Classical control analysis

by Dr. J A Rossiter

The focus of these chapters is on classic control analysis methods, that is root-loci and frequency response. These apply for single input single output loops. There is discussion of the foundation for these methods followed by several examples showing efficient metchanisms for using them in practice. There is also discussion and demonstration of the MATLAB tools available to support control analysis.

 

LIST OF CHAPTERS ON CONTROL ANALYSIS METHODS

  1. ROOT-LOCI
  2. BODE DIAGRAMS AND FREQUENCY RESPONSE
  3. NYQUIST DIAGRAMS AND STABILITY CRITERIA
  4. GAIN AND PHASE MARGINS AND LEAD/LAG COMPENSATION

 

ROOT-LOCI

Root-loci 1 - What is a root-loci?

Introduces the concept of root-loci, that is a picture showing how closed-loop pole positions vary as compensator gain is varied assuming no changes in the loop poles and zeros. Uses numerical examples to demonstrate how root-loci could be computed analytically for simple examples.

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

Builds on the concept of root-loci introduced in video 1, that is a picture showing how closed-loop pole positions vary as compensator gain is varied. Uses MATLAB to show how the pole positions and corresponding closed-loop behaviours can be computed and compared efficiently for various choices of gain.

Root-loci 3 - Trial and error design with MATLAB

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

Root-loci 4 - Tutorial on compensator gain selection by trial and error using MATLAB

Tutorial to consolidate introductory concepts covered in videos 1-3. Without recourse to formal or detailed analysis, gives questions on gain selection to achieve specified closed-loop pole positions. Students use MATLAB tools and trial and error to determine the solutions. Demonstrations are given in real time on MATLAB.

Root-loci 5 - Introduction to rules for sketching root-loci

Gives an overview of the foundations for rules that are used for forming root-loci sketches. Main emphasis is introducing the underpinning closed-loop algebra that is used.

Root-loci 6 - Start and end points

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

Root-loci 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 video shows why that is the case and also how the asymptotes can be computed/sketched using just a few lines of elementry algebra. Numerical examples illustrate the required computations.

Root-loci 8 - Real axis is on the loci

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

Root-loci 9 - Worked examples using all the 5 sketching rules

Presents a number of worked examples illustrating the use of the rules and why sketching is a useful skill. Uses MATLAB to check results and reinforce how MATLAB can be used to plot root-loci.

Root-loci 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 he 5 basic rules introduced in videos 5-8. Worked solutions are included.

Root-loci 11 - using root-loci for proportional design

Indicates how root-loci can be used to indicate achieveable performance and to select the desired value of gain. The focus here is on simple paper and pen computations and estimation and it is shown how relatively crude estimation, based on root-loci sketches can give values of compensator gain very close to the ideal answer. Similar concepts were covered, but using MATLAB tools, in videos 2-4.

Root-loci 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.

Root-loci 13 - analysing impact of lag compensators using root-loci

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?

Root-loci 14 - analysing impact of lead compensators using root-loci

Indicates how root-loci can be used to analyse the impact of lead compensators on achievable closed-loop poles positions. Includes some unstable open-loop examples.


Note there is a typo around 6 min where the lead is give as K=(s+4)/(s+2) when it should be (s+2)/(s+4).

Root-loci 15 - basic rules for positive feedback

This video discusses how the rules change for positive feedback as opposed to negative feedback - differences are subtle but importnat. Also 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.

 

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BODE DIAGRAMS AND FREQUENCY RESPONSE

Bode diagrams 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.

Bode diagrams 2 - frequency response gain and phase for transfer functions

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

Bode diagram 3 - efficient computation of frequency response

Building on the definition for system gain and phase in terms of transfer function parameters, 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.

Bode diagram 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.

Bode diagram 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.

Bode diagrams 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.

Bode diagrams 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.

Bode diagrams 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.

Bode diagrams 9 - sketching for multiple simple factors

Develops Bode diagrams for systems comprising mutliple simple poles, zeros and integrators from first principles. Demonstrates how rules of logarithms allow simple insights into the construction of Bode diagrams, but recognises that albeit conceptually simple, the method is cumbersome.

Bode diagrams 10 - sketching with asymptotic information

Builds on the previous video by showing 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 suprisingly accurate Bode diagram sketching for systems with multiple simple poles and zeros and requires minimal extra computations.

Bode diagrams 11 - tutorial sheet on sketching with asymptotic methods and MATLAB

Demonstrates, through several examples, how simple asymptotic information and a few explicit computations can capture a fairly accurate bode diagram which thus is useful for insight into any subsequent design. Also demonstrates the use of MATLAB to form exact plots and shows how these compare to the hand drawn sketches.

Bode diagrams 12 - lag compensator

Gives a detailed analysis of the bode diagram of a lag compensator. Core information is the ratio of pole to zero. [Warning: includes a minor typo on slide 9 - high frequency gain should be K]

Bode diagrams 13 - impact of lag compensator

Builds on analysis of the bode diagram of a lag compensator and properties of Bode diagrams to show how compensation with a Lag affects the Bode diagram of a system, that is compares the Bode diagrams of G(s) and G(s)M(s). Uses very simple observations and computations so that the compensated sketch can be done by inspection. [Warning: includes a minor typo on slide 6 - high frequency gain should be K]

Bode diagrams 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. [Warning: includes a minor verbal typo on penultimate slide where geometric mean is described as sqrt(2) rather than sqrt(1.5). ]

Bode diagrams 15 - impact of lead compensator

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

Bode diagrams 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.

Bode diagrams 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.

Bode diagrams 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.

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NYQUIST DIAGRAMS AND STABILITY CRITERIA

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.

Nyquist 1 - what is a Nyquist diagram?

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

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.

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)!]

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

 

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GAIN AND PHASE MARGINS AND LEAD/LAG COMPENSATION (in progress)

Margins 1 - motivation and illustration of impact

Illustration of how the position of the Nyquist diagam relative to the -1 point tends to be directly related to the closed-loop behaviour. Uses several examples to show that being close to -1 tends to result in poor behaviour and also indicates that some formal measure of distance from -1 could be useful.

Margins 2 - definition of gain margin

Introduces a definition of the distance of the Nyquist plot from the -1 point, that is the gain margin. Gives examples and pictures to help students understand this visually and a number of numerical examples to emphasise the procedure for computing the gain margin.

Margins 3 - definition of phase margin

Introduces a definition of the distance of the Nyquist plot from the -1 point, that is the phase margin. Gives examples and pictures to help students understand this visually and a number of numerical examples to emphasise the procedure for computing the phase margin.

Margins 4 - using the bode diagram and MATLAB

Shows how gain and phase margins can be deduced directly from the Bode diagram and indeed can be estimated by inspection. Links margins to closed-loop stability to give visual insight into what from of Bode diagram is 'good' and what form is usually 'bad'. Demonstrates MATLAB tools which compute and illustrate gain and phase margins.

Margins 5 - tutorial sheet

Goes through a number of examples, to demonstrate the computation of gain and phase margins. Some examples are analytic and some make use of Bode diagrams. [WARNING: minor typo at about 11min 40 sec where a superscript is wrong side of a bracket - should be (4-5.642)=-31.8 ]

Margins 6 - effect of changing compensator gain on the gain margin

Shows how change in compensator gain has a very simple affect on the gain margin. Presents simple formulae for this effect and several illustrations. Emphasises the use of Bode diagrams for margin computation and also shows how to achieve a specified gain margin with an elementary computation.

Margins 7 - effect of changing compensator gain on the phase margin

Shows how change in compensator gain has a non-simple affect on the phase margin, but by using the Bode diagram, the affect is obvious. Uses the phase margin definition to show how it is very simple to specify the required gain to achieved a desired phase margin. Examples demonstrate this both analytically and using Bode diagrams, the latter being more pragmatic for many systems.

Margins 8 - example designs changing compensator gain to achived desired phase margin

Develops the previous two videos by giving a number of worked examples showing how to achieve a desired phase margin just by changes in gain. Uses analytic methods, Bode diagrams and MATLAB tools.

Margins 9 - the affect of lag compensators on margins

Reviews the impact of a lag compensator on the Bode diagram and hence shows how this affects the margins. This insight is used to develop good and bad practice in lag compensator design. The video finishes with a mechanistic rule base for lag compensator design - something that is useful for very rapid rough tuning (but not necessarily a final design).

Margins 10 - mechanistic lag compensation design with MATLAB

Shows how MATLAB tools can be used quickly and efficiently to implement, and illustrate, the mechanistic design procedure for a lag compensator. Designs are based on a target phase margin and desired steady-state gain recovery. Further fine tuning would be needed in practice.

Margin 11 - the affect of lead compensators on margins

Reviews the impact of a lead compensator on the Bode diagram and hence shows how this affects the margins. This insight is used to develop good and bad practice in lead compensator design. The video finishes with a mechanistic rule base for lead compensator design - something that is useful for very rapid rough tuning (but not necessarily a final design).

Margins 12 - mechanistic lead compensation design with MATLAB

Shows how MATLAB tools can be used quickly and efficiently to implement, and illustrate, the mechanistic design procedure for a lead compensator. Designs are based on a target gain cross over frequency and a target phase margin. Further fine tuning would be needed in practice. [Two obvious typos: (i) on 5min 30 (author writes square root of beta instead of just beta) and (ii) around 13min 30 (author uses a cross over frequency of 9.75 in lead design as opposed to 9.43)].

Margins 13 - affect of lead-lag compensation on margins

Reviews the impact of lead and lag compensators and hence presents an argument for compensators which include both these components. This insight is used to propose and illustrate a simple mechanistic design procedure for lead-lag compensators, assuming that the specification includes three objectives: (i) gain cross over frequency; (ii) phase margin and (iii) low frequency gain characteristics.

Margins 14 - lead-lag compensation with MATLAB

Shows how MATLAB tools can be used quickly and efficiently to implement, and illustrate, the mechanistic design procedure for a lead-lag compensator. Designs are based on a target gain cross over frequency and a target phase margin. Further fine tuning would be needed in practice.

Margins 15 - what is an ideal phase margin?

Presents analysis which explains the basis for the use of a 60 degree phase margin as a good target. Illustrates the limitations of this assumption through numerous examples.

Margins 16 - exam question 1 on margins and and compensators

Presents a typical examination questions for students to attempt. Covers basic analysis tools of Nyquist, Bode and root-loci and analysis of potential lead/lag compensators. Also gives a worked solution. [Silly typo in construction of Bode gain plot - asymptote drawn to w=root(3) rather than w=3.]

Margins 17 - exam question 2 on margins and compensators

Presents a typical examination questions for students to attempt. Covers basic analysis tools of Nyquist, Bode and root-loci and analysis of potential lead/lag compensators. Also gives a worked solution.


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