root locus diagram

[Turbo46]

Hi<p>Does anyone know what a (Root locus diagram) is? Its used in control theory and its to do with laplace transforms?<p>cheers Turbo46

[Engineer1138]

Are you just curious or do you need to use it for something1
?
Root locus plot shows the poles and zeroes of the transfer function of a system. It's a method to analyze system stability & performance and can be used to design a compensator (system controller) that does, um, what you want it to do.<p>It's kind of a hard question to answer unless I know just what you're trying to do.

[Engineer1138]

Are you just curious or do you need to use it for something?<p>Root locus plot shows the poles and zeroes of the transfer function of a system. It's a method to analyze system stability & performance and can be used to design a compensator (system controller) that does, um, what you want it to do.<p>It's kind of a hard question to answer unless I know just what you're trying to do.

[haklesup]

A root locus is the frequency domain equivelent of a Bode plot (time domain).<p>When you do a Laplace or Forier transform you are converting a complicated math expression (usually with lots of calculus) in the time domain to a simpler algebreic expression in the frequency domain. Once converted, it is vastly easier math to work several equasions than if you had to work out the calculus in the time domain. <p>In the time domain, equasions are a function of Time (t). In the frequency domain the expressions are a function of frequncy (often called s for a reason I forget)<p>Anyway, if you plot these frequency domain expressions on an XY graph you'll find that in some places the expression = 0 (zeros) and in other places it = infinity (poles). From this Root Locus graph you can determine at what frequency the amplitude will fall to 0 or will become wildly unstable (at the poles)and in what ranges it is stable. No you can know what to avoid in the system you are modeling (yes, it is applied to more than electronics).<p>As hard as the transform and plot sounds, the real hard part is deriving the time domain calculus expressions that describe the system in the first place. This is 3rd year BSEE stuff, communication systems was the hardest class I ever took. It used every bit of math I was ever taught in every problem. (thus I am not a communications engineer)<p>Lets say you had this long calculus expression that described how an electrical signal is changing throughout time. Now imagine that you want to predict the output of a circuit (a filter perhaps) when this input is applied. First you work out an expression for the ckt then you multiply them right. No, really hard math. First you convert them to algebra, do 11th grade math then convert back to the time domain. <p>Now imagine that you want to know what happens to the output when you change the frequency of this signal. You would plot the (frequency domain)expression(X-Y)graph and see if the frequency you want is still stable over the range you need it to be.<p>Or you can make a real world measurement with a spectrum analyzer and see the same graph in the time domain which is known as a Bode plot.<p>I hope this helps (and I hope I'm correct, it's been years)

[MrAl]

Hello there,<p>I'd like to add a little too <p>
The most intuitive view of a root locus plot is probably this:<p>"The line(s) in the s plane formed by the plot of all
possible roots of the root locus equation while some
chosen parameter is varied."<p>Once you have a whole bunch of roots plotted, you can almost
connect the dots to form the line that represents the locus
of roots. It's sort of a connect-the-dots thing with this view <p>Another view is this:
"The root locations migrate about the s plane as some parameter is
varied. Knowing these locations and how they change allows one
to adjust the performance of a system as well as determine
stability."<p>In other words, once you know the root locations and how they
migrate with a given parameter variation you can understand
how the system behaves over a wide range of the parameter and
make corrections if necessary.<p>Unstable points result in time domain terms such as:
e^(at) * cos wt (or similar)
where 'a' is positive and therefore leads to an infinite
response at some point in time so you know it's unstable.<p>The only reason Laplace transforms are involved is really
just to transform the transfer equation into the complex
frequency domain (s=a+bj) and so allows finding the roots
for using this technique of analysis.<p>An example is where some gain in the system is uncontrollable
to some degree and it's desired to find out how this gain
affects the closed loop response when it varies from say
0 to 100. If it's ok from 0 to 50 but after 50 then one root
moves over to the right half plane, this system could become unstable
and so the gain would have to be limited to less then 50 or
some other change made.<p>Here's something to check out, complete with flow diagrams
and all...<p>http://www.engin.umich.edu/group/ctm/rlocus/rlocus.html<p>
Take care,
Al

[bridgen]

<blockquote><font size="1" face="Verdana, Helvetica, sans-serif">quote:</font><hr> Its that Los Alamos requirement to SIMULATE, mathematically, that was the prime technological driver for the developement of electronic computing...parallel along with decryption of the
Enigma and Blue Code<hr></blockquote><p>The Enigma code was broken at Bletchley Park, Buckinghamshire, England.

[rshayes]

The root locus plot is one method of estimating the stability of a feedback system. It is usually used when a feedback system can be mathematically described, and the roots evaluated. The basic idea is to keep the roots out of the right hand plane. In some systems, such as phase locked loops, the system is unstable if the gain is either too low or too high. A root locus plot indicates the limits which the gain must fall between if the system is to be stable.<p>Bode plots can also be used for this purpose. Here the objective is to keep the gain less than 1 at the point where the total phase shift around the loop equals 360 degrees. Bode plots can be done with either computed or measured data.<p>Nyquist plots are also used. Here the objective is to keep from going around the point -1,0. Again, either measured or computed data can be used.<p>Any of these plots can also be generated using a computer. Computers have enough capacity to allow "brute force" solutions to the same problem. SPICE does not compute transfer functions. It basically does a numerical integration from a set of initial conditions using an iteration process. If the system is unstable, SPICE will probably not converge to a solution. It may not converge for other reasons as well. This is a go-no-go process and doesn't give you any information on possible ways to fix the problem. The various graphical methods do.<p>Computers solve this type of problem by doing millions of very simple computations until the answer appears to be correct. They are only as good as the relationship between the model that they are using and the actual system. SPICE can calculate to five or six digits what will happen if a particular part is used. It won't tell you that the part you actually have is not the same as the assumed part. Consider 5% resistance tolerances or 3 to 1 beta ranges on transistors.<p>Computer can produce mountains of paper that appear to be very accurate. They may not detect that the wrong information was fed in (like a sign error in one term of a complex transfer function.<p>Models have limits of validity, and these limits are often not clearly labeled. NASA had a model for the effects of debree impacts. It was derived by fitting a mathematical relationship to measurements of previous impacts. Once this was incorporated into a computer program, it was assumed to be accurate. There was nothing in the mathematical function to indicate that the model was based solely on small particles. The results for large particles were totally erroneous, but were undoubtly expressed to four or five significant figures. When a large chunk of foam was actually fired at a wing section from a space shuttle, it blew a hole in it, contrary to the model's prediction.<p>Incidently, the first large electronic computer in this country was not developed for the Manhatten Project. The original application was to compute the trajectory of artillary shells. It was hijacked by the Manhatten Project as soon as it was completed. This probably saved the jobs of hundreds of people who did the same work with desk calculators.<p>[ July 22, 2004: Message edited by: stephen ]</p>

[rshayes]

The computer built for calculating gunnery tables was the ENIAC. By present standards, it was unreliable. It had 18,800 tubes, and an article was published in the October 1947 edition of "Electronics" tabulating the causes of failures of the 644 tubes that failed in one particular year. The machine was in service from 1945 to 1955. One of its upgrades was to add a 100 word core memory. This was probably four to five hundred bytes. It was probably programmed by jumpers on a patchboard. There is some information on this computer at http://ftp.arl.mil/~mike/comphist/eniac-story.html.<p>The problems with desk calculators may never have been solved. In 1962, I was in a program where 26 people shared about a dozen of these machines. Within a few weeks, about half of them would be down at any given time, usually jammed. The Fridens were the worst, followed by the SCMs (Smith Corona Marchant). The Monroes rarely jammed. Los Alamos used Fridens and SCMs, so I expect Feynman was kept busy. Some information on this is at http://www.childrenofthemanhattanprojec ... -06c18.htm.<p>The advanced computing capacity at Los Alamos was apparently an array of IBM card machines. Each machine performed a few operations, with decks of cards being moved from machine to machine. The code breaking at Pearl Harbor was also based on IBM card machines.<p>As I understand it, at the same time that Bletchley Park broke the Enigma code using its computers a Polish mathematician also broke it using a pencil and paper. Computers are very good at immense quantities of grunt work, but they don't think.<p>[ August 12, 2004: Message edited by: stephen ]</p>

[Lin Farquhar]

just for the record, re the first computer, ENIGMA etc, have a look at
http://www.picotech.com/applications/colossus.html
Sorry guys, it wasn't ENIAC.

[rshayes]

From the description on that web site, I doubt that Colossus could really be considered a computer. It seems to have been more of an electronic implementation of an Enigma decoding machine rather than any kind of general purpose computer. The United States built a machine to decode one of the Japanese codes (Magic), but no one has claimed that it was a computer, even though it predated Colossus and did a similar function. <p>Apparently ten of the Colossus machines were constructed, using 1500 tubes apiece. These would have been hand wired, and very expensive to build. When the war ended, the Eniac was just becoming operational. There were at least two other digital computers at that time, one at Harvard and one at Bell Laboratories. These were both relay machines, and probably worked at about ten percent of the speed of Colossus. Thus the various Colossus machines, if they were computers, represented about eighty percent of the world's computing capacity. This would have been practically a national treasure. Yet the British government made no attempt to use them for any other purpose, but dismantled them within months of the end of the war.<p>At least one mechanical digital calculator was built in France before the French Revolution. Babbage built parts of a mechanical programmable calculator that could be considered the first computer. The Bel Laboratories and Harvard relay machines used similar concepts with a different technology, relays. The ENIAC used a different technology, vacuum tubes, but used the same basic operations and concepts. Subsequent machines used transistors and integrated circuits, but the principles of operation have much in common. All of these machines take numeric data in, operate on it with algorithims set by programming, and generate numeric data as output. Colossus does not fit this pattern.<p>Colossus seems to have been more of a specialized correlating and tabulating machine rather than a computer. Bletchley Park had thousands of very bright people, some of whom are considered pioneers in the computing field. Given a toy like a computer, this type of person tends to use it for something, even if it is only seeing how many places that pi can be computed to or finding a few new prime numbers. Yet nobody seems to have found any use like this for the Colossus machines.<p>In some ways the relay computers were more advanced than ENIAC. The Bell Laboratories machine was intended for network analysis and apparently worked with complex numbers.

[Will]

' . . .obody seems to have found any use for the Colossus machines . . . In fact, as far as I believe, it was Alan Turing who envisaged the requirement for the Colossus machine - he had hundreds of clerical type decoder people working for him and they achieved decoding by essentially writing down every potential arrangment of the code characters and looking for repeats - Turing recognised that this was an OK way of doing it but realised that if it code be speeded up electronically then that would make an enormous difference so he and/or others appealed to the PO Research Establishment for help and were very lucky to get Tommy Flowers, who seemed to possess a unique talent for listening to other peoples ideas and implementing them elctronically. While Clossus was being designed/built Turing realised that he had specified a machine which, in a slightly modified form, would be capable of accepting sequential instructions to perform required tasks. It has been recorded that he spent a lot of his spare time writing programs for this (Non-esistent) machine - Eventually (I am not aware of exact details of this ) he became involved with a group who built his specified machine. That group was the English firm Ferranti who then produced the Atlas computer. The Atlas computer was claimed as some sort of first or original step in the manufacture of electronic computers.