0
$\begingroup$

I have two state space systems. Now I want to compute the state space system where the first state space system is the input of the other...

$$M_1 = \begin{cases}\dot{x}_1 = A_1 x_1 + B_1 u_1 \\ y_1 = C_1 x_1 + D_1 u_1 \end{cases}$$ $$M_2 = \begin{cases}\dot{x}_2 = A_2 x_2 + B_2 u_2 \\ y_2 = C_2 x_2 + D_2 u_2 \end{cases}$$

          +   u1
      r --->O---->[ M1 ]----+---> y1
          - |               | 
         y2 +-----[ M2 ]<---+ u2 

Where $u_1 = r - y_2$ and $u_2 = y_1$.

How do you compute the new $A,B,C,D$ matrices?

When I have $A_1 = B_1 = C_1 = D_1 = 1$ and $A_2 = B_2 = C_2 = D_2 = 2$ I should obtain

$A = \begin{bmatrix} 1/3 & -2/3 \\ 2/3 & 2/3 \end{bmatrix}, B = \begin{bmatrix} 1/3 \\ 2/3 \end{bmatrix}, C = \begin{bmatrix} 1/3 & -2/3 \end{bmatrix}, D = 1/3$

$\endgroup$
1
$\begingroup$

Write $u_1=r-y_2$ and $u_2 = y_1$, then

\begin{align*} y_1 &= C_1 x_1 + D_1 (r - y_2)\\ &= C_1 x_1 - D_1 C_2 x_2 - D_1 D_2 u_2 + D_1 r \\ (I + D_1 D_2) y_1 &= \begin{bmatrix} C_1 & -D_1 C_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + D_1 r \\ y_1 &= \begin{bmatrix} R C_1 & -R D_1 C_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + R D_1 r \\ \end{align*}

where $R = (I + D_1 D_2)^{-1}$ provided that it exists (the system is admissible).

\begin{align*} \dot{x}_1 &= A_1 x_1 + B_1 (r - y_2) \\ &= A_1 x_1 - B_1 C_2 x_2 - B_1 D_2 u_2 + B_1 r \\ &= \begin{bmatrix} A_1 - B_1 D_2 R C_1 & -B_1 C_2 + B_1 D_2 R D_1 C_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + (- B_1 D_2 R D_1 + B_1) r \\ \dot{x}_2 &= A_2 x_2 + B_2 y_1 \\ &= \begin{bmatrix} B_2 R C_1 & A_2 - B_2 R D_1 C_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + B_2 R D_1 r \end{align*}

Therefore, the overall system can be written as

\begin{align*} \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &= \begin{bmatrix} A_1 - B_1 D_2 R C_1 & B_1 (D_2 R D_1 - I) C_2 \\ B_2 R C_1 & A_2 - B_2 R D_1 C_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} B_1(I - D_2 R D_1) \\ B_2 R D_1 \end{bmatrix} r \\ y_1 &= \begin{bmatrix} R C_1 & -R D_1 C_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + R D_1 r \end{align*}

$\endgroup$
  • $\begingroup$ question btw... what does "the system is admissible" mean? $\endgroup$ – WG- Sep 24 '14 at 19:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.