzpicture}
@@ -992,11 +994,7 @@ Once the generalized plant is obtained, the $\mathcal{H}_\infty$ synthesis probl
 #+caption: General Control Configuration
 #+RESULTS:
 [[file:figs/general_control_names.png]]
-
+#+end_important
-Note that the closed-loop transfer function from $w$ to $z$ is:
-\begin{equation}
-  \frac{z}{w} = P_{11} + P_{12} K \big( I - P_{22} K \big)^{-1} P_{21} \triangleq F_l(P, K)
-\end{equation}
 
 Using Matlab, the $\mathcal{H}_\infty$ Synthesis applied on a Generalized plant can be applied using the =hinfsyn= command ([[https://www.mathworks.com/help/robust/ref/hinfsyn.html][documentation]]):
 #+begin_src matlab :eval no :tangoe no
@@ -1006,13 +1004,15 @@ where:
 - =P= is the generalized plant transfer function matrix
 - =nmeas= is the number of sensed output (size of $v$)
 - =ncont= is the number of control signals (size of $u$)
-- =K= obtained controller that minimized the $\mathcal{H}_\infty$ norm from $w$ to $z$
+- =K= obtained controller (of size =ncont x nmeas=) that minimizes the $\mathcal{H}_\infty$ norm from $w$ to $z$.
+
+Note that the general control configure of Figure [[fig:general_control_names]], as its name implies, is quite /general/ and can represent feedback control as well as feedforward control architectures.
 
 ** From a Classical Feedback Architecture to a Generalized Plant
 <<sec:generalized_plant_derivation>>
 
 The procedure to convert a typical control architecture as the one shown in Figure [[fig:classical_feedback_tracking]] to a generalized Plant is as follows:
-1. Define signals ($w$, $z$, $u$ and $v$) of the generalized plant
+1. Define signals of the generalized plant: $w$, $z$, $u$ and $v$
 2. Remove $K$ and rearrange the inputs and outputs to match the generalized configuration shown in Figure [[fig:general_plant]]
 
 #+begin_src latex :file classical_feedback_tracking.pdf
@@ -1060,8 +1060,10 @@ The procedure to convert a typical control architecture as the one shown in Figu
 #+end_src
 
 #+begin_exercice
-  1. Convert the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]] to a generalized configuration
+  Consider the feedback control architecture shown in Figure [[fig:classical_feedback_tracking]].
-  2. Compute the transfer function matrix using Matlab as a function or $K$ and $G$
+  Suppose we want to design $K$ using the general $\mathcal{H}_\infty$ synthesis, and suppose the signals to be minimized are the control input $u$ and the tracking error $\epsilon$.
+  1. Convert the control architecture to a generalized configuration
+  2. Compute the transfer function matrix of the generalized plant $P$ using Matlab as a function or $K$ and $G$
 
   #+name: fig:classical_feedback_tracking
   #+caption: Classical Feedback Control Architecture (Tracking)