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(a) A plant with the following transfer function \[ G(s)=\frac{s+8}{s^{2}-2 s-3} \] is connected in series with a controller, \( K(s) \), and the loop is closed by negative unity feedback. The initial controller design is purely a positive scalar gain, \( k \). A modified controller, \( K_{1}(s) \) is proposed, where \[ K_{1}(s)=\frac{k\left(s^{2}+21 s+20\right)}{(s+8)(s+50)^{2}} \] (i) Plot the locus of the closed loop poles as \( k \) varies from zero to infinity. Clearly identify each plotting rule used. [6 marks] (ii) Calculate the value of \( k \) required to ensure that the system has a closed loop pole at \( s=-17 \).

To compare the relative performance of the two controllers in terms of settling time, we need to compare the location of their closed loop poles. The