Question

A laboratory experimenttransfer functioncan be described as a system with anopen-loop P(s)=\frac{1}{(s+2)(s+5)} Consider a Proportional-Integral (PI) controller C(s) with the transfer function C(s)=\frac{K\left(s+Z_{1}\right)}{s} For the controller C(s) defined above with z, = 4: Determine the open loop transfer function G(s) = C(s)P(s), and identify the open-loop poles and open-loop zeros of the system. For the root locus of G(s), determine the number of asymptotes and-where they meet, and calculate the location of any double point(s). Sketch the root-locus for this system, using the information derived in(i)-(ii) and comment on the characteristics of this system for different-values of K, e.g. for small K and for larger values of K. : You may find it useful to know that the equation s3 + 9.5s² + 28s +20 = 0 has the solutions x, = -4.23 + 1.14i, x2 = -4.23 – 1.14i,and x3 = -1.04.

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