Introduction
While analysing a control system, the very first
investigation that needs to be done is, whether the system is stable or not since
an unstable system can never perform the task it is made for. However, the
determination of stability of a system is necessary but not sufficient, as a
stable system with low damping, large settling time etc. is still undesirable.
The Routh criterion gives satisfactory answer to the
question of absolute stability but its
application in determining the relative stability is difficult. Therefore,
another method called Root Locus technique is used which
enables the control system designer to visualize the effects of varying various
system parameters on absolute and relative stability of the system.
What is Root Locus ?
The root locus
is defined as the locus of the roots of the characteristic equation as the gain
K is varied from zero to infinity.
Though, the variable can be any other quantity like
damping ratio but usually gain K is the variable.
Let
(S+3)(S+2)+K(S+1) = 0 , be the characteristic equation.
If you vary K from zero to infinity and mark the
corresponding S-values obtained for each value of K and then join them, you get
the root locus.
Rules for drawing Root Locus
Let, NP=
number of open-loop (OL) poles and
NZ= number of OL zeros.
(1) The root locus is always symmetrical about the real
axis.
(2) Total number of root locus branches is equal to
the maximum of the number of OL poles and OL zeros.
i.e Total number
of root locus branches = max ( NP , NZ )
(3) If NP > NZ , ( NP-NZ
) branches end at infinity with K=∞. If NP < NZ , ( NZ-NP
) branches come from infinity with K=0. If NP = NZ , all
branches start from OL poles and end at OL zeros.
(4) As gain K increases from zero to infinity, each
branch of the root locus starts from an OL pole with K=0 and ends either at an OL
zero or at infinity with K= ∞ .
(5) The branches of the root locus which tend to
infinity, do so along straight lines
called asymptotes.
(6) The asymptotes cross the real axis at a common
point known as centroid and its value
is calculated as
(7) The asymptotes cutting real axis at centroid,
makes certain angles with the +ve direction of the real axis in anticlockwise
direction. The angles are calculated as
(8) A point on real axis lies on root locus, if the
number of OL poles+OL zeros , to
the right of the point is odd.
(9) The breakaway
points (points at which two or more root locus branches meet) of the root
locus are the solutions of (dK/ds)=0
Note :- Not all roots of the equation (dK/ds)=0
correspond to the actual
breakaway points. Only those roots which satisfy angle criterion are the breakaway
points.
(10) Number of root locus branches approaching a breakaway point = Number of root locus branches leaving that breakaway point.
(11) The root locus branches leave a breakaway point at angles given by ±1800/r
Where, r= number
of root locus branches approaching a breakaway point
(12) The angle
of departure (angle which a root locus branch starting from an open loop
pole, makes with a line parallel to the real axis, in the counter clockwise
direction) is given by
ΦA = ±1800+ Ï´
Where,
ϴ = Σangles subtended by OL zeros - Σangles
subtended by OL poles
(13) The angle of arrival (angle which a root locus branch ending at an open loop zero, makes with a line parallel to the real axis, in the counter clockwise direction) is given by
ΦA = ±1800- Ï´
Where,
ϴ = Σangles subtended by OL zeros - Σangles
subtended by OL poles
(14) Intersection of the root locus branches with the
imaginary axis (jw-axis) can be determined as :-
Method-1 :- By use of
Routh criterion.
Method-2 :- By simply
replacing S with jw in characteristic equation and solving it. The equation
will have two variables, gain K and frequency w. From solution values of K, take
only those values of K which are greater than zero because K cannot be -ve.
(15) The value of gain K at any point S on the root
locus can be found out as
This is nothing but another form of magnitude criterion.
Root Locus Stability Condition
All values of gain K for which root locus lies to the right side
of jw-axis in s-plane, the system is unstable.
Ex:- Let our root locus
be as shown below.
Let the value of gain K at the intersection of root locus with
jw-axis be 210.
Then from root locus, it is noted that for K>210, the two
branches have positive real parts, therefore, the system is unstable for
K>210 or stable for K<210.
At K=210, the root locus is on jw-axis, therefore, system is
marginally stable.
