Q$\mathrm{.}5/$ Consider a plant with a nominal transfer function model given by:

$4$

$G(s)=\frac{4s+1}{(s-0\mathrm{.}01)(s^2+0\mathrm{.}1s+1)}$

You are required to design an $H_{}$ controller with transfer function $K(s)$ that will operate on the plant as illustrated in Figure $3.$ Given the performance specifications:

The desired closed loop bandwidth: $w_B=10ra\frac{d}{s}.$

The desired steady state error: $0\mathrm{.}001.$

The desired bound on sensitivity $|S(jw)|$ for high frequencies: $2.$

The multiplicative uncertainty $W_m(s)=\frac{2s+1}{10s+2}$ and the control signal is weighted by $W_u=0\mathrm{.}001.$

a$)$ Find a suitable performance weighting function for the system.

b$)$ Use Lower Fractional Transformation $($LFT$)$ technique to find the generalized plant $P.$

c$)$ Use Upper Fractional Transformation $($UFT$)$ technique to find the matrix $N.$

d$)$ What are the necessary conditions for robust stability and performance.