Symmetric root$-$locus

In lecture, we have studied the symmetric root$-$locus for "single$-$input, single$-$output" systems, that is$,$ systems of the kind

$\frac{d}{dt}x=Ax+Bu$

$y=Cx$

where $u$ and $y$ are a single input and output, respectively, and the performance index that needs optimizing is

${\displaystyle \underset{0}{\overset{}{}}}{y}^2+u^2.$

$($a$)$ We now consider the case when $y$ consists of more than one output, such that the equivalent criterion to be optimized is

${\displaystyle \underset{0}{\overset{}{}}}{y}^{T}y+u^2,$

and so the transfer function $h(s)=C(sI-A{)}^{-1}B$ has a single input but possibly many outputs. Can you compute the locus of the closed$-$loop poles using the symmetric root$-$locus? Can you extract the optimal controller gain based solely on that symmetric root$-$locus? Detail the steps you will need to follow to reach that goal.

$($b$)$ Numerical application: Draw the symmetric root$-$locus for the system where

$A=\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& -1& 0\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right],C=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right].$

and compute the optimal control gain for $=1.$

$($c$)$ We now consider the case when the output $y$ consists of one output only, but the input $u$ my have several entries, and the criterion to be optimized is

${\displaystyle \underset{0}{\overset{}{}}}{y}^2+u^{T}u,$

so the transfer function $h(s)=C(sI-A{)}^{-1}B$ has a single output and possibly many inputs. Can you compute the locus of the closed$-$loop poles using the symmetric root$-$locus? Can you extract the optimal controller gain based solely on that symmetric root$-$locus? Detail the steps you will need to follow to reach that goal.

$($d$)$ Numerical application: Draw the symmetric root$-$locus for the system where

$A=\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& -1& 0\end{array}\right],B=\left[\begin{array}{cc}0& 0\\ 0& 1\\ 0& 0\\ 1& 0\end{array}\right],C=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right].$

Can you compute the optimal control gain for $=1$ directly from the symmetric root$-$locus?