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Thomas Dehaeze
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content/zettels/norms.md
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Backlinks:
- [Multivariable Control]({{< relref "multivariable_control" >}})
Tags
:
\\[ \SI{1}{\meter\per\second} \\]
Resources:
- ([Skogestad and Postlethwaite 2007](#org44811fa))
- ([Toivonen 2002](#orgfbd38d8))
- ([Zhang 2011](#orgc3b14cc))
- ([Skogestad and Postlethwaite 2007](#org4fdbcff))
- ([Toivonen 2002](#org4782daf))
- ([Zhang 2011](#org9b9c22a))
## Definition {#definition}
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A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real number, denoted \\(\\|e\\|\\), that satisfies the following properties:
@@ -47,7 +45,7 @@ A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real nu
## Matrix Norms {#matrix-norms}
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A norm on a matrix \\(\\|A\\|\\) is a matrix norm if, in addition to the four norm properties, it also satisfies the multiplicative property:
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### \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty--norm}
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Consider a proper linear stable system \\(G(s)\\).
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### \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2--norm}
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Consider a strictly proper system \\(G(s)\\).
@@ -178,17 +176,17 @@ In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as fo
The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}).
As explained in ([Monkhorst 2004](#orgc4a9d92)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
As explained in ([Monkhorst 2004](#orgb605c51)), 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="orgc4a9d92"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
<a id="orgb605c51"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
<a id="org44811fa"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
<a id="org4fdbcff"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
<a id="orgfbd38d8"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
<a id="org4782daf"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
<a id="orgc3b14cc"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
<a id="org9b9c22a"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.