Inverse of State-space representation (control)

Question

Ask two questions from a paper (2012 ACC):

Consider the plant:

Let X be the stabilizing solution of the Riccati equation:

where .
Define the LQR gain by .

The transfer matrix has a left spectral factorization ,
where WL is given by

Questions:

1. If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)

2. (Basic question) Why does the Riccati equation is $(A, B_2, C_1)$ not $(A, B_1, C_1)$ or $(A, B_2, C_2)$?

Answer 1

First question is about inverting a state space system that represents the dynamics from $u$ to $y$ such that the representation is from $y$ to $u$.

If you write down the equations explicitly:

$$ \begin{align} \dot x &= Ax+Bu\\ y &= Cx+Du \end{align} $$ Now assuming $D$ is invertible, we get

$$ \begin{align} \dot x &= Ax+B(-D^{-1}Cx + D^{-1}y)\\ u &= -D^{-1}Cx + D^{-1}y \end{align} $$ So the state space from $y$ to $u$ is given by the realization $$ G^{-1}(s) = \begin{bmatrix}A-BD^{-1}C &BD^{-1}\\-CD^{-1}&D^{-1}\end{bmatrix} $$

Then you can see the similarity between this inverse and $W_L$.

For the second question, in Linear Quadratic Cost function you would like to add the cost of the input to your cost function. In this formulation, often $B_1$ is the disturbance input to the system and $B_2$ models the input matrix.

Hence you want the second input appearing on the performance channel.