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

We have n=3 poles at s = -2,
-1 ± 1j. We have m=1 finite zero at s = -1. So there exists q=2 zeros
as s goes to infinity (q = n-m = 3-1 = 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)= s + 1, and D(s)= s^{3} + 4 s^{2}
+ 6 s + 4.

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

or D(s)+KN(s) = s^{3} + 4 s^{2} + 6 s + 4+ K( s + 1 ) = 0

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

The open loop transfer function, G(s)H(s), has 3 poles, therefore the locus has
3 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 K→∞ 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=2 zeros at infinity. (We have n=3 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:

-1 and -2

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

In the open loop transfer function, G(s)H(s), we have n=3 finite poles, and m=1
finite zero, therefore we have q=n-m=2 zeros at infinity.

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

There exists 3 poles at s = -2,
-1 ± 1j, ...so sum of poles=-4.

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

(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 ((-4)-(-1))/2 = -3/2 = -1.5 (highlighted
by five pointed star).

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

2 s^{3}
+ 7 s^{2} + 8 s + 2 = 0. (details below*)

This polynomial has 3
roots at s = -1.6 ±0.65j, -0.34.

From these 3 roots, there exists 1 real root
at s = -0.34. These are highlighted on the diagram above (with squares
or diamonds.)

None of the roots are on the locus.

* N(s) and D(s)
are numerator and denominator polylnomials of G(s)H(s), and the tick mark, ', denotes
differentiation.

N(s) = s + 1

N'(s) = 1

D(s)= s^{3} + 4 s^{2}
+ 6 s + 4

D'(s)= 3 s^{2} + 8 s + 6

N(s)D'(s)= 3 s^{3} + 11
s^{2} + 14 s + 6

N'(s)D(s)= s^{3} + 4 s^{2} + 6 s + 4

N(s)D'(s)-N'(s)D(s)= 2 s^{3} + 7 s^{2} + 8 s + 2

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.

Find angle of departure from pole at -1+1j

θ_{z1} =angle((Departing
pole)- (zero at -1) ).

θ_{z1} =angle((-1+1j) - (-1)) = angle(0+1j) =
90°

θ_{p1} =angle((Departing pole)- (pole at -2) ).

θ_{p1}
=angle((-1+1j) - (-2)) = angle(1+1j) = 45°

θ_{p3} =angle((-1+1j) - (-1-1j))
= angle(0+2j) = 90°

Angle of Departure is equal to:

θ_{depart}
= 180° + sum(angle to zeros) - sum(angle to poles).

θ_{depart}
= 180° + 90 - 135.

θ_{depart} = 135°

This angle is shown in gray.

It may be hard to see if it is near 0°.

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

Locus does not cross imaginary axis.

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

or D(s)+KN(s)
= s^{3} + 4 s^{2} + 6 s + 4+ K( s + 1 ) = 0

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

For example with K=2.60256, then the characteristic equation is

D(s)+KN(s)
= s^{3} + 4 s^{2} + 6 s + 4 + 2.6026( s + 1 ) = 0, or

s^{3}
+ 4 s^{2} + 8.6026 s + 6.6026= 0

This equation has 3 roots at s =
-1.4 ± 1.8j, -1.3. 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^{3} + 4 s^{2} + 6 s + 4 ) / ( 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= -1.2
+ 1.3j (marked by asterisk),

then D(s)=0.285 + -1.17j, N(s)=-0.225 + 1.27j,

and K=-D(s)/N(s)=0.929 + 0.0603j.

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

(see note below), pick real part of K (0.929)

For this K
there exist 3 closed loop poles at s = -1.2 ± 1.3j, -1.6Note: 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.