Modelling and control by Anthony Rossiter

# ROOT-LOCI ANALYSIS METHODS FOR FEEDBACK LOOPS

This is a section in the chapter on classic control analysis methods. Use the left hand toolbar to access the other chapters and themes. It is implicit in several of these chapters that students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials and Laplace. More information on these can be found in the Mathematics theme on the left hand toobar. This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also a tutorial sheet for users to test themselves.

## Tutorial sheets

Two PDF files provide a number of questions students can use to test their understanding. Students are recommended to use MATLAB, or equivalent software, to test their solutions. The notes and videos also provide mini questions and worked examples.

## Root-loci 1 - What is a root-loci?

Introduces the concept of root-loci, that is a picture showing how closed-loop pole positions vary as compensator gain is varied assuming no changes in the loop poles and zeros. Uses numerical examples to demonstrate how root-loci could be computed analytically for simple examples.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Which statement is correct?
A. Root-loci are pole positions for on an open-loop system transfer function.
B. Root-loci are fixed root positions for the closed-loop system.
C. Root-loci describe all possible closed-loop root positions for variable positive compensator gain.
D. None of the above.

## Root-loci 2 - The impact of changing compensator gain on closed-loop poles and behaviour

Builds on the concept of root-loci introduced in video 1, that is a picture showing how closed-loop pole positions vary as compensator gain is varied. Uses MATLAB to show how the pole positions and corresponding closed-loop behaviours can be computed and compared efficiently for various choices of gain.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Which statement is most important? An understanding of root-loci is important because:
A. Closed-loop poles can be directly linked to expected closed-loop behaviour.
B. The loci give an overview of the range of behaviours that are possible.
C. An understanding of root-loci gives insight into compensator design.
D. None of the above.

## Root-loci 3 - Trial and error design with MATLAB

Demonstrates how MATLAB tools can be used quickly and easily to select a suitable compensator gain to meet specified criteria on the cclosed-loop pole positions, assuming no changes in the open-loop poles and zeros.

• A talk through video is on youtube.

## Root-loci 4 - Tutorial on compensator gain selection by trial and error using MATLAB

Tutorial to consolidate introductory concepts covered in videos 1-3. Without recourse to formal or detailed analysis, gives questions on gain selection to achieve specified closed-loop pole positions. Students use MATLAB tools and trial and error to determine the solutions. Demonstrations are given in real time on MATLAB.

• A talk through video is on youtube.

## Root-loci 5 - Introduction to rules for sketching root-loci

Gives an overview of the foundations for rules that are used for forming root-loci sketches. Main emphasis is introducing the underpinning closed-loop algebra that is used.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

How many different ways are there to give a condition for a closed-loop pole?
A. One, that is define the closed-loop pole polynomial.
B. Two.
C. Three.
D. None of the above.

## Root-loci 6 - Start and end points

Shows how the start and end points for root-loci can be determined using relatively trivial computations. Numerical examples illustrate the required computations.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## Root-loci 7 - Computing asymptotes

For strictly proper systems, as gain increases some closed-loop poles will tend to very large values in specified asymptotic directions. This video shows why that is the case and also how the asymptotes can be computed/sketched using just a few lines of elementry algebra. Numerical examples illustrate the required computations.

• A summary of key facts and derivations is given in the following two files: directions and centroid.
• A talk through video is on youtube.

#### Quick test question

Which statement is true?
A. The number of asymptotes matches the number of open-loop poles.
B. The difference between the number of open-poles and open-loop zeros gives the number of asymptotes.
C. The centroid is given from [sum(zeros)-sum(poles)]/(number of poles).
D. None of the above.

## Root-loci 8 - Real axis is on the loci

Parts of the real axis are nearly always on the root-loci and it can be very insightful to mark these domains. This video shows how this is done by inspection and reinforces with numerical examples.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Which statement is true?
A. A part of the real axis is on the loci if an odd number of open-loop poles are to the right.
B. A part of the real axis is on the loci if an even number of poles and zeros are to the right.
C. When testing for which part of the axis is on the loci, complex poles and zeros can be ignored.
D. None of the above.

## Root-loci 9 - Worked examples using all the 5 sketching rules

Presents a number of worked examples illustrating the use of the rules and why sketching is a useful skill. Uses MATLAB to check results and reinforce how MATLAB can be used to plot root-loci.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

## Root-loci 10 - tutorial sheet on using basic rules for sketching

This video gives a number of tutorial questions for students to try. Students are asked to sketch root-loci using he 5 basic rules introduced in videos 5-8. Worked solutions are included.

• Talk through video is on youtube.

## Root-loci 11 - using root-loci for proportional design

Indicates how root-loci can be used to indicate achieveable performance and to select the desired value of gain. The focus here is on simple paper and pen computations and estimation and it is shown how relatively crude estimation, based on root-loci sketches can give values of compensator gain very close to the ideal answer. Similar concepts were covered, but using MATLAB tools, in videos 2-4.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

## Root-loci 12 - tutorial on using root-loci for proportional design

Tutorial questions on using root-loci sketches for gain selection to achieve specified performance. The focus is on simple paper and pen computations and estimation. Worked solutions are also provided.

• Talk through video is on youtube.

## Root-loci 13 - analysing impact of lag compensators using root-loci

Indicates how root-loci can be used to analyse the impact of lag compensators on achievable closed-loop poles positions. When is a lag design useful and when is it not?

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

#### Quick test question

A lag compensator does what?
A. Moves the centroid to the right.
B. Moves the centroid to the left.
C. Moves the loci further into the left half plane.
D. None of the above.

## Root-loci 14 - analysing impact of lead compensators using root-loci

Indicates how root-loci can be used to analyse the impact of lead compensators on achievable closed-loop poles positions. Includes some unstable open-loop examples.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

#### Quick test question

A. Moves the centroid to the right.
B. Moves the centroid to the left.
C. Moves the loci further into the right half plane.
D. None of the above.

## Root-loci 15 - basic rules for positive feedback

This video discusses how the rules change for positive feedback as opposed to negative feedback - differences are subtle but importnat. Also demonstrates, through examples, occasions where a system connected with negative feedback still needs the positive feedback root-loci rules in order to generate the correct root-loci sketch.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

#### Quick test question

Which of the following are true when comparing positive feedback to negative feedback?
A. The centroid of the asymptotes changes.
B. The rule for assessing which parts of the real axis are on the loci changes.
C. The number of asymptotes changes.
D. None of the above.

## Root-loci 16 - breakaway points

A breakaway point is where the loci meet up and separate from or join the real axis. It is useful to identify these points even for a rough sketch; generally these points will be estimated and a computer used if a more precise answer is needed.

• A summary of key facts and derivations is given in the pdf file.

## Root-loci 17 - angles of arrival and departure

The direction of loci when they leave open-loop poles or arrive at open-loop zeros can have a significant impact on the implied or achievable closed-loop damping. This document gives a quick summary of how such angles can be computed by hand.

• A summary of key facts and derivations is given in the pdf file.

## Tutorial sheets

Two PDF files provide a number of questions students can use to test their understanding. Students are recommended to use MATLAB, or equivalent software, to test their solutions.  