Root locus

In control theory, the root locus is the locus of the poles and zeros of a transfer function as the system gain "K" is varied on some interval. The root locus is a useful tool for analyzing single input single output (SISO) linear dynamic systems. A system is "stable" if all of its poles are in the left-hand side of the "s"-plane (for continuous systems) or inside the unit circle of the "z"-plane (for discrete systems).

Example

Suppose there is a motor with a transfer function expression "P"("s"), and a controller with both an adjustable gain "K" and a transfer function expression "C"("s"). A unity feedback loop is constructed to complete this feedback system. For this system, the overall transfer function is given by :$T(s)\; =\; frac\{K\; C(s)P(s)\}\{1+K\; C(s)P(s)\}$. Thus the closed-loop poles (roots of the characteristic equation) of the transfer function are the solutions to the equation 1+ "KC"("s")"P"("s") = 0. The principal feature of this equation is that roots may be found wherever KCP = -1. The variability of K (that's the gain you can choose for the controller) removes amplitude from the equation, meaning the complex valued evaluation of the polynomial in s K(s)C(s) needs to have net phase of 180 deg, wherever there is a closed loop pole. We are solving a root cracking problem using angles alone! So there is no computation per-se, only geometry. The geometrical construction adds angle contributions from the vectors extending from each of the poles of "KC" to a prospective closed loop root (pole) and subtracts the angle contributions from similar vectors extending from the zeros, requiring the sum be 180. The vector formulation arises from the fact that each polynomial term in the factored "CP,"(s-a) for example, represents the vector from "a" which is one of the roots, to "s" which is the prospective closed loop pole we are seeking. Thus the entire polynomial is the product of these terms, and according to vector mathematics the angles add (or subtract, for terms in the denominator) and lengths multiply (or divide). So to test a point for inclusion on the root locus, all you do is add the angles to all the open loop poles and zeros. Indeed a form of protractor, the "spirule" was once used to draw exact root loci.

From the function "T"("s"), we can also see that the zeros of the open loop system ("CP") are also the zeros of the closed loop system. It is important to note that the root locus only gives the location of closed loop poles as the gain "K" is varied, given the open loop transfer function. The zeros of a system can not be moved.

Using a few basic rules, the root locus method can plot the overall shape of the path (locus) traversed by the roots as the value of "K" varies. The plot of the root locus then gives an idea of the stability and dynamics of this feedback system for different values of "k".

The method is due to Walter R. Evans (AIEE Transactions, 1948)Fact|date=January 2008.

z-plane vs. s-plane

Root locus can also be computed in the "z"-plane, the discrete counterpart of the "s"-plane. An equation ($z\; =\; e^\{sT\}$) maps continuous "s"-plane poles (not zeros) into the "z"-domain, where "T" is the sample period. The stable, left half "s"-plane maps as the unit circle into the "z"-plane, with the "s"-plane origin equating to "z"=1 (because $e^0=1$). A diagonal line of constant damping in the "s"-plane maps around a spiral from (1,0) in the "z" plane as it curves in toward the origin. Note also that the Nyquist aliasing criteria is expressed graphically in the z-plane by the -x axis, where ($wn*T\; =\; pi$). The line of constant damping just described spirals in indefinitely but in sampled data systems, frequency content is aliased down to lower frequencies by integral multiples of the Nyquist frequency. That is, the sampled response appears as a lower frequency and better damped as well since the root in the z-plane maps equally well to the first loop of a different, better damped spiral curve of constant damping. Many other interesting and relevant mapping properties can be described, not least that z-plane controllers, having the property that they may be directly implemented from the z-plane transfer function (zero/pole ratio of polynomials), can be imagined graphically on a z-plane plot of the open loop transfer function, and immediately analyzed utilizing root locus.

Since root locus is a graphical angle technique, root locus rules work the same in the "z" and "s" planes.

The idea of a root locus can be applied to many systems where a single parameter K is varied. For example, it is useful to sweep any system parameter for which the exact value is uncertain, in order to determine its behavior.

ee also

*Phase margin
*Routh-Hurwitz stability criterion
*Nyquist stability criterion
*Gain and phase margin

References

External links

* [http://www.engin.umich.edu/group/ctm/rlocus/rlocus.html Carnegie Mellon / University of Michigan Tutorial]
* [http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node46.html The root-locus method: Drawing by hand techniques]
* [http://wikis.controltheorypro.com/index.php?title=Root_Locus Root Locus at ControlTheoryPro.com]
* http://www.roymech.co.uk/Related/Control/root_locus.html Root Locus Analysis of Control Systemssteps for the formation of root locus plot.1. it is symmetrical about real axis.2.locate poles and zeros of the given open/closed loop transfer function 3.find out the centroid by using the foermula centroid= (sum of real parts of poles -sum of real parts of zeros)/No of poles - No of zeros4.find out the angle of departure and arrival if there are any complex poles and complex zeros.5.use the Routh's Stability Criterion to find intersection point of Root Locus on the imaginary axis