Modelling and control by Anthony Rossiter

# NYQUIST DIAGRAMS AND STABILITY CRITERIA

This is a section in the chapter on introduction to the importance and impact of feedback. Use the left hand toolbar to access the other chapters and themes.

This chapter is split into two clear parts. The first part (videos 1-7) focuses on the sketching of Nyquist diagrams whereas the second part then shows how there is a strong link between Nyquist diagrams and closed-loop behaviours.

 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 tutorial sheets for users to test themselves.

## Nyquist 1 - what is a Nyquist diagram?

Gives the definition of a Nyquist diagram and demonstrates plotting by enumerating frequency response data explicitly.

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

#### Quick test question

A nyquist diagram
A. Is a plot of the closed-loop transfer function on an Argand diagram.
B. Is a plot of the closed-loop frequency response on an Argand diagram.
C. Is a plot of the open-loop frequency response on an Argand diagram.
D. None of the above.

## Nyquist 2 - sketching from gain and phase information

Introduces the idea that an effective means of sketching of a Nyquist diagram is to transcribe frequency response gain and phase information. A few useful insights are presented to allow viewers to form sketches quickly from key trends in the gain and phase.

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

#### Quick test question

An easy way of producing a Nyquist diagram is to:
A. Tabulate gain and phase for many frequencies and then form the plots by marking all these points.
B. Tabulate a few points only sketch the diagram based on these.
C. Sketch the Bode diagram and transcribe information from this.
D. None of the above.

## Nyquist 3 -illustrations of sketching from gain and phase information

Builds on previous video by giving a number of illustrations of how trends in the gain and phase plots can be used to produce a sketch of the Nyquist diagram relatively quickly. Also illustrates how relatively small changes in pole or zero positions can have substantial impacts on the overall shape. Shows how MATLAB can be used to check working. [Note TWO small errors: (I) in voice over on slide 8 - says anti-clockwise when clearly the direction on the diagram is clockwise. (ii) from 16.30-20min video writes quadrant 2 where clearly it should be writing quadrant 4 (sketches are correct though)!]

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

## Nyquist 4 - sketching for systems with integrators

Develops videos 1-3 by showing how sketching rules need to be modified slightly when a system includes a single integrator. Gives a number of worked examples and then compares answers with those obtained on MATLAB.

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

## Nyquist 5 - estimating the initial quadrant

While sketching is intended to be used only when this can be done quickly, or to develop insight, there are times when the initial quadrant of a Nyquist diagram is not obvious. Nevertheless, this information can be critical to the efficacy of the plot for later design and hence this video gives some simple techniques for estimating the initial quadrant correctly, with minimal computation.

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

## Nyquist 6 - dealing with RHP factors and delays

RHP factors were discussed extensively in the series on Bode diagrams. Consequently this video reinforces those messages through a few numerical illustrations of sketching Nyquist diagrams from first principles for systems with RHP factors. For completeness, the video also demonstrates the impact that input/output delay will have on a Nyqust diagram, although it is noted it would be difficult in general to form a good sketch for a system with a delay.

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

#### Quick test question

When sketching a Nyquist diagram for a system with delay:
A. Add a fixed phase rotation to all points on the graph.
B. Add a spiral near the origin.
C. Procede with great caution.
D. None of the above.

## Nyquist 7 - tutorial sheet on sketching of Nyquist diagrams

Gives a number of examples for students to attempt by themselves. Also includes worked solutions.

## Nyquist 8 - the link between Nyquist diagrams and closed-loop behaviour

Uses MATLAB demonstrations to show how the shape of the Nyquist diagram (for the loop transfer function) and in particular its proximity to the minus one point seems to have a very strong relationship with the corresponding closed-loop performance. Motivates further study of the potential uses of Nyquist diagrams for analysis and design.

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

## Nyquist 9 - Nyquist diagrams as a mapping of the D-contour

Introduces the D-contour and its relevance to frequency response diagrams. Shows how the Nyquist diagram is extended when considered as a mapping of the D-contour. Introduces key properties of the complete Nyquist diagram such as symmetry, conformal mappings, right hand turns and rotation where frequency is near zero.

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

## Nyquist 10 - Sketching complete Nyquist diagrams

Uses the properties associated to the Nyquist diagram as a mapping of the D-contour. Shows through several examples how these properties allow a rapid production of the complete Nyquist diagram, assuming one already has the sketch associated to positive frequencies. Includes some examples with integrators.

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

## Nyquist 11 - mapping of the D contour and the concept of encirclements

Introduces the concept of encirclements, and how to count them, followed by the association to Nyquist diagram. Uses examples to show the key difference between LHP and RHP factors when mapped under the D contour which later is central to the Nyquist stability criteria.

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

## Nyquist 12 - the Nyquist stability criteria

Introduces the stability criteria using a simple derivation of how encirclements of the -1 point in the Nyquist diagram for the open-loop system is related to closed-loop stability, for unity negative feedback.

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

## Nyquist 13 - applying the Nyquist stability criteria

Gives a number of numerical examples. Shows how the stability criteria can be used to infer closed-loop stability from open-loop Nyquist diagrams. Focus is on systems without integrators.

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

## Nyquist 14 - applying the Nyquist stability criteria to systems with integrators

Gives a number of numerical examples which include integrators. Shows how the stability criteria can be used to infer closed-loop stability from open-loop Nyquist diagrams. The inclusion of integrators cmplicates the computation of encirclements and how hence the video gives several examples of how to do this correctly.

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

## Nyquist 15 - tutorial sheet on Nyquist stability criteria

Gives a number of typical tutorial questions for students to try by themselves. Worked solutions are provided for several of these.