diff --git a/index.html b/index.html index 4bd8178..31eece2 100644 --- a/index.html +++ b/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Robust Control - \(\mathcal{H}_\infty\) Synthesis @@ -30,29 +30,29 @@

Table of Contents

-
-

1 Introduction to the Control Methodology - Model Based Control

+
+

1 Introduction to the Control Methodology - Model Based Control

-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:

    @@ -66,7 +66,7 @@ It consists of three steps:
-
+

control-procedure.png

Figure 1: Typical Methodoly for Model Based Control

@@ -78,8 +78,8 @@ In this document, we will mainly focus on steps 2 and 3.
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-

2 Some Background: From Classical Control to Robust Control

+
+

2 Some Background: From Classical Control to Robust Control

Classical Control (1930) @@ -156,10 +156,10 @@ Robust Control (1980)

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-

3 The \(\mathcal{H}_\infty\) Norm

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+

3 The \(\mathcal{H}_\infty\) Norm

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+

The \(\mathcal{H}_\infty\) norm is defined as the peak of the maximum singular value of the frequency response

@@ -176,7 +176,7 @@ For a SISO system \(G(s)\), it is simply the peak value of \(|G(j\omega)|\) as a
-
+

Let’s define a plant dynamics:

@@ -201,12 +201,12 @@ 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 2. +The magnitude \(|G(j\omega)|\) of the plant \(G(s)\) as a function of frequency is shown in Figure 2. 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.

-
+

hinfinity_norm_siso_bode.png

Figure 2: Example of the \(\mathcal{H}_\infty\) norm of a SISO system

@@ -216,46 +216,46 @@ The maximum value of the magnitude over all frequencies does correspond to the \
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4 \(\mathcal{H}_\infty\) Synthesis

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

Optimization problem: -\(\hinf\) 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 \(\hinf\) norms of defined transfer functions are minimized. +\(\mathcal{H}_\infty\) 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:

    -
  1. Write the problem as standard \(\hinf\) problem
    2. -
  2. Translate the specifications as \(\hinf\) norms
    3. +
  3. Write the problem as standard \(\mathcal{H}_\infty\) problem
    4. +
  4. Translate the specifications as \(\mathcal{H}_\infty\) norms
  5. Make the synthesis and analyze the obtain controller
  6. Reduce the order of the controller for implementation

-Many ways to use the \(\hinf\) Synthesis: +Many ways to use the \(\mathcal{H}_\infty\) Synthesis:

    -
  • Traditional \(\hinf\) Synthesis
    • +
  • Traditional \(\mathcal{H}_\infty\) Synthesis
  • Mixed Sensitivity Loop Shaping
    • -
  • Fixed-Structure \(\hinf\) Synthesis
    • -
  • Signal Based \(\hinf\) Synthesis
    • +
  • Fixed-Structure \(\mathcal{H}_\infty\) Synthesis
    • +
  • Signal Based \(\mathcal{H}_\infty\) Synthesis
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-

5 The Generalized Plant

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+

5 The Generalized Plant

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+

general_plant.png

- +
@@ -303,10 +303,10 @@ The maximum value of the magnitude over all frequencies does correspond to the \ -
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6 Problem Formulation

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+

6 Problem Formulation

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The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing controllers \(K\) which minimize

@@ -317,7 +317,7 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
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general_control_names.png

Figure 4: General Control Configuration

@@ -326,17 +326,17 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro
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7 Classical feedback control and closed loop transfer functions

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7 Classical feedback control and closed loop transfer functions

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+

classical_feedback.png

Figure 5: Classical Feedback Architecture

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Table 1: Notations for the general configuration
+
@@ -390,8 +390,8 @@ The \(\mathcal{H}_\infty\) Synthesis objective is to find all stabilizing contro -
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8 From a Classical Feedback Architecture to a Generalized Plant

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8 From a Classical Feedback Architecture to a Generalized Plant

The procedure is: @@ -401,13 +401,13 @@ The procedure is:

  • Remove \(K\) and rearrange the inputs and outputs
    • -
    +

    -Let’s find the Generalized plant of corresponding to the tracking control architecture shown in Figure 6 +Let’s find the Generalized plant of corresponding to the tracking control architecture shown in Figure 6

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    +

    classical_feedback_tracking.png

    Figure 6: Classical Feedback Control Architecture (Tracking)

    @@ -425,11 +425,11 @@ 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 7. +We obtain the generalized plant shown in Figure 7.

    -
    +

    mixed_sensitivity_ref_tracking.png

    Figure 7: Generalized plant of the Classical Feedback Control Architecture (Tracking)

    @@ -449,12 +449,12 @@ Using Matlab, the generalized plant can be defined as follows:
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    -

    9 Modern Interpretation of the Control Specifications

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    +

    9 Modern Interpretation of the Control Specifications

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    -

    9.1 Introduction

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    +

    9.1 Introduction

    • Reference tracking Overshoot, Static error, Setling time @@ -488,8 +488,8 @@ Using Matlab, the generalized plant can be defined as follows:
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    -

    10 Resources

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    +

    10 Resources

    @@ -503,7 +503,7 @@ Using Matlab, the generalized plant can be defined as follows:

    Author: Dehaeze Thomas

    -

    Created: 2020-11-25 mer. 19:37

    +

    Created: 2020-11-25 mer. 19:38

    diff --git a/index.org b/index.org index 4b21d25..10af179 100644 --- a/index.org +++ b/index.org @@ -56,7 +56,7 @@ It consists of three steps: - _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 $\hinf$ Norm + # - $\Longrightarrow$ We will use the $\mathcal{H}_\infty$ Norm 3. *Synthesis*: research of $K$ that satisfies the specifications for the model of the system #+begin_src latex :file control-procedure.pdf @@ -195,19 +195,19 @@ The maximum value of the magnitude over all frequencies does correspond to the $ * $\mathcal{H}_\infty$ Synthesis *Optimization problem*: -$\hinf$ 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 $\hinf$ norms of defined transfer functions are minimized. +$\mathcal{H}_\infty$ 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*: -1. Write the problem as standard $\hinf$ problem -2. Translate the specifications as $\hinf$ norms +1. Write the problem as standard $\mathcal{H}_\infty$ problem +2. Translate the specifications as $\mathcal{H}_\infty$ norms 3. Make the synthesis and analyze the obtain controller 4. Reduce the order of the controller for implementation -*Many ways to use the $\hinf$ Synthesis*: -- Traditional $\hinf$ Synthesis +*Many ways to use the $\mathcal{H}_\infty$ Synthesis*: +- Traditional $\mathcal{H}_\infty$ Synthesis - Mixed Sensitivity Loop Shaping -- Fixed-Structure $\hinf$ Synthesis -- Signal Based $\hinf$ Synthesis +- Fixed-Structure $\mathcal{H}_\infty$ Synthesis +- Signal Based $\mathcal{H}_\infty$ Synthesis * The Generalized Plant #+begin_src latex :file general_plant.pdf
    Table 2: Notations for the Classical Feedback Architecture