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)= s2 - 1 s - 2.
Characteristic Equation is 1+KG(s)H(s)=0, or 1+KN(s)/D(s)=0,
or D(s)+KN(s) = s2 - 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 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=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
s2 + 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)= s2 - 1 s - 2
D'(s)= 2 s - 1
N(s)D'(s)= 2 s2 + 5 s - 3
N'(s)D(s)= s2 - 1 s - 2
N(s)D'(s)-N'(s)D(s)= s2 + 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) = s2 - 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) = s2 - 1 s - 2 + 7.1593( s + 3 ) = 0, or
s2 + 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) = -( s2 - 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.
© Copyright 2005-2007
Erik Cheever This page may be freely used for educational
purposes.
Erik Cheever Department of Engineering Swarthmore College
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