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'''''H''<sub>∞</sub>''' (i.e. '''"''H''-infinity"''') '''methods''' are used in [[control theory]] to synthesize controllers achieving robust performance or stabilization.  To use ''H''<sub>∞</sub> methods, a control designer expresses the control problem as a [[mathematical optimization]] problem and then finds the controller that solves this.  ''H''<sub>∞</sub> techniques have the advantage over classical control techniques in that they are readily applicable to problems involving multivariable systems with cross-coupling between channels; disadvantages of ''H''<sub>∞</sub> techniques include the level of mathematical understanding needed to apply them successfully and the need for a reasonably good model of the system to be controlled.  Problem formulation is important, since any controller synthesized will only be 'optimal' in the formulated sense: optimizing the wrong thing often makes things worse rather than better.  Also, non-linear constraints such as saturation are generally not well-handled.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


The term '''''H''<sub>∞</sub>''' comes from the name of the mathematical space over which the optimization takes place: ''H''<sub>∞</sub> is the space of [[matrix (mathematics)|matrix]]-valued functions that are [[analytic function|analytic]] and bounded in the open right-half of the [[complex plane]] defined by Re(''s'')&nbsp;>&nbsp;0; the ''H''<sub>∞</sub> norm is the maximum singular value of the function over that space.  (This can be interpreted as a maximum gain in any direction and at any frequency; for [[Single-Input and Single-Output|SISO]] systems, this is effectively the maximum magnitude of the frequency response.) ''H''<sub>∞</sub> techniques can be used to minimize the closed loop impact of a perturbation: depending on the problem formulation, the impact will either be measured in terms of stabilization or performance.
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Simultaneously optimizing robust performance and robust stabilization is difficult.  One method that comes close to achieving this is [[H-infinity loop-shaping|''H''<sub>∞</sub> loop-shaping]], which allows the control designer to apply classical loop-shaping concepts to the multivariable frequency response to get good robust performance, and then optimizes the response near the system bandwidth to achieve good robust stabilization.
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Commercial software is available to support ''H''<sub>∞</sub> controller synthesis.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


== Problem formulation ==
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


First, the process has to be represented according to the following standard configuration:
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


[[Image:H-infty plant representation.png]]
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


The plant ''P'' has two inputs, the exogenous input ''w'', that includes reference signal and disturbances, and the manipulated variables ''u''. There are two outputs, the error signals ''z'' that we want to minimize, and the measured variables ''v'', that we use to control the system. ''v'' is used in ''K'' to calculate the manipulated variable ''u''. Notice that all these are generally [[vector (geometry)|vectors]], whereas '''P''' and '''K''' are [[matrix (mathematics)|matrices]].
==Demos==


In formulae, the system is:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


:<math>\begin{bmatrix} z\\ v \end{bmatrix} = \mathbf{P}(s)\, \begin{bmatrix} w\\ u\end{bmatrix} = \begin{bmatrix}P_{11}(s) & P_{12}(s)\\P_{21}(s) & P_{22}(s)\end{bmatrix} \, \begin{bmatrix} w\\ u\end{bmatrix}</math>


:<math>u = \mathbf{K}(s) \, v</math>
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


It is therefore possible to express the dependency of ''z'' on ''w'' as:
==Test pages ==


:<math>z=F_\ell(\mathbf{P},\mathbf{K})\,w</math>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
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*[[Help:Formula]]


Called the ''lower linear fractional transformation'',  <math>F_\ell</math> is defined (the subscript comes from ''lower''):
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>F_\ell(\mathbf{P},\mathbf{K}) = P_{11} + P_{12}\,\mathbf{K}\,(I-P_{22}\,\mathbf{K})^{-1}\,P_{21}</math>
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
Therefore, the objective of <math>\mathcal{H}_\infty</math> control design is to find a controller <math>\mathbf{K}</math> such that <math>F_\ell(\mathbf{P},\mathbf{K})</math> is minimised according to the <math>\mathcal{H}_\infty</math> norm. The same definition applies to <math>\mathcal{H}_2</math> control design. The infinity norm of the transfer function matrix <math>F_\ell(\mathbf{P},\mathbf{K})</math> is defined as:
 
:<math>||F_\ell(\mathbf{P},\mathbf{K})||_\infty = \sup_\omega \bar{\sigma}(F_\ell(\mathbf{P},\mathbf{K})(j\omega))</math>
 
where <math>\bar{\sigma}</math> is the maximum [[singular value]] of the matrix <math>F_\ell(\mathbf{P},\mathbf{K})(j\omega)</math>.
 
The achievable ''H''<sub>∞</sub> norm of the closed loop system is mainly given through the matrix ''D''<sub>11</sub> (when the system ''P'' is given in the form (''A'', ''B''<sub>1</sub>, ''B''<sub>2</sub>, ''C''<sub>1</sub>, ''C''<sub>2</sub>, ''D''<sub>11</sub>, ''D''<sub>12</sub>, ''D''<sub>22</sub>, ''D''<sub>21</sub>)). There are several ways to come to an ''H''<sub>∞</sub> controller:
* A [[Youla-Kucera parametrization]] of the closed loop often leads to very high-order controller.
* [[Riccati equation|Riccati]]-based approaches solve 2 [[Riccati equation]]s to find the controller, but require several simplifying assumptions.
* An optimization-based reformulation of the Riccati equation uses [[Linear matrix inequality|Linear matrix inequalities]] and requires fewer assumptions.
 
== See also ==
*[[Hardy space]]
*[[H square]]
*[[H-infinity loop-shaping]]
*[[Linear-quadratic-Gaussian control]] (LQG)
 
== References ==
{{refbegin}}
* {{citation|last1=Skogestad|first1=Sigurd|last2=Postlethwaite|first2=Ian|title=Multivariable Feedback Control: Analysis and Design|year=1996|publisher=Wiley|isbn=0-471-94277-4}}
* {{citation|last1=Skogestad|first1=Sigurd|last2=Postlethwaite|first2=Ian|title=Multivariable Feedback Control: Analysis and Design|year=2005|publisher=Wiley|isbn=0-470-01167-X|edition=2nd|url=http://www.nt.ntnu.no/users/skoge/book/}}
* {{citation|last=Simon|first=Dan|title=Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches|year=2006|publisher=Wiley|url=http://academic.csuohio.edu/simond/estimation/}}
* {{citation|last1=Green|first1=M.|last2=Limebeer|first2=D.|title=Linear Robust Control|year=1995|publisher=Prentice Hall|url=http://www3.imperial.ac.uk/portal/pls/portallive/docs/1/7287085.PDF}}
* {{Citation
| author = V. Barbu and S. S. Sritharan
| year = 1998
| title = H-infinity Control of Fluid Dynamics
| journal = Proceedings of the Royal Royal Society of London, Ser. A.
| volume = 545
|pages= 3009–3033
|url= http://www.nps.edu/Academics/Schools/GSEAS/SRI/R19.pdf
| postscript = .
}}
 
 
{{refend}}
 
[[Category:Control theory]]
[[Category:Hardy spaces]]
 
[[de:H-unendlich-Regelung]]
[[ar:تحكم إتش إنفينتي]]
[[fr:Hinfini]]
[[ja:H∞制御理論]]
[[pl:H-nieskończoność]]
[[ru:H∞-управление]]
[[uk:H∞-керування]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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