Problem #$1\left[\begin{array}{cc}{M}_2(s)& N_2(s)\\ -Y_1(s)& x_1(s)\end{array}\right]=\left[\begin{array}{ccc}A+H{C}_2& H& B_2+H{D}_{22}\\ C_2& \text{I}& D_{22}\\ -F& 0& \text{I}\end{array}\right]$

and

by choosing F and H such that $A+B_{2}F$ and $A+H{C}_2$ are stable.

Show that

$\left[\begin{array}{cc}{M}_2(s)& N_2(s)\\ -Y_1(s)& x_1(s)\end{array}\right]\left[\begin{array}{cc}{x}_2(s)& -N_1(s)\\ Y_2(s)& M_1(s)\end{array}\right]=\left[\begin{array}{cc}\text{I}& 0\\ 0& \text{I}\end{array}\right]$

and

$M_2(s{)}^{-1}{N}_2(s)=G_{22}(s)$ Problem #$2$

With Youla's controller parametrization, the feedback control system block diagram in

Problem#$1$ can be redrawn as the following

Find the closed$-$loop transfer function of the system using linear fractional transformation based

on $T(s)$ and $Q(s).$

Show that $T_{22}(s)=0$ and explain why this equality is a significant feature of Youla's controller

parametrization.

Consider the following feedback control system,

wh$...$

$G(s)=\left[\begin{array}{cc}{G}_{11}(s)& G_{12}(s)\\ G_{21}(s)& G_{22}(s)\end{array}\right]=\left[\begin{array}{ccc}A& B_1& B_2\\ C_1& D_{11}& D_{12}\\ C_2& D_{21}& D_{22}\end{array}\right]$

Assume that the realization of $G(s)$ shown above is minimal and the subsystem $G_{22}(s)$ is

stabilizable and detectable. We can construct the following proper stable realizations

$\left[\begin{array}{cc}{M}_2(s)& N_2(s)\\ -Y_1(s)& x_1(s)\end{array}\right]=\left[\begin{array}{ccc}A+H{C}_2& H& B_2+H{D}_{22}\\ C_2& \text{I}& D_{22}\\ -F& 0& \text{I}\end{array}\right]$

and

$\left[\begin{array}{cc}{x}_2(s)& -N_1(s)\\ Y_2(s)& M_1(s)\end{array}\right]=\left[\begin{array}{ccc}A+B_{2}F& H& B_2\\ -(C_2+D_{22}(F))& \text{I}& -D_{22}\\ F& 0& \text{I}\end{array}\right]$

by choosing F and H such that $A+B_{2}F$ and $A+H{C}_2$ are stable.

Show that