Update content - 2026-10-08

content/article/stoev17_tensor_method_mimo_decoup_contr.md

@@ -30,7 +30,7 @@ Because their are strong non-intuitive aspect of MIMO loop-shaping, the followin
 -   **[Interaction Analysis]({{< relref "interaction_analysis.md" >}})**.
     The goal is to identify two sided interactions in the plant dynamics.
     If there is no two sided interaction, then feedback design becomes a standard multi-loop SISO design problem.
-    Two measured of the plant interaction are [Relative Gain Array]({{< relref "relative_gain_array.md" >}}) and [Structured Singular Value]({{< relref "structured_singular_value.md" >}}).
+    Two measured of the plant interaction are Relative Gain Array and Structured Singular Value.
 -   **Decoupling transformations**.
     To reduce interaction, one may redefine the input and output of the plant using a decoupling transformation.
     For motion systems, most transformations are found on the basis of **kinematic model**.

content/inbook/albertos04_decen_decoup_contr.md

@@ -48,7 +48,7 @@ The multi-loop control may not work in strongly coupled systems.
 Therefore, a methodology the access the degree of interaction between the loops is needed.
 
 
-### [Relative Gain Array]({{< relref "relative_gain_array.md" >}}) {#relative-gain-array--relative-gain-array-dot-md}
+### Relative Gain Array {#relative-gain-array}
 
 The Relative Gain Array (RGA) \\(\Lambda(s)\\) is defined as:
 

content/zettels/continuous_transfer_functions.md

@@ -6,10 +6,244 @@ category = "Signals and Systems"
 +++
 
 Tags
-:
+: [Discrete Transfer Functions]({{< relref "discrete_transfer_functions.md" >}})
+
+Classical transfer functions in the Laplace domain, with the meaning of their parameters.
+In all the examples, the frequency of interest is \\(\omega\_0 = 2\pi\\) rad/s (1 Hz) and is shown with a dashed line.
+To obtain the equivalent digital filter, see [Discrete Transfer Functions]({{< relref "discrete_transfer_functions.md" >}}).
 
 
-## Typical Transfer Functions {#typical-transfer-functions}
+## First Order Low Pass Filter {#first-order-low-pass-filter}
+
+\begin{equation}
+  H(s) = \frac{1}{1 + s/\omega\_0}
+\end{equation}
+
+Parameters:
+
+-   \\(\omega\_0\\): cut-off frequency in [rad/s]
+
+Characteristics:
+
+-   Low frequency gain of \\(1\\)
+-   At \\(\omega\_0\\): gain of \\(1/\sqrt{2}\\) (-3 dB) and phase of -45 deg
+-   High frequency roll-off of -20 dB/dec, phase tends to -90 deg
+
+<!--listend-->
+
+```matlab
+w0 = 2*pi; % Cut-off frequency [rad/s]
+
+H = 1/(1 + s/w0);
+```
+
+<a id="figure--fig:filter-low-pass-first-order"></a>
+
+{{< figure src="/ox-hugo/filter_low_pass_first_order.png" caption="<span class='figure-number'>Figure 1: </span>First order low pass filter, effect of \\(\omega\_0\\) (dashed: \\(\omega\_0 = 2\pi\\) rad/s)" >}}
+
+
+## First Order High Pass Filter {#first-order-high-pass-filter}
+
+\begin{equation}
+  H(s) = \frac{s/\omega\_0}{1 + s/\omega\_0}
+\end{equation}
+
+Parameters:
+
+-   \\(\omega\_0\\): cut-off frequency in [rad/s]
+
+Characteristics:
+
+-   High frequency gain of \\(1\\)
+-   At \\(\omega\_0\\): gain of \\(1/\sqrt{2}\\) (-3 dB) and phase of +45 deg
+-   Low frequency slope of +20 dB/dec, phase tends to +90 deg
+
+<!--listend-->
+
+```matlab
+w0 = 2*pi; % Cut-off frequency [rad/s]
+
+H = (s/w0)/(1 + s/w0);
+```
+
+<a id="figure--fig:filter-high-pass-first-order"></a>
+
+{{< figure src="/ox-hugo/filter_high_pass_first_order.png" caption="<span class='figure-number'>Figure 2: </span>First order high pass filter, effect of \\(\omega\_0\\) (dashed: \\(\omega\_0 = 2\pi\\) rad/s)" >}}
+
+
+## Second Order Low Pass Filter {#second-order-low-pass-filter}
+
+\begin{equation}
+  H(s) = \frac{1}{1 + \frac{2 \zeta}{\omega\_0} s + \frac{s^2}{\omega\_0^2}}
+\end{equation}
+
+Parameters:
+
+-   \\(\omega\_0\\): natural frequency in [rad/s]
+-   \\(\zeta\\): damping ratio
+
+Characteristics:
+
+-   Low frequency gain of \\(1\\)
+-   High frequency roll-off of -40 dB/dec, phase tends to -180 deg
+-   Phase of -90 deg at \\(\omega\_0\\), whatever the damping
+-   Gain at \\(\omega\_0\\) equal to \\(1/(2\zeta)\\)
+-   For \\(\zeta < 1/\sqrt{2}\\) there is a resonance peak, for \\(\zeta \geq 1/\sqrt{2}\\) the magnitude is monotonically decreasing
+-   Smaller \\(\zeta\\) gives a sharper phase transition around \\(\omega\_0\\)
+
+<!--listend-->
+
+```matlab
+w0 = 2*pi; % Natural frequency [rad/s]
+zeta = 0.1;  % Damping ratio
+
+H = 1/(1 + 2*zeta/w0*s + s^2/w0^2);
+```
+
+<a id="figure--fig:filter-low-pass-second-order"></a>
+
+{{< figure src="/ox-hugo/filter_low_pass_second_order.png" caption="<span class='figure-number'>Figure 3: </span>Second order low pass filter, effect of the damping ratio \\(\zeta\\)" >}}
+
+
+## Second Order High Pass Filter {#second-order-high-pass-filter}
+
+\begin{equation}
+  H(s) = \frac{s^2/\omega\_0^2}{1 + \frac{2 \zeta}{\omega\_0} s + \frac{s^2}{\omega\_0^2}}
+\end{equation}
+
+Parameters:
+
+-   \\(\omega\_0\\): natural frequency in [rad/s]
+-   \\(\zeta\\): damping ratio
+
+Characteristics:
+
+-   High frequency gain of \\(1\\)
+-   Low frequency slope of +40 dB/dec, phase tends to +180 deg
+-   Phase of +90 deg at \\(\omega\_0\\), whatever the damping
+-   Gain at \\(\omega\_0\\) equal to \\(1/(2\zeta)\\), resonance peak for \\(\zeta < 1/\sqrt{2}\\)
+
+<!--listend-->
+
+```matlab
+w0 = 2*pi; % Natural frequency [rad/s]
+zeta = 0.1;  % Damping ratio
+
+H = (s^2/w0^2)/(1 + 2*zeta/w0*s + s^2/w0^2);
+```
+
+<a id="figure--fig:filter-high-pass-second-order"></a>
+
+{{< figure src="/ox-hugo/filter_high_pass_second_order.png" caption="<span class='figure-number'>Figure 4: </span>Second order high pass filter, effect of the damping ratio \\(\zeta\\)" >}}
+
+
+## Notch Filter {#notch-filter}
+
+\begin{equation}
+  H(s) = \frac{s^2 + 2 g\_c \zeta \omega\_n s + \omega\_n^2}{s^2 + 2 \zeta \omega\_n s + \omega\_n^2}
+\end{equation}
+
+Parameters:
+
+-   \\(\omega\_n\\): frequency of the notch in [rad/s]
+-   \\(g\_c\\): gain at the notch frequency (\\(g\_c < 1\\) gives a notch, \\(g\_c > 1\\) gives a peak)
+-   \\(\zeta\\): damping ratio, sets the width of the notch
+
+Characteristics:
+
+-   Unity gain at low and high frequency
+-   Gain of exactly \\(g\_c\\) at \\(\omega\_n\\), with a phase of 0 deg
+-   For \\(g\_c < 1\\): phase is negative below \\(\omega\_n\\) and positive above
+-   Larger \\(\zeta\\) gives a wider notch, and more phase lag/lead around it
+
+<!--listend-->
+
+```matlab
+wn = 2*pi; % Notch frequency [rad/s]
+gc = 0.02; % Gain at the notch frequency
+zeta = 0.2;  % Damping ratio (notch width)
+
+H = (s^2 + 2*gc*zeta*wn*s + wn^2)/(s^2 + 2*zeta*wn*s + wn^2);
+```
+
+<a id="figure--fig:filter-notch-gc"></a>
+
+{{< figure src="/ox-hugo/filter_notch_gc.png" caption="<span class='figure-number'>Figure 5: </span>Notch filter, effect of the gain at the notch frequency \\(g\_c\\)" >}}
+
+<a id="figure--fig:filter-notch-zeta"></a>
+
+{{< figure src="/ox-hugo/filter_notch_zeta.png" caption="<span class='figure-number'>Figure 6: </span>Notch filter, effect of the damping ratio \\(\zeta\\) (notch width)" >}}
+
+
+## Lead Filter {#lead-filter}
+
+A lead filter is usually used to add some phase lead at a particular frequency, at the expense of a reduction of the gain at low frequency and increase of the gain at high frequency.
+
+\begin{equation}
+  H(s) = \frac{1}{\sqrt{a}} \frac{1 + \frac{s}{\omega\_c/\sqrt{a}}}{1 + \frac{s}{\omega\_c \sqrt{a}}}, \quad a > 1
+\end{equation}
+
+Parameters:
+
+-   \\(\omega\_c\\): frequency at which the phase lead is maximum, in [rad/s]
+-   \\(a\\): sets the amount of phase lead, and the gain increase between low and high frequency
+
+Characteristics:
+
+-   Zero at \\(\omega\_c/\sqrt{a}\\) and pole at \\(\omega\_c \sqrt{a}\\), geometrically centered on \\(\omega\_c\\)
+-   Unity gain at \\(\omega\_c\\), low frequency gain of \\(1/\sqrt{a}\\), high frequency gain of \\(\sqrt{a}\\)
+-   Maximum phase lead, obtained at \\(\omega\_c\\), equal to (Figure [Figure 8](#figure--fig:filter-lead-effect-a-phase)):
+    \\[ \angle H(j\omega\_c) = \tan^{-1}(\sqrt{a}) - \tan^{-1}(1/\sqrt{a}) \\]
+
+<!--listend-->
+
+```matlab
+a  = 3;    % Amount of phase lead
+wc = 2*pi; % Frequency of maximum phase lead [rad/s]
+
+H = 1/sqrt(a)*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a)));
+```
+
+<a id="figure--fig:filter-lead"></a>
+
+{{< figure src="/ox-hugo/filter_lead.png" caption="<span class='figure-number'>Figure 7: </span>Lead filter, effect of the parameter \\(a\\)" >}}
+
+<a id="figure--fig:filter-lead-effect-a-phase"></a>
+
+{{< figure src="/ox-hugo/filter_lead_effect_a_phase.png" caption="<span class='figure-number'>Figure 8: </span>Maximum phase lead as a function of \\(a\\)" >}}
+
+
+## Lag Filter {#lag-filter}
+
+A lag filter is usually used to add low frequency gain, at the expense of some phase lag.
+
+\begin{equation}
+  H(s) = a \cdot \frac{1 + \frac{s}{\omega\_c \sqrt{a}}}{1 + \frac{s}{\omega\_c/\sqrt{a}}}, \quad a > 1
+\end{equation}
+
+Parameters:
+
+-   \\(\omega\_c\\): frequency at which the phase lag is maximum, in [rad/s]
+-   \\(a\\): sets the amount of phase lag, and the gain increase at low frequency (the low frequency gain is \\(a\\))
+
+Characteristics:
+
+-   Zero at \\(\omega\_c \sqrt{a}\\) and pole at \\(\omega\_c/\sqrt{a}\\)
+-   Low frequency gain of \\(a\\), high frequency gain of \\(1\\) (unchanged), gain of \\(\sqrt{a}\\) at \\(\omega\_c\\)
+-   It is used to increase the low frequency gain (e.g. instead of a pure integrator), \\(\omega\_c\\) should be kept well below the crossover frequency to limit the phase lag there
+
+<!--listend-->
+
+```matlab
+a  = 3;    % Amount of phase lag
+wc = 2*pi; % Frequency of maximum phase lag [rad/s]
+
+H = a*(1 + s/(wc*sqrt(a)))/(1 + s/(wc/sqrt(a)));
+```
+
+<a id="figure--fig:filter-lag"></a>
+
+{{< figure src="/ox-hugo/filter_lag.png" caption="<span class='figure-number'>Figure 9: </span>Lag filter, effect of the parameter \\(a\\)" >}}
 
 
 ## Bibliography {#bibliography}

content/zettels/decoupled_control.md

@@ -10,6 +10,88 @@ Tags
 : [Multivariable Control]({{< relref "multivariable_control.md" >}})
 
 
+## Decoupling Control of Parallel Kinematics Robots {#decoupling-control-of-parallel-kinematics-robots}
+
+Consider a rigid body (mass matrix \\(\bm{M}\\), stiffness matrix \\(\bm{K}\\)) suspended by actuators, with their forces \\(\bm{\tau}\\) and the change of length of the struts \\(\bm{\mathcal{L}}\\).
+The Jacobian matrix \\(\bm{J}\\) relates the pose \\(\bm{\mathcal{X}}\\) of the body to the strut lengths, \\(\bm{\mathcal{L}} = \bm{J} \bm{\mathcal{X}}\\), and the actuator forces to the wrench \\(\bm{\mathcal{F}}\\) applied on the body, \\(\bm{\mathcal{F}} = \bm{J}^{\intercal} \bm{\tau}\\).
+
+The transfer function in the actuator frame is:
+
+\begin{equation}
+  \bm{G}\_{\\{\mathcal{L}\\}}(s) = \frac{\bm{\mathcal{L}}}{\bm{\tau}} = \bm{J} \left( \bm{M} s^2 + \bm{C} s + \bm{K} \right)^{-1} \bm{J}^{\intercal}
+\end{equation}
+
+Depending on the frequency range, the dynamics is dominated by the stiffness (low frequency) or the inertia (high frequency).
+The best frame to decouple the system is not the same in the two cases.
+
+
+### Control at the Center Of Mass {#control-at-the-center-of-mass}
+
+For soft systems (for instance voice coil based), it makes sense to work at the Center of Mass (CoM).
+Above the "suspension modes", the dynamics is dominated by inertia.
+Therefore, the transfer function from forces and torques applied at the CoM to the translation and rotation of the suspended mass evaluated at the CoM is decoupled.
+It is equal to the inverse of the mass matrix (diagonal when evaluated at the CoM) times \\(1/s^2\\):
+
+\begin{equation}
+  \bm{G}\_{\\{CoM\\}}(s) = \frac{\bm{\mathcal{X}}\_{\\{CoM\\}}}{\bm{\mathcal{F}}\_{\\{CoM\\}}} \approx \frac{1}{s^2} \bm{M}^{-1}
+\end{equation}
+
+To work at the CoM, the plant is pre- and post-multiplied by the Jacobian matrices as shown in Figure [Figure 1](#figure--fig:detail-control-decoupling-control-jacobian):
+
+\begin{equation}
+  \bm{G}\_{\\{CoM\\}}(s) = \bm{J}\_{\\{CoM\\}}^{-1} \\, \bm{G}\_{\\{\mathcal{L}\\}}(s) \\, \bm{J}\_{\\{CoM\\}}^{-\intercal}
+\end{equation}
+
+<a id="figure--fig:detail-control-decoupling-control-jacobian"></a>
+
+{{< figure src="/ox-hugo/detail_control_decoupling_control_jacobian.png" caption="<span class='figure-number'>Figure 1: </span>Plant with inputs \\(\bm{\tau}\\) and outputs \\(\bm{\mathcal{L}}\\), and the Jacobian matrices used to work with forces and displacements at the CoM" >}}
+
+The animations below show a 3DoF suspended body above its suspension modes (1 Hz), when a force or a torque is applied at the CoM.
+The three actuator forces are computed with \\(\bm{\tau} = \bm{J}\_{\\{CoM\\}}^{-\intercal} \bm{\mathcal{F}}\_{\\{CoM\\}}\\) (the darker the actuator, the larger its force).
+
+**Force along \\(x\\) applied at the CoM**: the body only translates along \\(x\\).
+
+<a id="figure--fig:decoupled-control-com-Fx"></a>
+
+{{< figure src="/ox-hugo/decoupled_control_com_Fx.gif" caption="<span class='figure-number'>Figure 2: </span>Force along \\(x\\) applied at the CoM (1 Hz): pure translation" >}}
+
+**Torque about \\(z\\) applied at the CoM**: the body only rotates around its CoM.
+
+<a id="figure--fig:decoupled-control-com-Mz"></a>
+
+{{< figure src="/ox-hugo/decoupled_control_com_Mz.gif" caption="<span class='figure-number'>Figure 3: </span>Torque about \\(z\\) applied at the CoM (1 Hz): pure rotation" >}}
+
+
+### Control in the Actuator Frame {#control-in-the-actuator-frame}
+
+For stiff parallel systems, the dynamics from \\(\bm{\tau}\\) to \\(\bm{\mathcal{L}}\\) is well decoupled at low frequency (well below the first resonance frequency).
+At low frequency, the dynamics is dominated by the stiffness: with \\(\bm{K} = \bm{J}^{\intercal} \mathcal{K} \bm{J}\\), \\(\mathcal{k}\\) being the diagonal matrix of the actuator stiffnesses, the transfer function from \\(\bm{\tau}\\) to \\(\bm{\mathcal{L}}\\) is diagonal:
+
+\begin{equation}
+  \bm{G}\_{\\{\mathcal{L}\\}}(0) = \bm{J} \bm{K}^{-1} \bm{J}^{\intercal} = \mathcal{K}^{-1}
+\end{equation}
+
+Each actuator then only changes the length of its own strut.
+The control can be designed independently for each actuator, using the length of the strut as the sensor.
+
+Below the suspension modes, actuating one actuator moves the body, but the lengths of the two other struts do not change.
+Therefore, the plant in the frame of the actuators is decoupled.
+
+<a id="figure--fig:decoupled-control-low-freq-act-1"></a>
+
+{{< figure src="/ox-hugo/decoupled_control_low_freq_act_1.gif" caption="<span class='figure-number'>Figure 4: </span>Left actuator at 0.1 Hz (below the suspension modes): only the strut of the actuated direction changes length" >}}
+
+<a id="figure--fig:decoupled-control-low-freq-act-2"></a>
+
+{{< figure src="/ox-hugo/decoupled_control_low_freq_act_2.gif" caption="<span class='figure-number'>Figure 5: </span>Bottom left actuator at 0.1 Hz (below the suspension modes): only the strut of the actuated direction changes length" >}}
+
+At higher frequency, the inertia becomes important and the system is coupled: using only the first actuator also excites the other directions and the other struts change length.
+
+<a id="figure--fig:decoupled-control-high-freq-act-1"></a>
+
+{{< figure src="/ox-hugo/decoupled_control_high_freq_act_1.gif" caption="<span class='figure-number'>Figure 6: </span>Left actuator at 1 Hz (above the suspension modes): the system is coupled, the body translates and rotates" >}}
+
+
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">

content/zettels/interaction_analysis.md

@@ -11,8 +11,59 @@ Tags
 
 Two main ways to evaluate the interaction:
 
--   [Relative Gain Array]({{< relref "relative_gain_array.md" >}})
+-   Relative Gain Array
--   [Structured Singular Value]({{< relref "structured_singular_value.md" >}})
+-   Structured Singular Value
+
+
+## Relative Gain Array {#relative-gain-array}
+
+Imagine a 2x2 plant:
+\\[ G(s) = \begin{bmatrix} g\_{11}(s) & g\_{12}(s)\\\ g\_{21}(s) & g\_{22}(s) \end{bmatrix} \\]
+
+Now suppose a controller \\(k\_{1}(s)\\) is closed on the first loop.
+The new transfer function from \\(u\_2\\) to \\(y\_2\\) (while the first loop is closed) is :
+
+\begin{equation}
+y\_2 = g\_{22} \left[ 1 - \underbrace{\frac{g\_{21}g\_{12}}{g\_{11}g\_{22}}}\_{\phi} \underbrace{\frac{g\_{11} k\_{11}}{1  + g\_{11}k\_{11}}}\_{T\_{11}} \right]
+\end{equation}
+
+\\(\phi\\) is called the **interaction index**.
+\\(T\_{11}\\) is the complementary sensitivity of the first loop (equal to 1 in the bandwidth of the first controller).
+Therefore, we want \\(\phi \approx 0\\) to have no interaction.
+
+Similarly, the **relative gain** is defined as:
+\\[ \Lambda = \frac{1}{1 - \phi} \\]
+And we want \\(\Lambda \approx 1\\) to have no interaction.
+
+Note that the scaling of inputs or outputs of the MIMO plant has no effect on \\(\phi\\) or \\(\Lambda\\).
+
+The **relative gain array** is defined as:
+\\[ \Lambda(G) = G \star (G^{-1})^{T} \\]
+where \\(\star\\) means element wise multiplication.
+
+```matlab
+RGA = G.*pinv(G.')
+```
+
+
+## RGA-number {#rga-number}
+
+The RGA-number is a scalar measure of how far the RGA is from the identity matrix (i.e. from a plant without interaction):
+
+\begin{equation}
+  \text{RGA-number} = \\| \Lambda(G) - I \\|\_{\text{sum}} = \sum\_{i,j} \left| \Lambda\_{ij} - \delta\_{ij} \right|
+\end{equation}
+
+with \\(\delta\_{ij}\\) the Kronecker delta, and \\(\\|\cdot\\|\_{\text{sum}}\\) the sum of the absolute values of all the elements.
+
+The RGA-number should be small around the crossover frequency of the decentralized controller, as this is where the interaction matters most.
+
+```matlab
+RGA = G.*pinv(G.');
+RGA_number = sum(sum(abs(RGA - eye(size(RGA)))));
+```
+
+For a plant evaluated over frequency, this is computed at each frequency point of the frequency response.
 
 
 ## Bibliography {#bibliography}

content/zettels/model_predictive_control.md

@@ -1,16 +0,0 @@
-+++
-title = "Model Predictive Control"
-author = ["Dehaeze Thomas"]
-draft = false
-category = "Control Theory"
-subcategory = "Advanced Control"
-+++
-
-Tags
-:
-
-
-## Bibliography {#bibliography}
-
-<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
-</div>

content/zettels/test.md

@@ -465,7 +465,7 @@ Numbering can be continued by using `+n` option as shown below.
 ```
 <div class="src-block-caption">
   <span class="src-block-number"><a href="#code-snippet--lst:tikz-test-general-control-names">Code Snippet 4</a>:</span>
-  Tikz code that is used to generate <a href="#org4a89346">5</a>
+  Tikz code that is used to generate <a href="#org587cbdd">5</a>
 </div>
 
 <a id="figure--fig:test-general-control-names"></a>
@@ -632,9 +632,21 @@ It is approximately **12,742 km**
 
 ## Videos {#videos}
 
+
+### Youtube link {#youtube-link}
+
 {{< youtube SzA2YODtgK4 >}}
 
 
+### GIF generated from Python {#gif-generated-from-python}
+
+Animation of a 3DoF rigid body excited by the left actuator at 1 Hz, generated by the Python block below.
+
+<a id="figure--fig:coupling-high-freq-1Hz"></a>
+
+{{< figure src="/ox-hugo/coupling_high_freq_1Hz.gif" caption="<span class='figure-number'>Figure 7: </span>GIF generated from Python" >}}
+
+
 ## Bibliography {#bibliography}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">