Classical control design techniques
Section two: Frequency Response and Bode Diagrams
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:
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
12. Lag compensator
Gives a detailed analysis of the bode diagram of a lag compensator. Core information is the ratio of pole to zero.
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.
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.
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.
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.
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.
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.