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Root locus

Root locus

Overview
Root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly the gain of a feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

 system. This is a technique used in the field of control systems developed by Walter R. Evans
Walter R. Evans
Walter Richard Evans was a noted American control theorist and the inventor of the root locus method in 1948. He was the recipient of the 1987 American Society of Mechanical Engineers Rufus Oldenburger Medal and the 1988 AACC Richard E. Bellman Control Heritage Award.-Biography:He was born on...

.

In addition to determining the stability of the system, the root locus can be used to design for the damping ratio
Damping ratio
[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

 and natural frequency of a feedback system.
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Encyclopedia
Root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly the gain of a feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

 system. This is a technique used in the field of control systems developed by Walter R. Evans
Walter R. Evans
Walter Richard Evans was a noted American control theorist and the inventor of the root locus method in 1948. He was the recipient of the 1987 American Society of Mechanical Engineers Rufus Oldenburger Medal and the 1988 AACC Richard E. Bellman Control Heritage Award.-Biography:He was born on...

.

Uses


In addition to determining the stability of the system, the root locus can be used to design for the damping ratio
Damping ratio
[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

 and natural frequency of a feedback system. Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arcs whose center points coincide with the origin. By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency a gain, K, can be calculated and implemented in the controller. More elaborate techniques of controller design using the root locus are available in most control textbooks: for instance, lag, lead
Lead-lag compensator
A lead–lag compensator is a component in a control system that improves an undesirable frequency response in a feedback and control system. It is a fundamental building block in classical control theory.- Applications :...

, PI, PD and PID
PID controller
A proportional–integral–derivative controller is a generic control loop feedback mechanism widely used in industrial control systems – a PID is the most commonly used feedback controller. A PID controller calculates an "error" value as the difference between a measured process variable and a...

 controllers can be designed approximately with this technique.

The definition of the damping ratio
Damping ratio
[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

 and natural frequency presumes that the overall feedback system is well approximated by a second order system, that is, the system has a dominant pair of poles. This often doesn't happen and so it's good practice to simulate the final design to check if the project goals are satisfied.


Example


Suppose there is a plant
Plant (control theory)
A plant in control theory is the combination of process and actuator....

 (process) with a transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

 expression P(s), and a forward controller with both an adjustable gain K and output expression C(s) as shown in the block diagram below.
A unity feedback loop is constructed to complete this feedback system. For this system, the overall transfer function is given by


Thus the closed-loop pole
Closed-loop pole
Closed-loop poles are the positions of the poles of a closed-loop transfer function in the s-plane. The open-loop transfer function is equal to the product of all transfer function blocks in the forward path in the block diagram...

s (roots of the characteristic equation) of the transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

 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, the gain for the controller, removes amplitude from the equation, meaning the complex valued evaluation of the polynomial in s C(s)P(s) needs to have net phase of 180 deg, wherever there is a closed loop pole. 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.

Sketching root locus

  • Mark open-loop poles and zeros
  • Mark real axis portion to the left of an odd number of poles and zeros
  • Find asymptote
    Asymptote
    In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

    s


Let P be the number of poles and Z be the number of zeros:
number of asymptotes

The asymptotes intersect the real axis at and depart at angle given by:



where is the sum of all the locations of the poles, and is the sum of all the locations of the explicit zeros.
  • Phase condition on test point to find angle of deparature
  • Compute breakaway/break-in points


The breakaway points are located at the roots of the following equation:


Once you solve for z, the real roots give you the breakaway/reentry points. Complex roots correspond to a lack of breakaway/reentry.

The break-away (break-in) points are obtained by solving a polynomial equation

z-plane versus s-plane


The root locus can also be computed in the z-plane
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....

, the discrete counterpart of the s-plane. An equation (z = esT) maps continuous s-plane poles (not zeros) into the z-domain, where T is the sampling period. The stable, left half s-plane maps into the interior of the unit circle of the z-plane, with the s-plane origin equating to |z| = 1 (because e0 = 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
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

 criteria is expressed graphically in the z-plane by the x-axis, where (wnT = π). 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
Nyquist frequency
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...

. 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 polynomialls), 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.

See also

  • Phase margin
    Phase margin
    In electronic amplifiers, phase margin is the difference between the phase, measured in degrees, of an amplifier's output signal and 180°, as a function of frequency. The PM is taken as positive at frequencies below where the open-loop phase first crosses 180°, i.e. the signal becomes inverted,...

  • Routh–Hurwitz stability criterion
  • Nyquist stability criterion
    Nyquist stability criterion
    When designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system...

  • Gain and phase margin
  • Bode plot
    Bode plot
    A Bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response...


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