by Dr. J A Rossiter
The focus of these chapters is on classic control analysis methods, that is rootloci 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.
 ROOTLOCI
 BODE DIAGRAMS AND FREQUENCY RESPONSE
 NYQUIST DIAGRAMS AND STABILITY CRITERIA
 GAIN AND PHASE MARGINS AND LEAD/LAG COMPENSATION
Rootloci 1  What is a rootloci?
Introduces the concept of rootloci, that is a picture showing how closedloop pole positions vary as compensator gain is varied assuming no changes in the loop poles and zeros. Uses numerical examples to demonstrate how rootloci could be computed analytically for simple examples.

Rootloci 2  The impact of changing compensator gain on closedloop poles and behaviour
Builds on the concept of rootloci introduced in video 1, that is a picture showing how closedloop pole positions vary as compensator gain is varied. Uses MATLAB to show how the pole positions and corresponding closedloop behaviours can be computed and compared efficiently for various choices of gain.

Rootloci 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 cclosedloop pole positions, assuming no changes in the openloop poles and zeros.

Rootloci 4  Tutorial on compensator gain selection by trial and error using MATLAB
Tutorial to consolidate introductory concepts covered in videos 13. Without recourse to formal or detailed analysis, gives questions on gain selection to achieve specified closedloop pole positions. Students use MATLAB tools and trial and error to determine the solutions.
Demonstrations are given in real time on MATLAB.

Rootloci 5  Introduction to rules for sketching rootloci
Gives an overview of the foundations for rules that are used for forming rootloci sketches. Main emphasis is introducing the underpinning closedloop algebra that is used.

Rootloci 6  Start and end points
Shows how the start and end points for rootloci can be determined using relatively trivial computations. Numerical examples illustrate the required computations.

Rootloci 7  Computing asymptotes
For strictly proper systems, as gain increases some closedloop 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.

Rootloci 8  Real axis is on the loci
Parts of the real axis are nearly always on the rootloci and it can be very insightful to mark these domains. This video shows how this is done by inspection and reinforces with numerical examples.

Rootloci 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 rootloci.

Rootloci 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 rootloci using he 5 basic rules introduced in videos 58. Worked solutions are included.

Rootloci 11  using rootloci for proportional design
Indicates how rootloci 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 rootloci sketches can give values of compensator gain very close to the ideal answer. Similar concepts were covered, but using MATLAB tools, in videos 24.

Rootloci 12  tutorial on using rootloci for proportional design
Tutorial questions on using rootloci sketches for gain selection to achieve specified performance. The focus is on simple paper and pen computations and estimation. Worked solutions are also provided.

Rootloci 13  analysing impact of lag compensators using rootloci
Indicates how rootloci can be used to analyse the impact of lag compensators on achievable closedloop poles positions. When is a lag design useful and when is it not?

Rootloci 14  analysing impact of lead compensators using rootloci
Indicates how rootloci can be used to analyse the impact of lead compensators on achievable closedloop poles positions. Includes some unstable openloop 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).

Rootloci 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 rootloci rules in order to generate the correct rootloci sketch.

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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  leadlag compensator
Gives a detailed analysis of the bode diagram of a leadlag 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 underdamped modes, and investigates the associated Bode diagrams. Shows that underdamped 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 openloop bandwidth and the expected bandwidth of the same system when connected with unity negative feedback.

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This chapter is split into two clear parts. The first part (videos 17) focuses on the sketching of Nyquist diagrams whereas the second part then shows how there is a strong link between Nyquist diagrams and closedloop 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 anticlockwise when clearly the direction on the diagram is clockwise. (ii) from 16.3020min 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 13 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 closedloop 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 closedloop performance. Motivates further study of the potential uses of Nyquist diagrams for analysis and design.

Nyquist 9  Nyquist diagrams as a mapping of the Dcontour
Introduces the Dcontour and its relevance to frequency response diagrams. Shows how the Nyquist diagram is extended when considered as a mapping of the Dcontour. 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 Dcontour. 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 openloop system is related to closedloop 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 closedloop stability from openloop 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 closedloop stability from openloop 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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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 closedloop 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 closedloop 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 (45.64^{2})=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 nonsimple 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 steadystate 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 leadlag 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 leadlag compensators, assuming that the specification includes three objectives: (i) gain cross over frequency; (ii) phase margin and (iii) low frequency gain characteristics.

Margins 14  leadlag compensation with MATLAB Shows how MATLAB tools can be used quickly and efficiently to implement, and illustrate, the mechanistic design procedure for a leadlag 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 rootloci 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 rootloci and analysis of potential lead/lag compensators. Also gives a worked solution.


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