Root locus

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 system. This is a technique used in the field of control systems developed by Walter R. Evans.

Contents

Uses

In addition to determining the stability of the system, the root locus can be used to design for the damping ratio 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, PI, PD and PID controllers can be designed approximately with this technique.

The definition of the damping ratio 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 (process) with a transfer function 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

T(s) = \frac{Y(s)}{X(s)} = \frac{K C(s)P(s)}{1%2BK 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, 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.[1][2]

Sketching root locus

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

P - Z = number of asymptotes

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

\phi_l = \frac{180^\circ %2B (l - 1)360^\circ}{P-Z}, l = 1, 2, ..., P - Z
\alpha = \frac{\sum_P - \sum_Z}{P - Z}

where \sum_P is the sum of all the locations of the poles, and \sum_Z is the sum of all the locations of the explicit zeros.

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

\frac{-G(s)H(s)}{ds} = 0\text{ or }\frac{\overline{GH}(z)}{dz} = 0

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, 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 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. 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

References

  1. ^ Evans, W. R. (January 1948), "Graphical Analysis of Control Systems", Trans. AIEE 67 (1): 547–551, doi:10.1109/T-AIEE.1948.5059708, ISSN 0096-3860 
  2. ^ Evans, W. R. (January 1950), "Control Systems Synthesis by Root Locus Method", Trans. AIEE 69 (1): 66–69, doi:10.1109/T-AIEE.1950.5060121, ISSN 0096-3860 

External links