How to derive H2 transfer function (w -> z)?

Question

Give a general feedback system:

1. The dynamics of G:
   

2. The dynamics of K:

   

3. Suppose the A of the closed loop system；

My questions is: how to prove the transfer function T(s): w -> z:

I know this is the form of C*inv(sI - A)*B + D, but how to specify them?

Answer 1

The following assumes $D_k=0$:

The simplifying equations are $u=C_k x_k$ and $y=C_2 x + D_{21}w + D_{22}u = C_2 x + D_{22} C_k x_k + D_{21}w$.

Substituting this into the equations for $\dot{x}, \dot{x_k}$ and $z$ gives \begin{eqnarray} \dot{x} &=& A x + B_2 C_k x_k + B_1 w \\ \dot{x_k} &=& B_k C_2 x + (A_k +B_k D_{22} C_k)x_k + B_k D_{21} w \\ z &=& C_1 x + D_{12} C_k x_k + D_{11} w \end{eqnarray} or in matrix notation: \begin{eqnarray} \dot{\bar{x}} &=& \begin{bmatrix} A & B_2 C_k \\ B_k C_2 & A_k +B_k D_{22} C_k \end{bmatrix} \bar{x} + \begin{bmatrix} B_1 \\ B_k D_{21} \end{bmatrix} w \\ z &=& \begin{bmatrix} C_1 & D_{12} C_k \end{bmatrix} \bar{x} + D_{11} w \end{eqnarray} If $D_k = 0$, we have $\begin{bmatrix} I & 0 \\ -D_{22} & I \end{bmatrix}^{-1} = \begin{bmatrix} I & 0 \\ D_{22} & I \end{bmatrix}$, and so $\begin{bmatrix} B_2 & 0 \\ 0 & B_k \end{bmatrix} \begin{bmatrix} I & 0 \\ -D_{22} & I \end{bmatrix}^{-1} \begin{bmatrix} 0 & C_k \\ C_2 & 0 \end{bmatrix}^{-1} = \begin{bmatrix} 0 & B_2 C_k \\ B_k C_2 & B_k D_{22} C_k \end{bmatrix}$. This gives the $\bar{A}$ above (adding $A,A_k$ in the formula for $\dot{\bar{x}}$, of course), from which we get the desired result.