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 Nyquist diagrams. What are Nyquist diagrams? How do I sketch these and how do I use them for closed-loop analysis? What insights do they give which can be used to support systematic compensator design and specifically, lead and lag compensation?

The first part (numbered 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.

A relatively quick overview video introducing the core topics is here: Introduction to Nyquist diagrams.

1. What is a Nyquist diagram?

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

A talk through video is on YouTube. View the notes (PDF, 512 KB).

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 talk through video is on YouTube. View the notes (PDF, 480 KB).

3. Illustrations of sketching from gain and phase information

Gives 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 impact of small changes in pole or zero positions. Uses MATLAB to check working.

A talk through video is on YouTube. Two obvious errors: (a) on slide 8 voice says anti-clockwise instead of clockwise. (b) 16.30-20min video writes quadrant 2 instead of quadrant 4.

View the notes (PDF, 482 KB).

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 talk through video is on YouTube. View the notes (PDF, 596 KB).

5. Estimating the initial quadrant

Sketching is used only when this can be done quickly, or to develop insight, but there are times when the initial quadrant of a Nyquist diagram is not obvious. This information can be critical to the efficacy of the plot and hence this video gives some simple techniques for estimating the initial quadrant correctly.

A talk through video is on YouTube. View the notes (PDF, 487 KB).

6. Dealing with RHP factors and delays

RHP factors were discussed extensively in the series on Bode diagrams. This video reinforces those messages through numerical illustrations of sketching Nyquist diagrams from first principles for systems with RHP factors. Also demonstrates the impact that input/output delay will have on a Nyquist diagram.

A talk through video is on YouTube. View the notes (PDF, 493 KB).

7. Tutorial sheet on sketching of Nyquist diagrams

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

Tutorial sheet for self study (PDF, 493 KB).

A talk through video is on YouTube.

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 talk through video is on YouTube. View the notes (PDF, 507 KB).

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 talk through video is on YouTube. View the notes (PDF, 533 KB).

10. Sketching complete Nyquist diagrams

Uses the properties associated to the Nyquist diagram as a mapping of the D-contour. Shows through 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 talk through video is on YouTube. View the notes (PDF, 453 KB).

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 talk through video is on YouTube. View the notes (PDF, 488 KB).

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 talk through video is on YouTube. View the notes (PDF, 432 KB).

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 talk through video is on YouTube. View the notes (PDF, 454 KB).

14. Applying the Nyquist stability criteria to systems with integrators

Gives a number of numerical examples to show how the stability criteria can be used to infer closed-loop stability from open-loop Nyquist diagrams. The inclusion of integrators complicates the computation of encirclements and how hence the video gives several examples.

A talk through video is on YouTube. View the notes (PDF, 545 KB).

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.

A tutorial sheet (PDF, 452 KB).

A talk through video is on YouTube.