The Root Locus Analysis Eng R. L. Nkumbwa MSc, MBA, BEng, REng. Copperbelt University.

The Root Locus Analysis

Eng R. L. Nkumbwa

MSc, MBA, BEng, REng.

Copperbelt University

04/20/23 Eng R. L. Nkumbwa@CBU-20102

Stability of Control Systems

Its all about Stability…




Auto-Pilot or Fly-by-Wire Systems

Let us consider the short period approximate model of the Fly Zambezi 727 aircraft landing at Lusaka International Airport.



Where δe is the elevator input,

Take the output as θ, input is δe, then form the transfer function is of the form;



For the Zambezi 727 (40Kft, M = 0.8) the Transfer Function reduces to:



Such that, the dominant roots have a frequency of approximately 1 rad/sec and damping of about 0.4 as shown on the pole-zero map below:



As the plane continue navigating the sky, we need to know and analyze where the poles are 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.


Root Locus Analysis Intro

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

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



Another important study of the Control systems is the investigation of the trajectories of the roots of the characteristic equation or simply the Root Locus – When certain system parameters vary.

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

Wade R. Evans in 1948



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 Response Analysis Tool (time tool) of ACSYS can be used.



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.


Root Locus Example

Consider an illustrative example for the Radio Volume control in the Course Text Book by Nkumbwa on page 75.

It illustrates how root locus is applied in volume control of radio systems.


Root Locus Example: three poles



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.


Defining Root Locus

To start with, let’s make sure we’re clear on exactly what we mean by the words “Root Locus plot.”

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

identity when it is substituted for one variable.”

Roots of this equation are the closed-loop poles of the feedback system.


Defining Root Locus

Then, what is a Locus? “The set of all points whose location is

determined by stated conditions.” The “stated conditions” here are that 1 + kL (s) =

0 for some value of k, and the “points” whose 0 locations matter to us are points in the s-plane.


Defining Root Locus

Now, what is a Root Locus? The set of all points in the s-plane that satisfy the

equation 1 + kL (s) = 0 for some 0 value of k. 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.


Defining Root Locus

The root- locus technique is a graphical method for sketching the locus of the roots in the s-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 the parameter being considered.


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 questions

will help us understand Root Locus technique.


Pole and Zero Locations by R-Locus

Let's say we have a closed-loop transfer function for a particular system:



Where N is the numerator polynomial and D is the denominator polynomial of the transfer functions, respectively.

Now, we know that to find the poles of the equation, we must set the denominator to 0, and solve the characteristic equation.



In other words, the locations of the poles of a specific equation must satisfy the following relationship:



And from the above equation we can manipulate an equation such as:



And finally by converting to polar coordinates, we get:


Equation for all Gain Values

Now we have 2 equations that govern the locations of the poles of a system for all gain values:

The Magnitude Equation


Equation for all Gain Values

The Angle Equation


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 the poles move towards a zero.



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 poles will approach.


Note

Remember that, Poles are marked on the graph with an 'X' and zeros are marked with an 'O‘ by common convention.



We can start drawing the root-locus by first placing 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 and denominator of the system transfer function.



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.



Now, a root-locus line starts at every pole. Therefore, any place that two poles appear to be

connected 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.


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.



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.



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.


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.



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 given by



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

point given by



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:



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


Root Locus Example

A single- loop feedback system has a characteristic equation as follows:


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:


Root Locus Example


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.


Root Locus Example

The intersection of the asymptotes is:


Root Locus Example

The angles of the asymptotes are:


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 of the root locus:


Root Locus Example


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 found accurately to give quantitative design information.


Compensator design using the root locus

The root locus typically allows us to choose the proper 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 that point on the root locus.

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


Possible Root Locus


Possible Response Options


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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The Root Locus Analysis Eng R. L. Nkumbwa MSc, MBA, BEng, REng. Copperbelt University.

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