From 0d13665c50185c3c6738ad29e82f81affe761272 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 2 Dec 2020 17:01:03 +0100 Subject: [PATCH] Re-worked first section --- index.html | 662 ++++++++++++++++++++++++++++------------------------- index.org | 83 +++++-- 2 files changed, 413 insertions(+), 332 deletions(-) diff --git a/index.html b/index.html index 67d77ce..9f7e864 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + A brief and practical introduction to \(\mathcal{H}_\infty\) Control @@ -30,56 +30,56 @@

Table of Contents

@@ -103,80 +103,114 @@ When possible, Matlab scripts used for the example/exercises are provided such t The general structure of this document is as follows:

-
-

1 Introduction to Model Based Control

+
+

1 Introduction to Model Based Control

- +

-
-
-

1.1 Model Based Control - Methodology

+
+

1.1 Model Based Control - Methodology

- + +

+ +

+The typical methodology for Model Based Control techniques 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:

    -
  1. Identification or modeling: \(\Longrightarrow\) mathematical model
    2. +
  2. Identification or modeling: a mathematical model \(G(s)\) representing the plant dynamics is obtained
  3. Translate the specifications into mathematical criteria:
    • Specifications: Response Time, Noise Rejection, Maximum input amplitude, Robustness, …
      • -
    • Mathematical Criteria: Cost Function, Shape of TF
      • +
    • Mathematical Criteria: Cost Function, Shape of transfer function, Phase/Gain margin, Roll-Off, …
    4. -
  4. Synthesis: research of \(K\) that satisfies the specifications for the model of the system
    5. +
  5. Synthesis: research of a controller \(K(s)\) that satisfies the specifications for the model of the system
-
+

control-procedure.png

Figure 1: Typical Methodoly for Model Based Control

-In this document, we will mainly focus on steps 2 and 3. +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 Section 2, steps 2 and 3 will be described for a control techniques called classical (open-)loop shaping.

-Step 2 will be discussed in Section 4. -There are two main methods for the controller synthesis (step 3): +Then, steps 2 and 3 for the \(\mathcal{H}_\infty\) Loop Shaping of closed-loop transfer functions will be discussed in Sections 4, 5 and 6.

-
    -
  • open loop shaping discussed in Section 2
    • -
  • closed loop shaping discussed in Sections 4 and 6
    • -
-
-

1.2 From Classical Control to Robust Control

+
+

1.2 From Classical Control to 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 lead 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 1. +

+ +

+Note that in parallel, there have been numerous other developments, including non-linear control, adaptive control, machine-learning control just to name a few. +

+ +
@@ -239,6 +273,13 @@ There are two main methods for the controller synthesis (step 3): + + + + + + + @@ -305,11 +346,18 @@ There are two main methods for the controller synthesis (step 3): + + + + + + +
Table 1: Table summurazing the main differences between classical, modern and robust control
  Disk margin
   Singular Value Decomposition
Difficult Rejection of Perturbations Need a reasonably good model of the system
 No clear way to limit input usage  
-
+

robustness_performance.png

Figure 2: 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.

@@ -317,27 +365,27 @@ There are two main methods for the controller synthesis (step 3):
-
-

1.3 Example System

+
+

1.3 Example System

- +

-Let’s consider the model shown in Figure 3. +Let’s consider the model shown in Figure 3. 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 3: 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.

- +
@@ -423,7 +471,7 @@ The notations used are listed in Table 2.
Table 2: Example system variables
-
+

Derive the following open-loop transfer functions:

@@ -456,11 +504,11 @@ You can follow this generic procedure:

-Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure 3 into a classical feedback form as shown in Figure 7. +Having obtained \(G(s)\) and \(G_d(s)\), we can transform the system shown in Figure 3 into a classical feedback form as shown in Figure 7.

-
+

classical_feedback_test_system.png

Figure 4: Block diagram corresponding to the example system

@@ -478,7 +526,7 @@ 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 5 and 6). +And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown in Figures 5 and 6). 

4: G = 1/(m*s^2 + c*s + k); % Plant
@@ -487,14 +535,14 @@ And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown
-
+

bode_plot_example_afm.png

Figure 5: Bode plot of the plant \(G(s)\)

-
+

bode_plot_example_Gd.png

Figure 6: Magnitude of the disturbance transfer function \(G_d(s)\)

@@ -503,44 +551,44 @@ 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 Loop Shaping

+
+

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 7): +The Loop Gain \(L(s)\) usually refers to as the product of the controller and the plant (“Gain around the loop”, see Figure 7):

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

open_loop_shaping.png

Figure 7: Classical Feedback Architecture

@@ -565,11 +613,11 @@ 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 (an example is shown in Figure 8). +\(K(s)\) then consists of a combination of leads, lags, notches, etc. such that \(L(s)\) has the wanted shape (an example is shown in Figure 8).

-
+

open_loop_shaping_shape.png

Figure 8: Typical Wanted Shape for the Loop Gain \(L(s)\)

@@ -577,14 +625,14 @@ The Open Loop shape is usually done manually has the loop gain \(L(s)\) depends
-
-

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:

@@ -596,7 +644,7 @@ Let’s take our example system and try to apply the Open-Loop shaping strat
-
+

Using SISOTOOL, design a controller that fulfill the specifications.

@@ -613,7 +661,7 @@ 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 9. +The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure 9. 

K = 14e8 * ... % Gain
@@ -624,7 +672,7 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
-
+

loop_gain_manual_afm.png

Figure 9: Bode Plot of the obtained Loop Gain \(L(s) = G(s) K(s)\)

@@ -672,11 +720,11 @@ 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

- +

@@ -703,7 +751,7 @@ where:

  • K is the synthesize controller
    • -
    +

    Matlab documentation of loopsyn (link).

    @@ -712,11 +760,11 @@ 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

    - +

    @@ -747,7 +795,7 @@ The \(\mathcal{H}_\infty\) optimal open loop shaping synthesis is performed usin

    -
    +

    It is always important to analyze the controller after the synthesis is performed.

    @@ -759,7 +807,7 @@ In the end, a synthesize controller is just a combination of low pass filters, h

    -Let’s briefly analyze the obtained controller which bode plot is shown in Figure 10: +Let’s briefly analyze the obtained controller which bode plot is shown in Figure 10:

    • two integrators are used at low frequency to have the wanted low frequency high gain
      • @@ -768,28 +816,28 @@ Let’s briefly analyze the obtained controller which bode plot is shown in
    -
    +

    open_loop_shaping_hinf_K.png

    Figure 10: Obtained controller \(K\) using the open-loop \(\mathcal{H}_\infty\) shaping

    -The obtained Loop Gain is shown in Figure 11 and matches the specified one by a factor \(\gamma\). +The obtained Loop Gain is shown in Figure 11 and matches the specified one by a factor \(\gamma\).

    -
    +

    open_loop_shaping_hinf_L.png

    Figure 11: 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.

    - +
    @@ -830,29 +878,29 @@ Let’s now compare the obtained stability margins of the \(\mathcal{H}_\inf -
    -

    3 A first Step into the \(\mathcal{H}_\infty\) world

    +
    +

    3 A first Step into 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

    @@ -869,7 +917,7 @@ For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a
    -
    +

    Let’s compute the \(\mathcal{H}_\infty\) norm of our test plant \(G(s)\) using the hinfnorm function:

    @@ -884,11 +932,11 @@ Let’s compute the \(\mathcal{H}_\infty\) norm of our test plant \(G(s)\) u

    -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 12. +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 12.

    -
    +

    hinfinity_norm_siso_bode.png

    Figure 12: Example of the \(\mathcal{H}_\infty\) norm of a SISO system

    @@ -898,14 +946,14 @@ We can see that the \(\mathcal{H}_\infty\) norm of \(G(s)\) does corresponds to
    -
    -

    3.2 \(\mathcal{H}_\infty\) Synthesis

    +
    +

    3.2 \(\mathcal{H}_\infty\) Synthesis

    - +

    -
    +

    \(\mathcal{H}_\infty\) 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.

    @@ -941,11 +989,11 @@ Note that there are many ways to use the \(\mathcal{H}_\infty\) Synthesis:
    -
    -

    3.3 The Generalized Plant

    +
    +

    3.3 The Generalized Plant

    - +

    @@ -955,9 +1003,9 @@ It makes things much easier for the following steps.

    -The generalized plant, usually noted \(P(s)\), is shown in Figure 13. +The generalized plant, usually noted \(P(s)\), is shown in Figure 13. It has two inputs and two outputs (both could contains many signals). -The meaning of the inputs and outputs are summarized in Table 4. +The meaning of the inputs and outputs are summarized in Table 4.

    @@ -970,14 +1018,14 @@ It can indeed represent feedback as well as feedforward control architectures. \end{equation} -

    +

    general_plant.png

    Figure 13: Inputs and Outputs of the generalized Plant

    -
    -
    Table 3: Comparison of the characteristics obtained with the two methods
    +
    +
    @@ -1023,18 +1071,18 @@ It can indeed represent feedback as well as feedforward control architectures. -
    -

    3.4 The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant

    +
    +

    3.4 The \(\mathcal{H}_\infty\) Synthesis applied on the Generalized plant

    - +

    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
    @@ -1049,7 +1097,7 @@ After \(K\) is found, the system is robustified by adjusting the response
    -
    +

    general_control_names.png

    Figure 14: General Control Configuration

    @@ -1081,29 +1129,29 @@ where:
    -
    -

    3.5 From a Classical Feedback Architecture to a Generalized Plant

    +
    +

    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 15 to a generalized Plant is as follows: +The procedure to convert a typical control architecture as the one shown in Figure 15 to a generalized Plant is as follows:

    1. Define signals (\(w\), \(z\), \(u\) and \(v\)) of the generalized plant
      2. -
    2. Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration shown in Figure 13
      3. +
    3. Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration shown in Figure 13
    -
    +
      -
    1. Convert the tracking control architecture shown in Figure 15 to a generalized configuration
      2. +
    2. Convert the tracking control architecture shown in Figure 15 to a generalized configuration
    3. Compute the transfer function matrix using Matlab as a function or \(K\) and \(G\)
    -
    +

    classical_feedback_tracking.png

    Figure 15: Classical Feedback Control Architecture (Tracking)

    @@ -1122,17 +1170,17 @@ Usually, we want to minimize the tracking errors \(\epsilon\) and the control si

    -Then, Remove \(K\) and rearrange the inputs and outputs as in Figure 13. +Then, Remove \(K\) and rearrange the inputs and outputs as in Figure 13.

    Answer

    -The obtained generalized plant shown in Figure 16. +The obtained generalized plant shown in Figure 16.

    -
    +

    mixed_sensitivity_ref_tracking.png

    Figure 16: Generalized plant of the Classical Feedback Control Architecture (Tracking)

    @@ -1156,21 +1204,21 @@ P.OutputName = {'e', 'u
    -
    -

    4 Modern Interpretation of Control Specifications

    +
    +

    4 Modern Interpretation of Control Specifications

    - +

    -As shown in Section 2, the loop gain \(L(s) = G(s) K(s)\) is a useful and easy tool when manually designing controllers. +As shown in Section 2, the loop gain \(L(s) = G(s) K(s)\) is a useful and easy tool when manually designing controllers. This is mainly due to the fact that \(L(s)\) is very easy to shape as it depends linearly on \(K(s)\). Moreover, important quantities such as the stability margins and the control bandwidth can be estimated from the shape/phase of \(L(s)\).

    @@ -1180,11 +1228,11 @@ However, the loop gain \(L(s)\) does not directly give the performances o

    -If we consider the feedback system shown in Figure 17, we can link to the following specifications to closed-loop transfer functions. -This is summarized in Table 5. +If we consider the feedback system shown in Figure 17, we can link to the following specifications to closed-loop transfer functions. +This is summarized in Table 5.

    -
    Table 4: Notations for the general configuration
    +
    @@ -1226,33 +1274,33 @@ This is summarized in Table 5. - +
    Table 5: Typical Specification and associated closed-loop transfer function
    Robustness (stability margins)Module margin (see Section 4.3)Module margin (see Section 4.3)
    -
    +

    gang_of_four_feedback.png

    Figure 17: Simple Feedback Architecture

    -
    -

    4.1 Closed Loop Transfer Functions

    +
    +

    4.1 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 \(K(s)\).

    -
    +

    -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 17. +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 17.

    Hint @@ -1289,9 +1337,9 @@ The following equations should be obtained:
    -
    +

    -We can see that they are 4 different transfer functions describing the behavior of the system in Figure 17. +We can see that they are 4 different transfer functions describing the behavior of the system in Figure 17. These called the Gang of Four:

    \begin{align} @@ -1303,7 +1351,7 @@ These called the Gang of Four:
    -
    +

    If a feedforward controller is included, a Gang of Six transfer functions can be defined. More on that in this short video. @@ -1327,11 +1375,11 @@ Similarly, to reduce the effect of measurement noise \(n\) on the output \(y\),

    -
    -

    4.2 Sensitivity Function

    +
    +

    4.2 Sensitivity Function

    - +

    @@ -1342,14 +1390,14 @@ In this section, we will see how the shape of the sensitivity function will impa

    Suppose we have developed a “reference” controller \(K_r(s)\) and made three small changes to obtained three controllers \(K_1(s)\), \(K_2(s)\) and \(K_3(s)\). -The obtained sensitivity functions are shown in Figure 18 and the corresponding step responses are shown in Figure 19. +The obtained sensitivity functions are shown in Figure 18 and the corresponding step responses are shown in Figure 19.

    -The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table 6. +The comparison of the sensitivity functions shapes and their effect on the step response is summarized in Table 6.

    - +
    @@ -1388,20 +1436,20 @@ The comparison of the sensitivity functions shapes and their effect on the step
    Table 6: Comparison of the sensitivity function shape and the corresponding step response for the three controller variations
    -
    +

    sensitivity_shape_effect.png

    Figure 18: Sensitivity function magnitude \(|S(j\omega)|\) corresponding to the reference controller \(K_r(s)\) and the three modified controllers \(K_i(s)\)

    -
    +

    sensitivity_shape_effect_step.png

    Figure 19: Step response (response from \(r\) to \(y\)) for the different controllers

    -
    +
    Closed-Loop Bandwidth

    The closed-loop bandwidth \(\omega_b\) is the frequency where \(|S(j\omega)|\) first crosses \(1/\sqrt{2} = -3dB\) from below. @@ -1414,9 +1462,9 @@ In general, a large bandwidth corresponds to a faster rise time.

    -
    +

    -From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure 20. +From the simple analysis above, we can draw a first estimation of the wanted shape for the sensitivity function in Figure 20.

    @@ -1431,7 +1479,7 @@ This will become clear in the next section about the module margin. -

    +

    h-infinity-spec-S.png

    Figure 20: Typical wanted shape of the Sensitivity transfer function

    @@ -1441,18 +1489,18 @@ This will become clear in the next section about the module margin.
    -
    -

    4.3 Robustness: Module Margin

    +
    +

    4.3 Robustness: Module Margin

    - +

    Let’s start by an example demonstrating why the phase and gain margins might not be good indicators of robustness.

    -
    +

    Let’s consider the following plant \(G_t(s)\):

    @@ -1466,7 +1514,7 @@ Gt = 1/k*(s

    -Let’s say we have designed a controller \(K_t(s)\) that gives the loop gain shown in Figure 21. +Let’s say we have designed a controller \(K_t(s)\) that gives the loop gain shown in Figure 21.

    @@ -1475,7 +1523,7 @@ Let’s say we have designed a controller \(K_t(s)\) that gives the loop gai

    -The following characteristics can be determined from Figure 21: +The following characteristics can be determined from Figure 21:

    • bandwidth of \(\approx 10\, \text{Hz}\)
      • @@ -1488,7 +1536,7 @@ This might indicate very good robustness properties of the closed-loop system.

      -
      +

      phase_gain_margin_model_plant.png

      Figure 21: Bode plot of the obtained Loop Gain \(L(s)\)

      @@ -1504,7 +1552,7 @@ Now let’s suppose the “real” plant \(G_r(s)\) as a slightly lo

      -The obtained “real” loop gain is shown in Figure 22. +The obtained “real” loop gain is shown in Figure 22. At a frequency little bit above 100Hz, the phase of the loop gain reaches -180 degrees while its magnitude is more than one which indicated instability.

      @@ -1522,7 +1570,7 @@ It is confirmed by checking the stability of the closed loop system: -
      +

      phase_gain_margin_real_plant.png

      Figure 22: Bode plots of \(L(s)\) (loop gain corresponding the nominal plant) and \(L_r(s)\) (loop gain corresponding to the real plant)

      @@ -1542,7 +1590,7 @@ This is due to the fact that the gain and phase margin are robustness indicators Let’s now determine a new robustness indicator based on the Nyquist Stability Criteria.

      -
      +
      Nyquist Stability Criteria (for stable systems)
      If the open-loop transfer function \(L(s)\) is stable, then the closed-loop system is unstable for any encirclement of the point \(−1\) on the Nyquist plot.
      @@ -1551,7 +1599,7 @@ Let’s now determine a new robustness indicator based on the Nyquist Stabil
      -
      +

      For more information about the general Nyquist Stability Criteria, you may want to look at this video.

      @@ -1563,21 +1611,21 @@ From the Nyquist stability criteria, it is clear that we want \(L(j\omega)\) to This minimum distance is called the module margin.

      -
      +
      Module Margin
      The Module Margin \(\Delta M\) is defined as the minimum distance between the point \(-1\) and the loop gain \(L(j\omega)\) in the complex plane.
      -
      +

      -A typical Nyquist plot is shown in Figure 23. +A typical Nyquist plot is shown in Figure 23. The gain, phase and module margins are graphically shown to have an idea of what they represent.

      -
      +

      module_margin_example.png

      Figure 23: Nyquist plot with visual indication of the Gain margin \(\Delta G\), Phase margin \(\Delta \phi\) and Module margin \(\Delta M\)

      @@ -1586,7 +1634,7 @@ The gain, phase and module margins are graphically shown to have an idea of what

      -As expected from Figure 23, there is a close relationship between the module margin and the gain and phase margins. +As expected from Figure 23, there is a close relationship between the module margin and the gain and phase margins. We can indeed show that for a given value of the module margin \(\Delta M\), we have:

      \begin{equation} @@ -1605,7 +1653,7 @@ Let’s now try to express the Module margin \(\Delta M\) as an \(\mathcal{H &= \frac{1}{\|S\|_\infty} \end{align*} -
      +

      The \(\mathcal{H}_\infty\) norm of the sensitivity function \(\|S\|_\infty\) is a measure of the Module margin \(\Delta M\) and therefore an indicator of the system robustness.

      @@ -1625,14 +1673,14 @@ Note that this is why large peak value of \(|S(j\omega)|\) usually indicate robu And we know understand why setting an upper bound on the magnitude of \(S\) is generally a good idea.

      -
      +

      Typical, we require \(\|S\|_\infty < 2 (6dB)\) which implies \(\Delta G \ge 2\) and \(\Delta \phi \ge 29^o\)

      -
      +

      To learn more about module/disk margin, you can check out this video.

      @@ -1641,11 +1689,11 @@ To learn more about module/disk margin, you can check out -

      4.4 Other Requirements

      +
      +

      4.4 Other Requirements

      - +

      @@ -1674,7 +1722,7 @@ with: Noise Attenuation: typical wanted shape for \(T\)

      - +
      @@ -1727,11 +1775,11 @@ Noise Attenuation: typical wanted shape for \(T\) -
      -

      5 \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions

      +
      +

      5 \(\mathcal{H}_\infty\) Shaping of closed-loop transfer functions

      - +

      In the previous sections, we have seen that the performances of the system depends on the shape of the closed-loop transfer function. @@ -1748,22 +1796,22 @@ But don’t worry, the \(\mathcal{H}_\infty\) synthesis will do this job for

      This -Section 5.1 -Section 5.2 -Section 5.3 -Section 5.4 +Section 5.1 +Section 5.2 +Section 5.3 +Section 5.4

      -
      -

      5.1 How to Shape closed-loop transfer function? Using Weighting Functions!

      +
      +

      5.1 How to Shape closed-loop transfer function? Using Weighting Functions!

      - +

      -If the \(\mathcal{H}_\infty\) synthesis is applied on the generalized plant \(P(s)\) shown in Figure 24, it will generate a controller \(K(s)\) such that the \(\mathcal{H}_\infty\) norm of closed-loop transfer function from \(r\) to \(\epsilon\) is minimized. +If the \(\mathcal{H}_\infty\) synthesis is applied on the generalized plant \(P(s)\) shown in Figure 24, it will generate a controller \(K(s)\) such that the \(\mathcal{H}_\infty\) norm of closed-loop transfer function from \(r\) to \(\epsilon\) is minimized. This closed-loop transfer function actually correspond to the sensitivity function. Therefore, it will minimize the the \(\mathcal{H}_\infty\) norm of the sensitivity function: \(\|S\|_\infty\).

      @@ -1773,16 +1821,16 @@ However, as the \(\mathcal{H}_\infty\) norm is the maximum peak value of the tra

      -
      +

      loop_shaping_S_without_W.png

      Figure 24: Generalized Plant

      -
      +

      -The trick is to include a weighting function \(W_S(s)\) in the generalized plant as shown in Figure 25. +The trick is to include a weighting function \(W_S(s)\) in the generalized plant as shown in Figure 25.

      @@ -1800,7 +1848,7 @@ Let’s now show how this is equivalent as shaping the sensitivity fu \Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega \label{eq:sensitivity_shaping} \end{align} -

      +

      As shown in Equation \eqref{eq:sensitivity_shaping}, the \(\mathcal{H}_\infty\) synthesis applying on the weighted generalized plant allows to shape the magnitude of the sensitivity transfer function.

      @@ -1812,15 +1860,15 @@ Therefore, the choice of the weighting function \(W_s(s)\) is very important: it
      -
      +

      loop_shaping_S_with_W.png

      Figure 25: Weighted Generalized Plant

      -
      +

      -Using matlab, compute the weighted generalized plant shown in Figure 26 as a function of \(G(s)\) and \(W_S(s)\). +Using matlab, compute the weighted generalized plant shown in Figure 26 as a function of \(G(s)\) and \(W_S(s)\).

      Hint @@ -1858,18 +1906,18 @@ The second solution is however more general, and can also be used when weights a
      -
      -

      5.2 Design of Weighting Functions

      +
      +

      5.2 Design of Weighting Functions

      - +

      Weighting function included in the generalized plant 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
      @@ -1894,9 +1942,9 @@ with:
    • hfgain: high frequency gain
      • -
      +

      -The Matlab code below produces a weighting function with the following characteristics (Figure 26): +The Matlab code below produces a weighting function with the following characteristics (Figure 26):

      • Low frequency gain of 100
        • @@ -1910,7 +1958,7 @@ The Matlab code below produces a weighting function with the following character
      -
      +

      first_order_weight.png

      Figure 26: Obtained Magnitude of the Weighting Function

      @@ -1918,7 +1966,7 @@ The Matlab code below produces a weighting function with the following character
      -
      +

      Quite often, higher orders weights are required.

      @@ -1950,7 +1998,7 @@ A Matlab function implementing Equation \eqref{eq:weight_formula_advanced} is sh

      -
      function [W] = generateWeight(args)
+
      function [W] = generateWeight(args)
     arguments
         args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
         args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
@@ -1985,11 +2033,11 @@ W3 = generateWeight('G0', 1e2, 27.
+The obtained shapes are shown in Figure 27.
 
      
 
 
-
      
+
      
 
      high_order_weight.png
 
      
 
      Figure 27: Higher order weights using Equation \eqref{eq:weight_formula_advanced}
      
@@ -1999,11 +2047,11 @@ The obtained shapes are shown in Figure 27.
 
      
 
      
 
-
      
-
      5.3 Shaping the Sensitivity Function
      
+
      
+
      5.3 Shaping the Sensitivity Function
      
 
      
 
      
-
+
 
      
 
 
      
@@ -2016,17 +2064,17 @@ Let’s design a controller using the \(\mathcal{H}_\infty\) synthesis that
 
 
 
      
-As usual, the plant used is the one presented in Section 1.3.
+As usual, the plant used is the one presented in Section 1.3.
 
      
 
-
      
+
      
 
      
 Translate the requirements as upper bounds on the Sensitivity function and design the corresponding Weight using Matlab.
 
      
 
 
      Hint
 
      
-The typical wanted upper bound of the sensitivity function is shown in Figure 28.
+The typical wanted upper bound of the sensitivity function is shown in Figure 28.
 
      
 
 
      
@@ -2044,7 +2092,7 @@ Remember that the wanted upper bound of the sensitivity function is defined by t
 
      
 
 
-
      
+
      
 
      h-infinity-spec-S.png
 
      
 
      Figure 28: Typical wanted shape of the Sensitivity transfer function
      
@@ -2109,7 +2157,7 @@ And the \(\mathcal{H}_\infty\) synthesis is performed on the weighted gen
      -
      +
       Test bounds:  0.5 <=  gamma  <=  0.51
 
  gamma        X>=0        Y>=0       rho(XY)<1    p/f
@@ -2130,10 +2178,10 @@ Best performance (actual): 0.503
 \end{aligned}
 
 
      
-This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure 29.
+This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure 29.
 
      
 
-
      
+
      
 
      
 Having \(\gamma < 1\) means that the \(\mathcal{H}_\infty\) synthesis found a controller such that the specified closed-loop transfer functions are bellow the specified upper bounds.
 
      
@@ -2146,7 +2194,7 @@ It just means that at some frequency, one of the closed-loop transfer functions
 
      
 
 
-
      
+
      
 
      results_sensitivity_hinf.png
 
      
 
      Figure 29: Weighting function and obtained closed-loop sensitivity
      
@@ -2154,16 +2202,16 @@ It just means that at some frequency, one of the closed-loop transfer functions
 
      
 
      
 
-
      
-
      5.4 Shaping multiple closed-loop transfer functions
      
+
      
+
      5.4 Shaping multiple closed-loop transfer functions
      
 
      
 
      
-
+
 
      
 
 
      
-As was shown in Section 4, depending on the specifications, up to four closed-loop transfer function may be shaped (the Gang of four).
-This was summarized in Table 7.
+As was shown in Section 4, depending on the specifications, up to four closed-loop transfer function may be shaped (the Gang of four).
+This was summarized in Table 7.
 
      
 
 
      
@@ -2171,7 +2219,7 @@ For instance to limit the control input \(u\), \(KS\) should be shaped while to
 
      
 
 
      
-When multiple closed-loop transfer function are shaped at the same time, it is refereed to as “Mixed-Sensitivity \(\mathcal{H}_\infty\) Control” and is the subject of Section 6.
+When multiple closed-loop transfer function are shaped at the same time, it is refereed to as “Mixed-Sensitivity \(\mathcal{H}_\infty\) Control” and is the subject of Section 6.
 
      
 
 
      
@@ -2179,14 +2227,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 
      
 
      Shaping of S and KS
 
-
      
+
      
 
      general_conf_shaping_S_KS.png
 
      
 
      Figure 30: Generalized Plant to shape \(S\) and \(KS\)
      
 
      
 
 
      
-
      P = [W1 -G*W1
+
      P = [W1 -G*W1
      0   W2
      1  -G];
 
      
@@ -2199,14 +2247,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 
      
 
      Shaping of S and T
 
-
      
+
      
 
      general_conf_shaping_S_T.png
 
      
 
      Figure 31: Generalized Plant to shape \(S\) and \(T\)
      
 
      
 
 
      
-
      P = [W1 -G*W1
+
      P = [W1 -G*W1
      0   G*W2
      1   -G];
 
      
@@ -2219,14 +2267,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 
      
 
      Shaping of S and GS
 
-
      
+
      
 
      general_conf_shaping_S_GS.png
 
      
 
      Figure 32: Generalized Plant to shape \(S\) and \(GS\)
      
 
      
 
 
      
-
      P = [W1   -W1
+
      P = [W1   -W1
      G*W2 -G*W2
      G    -G];
 
      
@@ -2238,14 +2286,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 
      
 
      Shaping of S, T and KS
 
-
      
+
      
 
      general_conf_shaping_S_T_KS.png
 
      
 
      Figure 33: Generalized Plant to shape \(S\), \(T\) and \(KS\)
      
 
      
 
 
      
-
      P = [W1 -G*W1
+
      P = [W1 -G*W1
      0   W2
      0   G*W3
      1   -G];
@@ -2260,14 +2308,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 
      
 
      Shaping of S, T and GS
 
-
      
+
      
 
      general_conf_shaping_S_T_GS.png
 
      
 
      Figure 34: Generalized Plant to shape \(S\), \(T\) and \(GS\)
      
 
      
 
 
      
-
      P = [W1   -W1
+
      P = [W1   -W1
      G*W2 -G*W2
      0     W3
      G    -G];
@@ -2282,14 +2330,14 @@ Depending on the closed-loop transfer function being shaped, different general c
 
      
 
      Shaping of S, T, KS and GS
 
-
      
+
      
 
      general_conf_shaping_S_T_KS_GS.png
 
      
 
      Figure 35: Generalized Plant to shape \(S\), \(T\), \(KS\) and \(GS\)
      
 
      
 
 
      
-
      P = [ W1  -W1*G*W3 -G*W1
+
      P = [ W1  -W1*G*W3 -G*W1
       0    0        W2
       1   -G*W3    -G];
 
      
@@ -2313,7 +2361,7 @@ When shaping multiple closed-loop transfer functions, one should be verify caref
 
 
 
-
      
+
      
 
      
 Mathematical relations are linking the closed-loop transfer functions.
 For instance, the sensitivity function \(S(s)\) and the complementary sensitivity function \(T(s)\) as link by the following well known relation:
@@ -2346,7 +2394,7 @@ The control bandwidth is clearly limited by physical constrains such as sampling
 Similar relationship can be found for \(T\), \(KS\) and \(GS\).
 
      
 
-
      
+
      
 
      
 Determine the approximate norms of \(T\), \(KS\) and \(GS\) for large loop gains (\(|G(j\omega) K(j\omega)| \gg 1\)) and small loop gains (\(|G(j\omega) K(j\omega)| \ll 1\)).
 
      
@@ -2364,7 +2412,7 @@ You can follows this procedure for \(T\), \(KS\) and \(GS\):
 
 
      Answer
 
      
-The obtained constrains are shown in Figure 36.
+The obtained constrains are shown in Figure 36.
 
      
 
      
 
@@ -2377,17 +2425,17 @@ However, in some frequency bands, the norms do not depend on the controller and
 
 
      
 Therefore the weighting functions should only focus on certainty frequency range depending on the transfer function being shaped.
-These regions are summarized in Figure 36.
+These regions are summarized in Figure 36.
 
      
 
 
-
      
+
      
 
      h-infinity-4-blocs-constrains.png
 
      
 
      Figure 36: Shaping the Gang of Four: Limitations
      
 
      
 
-
      
+
      
 
      
 The order (resp. number of state) of the controller given by the \(\mathcal{H}_\infty\) synthesis is equal to the order (resp. number of state) of the weighted generalized plant.
 It is thus equal to the sum of the number of state of the non-weighted generalized plant and the number of state of all the weighting functions.
@@ -2406,16 +2454,16 @@ Two approaches can be used to obtain controllers with reasonable order:
 
      
 
      
 
-
      
-
      6 Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example
      
+
      
+
      6 Mixed-Sensitivity \(\mathcal{H}_\infty\) Control - Example
      
 
      
 
      
-
+
 
      
 
      
 
-
      
-
      6.1 Control Problem
      
+
      
+
      6.1 Control Problem
      
 
      
 
        
 
      • [ ] Control Diagram
        • 
@@ -2468,8 +2516,8 @@ d(t>0.5) = 5e-4;
 
      
 
      
 
-
      
-
      6.2 Control Design Procedure
      
+
      
+
      6.2 Control Design Procedure
      Table 7: Typical Specifications and corresponding wanted norms of open and closed loop tansfer functions
      @@ -2516,7 +2564,7 @@ d(t>0.5) = 5e-4;
      -
      +

      mixed_sensitivity_control_schematic.png

      Figure 37: Generalized Plant used for the Mixed Sensitivity Synthesis

      @@ -2538,11 +2586,11 @@ d(t>0.5) = 5e-4;
      -
      -

      6.3 Step 1 - Shaping of \(S\)

      +
      +

      6.3 Step 1 - Shaping of \(S\)

      - +

      @@ -2574,7 +2622,7 @@ W3 = tf(0.1);

      -
      +
       K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
 
   Test bounds:  0.5 <=  gamma  <=  0.552
@@ -2635,11 +2683,11 @@ xlabel('Time [s]'); ylabel(
-
      6.4 Step 2 - Shaping of \(KS\)
      
+
      
+
      6.4 Step 2 - Shaping of \(KS\)
      
 
      
 
      
-
+
 
      
 
 
      
@@ -2660,7 +2708,7 @@ xlabel('Time [s]'); ylabel(
+
       K1 = hinfsyn(Pw, 1, 1, 'Display', 'on');
 
   Test bounds:  0.51 <=  gamma  <=  1.2
@@ -2719,11 +2767,11 @@ xlabel('Time [s]'); ylabel(
-
      6.5 Step 3 - Shaping of \(T\)
      
+
      
+
      6.5 Step 3 - Shaping of \(T\)
      
 
      
 
      
-
+
 
      
 
 
      
@@ -2744,7 +2792,7 @@ xlabel('Time [s]'); ylabel(
+
       K3 = hinfsyn(Pw, 1, 1, 'Display', 'on');
 
   Test bounds:  0.578 <=  gamma  <=  1.66
@@ -2806,18 +2854,18 @@ xlabel('Time [s]'); ylabel(
-
      7 Conclusion
      
+
      
+
      7 Conclusion
      
 
      
 
      
-
+
 
      
 
      
 
      
 
-
      
-
      Resources
      
-
      
+
      
+
      Resources
      
+
      
 
      
 
      
 
      
@@ -2830,7 +2878,7 @@ xlabel('Time [s]'); ylabel(
 
      Author: Dehaeze Thomas
      
-
      Created: 2020-12-02 mer. 11:00
      
+
      Created: 2020-12-02 mer. 16:46
      
 
      
 
 
diff --git a/index.org b/index.org
index 0867d17..e259d64 100644
--- a/index.org
+++ b/index.org
@@ -78,22 +78,21 @@ 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
 <>
 
-The typical methodology when applying Model Based Control to a plant is schematically shown in Figure [[fig:control-procedure]].
+The typical methodology for *Model Based Control* techniques is schematically shown in Figure [[fig:control-procedure]].
+
 It consists of three steps:
-1. *Identification or modeling*: $\Longrightarrow$ mathematical model
+1. *Identification or modeling*: a mathematical model $G(s)$ representing the plant dynamics is obtained
 2. *Translate the specifications into mathematical criteria*:
    - _Specifications_: Response Time, Noise Rejection, Maximum input amplitude, Robustness, ...
-   - _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
+   - _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
 
 #+begin_src latex :file control-procedure.pdf
   \begin{tikzpicture}
@@ -125,20 +124,44 @@ It consists of three steps:
 #+RESULTS:
 [[file:figs/control-procedure.png]]
 
-In this document, we will mainly focus on steps 2 and 3.
+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.
 
-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]]
+
+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]].
 
 ** From Classical Control to 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
-|                         |        *Classical Control*         |           *Modern Control*           |                            *Robust Control*                             |
 |                      |                                 |                                   |                                                                      |
+|                         |        *Classical Control*         |           *Modern Control*           |                            *Robust Control*                             |
 |-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
 | *Date*                  |               1930-                |                1960-                 |                                  1980-                                  |
 |-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
@@ -147,6 +170,7 @@ There are two main methods for the controller synthesis (step 3):
 |                         |             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                            |
@@ -159,6 +183,7 @@ There are two main methods for the controller synthesis (step 3):
 |-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
 | *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}
@@ -229,17 +254,23 @@ There are two main methods for the controller synthesis (step 3):
   \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
 <>
 
-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]].
+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]].
 
 #+begin_src latex :file mech_sys_1dof_inertial_contr.pdf
   \begin{tikzpicture}
@@ -319,7 +350,7 @@ You can follow this generic procedure:
 #+HTML: 
 #+end_exercice
 
-Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure [[fig:mech_sys_1dof_inertial_contr]] into a classical feedback form as shown in Figure [[fig:open_loop_shaping]].
+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]].
 
 #+begin_src latex :file classical_feedback_test_system.pdf
   \begin{tikzpicture}
@@ -341,7 +372,7 @@ 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
+#+caption: Block diagram corresponding to the example system of Figure [[fig:mech_sys_1dof_inertial_contr]]
 #+RESULTS:
 [[file:figs/classical_feedback_test_system.png]]
 
@@ -353,12 +384,14 @@ Let's define the system parameters on Matlab.
   m = 10; % Mass [kg]
 #+end_src
 
-And now the system dynamics $G(s)$ and $G_d(s)$ (their bode plots are shown in Figures [[fig:bode_plot_example_afm]] and [[fig:bode_plot_example_Gd]]).
+And now the system dynamics $G(s)$ and $G_d(s)$.
 #+begin_src matlab +n -r
   G = 1/(m*s^2 + c*s + k); % Plant
   Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
 #+end_src
 
+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);