Root Locus

ROOT LOCUS

Definition:

In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of control systems developed by Walter R. Evans which can determine stability of the system. The root locus plots the poles of the closed loop transfer function as a function of a gain parameter.

Uses:

In addition to determining the stability of the system, the root locus can be used to design 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.

Rules of Sketching Root locus:

• Transfer Function of the system
• Rules
  1. Symmetry
  2. Number of Branches
  3. Starting and Ending Points
  4. Locus on Real Axis
  5. Asymptotes as |s|→∞
  6. Break-Away and -In Points
  7. Angle of Departure
  8. Angle of Arrival
  9. Locus Crosses Imaginary Axis
• Given Gain "K," Determine Location of Poles
• Given Pole Location,  Determine Value of "K"

Closed loop Transfer Function:

The closed loop transfer function of the system shown is

                                

                                             

    

                                           

So the characteristic equation is

                                     

Here is an Example of Sketching Root locus.

Transfer function

Transfer Function Info

For the open loop transfer function, G(s)H(s):

We have n=2 poles at s = 0, -3.  We have m=0 finite zeros.  So there exists q=2 zeros as s goes to infinity (q = n-m = 2-0 = 2).

We can rewrite the open loop transfer function as G(s)H(s)=N(s)/D(s) where N(s) is the numerator polynomial, and D(s) is the denominator polynomial.

N(s)= 1, and D(s)= s² + 3 s.

Characteristic Equation is 1+KG(s)H(s)=0, or 1+KN(s)/D(s)=0,

or D(s)+KN(s) = s² + 3 s+ K( 1 ) = 0

Completed Root Locus

Root Locus Symmetry

As you can see, the locus is symmetric about the real axis

Number of Branches

The open loop transfer function, G(s)H(s), has 2 poles, therefore the locus has 2 branches. Each branch is displayed in a different color.

Start/End Points

Root locus starts (K=0) at poles of open loop transfer function, G(s)H(s).  These are shown by an "x" on the diagram above

As K→∞ the location of closed loop poles move to the zeros of the open loop transfer function, G(s)H(s).  Don't forget we have we also have q=n-m=2 zeros at infinity.  (We have n=2 finite poles, and m=0 finite zeros).

Locus on Real Axis

The root locus exists on real axis to left of an odd number of poles and zeros of open loop transfer function, G(s)H(s), that are on the real axis.   These real pole and zero locations are highlighted on diagram, along with the portion of the locus that exists on the real axis.

Root locus exists on real axis between:
0 and -3

... because on the real axis, we have 2 poles at s = -3, 0, and we have no zeros.

Asymptotes as |s| goes to infinity

In the open loop transfer function, G(s)H(s), we have n=2 finite poles, and m=0 finite zeros, therefore we have q=n-m=2 zeros at infinity.

Angle of asymptotes at odd multiples of ±180°/q, (i.e., ±90°)

There exists 2 poles at s = 0, -3, ...so sum of poles=-3.
There exists 0 zeros, ...so sum of zeros=0.
(Any imaginary components of poles and zeros cancel when summed because they appear as complex conjugate pairs.)

Intersect of asymptotes is at ((sum of poles)-(sum of zeros))/q = -1.5.
Intersect is at ((-3)-(0))/2 = -3/2 = -1.5 (highlighted by five pointed star).

Break-Out and In Points on Real Axis

Break Out (or Break In) points occur where N(s)D'(s)-N'(s)D(s)=0, or 2 s + 3 = 0. (details below*)

This polynomial has 1 root at s = -1.5.

From these 1 root, there exists 1 real root at s = -1.5.  These are highlighted on the diagram above (with squares or diamonds.)

These roots are all on the locus (i.e., K>0), and are highlighted with squares.

* N(s) and D(s) are numerator and denominator polynomials of G(s)H(s), and the tick mark, ', denotes differentiation.
N(s) = 1
N'(s) = 0
D(s)= s² + 3 s
D'(s)= 2 s + 3
N(s)D'(s)= 2 s + 3
N'(s)D(s)= 0
N(s)D'(s)-N'(s)D(s)= 2 s + 3

Here we used N(s)D'(s)-N'(s)D(s)=0, but we could multiply by -1 and use N'(s)D(s)-N(s)D'(s)=0.

Angle of Departure

No complex poles in loop gain, so no angles of departure.

Angle of Arrival

No complex zeros in loop gain, so no angles of arrival.

Cross Imag. Axis

Locus crosses imaginary axis at 1 value of K.  These values are normally determined by using Routh's method.  This program does it numerically, and so is only an estimate.

Locus crosses where K = 0, corresponding to crossing imaginary axis at s=0.

These crossings are shown on plot.

Changing K Changes Closed Loop Poles

Characteristic Equation is 1+KG(s)H(s)=0, or 1+KN(s)/D(s)=0,
or D(s)+KN(s) = s² + 3 s+ K( 1 ) = 0

So, by choosing K we determine the characteristic equation whose roots are the closed loop poles.

For example with K=2.25225, then the characteristic equation is
D(s)+KN(s) = s² + 3 s + 2.2522( 1 ) = 0, or
s² + 3 s + 2.2522= 0

This equation has 2 roots at s = -1.5 ±0.047j.  These are shown by the large dots on the root locus plot

Choose Pole Location and Find K

Characteristic Equation is 1+KG(s)H(s)=0, or 1+KN(s)/D(s)=0, or
K = -D(s)/N(s) = -( s² + 3 s ) / ( 1 )
We can pick a value of s on the locus, and find K=-D(s)/N(s).

For example if we choose s= -1.6 + 1.6j (marked by asterisk),
then D(s)=-4.87 + -0.243j, N(s)= 1 + 0j,
and K=-D(s)/N(s)= 4.87 + 0.243j.
This s value is not exactly on the locus, so K is complex, (see note below), pick real part of K ( 4.87)

For this K there exist 2 closed loop poles at s = -1.5 ± 1.6j.  These poles are highlighted on the diagram with dots, the value of "s" that was originally specified is shown by an asterisk.

Note: it is often difficult to choose a value of s that is precisely on the locus, but we can pick a point that is close.  If the value is not exactly on the locus, then the calculated value of K will be complex instead of real. Just ignore the the imaginary part of K (which will be small). 

Note also that only one pole location was chosen and this determines the value of K. If the system has more than one closed loop pole, the location of the other poles are determine solely by K, and may be in undesirable locations.

Apart From Avionics:

control application in avionics

References:

http://en.wikipedia.org/wiki/Root_locus

http://lpsa.swarthmore.edu/Root_Locus/DeriveRootLocusRules.html