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Add many things about loop shaping

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Thomas Dehaeze 2020-11-30 17:44:13 +01:00
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commit b24a473b35
24 changed files with 3025 additions and 326 deletions
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@ -1,4 +1,4 @@
#+TITLE: Robust Control - $\mathcal{H}_\infty$ Synthesis #+TITLE: A brief and practical introduction to $\mathcal{H}_\infty$ Control
:DRAWER: :DRAWER:
#+STARTUP: overview #+STARTUP: overview
@ -31,6 +31,7 @@
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100 #+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results file raw replace #+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no #+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :tangle no
#+PROPERTY: header-args:latex+ :eval no-export #+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results #+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes #+PROPERTY: header-args:latex+ :mkdirp yes
@ -48,8 +49,15 @@
<<matlab-init>> <<matlab-init>>
#+end_src #+end_src
#+begin_src matlab :tangle no
addpath('matlab')
#+end_src
* Introduction to the Control Methodology - Model Based Control * Introduction to the Control Methodology - Model Based Control
** Control Methodology <<sec:model_based_control>>
** Model Based Control - Methodology
<<sec:model_based_control_methodology>>
The typical methodology when applying Model Based Control to a plant is schematically shown in Figure [[fig:control-procedure]]. The typical methodology when applying Model Based Control to a plant is schematically shown in Figure [[fig:control-procedure]].
It consists of three steps: It consists of three steps:
@ -94,6 +102,7 @@ It consists of three steps:
In this document, we will mainly focus on steps 2 and 3. In this document, we will mainly focus on steps 2 and 3.
** Some Background: From Classical Control to Robust Control ** Some Background: From Classical Control to Robust Control
<<sec:comp_classical_modern_robust_control>>
#+name: tab:comparison_control_methods #+name: tab:comparison_control_methods
#+caption: Table summurazing the main differences between classical, modern and robust control #+caption: Table summurazing the main differences between classical, modern and robust control
@ -122,6 +131,7 @@ In this document, we will mainly focus on steps 2 and 3.
| | Only SISO | Difficult Rejection of Perturbations | Need a reasonably good model of the system | | | Only SISO | Difficult Rejection of Perturbations | Need a reasonably good model of the system |
** Example System ** Example System
<<sec:example_system>>
Let's consider the model shown in Figure [[fig:mech_sys_1dof_inertial_contr]]. Let's consider the model shown in Figure [[fig:mech_sys_1dof_inertial_contr]].
It could represent a suspension system with a payload to position or isolate using an force actuator and an inertial sensor. It could represent a suspension system with a payload to position or isolate using an force actuator and an inertial sensor.
@ -204,6 +214,20 @@ You can follow this generic procedure:
\end{align} \end{align}
#+HTML: </details> #+HTML: </details>
#+end_exercice #+end_exercice
Hi Musa,
Thank you very much for sharing this awesome package.
For a long time, I am dreaming of being abble to export source blocks to HTML tha are surounded by <details> blocks.
For now, I am manually adding #+HTML: <details><summary>Code</summary> and #+HTML: </details> around the source blocks I want to hide...
This is a very simple solution, but not so elegent nor practical.
Do you have any idea if it would be easy to extend to org-mode export of source blocks to add such functionallity?
Similarly, I would love to be able to export a <span> block with the name of the file corresponding to the source block.
For instance, if a particular source block is tangled to script.sh, it would be so nice to display the filename when exporting!
Thanks in advance
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]]. 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]].
@ -302,10 +326,17 @@ And now the system dynamics $G(s)$ and $G_d(s)$ (their bode plots are shown in F
[[file:figs/bode_plot_example_Gd.png]] [[file:figs/bode_plot_example_Gd.png]]
* Classical Open Loop Shaping * Classical Open Loop Shaping
** Introduction to Open Loop Shaping <<sec:open_loop_shaping>>
** Introduction to Loop Shaping
<<sec:open_loop_shaping_introduction>>
#+begin_definition #+begin_definition
The *Loop Gain* $L(s)$ usually refers to as the product of the controller and the plant (Figure [[fig:open_loop_shaping]]): *Loop Shaping* refers to a design procedure that involves explicitly shaping the magnitude of the *Loop Transfer Function* $L(s)$.
#+end_definition
#+begin_definition
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]]):
\begin{equation} \begin{equation}
L(s) = G(s) \cdot K(s) \label{eq:loop_gain} L(s) = G(s) \cdot K(s) \label{eq:loop_gain}
\end{equation} \end{equation}
@ -336,11 +367,7 @@ The *Loop Gain* $L(s)$ usually refers to as the product of the controller and th
[[file:figs/open_loop_shaping.png]] [[file:figs/open_loop_shaping.png]]
#+end_definition #+end_definition
#+begin_definition This synthesis method is widely used as many characteristics of the closed-loop system depend on the shape of the open loop gain $L(s)$ such as:
*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 - *Performance*: $L$ large
- *Good disturbance rejection*: $L$ large - *Good disturbance rejection*: $L$ large
- *Limitation of measurement noise on plant output*: $L$ small - *Limitation of measurement noise on plant output*: $L$ small
@ -350,9 +377,52 @@ This synthesis method is widely used as many characteristics of the closed-loop
The Open Loop shape is usually done manually has the loop gain $L(s)$ depends linearly on $K(s)$ eqref:eq:loop_gain. 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. $K(s)$ 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]]).
#+begin_src latex :file open_loop_shaping_shape.pdf
\begin{tikzpicture}
% Phase Axis
\draw[->] (-0.3, -0.5) -- ++(8, 0) node[above]{$\omega$}; \draw[<-] (0, 0)
node[left]{$\angle L(j\omega)$} -- ++(0, -2.3);
% Gain Axis
\draw[->] (-0.3, 2) -- ++(8, 0) node[above]{$\omega$}; \draw[->] (0, 0.5) --
++(0, 3) node[left]{$\left|L(j\omega)\right|$};
% Gain Slopes
\draw[shift={(0,2)}] (0.5, 1.25) -- node[midway, above]{$-2$} (2, 0.5) --
node[midway, above]{$-1$} (6, -0.5) -- node[midway, below left]{$-2$} (7.5,
-1.25);
% Forbiden region
\path[shift={(0,1.8)}, fill=red!50!white] (0.5, 1.25) -- (2, 0.5) -| coordinate[near start](lfshaping) cycle;
\path[shift={(0,2.2)}, fill=red!50!white] (6, -0.5) -- (7.5, -1.25) |- coordinate[near end](hfshaping) cycle;
\draw[<-] (lfshaping) -- ++(0, -0.8) node[below, align=center]{Reference\\Tracking};
\draw[<-] (hfshaping) -- ++(0, 0.8) node[above, align=center]{Noise\\Rejection};
% Crossover frequency
\node[below] (wc) at (4,2){$\omega_c$};
\draw[<-] (wc.south) -- ++(0, -0.4) node[below, align=center]{Bandwidth};
% Phase
\draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4, -1.25)[out=0, in=-180] to
(6, -2) -- (7.5, -2); \draw[] (0.5, -2) -- (2, -2)[out=0, in=-180] to (4,
-1.25)[out=0, in=-180] to (6, -2) -- (7.5, -2);
% Phase Margin
\draw[->, dashed] (4, -2) -- (4, -1.25) node[above]{Phase Margin};
\draw[dashed] (0, -2) node[left]{$-\pi$} -- (7.5, -2);
\end{tikzpicture}
#+end_src
#+name: fig:open_loop_shaping_shape
#+caption: Typical Wanted Shape for the Loop Gain $L(s)$
#+RESULTS:
[[file:figs/open_loop_shaping_shape.png]]
** Example of Open Loop Shaping ** Example of Open Loop Shaping
<<sec:loop_shaping_example>>
#+begin_exampl #+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: Let's take our example system and try to apply the Open-Loop shaping strategy to design a controller that fulfils the following specifications:
@ -421,23 +491,24 @@ And we can verify that we have the wanted stability margins:
#+end_src #+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Manual Method'}, ' %.1f '); data2orgtable([Gm; Pm; Wc/2/pi], {'Gain Margin $> 3$ [dB]', 'Phase Margin $> 30$ [deg]', 'Crossover $\approx 10$ [Hz]'}, {'Requirements', 'Manual Method'}, ' %.1f ');
#+end_src #+end_src
#+RESULTS: #+RESULTS:
| | Manual Method | | Requirements | Manual Method |
|-----------------------------+---------------| |-----------------------------+---------------|
| Gain Margin $> 3$ [dB] | 3.1 | | Gain Margin $> 3$ [dB] | 3.1 |
| Phase Margin $> 30$ [deg] | 35.4 | | Phase Margin $> 30$ [deg] | 35.4 |
| Crossover $\approx 10$ [Hz] | 10.1 | | Crossover $\approx 10$ [Hz] | 10.1 |
** $\mathcal{H}_\infty$ Loop Shaping Synthesis ** $\mathcal{H}_\infty$ Loop Shaping Synthesis
<<sec:h_infinity_open_loop_shaping>>
The Open Loop Shaping synthesis can be performed using the $\mathcal{H}_\infty$ 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. Even though we will not go into details, we will provide one example.
Using Matlab, the $\mathcal{H}_\infty$ synthesis of a controller based on the wanted open loop shape can be performed using the =loopsyn= command: Using Matlab, the $\mathcal{H}_\infty$ Loop Shaping Synthesis can be performed using the =loopsyn= command:
#+begin_src matlab :eval no #+begin_src matlab :eval no
K = loopsyn(G, Gd); K = loopsyn(G, Gd);
#+end_src #+end_src
@ -451,6 +522,7 @@ where:
#+end_seealso #+end_seealso
** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis ** Example of the $\mathcal{H}_\infty$ Loop Shaping Synthesis
<<sec:h_infinity_open_loop_shaping_example>>
Let's reuse the previous plant. Let's reuse the previous plant.
@ -465,7 +537,7 @@ Translate the specification into the wanted shape of the open loop gain.
(1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead (1 + s/(2*pi*10/sqrt(3)))/(1 + s/(2*pi*10*sqrt(3))); % Lead
#+end_src #+end_src
The $\mathcal{H}_\infty$ optimal open loop shaping is performed using the =loopsyn= command: The $\mathcal{H}_\infty$ optimal open loop shaping synthesis is performed using the =loopsyn= command:
#+begin_src matlab #+begin_src matlab
[K, ~, GAM] = loopsyn(G, Lw); [K, ~, GAM] = loopsyn(G, Lw);
#+end_src #+end_src
@ -575,8 +647,11 @@ Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ con
| Phase Margin $> 30$ [deg] | 35.4 | 54.7 | | Phase Margin $> 30$ [deg] | 35.4 | 54.7 |
| Crossover $\approx 10$ [Hz] | 10.1 | 9.9 | | Crossover $\approx 10$ [Hz] | 10.1 | 9.9 |
* First Step in the $\mathcal{H}_\infty$ world * First Steps in the $\mathcal{H}_\infty$ world
<<sec:h_infinity_introduction>>
** The $\mathcal{H}_\infty$ Norm ** The $\mathcal{H}_\infty$ Norm
<<sec:h_infinity_norm>>
#+begin_definition #+begin_definition
The $\mathcal{H}_\infty$ norm is defined as the peak of the maximum singular value of the frequency response The $\mathcal{H}_\infty$ norm is defined as the peak of the maximum singular value of the frequency response
@ -591,7 +666,7 @@ Let's now compare the obtained stability margins of the $\mathcal{H}_\infty$ con
#+end_definition #+end_definition
#+begin_exampl #+begin_exampl
And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function: Let's compute the $\mathcal{H}_\infty$ norm of our test plant $G(s)$ using the =hinfnorm= function:
#+begin_src matlab :results value replace #+begin_src matlab :results value replace
hinfnorm(G) hinfnorm(G)
#+end_src #+end_src
@ -599,8 +674,7 @@ And compute its $\mathcal{H}_\infty$ norm using the =hinfnorm= function:
#+RESULTS: #+RESULTS:
: 7.9216e-06 : 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]]. 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]].
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 #+begin_src matlab :exports none
freqs = logspace(0, 3, 1000); freqs = logspace(0, 3, 1000);
@ -626,23 +700,47 @@ The maximum value of the magnitude over all frequencies does correspond to the $
#+end_exampl #+end_exampl
** $\mathcal{H}_\infty$ Synthesis ** $\mathcal{H}_\infty$ Synthesis
<<sec:h_infinity_synthesis>>
*Optimization problem*: #+begin_definition
$\mathcal{H}_\infty$ 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. $\mathcal{H}_\infty$ 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.
#+end_definition
*Engineer work*: 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.
Then applying the $\mathcal{H}_\infty$ synthesis to a plant, the engineer work usually consists of the following steps
1. Write the problem as standard $\mathcal{H}_\infty$ problem 1. Write the problem as standard $\mathcal{H}_\infty$ problem
2. Translate the specifications as $\mathcal{H}_\infty$ norms 2. Translate the specifications as $\mathcal{H}_\infty$ norms of transfer functions
3. Make the synthesis and analyze the obtain controller 3. Make the synthesis and analyze the obtain controller
4. Reduce the order of the controller for implementation 4. Reduce the order of the controller for implementation
*Many ways to use the $\mathcal{H}_\infty$ Synthesis*:
- Traditional $\mathcal{H}_\infty$ Synthesis Note that there are many ways to use the $\mathcal{H}_\infty$ Synthesis:
- Mixed Sensitivity Loop Shaping - Traditional $\mathcal{H}_\infty$ Synthesis (=hinfsyn= [[https://www.mathworks.com/help/robust/ref/hinfsyn.html][doc]])
- Fixed-Structure $\mathcal{H}_\infty$ Synthesis - Open Loop Shaping $\mathcal{H}_\infty$ Synthesis (=loopsyn= [[https://www.mathworks.com/help/robust/ref/loopsyn.html][doc]])
- Mixed Sensitivity Loop Shaping (=mixsyn= [[https://www.mathworks.com/help/robust/ref/lti.mixsyn.html][doc]])
- Fixed-Structure $\mathcal{H}_\infty$ Synthesis (=hinfstruct= [[https://www.mathworks.com/help/robust/ref/lti.hinfstruct.html][doc]])
- Signal Based $\mathcal{H}_\infty$ Synthesis - Signal Based $\mathcal{H}_\infty$ Synthesis
** The Generalized Plant ** The Generalized Plant
<<sec:generalized_plant>>
The first step when applying the $\mathcal{H}_\infty$ synthesis is usually to write the problem as a standard $\mathcal{H}_\infty$ problem.
This consist of deriving the *Generalized Plant* for the current problem.
It makes things much easier for the following steps.
The generalized plant, usually noted $P(s)$, is shown in Figure [[fig:general_plant]].
It has two inputs and two outputs (both could contains many signals).
The meaning of the inputs and outputs are summarized in Table [[tab:notation_general]].
Note that this generalized plant is as its name implies, quite /general/.
It can indeed represent feedback as well as feedforward control architectures.
\begin{equation}
\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}
#+begin_src latex :file general_plant.pdf #+begin_src latex :file general_plant.pdf
\begin{tikzpicture} \begin{tikzpicture}
\node[block={2.0cm}{2.0cm}] (P) {$P$}; \node[block={2.0cm}{2.0cm}] (P) {$P$};
@ -663,9 +761,12 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src
#+name: fig:general_plant
#+caption: Inputs and Outputs of the generalized Plant
#+RESULTS: #+RESULTS:
[[file:figs/general_plant.png]] [[file:figs/general_plant.png]]
#+begin_important
#+name: tab:notation_general #+name: tab:notation_general
#+caption: Notations for the general configuration #+caption: Notations for the general configuration
| Notation | Meaning | | Notation | Meaning |
@ -675,50 +776,68 @@ $\mathcal{H}_\infty$ synthesis is a method that uses an *algorithm* (LMI optimiz
| $z$ | Exogenous outputs: signals to be minimized | | $z$ | Exogenous outputs: signals to be minimized |
| $v$ | Controller inputs: measurements | | $v$ | Controller inputs: measurements |
| $u$ | Control signals | | $u$ | Control signals |
#+end_important
\begin{equation} ** The General Synthesis Problem Formulation
\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} <<sec:h_infinity_general_synthesis>>
\end{equation}
** From a Classical Feedback Architecture to a Generalized Plant Once the generalized plant is obtained, the $\mathcal{H}_\infty$ synthesis problem can be stated as follows:
#+begin_src latex :file classical_feedback.pdf #+begin_important
- $\mathcal{H}_\infty$ Synthesis applied on the generalized plant ::
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.
After $K$ is found, the system is /robustified/ by adjusting the response around the unity gain frequency to increase stability margins.
#+end_important
#+begin_src latex :file general_control_names.pdf
\begin{tikzpicture} \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] (addu){};
\node[block, right=0.8 of addu] (G){$G(s)$};
\draw[<-] (addfb.west) -- ++(-0.8, 0) node[above right]{$r$}; % Blocs
\draw[->] (addfb.east) -- (K.west) node[above left]{$\epsilon$}; \node[block={2.0cm}{2.0cm}] (P) {$P$};
\draw[->] (K.east) -- (addu.west) node[above left]{$u$}; \node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$};
\draw[->] (addu.east) -- (G.west);
\draw[<-] (addu.north) -- ++(0, 0.8) node[below right]{$d$}; % Input and outputs coordinates
\draw[->] (G.east) -- ++(1.2, 0); \coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$);
\draw[->] ($(G.east) + (0.6, 0)$) node[branch]{} node[above]{$y$} -- ++(0, -0.8) -| (addfb.south); \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{tikzpicture}
#+end_src #+end_src
#+name: fig:classical_feedback #+name: fig:general_control_names
#+caption: Classical Feedback Architecture #+caption: General Control Configuration
#+RESULTS: #+RESULTS:
[[file:figs/classical_feedback.png]] [[file:figs/general_control_names.png]]
#+name: table:notation_conventional Note that the closed-loop transfer function from $w$ to $z$ is:
#+caption: Notations for the Classical Feedback Architecture \begin{equation}
| Notation | Meaning | \frac{z}{w} = P_{11} + P_{12} K \big( I - P_{22} K \big)^{-1} P_{21} \triangleq F_l(P, K)
|------------+-------------------| \end{equation}
| $G$ | Plant model |
| $K$ | Controller |
| $r$ | Reference inputs |
| $y$ | Plant outputs |
| $u$ | Control signals |
| $d$ | Input Disturbance |
| $\epsilon$ | Tracking Error |
The procedure is: 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]]):
1. define signals of the generalized plant #+begin_src matlab :eval no
2. Remove $K$ and rearrange the inputs and outputs 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_src latex :file classical_feedback_tracking.pdf
\begin{tikzpicture} \begin{tikzpicture}
@ -765,12 +884,13 @@ The procedure is:
#+end_src #+end_src
#+begin_exercice #+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 #+name: fig:classical_feedback_tracking
#+caption: Classical Feedback Control Architecture (Tracking) #+caption: Classical Feedback Control Architecture (Tracking)
[[file:figs/classical_feedback_tracking.png]] [[file:figs/classical_feedback_tracking.png]]
#+HTML: <details><summary>Hint</summary>
First, define the signals of the generalized plant: First, define the signals of the generalized plant:
- Exogenous inputs: $w = r$ - Exogenous inputs: $w = r$
- Signals to be minimized: $z_1 = \epsilon$, $z_2 = u$ - Signals to be minimized: $z_1 = \epsilon$, $z_2 = u$
@ -778,76 +898,550 @@ The procedure is:
- Control inputs: $u$ - Control inputs: $u$
Then, Remove $K$ and rearrange the inputs and outputs. 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 #+name: fig:mixed_sensitivity_ref_tracking
#+caption: Generalized plant of the Classical Feedback Control Architecture (Tracking) #+caption: Generalized plant of the Classical Feedback Control Architecture (Tracking)
[[file:figs/mixed_sensitivity_ref_tracking.png]] [[file:figs/mixed_sensitivity_ref_tracking.png]]
#+HTML: </details>
#+end_exercice
Using Matlab, the generalized plant can be defined as follows: Using Matlab, the generalized plant can be defined as follows:
#+begin_src matlab :tangle no :eval no #+begin_src matlab :tangle no :eval no
P = [1 -G; P = [1 -G;
0 1; 0 1;
1 -G] 1 -G]
#+end_src P.InputName = {'w', 'u'};
#+end_exercice P.OutputName = {'e', 'u', 'v'};
** 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 #+end_src
#+name: fig:general_control_names
#+caption: General Control Configuration
#+RESULTS:
[[file:figs/general_control_names.png]]
* Modern Interpretation of the Control Specifications * Modern Interpretation of the Control Specifications
** Introduction <<sec:modern_interpretation_specification>>
- *Reference tracking* Overshoot, Static error, Setling time ** Introduction
- $S(s) = T_{r \rightarrow \epsilon}$ 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* - *Disturbances rejection*
- From $d$ to $y$
- $G(s) S(s) = T_{d \rightarrow \epsilon}$ - $G(s) S(s) = T_{d \rightarrow \epsilon}$
- *Measurement noise filtering* - *Measurement noise filtering*
- From $n$ to $y$
- $T(s) = T_{n \rightarrow \epsilon}$ - $T(s) = T_{n \rightarrow \epsilon}$
- *Small command amplitude* - *Small command amplitude*
- From $n, r, d$ to $u$
- $K(s) S(s) = T_{r \rightarrow u}$ - $K(s) S(s) = T_{r \rightarrow u}$
- *Stability* - *Stability*
- $S(s)$, $T(s)$, $K(s)S(s)$, $G(s)S(s)$ - $S(s)$, $T(s)$, $K(s)S(s)$, $G(s)S(s)$
- *Robustness to plant uncertainty* (stability margins) - *Robustness to plant uncertainty* (stability margins)
- *Controller implementation* - *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 * Resources

20
matlab/generateWeight.m Normal file
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@ -0,0 +1,20 @@
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