Update Content - 2022-03-15

content/zettels/norms.md

@@ -1,6 +1,6 @@
 +++
 title = "Systems and Signals Norms"
-author = ["Thomas Dehaeze"]
+author = ["Dehaeze Thomas"]
 draft = false
 +++
 
@@ -11,15 +11,14 @@ Tags
 
 Resources:
 
--   ([Skogestad and Postlethwaite 2007](#orga6846b3))
--   ([Toivonen 2002](#org300cd1c))
--   ([Zhang 2011](#org037ea69))
+-   (NO_ITEM_DATA:skogestad05_multiv_feedb_contr)
+-   (<a href="#citeproc_bib_item_2">Toivonen 2002</a>)
+-   (<a href="#citeproc_bib_item_3">Zhang 2011</a>)
 
 
 ## Definition {#definition}
 
 <div class="definition">
-  <div></div>
 
 A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real number, denoted \\(\\|e\\|\\), that satisfies the following properties:
 
@@ -46,7 +45,6 @@ A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real nu
 ## Matrix Norms {#matrix-norms}
 
 <div class="definition">
-  <div></div>
 
 A norm on a matrix \\(\\|A\\|\\) is a matrix norm if, in addition to the four norm properties, it also satisfies the multiplicative property:
 \\[ \\|AB\\| \le \\|A\\| \cdot \\|B\\| \\]
@@ -137,10 +135,9 @@ We now consider which system norms result from the definition of input classes a
 ## System Norms {#system-norms}
 
 
-### \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty--norm}
+### \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty-norm}
 
 <div class="exampl">
-  <div></div>
 
 Consider a proper linear stable system \\(G(s)\\).
 The \\(\mathcal{H}\_\infty\\) norm is the peak value of its maximum singular value:
@@ -155,16 +152,15 @@ In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as fo
     \\[ \\|G(s)\\|\_\infty = \max\_{d(t)} \frac{\\|e(t)\\|\_2 \neq 0}{\\|d(t)\\|\_2} = \max\_{\\|d(t)\\|\_2 = 1} \\|e(t)\\|\_2 \\]
 
 
-### \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2--norm}
+### \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2-norm}
 
 <div class="exampl">
-  <div></div>
 
 Consider a strictly proper system \\(G(s)\\).
 The \\(\mathcal{H}\_2\\) norm is:
 
 \begin{align\*}
-\\|G(s)\\|\_2 &\triangleq \sqrt{\frac{1}{2\pi} \int\_{-\infty}^{\infty} \text{tr}\left(G(j\omega)^HG(j\omega)\right) d\omega} \\\\\\
+\\|G(s)\\|\_2 &\triangleq \sqrt{\frac{1}{2\pi} \int\_{-\infty}^{\infty} \text{tr}\left(G(j\omega)^HG(j\omega)\right) d\omega} \\\\
            &= \sqrt{\frac{1}{2\pi} \int\_{-\infty}^{\infty} \sum\_i {\sigma\_i}^2(G(j\omega)) d\omega}
 \end{align\*}
 
@@ -174,20 +170,18 @@ In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as fo
 
 -   it is a measure of the expected RMS value of the output to white noise excitation
 
-The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}).
+The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}}).
 
-As explained in ([Monkhorst 2004](#org16354b5)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
+As explained in (<a href="#citeproc_bib_item_1">Monkhorst 2004</a>), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
 
 > The squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
 
 
-
 ## Bibliography {#bibliography}
 
-<a id="org16354b5"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
-
-<a id="orga6846b3"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
-
-<a id="org300cd1c"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
-
-<a id="org037ea69"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+  <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_2"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.</div>
+  <div class="csl-entry"><a id="citeproc_bib_item_3"></a>Zhang, Weidong. 2011. <i>Quantitative Process Control Theory</i>. CRC Press.</div>
+  <div class="csl-entry">NO_ITEM_DATA:skogestad05_multiv_feedb_contr</div>
+</div>