Lecture 13
Sketching root locus
(Problem formulation)
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Plot in the complex s plane the locations of all roots of the equation
as K varies from 0 to infinity.
This plot is called the (positive) “root locus”.
(phase and magnitude conditions)
Note that if a point in the complex plane lays on the root locus, it has to satisfy
which implies that these conditions hold:
Main features of root locus
Number of branches
Open loop poles (starting points for K=0)
Open loop zeros (limiting points for K infinity) Parts of real line that belong to root locus
Asymptotes
Breakaway point (branches intersect)
Intersections with imaginary axis
Angles of departure or arrival at poles/zeroes
Number of branches
The root locus will have L branches, where L is the maximum between numbers of poles/zeros of F(s).
If F is proper, then L is equal to the number of poles.
F does not need to be proper in general, as it does not correspond to a model of physical system.
Characteristic equation is:
We consider this example in detail.
Consider the polynomial
3 poles at: 1 zero at:
Root locus has max{3,1}=3 branches in this example.
Open loop poles/zeroes
For very small values of K, root locus contains points close to the poles of F(s):
Zeroes of characteristic polynomial are the poles of the transfer function!
We can say that branches “emanate from” open loop poles (poles as “sources”).
For large values of K, root locus is close to zeros of F(s):
We can say that zeroes are limits of branches of root locus as K grows to infinity (zeroes as “sinks”).
We enter the poles and zeros of F(s)
Poles are denoted as crosses and zeros with circles.
Real line segments on the root locus
We can quickly determine which parts of real axis belong to the root locus because of the phase condition:
A point on the real axis is a part of root locus iff it is to the left of an odd number of real poles and zeros
Complex conjugate poles/zeros
are irrelevant.
Phase condition does not hold when we are on the right
Phase condition holds when we are on the left
Red lines belong to root locus
If n>m then the root locus has n-m branches that approach infinity along asymptotes that intersect the real axis at
and with angles
Situation m>n is discussed in lecture notes.
Sketch of proof (see Ogata):
AsK∞,F(s)0oneachbranch.Ofthenbranches,mterminateat zeros of numerator M(s). The remaining n-m branches must therefore stretch indefinitely. Since n>m, F(s)0 as |s|∞.
For large s, root locus approaches root locus of but with origin shifted to :
For large s:
Sketch of proof
The phase condition for is
or equivalently
Several typical cases
n-m=1 n-m=2
Since n-m=2, asymptotes are:
Points where branches intersect (repeated roots of characteristic equation)
Consider a function f and suppose that Then, we have
Formula for repeated roots:
Consider
K at which repeated roots occur:
Alternative approach:
We can alternatively consider:
Points where branches intersect can be
obtained alternatively from
NOTE: these points have to be on the root locus!
𝑑𝑑𝑑𝑑=0 −0.02𝑠𝑠 − 0.41𝑠𝑠 − 2.3𝑠𝑠 − 2.5 = 0
Intersections with imaginary axis
Intersections with imaginary axis
We can first compute the Routh array as a function of K.
Then, we look for values of K for which some elements in the first column are equal to zero.
Those values of K yield poles with zero real parts.
With those values of K, we can find the purely imaginary poles of the closed loop, i.e. intersections with the imaginary axis.
We computed the Routh array for the example in the last lecture:
Hence, no intersections with the imaginary axis.
Arrival/departure angles
Phase condition
We apply the phase condition when the test point coincides
with a pole/zero for which we want to compute departure/arrival angle.
In this example, since all poles and zeroes are on the real axis it is trivial to compute departure/arrival angles.
Example (completed root locus)
Plot the root locus for:
Conclusions
We showed how to construct root locus of arbitrary systems via its main features:
– Number of branches; open loop poles/zeroes.
– Segments of real line on root locus.
– Asymptotes.
– Intersections of branches.
– Intersections with imaginary axis.
– Multiple roots.
– Angles of arrival/departure.
Thank you for your attention.
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