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Experiment 2 Consider the system shown in the figure below. R(s) E(s) C(s) Where, K G(s) = H(s) = s² - 6s + 16 s² + 4s + 10 (s+4)(s+7) In this experiment we are going to investigate the effect of increasing system gain K on the resulting response, closed-loop pole locations, and the stability of the system. a) For some K value, find the closed-loop transfer function, and the closed-loop poles. b) Determine from the location of those poles, whether your system is stable. Output a message like: "Stable / Unstable at < Kvalue> with largest pole at < polevalue>" c) Have your program run in a loop for K values ranging from 0 to 40 (steps of 0.1). 3 d) Determine the gain Kcr, that takes the system to critical oscillations. To do this, determine the K value between that resulting in stable and unstable system. e) Plot the step response and pole locations for a value of K = Kcr - 2. f) Plot the step response and pole locations for a value of K = Kcr. Hint: use the Matlab command "step(Ts,tfinal)" where Ts is your closed-loop transfer function and tfinal is the final time value in seconds. g) Plot the step response and pole locations for a value of K = Kcr + 2. h) Plot the system poles (on one graph) for 40 values of K, starting with K = 1 to K = 40, to create a path of the system pole changes. i) Answer the questions in your worksheet. G(s) H(s)
a) Modify your program from experiment 2 to automatically determine the critical gain K, and report a message such as: "Change in stability at K =< Kvalue >" where Kvalue is the mean of K values either side of the stability boundary. b) Now modify your program report a message such as "Stability region: K<or > < Kvalue >"
Useful MATLAB instructions The following (previously used) MATLAB commands might be useful in this lab: conj, poly, tf, step, feedback, roots, zero, pole, tfdata, pzmap. A concise description of each MATLAB command can be obtained by using the "help" command, e.g., "help conj".
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