(Solved): A plant with the following transfer function \[ G(s)=\frac{s+8}{s^{2}-2 s-3} ...

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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 \[ K_{1}(s)=\frac{k\left(s^{2}+21 s+20\right)}{(s+8)(s+50)^{2}} \] (c) Assuming that both controller gains were selected such that both systems have a closed loop pole at \( s=-17 \), compare their relative performance in terms of: (i) Settling time (i.e. the time taken to attain a near steady state response). [2 marks] (ii) Steady state error to a unit step input.