Chapter four

This chapter is on the theme of linear feedback control. For example with G(s) representing a system, M(s) a compensator and d an input disturbance signal:

This section focuses on the root-loci techniques for analysis of the expected closed-loop behaviour.

It looks at the questions, what are root-loci and how are they useful? How can these be used for system design of a compensator M(s)?

These are relatively quick overview videos introducing the core topics:

• What is a root-loci?

• Sketching rules for root-loci

• Sketching with positive feedback

• Gain design using root-loci

• Compensator design with root-loci

Detailed resources can be found below.

1. What is a root-loci?

This section introduces the concept of root-loci, showing how closed-loop pole positions vary as compensator gain is varied assuming no changes in the loop poles and zeros. It uses numerical examples to demonstrate how root-loci could be computed analytically for simple examples.

A talk through video is on YouTube. View the notes (PDF, 819KB).

2. The impact of changing compensator gain on closed-loop poles and behaviour

This section builds on the concept of root-loci. It 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 talk through video is on YouTube. View the notes (PDF, 644KB).

3. Trial and error design with MATLAB

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

A talk through video is on YouTube. View the notes (PDF, 791KB).

4. Tutorial on compensator gain selection by trial and error using MATLAB

A tutorial to consolidate introductory concepts. Without recourse to formal or detailed analysis, this gives questions on gain selection to achieve specified closed-loop pole positions. It demonstrates MATLAB tools in real-time and trial and error to determine the solutions.

A talk through video is on YouTube. View the notes (PDF, 804KB).

5. Introduction to rules for sketching root-loci

This section gives an overview of the foundations for rules that are used for forming root-loci sketches. The main emphasis is on introducing the underpinning closed-loop algebra that is used.

A talk through video is on YouTube. View the notes (PDF, 667KB).

6. Start and end points

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

A talk through video is on YouTube. View the notes (PDF, 582KB).

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 section shows why that is the case and also how the asymptotes can be computed and sketched using just a few lines of elementary algebra.

A talk through video is on YouTube. View the notes on asymptote directions (PDF, 588KB) and asymptote centroid (PDF, 674KB).

8. Real axis is on the loci

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

A talk through video is on YouTube. View the notes (PDF, 567KB).

9. Worked examples using all the five sketching rules

This section presents a number of worked examples, illustrating the use of the rules and why sketching is a useful skill. It uses MATLAB to check results and reinforce how MATLAB can be used to plot root-loci.

A talk through video is on YouTube. View the notes (PDF, 601KB).

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 the five basic rules introduced in previous sections. Worked solutions are included.

A talk through video is on YouTube.

11. Using root-loci for proportional design

Root-loci can be used to indicate achievable performance and to select the desired value of gain. Simple paper and pen sketches and relatively crude estimations can give sensible values of compensator gain.

A talk through video is on YouTube. View the notes (PDF, 713KB).

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.

A talk through video is on YouTube.

13. Analysing the impact of lag compensators using root-loci

This section 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 talk through video is on YouTube. View the notes (PDF, 682KB).

14. Analysing the impact of lead compensators using root-loci

This section indicates how root-loci can be used to analyse the impact of lead compensators on achievable closed-loop poles positions. It includes some unstable open-loop examples.

A talk through video is on YouTube. View the notes (PDF, 687KB).

15. Basic rules for positive feedback

The rules change for positive feedback. This section 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 talk through video is on YouTube. View the notes (PDF, 681KB).

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.

View the notes (PDF, 561KB).

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.

View the notes (PDF, 656KB).

18. Systematic design

A much longer document outlining formal specifications underpinning systematic design. It looks at how these specifications can be used with root-loci, frequency response and MATLAB to undertake design of lead and lag compensators.

View the notes (PDF, 1.38MB).