Add some examples and comments about loop shaping
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figs/open_loop_shaping_hinf_K.pdf
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@ -38,7 +38,7 @@
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
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* Introduction :ignore:
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* TODO Introduction :ignore:
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* Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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@ -95,53 +95,36 @@ In this document, we will mainly focus on steps 2 and 3.
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** Some Background: From Classical Control to Robust Control
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Classical Control (1930)
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- Tools:
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- TF (input-output)
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- Nyquist, Bode, Black, \ldots
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- P-PI-PID, Phase lead-lag, \ldots
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- Advantages:
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- Stability
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- Performances
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- Robustness
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- Disadvantages:
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- Manual Method
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- Only SISO
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Modern Control (1960)
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- Tools:
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- State Space
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- Optimal Command
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- LQR, LQG
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- Advantages:
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- Automatic Synthesis
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- MIMO
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- Optimisation problem
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- Disadvantages:
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- Robustness
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- Rejection of Perturbations
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Robust Control (1980)
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- Tools:
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- Disk Margin
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- Systems and Signals norms ($\mathcal{H}_\infty$ and $\mathcal{H}_2$ norms)
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- Closed Loop Transfer Functions
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- Loop Shaping
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- Advantages:
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- Stability
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- Performances
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- Robustness
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- Automatic Synthesis
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- MIMO
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- Optimization Problem
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- Disadvantages:
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- Requires the knowledge of specific tools
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- Need a reasonably good model of the system
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#+name: tab:comparison_control_methods
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#+caption: Table summurazing the main differences between classical, modern and robust control
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| | *Classical Control* | *Modern Control* | *Robust Control* |
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| <l> | <c> | <c> | <c> |
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|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
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| *Date* | 1930- | 1960- | 1980- |
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|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
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| *Tools* | Transfer Functions | State Space formulation | Systems and Signals Norms ($\mathcal{H}_\infty$, $\mathcal{H}_2$ Norms) |
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| | Nyquist Plots | Riccati Equations | Closed Loop Transfer Functions |
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| | Bode Plots | | Open/Closed Loop Shaping |
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| | Phase and Gain margins | | Weighting Functions |
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| | | | Disk margin |
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|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
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| *Control Architectures* | Proportional, Integral, Derivative | Full State Feedback | General Control Configuration |
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| | Leads, Lags | LQR, LQG | |
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| | | Kalman Filters | |
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|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
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| *Advantages* | Study Stability | Automatic Synthesis | Automatic Synthesis |
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| | Simple | MIMO | MIMO |
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| | Natural | Optimization Problem | Optimization Problem |
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| | | | Guaranteed Robustness |
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| | | | Easy specification of performances |
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|-------------------------+------------------------------------+--------------------------------------+-------------------------------------------------------------------------|
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| *Disadvantages* | Manual Method | No Guaranteed Robustness | Required knowledge of specific tools |
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| | Only SISO | Difficult Rejection of Perturbations | Need a reasonably good model of the system |
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** Example System
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Let's consider the test-system shown in Figure [[fig:mech_sys_1dof_inertial_contr]].
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Let's consider the model shown in Figure [[fig:mech_sys_1dof_inertial_contr]].
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It could represent a suspension system with a payload to position or isolate using an force actuator and an inertial sensor.
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The notations used are listed in Table [[tab:example_notations]].
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#+begin_src latex :file mech_sys_1dof_inertial_contr.pdf
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@ -187,9 +170,9 @@ The notations used are listed in Table [[tab:example_notations]].
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#+caption: Example system variables
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| *Notation* | *Description* | *Value* | *Unit* |
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|--------------------+----------------------------------------------------------------+----------------+-----------|
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| $m$ | Payload's mass to position / isolate | | [kg] |
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| $k$ | Stiffness of the suspension system | | [N/m] |
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| $c$ | Damping coefficient of the suspension system | | [N/(m/s)] |
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| $m$ | Payload's mass to position / isolate | $10$ | [kg] |
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| $k$ | Stiffness of the suspension system | $10^6$ | [N/m] |
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| $c$ | Damping coefficient of the suspension system | $400$ | [N/(m/s)] |
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| $y$ | Payload absolute displacement (measured by an inertial sensor) | | [m] |
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| $d$ | Ground displacement, it acts as a disturbance | | [m] |
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| $u$ | Actuator force | | [N] |
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@ -204,13 +187,22 @@ Derive the following open-loop transfer functions:
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G_d(s) &= \frac{y}{d}
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\end{align}
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*Hint:* You can follow this generic procedure:
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#+HTML: <details><summary>Hint</summary>
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You can follow this generic procedure:
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1. List all applied forces ot the mass: Actuator force, Stiffness force (Hooke's law), ...
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2. Apply the Newton's Second Law on the payload
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\[ m \ddot{y} = \Sigma F \]
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3. Transform the differential equations into the Laplace domain:
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\[ \frac{d\ \cdot}{dt} \Leftrightarrow \cdot \times s \]
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4. Write $y(s)$ as a function of $u(s)$ and $w(s)$
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#+HTML: </details>
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#+HTML: <details><summary>Results</summary>
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\begin{align}
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G(s) &= \frac{1}{m s^2 + cs + k} \\
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G_d(s) &= \frac{cs + k}{m s^2 + cs + k}
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\end{align}
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#+HTML: </details>
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#+end_exercice
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Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure [[fig:mech_sys_1dof_inertial_contr]] into a classical feedback form as shown in Figure [[fig:open_loop_shaping]].
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@ -239,75 +231,22 @@ Having obtained $G(s)$ and $G_d(s)$, we can transform the system shown in Figure
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#+RESULTS:
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[[file:figs/classical_feedback_test_system.png]]
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#+begin_src matlab
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Let's define the system parameters on Matlab.
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#+begin_src matlab +n
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k = 1e6; % Stiffness [N/m]
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c = 4e2; % Damping [N/(m/s)]
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m = 16; % Mass [kg]
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m = 10; % Mass [kg]
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#+end_src
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#+begin_src matlab
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G = 1/(m*s^2 + c*s + k);
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Gd = (c*s + k)/(m*s^2 + c*s + k);
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And now the system dynamics $G(s)$ and $G_d(s)$ (their bode plots are shown in Figures [[fig:bode_plot_example_afm]] and [[fig:bode_plot_example_Gd]]).
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#+begin_src matlab +n -r
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G = 1/(m*s^2 + c*s + k); % Plant
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Gd = (c*s + k)/(m*s^2 + c*s + k); % Disturbance
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#+end_src
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* Classical Open Loop Shaping
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** Introduction ot Open Loop Shaping
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Usually, the controller $K(s)$ is designed such that the loop gain $L(s)$ has desirable shape.
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This technique is called *Open Loop Shaping*.
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*************** TODO Explain why the Loop gain si an important "value"
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For instance example all the specifications can usually be explained in terms of the open loop gain.
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*************** END
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#+begin_src latex :file open_loop_shaping.pdf
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\begin{tikzpicture}
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\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
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\node[block, right=0.8 of addsub] (K) {$K(s)$};
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\node[below] at (K.south) {Controller};
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\node[block, right=0.8 of K] (G) {$G(s)$};
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\node[below] at (G.south) {Plant};
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\draw[<-] (addsub.west) -- ++(-0.8, 0) node[above right]{$r$};
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\draw[->] (addsub) -- (K.west) node[above left]{$\epsilon$};
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\draw[->] (K.east) -- (G.west) node[above left]{$u$};
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\draw[->] (G.east) -- ++(0.8, 0) node[above left]{$y$};
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\draw[] ($(G.east) + (0.5, 0)$) -- ++(0, -1.4);
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\draw[->] ($(G.east) + (0.5, -1.4)$) -| (addsub.south);
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\draw [decoration={brace, raise=5pt}, decorate] (K.north west) -- node[above=6pt]{$L(s)$} (G.north east);
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\end{tikzpicture}
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#+end_src
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#+name: fig:open_loop_shaping
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#+caption: Classical Feedback Architecture
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#+RESULTS:
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[[file:figs/open_loop_shaping.png]]
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This is usually done manually has the loop gain $L(s)$ depends linearly of $K(s)$:
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\begin{equation}
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L(s) = G(s) K(s)
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\end{equation}
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- where $L(s)$ is called the *Loop Gain Transfer Function*
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$K(s)$ then consists of a combination of leads, lags, notches, etc. such that its product with $G(s)$ has wanted shape.
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** Example of Open Loop Shaping
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#+begin_src matlab
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k = 1e-6;
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m = 10;
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c = 10;
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G =
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(1, 4, 1000);
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freqs = logspace(0, 3, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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@ -340,10 +279,87 @@ $K(s)$ then consists of a combination of leads, lags, notches, etc. such that it
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#+RESULTS:
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[[file:figs/bode_plot_example_afm.png]]
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Specifications:
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- *Performance*: Bandwidth of approximately 50Hz
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- *Noise Attenuation*: Roll-off of -40dB/decade past 250Hz
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- *Robustness*: Gain margin > 5dB and Phase margin > 40 deg
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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hold off;
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xlim([freqs(1), freqs(end)]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/bode_plot_example_Gd.pdf', 'width', 'wide', 'height', 'small');
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#+end_src
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#+name: fig:bode_plot_example_Gd
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#+caption: Magnitude of the disturbance transfer function $G_d(s)$
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#+RESULTS:
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[[file:figs/bode_plot_example_Gd.png]]
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|
||||
* Classical Open Loop Shaping
|
||||
** Introduction to Open Loop Shaping
|
||||
|
||||
#+begin_definition
|
||||
The *Loop Gain* $L(s)$ usually refers to as the product of the controller and the plant (Figure [[fig:open_loop_shaping]]):
|
||||
\begin{equation}
|
||||
L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
|
||||
\end{equation}
|
||||
|
||||
#+begin_src latex :file open_loop_shaping.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[addb={+}{}{}{}{-}] (addsub) at (0, 0){};
|
||||
|
||||
\node[block, right=0.8 of addsub] (K) {$K(s)$};
|
||||
\node[below] at (K.south) {Controller};
|
||||
\node[block, right=0.8 of K] (G) {$G(s)$};
|
||||
\node[below] at (G.south) {Plant};
|
||||
|
||||
\draw[<-] (addsub.west) -- ++(-0.8, 0) node[above right]{$r$};
|
||||
|
||||
\draw[->] (addsub) -- (K.west) node[above left]{$\epsilon$};
|
||||
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
|
||||
\draw[->] (G.east) -- ++(0.8, 0) node[above left]{$y$};
|
||||
\draw[] ($(G.east) + (0.5, 0)$) -- ++(0, -1.4);
|
||||
\draw[->] ($(G.east) + (0.5, -1.4)$) -| (addsub.south);
|
||||
|
||||
\draw [decoration={brace, raise=5pt}, decorate] (K.north west) -- node[above=6pt]{$L(s)$} (G.north east);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:open_loop_shaping
|
||||
#+caption: Classical Feedback Architecture
|
||||
[[file:figs/open_loop_shaping.png]]
|
||||
#+end_definition
|
||||
|
||||
#+begin_definition
|
||||
*Open Loop Shaping* refers to a control design technique where the controller $K(s)$ is designed such that the *Open Loop Gain* $L(s)$ has desirable shape.
|
||||
#+end_definition
|
||||
|
||||
This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$:
|
||||
- *Performance*: $L$ large
|
||||
- *Good disturbance rejection*: $L$ large
|
||||
- *Limitation of measurement noise on plant output*: $L$ small
|
||||
- *Small magnitude of input signal*: $K$ and $L$ small
|
||||
- *Nominal stability*: $L$ small (RHP zeros and time delays)
|
||||
- *Robust stability*: $L$ small (neglected dynamics)
|
||||
|
||||
The Open Loop shape is usually done manually has the loop gain $L(s)$ depends linearly on $K(s)$ eqref:eq:loop_gain.
|
||||
|
||||
$K(s)$ then consists of a combination of leads, lags, notches, etc. such that $L(s)$ has the wanted shape.
|
||||
|
||||
** Example of Open Loop Shaping
|
||||
|
||||
#+begin_exampl
|
||||
Let's take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:
|
||||
- *Performance*: Bandwidth of approximately 10Hz
|
||||
- *Noise Attenuation*: Roll-off of -40dB/decade past 30Hz
|
||||
- *Robustness*: Gain margin > 3dB and Phase margin > 30 deg
|
||||
#+end_exampl
|
||||
|
||||
#+begin_exercice
|
||||
Using =SISOTOOL=, design a controller that fulfill the specifications.
|
||||
@ -357,13 +373,14 @@ 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 [[fig:loop_gain_manual_afm]].
|
||||
#+begin_src matlab
|
||||
K = 6e4 * ... % Gain
|
||||
K = 14e8 * ... % Gain
|
||||
1/(s^2) * ... % Double Integrator
|
||||
(1 + s/111)/(1 + s/888); % Lead
|
||||
1/(1 + s/2/pi/40) * ... % Low Pass Filter
|
||||
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 4, 1000);
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
@ -375,7 +392,7 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
||||
hold off;
|
||||
ylim([1e-5, 1e1])
|
||||
ylim([1e-4, 1e1])
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
@ -390,7 +407,7 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/loop_gain_manual_afm.pdf', 'width', 'wide', 'height', 'tall');
|
||||
exportFig('figs/loop_gain_manual_afm.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:loop_gain_manual_afm
|
||||
@ -399,22 +416,23 @@ The obtained controller is shown below, and the bode plot of the Loop Gain is sh
|
||||
[[file:figs/loop_gain_manual_afm.png]]
|
||||
|
||||
And we can verify that we have the wanted stability margins:
|
||||
#+begin_src matlab :results output replace
|
||||
#+begin_src matlab
|
||||
[Gm, Pm, ~, Wc] = margin(G*K)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
||||
data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin [dB]', 'Phase Margin [deg]', 'Crossover [Hz]'}, {'Value'}, ' %.1f ');
|
||||
data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Manual Method'}, ' %.1f ');
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
| | Value |
|
||||
|--------------------+-------|
|
||||
| Gain Margin [dB] | 7.2 |
|
||||
| Phase Margin [deg] | 48.1 |
|
||||
| Crossover [Hz] | 50.7 |
|
||||
| | Manual Method |
|
||||
|-----------------------------+---------------|
|
||||
| Gain Margin $> 3$ [dB] | 3.1 |
|
||||
| Phase Margin $> 30$ [deg] | 35.4 |
|
||||
| Crossover $\approx 10$ [Hz] | 10.1 |
|
||||
|
||||
** $\mathcal{H}_\infty$ Loop Shaping Synthesis
|
||||
|
||||
The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ Synthesis.
|
||||
|
||||
Even though we will not go into details, we will provide one example.
|
||||
@ -434,45 +452,88 @@ where:
|
||||
|
||||
** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
|
||||
|
||||
Let's re-use the previous plant.
|
||||
Let's reuse the previous plant.
|
||||
|
||||
Translate the specification into the wanted shape of the open loop gain.
|
||||
- *Performance*: Bandwidth of approximately 10Hz: $|L_w(j2 \pi 10)| = 1$
|
||||
- *Noise Attenuation*: Roll-off of -40dB/decade past 30Hz
|
||||
- *Robustness*: Gain margin > 3dB and Phase margin > 30 deg
|
||||
|
||||
#+begin_src matlab
|
||||
G = tf(16,[1 0.16 16]);
|
||||
|
||||
Gd = 3.7e4*1/s*(1 + s/2/pi/20)/(1 + s/2/pi/220)*1/(s + s/2/pi/500);
|
||||
Lw = 2.3e3 * ...
|
||||
1/(s^2) * ... % Double Integrator
|
||||
(1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead
|
||||
#+end_src
|
||||
|
||||
The $\mathcal{H}_\infty$ optimal open loop shaping is performed using the =loopsyn= command:
|
||||
#+begin_src matlab
|
||||
[K, ~, GAM] = loopsyn(G, Lw);
|
||||
#+end_src
|
||||
|
||||
The Bode plot of the obtained controller is shown in Figure [[fig:open_loop_shaping_hinf_K]].
|
||||
|
||||
#+begin_important
|
||||
It is always important to analyze the controller after the synthesis is performed.
|
||||
|
||||
In the end, a synthesize controller is just a combination of low pass filters, high pass filters, notches, leads, etc.
|
||||
#+end_important
|
||||
|
||||
Let's briefly analyze this controller:
|
||||
- two integrators are used at low frequency to have the wanted low frequency high gain
|
||||
- a lead is added centered with the crossover frequency to increase the phase margin
|
||||
- a notch is added at the resonance of the plant to increase the gain margin (this is very typical of $\mathcal{H}_\infty$ controllers, and can be an issue, more info on that latter)
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
bodeFig({Gd}, struct('phase', true))
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
plot(freqs, abs(squeeze(freqresp(K, freqs, 'Hz'))));
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
||||
hold off;
|
||||
|
||||
ax2 = nexttile;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(K, freqs, 'Hz'))));
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
||||
yticks(-360:90:360); ylim([-180, 90]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
[K,CL,GAM,INFO] = loopsyn(G, Gd);
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/open_loop_shaping_hinf_K.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
bodeFig({K})
|
||||
#+end_src
|
||||
#+name: fig:open_loop_shaping_hinf_K
|
||||
#+caption: Obtained controller $K$ using the open-loop $\mathcal{H}_\infty$ shaping
|
||||
#+RESULTS:
|
||||
[[file:figs/open_loop_shaping_hinf_K.png]]
|
||||
|
||||
The obtained Loop Gain is shown in Figure [[fig:open_loop_shaping_hinf_L]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(1, 4, 1000);
|
||||
freqs = logspace(0, 3, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile([2,1]);
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))));
|
||||
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz'))), 'k--');
|
||||
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz')))*GAM, 'k-.');
|
||||
plot(freqs, abs(squeeze(freqresp(Gd, freqs, 'Hz')))/GAM, 'k-.');
|
||||
plot(freqs, abs(squeeze(freqresp(G*K, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
|
||||
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz'))), 'k--', 'DisplayName', '$L_w(s)$');
|
||||
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz')))*GAM, 'k-.', 'DisplayName', '$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$');
|
||||
plot(freqs, abs(squeeze(freqresp(Lw, freqs, 'Hz')))/GAM, 'k-.', 'HandleVisibility', 'off');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
||||
hold off;
|
||||
ylim([1e-5, 1e1])
|
||||
legend('location', 'northeast');
|
||||
ylim([1e-4, 1e2]);
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
@ -486,8 +547,36 @@ Translate the specification into the wanted shape of the open loop gain.
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/open_loop_shaping_hinf_L.pdf', 'width', 'wide', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
* The $\mathcal{H}_\infty$ Norm
|
||||
#+name: fig:open_loop_shaping_hinf_L
|
||||
#+caption: Obtained Open Loop Gain $L(s) = G(s) K(s)$ and comparison with the wanted Loop gain $L_w$
|
||||
#+RESULTS:
|
||||
[[file:figs/open_loop_shaping_hinf_L.png]]
|
||||
|
||||
Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ controller and of the manually developed controller in Table [[tab:open_loop_shaping_compare]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
[Gm_2, Pm_2, ~, Wc_2] = margin(G*K)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
||||
data2orgtable([Gm, Gm_2; Pm, Pm_2; Wc/2/pi, Wc_2/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Specifications', 'Manual Method', '$\mathcal{H}_\infty$ Method'}, ' %.1f ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:open_loop_shaping_compare
|
||||
#+caption: Comparison of the characteristics obtained with the two methods
|
||||
#+RESULTS:
|
||||
| Specifications | Manual Method | $\mathcal{H}_\infty$ Method |
|
||||
|-----------------------------+---------------+-----------------------------|
|
||||
| Gain Margin $> 3$ [dB] | 3.1 | 31.7 |
|
||||
| Phase Margin $> 30$ [deg] | 35.4 | 54.7 |
|
||||
| Crossover $\approx 10$ [Hz] | 10.1 | 9.9 |
|
||||
|
||||
* First Step in the $\mathcal{H}_\infty$ world
|
||||
** The $\mathcal{H}_\infty$ Norm
|
||||
|
||||
#+begin_definition
|
||||
The $\mathcal{H}_\infty$ norm is defined as the peak of the maximum singular value of the frequency response
|
||||
@ -502,31 +591,24 @@ Translate the specification into the wanted shape of the open loop gain.
|
||||
#+end_definition
|
||||
|
||||
#+begin_exampl
|
||||
Let's define a plant dynamics:
|
||||
#+begin_src matlab
|
||||
w0 = 2*pi; k = 1e6; xi = 0.04;
|
||||
|
||||
G = 1/k/(s^2/w0^2 + 2*xi*s/w0 + 1);
|
||||
#+end_src
|
||||
|
||||
And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function:
|
||||
#+begin_src matlab :results value replace
|
||||
hinfnorm(G)
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: 1.0013e-05
|
||||
: 7.9216e-06
|
||||
|
||||
The magnitude $|G(j\omega)|$ of the plant $G(s)$ as a function of frequency is shown in Figure [[fig:hinfinity_norm_siso_bode]].
|
||||
The maximum value of the magnitude over all frequencies does correspond to the $\mathcal{H}_\infty$ norm of $G(s)$ as Equation eqref:eq:hinf_norm_siso implies.
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-1, 1, 1000);
|
||||
freqs = logspace(0, 3, 1000);
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-');
|
||||
plot([0.5, 2], [hinfnorm(G) hinfnorm(G)], 'k--');
|
||||
text(2, hinfnorm(G), '$\quad \|G\|_\infty$')
|
||||
plot([20, 100], [hinfnorm(G) hinfnorm(G)], 'k--');
|
||||
text(100, hinfnorm(G), '$\quad \|G\|_\infty$')
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude $|G(j\omega)|$');
|
||||
@ -543,7 +625,7 @@ The maximum value of the magnitude over all frequencies does correspond to the $
|
||||
[[file:figs/hinfinity_norm_siso_bode.png]]
|
||||
#+end_exampl
|
||||
|
||||
* $\mathcal{H}_\infty$ Synthesis
|
||||
** $\mathcal{H}_\infty$ Synthesis
|
||||
|
||||
*Optimization problem*:
|
||||
$\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimization, Riccati equation) to find a controller of the same order as the system so that the $\mathcal{H}_\infty$ norms of defined transfer functions are minimized.
|
||||
@ -560,7 +642,7 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
|
||||
- Fixed-Structure $\mathcal{H}_\infty$ Synthesis
|
||||
- Signal Based $\mathcal{H}_\infty$ Synthesis
|
||||
|
||||
* The Generalized Plant
|
||||
** The Generalized Plant
|
||||
#+begin_src latex :file general_plant.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[block={2.0cm}{2.0cm}] (P) {$P$};
|
||||
@ -598,45 +680,7 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
|
||||
\begin{bmatrix} z \\ v \end{bmatrix} = P \begin{bmatrix} w \\ u \end{bmatrix} = \begin{bmatrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{bmatrix} \begin{bmatrix} w \\ u \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
* Problem Formulation
|
||||
|
||||
#+begin_important
|
||||
The $\mathcal{H}_\infty$ Synthesis objective is to find all stabilizing controllers $K$ which minimize
|
||||
\begin{equation}
|
||||
\| F_l(P, K) \|_\infty = \max_{\omega} \overline{\sigma} \big( F_l(P, K)(j\omega) \big)
|
||||
\end{equation}
|
||||
|
||||
#+end_important
|
||||
|
||||
#+begin_src latex :file general_control_names.pdf
|
||||
\begin{tikzpicture}
|
||||
|
||||
% Blocs
|
||||
\node[block={2.0cm}{2.0cm}] (P) {$P$};
|
||||
\node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$};
|
||||
|
||||
% Input and outputs coordinates
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
|
||||
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
|
||||
|
||||
% Connections and labels
|
||||
\draw[<-] (inputw) node[above left, align=right]{(weighted)\\exogenous inputs\\$w$} -- ++(-1.5, 0);
|
||||
\draw[<-] (inputu) -- ++(-0.8, 0) |- node[left, near start, align=right]{control signals\\$u$} (K.west);
|
||||
|
||||
\draw[->] (outputz) node[above right, align=left]{(weighted)\\exogenous outputs\\$z$} -- ++(1.5, 0);
|
||||
\draw[->] (outputv) -- ++(0.8, 0) |- node[right, near start, align=left]{sensed output\\$v$} (K.east);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:general_control_names
|
||||
#+caption: General Control Configuration
|
||||
#+RESULTS:
|
||||
[[file:figs/general_control_names.png]]
|
||||
|
||||
|
||||
* Classical feedback control and closed loop transfer functions
|
||||
** From a Classical Feedback Architecture to a Generalized Plant
|
||||
|
||||
#+begin_src latex :file classical_feedback.pdf
|
||||
\begin{tikzpicture}
|
||||
@ -672,7 +716,6 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
|
||||
| $d$ | Input Disturbance |
|
||||
| $\epsilon$ | Tracking Error |
|
||||
|
||||
* From a Classical Feedback Architecture to a Generalized Plant
|
||||
The procedure is:
|
||||
1. define signals of the generalized plant
|
||||
2. Remove $K$ and rearrange the inputs and outputs
|
||||
@ -716,12 +759,12 @@ The procedure is:
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(G.south-|start.west) ($(z1.north west)+(-0.4, 0)$)}, inner sep=6pt, draw, dashed, fill=black!20!white] (P) {};
|
||||
\node[below right] at (P.north west) {Generalized Plant $P(s)$};
|
||||
\node[below] at (P.north) {Generalized Plant $P(s)$};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+begin_exampl
|
||||
#+begin_exercice
|
||||
Let's find the Generalized plant of corresponding to the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]]
|
||||
|
||||
#+name: fig:classical_feedback_tracking
|
||||
@ -747,7 +790,46 @@ The procedure is:
|
||||
0 1;
|
||||
1 -G]
|
||||
#+end_src
|
||||
#+end_exampl
|
||||
#+end_exercice
|
||||
|
||||
|
||||
** The General Synthesis Problem Formulation
|
||||
|
||||
#+begin_important
|
||||
The $\mathcal{H}_\infty$ Synthesis objective is to find all stabilizing controllers $K$ which minimize
|
||||
\begin{equation}
|
||||
\| F_l(P, K) \|_\infty = \max_{\omega} \overline{\sigma} \big( F_l(P, K)(j\omega) \big)
|
||||
\end{equation}
|
||||
|
||||
#+end_important
|
||||
|
||||
#+begin_src latex :file general_control_names.pdf
|
||||
\begin{tikzpicture}
|
||||
|
||||
% Blocs
|
||||
\node[block={2.0cm}{2.0cm}] (P) {$P$};
|
||||
\node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$};
|
||||
|
||||
% Input and outputs coordinates
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
|
||||
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
|
||||
|
||||
% Connections and labels
|
||||
\draw[<-] (inputw) node[above left, align=right]{(weighted)\\exogenous inputs\\$w$} -- ++(-1.5, 0);
|
||||
\draw[<-] (inputu) -- ++(-0.8, 0) |- node[left, near start, align=right]{control signals\\$u$} (K.west);
|
||||
|
||||
\draw[->] (outputz) node[above right, align=left]{(weighted)\\exogenous outputs\\$z$} -- ++(1.5, 0);
|
||||
\draw[->] (outputv) -- ++(0.8, 0) |- node[right, near start, align=left]{sensed output\\$v$} (K.east);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:general_control_names
|
||||
#+caption: General Control Configuration
|
||||
#+RESULTS:
|
||||
[[file:figs/general_control_names.png]]
|
||||
|
||||
|
||||
* Modern Interpretation of the Control Specifications
|
||||
** Introduction
|
||||
|
||||
Loading…
Reference in New Issue
Block a user