Lecture4 Root Locus Method

8/3/2019 Lecture4 Root Locus Method

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The Root Locus Analysis

Eng R. L. Nkumbwa

MSc, MBA, BEng, REng.

Copperbelt University


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04/29/12 Eng R. L. Nkumbwa@CBU-20102

Stability of Control Systems

Its all about Stability


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Auto-Pilot or Fly-by-Wire Systems

Let us consider the short period approximate

model of the Fly Zambezi 727 aircraft landingat Lusaka International Airport.


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Where e is the elevator input,

Take the output as , input is e, then formthe transfer function is of the form;


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For the Zambezi 727 (40Kft, M = 0.8) the

Transfer Function reduces to:


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Such that, the dominant roots have a frequency

of approximately 1 rad/sec and damping of about0.4 as shown on the pole-zero map below:


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As the plane continue navigating the sky, we

need to know and analyze where the polesare going as a function of the input command

constant in the above pole-zero map.

How do we know where the poles moves as

the Zambezi 727 system gain changes? This is where Root Locus comes to address

the problem and provide the solutions.


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Root Locus Analysis Intro

In Control Systems I and other previous chapter,we have demonstrated the importance of the polesand zeros of the closed loop transfer function ofthe linear control system on the dynamicperformance of the system.

The roots of the characteristic equation which arethe poles of the closed loop transfer function,determine the absolute and relative stability oflinear SISO Systems.


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Another important study of the Control systemsis the investigation of the trajectories of theroots of the characteristic equation or simplythe Root Locus When certain systemparameters vary.

The first basic properties of the root loci andthe systematic construction are due to

Wade R. Evans in 1948


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In general, root locus may be sketched by

following some simple rules and properties.

For plotting the root locus accurately the

MATLAB root locus tool in the Control System

Toolbox (control) or in the Time ResponseAnalysis Tool (time tool) of ACSYS can be used.


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The root locus technique is not confined only to

the study of control systems.

In general, the method can be applied to study

the behavior of roots of any algebraic equation

with one or more variable parameters.


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Root Locus Example

Consider an illustrative example for the

Radio Volume control in the Course TextBook by Nkumbwa on page 75.

It illustrates how root locus is applied in

volume control of radio systems.


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Root Locus Example: three poles


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General root locus is hard to determine by hand

and requires Matlab tools such as:rlocus (num,den)

To obtain full result, we can get some important

insights by developing a short set of plotting

rules.


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Defining Root Locus

To start with, lets make sure were clear onexactly what we mean by the words RootLocus plot.

So, what is a Root? A number that reduces an equation to an

identity when it is substituted for onevariable.Roots of this equation are the closed-loop

poles of the feedback system.


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Defining Root Locus

Then, what is a Locus?

The set of all points whose location isdetermined by stated conditions.

The stated conditions here are that 1 + kL (s) =

0for some value ofk, and the points whose 0

locations matter to us are points in the s-plane.


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Defining Root Locus

Now, what is a Root Locus?

The set of all points in the s-plane that satisfy theequation 1 + kL (s) = 0for some 0value ofk.

Root locus is a graphical presentation of the

closed- loop poles as a system parameter is

varied.Root locus is a powerful method of analysis and

design for stability and transient response.


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Defining Root Locus

The root- locus technique is a graphical

method for sketching the locus of the roots in thes-plane as a parameter is varied.

In fact, the root- locus method provides the

engineer with a measure of the sensitivity of the

roots of the system to a variation in theparameter being considered.


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Some Root Locus Basic Questions

What points are on the root locus?

Where does the root locus start?Where does the root locus end?When/where is the locus on the real line? Etc

Answering these and many more questionswill help us understand Root Locustechnique.


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Pole and Zero Locations by R-Locus

Let's say we have a closed-loop transfer

function for a particular system:


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Where N is the numerator polynomial and D

is the denominator polynomial of the transferfunctions, respectively.

Now, we know that to find the poles of the

equation, we must set the denominator to 0,and solve the characteristic equation.


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In other words, the locations of the poles of a

specific equation must satisfy the followingrelationship:


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And from the above equation we can

manipulate an equation such as:


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And finally by converting to polar coordinates,

we get:


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Equation for all Gain Values

Now we have 2 equations that govern the

locations of the poles of a system forall gainvalues:

The Magnitude Equation


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Equation for all Gain Values

The Angle Equation


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Root-Locus Design Procedure

In laplace transform domain, when the gain is

small the poles start at the poles of the open loop

transfer function.

When gain becomes infinity, the poles move to

overlap the zeros of the system.

This means that on a root-locus graph, all thepoles move towards a zero.


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Only one pole may move towards one zero

and this means that there must be the same

number of poles as zeros.

If there are fewer zeros than poles in the

transfer function, there are a number of

implicit zeros located at infinity that the poleswill approach.


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Note

Remember that, Poles are marked on the

graph with an 'X' and zeros are marked with

an 'O by common convention.


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We can start drawing the root-locus by firstplacing the roots of b(s) on the graph with an

'X'.Next, we place the roots of a(s) on the graph,

and mark them with an 'O'.

Where b(s) and a(s) are the numerator anddenominator of the system transfer function.


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Next, we examine the real-axis.

Starting from the right-hand side of the graph

and traveling to the left, we draw a root-locus

line on the real-axis at every point to the left of

an odd number of poles on the real-axis.


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Now, a root-locus line starts at every pole.

Therefore, any place that two poles appear to beconnected by a root locus line on the real-axis,

the two poles actually move towards each other,

and then they "breakaway", and move off the

axis. The point where the poles break off the axis is

called the breakaway point.


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Note

It is important to note that the s-plane is

symmetrical about the real axis, so whatever is

drawn on the top half of the S-plane, must be

drawn in mirror-image on the bottom-half plane.


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Once a pole breaks away from the real axis,

they can either travel out towards infinity (to

meet an implicit zero) or they can travel to

meet an explicit zero, or they can re-join the

real-axis to meet a zero that is located on the

real-axis.


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If a pole is traveling towards infinity, it always

follows an asymptote.

The number of asymptotes is equal to the

number of implicit zeros at infinity.


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Root-Locus Construction Rules

Rule 1: Starting Point (K=0) The root locus starts at open loop poles. Or there is

one branch of the root-locus for every root of b(s).

Rule 2: Terminating Point (K=infinity) The root locus terminates at open loop zeros which

include those at infinity.Rule 3: Number of Distinct Root Loci

There will be as many root loci as the highest number

of finite open loop poles or zeros.


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Rule 4: Symmetry of the Root Loci The root loci are symmetrical with respect to the

real axis and all complex roots are conjugate.

Rule 5: Angle of Asymptotes The root loci are asymptotic to straight lines at

large values and the angle of asymptotes is givenby


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Rule 6: Asymptotic Intersection The asymptotes intersects the real axis at the point

given by


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Rule 7: Root Locus Location on the Real

Axis The root loci may be found on portions of the real axis

to the left of an old number of open loop poles and

zeros.

Rule 8: Locus Breakaway Point The points at which the root locus break away can be

calculated by the following:


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Rule 9: Angle of Departure and Arrival Find the formula

Rule 10: Point of Intersection with the

Imaginary Axis Find the formula

Rule 11: Determination of K Find the formula

And many more rules and equations


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Root Locus Example

A single- loop feedback system has a

characteristic equation as follows:


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Root Locus Example

We wish to sketch the root locus in order to

determine the effect of the gain K. The poles

and the zeros are located in the s-plane as:


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Root Locus Example


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Root Locus Example

The root loci on the real axis must be located

to the left of an odd number of poles and

zeros and are therefore located as shown on

the figure above in heavy lines.


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Root Locus Example

The intersection of the asymptotes is:


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Root Locus Example

The angles of the asymptotes are:


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Root Locus Example

There are three asymptotes, since the number of

poles minus the number of zeros, n m = 3.

Also, we note that the root loci must begin at

poles, and therefore two loci must leave the

double pole at s = - 4. Then, with the asymptotes

as sketched below, we may sketch the form ofthe root locus:


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Root Locus Example


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Compensator design using the root locus

The root locus graphically displays both transient

response and stability information.

The locus can be sketched quickly to get a

general idea of the changes in transient

response generated by changes in gain.

Specific points on the locus can also be foundaccurately to give quantitative design

information.


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Compensator design using the root locus

The root locus typically allows us to choose theproper loop gain to meet a transient response

specification.As the gain is varied, we move through different

regions of response. Setting the gain at a particular value yields the

transient response dictated by the poles at thatpoint on the root locus.

Thus, we are limited to those responses thatexist along the root locus.


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Possible Root Locus


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Possible Response Options


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Wrap Up

Root Locus is a very important techniques

that can be used for compensation design of

various control systems

Do further research on this topic


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Lecture4 Root Locus Method

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