Add many things about loop shaping
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@@ -1,4 +1,4 @@
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#+TITLE: Robust Control - $\mathcal{H}_\infty$ Synthesis
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#+TITLE: A brief and practical introduction to $\mathcal{H}_\infty$ Control
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:DRAWER:
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#+STARTUP: overview
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@@ -31,6 +31,7 @@
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :tangle no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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@@ -48,8 +49,15 @@
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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addpath('matlab')
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#+end_src
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* Introduction to the Control Methodology - Model Based Control
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** Control Methodology
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<<sec:model_based_control>>
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** Model Based Control - Methodology
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<<sec:model_based_control_methodology>>
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The typical methodology when applying Model Based Control to a plant is schematically shown in Figure [[fig:control-procedure]].
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It consists of three steps:
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@@ -94,6 +102,7 @@ It consists of three steps:
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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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<<sec:comp_classical_modern_robust_control>>
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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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@@ -122,6 +131,7 @@ In this document, we will mainly focus on steps 2 and 3.
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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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<<sec:example_system>>
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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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@@ -204,6 +214,20 @@ You can follow this generic procedure:
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\end{align}
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#+HTML: </details>
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#+end_exercice
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Hi Musa,
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Thank you very much for sharing this awesome package.
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For a long time, I am dreaming of being abble to export source blocks to HTML tha are surounded by <details> blocks.
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For now, I am manually adding #+HTML: <details><summary>Code</summary> and #+HTML: </details> around the source blocks I want to hide...
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This is a very simple solution, but not so elegent nor practical.
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Do you have any idea if it would be easy to extend to org-mode export of source blocks to add such functionallity?
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Similarly, I would love to be able to export a <span> block with the name of the file corresponding to the source block.
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For instance, if a particular source block is tangled to script.sh, it would be so nice to display the filename when exporting!
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Thanks in advance
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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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@@ -302,10 +326,17 @@ And now the system dynamics $G(s)$ and $G_d(s)$ (their bode plots are shown in F
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[[file:figs/bode_plot_example_Gd.png]]
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* Classical Open Loop Shaping
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** Introduction to Open Loop Shaping
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<<sec:open_loop_shaping>>
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** Introduction to Loop Shaping
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<<sec:open_loop_shaping_introduction>>
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#+begin_definition
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The *Loop Gain* $L(s)$ usually refers to as the product of the controller and the plant (Figure [[fig:open_loop_shaping]]):
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*Loop Shaping* refers to a design procedure that involves explicitly shaping the magnitude of the *Loop Transfer Function* $L(s)$.
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#+end_definition
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#+begin_definition
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The *Loop Gain* $L(s)$ usually refers to as the product of the controller and the plant ("Gain around the loop", see Figure [[fig:open_loop_shaping]]):
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\begin{equation}
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L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
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\end{equation}
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@@ -336,11 +367,7 @@ The *Loop Gain* $L(s)$ usually refers to as the product of the controller and th
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[[file:figs/open_loop_shaping.png]]
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#+end_definition
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#+begin_definition
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*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.
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#+end_definition
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This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$:
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This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$ such as:
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- *Performance*: $L$ large
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- *Good disturbance rejection*: $L$ large
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- *Limitation of measurement noise on plant output*: $L$ small
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@@ -350,9 +377,52 @@ This synthesis method is widely used as many characteristics of the closed-loop
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The Open Loop shape is usually done manually has the loop gain $L(s)$ depends linearly on $K(s)$ eqref:eq:loop_gain.
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$K(s)$ then consists of a combination of leads, lags, notches, etc. such that $L(s)$ has the wanted shape.
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$K(s)$ then consists of a combination of leads, lags, notches, etc. such that $L(s)$ has the wanted shape (an example is shown in Figure [[fig:open_loop_shaping_shape]]).
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#+begin_src latex :file open_loop_shaping_shape.pdf
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\begin{tikzpicture}
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% Phase Axis
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\draw[->] (-0.3, -0.5) -- ++(8, 0) node[above]{$\omega$}; \draw[<-] (0, 0)
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node[left]{$\angle L(j\omega)$} -- ++(0, -2.3);
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% Gain Axis
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\draw[->] (-0.3, 2) -- ++(8, 0) node[above]{$\omega$}; \draw[->] (0, 0.5) --
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++(0, 3) node[left]{$\left|L(j\omega)\right|$};
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% Gain Slopes
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\draw[shift={(0,2)}] (0.5, 1.25) -- node[midway, above]{$-2$} (2, 0.5) --
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node[midway, above]{$-1$} (6, -0.5) -- node[midway, below left]{$-2$} (7.5,
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-1.25);
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% Forbiden region
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\path[shift={(0,1.8)}, fill=red!50!white] (0.5, 1.25) -- (2, 0.5) -| coordinate[near start](lfshaping) cycle;
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\path[shift={(0,2.2)}, fill=red!50!white] (6, -0.5) -- (7.5, -1.25) |- coordinate[near end](hfshaping) cycle;
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\draw[<-] (lfshaping) -- ++(0, -0.8) node[below, align=center]{Reference\\Tracking};
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\draw[<-] (hfshaping) -- ++(0, 0.8) node[above, align=center]{Noise\\Rejection};
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% Crossover frequency
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\node[below] (wc) at (4,2){$\omega_c$};
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\draw[<-] (wc.south) -- ++(0, -0.4) node[below, align=center]{Bandwidth};
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% Phase
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\draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4, -1.25)[out=0, in=-180] to
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(6, -2) -- (7.5, -2); \draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4,
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-1.25)[out=0, in=-180] to (6, -2) -- (7.5, -2);
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% Phase Margin
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\draw[->, dashed] (4, -2) -- (4, -1.25) node[above]{Phase Margin};
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\draw[dashed] (0, -2) node[left]{$-\pi$} -- (7.5, -2);
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\end{tikzpicture}
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#+end_src
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#+name: fig:open_loop_shaping_shape
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#+caption: Typical Wanted Shape for the Loop Gain $L(s)$
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#+RESULTS:
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[[file:figs/open_loop_shaping_shape.png]]
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** Example of Open Loop Shaping
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<<sec:loop_shaping_example>>
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#+begin_exampl
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Let's take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:
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@@ -421,23 +491,24 @@ And we can verify that we have the wanted stability margins:
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Manual Method'}, ' %.1f ');
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data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Requirements', 'Manual Method'}, ' %.1f ');
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#+end_src
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#+RESULTS:
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| | Manual Method |
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| Requirements | Manual Method |
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|-----------------------------+---------------|
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| Gain Margin $> 3$ [dB] | 3.1 |
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| Phase Margin $> 30$ [deg] | 35.4 |
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| Crossover $\approx 10$ [Hz] | 10.1 |
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** $\mathcal{H}_\infty$ Loop Shaping Synthesis
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<<sec:h_infinity_open_loop_shaping>>
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The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ Synthesis.
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Even though we will not go into details, we will provide one example.
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Using Matlab, the $\mathcal{H}_\infty$ synthesis of a controller based on the wanted open loop shape can be performed using the =loopsyn= command:
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Using Matlab, the $\mathcal{H}_\infty$ Loop Shaping Synthesis can be performed using the =loopsyn= command:
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#+begin_src matlab :eval no
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K = loopsyn(G, Gd);
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#+end_src
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@@ -451,6 +522,7 @@ where:
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#+end_seealso
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** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
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<<sec:h_infinity_open_loop_shaping_example>>
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Let's reuse the previous plant.
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@@ -465,7 +537,7 @@ Translate the specification into the wanted shape of the open loop gain.
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(1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead
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#+end_src
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The $\mathcal{H}_\infty$ optimal open loop shaping is performed using the =loopsyn= command:
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The $\mathcal{H}_\infty$ optimal open loop shaping synthesis is performed using the =loopsyn= command:
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#+begin_src matlab
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[K, ~, GAM] = loopsyn(G, Lw);
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#+end_src
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@@ -575,8 +647,11 @@ Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ con
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| Phase Margin $> 30$ [deg] | 35.4 | 54.7 |
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| Crossover $\approx 10$ [Hz] | 10.1 | 9.9 |
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* First Step in the $\mathcal{H}_\infty$ world
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* First Steps in the $\mathcal{H}_\infty$ world
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<<sec:h_infinity_introduction>>
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** The $\mathcal{H}_\infty$ Norm
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<<sec:h_infinity_norm>>
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#+begin_definition
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The $\mathcal{H}_\infty$ norm is defined as the peak of the maximum singular value of the frequency response
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@@ -591,7 +666,7 @@ Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ con
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#+end_definition
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#+begin_exampl
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And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function:
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Let's compute the $\mathcal{H}_\infty$ norm of our test plant $G(s)$ using the =hinfnorm= function:
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#+begin_src matlab :results value replace
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hinfnorm(G)
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#+end_src
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@@ -599,8 +674,7 @@ And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function:
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#+RESULTS:
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: 7.9216e-06
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The magnitude $|G(j\omega)|$ of the plant $G(s)$ as a function of frequency is shown in Figure [[fig:hinfinity_norm_siso_bode]].
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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.
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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 [[fig:hinfinity_norm_siso_bode]].
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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@@ -626,23 +700,47 @@ The maximum value of the magnitude over all frequencies does correspond to the $
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#+end_exampl
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** $\mathcal{H}_\infty$ Synthesis
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<<sec:h_infinity_synthesis>>
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*Optimization problem*:
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$\mathcal{H}_\infty$ 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.
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#+begin_definition
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$\mathcal{H}_\infty$ 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.
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#+end_definition
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*Engineer work*:
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Why optimizing the $\mathcal{H}_\infty$ norm of transfer functions is a pertinent choice will become clear when we will translate the typical control specifications into the $\mathcal{H}_\infty$ norm of transfer functions.
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Then applying the $\mathcal{H}_\infty$ synthesis to a plant, the engineer work usually consists of the following steps
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1. Write the problem as standard $\mathcal{H}_\infty$ problem
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2. Translate the specifications as $\mathcal{H}_\infty$ norms
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2. Translate the specifications as $\mathcal{H}_\infty$ norms of transfer functions
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3. Make the synthesis and analyze the obtain controller
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4. Reduce the order of the controller for implementation
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*Many ways to use the $\mathcal{H}_\infty$ Synthesis*:
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- Traditional $\mathcal{H}_\infty$ Synthesis
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- Mixed Sensitivity Loop Shaping
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- Fixed-Structure $\mathcal{H}_\infty$ Synthesis
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Note that there are many ways to use the $\mathcal{H}_\infty$ Synthesis:
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- Traditional $\mathcal{H}_\infty$ Synthesis (=hinfsyn= [[https://www.mathworks.com/help/robust/ref/hinfsyn.html][doc]])
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- Open Loop Shaping $\mathcal{H}_\infty$ Synthesis (=loopsyn= [[https://www.mathworks.com/help/robust/ref/loopsyn.html][doc]])
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- Mixed Sensitivity Loop Shaping (=mixsyn= [[https://www.mathworks.com/help/robust/ref/lti.mixsyn.html][doc]])
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- Fixed-Structure $\mathcal{H}_\infty$ Synthesis (=hinfstruct= [[https://www.mathworks.com/help/robust/ref/lti.hinfstruct.html][doc]])
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- Signal Based $\mathcal{H}_\infty$ Synthesis
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** The Generalized Plant
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<<sec:generalized_plant>>
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The first step when applying the $\mathcal{H}_\infty$ synthesis is usually to write the problem as a standard $\mathcal{H}_\infty$ problem.
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This consist of deriving the *Generalized Plant* for the current problem.
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It makes things much easier for the following steps.
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The generalized plant, usually noted $P(s)$, is shown in Figure [[fig:general_plant]].
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It has two inputs and two outputs (both could contains many signals).
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The meaning of the inputs and outputs are summarized in Table [[tab:notation_general]].
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Note that this generalized plant is as its name implies, quite /general/.
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It can indeed represent feedback as well as feedforward control architectures.
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\begin{equation}
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\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}
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\end{equation}
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#+begin_src latex :file general_plant.pdf
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\begin{tikzpicture}
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\node[block={2.0cm}{2.0cm}] (P) {$P$};
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@@ -663,9 +761,12 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
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\end{tikzpicture}
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#+end_src
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#+name: fig:general_plant
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#+caption: Inputs and Outputs of the generalized Plant
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#+RESULTS:
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[[file:figs/general_plant.png]]
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#+begin_important
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#+name: tab:notation_general
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#+caption: Notations for the general configuration
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| Notation | Meaning |
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@@ -675,50 +776,68 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
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| $z$ | Exogenous outputs: signals to be minimized |
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| $v$ | Controller inputs: measurements |
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| $u$ | Control signals |
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#+end_important
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\begin{equation}
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\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}
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\end{equation}
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** The General Synthesis Problem Formulation
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<<sec:h_infinity_general_synthesis>>
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** From a Classical Feedback Architecture to a Generalized Plant
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Once the generalized plant is obtained, the $\mathcal{H}_\infty$ synthesis problem can be stated as follows:
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#+begin_src latex :file classical_feedback.pdf
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#+begin_important
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- $\mathcal{H}_\infty$ Synthesis applied on the generalized plant ::
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Find a stabilizing controller $K$ that, using the sensed output $v$, generates a control signal $u$ such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer function from $w$ to $z$ is minimized.
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After $K$ is found, the system is /robustified/ by adjusting the response around the unity gain frequency to increase stability margins.
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#+end_important
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#+begin_src latex :file general_control_names.pdf
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\begin{tikzpicture}
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\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
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\node[block, right=0.8 of addfb] (K){$K(s)$};
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\node[addb={+}{}{}{}{}, right=0.8 of K] (addu){};
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\node[block, right=0.8 of addu] (G){$G(s)$};
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\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
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\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
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\draw[->] (K.east) -- (addu.west) node[above left]{$u$};
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\draw[->] (addu.east) -- (G.west);
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\draw[<-] (addu.north) -- ++(0, 0.8) node[below right]{$d$};
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\draw[->] (G.east) -- ++(1.2, 0);
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\draw[->] ($(G.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -0.8) -| (addfb.south);
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% Blocs
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\node[block={2.0cm}{2.0cm}] (P) {$P$};
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\node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$};
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% Input and outputs coordinates
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\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
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\coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$);
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\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$);
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\coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$);
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% Connections and labels
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\draw[<-] (inputw) node[above left, align=right]{(weighted)\\exogenous inputs\\$w$} -- ++(-1.5, 0);
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\draw[<-] (inputu) -- ++(-0.8, 0) |- node[left, near start, align=right]{control signals\\$u$} (K.west);
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\draw[->] (outputz) node[above right, align=left]{(weighted)\\exogenous outputs\\$z$} -- ++(1.5, 0);
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\draw[->] (outputv) -- ++(0.8, 0) |- node[right, near start, align=left]{sensed output\\$v$} (K.east);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:classical_feedback
|
||||
#+caption: Classical Feedback Architecture
|
||||
#+name: fig:general_control_names
|
||||
#+caption: General Control Configuration
|
||||
#+RESULTS:
|
||||
[[file:figs/classical_feedback.png]]
|
||||
[[file:figs/general_control_names.png]]
|
||||
|
||||
#+name: table:notation_conventional
|
||||
#+caption: Notations for the Classical Feedback Architecture
|
||||
| Notation | Meaning |
|
||||
|------------+-------------------|
|
||||
| $G$ | Plant model |
|
||||
| $K$ | Controller |
|
||||
| $r$ | Reference inputs |
|
||||
| $y$ | Plant outputs |
|
||||
| $u$ | Control signals |
|
||||
| $d$ | Input Disturbance |
|
||||
| $\epsilon$ | Tracking Error |
|
||||
Note that the closed-loop transfer function from $w$ to $z$ is:
|
||||
\begin{equation}
|
||||
\frac{z}{w} = P_{11} + P_{12} K \big( I - P_{22} K \big)^{-1} P_{21} \triangleq F_l(P, K)
|
||||
\end{equation}
|
||||
|
||||
The procedure is:
|
||||
1. define signals of the generalized plant
|
||||
2. Remove $K$ and rearrange the inputs and outputs
|
||||
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
|
||||
K = hinfsyn(P, nmeas, ncont);
|
||||
#+end_src
|
||||
where:
|
||||
- =P= is the generalized plant transfer function matrix
|
||||
- =nmeas= is the number of sensed output (size of $v$)
|
||||
- =ncont= is the number of control signals (size of $u$)
|
||||
- =K= obtained controller that minimized the $\mathcal{H}_\infty$ norm from $w$ to $z$
|
||||
|
||||
** From a Classical Feedback Architecture to a Generalized Plant
|
||||
<<sec:generalized_plant_derivation>>
|
||||
|
||||
The procedure to convert a typical control architecture as the one shown in Figure [[fig:classical_feedback_tracking]] 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
|
||||
|
||||
#+begin_src latex :file classical_feedback_tracking.pdf
|
||||
\begin{tikzpicture}
|
||||
@@ -765,12 +884,13 @@ The procedure is:
|
||||
#+end_src
|
||||
|
||||
#+begin_exercice
|
||||
Let's find the Generalized plant of corresponding to the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]]
|
||||
Compute the Generalized plant of corresponding to the tracking control architecture shown in Figure [[fig:classical_feedback_tracking]]
|
||||
|
||||
#+name: fig:classical_feedback_tracking
|
||||
#+caption: Classical Feedback Control Architecture (Tracking)
|
||||
[[file:figs/classical_feedback_tracking.png]]
|
||||
|
||||
#+HTML: <details><summary>Hint</summary>
|
||||
First, define the signals of the generalized plant:
|
||||
- Exogenous inputs: $w = r$
|
||||
- Signals to be minimized: $z_1 = \epsilon$, $z_2 = u$
|
||||
@@ -778,76 +898,550 @@ The procedure is:
|
||||
- Control inputs: $u$
|
||||
|
||||
Then, Remove $K$ and rearrange the inputs and outputs.
|
||||
We obtain the generalized plant shown in Figure [[fig:mixed_sensitivity_ref_tracking]].
|
||||
#+HTML: </details>
|
||||
|
||||
#+HTML: <details><summary>Answer</summary>
|
||||
The obtained generalized plant shown in Figure [[fig:mixed_sensitivity_ref_tracking]].
|
||||
|
||||
#+name: fig:mixed_sensitivity_ref_tracking
|
||||
#+caption: Generalized plant of the Classical Feedback Control Architecture (Tracking)
|
||||
[[file:figs/mixed_sensitivity_ref_tracking.png]]
|
||||
|
||||
Using Matlab, the generalized plant can be defined as follows:
|
||||
#+begin_src matlab :tangle no :eval no
|
||||
P = [1 -G;
|
||||
0 1;
|
||||
1 -G]
|
||||
#+end_src
|
||||
#+HTML: </details>
|
||||
#+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}
|
||||
Using Matlab, the generalized plant can be defined as follows:
|
||||
#+begin_src matlab :tangle no :eval no
|
||||
P = [1 -G;
|
||||
0 1;
|
||||
1 -G]
|
||||
P.InputName = {'w', 'u'};
|
||||
P.OutputName = {'e', 'u', 'v'};
|
||||
#+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
|
||||
<<sec:modern_interpretation_specification>>
|
||||
|
||||
- *Reference tracking* Overshoot, Static error, Setling time
|
||||
- $S(s) = T_{r \rightarrow \epsilon}$
|
||||
** Introduction
|
||||
The
|
||||
|
||||
#+name: fig:gang_of_four_feedback
|
||||
#+caption: Simple Feedback Architecture
|
||||
#+RESULTS:
|
||||
[[file:figs/gang_of_four_feedback.png]]
|
||||
|
||||
- *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
|
||||
<<sec:losed_loop_tf>>
|
||||
|
||||
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_src latex :file gang_of_four_feedback.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
||||
\node[block, right=0.8 of addfb] (K){$K(s)$};
|
||||
\node[addb, right=0.8 of K] (addd){};
|
||||
\node[block, right=0.8 of addd] (G){$G(s)$};
|
||||
\node[addb, below right=0.4 and 0.2 of G] (addn){};
|
||||
|
||||
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$};
|
||||
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$};
|
||||
\draw[->] (K.east) -- (addd.west);
|
||||
\draw[<-] (addd.north) -- ++(0, 0.6) node[below right]{$d$};
|
||||
\draw[->] (addd.east) -- (G.west) node[above left]{$u$};
|
||||
\draw[->] (G.east) -- ++(1.6, 0) node[above left]{$y$};
|
||||
\draw[->] (G-|addn) node[branch]{} -- (addn.north);
|
||||
\draw[<-] (addn.east) -- ++(0.8, 0) node[above left]{$n$};
|
||||
\draw[->] (addn.west) -| (addfb.south);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:gang_of_four_feedback
|
||||
#+caption: Simple Feedback Architecture
|
||||
#+RESULTS:
|
||||
[[file:figs/gang_of_four_feedback.png]]
|
||||
|
||||
#+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]].
|
||||
|
||||
#+HTML: <details><summary>Hint</summary>
|
||||
Take one of the output (e.g. $y$), and write it as a function of the inputs $[d, r, n]$ going step by step around the loop:
|
||||
\begin{aligned}
|
||||
y &= G u \\
|
||||
&= G (d + K \epsilon) \\
|
||||
&= G \big(d + K (r - n - y) \big) \\
|
||||
&= G d + GK r - GK n - GK y
|
||||
\end{aligned}
|
||||
|
||||
Isolate $y$ at the right hand side, and finally obtain:
|
||||
\[ y = \frac{GK}{1+ GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \]
|
||||
|
||||
Do the same procedure for $u$ and $\epsilon$
|
||||
#+HTML: </details>
|
||||
|
||||
#+HTML: <details><summary>Anwser</summary>
|
||||
The following equations should be obtained:
|
||||
\begin{align}
|
||||
y &= \frac{GK}{1 + GK} r + \frac{G}{1 + GK} d - \frac{GK}{1 + GK} n \\
|
||||
\epsilon &= \frac{1 }{1 + GK} r - \frac{G}{1 + GK} d - \frac{G }{1 + GK} n \\
|
||||
u &= \frac{K }{1 + GK} r - \frac{1}{1 + GK} d - \frac{K }{1 + GK} n
|
||||
\end{align}
|
||||
#+HTML: </details>
|
||||
#+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{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}
|
||||
|
||||
#+begin_seealso
|
||||
If a feedforward controller is included, a *Gang of Six* transfer functions can be defined.
|
||||
More on that in this [[https://www.youtube.com/watch?v=b_8v8scghh8][short video]].
|
||||
#+end_seealso
|
||||
|
||||
And we have:
|
||||
\begin{align}
|
||||
\epsilon &= S r - GS d - GS n \\
|
||||
y &= T r + GS d - T n \\
|
||||
u &= KS r - S d - KS n
|
||||
\end{align}
|
||||
|
||||
** Sensitivity Transfer Function
|
||||
<<sec:sensitivity_transfer_functions>>
|
||||
|
||||
#+begin_src matlab
|
||||
K1 = 14e8 * ... % Gain
|
||||
1/(s^2) * ... % Double Integrator
|
||||
(1 + s/(2*pi*10/sqrt(8)))/(1 + s/(2*pi*10*sqrt(8))); % Lead
|
||||
|
||||
K2 = 1e8 * ... % Gain
|
||||
1/(s^2) * ... % Double Integrator
|
||||
(1 + s/(2*pi*1/sqrt(8)))/(1 + s/(2*pi*1*sqrt(8))); % Lead
|
||||
|
||||
K3 = 1e8 * ... % Gain
|
||||
1/(s^2) * ... % Double Integrator
|
||||
(1 + s/(2*pi*1/sqrt(2)))/(1 + s/(2*pi*1*sqrt(2))); % Lead
|
||||
|
||||
S1 = 1/(1 + K1*G);
|
||||
S2 = 1/(1 + K2*G);
|
||||
S3 = 1/(1 + K3*G);
|
||||
|
||||
T1 = K1*G/(1 + K1*G);
|
||||
T2 = K2*G/(1 + K2*G);
|
||||
T3 = K3*G/(1 + K3*G);
|
||||
|
||||
bodeFig({S1, S2, S3})
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(S1, freqs, 'Hz'))), 'DisplayName', '$L(s)$');
|
||||
plot(freqs, abs(squeeze(freqresp(S2, freqs, 'Hz'))), 'DisplayName', '$L_w(s)$');
|
||||
plot(freqs, abs(squeeze(freqresp(S3, freqs, 'Hz'))), 'DisplayName', '$L_w(s) / \gamma$, $L_w(s) \cdot \gamma$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frquency [Hz]'); ylabel('Sensitivity Magnitude');
|
||||
hold off;
|
||||
|
||||
ax2 = nexttile;
|
||||
t = linspace(0, 1, 1000);
|
||||
y1 = step(T1, t);
|
||||
y2 = step(T2, t);
|
||||
y3 = step(T3, t);
|
||||
hold on;
|
||||
plot(t, y1)
|
||||
plot(t, y2)
|
||||
plot(t, y3)
|
||||
hold off
|
||||
xlabel('Time [s]'); ylabel('Step Response');
|
||||
#+end_src
|
||||
|
||||
#+begin_src latex :file h-infinity-spec-S.pdf
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[%
|
||||
width=8cm,
|
||||
height=4cm,
|
||||
at={(0,0)},
|
||||
xmode=log,
|
||||
xmin=0.01,
|
||||
xmax=10000,
|
||||
ymin=-80,
|
||||
ymax=40,
|
||||
ylabel={Magnitude [dB]},
|
||||
xlabel={Frequency [Hz]},
|
||||
ytick={40, 20, 0, -20, -40, -60, -80},
|
||||
xminorgrids,
|
||||
yminorgrids
|
||||
]
|
||||
\addplot [thick, color=black, forget plot]
|
||||
table[row sep=crcr]{%
|
||||
0.01 -60\\
|
||||
0.1 -60\\
|
||||
190 6\\
|
||||
10000 6\\
|
||||
};
|
||||
\draw[<-] (0.05, -60) -- (0.1, -70);
|
||||
\draw (0.1, -70) -- (2, -70) node[right, fill=white, draw]{$\SI{-60}{\decibel} \rightarrow$ \footnotesize{Static error}};
|
||||
\draw[<-] (1, -40) -- (10, -40) node[right, fill=white, draw]{$\SI{20}{\decibel/dec} \rightarrow$ \footnotesize{Ref. track.}};
|
||||
\draw[<-] (100, 0) -- (3, 0) node[left, fill=white, draw]{$\omega_c \rightarrow$ \footnotesize{Speed}};
|
||||
|
||||
\draw[<-] (300, 6) -- (200, 20);
|
||||
\draw (200, 20) -- (10, 20) node[left, fill=white, draw]{$\SI{6}{\decibel} \rightarrow$ \footnotesize{Module margin}};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:h-infinity-spec-S
|
||||
#+caption: Typical wanted shape of the Sensitivity transfer function
|
||||
#+RESULTS:
|
||||
[[file:figs/h-infinity-spec-S.png]]
|
||||
|
||||
** Robustness: Module Margin
|
||||
<<sec:module_margin>>
|
||||
|
||||
- [ ] Definition of Module margin
|
||||
- [ ] Why it represents robustness
|
||||
- [ ] Example
|
||||
|
||||
\[ M_S < 2 \Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o \]
|
||||
|
||||
** How to *Shape* transfer function? Using of Weighting Functions!
|
||||
<<sec:weighting_functions>>
|
||||
|
||||
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
|
||||
\begin{tikzpicture}
|
||||
\node[block] (G) {$G(s)$};
|
||||
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
|
||||
\coordinate[above right=1.0 and 1.4 of addw] (epsilon);
|
||||
|
||||
\coordinate[] (w) at ($(epsilon-|G.west)+(-1.0, 0)$);
|
||||
|
||||
\node[block, below left=0.8 and 0 of addw] (K) {$K(s)$};
|
||||
|
||||
% Connections
|
||||
\draw[->] (G.east) -- (addw.west);
|
||||
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (epsilon) node[above left](z1){$\epsilon$};
|
||||
|
||||
\draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
|
||||
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.4, 0)$) -- (G.west);
|
||||
|
||||
\draw[->] (w) node[above]{$w = r$} -| (addw.north);
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(G.south west) ($(z1.north west)+(-0.4, 0)$)}, inner sep=12pt, draw, dashed, fill=black!20!white] (P) {};
|
||||
\node[below] at (P.north) {Generalized Plant $P(s)$};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:loop_shaping_S_without_W
|
||||
#+caption: Generalized Plant
|
||||
#+RESULTS:
|
||||
[[file:figs/loop_shaping_S_without_W.png]]
|
||||
|
||||
If the $\mathcal{H}_\infty$ synthesis is directly applied on the generalized plant $P(s)$ shown in Figure [[fig:loop_shaping_S_without_W]], if will minimize the $\mathcal{H}_\infty$ norm of transfer function from $r$ to $\epsilon$ (the sensitivity transfer function).
|
||||
|
||||
However, as the $\mathcal{H}_\infty$ norm is the maximum peak value of the transfer function's magnitude, it does not allow to *shape* the norm over all frequencies.
|
||||
|
||||
|
||||
|
||||
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_important
|
||||
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.
|
||||
Its inverse magnitude will define the frequency dependent upper bound of the sensitivity transfer function magnitude.
|
||||
#+end_important
|
||||
|
||||
#+begin_src latex :file loop_shaping_S_with_W.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[block] (G) {$G(s)$};
|
||||
\node[addb={+}{-}{}{}{}, right=0.6 of G] (addw) {};
|
||||
\node[block, above right=1.0 and 1.0 of addw] (Ws) {$W_s(s)$};
|
||||
\coordinate[right=0.8 of Ws] (epsilon);
|
||||
|
||||
\coordinate[] (w) at ($(epsilon-|G.west)+(-1.0, 0)$);
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(G.south west) (Ws.north east)}, inner sep=8pt, draw, dashed, fill=black!20!white] (P) {};
|
||||
\node[above] at (P.north) {Weighted Generalized Plant $\tilde{P}(s)$};
|
||||
\end{scope}
|
||||
|
||||
\node[block, below=0.4 of P] (K) {$K(s)$};
|
||||
|
||||
% Connections
|
||||
\draw[->] (G.east) -- (addw.west);
|
||||
\draw[->] ($(addw.east)+(0.4, 0)$)node[branch]{} |- (Ws.west)node[above left]{$\epsilon$};
|
||||
\draw[->] (Ws.east) -- (epsilon) node[above left](z1){$\tilde{\epsilon}$};
|
||||
|
||||
\draw[->] (addw.east) -- (addw-|z1) |- node[near start, right]{$v$} (K.east);
|
||||
\draw[->] (K.west) -| node[near end, left]{$u$} ($(G-|w)+(0.4, 0)$) -- (G.west);
|
||||
|
||||
\draw[->] (w) node[above]{$w = r$} -| (addw.north);
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:loop_shaping_S_with_W
|
||||
#+caption: Weighted Generalized Plant
|
||||
#+RESULTS:
|
||||
[[file:figs/loop_shaping_S_with_W.png]]
|
||||
|
||||
Once the weighting function is designed, it should be added to the generalized plant as shown in Figure [[fig:loop_shaping_S_with_W]].
|
||||
|
||||
The weighted generalized plant can be defined in Matlab by either re-defining all the inputs or by pre-multiplying the (non-weighted) generalized plant by a block-diagonal MIMO transfer function containing the weights for the outputs $z$ and =1= for the outputs $v$.
|
||||
|
||||
#+begin_src matlab :tangle no :eval no
|
||||
Pw = [Ws -Ws*G;
|
||||
1 -G]
|
||||
|
||||
% Alternative
|
||||
Pw = blkdiag(Ws, 1)*P;
|
||||
#+end_src
|
||||
|
||||
** Design of Weighting Functions
|
||||
<<sec:weighting_functions_design>>
|
||||
|
||||
Weighting function used must be *proper*, *stable* and *minimum phase* transfer functions.
|
||||
- proper ::
|
||||
more poles than zeros, this implies $\lim_{\omega \to \infty} |W(j\omega)| < \infty$
|
||||
- stable ::
|
||||
no poles in the right half plane
|
||||
- minimum phase ::
|
||||
no zeros in the right half plane
|
||||
|
||||
Matlab is providing the =makeweight= function that creates a first-order weights by specifying the low frequency gain, high frequency gain, and a gain at a specific frequency:
|
||||
#+begin_src matlab :tangle no :eval no
|
||||
W = makeweight(dcgain,[freq,mag],hfgain)
|
||||
#+end_src
|
||||
with:
|
||||
- =dcgain=
|
||||
- =freq=
|
||||
- =mag=
|
||||
- =hfgain=
|
||||
|
||||
#+begin_exampl
|
||||
The Matlab code below produces a weighting function with a magnitude shape shown in Figure [[fig:first_order_weight]].
|
||||
|
||||
#+begin_src matlab
|
||||
Ws = makeweight(1e2, [2*pi*10, 1], 1/2);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-2, 2, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(Ws, freqs, 'Hz'))), 'k-');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frquency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/first_order_weight.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:first_order_weight
|
||||
#+caption: Obtained Magnitude of the Weighting Function
|
||||
#+RESULTS:
|
||||
[[file:figs/first_order_weight.png]]
|
||||
#+end_exampl
|
||||
|
||||
#+begin_seealso
|
||||
Quite often, higher orders weights are required.
|
||||
|
||||
In such case, the following formula can be used the design of these weights:
|
||||
|
||||
\begin{equation}
|
||||
W(s) = \left( \frac{
|
||||
\frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
|
||||
}{
|
||||
\left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
|
||||
}\right)^n \label{eq:weight_formula_advanced}
|
||||
\end{equation}
|
||||
|
||||
The parameters permit to specify:
|
||||
- the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
|
||||
- the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
|
||||
- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
|
||||
- the absolute slope between high and low frequency: $n$
|
||||
|
||||
A Matlab function implementing Equation eqref:eq:weight_formula_advanced is shown below:
|
||||
|
||||
#+name: lst:generateWeight
|
||||
#+caption: Matlab Function that can be used to generate Weighting functions
|
||||
#+begin_src matlab :tangle matlab/generateWeight.m :comments none :eval no
|
||||
function [W] = generateWeight(args)
|
||||
arguments
|
||||
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
||||
args.G1 (1,1) double {mustBeNumeric, mustBePositive} = 10
|
||||
args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
args.wc (1,1) double {mustBeNumeric, mustBePositive} = 2*pi
|
||||
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
|
||||
end
|
||||
|
||||
if (args.Gc <= args.G0 && args.Gc <= args.G1) || (args.Gc >= args.G0 && args.Gc >= args.G1)
|
||||
eid = 'value:range';
|
||||
msg = 'Gc must be between G0 and G1';
|
||||
throwAsCaller(MException(eid,msg))
|
||||
end
|
||||
|
||||
s = zpk('s');
|
||||
|
||||
W = (((1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (args.G0/args.Gc)^(1/args.n))/((1/args.G1)^(1/args.n)*(1/args.wc)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.G1)^(2/args.n)))*s + (1/args.Gc)^(1/args.n)))^args.n;
|
||||
|
||||
end
|
||||
#+end_src
|
||||
|
||||
Let's use this function to generate three weights with the same high and low frequency gains, but but different slopes.
|
||||
|
||||
#+begin_src matlab
|
||||
W1 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 1);
|
||||
W2 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
|
||||
W3 = generateWeight('G0', 1e2, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 3);
|
||||
#+end_src
|
||||
|
||||
The obtained shapes are shown in Figure [[fig:high_order_weight]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
freqs = logspace(-2, 2, 1000);
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), ...
|
||||
'DisplayName', '$n = 1$');
|
||||
plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), ...
|
||||
'DisplayName', '$n = 2$');
|
||||
plot(freqs, abs(squeeze(freqresp(W3, freqs, 'Hz'))), ...
|
||||
'DisplayName', '$n = 3$');
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frquency [Hz]'); ylabel('Magnitude');
|
||||
legend('location', 'northeast');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/high_order_weight.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:high_order_weight
|
||||
#+caption: Higher order weights using Equation eqref:eq:weight_formula_advanced
|
||||
#+RESULTS:
|
||||
[[file:figs/high_order_weight.png]]
|
||||
#+end_seealso
|
||||
|
||||
** Sensitivity Function Shaping - Example
|
||||
<<sec:sensitivity_shaping_example>>
|
||||
|
||||
- Robustness: Module margin > 2 ($\Rightarrow \text{GM} > 2 \text{ and } \text{PM} > 29^o$)
|
||||
- Bandwidth:
|
||||
- Slope of -2
|
||||
|
||||
First, the weighting functions is generated.
|
||||
#+begin_src matlab
|
||||
Ws = generateWeight('G0', 1e3, 'G1', 1/2, 'Gc', 1, 'wc', 2*pi*10, 'n', 2);
|
||||
#+end_src
|
||||
|
||||
It is then added to the generalized plant.
|
||||
#+begin_src matlab
|
||||
Pw = blkdiag(Ws, 1)*P;
|
||||
#+end_src
|
||||
|
||||
And the $\mathcal{H}_\infty$ synthesis is performed.
|
||||
#+begin_src matlab :results output replace
|
||||
K = hinfsyn(Pw, 1, 1, 'Display', 'on');
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
#+begin_example
|
||||
K = hinfsyn(Pw, 1, 1, 'Display', 'on');
|
||||
|
||||
Test bounds: 0.5 <= gamma <= 0.51
|
||||
|
||||
gamma X>=0 Y>=0 rho(XY)<1 p/f
|
||||
5.05e-01 0.0e+00 0.0e+00 4.497e-28 p
|
||||
Limiting gains...
|
||||
5.05e-01 0.0e+00 0.0e+00 0.000e+00 p
|
||||
5.05e-01 -1.8e+01 # -2.9e-15 1.514e-15 f
|
||||
|
||||
Best performance (actual): 0.504
|
||||
#+end_example
|
||||
|
||||
The obtained $\gamma \approx 0.5$ means that it found a controller $K(s)$ that stabilize the closed-loop system, and such that:
|
||||
\begin{aligned}
|
||||
& \| W_s(s) S(s) \|_\infty < 0.5 \\
|
||||
& \Leftrightarrow |S(j\omega)| < \frac{0.5}{|W_s(j\omega)|} \quad \forall \omega
|
||||
\end{aligned}
|
||||
|
||||
This is indeed what we can see by comparing $|S|$ and $|W_S|$ in Figure [[fig:results_sensitivity_hinf]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, 1./abs(squeeze(freqresp(Ws, freqs, 'Hz'))), 'k--', 'DisplayName', '$|W_s|^{-1}$');
|
||||
plot(freqs, abs(squeeze(freqresp(1/(1 + K*G), freqs, 'Hz'))), 'k-', 'DisplayName', '$|S|$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
legend('location', 'southeast', 'FontSize', 8);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/results_sensitivity_hinf.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:results_sensitivity_hinf
|
||||
#+caption: Weighting function and obtained closed-loop sensitivity
|
||||
#+RESULTS:
|
||||
[[file:figs/results_sensitivity_hinf.png]]
|
||||
|
||||
* $\mathcal{H}_\infty$ Mixed-Sensitivity Synthesis
|
||||
<<sec:h_infinity_mixed_sensitivity>>
|
||||
|
||||
** Problem
|
||||
|
||||
** Typical Procedure
|
||||
|
||||
** Step 1 - Shaping of the Sensitivity Function
|
||||
|
||||
** Step 2 - Shaping of
|
||||
|
||||
* Conclusion
|
||||
|
||||
* Resources
|
||||
|
||||
|
||||
Reference in new issue
Block a user