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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + -Robust Control - \(\mathcal{H}_\infty\) Synthesis +A brief and practical introduction to \(\mathcal{H}_\infty\) Control @@ -25,54 +25,77 @@ | HOME
-

Robust Control - \(\mathcal{H}_\infty\) Synthesis

+

A brief and practical introduction to \(\mathcal{H}_\infty\) Control

Table of Contents

-
-

1 Introduction to the Control Methodology - Model Based Control

+
+

1 Introduction to the Control Methodology - Model Based Control

+

+ +

-
-

1.1 Control Methodology

+ +
+

1.1 Model Based Control - Methodology

-The typical methodology when applying Model Based Control to a plant is schematically shown in Figure 1. + +

+ +

+The typical methodology when applying Model Based Control to a plant is schematically shown in Figure 1. It consists of three steps:

    @@ -86,7 +109,7 @@ It consists of three steps:
-
+

control-procedure.png

Figure 1: Typical Methodoly for Model Based Control

@@ -98,10 +121,14 @@ In this document, we will mainly focus on steps 2 and 3.
-
-

1.2 Some Background: From Classical Control to Robust Control

+
+

1.2 Some Background: From Classical Control to Robust Control

- +

+ +

+ +
@@ -242,23 +269,27 @@ In this document, we will mainly focus on steps 2 and 3. -
-

1.3 Example System

+
+

1.3 Example System

-Let’s consider the model shown in Figure 2. + +

+ +

+Let’s consider the model shown in Figure 2. 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 2. +The notations used are listed in Table 2.

-
+

mech_sys_1dof_inertial_contr.png

Figure 2: 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.

-
Table 1: Table summurazing the main differences between classical, modern and robust control
+
@@ -344,7 +375,7 @@ The notations used are listed in Table 2.
Table 2: Example system variables
-
+

Derive the following open-loop transfer functions:

@@ -375,13 +406,37 @@ You can follow this generic procedure:
+

+Hi Musa, +Thank you very much for sharing this awesome package. +For a long time, I am dreaming of being abble to export source blocks to HTML tha are surounded by <details> blocks. +

-Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure 2 into a classical feedback form as shown in Figure 6. +For now, I am manually adding #+HTML: <details><summary>Code</summary> and #+HTML: </details> around the source blocks I want to hide… +This is a very simple solution, but not so elegent nor practical. +

+ +

+Do you have any idea if it would be easy to extend to org-mode export of source blocks to add such functionallity?

-
+

+Similarly, I would love to be able to export a <span> block with the name of the file corresponding to the source block. +For instance, if a particular source block is tangled to script.sh, it would be so nice to display the filename when exporting! +

+ +

+Thanks in advance +

+ +

+Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure 2 into a classical feedback form as shown in Figure 6. +

+ + +

classical_feedback_test_system.png

Figure 3: Block diagram corresponding to the example system

@@ -399,7 +454,7 @@ Let’s define the system parameters on Matlab.

-And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown in Figures 4 and 5). +And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown in Figures 4 and 5). 

4: G = 1/(m*s^2 + c*s + k); % Plant
@@ -408,14 +463,14 @@ And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown
-
+

bode_plot_example_afm.png

Figure 4: Bode plot of the plant \(G(s)\)

-
+

bode_plot_example_Gd.png

Figure 5: Magnitude of the disturbance transfer function \(G_d(s)\)

@@ -424,23 +479,38 @@ And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown
-
-

2 Classical Open Loop Shaping

+
+

2 Classical Open Loop Shaping

-
-
-

2.1 Introduction to Open Loop Shaping

-
-

-The Loop Gain \(L(s)\) usually refers to as the product of the controller and the plant (Figure 6): + +

+
+ +
+

2.1 Introduction to Loop Shaping

+
+

+ +

+ +
+

+Loop Shaping refers to a design procedure that involves explicitly shaping the magnitude of the Loop Transfer Function \(L(s)\). +

+ +
+ +
+

+The Loop Gain \(L(s)\) usually refers to as the product of the controller and the plant (“Gain around the loop”, see Figure 6):

\begin{equation} L(s) = G(s) \cdot K(s) \label{eq:loop_gain} \end{equation} -
+

open_loop_shaping.png

Figure 6: Classical Feedback Architecture

@@ -448,15 +518,8 @@ The Loop Gain \(L(s)\) usually refers to as the product of the controller
-

-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. -

- -
- -

-This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain \(L(s)\): +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
    • @@ -472,15 +535,26 @@ The Open Loop shape is usually done manually has the loop gain \(L(s)\) depends

    -\(K(s)\) then consists of a combination of leads, lags, notches, etc. such that \(L(s)\) has the wanted shape. +\(K(s)\) then consists of a combination of leads, lags, notches, etc. such that \(L(s)\) has the wanted shape (an example is shown in Figure 7).

    + + +
    +

    open_loop_shaping_shape.png +

    +

    Figure 7: Typical Wanted Shape for the Loop Gain \(L(s)\)

    +
-
-

2.2 Example of Open Loop Shaping

+
+

2.2 Example of Open Loop Shaping

-
+

+ +

+ +

Let’s take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:

@@ -492,7 +566,7 @@ Let’s take our example system and try to apply the Open-Loop shaping strat
-
+

Using SISOTOOL, design a controller that fulfill the specifications.

@@ -509,7 +583,7 @@ In order to have the wanted Roll-off, two integrators are used, a lead is also a

-The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure 7. +The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure 8. 

K = 14e8 * ... % Gain
@@ -520,10 +594,10 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
-
+

loop_gain_manual_afm.png

-

Figure 7: Bode Plot of the obtained Loop Gain \(L(s) = G(s) K(s)\)

+

Figure 8: Bode Plot of the obtained Loop Gain \(L(s) = G(s) K(s)\)

@@ -544,7 +618,7 @@ And we can verify that we have the wanted stability margins: -  +Requirements Manual Method @@ -568,9 +642,13 @@ And we can verify that we have the wanted stability margins:

-
-

2.3 \(\mathcal{H}_\infty\) Loop Shaping Synthesis

+
+

2.3 \(\mathcal{H}_\infty\) Loop Shaping Synthesis

+

+ +

+

The Open Loop Shaping synthesis can be performed using the \(\mathcal{H}_\infty\) Synthesis.

@@ -580,7 +658,7 @@ 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: +Using Matlab, the \(\mathcal{H}_\infty\) Loop Shaping Synthesis can be performed using the loopsyn command: 

K = loopsyn(G, Gd);
@@ -595,7 +673,7 @@ where:
 
  • K is the synthesize controller
    • 
 
 
-
    
+
    
 
    
 Matlab documentation of loopsyn (link).
 
    
@@ -604,9 +682,13 @@ Matlab documentation of loopsyn (
-
    2.4 Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis
    
+
    
+
    2.4 Example of the \(\mathcal{H}_\infty\) Loop Shaping Synthesis
    
 
    
+
    
+
+
    
+
 
    
 Let’s reuse the previous plant.
 
    
@@ -628,7 +710,7 @@ Translate the specification into the wanted shape of the open loop gain.
 
    
 
 
    
-The \(\mathcal{H}_\infty\) optimal open loop shaping is performed using the loopsyn command:
+The \(\mathcal{H}_\infty\) optimal open loop shaping synthesis is performed using the loopsyn command:
 
    
 
    
 
    [K, ~, GAM] = loopsyn(G, Lw);
@@ -636,10 +718,10 @@ The \(\mathcal{H}_\infty\) optimal open loop shaping is performed using the 
 
 
    
-The Bode plot of the obtained controller is shown in Figure 8.
+The Bode plot of the obtained controller is shown in Figure 9.
 
    
 
-
    
+
    
 
    
 It is always important to analyze the controller after the synthesis is performed.
 
    
@@ -660,28 +742,28 @@ Let’s briefly analyze this controller:
 
 
 
-
    
+
    
 
    open_loop_shaping_hinf_K.png
 
    
-
    Figure 8: Obtained controller \(K\) using the open-loop \(\mathcal{H}_\infty\) shaping
    
+
    Figure 9: Obtained controller \(K\) using the open-loop \(\mathcal{H}_\infty\) shaping
    
 
    
 
 
    
-The obtained Loop Gain is shown in Figure 9.
+The obtained Loop Gain is shown in Figure 10.
 
    
 
 
-
    
+
    
 
    open_loop_shaping_hinf_L.png
 
    
-
    Figure 9: Obtained Open Loop Gain \(L(s) = G(s) K(s)\) and comparison with the wanted Loop gain \(L_w\)
    
+
    Figure 10: Obtained Open Loop Gain \(L(s) = G(s) K(s)\) and comparison with the wanted Loop gain \(L_w\)
    
 
    
 
 
    
-Let’s now compare the obtained stability margins of the \(\mathcal{H}_\infty\) controller and of the manually developed controller in Table 3.
+Let’s now compare the obtained stability margins of the \(\mathcal{H}_\infty\) controller and of the manually developed controller in Table 3.
 
    
 
-
+
    @@ -722,14 +804,22 @@ Let’s now compare the obtained stability margins of the \(\mathcal{H}_\inf
 
 
-
    
-
    3 First Step in the \(\mathcal{H}_\infty\) world
    
+
    
+
    3 First Steps in the \(\mathcal{H}_\infty\) world
    
 
    
+
    
+
+
    
 
    
-
    
-
    3.1 The \(\mathcal{H}_\infty\) Norm
    
+
+
    
+
    3.1 The \(\mathcal{H}_\infty\) Norm
    
 
    
-
    
+
    
+
+
    
+
+
    
 
    
 The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response
 
    
@@ -746,9 +836,9 @@ For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a
 
 
    
 
-
    
+
    
 
    
-And compute its \(\mathcal{H}_\infty\) norm using the hinfnorm function:
+Let’s compute the \(\mathcal{H}_\infty\) norm of our test plant \(G(s)\) using the hinfnorm function:
 
    
 
    
 
    hinfnorm(G)
@@ -761,61 +851,100 @@ And compute its \(\mathcal{H}_\infty\) norm using the hinfnorm func
 
 
 
    
-The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure 10.
-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.
+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 11.
 
    
 
 
-
    
+
    
 
    hinfinity_norm_siso_bode.png
 
    
-
    Figure 10: Example of the \(\mathcal{H}_\infty\) norm of a SISO system
    
+
    Figure 11: Example of the \(\mathcal{H}_\infty\) norm of a SISO system
    
 
    
 
 
    
 
    
 
    
 
-
    
-
    3.2 \(\mathcal{H}_\infty\) Synthesis
    
+
    
+
    3.2 \(\mathcal{H}_\infty\) 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.
+
 
    
 
+
    
 
    
-Engineer work:
+\(\mathcal{H}_\infty\) 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.
+
    
+
+
    
+
+
    
+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.
+
    
+
+
+
    
+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
      2. 
-
    2. Translate the specifications as \(\mathcal{H}_\infty\) norms
      3. 
+
    3. Translate the specifications as \(\mathcal{H}_\infty\) norms of transfer functions
      4. 
 
    4. Make the synthesis and analyze the obtain controller
      5. 
 
    5. Reduce the order of the controller for implementation
      6. 
 
    
 
+
 
    
-Many ways to use the \(\mathcal{H}_\infty\) Synthesis:
+Note that there are many ways to use the \(\mathcal{H}_\infty\) Synthesis:
 
    
 
      
-
    • Traditional \(\mathcal{H}_\infty\) Synthesis
      • 
-
    • Mixed Sensitivity Loop Shaping
      • 
-
    • Fixed-Structure \(\mathcal{H}_\infty\) Synthesis
      • 
+
    • Traditional \(\mathcal{H}_\infty\) Synthesis (hinfsyn doc)
      • 
+
    • Open Loop Shaping \(\mathcal{H}_\infty\) Synthesis (loopsyn doc)
      • 
+
    • Mixed Sensitivity Loop Shaping (mixsyn doc)
      • 
+
    • Fixed-Structure \(\mathcal{H}_\infty\) Synthesis (hinfstruct doc)
      • 
 
    • Signal Based \(\mathcal{H}_\infty\) Synthesis
      • 
 
    
 
    
 
    
 
-
    
-
    3.3 The Generalized Plant
    
+
    
+
    3.3 The Generalized Plant
    
 
    
+
    
+
+
    
 
-
    
+
    
+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 12.
+It has two inputs and two outputs (both could contains many signals).
+The meaning of the inputs and outputs are summarized in Table 4.
+
    
+
+
    
+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}
+\end{equation}
+
+
+
    
 
    general_plant.png
 
    
+
    Figure 12: Inputs and Outputs of the generalized Plant
    
 
    
 
-
    
 Table 3: Comparison of the characteristics obtained with the two methods
 
 

 
    
+
    
+
    @@ -857,94 +986,96 @@ The maximum value of the magnitude over all frequencies does correspond to the \
 
    
 Table 4: Notations for the general configuration
 
 

 
    
 
-\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}
-\end{equation}
+
    
 
    
 
    
 
-
    
-
    3.4 From a Classical Feedback Architecture to a Generalized Plant
    
+
    
+
    3.4 The General Synthesis Problem Formulation
    
 
    
-
-
    
-
    classical_feedback.png
+
    
+
 
    
-
    Figure 12: Classical Feedback Architecture
    
-
    
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-
-
-
-
-
-
-
-
-
-
-
-
-
    Table 5: Notations for the Classical Feedback Architecture



    NotationMeaning
    \(G\)Plant model
    \(K\)Controller
    \(r\)Reference inputs
    \(y\)Plant outputs
    \(u\)Control signals
    \(d\)Input Disturbance
    \(\epsilon\)Tracking Error
    
 
 
    
-The procedure is:
+Once the generalized plant is obtained, the \(\mathcal{H}_\infty\) synthesis problem can be stated as follows:
+
    
+
+
    
+
    
+
    \(\mathcal{H}_\infty\) 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.
+
    
+
+
    
+After \(K\) is found, the system is robustified by adjusting the response around the unity gain frequency to increase stability margins.
+
    
+
+
    
+
+
+
    
+
    general_control_names.png
+
    
+
    Figure 13: General Control Configuration
    
+
    
+
+
    
+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 (documentation):
+
    
+
    
+
    K = hinfsyn(P, nmeas, ncont);
+
    
+
    
+
    
+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 \(z\)
      • 
+
    
+
    
+
    
+
+
    
+
    3.5 From a Classical Feedback Architecture to a Generalized Plant
    
+
    
+
    
+
+
    
+
+
    
+The procedure to convert a typical control architecture as the one shown in Figure 14 to a generalized Plant is as follows:
 
    
 
      
-
    1. define signals of the generalized plant
      2. 
-
    2. Remove \(K\) and rearrange the inputs and outputs
      3. 
+
    3. Define signals (\(w\), \(z\), \(u\) and \(v\)) of the generalized plant
      4. 
+
    4. Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration
      5. 
 
    
 
-
    
+
    
 
    
-Let’s find the Generalized plant of corresponding to the tracking control architecture shown in Figure 13
+Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure 14
 
    
 
 
-
    
+
    
 
    classical_feedback_tracking.png
 
    
-
    Figure 13: Classical Feedback Control Architecture (Tracking)
    
+
    Figure 14: Classical Feedback Control Architecture (Tracking)
    
 
    
 
+
    Hint
 
    
 First, define the signals of the generalized plant:
 
    
@@ -957,14 +1088,22 @@ First, define the signals of the generalized plant:
 
 
    
 Then, Remove \(K\) and rearrange the inputs and outputs.
-We obtain the generalized plant shown in Figure 14.
+
    
+
    
+
+
    Answer
+
    
+The obtained generalized plant shown in Figure 15.
 
    
 
 
-
    
+
    
 
    mixed_sensitivity_ref_tracking.png
 
    
-
    Figure 14: Generalized plant of the Classical Feedback Control Architecture (Tracking)
    
+
    Figure 15: Generalized plant of the Classical Feedback Control Architecture (Tracking)
    
+
    
+
    
+
 
    
 
 
    
@@ -974,60 +1113,54 @@ Using Matlab, the generalized plant can be defined as follows:
 
    P = [1 -G;
      0  1;
      1 -G]
+P.InputName = {'w', 'u'};
+P.OutputName = {'e', 'u', 'v'};
 
    
 
    
-
 
    
 
    
 
    
 
-
-
    
-
    3.5 The General Synthesis Problem Formulation
    
-
    
-
    
-
    
-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}
-
-
    
-
-
-
    
-
    general_control_names.png
-
    
-
    Figure 15: General Control Configuration
    
-
    
-
    
-
    
-
    
-
-
-
    
-
    4 Modern Interpretation of the Control Specifications
    
+
    
+
    4 Modern Interpretation of the Control Specifications
    
 
    
+
    
+
+
    
 
    
-
    
-
    4.1 Introduction
    
+
+
    
+
    4.1 Introduction
    
 
    
+
    
+The
+
    
+
+
+
    
+
    gang_of_four_feedback.png
+
    
+
    Figure 16: Simple Feedback Architecture
    
+
    
+
 
      
-
    • Reference tracking Overshoot, Static error, Setling time
+
    • Reference tracking Overshoot, Static error, Settling time
 
        
-
      • \(S(s) = T_{r \rightarrow \epsilon}\)
        • 
+
      • From \(r\) to \(\epsilon\)
        • 
 
      • 
 
    • Disturbances rejection
 
        
+
      • From \(d\) to \(y\)
        • 
 
      • \(G(s) S(s) = T_{d \rightarrow \epsilon}\)
        • 
 
      • 
 
    • Measurement noise filtering
 
        
+
      • From \(n\) to \(y\)
        • 
 
      • \(T(s) = T_{n \rightarrow \epsilon}\)
        • 
 
      • 
 
    • Small command amplitude
 
        
+
      • From \(n, r, d\) to \(u\)
        • 
 
      • \(K(s) S(s) = T_{r \rightarrow u}\)
        • 
 
      • 
 
    • Stability
@@ -1037,18 +1170,501 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
 
    • Robustness to plant uncertainty (stability margins)
      • 
 
    • Controller implementation
      • 
 
    
+
    
+
    
+
+
    
+
    4.2 Closed Loop Transfer Functions
    
+
    
+
    
+
+
    
 
 
    
-**
+As the performances of a controlled system depend on the closed loop transfer functions, it is very important to derive these closed-loop transfer functions as a function of the plant \(G(s)\) and controller \(K(s)\).
+
    
+
+
+
    
+
    gang_of_four_feedback.png
+
    
+
    Figure 17: Simple Feedback Architecture
    
+
    
+
+
    
+
    
+Write the output signals \([\epsilon, u, y]\) as a function of the systems \(K(s), G(s)\) and of the input signals \([r, d, n]\) as shown in Figure 16.
+
    
+
+
    Hint
+
    
+Take one of the output (e.g. \(y\)), and write it as a function of the inputs \([d, r, n]\) going step by step around the loop:
+
    
+\begin{aligned}
+  y &= G u \\
+    &= G (d + K \epsilon) \\
+    &= G \big(d + K (r - n - y) \big) \\
+    &= G d + GK r - GK n - GK y
+\end{aligned}
+
+
    
+Isolate \(y\) at the right hand side, and finally obtain:
+\[ y = \frac{GK}{1+ GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \]
+
    
+
+
    
+Do the same procedure for \(u\) and \(\epsilon\)
+
    
+
    
+
+
    Anwser
+
    
+The following equations should be obtained:
+
    
+\begin{align}
+  y        &= \frac{GK}{1 + GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \\
+  \epsilon &= \frac{1 }{1 + GK} r - \frac{G}{1 + GK} d - \frac{G }{1 + GK} n \\
+  u        &= \frac{K }{1 + GK} r - \frac{1}{1 + GK} d - \frac{K }{1 + GK} n
+\end{align}
+
    
+
+
    
+
+
    
+We can see that their are 4 different transfer functions describing the behavior of the system in Figure 16.
+They are called the Gang of Four:
+
    
+\begin{align}
+  S  &= \frac{1 }{1 + GK}, \quad \text{the sensitivity function} \\
+  T  &= \frac{GK}{1 + GK}, \quad \text{the complementary sensitivity function} \\
+  GS &= \frac{G }{1 + GK}, \quad \text{the load disturbance sensitivity function} \\
+  KS &= \frac{K }{1 + GK}, \quad \text{the noise sensitivity function}
+\end{align}
+
+
    
+
    
+If a feedforward controller is included, a Gang of Six transfer functions can be defined.
+More on that in this short video.
+
    
+
+
    
+
+
    
+And we have:
+
    
+\begin{align}
+  \epsilon &= S  r - GS d - GS n \\
+  y        &= T  r + GS d - T  n \\
+  u        &= KS r - S  d - KS n
+\end{align}
+
    
+
    
+
+
    
+
    4.3 Sensitivity Transfer Function
    
+
    
+
    
+
+
    
+
+
    
+
    K1 = 14e8 * ... % Gain
+     1/(s^2) * ... % Double Integrator
+     (1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
+
+K2 = 1e8 * ... % Gain
+     1/(s^2) * ... % Double Integrator
+     (1 + s/(2*pi*1/sqrt(8)))/(1 + s/(2*pi*1*sqrt(8))); % Lead
+
+K3 = 1e8 * ... % Gain
+     1/(s^2) * ... % Double Integrator
+     (1 + s/(2*pi*1/sqrt(2)))/(1 + s/(2*pi*1*sqrt(2))); % Lead
+
+S1 = 1/(1 + K1*G);
+S2 = 1/(1 + K2*G);
+S3 = 1/(1 + K3*G);
+
+T1 = K1*G/(1 + K1*G);
+T2 = K2*G/(1 + K2*G);
+T3 = K3*G/(1 + K3*G);
+
+bodeFig({S1, S2, S3})
+
    
+
    
+
+
    
+
    freqs = logspace(-1, 2, 1000);
+
+figure;
+tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+ax1 = nexttile;
+hold on;
+plot(freqs, abs(squeeze(freqresp(S1, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
+plot(freqs, abs(squeeze(freqresp(S2, freqs, 'Hz'))), 'DisplayName', '$L_w(s)$');
+plot(freqs, abs(squeeze(freqresp(S3, freqs, 'Hz'))), 'DisplayName', '$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frquency [Hz]'); ylabel('Sensitivity Magnitude');
+hold off;
+
+ax2 = nexttile;
+t = linspace(0, 1, 1000);
+y1 = step(T1, t);
+y2 = step(T2, t);
+y3 = step(T3, t);
+hold on;
+plot(t, y1)
+plot(t, y2)
+plot(t, y3)
+hold off
+xlabel('Time [s]'); ylabel('Step Response');
+
    
+
    
+
+
+
    
+
    h-infinity-spec-S.png
+
    
+
    Figure 18: Typical wanted shape of the Sensitivity transfer function
    
+
    
+
    
+
    
+
+
    
+
    4.4 Robustness: Module Margin
    
+
    
+
    
+
+
    
+
+
      
+
    • [ ] Definition of Module margin
      • 
+
    • [ ] Why it represents robustness
      • 
+
    • [ ] Example
      • 
+
    
+
+
    
+\[ M_S < 2 \Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o \]
 
    
 
    
 
    
+
+
    
+
    4.5 How to Shape transfer function? Using of Weighting Functions!
    
+
    
+
    
+
+
    
+
+
    
+Let’s say we want to shape the sensitivity transfer function corresponding to the transfer function from \(r\) to \(\epsilon\) of the control architecture shown in Figure 19.
+
    
+
+
+
    
+
    loop_shaping_S_without_W.png
+
    
+
    Figure 19: Generalized Plant
    
 
    
 
-
    
-
    5 Resources
    
+
    
+If the \(\mathcal{H}_\infty\) synthesis is directly applied on the generalized plant \(P(s)\) shown in Figure 19, if will minimize the \(\mathcal{H}_\infty\) norm of transfer function from \(r\) to \(\epsilon\) (the sensitivity transfer function).
+
    
+
+
    
+However, as the \(\mathcal{H}_\infty\) norm is the maximum peak value of the transfer function’s magnitude, it does not allow to shape the norm over all frequencies.
+
    
+
+
+
+
    
+A trick is to include a weighting function in the generalized plant as shown in Figure 20.
+Applying the \(\mathcal{H}_\infty\) synthesis to the weighted generalized plant \(\tilde{P}(s)\) (Figure 20) will generate a controller \(K(s)\) that minimizes the \(\mathcal{H}_\infty\) norm between \(r\) and \(\tilde{\epsilon}\):
+
    
+\begin{equation}
+\begin{aligned}
+                  & \left\| \frac{\tilde{\epsilon}}{r} \right\|_\infty < \gamma (=1) \\
+  \Leftrightarrow & \left\| W_s(s) S(s) \right\|_\infty < 1 \\
+  \Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega \\
+  \Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega
+\end{aligned}\label{eq:sensitivity_shaping}
+\end{equation}
+
+
    
+
    
+As shown in Equation \eqref{eq:sensitivity_shaping}, the \(\mathcal{H}_\infty\) synthesis allows to shape the magnitude of the sensitivity transfer function.
+Therefore, the choice of the weighting function \(W_s(s)\) is very important.
+Its inverse magnitude will define the frequency dependent upper bound of the sensitivity transfer function magnitude.
+
    
+
+
    
+
+
+
    
+
    loop_shaping_S_with_W.png
+
    
+
    Figure 20: Weighted Generalized Plant
    
+
    
+
+
    
+Once the weighting function is designed, it should be added to the generalized plant as shown in Figure 20.
+
    
+
+
    
+The weighted generalized plant can be defined in Matlab by either re-defining all the inputs or by pre-multiplying the (non-weighted) generalized plant by a block-diagonal MIMO transfer function containing the weights for the outputs \(z\) and 1 for the outputs \(v\).
+
    
+
+
    
+
    Pw = [Ws -Ws*G;
+      1  -G]
+
+% Alternative
+Pw = blkdiag(Ws, 1)*P;
+
    
+
    
+
    
+
    
+
+
    
+
    4.6 Design of Weighting Functions
    
+
    
+
    
+
+
    
+
+
    
+Weighting function used must be proper, stable and minimum phase transfer functions.
+
    
+
    
+
    proper
    more poles than zeros, this implies \(\lim_{\omega \to \infty} |W(j\omega)| < \infty\)
    
+
    stable
    no poles in the right half plane
    
+
    minimum phase
    no zeros in the right half plane
    
+
    
+
+
    
+Matlab is providing the makeweight function that creates a first-order weights by specifying the low frequency gain, high frequency gain, and a gain at a specific frequency:
+
    
+
    
+
    W = makeweight(dcgain,[freq,mag],hfgain)
+
    
+
    
+
    
+with:
+
    
+
      
+
    • dcgain
      • 
+
    • freq
      • 
+
    • mag
      • 
+
    • hfgain
      • 
+
    
+
+
    
+
    
+The Matlab code below produces a weighting function with a magnitude shape shown in Figure 21.
+
    
+
+
    
+
    Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
+
    
+
    
+
+
+
    
+
    first_order_weight.png
+
    
+
    Figure 21: Obtained Magnitude of the Weighting Function
    
+
    
+
+
    
+
+
    
+
    
+Quite often, higher orders weights are required.
+
    
+
+
    
+In such case, the following formula can be used the design of these weights:
+
    
+
+\begin{equation}
+  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}}
+         }{
+           \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}
+\end{equation}
+
+
    
+The parameters permit to specify:
+
    
+
      
+
    • the low frequency gain: \(G_0 = lim_{\omega \to 0} |W(j\omega)|\)
      • 
+
    • the high frequency gain: \(G_\infty = lim_{\omega \to \infty} |W(j\omega)|\)
      • 
+
    • the absolute gain at \(\omega_0\): \(G_c = |W(j\omega_0)|\)
      • 
+
    • the absolute slope between high and low frequency: \(n\)
      • 
+
    
+
+
    
+A Matlab function implementing Equation \eqref{eq:weight_formula_advanced} is shown below:
+
    
+
+
    
+
    function [W] = generateWeight(args)
+    arguments
+        args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
+        args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
+        args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
+        args.wc (1,1) double {mustBeNumeric, mustBePositive} = 2*pi
+        args.n  (1,1) double {mustBeInteger, mustBePositive} = 1
+    end
+
+    if (args.Gc <= args.G0 && args.Gc <= args.G1) || (args.Gc >= args.G0 && args.Gc >= args.G1)
+        eid = 'value:range';
+        msg = 'Gc must be between G0 and G1';
+        throwAsCaller(MException(eid,msg))
+    end
+
+    s = zpk('s');
+
+    W = (((1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n))/((1/args.G1)^(1/args.n)*(1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
+
+end
+
    
+
    
+
+
    
+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 22.
+
    
+
+
+
    
+
    high_order_weight.png
+
    
+
    Figure 22: Higher order weights using Equation \eqref{eq:weight_formula_advanced}
    
+
    
+
+
    
+
    
+
    
+
+
    
+
    4.7 Sensitivity Function Shaping - Example
    
+
    
+
    
+
+
    
+
+
      
+
    • Robustness: Module margin > 2 (\(\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o\))
      • 
+
    • Bandwidth:
      • 
+
    • Slope of -2
      • 
+
    
+
+
    
+First, the weighting functions is generated.
+
    
+
    
+
    Ws = generateWeight('G0', 1e3, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
+
    
+
    
+
+
    
+It is then added to the generalized plant.
+
    
+
    
+
    Pw = blkdiag(Ws, 1)*P;
+
    
+
    
+
+
    
+And the \(\mathcal{H}_\infty\) synthesis is performed.
+
    
+
    
+
    K = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
    
+
    
+
+
    +K = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  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     4.497e-28     p
+  Limiting gains...
+  5.05e-01     0.0e+00     0.0e+00     0.000e+00     p
+  5.05e-01    -1.8e+01 #  -2.9e-15     1.514e-15     f
+
+  Best performance (actual): 0.504
+
    
+
+
    
+The obtained \(\gamma \approx 0.5\) means that it found a controller \(K(s)\) that stabilize the closed-loop system, and such that:
+
    
+\begin{aligned}
+  & \| W_s(s) S(s) \|_\infty < 0.5 \\
+  & \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
+\end{aligned}
+
+
    
+This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure 23.
+
    
+
+
+
    
+
    results_sensitivity_hinf.png
+
    
+
    Figure 23: Weighting function and obtained closed-loop sensitivity
    
+
    
+
    
+
    
+
    
+
+
    
+
    5 \(\mathcal{H}_\infty\) Mixed-Sensitivity Synthesis
    
 
    
 
    
+
+
    
+
    
+
+
    
+
    5.1 Problem
    
+
    
+
+
    
+
    5.2 Typical Procedure
    
+
    
+
+
    
+
    5.3 Step 1 - Shaping of the Sensitivity Function
    
+
    
+
+
    
+
    5.4 Step 2 - Shaping of
    
+
    
+
    
+
+
    
+
    6 Conclusion
    
+
    
+
+
    
+
    7 Resources
    
+
    
+
    
 
    
 
    
 
@@ -1060,7 +1676,7 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
 
    
 
    
 
    Author: Dehaeze Thomas
    
-
    Created: 2020-11-27 ven. 23:12
    
+
    Created: 2020-11-30 lun. 17:44
    
 
    
 
 
diff --git a/index.org b/index.org
index fe0d511..ddfecfc 100644
--- a/index.org
+++ b/index.org
@@ -1,4 +1,4 @@
-#+TITLE: Robust Control - $\mathcal{H}_\infty$ Synthesis
+#+TITLE: A brief and practical introduction to $\mathcal{H}_\infty$ Control
 :DRAWER:
 #+STARTUP: overview
 
@@ -31,6 +31,7 @@
 #+PROPERTY: header-args:latex+ :imoutoptions -quality 100
 #+PROPERTY: header-args:latex+ :results file raw replace
 #+PROPERTY: header-args:latex+ :buffer no
+#+PROPERTY: header-args:latex+ :tangle no
 #+PROPERTY: header-args:latex+ :eval no-export
 #+PROPERTY: header-args:latex+ :exports results
 #+PROPERTY: header-args:latex+ :mkdirp yes
@@ -48,8 +49,15 @@
   <>
 #+end_src
 
+#+begin_src matlab :tangle no
+  addpath('matlab')
+#+end_src
+
 * Introduction to the Control Methodology - Model Based Control
-** Control Methodology
+<>
+
+** Model Based 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:
@@ -94,6 +102,7 @@ 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
@@ -122,6 +131,7 @@ In this document, we will mainly focus on steps 2 and 3.
 |                         |             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.
@@ -204,6 +214,20 @@ You can follow this generic procedure:
 \end{align}
 #+HTML: 
 #+end_exercice
+Hi Musa,
+Thank you very much for sharing this awesome package.
+For a long time, I am dreaming of being abble to export source blocks to HTML tha are surounded by 
     blocks.
+
+For now, I am manually adding #+HTML: 
    Code and #+HTML: 
     around the source blocks I want to hide...
+This is a very simple solution, but not so elegent nor practical.
+
+Do you have any idea if it would be easy to extend to org-mode export of source blocks to add such functionallity?
+
+
+Similarly, I would love to be able to export a  block with the name of the file corresponding to the source block.
+For instance, if a particular source block is tangled to script.sh, it would be so nice to display the filename when exporting!
+
+Thanks in advance
 
 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]].
 
@@ -302,10 +326,17 @@ And now the system dynamics $G(s)$ and $G_d(s)$ (their bode plots are shown in F
 [[file:figs/bode_plot_example_Gd.png]]
 
 * Classical Open Loop Shaping
-** Introduction to Open Loop Shaping
+<>
+
+** Introduction to 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]]):
+*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* $L(s)$ 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}
@@ -336,11 +367,7 @@ The *Loop Gain* $L(s)$ usually refers to as the product of the controller and th
 [[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 $L(s)$:
+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
 - *Limitation of measurement noise on plant output*: $L$ small
@@ -350,9 +377,52 @@ This synthesis method is widely used as many characteristics of the closed-loop
 
 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.
+$K(s)$ 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}
+    % Phase Axis
+    \draw[->] (-0.3, -0.5) -- ++(8, 0) node[above]{$\omega$}; \draw[<-] (0, 0)
+    node[left]{$\angle L(j\omega)$} -- ++(0, -2.3);
+
+    % Gain Axis
+    \draw[->] (-0.3, 2) -- ++(8, 0) node[above]{$\omega$}; \draw[->] (0, 0.5) --
+    ++(0, 3) node[left]{$\left|L(j\omega)\right|$};
+
+    % Gain Slopes
+    \draw[shift={(0,2)}] (0.5, 1.25) -- node[midway, above]{$-2$} (2, 0.5) --
+    node[midway, above]{$-1$} (6, -0.5) -- node[midway, below left]{$-2$} (7.5,
+    -1.25);
+
+    % Forbiden region
+    \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]{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]{Bandwidth};
+
+    % Phase
+    \draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4, -1.25)[out=0, in=-180] to
+    (6, -2) -- (7.5, -2); \draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4,
+    -1.25)[out=0, in=-180] to (6, -2) -- (7.5, -2);
+
+    % 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
+
+#+name: fig:open_loop_shaping_shape
+#+caption: Typical Wanted Shape for the Loop Gain $L(s)$
+#+RESULTS:
+[[file:figs/open_loop_shaping_shape.png]]
 
 ** 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:
@@ -421,23 +491,24 @@ And we can verify that we have the wanted stability margins:
 #+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 ');
+  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:
-|                             | Manual Method |
+| 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
+<>
 
 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:
+Using Matlab, the $\mathcal{H}_\infty$ Loop Shaping Synthesis can be performed using the =loopsyn= command:
 #+begin_src matlab :eval no
   K = loopsyn(G, Gd);
 #+end_src
@@ -451,6 +522,7 @@ where:
 #+end_seealso
 
 ** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
+<>
 
 Let's reuse the previous plant.
 
@@ -465,7 +537,7 @@ Translate the specification into the wanted shape of the open loop gain.
        (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:
+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
@@ -575,8 +647,11 @@ Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ con
 | Phase Margin $> 30$ [deg]   |          35.4 |                        54.7 |
 | Crossover $\approx 10$ [Hz] |          10.1 |                         9.9 |
 
-* First Step in the $\mathcal{H}_\infty$ world
+* First Steps in the $\mathcal{H}_\infty$ world
+<>
+
 ** 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
@@ -591,7 +666,7 @@ Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ con
 #+end_definition
 
 #+begin_exampl
-And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function:
+Let's compute the $\mathcal{H}_\infty$ norm of our test plant $G(s)$ using the =hinfnorm= function:
 #+begin_src matlab :results value replace
   hinfnorm(G)
 #+end_src
@@ -599,8 +674,7 @@ And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function:
 #+RESULTS:
 : 7.9216e-06
 
-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.
+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]].
 
 #+begin_src matlab :exports none
   freqs = logspace(0, 3, 1000);
@@ -626,23 +700,47 @@ The maximum value of the magnitude over all frequencies does correspond to the $
 #+end_exampl
 
 ** $\mathcal{H}_\infty$ 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.
+#+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.
+#+end_definition
 
-*Engineer work*:
+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.
+
+
+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
-2. Translate the specifications as $\mathcal{H}_\infty$ norms
+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
 
-*Many ways to use the $\mathcal{H}_\infty$ Synthesis*:
-- Traditional $\mathcal{H}_\infty$ Synthesis
-- Mixed Sensitivity Loop Shaping
-- Fixed-Structure $\mathcal{H}_\infty$ Synthesis
+
+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]])
+- Open Loop Shaping $\mathcal{H}_\infty$ Synthesis (=loopsyn= [[https://www.mathworks.com/help/robust/ref/loopsyn.html][doc]])
+- Mixed Sensitivity Loop Shaping (=mixsyn= [[https://www.mathworks.com/help/robust/ref/lti.mixsyn.html][doc]])
+- Fixed-Structure $\mathcal{H}_\infty$ Synthesis (=hinfstruct= [[https://www.mathworks.com/help/robust/ref/lti.hinfstruct.html][doc]])
 - Signal Based $\mathcal{H}_\infty$ Synthesis
 
 ** The Generalized Plant
+<>
+
+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).
+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}
+\end{equation}
+
 #+begin_src latex :file general_plant.pdf
   \begin{tikzpicture}
     \node[block={2.0cm}{2.0cm}] (P) {$P$};
@@ -663,9 +761,12 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
   \end{tikzpicture}
 #+end_src
 
+#+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                                         |
@@ -675,50 +776,68 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
 | $z$      | Exogenous outputs: signals to be minimized      |
 | $v$      | Controller inputs: measurements                 |
 | $u$      | Control signals                                 |
+#+end_important
 
-\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}
-\end{equation}
+** The General Synthesis Problem Formulation
+<>
 
-** From a Classical Feedback Architecture to a Generalized Plant
+Once the generalized plant is obtained, the $\mathcal{H}_\infty$ synthesis problem can be stated as follows:
 
-#+begin_src latex :file classical_feedback.pdf
+#+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.
+
+  After $K$ is found, the system is /robustified/ by adjusting the response around the unity gain frequency to increase stability margins.
+#+end_important
+
+#+begin_src latex :file general_control_names.pdf
   \begin{tikzpicture}
-    \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
-    \node[block, right=0.8 of addfb] (K){$K(s)$};
-    \node[addb={+}{}{}{}{}, right=0.8 of K] (addu){};
-    \node[block, right=0.8 of addu] (G){$G(s)$};
 
-    \draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
-    \draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
-    \draw[->] (K.east) -- (addu.west) node[above left]{$u$};
-    \draw[->] (addu.east) -- (G.west);
-    \draw[<-] (addu.north) -- ++(0, 0.8) node[below right]{$d$};
-    \draw[->] (G.east) -- ++(1.2, 0);
-    \draw[->] ($(G.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -0.8) -| (addfb.south);
+    % 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:classical_feedback
-#+caption: Classical Feedback Architecture
+#+name: fig:general_control_names
+#+caption: General Control Configuration
 #+RESULTS:
-[[file:figs/classical_feedback.png]]
+[[file:figs/general_control_names.png]]
 
-#+name:       table:notation_conventional
-#+caption:    Notations for the Classical Feedback Architecture
-| Notation   | Meaning           |
-|------------+-------------------|
-| $G$        | Plant model       |
-| $K$        | Controller        |
-| $r$        | Reference inputs  |
-| $y$        | Plant outputs     |
-| $u$        | Control signals   |
-| $d$        | Input Disturbance |
-| $\epsilon$ | Tracking Error    |
+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}
 
-The procedure is:
-1. define signals of the generalized plant
-2. Remove $K$ and rearrange the inputs and outputs
+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
+  K = hinfsyn(P, nmeas, ncont);
+#+end_src
+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 $z$
+
+** From a Classical Feedback Architecture to a Generalized Plant
+<>
+
+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
+2. Remove $K$ and rearrange the inputs and outputs to match the generalized configuration
 
 #+begin_src latex :file classical_feedback_tracking.pdf
   \begin{tikzpicture}
@@ -765,12 +884,13 @@ The procedure is:
 #+end_src
 
 #+begin_exercice
-  Let's find the Generalized plant of corresponding to the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]]
+  Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]]
 
   #+name: fig:classical_feedback_tracking
   #+caption: Classical Feedback Control Architecture (Tracking)
   [[file:figs/classical_feedback_tracking.png]]
 
+#+HTML: 
    Hint
   First, define the signals of the generalized plant:
   - Exogenous inputs: $w = r$
   - Signals to be minimized: $z_1 = \epsilon$, $z_2 = u$
@@ -778,76 +898,550 @@ The procedure is:
   - Control inputs: $u$
 
   Then, Remove $K$ and rearrange the inputs and outputs.
-  We obtain the generalized plant shown in Figure [[fig:mixed_sensitivity_ref_tracking]].
+#+HTML: 
    
+
+#+HTML: 
    Answer
+  The obtained generalized plant shown in Figure [[fig:mixed_sensitivity_ref_tracking]].
 
   #+name: fig:mixed_sensitivity_ref_tracking
   #+caption: Generalized plant of the Classical Feedback Control Architecture (Tracking)
   [[file:figs/mixed_sensitivity_ref_tracking.png]]
-
-  Using Matlab, the generalized plant can be defined as follows:
-  #+begin_src matlab :tangle no :eval no
-    P = [1 -G;
-         0  1;
-         1 -G]
-  #+end_src
+#+HTML: 
    
 #+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}
+Using Matlab, the generalized plant can be defined as follows:
+#+begin_src matlab :tangle no :eval no
+  P = [1 -G;
+       0  1;
+       1 -G]
+  P.InputName = {'w', 'u'};
+  P.OutputName = {'e', 'u', 'v'};
 #+end_src
 
-#+name: fig:general_control_names
-#+caption: General Control Configuration
-#+RESULTS:
-[[file:figs/general_control_names.png]]
-
-
 * Modern Interpretation of the Control Specifications
-** Introduction
+<>
 
-- *Reference tracking* Overshoot, Static error, Setling time
-  - $S(s) = T_{r \rightarrow \epsilon}$
+** Introduction
+The
+
+#+name: fig:gang_of_four_feedback
+#+caption: Simple Feedback Architecture
+#+RESULTS:
+[[file:figs/gang_of_four_feedback.png]]
+
+- *Reference tracking* Overshoot, Static error, Settling time
+  - From $r$ to $\epsilon$
 - *Disturbances rejection*
+  - From $d$ to $y$
   - $G(s) S(s) = T_{d \rightarrow \epsilon}$
 - *Measurement noise filtering*
+  - From $n$ to $y$
   - $T(s) = T_{n \rightarrow \epsilon}$
 - *Small command amplitude*
+  - From $n, r, d$ to $u$
   - $K(s) S(s) = T_{r \rightarrow u}$
 - *Stability*
   - $S(s)$, $T(s)$, $K(s)S(s)$, $G(s)S(s)$
 - *Robustness to plant uncertainty* (stability margins)
 - *Controller implementation*
 
-**
+** Closed Loop Transfer Functions
+<>
+
+As the performances of a controlled system depend on the *closed* loop transfer functions, it is very important to derive these closed-loop transfer functions as a function of the plant $G(s)$ and controller $K(s)$.
+
+#+begin_src latex :file gang_of_four_feedback.pdf
+  \begin{tikzpicture}
+    \node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
+    \node[block, right=0.8 of addfb] (K){$K(s)$};
+    \node[addb,  right=0.8 of K] (addd){};
+    \node[block, right=0.8 of addd] (G){$G(s)$};
+    \node[addb,  below right=0.4 and 0.2 of G] (addn){};
+
+    \draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
+    \draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
+    \draw[->] (K.east) -- (addd.west);
+    \draw[<-] (addd.north) -- ++(0, 0.6) node[below right]{$d$};
+    \draw[->] (addd.east) -- (G.west) node[above left]{$u$};
+    \draw[->] (G.east) -- ++(1.6, 0) node[above left]{$y$};
+    \draw[->] (G-|addn) node[branch]{} -- (addn.north);
+    \draw[<-] (addn.east) -- ++(0.8, 0) node[above left]{$n$};
+    \draw[->] (addn.west) -| (addfb.south);
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:gang_of_four_feedback
+#+caption: Simple Feedback Architecture
+#+RESULTS:
+[[file:figs/gang_of_four_feedback.png]]
+
+#+begin_exercice
+Write the output signals $[\epsilon, u, y]$ as a function of the systems $K(s), G(s)$ and of the input signals $[r, d, n]$ as shown in Figure [[fig:gang_of_four_feedback]].
+
+#+HTML: 
    Hint
+Take one of the output (e.g. $y$), and write it as a function of the inputs $[d, r, n]$ going step by step around the loop:
+\begin{aligned}
+  y &= G u \\
+    &= G (d + K \epsilon) \\
+    &= G \big(d + K (r - n - y) \big) \\
+    &= G d + GK r - GK n - GK y
+\end{aligned}
+
+Isolate $y$ at the right hand side, and finally obtain:
+\[ y = \frac{GK}{1+ GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \]
+
+Do the same procedure for $u$ and $\epsilon$
+#+HTML: 
    
+
+#+HTML: 
    Anwser
+The following equations should be obtained:
+\begin{align}
+  y        &= \frac{GK}{1 + GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \\
+  \epsilon &= \frac{1 }{1 + GK} r - \frac{G}{1 + GK} d - \frac{G }{1 + GK} n \\
+  u        &= \frac{K }{1 + GK} r - \frac{1}{1 + GK} d - \frac{K }{1 + GK} n
+\end{align}
+#+HTML: 
    
+#+end_exercice
+
+We can see that their are 4 different transfer functions describing the behavior of the system in Figure [[fig:gang_of_four_feedback]].
+They are called the *Gang of Four*:
+\begin{align}
+  S  &= \frac{1 }{1 + GK}, \quad \text{the sensitivity function} \\
+  T  &= \frac{GK}{1 + GK}, \quad \text{the complementary sensitivity function} \\
+  GS &= \frac{G }{1 + GK}, \quad \text{the load disturbance sensitivity function} \\
+  KS &= \frac{K }{1 + GK}, \quad \text{the noise sensitivity function}
+\end{align}
+
+#+begin_seealso
+If a feedforward controller is included, a *Gang of Six* transfer functions can be defined.
+More on that in this [[https://www.youtube.com/watch?v=b_8v8scghh8][short video]].
+#+end_seealso
+
+And we have:
+\begin{align}
+  \epsilon &= S  r - GS d - GS n \\
+  y        &= T  r + GS d - T  n \\
+  u        &= KS r - S  d - KS n
+\end{align}
+
+** Sensitivity Transfer Function
+<>
+
+#+begin_src matlab
+  K1 = 14e8 * ... % Gain
+       1/(s^2) * ... % Double Integrator
+       (1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
+
+  K2 = 1e8 * ... % Gain
+       1/(s^2) * ... % Double Integrator
+       (1 + s/(2*pi*1/sqrt(8)))/(1 + s/(2*pi*1*sqrt(8))); % Lead
+
+  K3 = 1e8 * ... % Gain
+       1/(s^2) * ... % Double Integrator
+       (1 + s/(2*pi*1/sqrt(2)))/(1 + s/(2*pi*1*sqrt(2))); % Lead
+
+  S1 = 1/(1 + K1*G);
+  S2 = 1/(1 + K2*G);
+  S3 = 1/(1 + K3*G);
+
+  T1 = K1*G/(1 + K1*G);
+  T2 = K2*G/(1 + K2*G);
+  T3 = K3*G/(1 + K3*G);
+
+  bodeFig({S1, S2, S3})
+#+end_src
+
+#+begin_src matlab
+  freqs = logspace(-1, 2, 1000);
+
+  figure;
+  tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
+
+  ax1 = nexttile;
+  hold on;
+  plot(freqs, abs(squeeze(freqresp(S1, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
+  plot(freqs, abs(squeeze(freqresp(S2, freqs, 'Hz'))), 'DisplayName', '$L_w(s)$');
+  plot(freqs, abs(squeeze(freqresp(S3, freqs, 'Hz'))), 'DisplayName', '$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$');
+  hold off;
+  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+  xlabel('Frquency [Hz]'); ylabel('Sensitivity Magnitude');
+  hold off;
+
+  ax2 = nexttile;
+  t = linspace(0, 1, 1000);
+  y1 = step(T1, t);
+  y2 = step(T2, t);
+  y3 = step(T3, t);
+  hold on;
+  plot(t, y1)
+  plot(t, y2)
+  plot(t, y3)
+  hold off
+  xlabel('Time [s]'); ylabel('Step Response');
+#+end_src
+
+#+begin_src latex :file h-infinity-spec-S.pdf
+  \begin{tikzpicture}
+    \begin{axis}[%
+      width=8cm,
+      height=4cm,
+      at={(0,0)},
+      xmode=log,
+      xmin=0.01,
+      xmax=10000,
+      ymin=-80,
+      ymax=40,
+      ylabel={Magnitude [dB]},
+      xlabel={Frequency [Hz]},
+      ytick={40, 20, 0, -20, -40, -60, -80},
+      xminorgrids,
+      yminorgrids
+      ]
+      \addplot [thick, color=black, forget plot]
+      table[row sep=crcr]{%
+        0.01 -60\\
+        0.1  -60\\
+        190   6\\
+        10000 6\\
+      };
+      \draw[<-] (0.05, -60) -- (0.1, -70);
+      \draw (0.1, -70) -- (2, -70) node[right, fill=white, draw]{$\SI{-60}{\decibel} \rightarrow$ \footnotesize{Static error}};
+      \draw[<-] (1, -40) -- (10, -40) node[right, fill=white, draw]{$\SI{20}{\decibel/dec} \rightarrow$ \footnotesize{Ref. track.}};
+      \draw[<-] (100, 0) -- (3, 0) node[left, fill=white, draw]{$\omega_c \rightarrow$ \footnotesize{Speed}};
+
+      \draw[<-] (300, 6) -- (200, 20);
+      \draw (200, 20) -- (10, 20) node[left, fill=white, draw]{$\SI{6}{\decibel} \rightarrow$ \footnotesize{Module margin}};
+    \end{axis}
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:h-infinity-spec-S
+#+caption: Typical wanted shape of the Sensitivity transfer function
+#+RESULTS:
+[[file:figs/h-infinity-spec-S.png]]
+
+** Robustness: Module Margin
+<>
+
+- [ ] Definition of Module margin
+- [ ] Why it represents robustness
+- [ ] Example
+
+\[ M_S < 2 \Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o \]
+
+** How to *Shape* transfer function? Using of Weighting Functions!
+<>
+
+Let's say we want to shape the sensitivity transfer function corresponding to the transfer function from $r$ to $\epsilon$ of the control architecture shown in Figure [[fig:loop_shaping_S_without_W]].
+
+#+begin_src latex :file loop_shaping_S_without_W.pdf
+  \begin{tikzpicture}
+    \node[block] (G) {$G(s)$};
+    \node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
+    \coordinate[above right=1.0 and 1.4 of addw] (epsilon);
+
+    \coordinate[] (w) at ($(epsilon-|G.west)+(-1.0, 0)$);
+
+    \node[block, below left=0.8 and 0 of addw] (K) {$K(s)$};
+
+    % Connections
+    \draw[->] (G.east) -- (addw.west);
+    \draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (epsilon) node[above left](z1){$\epsilon$};
+
+    \draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
+    \draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.4, 0)$) -- (G.west);
+
+    \draw[->] (w) node[above]{$w = r$} -| (addw.north);
+
+    \begin{scope}[on background layer]
+      \node[fit={(G.south west) ($(z1.north west)+(-0.4, 0)$)}, inner sep=12pt, draw, dashed, fill=black!20!white] (P) {};
+      \node[below] at (P.north) {Generalized Plant $P(s)$};
+    \end{scope}
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:loop_shaping_S_without_W
+#+caption: Generalized Plant
+#+RESULTS:
+[[file:figs/loop_shaping_S_without_W.png]]
+
+If the $\mathcal{H}_\infty$ synthesis is directly applied on the generalized plant $P(s)$ shown in Figure [[fig:loop_shaping_S_without_W]], if will minimize the $\mathcal{H}_\infty$ norm of transfer function from $r$ to $\epsilon$ (the sensitivity transfer function).
+
+However, as the $\mathcal{H}_\infty$ norm is the maximum peak value of the transfer function's magnitude, it does not allow to *shape* the norm over all frequencies.
+
+
+
+A /trick/ is to include a *weighting function* in the generalized plant as shown in Figure [[fig:loop_shaping_S_with_W]].
+Applying the $\mathcal{H}_\infty$ synthesis to the /weighted/ generalized plant $\tilde{P}(s)$ (Figure [[fig:loop_shaping_S_with_W]]) will generate a controller $K(s)$ that minimizes the $\mathcal{H}_\infty$ norm between $r$ and $\tilde{\epsilon}$:
+\begin{equation}
+\begin{aligned}
+                  & \left\| \frac{\tilde{\epsilon}}{r} \right\|_\infty < \gamma (=1) \\
+  \Leftrightarrow & \left\| W_s(s) S(s) \right\|_\infty < 1 \\
+  \Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega \\
+  \Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega
+\end{aligned}\label{eq:sensitivity_shaping}
+\end{equation}
+
+#+begin_important
+  As shown in Equation eqref:eq:sensitivity_shaping, the $\mathcal{H}_\infty$ synthesis allows to *shape* the magnitude of the sensitivity transfer function.
+  Therefore, the choice of the weighting function $W_s(s)$ is very important.
+  Its inverse magnitude will define the frequency dependent upper bound of the sensitivity transfer function magnitude.
+#+end_important
+
+#+begin_src latex :file loop_shaping_S_with_W.pdf
+  \begin{tikzpicture}
+    \node[block] (G) {$G(s)$};
+    \node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
+    \node[block, above right=1.0 and 1.0 of addw] (Ws) {$W_s(s)$};
+    \coordinate[right=0.8 of Ws] (epsilon);
+
+    \coordinate[] (w) at ($(epsilon-|G.west)+(-1.0, 0)$);
+
+    \begin{scope}[on background layer]
+      \node[fit={(G.south west) (Ws.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
+      \node[above] at (P.north) {Weighted Generalized Plant $\tilde{P}(s)$};
+    \end{scope}
+
+    \node[block, below=0.4 of P] (K) {$K(s)$};
+
+    % Connections
+    \draw[->] (G.east) -- (addw.west);
+    \draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (Ws.west)node[above left]{$\epsilon$};
+    \draw[->] (Ws.east) -- (epsilon) node[above left](z1){$\tilde{\epsilon}$};
+
+    \draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
+    \draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.4, 0)$) -- (G.west);
+
+    \draw[->] (w) node[above]{$w = r$} -| (addw.north);
+  \end{tikzpicture}
+#+end_src
+
+#+name: fig:loop_shaping_S_with_W
+#+caption: Weighted Generalized Plant
+#+RESULTS:
+[[file:figs/loop_shaping_S_with_W.png]]
+
+Once the weighting function is designed, it should be added to the generalized plant as shown in Figure [[fig:loop_shaping_S_with_W]].
+
+The weighted generalized plant can be defined in Matlab by either re-defining all the inputs or by pre-multiplying the (non-weighted) generalized plant by a block-diagonal MIMO transfer function containing the weights for the outputs $z$ and =1= for the outputs $v$.
+
+#+begin_src matlab :tangle no :eval no
+  Pw = [Ws -Ws*G;
+        1  -G]
+
+  % Alternative
+  Pw = blkdiag(Ws, 1)*P;
+#+end_src
+
+** Design of Weighting Functions
+<>
+
+Weighting function used must be *proper*, *stable* and *minimum phase* transfer functions.
+- proper ::
+  more poles than zeros, this implies $\lim_{\omega \to \infty} |W(j\omega)| < \infty$
+- stable ::
+  no poles in the right half plane
+- minimum phase ::
+  no zeros in the right half plane
+
+Matlab is providing the =makeweight= function that creates a first-order weights by specifying the low frequency gain, high frequency gain, and a gain at a specific frequency:
+#+begin_src matlab :tangle no :eval no
+  W = makeweight(dcgain,[freq,mag],hfgain)
+#+end_src
+with:
+- =dcgain=
+- =freq=
+- =mag=
+- =hfgain=
+
+#+begin_exampl
+The Matlab code below produces a weighting function with a magnitude shape shown in Figure [[fig:first_order_weight]].
+
+#+begin_src matlab
+  Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
+#+end_src
+
+#+begin_src matlab :exports none
+  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('Frquency [Hz]'); ylabel('Magnitude');
+  hold off;
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/first_order_weight.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:first_order_weight
+#+caption: Obtained Magnitude of the Weighting Function
+#+RESULTS:
+[[file:figs/first_order_weight.png]]
+#+end_exampl
+
+#+begin_seealso
+Quite often, higher orders weights are required.
+
+In such case, the following formula can be used the design of these weights:
+
+\begin{equation}
+  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}}
+         }{
+           \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}
+\end{equation}
+
+The parameters permit to specify:
+- the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
+- the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
+- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
+- the absolute slope between high and low frequency: $n$
+
+A Matlab function implementing Equation eqref:eq:weight_formula_advanced is shown below:
+
+#+name: lst:generateWeight
+#+caption: Matlab Function that can be used to generate Weighting functions
+#+begin_src matlab :tangle matlab/generateWeight.m :comments none :eval no
+  function [W] = generateWeight(args)
+      arguments
+          args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
+          args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
+          args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
+          args.wc (1,1) double {mustBeNumeric, mustBePositive} = 2*pi
+          args.n  (1,1) double {mustBeInteger, mustBePositive} = 1
+      end
+
+      if (args.Gc <= args.G0 && args.Gc <= args.G1) || (args.Gc >= args.G0 && args.Gc >= args.G1)
+          eid = 'value:range';
+          msg = 'Gc must be between G0 and G1';
+          throwAsCaller(MException(eid,msg))
+      end
+
+      s = zpk('s');
+
+      W = (((1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n))/((1/args.G1)^(1/args.n)*(1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
+
+  end
+#+end_src
+
+Let's use this function to generate three weights with the same high and low frequency gains, but but different slopes.
+
+#+begin_src matlab
+  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);
+#+end_src
+
+The obtained shapes are shown in Figure [[fig:high_order_weight]].
+
+#+begin_src matlab :exports none
+  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('Frquency [Hz]'); ylabel('Magnitude');
+  legend('location', 'northeast');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/high_order_weight.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:high_order_weight
+#+caption: Higher order weights using Equation eqref:eq:weight_formula_advanced
+#+RESULTS:
+[[file:figs/high_order_weight.png]]
+#+end_seealso
+
+** Sensitivity Function Shaping - Example
+<>
+
+- Robustness: Module margin > 2 ($\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o$)
+- Bandwidth:
+- Slope of -2
+
+First, the weighting functions is generated.
+#+begin_src matlab
+  Ws = generateWeight('G0', 1e3, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
+#+end_src
+
+It is then added to the generalized plant.
+#+begin_src matlab
+  Pw = blkdiag(Ws, 1)*P;
+#+end_src
+
+And the $\mathcal{H}_\infty$ synthesis is performed.
+#+begin_src matlab :results output replace
+  K = hinfsyn(Pw, 1, 1, 'Display', 'on');
+#+end_src
+
+#+RESULTS:
+#+begin_example
+K = hinfsyn(Pw, 1, 1, 'Display', 'on');
+
+  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     4.497e-28     p
+  Limiting gains...
+  5.05e-01     0.0e+00     0.0e+00     0.000e+00     p
+  5.05e-01    -1.8e+01 #  -2.9e-15     1.514e-15     f
+
+  Best performance (actual): 0.504
+#+end_example
+
+The obtained $\gamma \approx 0.5$ means that it found a controller $K(s)$ that stabilize the closed-loop system, and such that:
+\begin{aligned}
+  & \| W_s(s) S(s) \|_\infty < 0.5 \\
+  & \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
+\end{aligned}
+
+This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:results_sensitivity_hinf]].
+
+#+begin_src matlab :exports none
+  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);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+  exportFig('figs/results_sensitivity_hinf.pdf', 'width', 'wide', 'height', 'normal');
+#+end_src
+
+#+name: fig:results_sensitivity_hinf
+#+caption: Weighting function and obtained closed-loop sensitivity
+#+RESULTS:
+[[file:figs/results_sensitivity_hinf.png]]
+
+* $\mathcal{H}_\infty$ Mixed-Sensitivity Synthesis
+<>
+
+** Problem
+
+** Typical Procedure
+
+** Step 1 - Shaping of the Sensitivity Function
+
+** Step 2 - Shaping of
+
+* Conclusion
 
 * Resources
 
diff --git a/matlab/generateWeight.m b/matlab/generateWeight.m
new file mode 100644
index 0000000..f28ec1a
--- /dev/null
+++ b/matlab/generateWeight.m
@@ -0,0 +1,20 @@
+function [W] = generateWeight(args)
+    arguments
+        args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
+        args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
+        args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
+        args.wc (1,1) double {mustBeNumeric, mustBePositive} = 2*pi
+        args.n  (1,1) double {mustBeInteger, mustBePositive} = 1
+    end
+
+    if (args.Gc <= args.G0 && args.Gc <= args.G1) || (args.Gc >= args.G0 && args.Gc >= args.G1)
+        eid = 'value:range';
+        msg = 'Gc must be between G0 and G1';
+        throwAsCaller(MException(eid,msg))
+    end
+
+    s = zpk('s');
+
+    W = (((1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n))/((1/args.G1)^(1/args.n)*(1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
+
+end