formulasearchengine - User contributions [en]

Schuler tuning

2013-08-27T10:58:41Z

24.91.233.200: Archived a copy of Schuler's paper in Webcitation

[[File:Erik Akkersdijk is solving a 3×3×3 Rubik's Cube in 10.50s.ogv|thumb|[[Erik Akkersdijk]] solving a 3×3×3 Rubik's Cube in 10.50 seconds.]]
[[File:Speedcubing-Twilightsojourn.jpg|thumb|A speedsolver completing a 3×3×3 Rubik's Cube.]]
'''Speedcubing''' (also known as '''speedsolving''') is the activity of solving a [[Rubik's Cube]] or related puzzle as quickly as possible. Here, solving is defined as performing a series of moves that transforms a scrambled puzzle into a state where each of the puzzle's six faces is one single, solid color.

Most cubes are sold commercially in variations of [[Pocket Cube|2×2×2]], [[Rubik's Cube|3×3×3]], [[Rubik's Revenge|4×4×4]], [[Professor's Cube|5×5×5]], [[V-Cube 6|6×6×6]], and [[V-Cube 7|7×7×7]], although variations of the puzzle have been designed with as many as 17 layers.<ref>{{cite web|url=https://www.youtube.com/watch?v=ihWyzvOM9pk|title=Over The Top - 17x17x17}}</ref> The current world record for a single solve of the [[Rubik's Cube|3×3×3]] in competition is 5.55 seconds set by [[Mats Valk]] during the Zonhoven Open 2013.<ref name="WCA records">{{cite web|url=http://www.worldcubeassociation.org/results/regions.php|title=Records|accessdate=2013-04-02}}</ref><ref>{{cite web|url=https://www.youtube.com/watch?v=vhTMm85G9GE|title=Kopie van Mats Valk official Rubik`s cube single WR: 5.55}}</ref>

Speedcubing is a popular activity among the international [[Rubik's Cube]] community. Members come together to hold competitions, work to develop new solving methods, and seek to perfect their technique. As a part of the community, puzzle builders try to invent new forms of [[combination puzzle]]s.

==History==
The [[Rubik's Cube]] was invented in 1974 by [[Hungarian people|Hungarian]] professor of [[architecture]] [[Ernő Rubik]]. A widespread international interest in the cube began in 1980, which soon developed into a global craze. On June 5, 1982, the first world championship was held in [[Budapest]], [[Hungary]]. The height of the craze began to fade away after 1983, but with the advent of the [[Internet]], [[Website|sites]] relating to speedcubing began to surface. Simultaneously spreading effective speedsolving methods and teaching people new to the cube to solve it for the first time, these sites brought in a new generation of cubers, created a growing international online community, and raised the profile of the art. Twenty years after the first World Championship, the 2002 Dutch Open competition was the first in a new wave of organized speedcubing events, which include regular national and international competitions.<ref>{{cite web|url=http://www.worldcubeassociation.org/results/competitions.php|title=Competitions|accessdate=2007-11-27}}</ref> There have been five more World Championships since [[Budapest]]'s 1982 competition, which are traditionally held every other year, the first held in [[Toronto]], [[Ontario]], in 2003. This new wave of speedcubing competitions have been organised by the [[World Cube Association]] founded by [[Ron van Bruchem]].

==Solving methods==
The standard [[Rubik's Cube]] can be solved using a number of methods, not all of which are intended for speedcubing. Although some methods employ a layer-by-layer system and algorithms, other significant (though less widely used) methods include corners-first methods, and the Roux method.

===CFOP system===
{{main|Fridrich Method}}
The CFOP (Cross - F2L - OLL - PLL) system, also known as the [[Fridrich Method]], was named after one of its inventors, [[Jessica Fridrich]], who finished 2nd in the 2003 Rubik's Cube World Championships. The first step of the method is to solve a cross-shaped arrangement of pieces on the first layer. The remainder of the first layer and all of the second layer are then solved together in what are referred to as "corner-edge pairs" or slots. Finally, the last layer is solved in two steps — first, all of the pieces in the layer are oriented to form a solid color (but without the individual pieces always being in their correct places on the cube). This step is referred to as orientation and is usually performed with a single set of algorithms known as OLL (Orientation of Last Layer). Then, all of those pieces are permuted to their correct spots. This is also usually performed as a single set of algorithms known as PLL (Permutation of Last Layer).

The CFOP system is a widely used speedcubing method. Its popularity stems from the speed at which it can be easily performed. Besides the first step, which can be planned during the customary 15-second inspection time, the entire solve of the cube consists of executing predefined algorithms based on the state of the cube. However, intuition can still be brought into play to decrease the total number of moves required to complete the solve, either by attempting to force the next step to be an easier case while completing the current step, or sometimes even solving more than one corner-edge pair simultaneously.

===Roux method===
The Roux method was invented by [[French people|French]] speedcuber [[Gilles Roux]]. The first step of the Roux method is to form a 3×2×1 block placed in the lower portion of the left layer. The second step is to create another 3×2×1 on the opposite side. The remaining four corners are then solved using a set of algorithms known as CMLL (Corners of the Last Layer, without regard to the M-slice), which leaves six edges and four centers that are solved in the last step.

This method is not as dependent on algorithm memorization as the [[Fridrich method|CFOP method]], since all but the third step is done with intuition as opposed to predefined sets of algorithms. The Roux method can more easily be performed without rotations (unlike the CFOP method) which means it is easier to look ahead (solving a collection of pieces while at the same time looking for the solution to the next step) while solving.

===ZZ method===
The ZZ method is a modern speedcubing method originally proposed by Zbigniew Zborowski in 2006. The method was designed specifically to achieve high turning speed by focusing on move ergonomics. The initial pre-planned step is called EOLine, and is the most distinctive hallmark of the ZZ method. It involves orienting all edges while placing two opposite down-face edges. The next step solves the remaining first two layers using only left, right and top face turns. On completion of the first two layers, the last layer's edges are all correctly oriented because of edge pre-orientation during EOLine. The last layer may be completed using a number of techniques including those used in the [[Fridrich method|CFOP method]]. An expert variant of this method (ZZ-a) allows the last layer to be completed in a single step with an average of just over 12 moves and knowledge of 177 algorithms.<ref name="Bernard Helmstetter's Move Count Statistics">{{cite web|url=http://www.ai.univ-paris8.fr/~bh/cube/|title=Rubik's Cube: Algorithms for the last layer|accessdate=2009-06-02}}</ref>

===Petrus method===
The Petrus method, named after its inventor [[Lars Petrus]], is considered to be more intuitive than the structured [[Fridrich method|CFOP method]]. The first step of the Petrus method is to solve a 2×2×2 block of the cube. This block is then extended to a solved 2×2×3 block. All edges are then oriented and the first and second layers are completed. Next, the top corners are put in the right place and the layer is oriented correctly (all stickers facing up) and finally the last edges are permuted (moved around). [[Lars Petrus]] developed this method to address what he felt were inherent inefficiencies in layer-by-layer approaches. This method is often used as the basis for fewest moves competition .

===Corners-first methods===
Corners-first methods involve solving the corners then finishing the edges with slice turns. Corners-first solutions were common in the 1980s, with one of the most popular methods that of 1982 world champion [[Minh Thai]]. Currently corners-first solutions are less common among speedsolvers. Dutch cuber Marc Waterman created a corners-first method in the cube craze, and averaged 16 seconds in the mid-late 1980s.

===The Fish Method===
The Fish Method is a relatively unknown method which orients and permutes three middle layer pieces before the corners and then the fourth edge piece. This method is not widely used and was used as a method for beginners to solving the cube. The creator of the Fish Method is Jacob Kersey who attained an average 52 seconds in 2012.

==Competitions==
[[File:Estonian Open 2009 - Anssi Vanhala.JPG|thumb|Anssi Vanhala solving a 3×3×3 Rubik's Cube with his feet in 36.72 seconds, at the 2009 Estonian Open.]]
According to the [[World Cube Association]] (WCA), competitors (in the same round) must solve cubes that are scrambled using a consistent set of moves (every competitor solves the same scramble). Currently, the official timer used in competitions is the [[StackMat timer]]. This device has touch-sensitive pads that are triggered by the speedcuber lifting their hands to start the time and placing their hands back on the pads after releasing the puzzle to stop the timer. In addition to the electronic timer, there are human judges with [[Stopwatch|stopwatches]] who time the 15-second inspection period before each solve, as well as solves which may take longer than 10 minutes. These judges also ensure that the competitors are following competition regulations.

Official competitions are currently being held in several categories.
{| class="wikitable" style="text-align:center"
! Category || Cube Type
|-
| speedsolving || [[Pocket Cube|2×2×2]], [[Rubik's Cube|3×3×3]], [[Rubik's Revenge|4×4×4]], [[Professor's Cube|5×5×5]], [[V-Cube 6|6×6×6]], [[V-Cube 7|7×7×7]]
|-
| one-handed solving || [[Rubik's Cube|3×3×3]]
|-
| blindfolded solving || [[Rubik's Cube|3×3×3]], [[Rubik's Revenge|4×4×4]], [[Professor's Cube|5×5×5]]
|-
| multiple-blindfolded solving || [[Rubik's Cube|3×3×3]]
|-
| solving with feet || [[Rubik's Cube|3×3×3]]
|-
| solving in fewest moves || [[Rubik's Cube|3×3×3]]
|}

[[File:Megaminx, Estonian Open 2011.jpg|thumb|Speedsolvers completing Megaminxes at the 2011 Estonian Open.]]
Competitions will often include events for speedsolving these other puzzles, as well:
*[[Megaminx]]
*[[Pyraminx]]
*[[Square One (puzzle)|Square-1]]
*[[Rubik's Clock]]
*[[Skewb]]

===World Rubik's Cube Championships===
The [[World Cube Association|WCA]] organizes the World Rubik's Cube Championship as the main international competition once every two years. The latest championship was held in Las Vegas, Nevada, from July 26-28, 2013.
<div align="center">
{| class="wikitable" style="text-align:center;"
!Championship
!Year
!Host
!Dates
!Nations
!Puzzles
!Events
!Winner
!Winning time(s)
!Ref
|-
| [[1982 World Rubik's Cube Championship|I]]
| 1982
| {{flagicon|Hungary}} [[Budapest]]
| June 5
| 19
| 1
| 1
| [[Minh Thai]]
| 22.95
|<ref name=budapest1982>{{cite web|title=World Rubik's Cube Championship 1982|publisher=World Cube Association|url=http://www.worldcubeassociation.org/results/c.php?i=WC1982|accessdate=2010-12-27}}</ref>
|-
| [[2003 World Rubik's Games Championship|II]]
| 2003
| {{flagicon|Canada}} [[Toronto]]
| August 23–24
| 15
| 9
| 13
| Dan Knights
| 20.00
|<ref name=toronto2003>{{cite web|title=World Rubik's Games Championship 2003|publisher=World Cube Association|url=http://www.worldcubeassociation.org/results/c.php?i=WC2003|accessdate=2010-12-27}}</ref>
|-
| III
| 2005
| {{flagicon|USA}} [[Lake Buena Vista, Florida|Lake Buena Vista]]
| November 5–6
| 16
| 9
| 15
| Jean Pons
| 15.10
|<ref name=lakebuenavista2005>{{cite web|title=Rubik's World Championship 2005|publisher=World Cube Association|url=http://www.worldcubeassociation.org/results/c.php?i=WC2005|accessdate=2010-12-27}}</ref>
|-
| IV
| 2007
| {{flagicon|Hungary}} [[Budapest]]
| October 5–7
| 28
| 10
| 17
| [[Yu Nakajima]]
| 12.46
|<ref name=budapest2007>{{cite web|title=World Rubik's Cube Championship 2007|publisher=World Cube Association|url=http://www.worldcubeassociation.org/results/c.php?i=WC2007|accessdate=2010-12-27}}</ref>
|-
| V
| 2009
| {{flagicon|Germany}} [[Düsseldorf]]
| October 9–11
| 32
| 12
| 19
| Breandan Vallance
| 10.74
|<ref name=dusseldorf2009>{{cite web|title=World Rubik's Cube Championship 2009|publisher=World Cube Association|url=http://www.worldcubeassociation.org/results/c.php?i=WC2009|accessdate=2010-12-27}}</ref>
|-
| VI
| 2011
| {{flagicon|Thailand}} [[Bangkok]]
| October 14–16
| 35
| 12
| 19
| Michał Pleskowicz
| 8.65
|<ref name=bangkok2011>{{cite web|title=World Rubik's Cube Championship 2011|publisher=World Cube Association|url=http://www.worldcubeassociation.org/results/c.php?i=WC2011|accessdate=2011-01-24}}</ref>
|-
| VII
| 2013
| {{flagicon|United States}} [[Las Vegas Valley|Las Vegas]]
| July 26-28
| 35
| 10
| 17
| [[Feliks Zemdegs]]
| 8.18
|<ref name=lasvegas2013>{{cite web|title=World Rubik's Cube Championship 2013|publisher=World Cube Association|url=http://www.worldcubeassociation.org/results/c.php?i=WC2013|accessdate=2013-07-29}}</ref>
|}
</div>

==World records==
The following are the official speedcubing world records that are approved by the [[World Cube Association]].<ref name="WCA records"/>

Note: For averages of 5 solves, the best time and the worst time are dropped, and the mean of the remaining 3 solves is taken. When only 3 solves are done, the mean of all 3 is taken as normal.
{| class="wikitable" style="text-align:center"
! Event !! Type !! Result (Min:Sec) !! Person !! Competition !! Result Details (Min:Sec)
|-
|rowspan="2"| [[Pocket Cube|2×2×2]]
| Single || 00:00.69 || {{flagicon|Italy}} Christian Kaserer || Trentin Open 2011 ||
|-
|Average || 00:01.71 || {{flagicon|USA}} Christopher Olson || Cubetcha 2013 || 00:04.31 / 00:01.64 / 00:01.71 / 00:01.68 / 00:01.47
|-
|rowspan="2"| [[Rubik's Cube|3×3×3]]
| Single || 00:05.55 || {{flagicon|Netherlands}} [[Mats Valk]] || Zonhoven Open 2013 ||
|-
| Average || 00:06.54 || {{flagicon|Australia}} [[Feliks Zemdegs]] || Melbourne Cube Day 2013 ||00:06.91 / 00:06.41 / 00:06.25 / 00:07.30 / 00:06.31
|-
|rowspan="2"| [[Rubik's Revenge|4×4×4]]
| Single || 00:24.66 || {{flagicon|Australia}} [[Feliks Zemdegs]] || Lifestyle Seasons Summer 2014 ||
|-
| Average || 00:28.15 || {{flagicon|Germany}} Sebastian Weyer || Frankfurt Cube Days 2012 || 00:27.90 / 00:27.13 / 00:29.63 / 00:26.46 / 00:29.41
|-
|rowspan="2"| [[Professor's Cube|5×5×5]]
| Single || 00:50.50 || {{flagicon|Australia}} [[Feliks Zemdegs]] || Australian Nationals 2013 ||
|-
| Average || 00:55.33 || {{flagicon|Australia}} [[Feliks Zemdegs]] || Melbourne Cube Day 2013 ||00:55.63 / 00:53.55 / 01:09.15 / 00:51.69 / 00:56.80
|-
|rowspan="2"| [[V-Cube 6|6×6×6]]
| Single || 01:40.86 || {{flagicon|USA}} Kevin Hays || Vancouver Summer 2013 ||
|-
| Average || 01:51.30 || {{flagicon|USA}} Kevin Hays || Vancouver Summer 2013 || 01:40.86 / 02:01.94 / 01:51.11
|-
|rowspan="2"| [[V-Cube 7|7×7×7]]
| Single || 02:40.11 || {{flagicon|Hungary}} Bence Barát || Austrian Big Cube Open 2013 ||
|-
| Average || 02:52.09 || {{flagicon|Australia}} [[Feliks Zemdegs]] || Australian Nationals 2013 ||02:54.27 / 02:42.44 / 02:59.55
|-
|rowspan="2"| [[Megaminx]]
| Single || 00:42.28 || {{flagicon|Sweden}} [[Simon Westlund]] || Danish Open 2011 ||
|-
| Average || 00:47.82 || {{flagicon|Hungary}} Bálint Bodor || Hungarian Open 2012 || 00:51.90 / 00:49.55 / 00:47.50 / 00:45.88 / 00:46.40
|-
|rowspan="2"| [[Pyraminx]]
| Single || 00:01.36 || {{flagicon|Denmark}} Oscar Roth Andersen || Danish Special 2013 ||
|-
| Average || 00:02.96 || {{flagicon|Denmark}} Oscar Roth Andersen || Danish Special 2013 || 00:03.38 / 00:01.36 / 00:03.00 / 00:02.86 / 00:03.02
|-
|rowspan="2"| [[Square One (puzzle)|Square-1]]
| Single || 00:07.41 || {{flagicon|Italy}} Andrea Santambrogio || Legnano Open 2013 ||
|-
| Average || 00:11.31 || {{flagicon|China}} Bingliang Li || Guiyang Open 2012 || 00:11.24 / 00:10.43 / 00:10.87 / 00:11.82 / 00:13.62
|-
|rowspan="2"| [[Rubik's Clock]]
| Single || 00:05.27 || {{flagicon|China}} Sam Zhixiao Wang || Shanghai Winter 2011 ||
|-
| Average || 00:06.79 || {{flagicon|USA}} Evan Liu || World Championship 2013 || 00:05.45 / 00:07.02 / 00:06.99 / 00:06.35 / DNF
|-
|rowspan="2"| [[Skewb]]
| Single || 00:04.27 || {{flagicon|Australia}} Jayden McNeill || Lifestyle Seasons Summer 2014 ||
|-
| Average || 00:07.34 || {{flagicon|Italy}} Marco Rota || Milan Winter Open 2014 || 00:05.66 / 00:11.25 / 00:06.31 / 00:07.41 / 00:08.31
|-
|rowspan="2"| 3×3×3: Blindfolded
| Single || 00:23.80 || {{flagicon|Poland}} Marcin Zalewski || Polish Nationals 2013 ||
|-
| Average || 00:28.87 || {{flagicon|Poland}} Marcin Kowalczyk || GLS Autumn Reda 2013 || 00:32.22 / 00:24.71 / 00:29.69
|-
| 4×4×4: Blindfolded
| Single || 02:30.62 || {{flagicon|Hungary}} Marcell Endrey || Slovenian Open 2013 ||
|-
| 5×5×5: Blindfolded
| Single || 06:06.41 || {{flagicon|Hungary}} Marcell Endrey || World Championship 2013 ||
|-
| 3×3×3: Multiple Blindfolded
| Single || 41/41 || {{flagicon|Poland}} Marcin Kowalczyk || SLS Swierklany 2013 || 54:14.00
|-
|rowspan="2"| 3×3×3: One-handed
| Single || 00:09.03 || {{flagicon|Australia}} [[Feliks Zemdegs]] || Lifestyle Seasons Summer 2014 ||
|-
| Average || 00:12.67 || {{flagicon|Poland}} Michał Pleskowicz || Cubing Spring Grudziadz 2012 || 00:12.15 / 00:14.53 / 00:13.27 / 00:12.58 / 00:10.77
|-
|rowspan="2"| 3×3×3: With feet
| Single || 00:27.93 || {{flagicon|Indonesia}} Fakhri Raihaan || Celebes 2012 ||
|-
| Average || 00:30.57 || {{flagicon|Brazil}} Gabriel Pereira Campanha || Valeparaibano 2013 || 00:32.25 / 00:28.01 / 00:30.45
|-
|rowspan="2"| 3×3×3: Fewest moves
| Single || 20 || {{flagicon|Japan}} Tomoaki Okayama || Czech Open 2012 ||
|-
| Average || 25.69|| {{flagicon|Germany}} Sébastien Auroux || Villa de Catral 2013 || 23 / 25 / 29
|}

==Lubrication==
Some speedcubers [[Lubrication|lubricate]] their cubes to prevent [[wrist]] and [[finger]] injury. [[Lubrication|Lubricating]] the cube also allows it to be manipulated more quickly than a non-lubricated cube. The [[World Cube Association|WCA]] allows [[lubrication]] for official competitions.

Some of the popular lubricants among speedcubers are:
*Lubix Cube Lubricant
*CRC Heavy Duty Silicone Spray
*D-39 Silicone Spray
*Cyclo Silicone Spray
*Maru Lubricant
*Traxxas 50K Differential Oil
*Cubesmith Lubricant

Checking a lubricant's [[Material safety data sheet|MSDS]] is often helpful in identifying cube-damaging ingredients.

==Terminology==
Below are some definitions of words generally used by the speedcubing community. For a more complete list of speedcubing terminology, see the [http://www.cubefreak.net/other/glossary.php cubefreak.net glossary].

;Algorithm : A predefined sequence of moves used to effect a specific change on the cube. Often referred to as ''alg'' or (less commonly) an ''algo''.
;BLD : Blindfolded solving, i.e. memorize, blindfold, then solve.
;Center piece : One of the six centers of the faces of the cube. The centers never move relative to each other on an NxNxN cube, where N is odd. On NxNxN cubes with N>3, every piece with only one sticker is referred to as a 'center piece', including those pieces that can move relative to each other.
;CLL : Corners of the Last Layer. This is the first of two steps of one of the methods of solving the last layer of the cube. In the process, edges may be unoriented. This is used in Corners First methods for the last layer, in which the first all corners are solved, followed by the edges (see ELL). CLL is also commonly used to solve the last layer of a 2x2x2 cube in one step.
;Corner piece : One of the 8 pieces with exactly three stickers, called a "corner" piece because a corner is exposed.
;Cubie : One of the mechanically independent pieces that make up a puzzle. The cubies do not include fixed center pieces, nor the central axis to which they are attached.
;Cycle : To rotate pieces' positions on the cube. e.g. a 3-cycle would make cubie set A-B-C become C-A-B.
;DNF : Did Not Finish, used in competition. e.g. when a piece pop occurs and the competitor decides not to continue the solving of the puzzle, or when a blindfolded solver stops the timer with the cube still unsolved.
;DNS : Did Not Start, used in competition when the competitor does not begin a solve, either by opting to skip it (common in Blindfold Cubing) or by not showing up when he or she is called, or not qualifying for the remaining (usually 3) solves of a certain round.
;Edge piece : One of the 12 pieces with exactly two stickers, called an "edge" piece because only one edge is exposed.
;ELL : Edges of the Last Layer. The second of two steps of one of the methods of solving the last layer of the cube, solving the edge pieces without disturbing the corner pieces (see CLL).
;F2B : First two blocks. This is used in the Roux method.
;F2L : First two layers. This is used in the CFOP (Fridrich), Petrus, and ZZ methods.
;Layer : One section of a cube consisting of a number of cubies that turn as a unit. (e.g. a standard [[Rubik's Cube]] has 3 ''layers'')
;LL : Last Layer. Usually refers to the top layer of the cube, but for the Roux method can refer to the middle layer between the left and right faces.
;Method : A combination of steps that can be used to solve a cube.
;Move : A turn of one of the six faces or three slices of the cube.
;''N''-look, also known as ''X''-Look: Refers to the number of algorithms needed to complete a step in a particular solving method, often the last layer, e.g. '4-look LL'.
;OLL : Orientation of the Last Layer, usually used in reference to the respective step of the CFOP method.
;Orient : To change the orientation of a piece.
;PB : Personal Best - personal record time to solve a puzzle. This can either be a single attempt or a [[Truncated mean|trimmed average]], depending on context.
;Permute : To relocate certain pieces in a way to achieve a desired result.
;PLL : Permutation of the Last Layer. Usually used in reference to the respective step of the CFOP method, in which case it would follow the OLL step.
;Pop : When, during a speedsolve, one or more cubies come out of the puzzle.
;Prime : A counter-clockwise move popularly denoted with a ', e.g. 'R Prime', denoted as R', R-, <math>R^{-1}</math>, or Ri. Also (less commonly) known as "inverse" or "inverted".
;Slice : The four center pieces and four edge pieces between two opposite layers of the cube.
;Two-Second Penalty, also known as +2 : A penalty of 2 seconds which is added to a solving time in official competitions when the cube is placed back on the timing pad with one or more misaligned faces. It can also be given in other cases, such as when the competitor starts the timer too slow or does not correctly stop the timer after finishing the solve.
;UWR : Unofficial World Record.
;WCA : [[World Cube Association]], the international governing body for official cube competitions.
;WR : World Record. Can also "World Rank" when referring to the rank of a person's record in a database.

==See also==
*[[Rubik's Cube]]
*[[World Cube Association]]
*[[Feliks Zemdegs]]
*[[Mats Valk]]

==Notes==
{{Reflist}}

==External links==
*[http://www.speedsolving.com/ Speedsolving.com]
*[http://www.speedsolving.com/wiki/index.php/Main_Page Speedsolving.com Wiki]
*[http://ws2.binghamton.edu/fridrich/system.html Fridrich Method]
*[http://grrroux.free.fr/method/Intro.html Roux Method]
*[http://cube.crider.co.uk/ ZZ Method]
*[http://www.lar5.com/cube/index.html Petrus Method]

{{Rubik's Cube}}

[[Category:Mechanical puzzles]]
[[Category:Puzzles]]
[[Category:Rubik's Cube]]

Planar algebra

2010-12-17T19:46:54Z

24.91.76.43: /* Definition */

[[File:Nyquist example.svg|thumb|300px|right|The Nyquist plot for <math>G(s)=\frac{1}{s^2+s+1}</math>.]]

In [[control theory]] and [[stability theory]], the '''Nyquist stability criterion''', discovered by Swedish-American electrical engineer [[Harry Nyquist]] at [[Bell Telephone Laboratories]] in 1932,<ref name="Nyquist">{{cite journal
| last = Nyquist
| first = H.
| title = Regeneration Theory
| journal = Bell System Tech. J.
| volume = 11
| issue = 1
| pages = 126–147
| publisher = American Tel. & Tel.
| location = USA
| date = January 1932
| url = http://www.alcatel-lucent.com/bstj/vol11-1932/articles/bstj11-1-126.pdf
| issn =
| doi =
| id =
| accessdate = December 5, 2012}} on [http://www.alcatel-lucent.com Alcatel-Lucent website]</ref> is a graphical technique for determining the [[stability criterion|stability of a system]]. Because it only looks at the [[Nyquist plot]] of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane singularities must be known). As a result, it can be applied to systems defined by non-[[rational function]]s, such as systems with delays. In contrast to [[Bode plot]]s, it can handle [[transfer function]]s with right half-plane singularities. In addition, there is a natural generalization to more complex systems with [[MIMO|multiple inputs and multiple outputs]], such as control systems for airplanes.

While Nyquist is one of the most general stability tests, it is still restricted to [[Linear system|linear]], [[Time-invariant system|time-invariant]] systems. Non-linear systems must use more complex [[stability criterion|stability criteria]], such as [[Lyapunov stability|Lyapunov]] or the [[circle criterion]]. While Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable. Techniques like Bode plots, while less general, are sometimes a more useful design tool.

==Background==
We consider a system whose open loop transfer function (OLTF) is <math>G(s)</math>; when placed in a closed loop with negative feedback <math>H(s)</math>, the closed loop transfer function (CLTF) then becomes <math>G/(1+GH)</math>. Stability can be determined by examining the roots of the polynomial <math>1+GH</math>, e.g. using the [[Routh array]], but this method is somewhat tedious. Conclusions can also be reached by examining the OLTF, using its [[Bode plot]]s or, as here, polar plot of the OLTF using the Nyquist criterion, as follows.

Any [[Laplace domain]] transfer function <math>\mathcal{T}(s)</math> can be expressed as the ratio of two polynomials:

:<math>\mathcal{T}(s) = \frac{N(s)}{D(s)}.</math>

The roots of <math>N(s)</math> are called the ''zeros'' of <math>\mathcal{T}(s)</math>, and the roots of<math>D(s)</math> are the ''poles'' of <math>\mathcal{T}(s)</math>. The poles of <math>\mathcal{T}(s)</math> are also said to be the roots of the "characteristic equation" <math>D(s) = 0</math>.

The stability of <math>\mathcal{T}(s)</math> is determined by the values of its poles: for stability, the real part of every pole must be negative. If <math>\mathcal{T}(s)</math> is formed by closing a negative unity feedback loop around the open-loop transfer function <math>G(s) = A(s)/B(s)</math>, then the roots of the characteristic equation are also the zeros of <math>1 + G(s)</math>, or simply the roots of <math>A(s) + B(s)=0</math>.

==Cauchy's argument principle==
From [[complex analysis]], specifically the [[argument principle]], we know that a contour <math>\Gamma_s</math> drawn in the complex <math>s</math> plane, encompassing but not passing through any number of zeros and poles of a function <math>F(s)</math>, can be mapped to another plane (the <math>F(s)</math> plane) by the function <math>F(s)</math>. The Nyquist plot of <math>F(s)</math>, which is the contour <math>\Gamma_{F(s)} = F(\Gamma_s)</math> will encircle the point <math>s={-1/k}</math> of the <math>F(s)</math> plane <math>N</math> times, where <math>N = Z - P</math>. Here are <math>Z</math> and <math>P</math> respectively the number of zeros of <math>1+kF(s)</math> and poles of <math>F(s)</math> inside the contour <math>\Gamma_s</math>. Note that we count encirclements in the <math>F(s)</math> plane in the same sense as the contour <math>\Gamma_s</math> and that encirclements in the opposite direction are ''negative'' encirclements. That is, we consider clockwise encirclements to be positive and counterclockwise encirclements to be negative.

Instead of Cauchy's argument principle, the original paper by [[Harry Nyquist]] in 1932 uses a less elegant approach. The approach explained here is similar to the approach used by Leroy MacColl (Fundamental theory of servomechanisms 1945) or by [[Hendrik Bode]] (Network analysis and feedback amplifier design 1945), both of whom also worked for [[Bell Laboratories]]. This approach appears in most modern textbooks on control theory.

==The Nyquist criterion==
We first construct '''The Nyquist Contour''', a contour that encompasses the right-half of the complex plane:
* a path traveling up the <math>j\omega</math> axis, from <math>0 - j\infty</math> to <math>0 + j\infty</math>.
* a semicircular arc, with radius <math>r \to \infty</math>, that starts at <math>0 + j\infty</math> and travels clock-wise to <math>0 - j\infty</math>.

The Nyquist Contour mapped through the function <math>1+G(s)</math> yields a plot of <math>1+G(s)</math> in the complex plane. By the Argument Principle, the number of clock-wise encirclements of the origin must be the number of zeros of <math>1+G(s)</math> in the right-half complex plane minus the poles of <math>1+G(s)</math> in the right-half complex plane. If instead,
the contour is mapped through the open-loop transfer function <math>G(s)</math>, the result is the [[Nyquist Plot]] of <math>G(s)</math>. By counting the resulting contour's encirclements of -1, we find the difference between the number of poles and zeros in the right-half complex plane of <math>1+G(s)</math>. Recalling that the zeros of <math>1+G(s)</math> are the poles of the closed-loop system, and noting that the poles of <math>1+G(s)</math> are same as the poles of <math>G(s)</math>, we now state '''The Nyquist Criterion''':

Given a Nyquist contour <math>\Gamma_s</math>, let <math>P</math> be the number of poles of <math>G(s)</math> encircled by <math>\Gamma_s</math>, and <math>Z</math> be the number of zeros of <math>1+G(s)</math> encircled by <math>\Gamma_s</math>. Alternatively, and more importantly, <math>Z</math> is the number of poles of the closed loop system in the right half plane. The resultant contour in the <math>G(s)</math>-plane, <math>\Gamma_{G(s)}</math> shall encircle (clock-wise) the point <math> (-1+j0) </math> <math>N</math> times such that <math>N = Z - P</math>.

If the system is originally open-loop unstable, feedback is necessary to stabilize the system. Right-half-plane (RHP) poles represent that instability. For closed-loop stability of a system, the number of closed-loop roots in the right half of the s-plane must be zero. Hence, the number of counter-clockwise encirclements about <math> -1+j0 </math> must be equal to the number of open-loop poles in the RHP. Any clockwise encirclements of the critical point by the open-loop frequency response (when judged from low frequency to high frequency) would indicate that the feedback control system would be destabilizing if the loop were closed. (Using RHP zeros to "cancel out" RHP poles does not remove the instability, but rather ensures that the system will remain unstable even in the presence of feedback, since the closed-loop roots travel between open-loop poles and zeros in the presence of feedback. In fact, the RHP zero can make the unstable pole unobservable and therefore not stabilizable through feedback.)

==The Nyquist criterion for systems with poles on the imaginary axis==
The above consideration was conducted with an assumption that the open-loop transfer function <math>G(s)</math> does not have any pole on the imaginary axis (i.e. poles of the form <math>0+j\omega</math>). This results from the requirement of the [[argument principle]] that the contour cannot pass through any pole of the mapping function. The most common case are systems with integrators (poles at zero).

To be able to analyze systems with poles on the imaginary axis, the Nyquist Contour can be modified to avoid passing through the point <math>0+j\omega</math>. One way to do it is to construct a semicircular arc with radius <math>r \to 0</math> around <math>0+j\omega</math>, that starts at <math>0 + j (\omega-r)</math> and travels anticlockwise to <math>0 + j (\omega+ r)</math>. Such a modification implies that the phasor <math>G(s)</math> travels along an arc of infinite radius by <math>-l\pi</math>, where <math>l</math> is the multiplicity of the pole on the imaginary axis.

==Mathematical Derivation==

Our goal is to, through this process, check for the stability of the transfer function of our unity feedback system with gain k, which is given by
:<math>T(s)=\frac{kG(s)}{1+kG(s)}</math>
That is, we would like to check whether the characteristic equation of the above transfer function, given by
:<math>D(s)=1+kG(s)=0</math>
has zeros outside the open left-half-plane (commonly initialized as the OLHP).

We suppose that we have a clockwise (i.e. negatively oriented) contour <math>\Gamma_s</math> enclosing the right hand plane, with indentations as needed to avoid passing through zeros or poles of the function <math>G(s)</math>. Cauchy's [[argument principle]] states that
: <math>-{{1}\over{2\pi i}} \oint_{\Gamma_s} {D'(s) \over D(s)}\, ds=N=Z-P </math>
Where <math>Z</math> denotes the number of zeros of <math>D(s)</math> enclosed by the contour and <math>P</math> denotes the number of poles of <math>D(s)</math> by the same contour. Rearranging, we have
<math>Z=N+P</math>, which is to say
:<math>Z=-{{1}\over{2\pi i}} \oint_{\Gamma_s} {D'(s) \over D(s)}\, ds + P</math>
We then note that <math>D(s)=1+kG(s)</math> has exactly the same poles as <math>G(s)</math>. Thus, we may find <math>P</math> by counting the poles of <math>G(s)</math> that appear within the contour, that is, within the open right half plane (ORHP).

We will now rearrange the above integral via substitution. That is, setting <math>u(s)=D(s)</math>, we have
:<math>N=-{{1}\over{2\pi i}} \oint_{\Gamma_s} {D'(s) \over D(s)}\, ds=-{{1}\over{2\pi i}} \oint_{u(\Gamma_s)} {1 \over u}\, du</math>
We then make a further substitution, setting <math>v(u) = {{u-1} \over {k}}</math>. This gives us
:<math>N=-{{1}\over{2\pi i}} \oint_{u(\Gamma_s)} {1 \over u}\, du=-{{1}\over{2\pi i}} \oint_{v(u(\Gamma_s))} {1 \over {v+1/k}}\, dv</math>
We now note that <math>v(u(\Gamma_s))={{D(\Gamma_s)-1} \over {k}}=G(\Gamma_s)</math> gives us the image of our contour under <math>G(s)</math>, which is to say our [[Nyquist Plot]]. We may further reduce the integral
:<math>N=-{{1}\over{2\pi i}} \oint_{G(\Gamma_s))} {1 \over {v+1/k}}\, dv</math>
by applying [[Cauchy's integral formula]]. In fact, we find that the above integral corresponds precisely to the number of times the Nyquist Plot encircles the point <math>-1/k</math> clockwise. Thus, we may finally state that
:<math>Z=N + P=\text{(number of times the Nyquist plot encircles -1/k clockwise)}+\text{(number of poles of G(s) in ORHP)}</math>
We thus find that <math>T(s)</math> as defined above corresponds to a stable unity-feedback system when <math>Z</math>, as evaluated above, is equal to 0.

==Summary==
* If the open-loop transfer function <math>G(s)</math> has a zero pole of multiplicity <math>l</math>, then the Nyquist plot has a discontinuity at <math>\omega = 0</math>. During further analysis it should be assumed that the phasor travels <math>l</math> times clock-wise along a semicircle of infinite radius. After applying this rule, the zero poles should be neglected, i.e. if there are no other unstable poles, then the open-loop transfer function <math>G(s)</math> should be considered stable.
* If the open-loop transfer function <math>G(s)</math> is stable, then the closed-loop system is unstable for ''any'' encirclement of the point -1.
* If the open-loop transfer function <math>G(s)</math> is ''unstable'', then there must be one ''counter'' clock-wise encirclement of -1 for each pole of <math>G(s)</math> in the right-half of the complex plane.
* The number of surplus encirclements (greater than N+P) is exactly the number of unstable poles of the closed-loop system
* However, if the graph happens to pass through the point <math>-1+j0</math>, then deciding upon even the [[marginal stability]] of the system becomes difficult and the only conclusion that can be drawn from the graph is that there exist zeros on the <math>j\omega</math> axis.

==See also==
* [[Nyquist Plot]]
* [[Bode plot]]
* [[Routh–Hurwitz stability criterion]]
* [[Control engineering]]
* [[Phase margin]]
* [[Barkhausen stability criterion]]
* [[Circle criterion]]

==References==
{{refbegin|2}}
*Faulkner, E.A. (1969): ''Introduction to the Theory of Linear Systems''; Chapman & Hall; ISBN 0-412-09400-2
*[[A. B. Pippard|Pippard, A.B.]] (1985): ''Response & Stability''; Cambridge University Press; ISBN 0-521-31994-3
*Gessing, R. (2004): ''Control fundamentals''; Silesian University of Technology; ISBN 83-7335-176-0
*Franklin, G. (2002): ''Feedback Control of Dynamic Systems''; Prentice Hall, ISBN 0-13-032393-4
{{refend}}

==Notes==
<references />

[[Category:Signal processing]]
[[Category:Classical control]]