From 5c01eca88c6d13272ead23f9bc2dd9878d066326 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Mon, 30 Nov 2020 19:42:57 +0100 Subject: [PATCH] Add general plant to shape S,T,KS,GS --- figs/general_conf_shaping_S_KS.pdf | Bin 0 -> 72955 bytes figs/general_conf_shaping_S_KS.png | Bin 0 -> 18546 bytes figs/general_conf_shaping_S_KS.svg | 228 ++++++ figs/general_conf_shaping_S_T.pdf | Bin 0 -> 72948 bytes figs/general_conf_shaping_S_T.png | Bin 0 -> 18552 bytes figs/general_conf_shaping_S_T.svg | 228 ++++++ figs/general_conf_shaping_S_T_KS.pdf | Bin 0 -> 73343 bytes figs/general_conf_shaping_S_T_KS.png | Bin 0 -> 21938 bytes figs/general_conf_shaping_S_T_KS.svg | 255 ++++++ figs/general_conf_shaping_S_T_KS_GS.pdf | Bin 0 -> 73595 bytes figs/general_conf_shaping_S_T_KS_GS.png | Bin 0 -> 25478 bytes figs/general_conf_shaping_S_T_KS_GS.svg | 266 +++++++ figs/h-infinity-4-blocs-constrains.pdf | Bin 0 -> 49193 bytes figs/h-infinity-4-blocs-constrains.png | Bin 0 -> 32501 bytes figs/h-infinity-4-blocs-constrains.svg | 174 +++++ figs/mixed_sensitivity_ref_tracking.pdf | Bin 54484 -> 73687 bytes figs/mixed_sensitivity_ref_tracking.png | Bin 16806 -> 19499 bytes figs/mixed_sensitivity_ref_tracking.svg | 222 +++--- figs/robustness_performance.pdf | Bin 0 -> 101864 bytes figs/robustness_performance.png | Bin 0 -> 47614 bytes figs/robustness_performance.svg | 989 ++++++++++++++++++++++++ index.html | 648 ++++++++++------ index.org | 474 +++++++++++- 23 files changed, 3128 insertions(+), 356 deletions(-) create mode 100644 figs/general_conf_shaping_S_KS.pdf create mode 100644 figs/general_conf_shaping_S_KS.png create mode 100644 figs/general_conf_shaping_S_KS.svg create mode 100644 figs/general_conf_shaping_S_T.pdf create mode 100644 figs/general_conf_shaping_S_T.png create mode 100644 figs/general_conf_shaping_S_T.svg create mode 100644 figs/general_conf_shaping_S_T_KS.pdf create mode 100644 figs/general_conf_shaping_S_T_KS.png create mode 100644 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/index.html b/index.html index c78c361..fece69b 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 @@ -30,72 +30,75 @@

Table of Contents

-
-

1 Introduction to the Control Methodology - Model Based Control

+
+

1 Introduction to the Control Methodology - Model Based Control

- +

-
-

1.1 Model Based 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:

    @@ -109,7 +112,7 @@ It consists of three steps:
-
+

control-procedure.png

Figure 1: Typical Methodoly for Model Based Control

@@ -121,14 +124,14 @@ 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

- +

- +
@@ -266,30 +269,37 @@ In this document, we will mainly focus on steps 2 and 3.
Table 1: Table summurazing the main differences between classical, modern and robust control
+ + +
+

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.

+
-
-

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 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 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.

+

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.

- +
@@ -375,7 +385,7 @@ The notations used are listed in Table 2.
Table 2: Example system variables
-
+

Derive the following open-loop transfer functions:

@@ -432,14 +442,14 @@ 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. +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 3: Block diagram corresponding to the example system

+

Figure 4: Block diagram corresponding to the example system

@@ -454,7 +464,7 @@ 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 5 and 6).

4: G = 1/(m*s^2 + c*s + k); % Plant
@@ -463,57 +473,57 @@ 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 \(G(s)\)

+

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

-
+

bode_plot_example_Gd.png

-

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

+

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

-
-

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 6): +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 6: Classical Feedback Architecture

+

Figure 7: Classical Feedback Architecture

@@ -535,26 +545,26 @@ 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 (an example is shown in Figure 7). +\(K(s)\) 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 7: Typical Wanted Shape for the Loop Gain \(L(s)\)

+

Figure 8: 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:

@@ -566,7 +576,7 @@ Let’s take our example system and try to apply the Open-Loop shaping strat
-
+

Using SISOTOOL, design a controller that fulfill the specifications.

@@ -583,7 +593,7 @@ 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 8. +The obtained controller is shown below, and the bode plot of the Loop Gain is shown in Figure 9.

K = 14e8 * ... % Gain
@@ -594,10 +604,10 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
 
-
+

loop_gain_manual_afm.png

-

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

+

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

@@ -642,11 +652,11 @@ 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

- +

@@ -673,7 +683,7 @@ where:

  • K is the synthesize controller
  • -
    +

    Matlab documentation of loopsyn (link).

    @@ -682,11 +692,11 @@ 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

    - +

    @@ -718,10 +728,10 @@ The \(\mathcal{H}_\infty\) optimal open loop shaping synthesis is performed usin

    -The Bode plot of the obtained controller is shown in Figure 9. +The Bode plot of the obtained controller is shown in Figure 10.

    -
    +

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

    @@ -742,28 +752,28 @@ Let’s briefly analyze this controller: -
    +

    open_loop_shaping_hinf_K.png

    -

    Figure 9: Obtained controller \(K\) using the open-loop \(\mathcal{H}_\infty\) shaping

    +

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

    -The obtained Loop Gain is shown in Figure 10. +The obtained Loop Gain is shown in Figure 11.

    -
    +

    open_loop_shaping_hinf_L.png

    -

    Figure 10: Obtained Open Loop Gain \(L(s) = G(s) K(s)\) and comparison with the wanted Loop gain \(L_w\)

    +

    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.

    - +
    @@ -804,22 +814,22 @@ Let’s now compare the obtained stability margins of the \(\mathcal{H}_\inf -
    -

    3 First Steps 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

    @@ -836,7 +846,7 @@ 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:

    @@ -851,28 +861,28 @@ 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 11. +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 11: Example of the \(\mathcal{H}_\infty\) norm of a SISO system

    +

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

    -
    -

    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.

    @@ -908,11 +918,11 @@ Note that there are many ways to use the \(\mathcal{H}_\infty\) Synthesis:
    -
    -

    3.3 The Generalized Plant

    +
    +

    3.3 The Generalized Plant

    - +

    @@ -922,9 +932,9 @@ It makes things much easier for the following steps.

    -The generalized plant, usually noted \(P(s)\), is shown in Figure 12. +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.

    @@ -937,14 +947,14 @@ It can indeed represent feedback as well as feedforward control architectures. \end{equation} -

    +

    general_plant.png

    -

    Figure 12: Inputs and Outputs of the generalized Plant

    +

    Figure 13: Inputs and Outputs of the generalized Plant

    -
    -
    Table 3: Comparison of the characteristics obtained with the two methods
    +
    +
    @@ -990,18 +1000,18 @@ It can indeed represent feedback as well as feedforward control architectures. -
    -

    3.4 The General Synthesis Problem Formulation

    +
    +

    3.4 The General Synthesis Problem Formulation

    - +

    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
    @@ -1016,10 +1026,10 @@ After \(K\) is found, the system is robustified by adjusting the response
    -
    +

    general_control_names.png

    -

    Figure 13: General Control Configuration

    +

    Figure 14: General Control Configuration

    @@ -1048,31 +1058,31 @@ 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 14 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. Remove \(K\) and rearrange the inputs and outputs to match the generalized configuration
    -
    +

    -Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure 14 +Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure 15

    -
    +

    classical_feedback_tracking.png

    -

    Figure 14: Classical Feedback Control Architecture (Tracking)

    +

    Figure 15: Classical Feedback Control Architecture (Tracking)

    Hint @@ -1093,14 +1103,14 @@ Then, Remove \(K\) and rearrange the inputs and outputs.
    Answer

    -The obtained generalized plant shown in Figure 15. +The obtained generalized plant shown in Figure 16.

    -
    +

    mixed_sensitivity_ref_tracking.png

    -

    Figure 15: Generalized plant of the Classical Feedback Control Architecture (Tracking)

    +

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

    @@ -1121,79 +1131,84 @@ P.OutputName = {'e', 'u
    -
    -

    4 Modern Interpretation of the Control Specifications

    +
    +

    4 Modern Interpretation of the Control Specifications

    - +

    -
    -

    4.1 Introduction

    +
    +

    4.1 Introduction

    -The +As shown in Section 2, the loop gain \(L(s) = G(s) K(s)\) is a useful and easy tool for the design of controllers by hand.

    - -
    -

    gang_of_four_feedback.png +

    +It is very easy to shape as it depends linearly on \(K(s)\). +Moreover, it gives information on important quantities such as the stability margins and the control bandwidth.

    -

    Figure 16: Simple Feedback Architecture

    -
    +

    +However, the loop gain \(L(s)\) does not directly give the performances of the closed-loop system. +The closed loop system behavior is determined by the closed-loop transfer functions. +

    + +

    +If we consider the feedback system shown in Figure 17, we can link to the following specifications to closed-loop transfer functions +

      -
    • Reference tracking Overshoot, Static error, Settling time +
    • 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
        -
      • \(S(s)\), \(T(s)\), \(K(s)S(s)\), \(G(s)S(s)\)
      • +
      • All closed-loop transfer functions must be stable
      • +
    • +
    • Robustness (stability margins) +
        +
      • Module margin (see Section 4.4)
    • -
    • Robustness to plant uncertainty (stability margins)
    • -
    • Controller implementation
    + + +
    +

    gang_of_four_feedback.png +

    +

    Figure 17: Simple Feedback Architecture

    +
    -
    -

    4.2 Closed Loop Transfer Functions

    +
    +

    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. +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 @@ -1230,9 +1245,10 @@ The following equations should be obtained:
    +

    -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: +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} S &= \frac{1 }{1 + GK}, \quad \text{the sensitivity function} \\ @@ -1241,7 +1257,9 @@ They are called the Gang of Four: 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. @@ -1257,14 +1275,19 @@ And we have: y &= T r + GS d - T n \\ u &= KS r - S d - KS n \end{align} + +

    +Thus, for reference tracking, we want to shape the closed-loop transfer function from \(r\) to \(\epsilon\), that is the sensitivity function \(S(s)\). +Similarly, to reduce the effect of measurement noise \(n\) on the output \(y\), we want to act on the complementary sensitivity function \(T(s)\). +

    -
    -

    4.3 Sensitivity Transfer Function

    +
    +

    4.3 Sensitivity Function

    - +

    @@ -1323,7 +1346,7 @@ xlabel('Time [s]'); ylabel( +

    h-infinity-spec-S.png

    Figure 18: Typical wanted shape of the Sensitivity transfer function

    @@ -1331,11 +1354,11 @@ xlabel('Time [s]'); ylabel( -

    4.4 Robustness: Module Margin

    +
    +

    4.4 Robustness: Module Margin

    - +

      @@ -1350,26 +1373,30 @@ xlabel('Time [s]'); ylabel( -

      4.5 How to Shape transfer function? Using of Weighting Functions!

      +
      +

      4.5 How to Shape transfer function? Using of Weighting Functions!

      - +

      +
        +
      • [ ] Maybe put this section in Previous chapter
      • +
      +

      -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. +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

      -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). +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).

      @@ -1379,19 +1406,17 @@ However, as the \(\mathcal{H}_\infty\) norm is the maximum peak value of the tra

      -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 \(\tilde{\epsilon}\): +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} +\begin{align} + & \left\| \frac{\tilde{\epsilon}}{r} \right\|_\infty < \gamma (=1)\nonumber \\ + \Leftrightarrow & \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\ + \Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega\nonumber \\ + \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 allows to shape the magnitude of the sensitivity transfer function. Therefore, the choice of the weighting function \(W_s(s)\) is very important. @@ -1401,14 +1426,14 @@ Its inverse magnitude will define the frequency dependent upper bound of the sen

      -
      +

      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. +Once the weighting function is designed, it should be added to the generalized plant as shown in Figure 20.

      @@ -1426,11 +1451,11 @@ Pw = blkdiag(Ws, 1)*P;

      -
      -

      4.6 Design of Weighting Functions

      +
      +

      4.6 Design of Weighting Functions

      - +

      @@ -1459,9 +1484,9 @@ with:

    • hfgain
    -
    +

    -The Matlab code below produces a weighting function with a magnitude shape shown in Figure 21. +The Matlab code below produces a weighting function with a magnitude shape shown in Figure 21.

    @@ -1470,7 +1495,7 @@ The Matlab code below produces a weighting function with a magnitude shape shown
    -
    +

    first_order_weight.png

    Figure 21: Obtained Magnitude of the Weighting Function

    @@ -1478,7 +1503,7 @@ The Matlab code below produces a weighting function with a magnitude shape shown
    -
    +

    Quite often, higher orders weights are required.

    @@ -1510,7 +1535,7 @@ 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
    @@ -1545,11 +1570,11 @@ W3 = generateWeight('G0', 1e2, 22.
    +The obtained shapes are shown in Figure 22.
     

    -
    +

    high_order_weight.png

    Figure 22: Higher order weights using Equation \eqref{eq:weight_formula_advanced}

    @@ -1559,11 +1584,11 @@ The obtained shapes are shown in Figure 22.
    -
    -

    4.7 Sensitivity Function Shaping - Example

    +
    +

    4.7 Sensitivity Function Shaping - Example

    - +

      @@ -1596,7 +1621,7 @@ And the \(\mathcal{H}_\infty\) synthesis is performed.
    -
    +
     K = hinfsyn(Pw, 1, 1, 'Display', 'on');
     
       Test bounds:  0.5 <=  gamma  <=  0.51
    @@ -1619,50 +1644,213 @@ The obtained \(\gamma \approx 0.5\) means that it found a controller \(K(s)\) th
     \end{aligned}
     
     

    -This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure 23. +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

    + +
    +

    4.8 Complementary Sensitivity Function

    -
    -

    5 \(\mathcal{H}_\infty\) Mixed-Sensitivity Synthesis

    + +
    +

    4.9 Summary

    +
    +
    Table 4: Notations for the general configuration
    + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    Table 5: Table caption
     Open-Loop ShapingClosed-Loop Shaping
    Reference Tracking\(L\) large\(S\) small
    Disturbance Rejection\(L\) large\(SG\) small
    Measurement Noise Filtering\(L\) small\(T\) small
    Small Command Amplitude\(K\) and \(L\) small\(KS\) small
    RobustnessPhase/Gain marginsModule margin: \(\Vert S\Vert_\infty\) small
    + + +
    +

    h-infinity-4-blocs-constrains.png +

    +

    Figure 24: Shaping the Gang of Four: Limitations

    +
    +
    +
    +
    + + +
    +

    5 \(\mathcal{H}_\infty\) Mixed-Sensitivity Synthesis

    - +

    -
    -

    5.1 Problem

    +
    +

    5.1 Problem

    -
    -

    5.2 Typical Procedure

    +
    +

    5.2 Typical Procedure

    -
    -

    5.3 Step 1 - Shaping of the Sensitivity Function

    +
    +

    5.3 Step 1 - Shaping of the Sensitivity Function

    -
    -

    5.4 Step 2 - Shaping of

    +
    +

    5.4 Step 2 - Shaping of

    +
    + +
    +

    5.5 General Configuration for various shaping

    +
    +
    Shaping of S and KS + +
    +

    general_conf_shaping_S_KS.png +

    +

    Figure 25: Generalized Plant to shape \(S\) and \(KS\)

    +
    + +
    +
    P = [W1 -G*W1
    +     0   W2
    +     1  -G];
    +
    +
    + +
      +
    • \(W_1(s)\) is used to shape \(S\)
    • +
    • \(W_2(s)\) is used to shape \(KS\)
    • +
    +
    +
    Shaping of S and T + +
    +

    general_conf_shaping_S_T.png +

    +

    Figure 26: Generalized Plant to shape \(S\) and \(T\)

    +
    + +
    +
    P = [W1 -G*W1
    +     0   G*W2
    +     1   -G];
    +
    +
    + +
      +
    • \(W_1\) is used to shape \(S\)
    • +
    • \(W_2\) is used to shape \(T\)
    • +
    +
    +
    Shaping of S, T and KS + +
    +

    general_conf_shaping_S_T_KS.png +

    +

    Figure 27: Generalized Plant to shape \(S\), \(T\) and \(KS\)

    +
    + +
    +
    P = [W1 -G*W1
    +     0   W2
    +     0   G*W3
    +     1   -G];
    +
    +
    + +
      +
    • \(W_1\) is used to shape \(S\)
    • +
    • \(W_2\) is used to shape \(KS\)
    • +
    • \(W_3\) is used to shape \(T\)
    • +
    +
    +
    Shaping of S, T, KS and GS + +
    +

    general_conf_shaping_S_T_KS_GS.png +

    +

    Figure 28: Generalized Plant to shape \(S\), \(T\), \(KS\) and \(GS\)

    +
    + +
    +
    P = [ W1  -W1*G*W3 -G*W1
    +      0    0        W2
    +      1   -G*W3    -G];
    +
    +
    + +
      +
    • \(W_1\) is used to shape \(S\)
    • +
    • \(W_2\) is used to shape \(KS\)
    • +
    • \(W_1W_3\) is used to shape \(GS\)
    • +
    • \(W_2W_3\) is used to shape \(T\)
    • +
    +
    +
    -
    -

    6 Conclusion

    +
    +

    6 Conclusion

    +
    +

    + +

    +
    -
    -

    7 Resources

    +
    +

    7 Resources

    @@ -1676,7 +1864,7 @@ This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure

    Author: Dehaeze Thomas

    -

    Created: 2020-11-30 lun. 17:44

    +

    Created: 2020-11-30 lun. 19:41

    diff --git a/index.org b/index.org index ddfecfc..7e385c8 100644 --- a/index.org +++ b/index.org @@ -130,6 +130,80 @@ In this document, we will mainly focus on steps 2 and 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 | +#+begin_src latex :file robustness_performance.pdf + \begin{tikzpicture} + % Scale + \def\yscale{0.8} + \def\xscale{1.0} + + % Colors + \def\colorstart{blue} + \def\colorend{red} + + % Axis + \draw [->] (-0.5,0) -- (10*\xscale,0) node[below left]{Robustness}; + \draw [->] (0,-0.5) -- (0,10*\yscale) node[below left, rotate=90, anchor=south east]{Performance}; + + % Color Bar + \shade[draw, bottom color=\colorstart, top color=\colorend, fill opacity=0.5] (10*\xscale, 1*\yscale) rectangle (11*\xscale, 9*\yscale); + \node[rotate=90, above] at (10*\xscale, 5*\yscale) {Required information on plant}; + \node[above] at (10.5*\xscale, 1*\yscale) {little}; + \node[below] at (10.5*\xscale, 9*\yscale) {large}; + + % =================================== + % Classical Control + % =================================== + % Control Types + \node[align=center] (pid) at (7.0*\xscale, 1.2*\yscale) {PID\\Lead-Lag}; + + \begin{scope}[on background layer] + % Control Families + \node[ellipse, draw, dashed, minimum width=3.0*\xscale cm, minimum height=2.0*\yscale cm, + fill=\colorstart!90!\colorend, fill opacity=0.5, text opacity=1] + (classicalcontrol) at (pid) {}; + \end{scope} + \node[above, align=center] at (classicalcontrol.north) {\textbf{Classical control} (1930)\\{\small SISO, Manual Method}}; + % =================================== + + + % =================================== + % Modern Control + % =================================== + % Control Types + \node[align=center] (lqg) at (2.0*\xscale, 7.5*\yscale) {LQR\\LQG}; + + \begin{scope}[on background layer] + \node[ellipse, draw, dashed, minimum width=2.0*\xscale cm, minimum height=2.0*\yscale cm, + fill=\colorstart!20!\colorend, fill opacity=0.5, text opacity=1] + (moderncontrol) at (lqg) {}; + \end{scope} + \node[above, align=center] at (moderncontrol.north) {\textbf{Modern control} (1960)\\{\small MIMO, Optimal}}; + % =================================== + + + % =================================== + % Robust Control + % =================================== + % Control Types + \node[align=center] (hinf) at (4.5*\xscale, 4.8*\yscale) {$H_\infty$\\$H_2$}; + \node[] (mu) at (5.5*\xscale, 4.8*\yscale) {$\mu$}; + + + \begin{scope}[on background layer] + \node[ellipse, draw, dashed, minimum width=3.0*\xscale cm, minimum height=2.5*\yscale cm, + shade, left color=\colorstart!50!\colorend, right color=\colorstart!10!\colorend, fill opacity=0.5, text opacity=1] + (robustcontrol) at ($0.5*(hinf)+0.5*(mu)$) {}; + \end{scope} + \node[above, align=center] at (robustcontrol.north) {\textbf{Robust control} (1990)\\{\small MIMO, Robust}}; + % =================================== + \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]] + ** Example System <> @@ -922,33 +996,28 @@ Using Matlab, the generalized plant can be defined as follows: <> ** Introduction -The -#+name: fig:gang_of_four_feedback -#+caption: Simple Feedback Architecture -#+RESULTS: -[[file:figs/gang_of_four_feedback.png]] +As shown in Section [[sec:open_loop_shaping]], the loop gain $L(s) = G(s) K(s)$ is a useful and easy tool for the design of controllers by hand. -- *Reference tracking* Overshoot, Static error, Settling time +It is very easy to shape as it depends linearly on $K(s)$. +Moreover, it gives information on important quantities such as the stability margins and the control bandwidth. + +However, the loop gain $L(s)$ does *not* directly give the performances of the closed-loop system. +The closed loop system behavior is determined by the *closed-loop transfer functions*. + +If we consider the feedback system shown in Figure [[fig:gang_of_four_feedback]], we can link to the following specifications to closed-loop transfer functions +- *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 $K(s)$. + - All closed-loop transfer functions must be stable +- *Robustness* (stability margins) + - Module margin (see Section [[sec:module_margin]]) #+begin_src latex :file gang_of_four_feedback.pdf \begin{tikzpicture} @@ -975,6 +1044,11 @@ As the performances of a controlled system depend on the *closed* loop transfer #+RESULTS: [[file:figs/gang_of_four_feedback.png]] +** 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_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]]. @@ -1003,14 +1077,16 @@ The following equations should be obtained: #+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_important +We can see that they are 4 different transfer functions describing the behavior of the system in Figure [[fig:gang_of_four_feedback]]. +These 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} +#+end_important #+begin_seealso If a feedforward controller is included, a *Gang of Six* transfer functions can be defined. @@ -1024,7 +1100,10 @@ And we have: u &= KS r - S d - KS n \end{align} -** Sensitivity Transfer Function +Thus, for reference tracking, we want to shape the /closed-loop/ transfer function from $r$ to $\epsilon$, that is the sensitivity function $S(s)$. +Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we want to act on the complementary sensitivity function $T(s)$. + +** Sensitivity Function <> #+begin_src matlab @@ -1132,6 +1211,8 @@ And we have: ** How to *Shape* transfer function? Using of Weighting Functions! <> +- [ ] Maybe put this section in Previous chapter + 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 @@ -1173,14 +1254,12 @@ However, as the $\mathcal{H}_\infty$ norm is the maximum peak value of the trans 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{align} + & \left\| \frac{\tilde{\epsilon}}{r} \right\|_\infty < \gamma (=1)\nonumber \\ + \Leftrightarrow & \left\| W_s(s) S(s) \right\|_\infty < 1\nonumber \\ + \Leftrightarrow & \left| W_s(j\omega) S(j\omega) \right| < 1 \quad \forall \omega\nonumber \\ + \Leftrightarrow & \left| S(j\omega) \right| < \frac{1}{\left| W_s(j\omega) \right|} \quad \forall \omega \label{eq:sensitivity_shaping} +\end{align} #+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. @@ -1430,6 +1509,77 @@ This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:re #+RESULTS: [[file:figs/results_sensitivity_hinf.png]] +** Complementary Sensitivity Function + + +** Summary + +#+name: tab:specification_modern +#+caption: Table caption +| | Open-Loop Shaping | Closed-Loop Shaping | +|-----------------------------+--------------------+--------------------------------------------| +| Reference Tracking | $L$ large | $S$ small | +| Disturbance Rejection | $L$ large | $GS$ small | +| Measurement Noise Filtering | $L$ small | $T$ small | +| Small Command Amplitude | $K$ and $L$ small | $KS$ small | +| Robustness | Phase/Gain margins | Module margin: $\Vert S\Vert_\infty$ small | + +#+begin_src latex :file h-infinity-4-blocs-constrains.pdf + \begin{tikzpicture} + \begin{scope}[shift={(0, 0)}] + \draw[] (2.5, 1.0) node[]{$S$}; + \draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5); + \draw[] (0.6, -0.5) node[]{$\sim GK^{-1}$}; + \draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5); + \draw[] (4.5, -0.5) node[]{$\sim 1$}; + \draw[fill=red!20] (2.5, 0.15) circle (0.15); + \draw[dashed] (-0.4, 0) -- (5.4, 0); + \draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3) to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0); + \draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4); + \end{scope} + + \begin{scope}[shift={(6.4, 0)}] + \draw[] (2.5, 1.0) node[]{$GS$}; + \draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5); + \draw[] (0.6, -0.5) node[]{$\sim K^{-1}$}; + \draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5); + \draw[] (4.5, -0.5) node[]{$\sim G$}; + \draw[dashed] (-0.4, 0) -- (5.4, 0); + \draw [] (0,-2) to[out=45,in=180+45] (1, -1) to[out=45, in=180] (2.5,-0.2) to[out=0,in=180-45] (4,-1) to[out=-45,in=180-45] (5, -2); + \draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4); + \end{scope} + + \begin{scope}[shift={(0, -4.4)}] + \draw[] (2.5, 1.0) node[]{$KS$}; + \draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5); + \draw[] (0.6, -1.8) node[]{$\sim G^{-1}$}; + \draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5); + \draw[] (4.5, -0.3) node[]{$\sim K$}; + \draw[dashed] (-0.4, 0) -- (5.4, 0); + \draw [] (0,-1.5) to[out=45,in=180+45] (1, -0.5) to[out=45, in=180] (2.5,0.3) to[out=0,in=180-45] (4,-0.5) to[out=-45,in=180-45] (5, -1.5); + \draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4); + \end{scope} + + \begin{scope}[shift={(6.4, -4.4)}] + \draw[] (2.5, 1.0) node[]{$T$}; + \draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5); + \draw[] (0.6, -0.5) node[]{$\sim 1$}; + \draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5); + \draw[] (4.5, -0.5) node[]{$\sim GK$}; + \draw[fill=red!20] (2.5, 0.15) circle (0.15); + \draw[dashed] (-0.4, 0) -- (5.4, 0); + \draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2); + \draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4); + \end{scope} + \end{tikzpicture} +#+end_src + +#+name: fig:h-infinity-4-blocs-constrains +#+caption: Shaping the Gang of Four: Limitations +#+RESULTS: +[[file:figs/h-infinity-4-blocs-constrains.png]] + + * $\mathcal{H}_\infty$ Mixed-Sensitivity Synthesis <> @@ -1441,7 +1591,275 @@ This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:re ** Step 2 - Shaping of +** General Configuration for various shaping +*** S KS :ignore: +#+HTML:
    Shaping of S and KS +#+begin_src latex :file general_conf_shaping_S_KS.pdf + \begin{tikzpicture} + % Blocs + \node[block] (G) {$G$}; + + \node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {}; + \node[block, above right=0.4 and 0.8 of addw] (W2) {$W_2$}; + \node[block, above=0.5 of W2] (W1) {$W_1$}; + + \coordinate (Gin) at ($(G.west)+(-0.5, 0)$); + + \begin{scope}[on background layer] + \node[fit={(Gin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {}; + \node[above] at (P.north) {Weighted Generalized Plant $P$}; + \end{scope} + + \node[block, below=0.6 of P] (K) {$K$}; + + \coordinate[right=0.8 of W1] (z); + \coordinate[above left=1.8 and 1.4 of G] (w); + + % Connections + \draw[->] (G.east) -- (addw.west); + \draw[->] ($(addw.east)+(0.2, 0)$)node[branch]{} |- (W1.west); + \draw[->] (Gin)node[branch]{} |- (W2.west); + + \draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$}; + \draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$}; + + \draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east); + \draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west); + + \draw[->] (w) node[above right]{$w$} -| (addw.north); + \end{tikzpicture} +#+end_src + +#+name: fig:general_conf_shaping_S_KS +#+caption: Generalized Plant to shape $S$ and $KS$ +#+RESULTS: +[[file:figs/general_conf_shaping_S_KS.png]] + +#+name: lst:general_plant_S_KS +#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_KS]] +#+begin_src matlab :eval no :tangle no + P = [W1 -G*W1 + 0 W2 + 1 -G]; +#+end_src + +- $W_1(s)$ is used to shape $S$ +- $W_2(s)$ is used to shape $KS$ +#+HTML:
    + +*** S T :ignore: +#+HTML:
    Shaping of S and T +#+begin_src latex :file general_conf_shaping_S_T.pdf + \begin{tikzpicture} + % Blocs + \node[block] (G) {$G$}; + + \node[addb={+}{-}{}{}{}, right=0.8 of G] (addw) {}; + \node[block, above right=0.4 and 0.8 of addw] (W2) {$W_2$}; + \node[block, above=0.5 of W2] (W1) {$W_1$}; + + \begin{scope}[on background layer] + \node[fit={(G.south west) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {}; + \node[above] at (P.north) {Weighted Generalized Plant $P$}; + \end{scope} + + \node[block, below=0.6 of P] (K) {$K$}; + + \coordinate[right=0.8 of W1] (z); + \coordinate[above left=1.8 and 0.8 of G] (w); + + % Connections + \draw[->] (G.east) -- (addw.west); + \draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west); + \draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W2.west); + + \draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$}; + \draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$}; + + \draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east); + \draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west); + + \draw[->] (w) node[above right]{$w$} -| (addw.north); + \end{tikzpicture} +#+end_src + +#+name: fig:general_conf_shaping_S_T +#+caption: Generalized Plant to shape $S$ and $T$ +#+RESULTS: +[[file:figs/general_conf_shaping_S_T.png]] + +#+name: lst:general_plant_S_T +#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T]] +#+begin_src matlab :eval no :tangle no + P = [W1 -G*W1 + 0 G*W2 + 1 -G]; +#+end_src + +- $W_1$ is used to shape $S$ +- $W_2$ is used to shape $T$ +#+HTML:
    + +*** S T KS :ignore: +#+HTML:
    Shaping of S, T and KS +#+begin_src latex :file general_conf_shaping_S_T_KS.pdf + \begin{tikzpicture} + % Blocs + \node[block] (G) {$G$}; + + \node[addb={+}{-}{}{}{}, right=0.8 of G] (addw) {}; + \node[block, above right=0.4 and 0.8 of addw] (W3) {$W_3$}; + \node[block, above=0.2 of W3] (W2) {$W_2$}; + \node[block, above=0.2 of W2] (W1) {$W_1$}; + + \coordinate (Gin) at ($(G.west)+(-0.5, 0)$); + + \begin{scope}[on background layer] + \node[fit={(Gin|-G.south) (W1.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {}; + \node[above] at (P.north) {Weighted Generalized Plant $P$}; + \end{scope} + + \node[block, below=0.6 of P] (K) {$K$}; + + \coordinate[right=0.8 of W1] (z); + \coordinate[above left=1.4 and 1.3 of G] (w); + + % Connections + \draw[->] (G.east) -- (addw.west); + \draw[->] ($(addw.east)+(0.3, 0)$)node[branch]{} |- (W1.west); + \draw[->] (Gin)node[branch]{} |- (W2.west); + \draw[->] ($(G.east)+(0.3, 0)$)node[branch]{} |- (W3.west); + + \draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$}; + \draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$}; + \draw[->] (W3.east) -- (W3-|z) node[above left](z3){$z_3$}; + + \draw[->] (addw.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east); + \draw[->] (K.west) -| node[near end, left]{$u$} (G-|w) -- (G.west); + + \draw[->] (w) node[above right]{$w$} -| (addw.north); + \end{tikzpicture} +#+end_src + +#+name: fig:general_conf_shaping_S_T_KS +#+caption: Generalized Plant to shape $S$, $T$ and $KS$ +#+RESULTS: +[[file:figs/general_conf_shaping_S_T_KS.png]] + +#+name: lst:general_plant_S_T_KS +#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T_KS]] +#+begin_src matlab :eval no :tangle no + P = [W1 -G*W1 + 0 W2 + 0 G*W3 + 1 -G]; +#+end_src + +- $W_1$ is used to shape $S$ +- $W_2$ is used to shape $KS$ +- $W_3$ is used to shape $T$ +#+HTML:
    + +*** S T KS GS :ignore: +#+HTML:
    Shaping of S, T, KS and GS +#+begin_src latex :file general_conf_shaping_S_T_KS_GS.pdf + \begin{tikzpicture} + % Blocs + \node[block] (G) {$G$}; + + \node[addb={+}{-}{}{}{}, right=0.6 of G] (addr) {}; + \node[addb, left=0.6 of G] (addd) {}; + \node[block, above right=0.4 and 0.8 of addr] (W2) {$W_2$}; + \node[block, above=0.5 of W2] (W1) {$W_1$}; + + \node[block, above left=0.7 and 0.8 of addd] (W3) {$W_3$}; + \node[block, above=0.5 of W3] (W4) {$1$}; + + \begin{scope}[on background layer] + \node[fit={(W3.west|-G.south) (W4.north -| W2.east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {}; + \node[above] at (P.north) {Weighted Generalized Plant $P$}; + \end{scope} + + \node[block, below=0.6 of P] (K) {$K$}; + + \coordinate[right=1.0 of W1] (z); + \coordinate[left=1.0 of W3] (w); + + % Connections + \draw[->] (G.east) -- (addr.west); + \draw[->] ($(addr.east)+( 0.2, 0)$)node[branch]{} |- (W1.west); + \draw[->] ($(addd.west)+(-0.4, 0)$)node[branch]{} |- (W2.west); + + \draw[->] (W1.east) -- (W1-|z) node[above left](z1){$z_1$}; + \draw[->] (W2.east) -- (W2-|z) node[above left](z2){$z_2$}; + + \draw[->] (addr.east) -- (addw-|z) |- node[near start, right]{$v$} (K.east); + \draw[->] (K.west) -| node[near end, left]{$u$} (addd-|w) -- (addd.west); + \draw[->] (addd.east) -- (G.west); + + \draw[->] (W3.east) -| (addd.north); + \draw[->] (W4.east) -| (addr.north); + + \draw[->] (w) node[above right]{$w_1$} -- (W3.west); + \draw[->] (w|-W4) node[above right]{$w_2$} -- (W4.west); + \end{tikzpicture} +#+end_src + +#+name: fig:general_conf_shaping_S_T_KS_GS +#+caption: Generalized Plant to shape $S$, $T$, $KS$ and $GS$ +#+RESULTS: +[[file:figs/general_conf_shaping_S_T_KS_GS.png]] + +#+name: lst:general_plant_S_T_KS_GS +#+caption: General Plant definition corresponding to Figure [[fig:general_conf_shaping_S_T_KS_GS]] +#+begin_src matlab :eval no :tangle no + P = [ W1 -W1*G*W3 -G*W1 + 0 0 W2 + 1 -G*W3 -G]; +#+end_src + +- $W_1$ is used to shape $S$ +- $W_2$ is used to shape $KS$ +- $W_1W_3$ is used to shape $GS$ +- $W_2W_3$ is used to shape $T$ +#+HTML:
    + * Conclusion +<> + +* Things to add :noexport: +** 2 blocs criterion - constrains +#+begin_src latex :file h-infinity-2-blocs-constrains.pdf + \begin{tikzpicture} + \begin{scope}[shift={(0, 0)}] + \draw[] (2.5, 1.0) node[]{$S$}; + \draw[fill=blue!20] (-0.2, -2.5) rectangle (1.4, 0.5); + \draw[] (0.6, -0.5) node[]{$\sim GK^{-1}$}; + \draw[fill=red!20] (3.6, -2.5) rectangle (5.2, 0.5); + \draw[] (4.5, -0.5) node[]{$\sim 1$}; + \draw[fill=red!20] (2.5, 0.15) circle (0.15); + \draw[dashed] (-0.4, 0) -- (5.4, 0); + \draw [] (0,-2) to[out=45,in=180+45] (2,0) to[out=45,in=180] (2.5,0.3) to[out=0,in=180] (3.5,0) to[out=0,in=180] (5, 0); + \draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4); + \end{scope} + + \begin{scope}[shift={(6.4, 0)}] + \draw[] (2.5, 1.0) node[]{$T$}; + \draw[fill=red!20] (-0.2, -2.5) rectangle (1.4, 0.5); + \draw[] (0.6, -0.5) node[]{$\sim 1$}; + \draw[fill=blue!20] (3.6, -2.5) rectangle (5.2, 0.5); + \draw[] (4.5, -0.5) node[]{$\sim GK$}; + \draw[fill=red!20] (2.5, 0.15) circle (0.15); + \draw[dashed] (-0.4, 0) -- (5.4, 0); + \draw [] (0,0) to[out=0,in=180] (1.5,0) to[out=0,in=180] (2.5,0.3) to[out=0,in=-45] (3,0) to[out=-45,in=180-45] (5, -2); + \draw[dashed] (-0.5, -2.7) rectangle (5.5, 1.4); + \end{scope} + \end{tikzpicture} +#+end_src + +#+RESULTS: +[[file:figs/h-infinity-2-blocs-constrains.png]] + * Resources

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