From 1065724fb5e02c82bad6a09d1ed50396a90f8cdc Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 1 Dec 2020 00:23:13 +0100 Subject: [PATCH] Add text about module margin --- figs/phase_gain_margin_model_plant.pdf | Bin 0 -> 52123 bytes figs/phase_gain_margin_model_plant.png | Bin 0 -> 77336 bytes figs/phase_gain_margin_real_plant.pdf | Bin 0 -> 68736 bytes figs/phase_gain_margin_real_plant.png | Bin 0 -> 83240 bytes index.html | 692 +++++++++++++++---------- index.org | 215 ++++++-- 6 files changed, 611 insertions(+), 296 deletions(-) create mode 100644 figs/phase_gain_margin_model_plant.pdf create mode 100644 figs/phase_gain_margin_model_plant.png create mode 100644 figs/phase_gain_margin_real_plant.pdf create mode 100644 figs/phase_gain_margin_real_plant.png diff --git a/figs/phase_gain_margin_model_plant.pdf b/figs/phase_gain_margin_model_plant.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c083218631e522e71473faca19215ed388723f11 GIT binary patch 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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 @@ -30,75 +30,89 @@

Table of Contents

-
-

1 Introduction to the Control Methodology - Model Based Control

+

+This document is structured as follows: +

+
    +
  • As \(\mathcal{H}_\infty\) Control is a model based control technique, a short introduction to model based control is given in Section 1
  • +
  • Classical open loop shaping method is presented in Section 2. +It is also shown that \(\mathcal{H}_\infty\) synthesis can be used for open loop shaping.
  • +
  • \(\mathcal{H}_\infty\) +Important concepts such as the \(\mathcal{H}_\infty\) norm and the generalized plant are introduced.
  • +
  • A
  • +
  • Finally, an complete example of the +is performed in Section 5.
  • +
+ +
+

1 Introduction to 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:

    @@ -112,7 +126,7 @@ It consists of three steps:
-
+

control-procedure.png

Figure 1: Typical Methodoly for Model Based Control

@@ -121,17 +135,26 @@ It consists of three steps:

In this document, we will mainly focus on steps 2 and 3.

+ +

+Step 2 will be discussed in Section 4. +There are two main methods for the controller synthesis (step 3): +

+
    +
  • open loop shaping discussed in Section 2
  • +
  • closed loop shaping discussed in Sections 4 and 5
  • +
-
-

1.2 Some Background: From Classical Control to Robust Control

+
+

1.2 From Classical Control to Robust Control

- +

- +
@@ -199,22 +222,15 @@ In this document, we will mainly focus on steps 2 and 3. - + - - - - - - - - - + + @@ -271,7 +287,7 @@ 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
Control Architectures Proportional, Integral, DerivativeFull State FeedbackFull State Feedback, LQR General Control Configuration
  Leads, LagsLQR, LQG 
  Kalman Filters Kalman Filters, LQGGeneralized Plant
-
+

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.

@@ -279,27 +295,27 @@ In this document, we will mainly focus on steps 2 and 3.
-
-

1.3 Example System

+
+

1.3 Example System

- +

-Let’s consider the model shown in Figure 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.

- +
@@ -385,7 +401,7 @@ The notations used are listed in Table 2.
Table 2: Example system variables
-
+

Derive the following open-loop transfer functions:

@@ -442,11 +458,11 @@ Thanks in advance

-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

@@ -464,7 +480,7 @@ 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
@@ -473,14 +489,14 @@ 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 \(G_d(s)\)

@@ -489,38 +505,38 @@ And now the system dynamics \(G(s)\) and \(G_d(s)\) (their bode plots are shown
-
-

2 Classical Open Loop Shaping

+
+

2 Classical Open Loop Shaping

- +

-
-

2.1 Introduction to 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

@@ -545,11 +561,11 @@ 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 \(L(s)\)

@@ -557,14 +573,14 @@ 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:

@@ -576,7 +592,7 @@ Let’s take our example system and try to apply the Open-Loop shaping strat
-
+

Using SISOTOOL, design a controller that fulfill the specifications.

@@ -593,7 +609,7 @@ 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
@@ -604,7 +620,7 @@ 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 \(L(s) = G(s) K(s)\)

@@ -652,11 +668,11 @@ 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

- +

@@ -683,7 +699,7 @@ where:

  • K is the synthesize controller
  • -
    +

    Matlab documentation of loopsyn (link).

    @@ -692,11 +708,11 @@ 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

    - +

    @@ -728,10 +744,10 @@ The \(\mathcal{H}_\infty\) optimal open loop shaping synthesis is performed usin

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

    @@ -752,28 +768,28 @@ Let’s briefly analyze this controller: -
    +

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

    -
    +

    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.

    - +
    @@ -814,22 +830,22 @@ 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

    @@ -846,7 +862,7 @@ 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:

    @@ -861,11 +877,11 @@ 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

    @@ -875,14 +891,14 @@ 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.

    @@ -918,11 +934,11 @@ Note that there are many ways to use the \(\mathcal{H}_\infty\) Synthesis:
    -
    -

    3.3 The Generalized Plant

    +
    +

    3.3 The Generalized Plant

    - +

    @@ -932,9 +948,9 @@ 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.

    @@ -947,14 +963,14 @@ 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
    +
    +
    @@ -1000,18 +1016,18 @@ It can indeed represent feedback as well as feedforward control architectures. -
    -

    3.4 The General Synthesis Problem Formulation

    +
    +

    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
    @@ -1026,7 +1042,7 @@ After \(K\) is found, the system is robustified by adjusting the response
    -
    +

    general_control_names.png

    Figure 14: General Control Configuration

    @@ -1058,28 +1074,28 @@ 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. 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 15 +Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure 15

    -
    +

    classical_feedback_tracking.png

    Figure 15: Classical Feedback Control Architecture (Tracking)

    @@ -1103,11 +1119,11 @@ Then, Remove \(K\) and rearrange the inputs and outputs.
    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)

    @@ -1131,63 +1147,85 @@ P.OutputName = {'e', 'u
    -
    -

    4 Modern Interpretation of the Control Specifications

    +
    +

    4 Modern Interpretation of the Control Specifications

    - +

    -
    -

    4.1 Introduction

    +
    +

    4.1 Introduction

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

    -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. +\(L(s)\) 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 \(L(s)\).

    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. +The closed loop system behavior is indeed 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 +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.

    -
      -
    • Reference tracking (Overshoot, Static error, Settling time, …) -
        -
      • From \(r\) to \(\epsilon\)
      • -
    • -
    • Disturbances rejection -
        -
      • From \(d\) to \(y\)
      • -
    • -
    • Measurement noise filtering -
        -
      • From \(n\) to \(y\)
      • -
    • -
    • Small command amplitude -
        -
      • From \(n, r, d\) to \(u\)
      • -
    • -
    • Stability -
        -
      • All closed-loop transfer functions must be stable
      • -
    • -
    • Robustness (stability margins) -
        -
      • Module margin (see Section 4.4)
      • -
    • -
    + +
    Table 4: Notations for the general configuration
    + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    Table 5: Typical Specification and associated closed-loop transfer function
    SpecificationClosed-Loop Transfer Function
    Reference TrackingFrom \(r\) to \(\epsilon\)
    Disturbance RejectionFrom \(d\) to \(y\)
    Measurement Noise FilteringFrom \(n\) to \(y\)
    Small Command AmplitudeFrom \(n,r,d\) to \(u\)
    StabilityAll closed-loop transfer function
    Robustness (stability margins)Module margin (see Section 4.4)
    -
    +

    gang_of_four_feedback.png

    Figure 17: Simple Feedback Architecture

    @@ -1195,20 +1233,20 @@ If we consider the feedback system shown in Figure 17,
    -
    -

    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 \(K(s)\).

    -
    +

    -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 @@ -1245,9 +1283,9 @@ 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} @@ -1259,7 +1297,7 @@ 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. @@ -1283,11 +1321,11 @@ Similarly, to reduce the effect of measurement noise \(n\) on the output \(y\),

    -
    -

    4.3 Sensitivity Function

    +
    +

    4.3 Sensitivity Function

    - +

    @@ -1346,7 +1384,7 @@ xlabel('Time [s]'); ylabel( +

    h-infinity-spec-S.png

    Figure 18: Typical wanted shape of the Sensitivity transfer function

    @@ -1354,30 +1392,152 @@ xlabel('Time [s]'); ylabel( -

    4.4 Robustness: Module Margin

    +
    +

    4.4 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: +

    +
    +
    w0 = 2*pi*100;
    +xi = 0.1;
    +k = 1e7;
    +
    +Gt = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1);
    +
    +
    + +

    +Let’s say we have designed a controller that gives the loop gain shown in Figure 19 where the following characteristics can be determined: +

    +
      +
    • bandwidth of \(\approx 10\, \text{Hz}\)
    • +
    • infinite gain margin (the phase of the loop-gain never reaches -180 degrees
    • +
    • more than 90 degrees of phase margin
    • +
    + +

    +This might indicate very good robustness properties of the closed-loop system. +

    + +
    +
    Kt = 1.2e6*(s + w0)/s;
    +
    +
    + + +
    +

    phase_gain_margin_model_plant.png +

    +

    Figure 19: Bode plot of the obtained Loop Gain \(L(s)\)

    +
    + + +

    +Now let’s suppose the “real” plant Gr as a slightly lower damping factor: +

    +
    +
    xi = 0.03;
    +Gr = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1);
    +
    +
    + +

    +The obtained loop gain is in Figure 20. +It is shown that 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 unstability. +

    + +

    +It is confirmed by checking the stability of the closed loop system: +

    +
    +
    isstable(feedback(Gp,K))
    +
    +
    + +
    +0
    +
    + + + +
    +

    phase_gain_margin_real_plant.png +

    +

    Figure 20: Bode plots of \(L(s)\) (loop gain corresponding the nominal plant) and \(L_r(s)\) (loop gain corresponding to the real plant)

    +
    + +

    +Therefore, even a small change of the plant parameter makes the system unstable even though both the gain margin and the phase margin for the nominal plant are excellent. +

    + +

    +This is due to the fact that the gain and phase margin are indicators to the robustness of the system to a pure change or gain or a pure change of phase but not a combination of both. +

    + +
    + +
    +

    +To learn more about module (or disk) margin, you can check out this video. +

    + +
    + +
      +
    • [ ] Example why the module margin might be a good alternative
    • +
    • [ ] Stability => L does not does at the left of point -1
    • +
    • [ ] => …
    • +
    + + +
    +

    +The module margin \(\Delta M\) is defined as the minimum distance between the point \(-1\) and \(L(j\omega) = G(j\omega) K(j\omega)\) in the complex plane. +

    + +
    + +

    +The module margin can be visually determined on the Nyquist plot (\(L(j\omega)\) from \(\omega = -\infty\) to \(\infty\) in the complex plane).

      -
    • [ ] Definition of Module margin
    • -
    • [ ] Why it represents robustness
    • -
    • [ ] Example
    • +
    • [ ] Example of Nyquist plot and module margin
    • +
    • [ ] Express the module margin as a function of \(\|S\|_\infty\)

    \[ M_S < 2 \Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o \]

    + +
    +

    +Large peak value of \(S\) indicate robustness problems. +

    + +

    +The wanted robustness of the closed-loop system can there be specified by setting a maximum value on \(\|S\|_\infty\). +

    + +
    -
    -

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

    +
    +

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

    - +

      @@ -1385,18 +1545,18 @@ xlabel('Time [s]'); ylabel(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 21.

      -
      +

      loop_shaping_S_without_W.png

      -

      Figure 19: Generalized Plant

      +

      Figure 21: 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 21, if will minimize the \(\mathcal{H}_\infty\) norm of transfer function from \(r\) to \(\epsilon\) (the sensitivity transfer function).

      @@ -1406,8 +1566,8 @@ 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 22. +Applying the \(\mathcal{H}_\infty\) synthesis to the weighted generalized plant \(\tilde{P}(s)\) (Figure 22) will generate a controller \(K(s)\) that minimizes the \(\mathcal{H}_\infty\) norm between \(r\) and \(\tilde{\epsilon}\):

      \begin{align} & \left\| \frac{\tilde{\epsilon}}{r} \right\|_\infty < \gamma (=1)\nonumber \\ @@ -1416,7 +1576,7 @@ Applying the \(\mathcal{H}_\infty\) synthesis to the weighted generalized \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. @@ -1426,14 +1586,14 @@ Its inverse magnitude will define the frequency dependent upper bound of the sen

      -
      +

      loop_shaping_S_with_W.png

      -

      Figure 20: Weighted Generalized Plant

      +

      Figure 22: 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 22.

      @@ -1451,11 +1611,11 @@ Pw = blkdiag(Ws, 1)*P;

      -
      -

      4.6 Design of Weighting Functions

      +
      +

      4.6 Design of Weighting Functions

      - +

      @@ -1484,9 +1644,9 @@ 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 23.

    @@ -1495,15 +1655,15 @@ 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

    +

    Figure 23: Obtained Magnitude of the Weighting Function

    -
    +

    Quite often, higher orders weights are required.

    @@ -1535,7 +1695,7 @@ 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
    @@ -1570,25 +1730,25 @@ W3 = generateWeight('G0', 1e2, 22.
    +The obtained shapes are shown in Figure 24.
     

    -
    +

    high_order_weight.png

    -

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

    +

    Figure 24: Higher order weights using Equation \eqref{eq:weight_formula_advanced}

    -
    -

    4.7 Sensitivity Function Shaping - Example

    +
    +

    4.7 Sensitivity Function Shaping - Example

    - +

      @@ -1621,7 +1781,7 @@ And the \(\mathcal{H}_\infty\) synthesis is performed.
    -
    +
     K = hinfsyn(Pw, 1, 1, 'Display', 'on');
     
       Test bounds:  0.5 <=  gamma  <=  0.51
    @@ -1644,28 +1804,28 @@ 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 25.

    -
    +

    results_sensitivity_hinf.png

    -

    Figure 23: Weighting function and obtained closed-loop sensitivity

    +

    Figure 25: Weighting function and obtained closed-loop sensitivity

    -
    -

    4.8 Complementary Sensitivity Function

    +
    +

    4.8 Complementary Sensitivity Function

    -
    -

    4.9 Summary

    +
    +

    4.9 Summary

    - - +
    Table 5: Table caption
    +@@ -1691,7 +1851,7 @@ This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure - + @@ -1715,53 +1875,53 @@ This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure -
    +

    h-infinity-4-blocs-constrains.png

    -

    Figure 24: Shaping the Gang of Four: Limitations

    +

    Figure 26: Shaping the Gang of Four: Limitations

    -
    -

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

    +
    +

    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

    +
    +

    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 \(KS\)

    +

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

    -
    P = [W1 -G*W1
    +
    P = [W1 -G*W1
          0   W2
          1  -G];
     
    @@ -1774,14 +1934,14 @@ This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure
    Shaping of S and T -
    +

    general_conf_shaping_S_T.png

    -

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

    +

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

    -26
    P = [W1 -G*W1
    +
    P = [W1 -G*W1
          0   G*W2
          1   -G];
     
    @@ -1794,14 +1954,14 @@ This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure
    Shaping of S, T and KS -
    +

    general_conf_shaping_S_T_KS.png

    -

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

    +

    Figure 29: Generalized Plant to shape \(S\), \(T\) and \(KS\)

    -27
    P = [W1 -G*W1
    +
    P = [W1 -G*W1
          0   W2
          0   G*W3
          1   -G];
    @@ -1816,14 +1976,14 @@ This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure 
     
    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 \(GS\)

    +

    Figure 30: Generalized Plant to shape \(S\), \(T\), \(KS\) and \(GS\)

    -28
    P = [ W1  -W1*G*W3 -G*W1
    +
    P = [ W1  -W1*G*W3 -G*W1
           0    0        W2
           1   -G*W3    -G];
     
    @@ -1840,17 +2000,17 @@ This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure
    -
    -

    6 Conclusion

    +
    -
    -

    7 Resources

    +
    +

    7 Resources

    @@ -1864,7 +2024,7 @@ This is indeed what we can see by comparing \(|S|\) and \(|W_S|\) in Figure

    Author: Dehaeze Thomas

    -

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

    +

    Created: 2020-12-01 mar. 00:23

    diff --git a/index.org b/index.org index 7e385c8..e1fceda 100644 --- a/index.org +++ b/index.org @@ -39,7 +39,18 @@ #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") :END: -* TODO Introduction :ignore: +* Introduction :ignore: + +This document is structured as follows: +- As $\mathcal{H}_\infty$ Control is a /model based/ control technique, a short introduction to model based control is given in Section [[sec:model_based_control]] +- Classical /open/ loop shaping method is presented in Section [[sec:open_loop_shaping]]. + It is also shown that $\mathcal{H}_\infty$ synthesis can be used for /open/ loop shaping. +- $\mathcal{H}_\infty$ + Important concepts such as the $\mathcal{H}_\infty$ norm and the generalized plant are introduced. +- A +- Finally, an complete example of the + is performed in Section [[sec:h_infinity_mixed_sensitivity]]. + * Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -53,7 +64,7 @@ addpath('matlab') #+end_src -* Introduction to the Control Methodology - Model Based Control +* Introduction to Model Based Control <> ** Model Based Control - Methodology @@ -101,7 +112,12 @@ It consists of three steps: In this document, we will mainly focus on steps 2 and 3. -** Some Background: From Classical Control to Robust Control +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]] + +** From Classical Control to Robust Control <> #+name: tab:comparison_control_methods @@ -117,9 +133,8 @@ In this document, we will mainly focus on steps 2 and 3. | | Phase and Gain margins | | Weighting Functions | | | | | Disk margin | |-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------| -| *Control Architectures* | Proportional, Integral, Derivative | Full State Feedback | General Control Configuration | -| | Leads, Lags | LQR, LQG | | -| | | Kalman Filters | | +| *Control Architectures* | Proportional, Integral, Derivative | Full State Feedback, LQR | General Control Configuration | +| | Leads, Lags | Kalman Filters, LQG | Generalized Plant | |-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------| | *Advantages* | Study Stability | Automatic Synthesis | Automatic Synthesis | | | Simple | MIMO | MIMO | @@ -508,7 +523,7 @@ Let's take our example system and try to apply the Open-Loop shaping strategy to #+begin_exercice Using =SISOTOOL=, design a controller that fulfill the specifications. -#+begin_src matlab :eval no +#+begin_src matlab :eval no :tangle no sisotool(G) #+end_src #+end_exercice @@ -583,7 +598,7 @@ The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ Even though we will not go into details, we will provide one example. Using Matlab, the $\mathcal{H}_\infty$ Loop Shaping Synthesis can be performed using the =loopsyn= command: -#+begin_src matlab :eval no +#+begin_src matlab :eval no :tangle no K = loopsyn(G, Gd); #+end_src where: @@ -852,7 +867,7 @@ It can indeed represent feedback as well as feedforward control architectures. | $u$ | Control signals | #+end_important -** The General Synthesis Problem Formulation +** 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: @@ -897,7 +912,7 @@ Note that the closed-loop transfer function from $w$ to $z$ is: \end{equation} Using Matlab, the $\mathcal{H}_\infty$ Synthesis applied on a Generalized plant can be applied using the =hinfsyn= command ([[https://www.mathworks.com/help/robust/ref/hinfsyn.html][documentation]]): -#+begin_src matlab :eval no +#+begin_src matlab :eval no :tangoe no K = hinfsyn(P, nmeas, ncont); #+end_src where: @@ -997,27 +1012,27 @@ Using Matlab, the generalized plant can be defined as follows: ** Introduction -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. +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 manual design of controllers. -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. +$L(s)$ 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 $L(s)$. 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*. +The closed loop system behavior is indeed 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 $y$ -- *Measurement noise filtering* - - From $n$ to $y$ -- *Small command amplitude* - - From $n, r, d$ to $u$ -- *Stability* - - All closed-loop transfer functions must be stable -- *Robustness* (stability margins) - - Module margin (see Section [[sec:module_margin]]) +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. +This is summarized in Table [[tab:spec_closed_loop_tf]]. + +#+name: tab:spec_closed_loop_tf +#+caption: Typical Specification and associated closed-loop transfer function +| Specification | Closed-Loop Transfer Function | +|--------------------------------+-----------------------------------------------| +| Reference Tracking | From $r$ to $\epsilon$ | +| Disturbance Rejection | From $d$ to $y$ | +| Measurement Noise Filtering | From $n$ to $y$ | +| Small Command Amplitude | From $n,r,d$ to $u$ | +| Stability | All closed-loop transfer function | +| Robustness (stability margins) | Module margin (see Section [[sec:module_margin]]) | #+begin_src latex :file gang_of_four_feedback.pdf \begin{tikzpicture} @@ -1202,12 +1217,152 @@ Similarly, to reduce the effect of measurement noise $n$ on the output $y$, we w ** Robustness: Module Margin <> -- [ ] Definition of Module margin -- [ ] Why it represents robustness -- [ ] Example +Let's start by an example demonstrating why the phase and gain margins might not be good indicators of robustness. + +#+begin_exampl +Let's consider the following plant: +#+begin_src matlab + w0 = 2*pi*100; + xi = 0.1; + k = 1e7; + + Gt = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1); +#+end_src + +Let's say we have designed a controller that gives the loop gain shown in Figure [[fig:phase_gain_margin_model_plant]] where the following characteristics can be determined: +- bandwidth of $\approx 10\, \text{Hz}$ +- infinite gain margin (the phase of the loop-gain never reaches -180 degrees +- more than 90 degrees of phase margin + +This might indicate very good robustness properties of the closed-loop system. + +#+begin_src matlab + Kt = 1.2e6*(s + w0)/s; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile([2,1]); + hold on; + plot(freqs, abs(squeeze(freqresp(Gt*Kt, freqs, 'Hz'))), 'DisplayName', '$L(s)$'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + hold off; + legend('location', 'northeast'); + + ax2 = nexttile; + hold on; + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gt*Kt, freqs, 'Hz'))))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + yticks(-360:90:360); ylim([-200, 0]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/phase_gain_margin_model_plant.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:phase_gain_margin_model_plant +#+caption: Bode plot of the obtained Loop Gain $L(s)$ +#+RESULTS: +[[file:figs/phase_gain_margin_model_plant.png]] + + +Now let's suppose the "real" plant =Gr= as a slightly lower damping factor: +#+begin_src matlab + xi = 0.03; + Gr = 1/k*(s/w0/4 + 1)/(s^2/w0^2 + 2*xi*s/w0 + 1); +#+end_src + +The obtained loop gain is in Figure [[fig:phase_gain_margin_real_plant]]. +It is shown that 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 unstability. + +It is confirmed by checking the stability of the closed loop system: +#+begin_src matlab :results value replace + isstable(feedback(Gp,K)) +#+end_src + +#+RESULTS: +: 0 + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + + ax1 = nexttile([2,1]); + hold on; + plot(freqs, abs(squeeze(freqresp(Gt*Kt, freqs, 'Hz'))), 'DisplayName', '$L(s)$'); + plot(freqs, abs(squeeze(freqresp(Gr*Kt, freqs, 'Hz'))), 'DisplayName', '$L_r(s)$'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Magnitude'); set(gca, 'XTickLabel',[]); + hold off; + legend('location', 'northeast'); + + ax2 = nexttile; + hold on; + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gt*Kt, freqs, 'Hz'))))); + plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Gr*Kt, freqs, 'Hz'))))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + yticks(-360:90:360); ylim([-200, 0]); + xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); + hold off; + + linkaxes([ax1,ax2],'x'); + xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/phase_gain_margin_real_plant.pdf', 'width', 'wide', 'height', 'normal'); +#+end_src + +#+name: fig:phase_gain_margin_real_plant +#+caption: Bode plots of $L(s)$ (loop gain corresponding the nominal plant) and $L_r(s)$ (loop gain corresponding to the real plant) +#+RESULTS: +[[file:figs/phase_gain_margin_real_plant.png]] + +Therefore, even a small change of the plant parameter makes the system unstable even though both the gain margin and the phase margin for the nominal plant are excellent. + +This is due to the fact that the gain and phase margin are indicators to the robustness of the system to a *pure* change or gain or a *pure* change of phase but not a combination of both. +#+end_exampl + +#+begin_seealso + To learn more about module (or disk) margin, you can check out [[https://www.youtube.com/watch?v=XazdN6eZF80][this]] video. +#+end_seealso + +- [ ] Example why the module margin might be a good alternative +- [ ] Stability => L does not does at the left of point -1 +- [ ] => ... + + +#+begin_definition + The *module margin* $\Delta M$ is defined as the *minimum distance* between the point $-1$ and $L(j\omega) = G(j\omega) K(j\omega)$ in the complex plane. +#+end_definition + +The module margin can be visually determined on the Nyquist plot ($L(j\omega)$ from $\omega = -\infty$ to $\infty$ in the complex plane). + +- [ ] Example of Nyquist plot and module margin +- [ ] Express the module margin as a function of $\|S\|_\infty$ \[ M_S < 2 \Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o \] +#+begin_important + Large peak value of $S$ indicate robustness problems. + + The wanted robustness of the closed-loop system can there be specified by setting a maximum value on $\|S\|_\infty$. +#+end_important + ** How to *Shape* transfer function? Using of Weighting Functions! <>
    Table 6: Table caption
    Disturbance Rejection \(L\) large\(SG\) small\(GS\) small