### 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.

### 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.

### 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.