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

We have n=2 poles at s = 2,
-1. We have m=1 finite zero at s = -3. So there exists q=1 zero as s
goes to infinity (q = n-m = 2-1 = 1).

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)= s + 3, and D(s)= s^{2} - 1 s -
2.

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

or D(s)+KN(s)
= s^{2} - 1 s - 2+ K( s + 3 ) = 0

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

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.

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 Kâ†’âˆž the location of
closed loop poles move to the zeros of the open loop transfer function, G(s)H(s).
Finite zeros are shown by a "o" on the diagram above. Don't forget we have we also
have q=n-m=1 zero at infinity. (We have n=2 finite poles, and m=1 finite zero).

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:

2 and -1

-3 and negative infinity

... because on the real
axis, we have 2 poles at s = -1, 2, and we have 1 zero at s = -3.

In the open loop transfer function, G(s)H(s), we have n=2 finite poles, and m=1
finite zero, therefore we have q=n-m=1 zero at infinity.

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

There exists 2 poles at s = 2,
-1, ...so sum of poles=1.

There exists 1 zero at s = -3, ...so sum of zeros=-3.

(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 = 4.

Intersect is at ((1)-(-3))/1 = 4/1 = 4 (highlighted by five
pointed star).

Since q=1, there is a single asymptote at ±180°

(on negative
real axis), so intersect of this asymptote

on the axis s not important (but it
is shown anyway).

Break Out (or Break In) points occur where N(s)D'(s)-N'(s)D(s)=0, or

s^{2}
+ 6 s - 1 = 0. (details below*)

This polynomial has 2 roots at s = -6.2,
0.16.

From these 2 roots, there exists 2 real roots at s = -6.2, 0.16.
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 polylnomials of G(s)H(s), and
the tick mark, ', denotes differentiation.

N(s) = s + 3

N'(s) = 1

D(s)=
s^{2} - 1 s - 2

D'(s)= 2 s - 1

N(s)D'(s)= 2 s^{2} + 5 s -
3

N'(s)D(s)= s^{2} - 1 s - 2

N(s)D'(s)-N'(s)D(s)= s^{2} +
6 s - 1

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.

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

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

Locus crosses imaginary axis at 2 values 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 = 0.646, 1, corresponding
to crossing imaginary axis at s=0, ±0.994j, respectively.

These crossings
are shown on plot.

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

or D(s)+KN(s)
= s^{2} - 1 s - 2+ K( s + 3 ) = 0

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

For example
with K=7.15931, then the characteristic equation is

D(s)+KN(s) = s^{2}
- 1 s - 2 + 7.1593( s + 3 ) = 0, or

s^{2} + 6.1593 s + 19.4779= 0

This equation has 2 roots at s = -3.1 ± 3.2j. These are shown by
the large dots on the root locus plot

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

K = -D(s)/N(s)
= -( s^{2} - 1 s - 2 ) / ( s + 3 )

We can pick a value of s on the locus,
and find K=-D(s)/N(s).

For example if we choose s= -3.2 + 3.3j (marked by
asterisk),

then D(s)=0.672 + -24.8j, N(s)=-0.234 + 3.32j,

and K=-D(s)/N(s)=
7.44 + -0.322j.

This s value is not exactly on the locus, so K is complex,

(see note below), pick real part of K ( 7.44)

For this K there exist 2 closed
loop poles at s = -3.2 ± 3.2j.

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 imaginary part. 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.