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 frequency response techniques for analysis of the expected closed-loop behaviour. What are frequency response and how are they useful? What is a Bode diagram and how are these sketched quickly? What is the impact on a Bode diagram of simple compensation such as lead and lag? Design is not covered in this section.

Some relatively quick overview videos introducing the core topics:

• Introduction to frequency response

• Sketching Bode diagrams using asymptotes

• Summary notes on Bode diagrams (PDF, 741 KB)

Detailed resources are below.

1. Basic concepts and illustration of frequency response

Introduces the concept of frequency response and uses examples to demonstrate how the gain and phase of the output change as the frequency of the input is changed. Gives definitions for gain and phase in terms of frequency response.

A talk through video is on YouTube. View the notes (PDF, 766 KB).

2. Frequency response gain and phase for transfer functions

Demonstrates how to solve for the frequency response parameters of a system from a transfer function model and hence shows that the gain and phase have simple analytic dependence upon the system parameters.

A talk through video is on YouTube. View the notes (PDF, 518 KB).

3. Efficient computation of frequency response

Building on the definition for system gain and phase, this video shows how using a factorised version of the transfer function enables the user to write down insightful expressions for gain and phase by inspection. Focus is on factors with LHP roots.

A talk through video is on YouTube. View the notes (PDF, 516 KB).

4. Frequency response with RHP poles and zeros

Students often make silly mistakes when computing the frequency response of systems with RHP factors. This video presents a simple approach for avoiding simple errors and getting the answer right first time.

A talk through video is on YouTube. View the notes (PDF, 476 KB).

5. Tutorial sheet on frequency response

Gives a number of tutorial questions on finding the frequency response for a number of alternative transfer functions for students to try. Also provides quick worked solutions.

A talk through video is on YouTube. View the notes (PDF, 524 KB).

6. Plotting frequency response

Introduces the plotting of frequency response information and illustrates the use of MATLAB to do so. Indicates the weaknesses of using linear graph scales for these plots.

A talk through video is on YouTube. View the notes (PDF, 470 KB).

7. What is a Bode diagram?

Tackles the weaknesses of simple graphical displays of frequency response information and thus introduces the definition of a Bode diagram which uses logarithmic scales. Discusses some key logarithmic values which help with Bode diagram interpretation.

A talk through video is on YouTube. View the notes (PDF, 574 KB).

8. Sketching for single simple factors

Develops Bode diagrams for simple poles, zeros and integrators from first principles. Introduces the concept of approximation and known values at key frequencies.

A talk through video is on YouTube. View the notes (PDF, 496 KB).

The MATLAB file bodeasymptote.m

9. Sketching for multiple simple factors

Develops Bode diagrams for systems comprising multiple simple poles, zeros and integrators. Demonstrates how rules of logarithms allow simple insights into the construction of Bode diagrams; albeit conceptually simple, the method is cumbersome.

A talk through video is on YouTube. View the notes (PDF, 471 KB).

The MATLAB file bodeasymptoteb.m

10. Sketching with asymptotic information

Shows how some asymptotic information in the Bode plot can be obtained with minimal or no computation. This asymptotic information can be used as the basis for surprisingly accurate Bode diagram sketching for systems with multiple simple poles and zeros and requires minimal extra computations.

A talk through video is on YouTube. View the notes (PDF, 526 KB).

11. Tutorial sheet on sketching with asymptotic methods and MATLAB

Demonstrates through examples how simple asymptotic information and a few computations can capture a fairly accurate bode diagram which is good enough for design. Also demonstrates the use of MATLAB to form plots.

A talk through video is on YouTube. View the notes (PDF, 551 KB).

12. Lag compensator

Gives a detailed analysis of the bode diagram of a lag compensator. Core information is the ratio of pole to zero.

A talk through video is on YouTube [minor typo on slide 9 - high frequency gain should be K]. View the notes (PDF, 592 KB).

13. Impact of lag compensator

Uses analysis of the bode diagram of a lag compensator to show how compensation with a Lag affects the Bode diagram, that is, compares the Bode diagrams of G(s) and G(s)M(s). Demonstrates that the compensated sketch can be done by inspection.

A talk through video is on YouTube [minor typo on slide 6 - high frequency gain should be K]. View the notes (PDF, 470 KB).

14. Lead compensator

Gives a detailed analysis of the bode diagram of a lead compensator and how this is affected by the pole/zero ratio.

A talk through video is on YouTube [minor verbal typo on penultimate slide where geometric mean is described as sqrt(2) rather than sqrt(1.5) ]. View the notes (PDF, 603 KB).

15. Impact of lead compensator

Uses analysis of the bode diagram of a lead compensator to show how compensation with a Lead affects the Bode diagram of a system, that is, compares the Bode diagrams of G(s) and G(s)M(s). Uses simple observations/computations so that the compensated sketch can be done by inspection.

A talk through video is on YouTube. View the notes (PDF, 468 KB).

16. Lead-lag compensator

Gives a detailed analysis of the bode diagram of a lead-lag compensator and emphasises key attributes and thus differences with a lead compensator. Also illustrates that a good sketch can be produced using just a few elementary observations at key corner frequencies.

A talk through video is on YouTube. View the notes (PDF, 460 KB).

17. Quadratic factors and resonance

Considers transfer functions which include complex poles, that is under-damped modes, and investigates the associated Bode diagrams. Shows that under-damped modes can lead to peaks in the gain plot; these peaks are evidence of resonance, that is frequencies where the gain is disproportionately high.

A talk through video is on YouTube. View the notes (PDF, 481 KB).

18. Bandwidth

Introduces possible definitions and interpretations of bandwidth and illustrates how this can be estimated from Bode gain plots. Also, illustrates links between open-loop bandwidth and the expected bandwidth of the same system when connected with unity negative feedback.

A talk through video is on YouTube. View the notes (PDF, 498 KB).