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Dr Ian R. Manchester

Dr Ian R. Manchester AMME 3500 : Root Locus

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Slide 7

Feedbackis important because of the ability tostabilize unstable systems, react to

disturbances, and reduce sensitivity to

changing system properties. But how does feedback affect system

properties? I.e. what happens to pole locations

under feedback?

Dr Ian R. Manchester AMME 3500 : Root Locus

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Slide 8Dr Ian R. Manchester AMME 3500 : Root Locus

For the moment let us examine aproportionalcontroller. (Other control structures such as integraland derivative terms may be lumped in to G(s)).

K

-

+

R(s) E(s) C(s)G(s)

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We can also determine the closed loop poles as afunction of the gain for the system

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The individual pole locations The root locus

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n(decreasing Tr)

Remember our descriptionof system specs as a

function of pole location.

So by increasing gain wecan reduce settling timeup to a point, beyond that

we will induce large

overshoot.

The system alwaysremainsstable

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We can easily derive theroot locus for a second

order system

What about for a general,possibly higher order,

control system?

Poles exist when thecharacteristic equation

(denominator) is zero

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Rule 1 : Number of Branches the n branches of the rootlocus start at the poles

For K=0, this suggests that the denominator must be zero(equivalent to the poles of the OL TF)

The number of branches in the root locus therefore equalsthe number of open loop poles

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Rule 2 : Symmetry - The root locus is symmetrical aboutthe real axis. This is a result of the fact that complex poles

will always occur in conjugate pairs.

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Rule 3 Real Axis Segments According to the angle criteria,points on the root locus will yield an angle of (2k+1)180o. On the real axis, angles from complex poles and zeros are

cancelled.

Poles and zeros to the left have an angle of 0o. This implies that roots will lie to the left of an odd number of

real-axis, finite open-loop poles and/or finite open-loop zeros.

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Consider the system atright

The closed loop transferfunction for this system

is given by

Difficult to evaluate theroot location as a

function of K

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Open loop poles andzeros

First plot the OL polesand zeros in the s-plane

This provides us withthe likely starting(poles) and ending(zeros) points for the

root locus

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Real axis segments Along the real axis, the

root locus is to the left

of an odd number of

poles and zeros

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Starting and end points The root locus will start

from the OL poles and

approach the OL zeros as K

approaches infinity

Even with a rough sketch,we can determine what the

root locus will look like

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Rule 5 Behaviour at infinity For larges and K, n-m of theloci are asymptotic to straight lines in thes-plane

The equations of the asymptotes are given by the real-axisintercept, a, and angle, a

Where k = 0, 1, 2, and the angle is given in radiansrelative to the positive real axis

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Why does this hold? We can write the characteristic equation as

This can be approximated by

For large s, this is the equation for a system with n-mpoles clustered at s=

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Here we have four OLpoles and one OL zero

We would thereforeexpect n-m = 3 distinct

asymptotes in the root

locus plot

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We can calculate theequations of theasymptotes, yielding

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For poles on the realaxis, the locus will

depart at 0o or 180o

For complex poles,the angle of departure

can be calculated by

considering the angle

criteria

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A similar approach canbe used to calculate the

angle of arrival of the

zeros

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We may also beinterested in the gain

at which the locus

crosses the imaginary

axis

This will determinethe gain with which

the system becomes

unstable

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All of this probably seems somewhatcomplicated

Fortunately, Matlab provides us with tools forplotting the root locus

It is still important to be able to sketch the rootlocus by hand because

This gives us an understanding to be applied todesigning controllers

It will probably appear on the exam

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As we saw previously, the specifications for asecond order system are often used indesigning a system

The resulting system performance must beevaluated in light of the true systemperformance

The root locus provides us with a tool withwhich we can design for a transient responseof interest

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We would usually follow these stepsSketch the root locusAssume the system is second order and find the

gain to meet the transient response specifications

Justify the second-order assumptions by findingthe location of all higher-order poles

If the assumptions are not justified, systemresponse should be simulated to ensure that it

meets the specifications

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Recall that for a second order system with nofinite zeros, the transient response parametersare approximated by

Rise time :Overshoot :Settling Time (2%) :

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Recall the systempresented earlier

Determine a value of thegain K to yield a 5%

percent overshoot For a second order

system, we could find K

explicitly

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Alternatively, we canexamine the Root Locus

x x

Im(s)

Re(s)

10 5 0

x

x

S=5+5.1j

=sin-1

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We can use Matlabto generate the root

locus

!% define the OL system!sys=tf(1,[1 10 0])!% plot the root locus!rlocus(sys)!

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We also need to verifythe resulting step

response

% set up the closed loop TF!

cl=51*sys/(1+51*sys)!% plot the step response!step(cl)!

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Consider this system This is a third order

system with an

additional pole

Determine a value of thegain K to yield a 5%

percent overshoot

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With the higher orderpoles, the 2nd order

assumptions are violated

However, we can usethe RL to guide our

design and iterate to

find a suitable solution

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The gain found basedon the 2nd order

assumption yields a

higher overshoot

We could then reducethe gain to reduce the

overshoot

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The preceding developments have beenpresented for a system in which the design

parameter is the forward path gain

In some instances, we may need to designsystems using other system parameters

In general, we can convert to a form in whichthe parameter of interest is in the required form

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Consider a system of thisform

The open loop transferfunction is no longer of

the familiar form KG(s)H

(s)

Rearrange to isolate p1Now we can sketch theroot locus as a function of

p1

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This results in thefollowing root locus as a

function of the

parameter p1

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We have looked at a graphical approach torepresenting the root positions as a function ofvariations in system parameters

We have presented rules for sketching the rootlocus given the open loop transfer function

We have begun looking at methods for usingthe root locus as a design tool

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NiseSections 8.1-8.6

Franklin & PowellSection 5.1-5.3

root locus 2012

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