留学生考试辅导 Lecture 13

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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