Controllability of parallel systems

Question

Can anyone give me a hint how to use Popov-Belevitch-Hautus (PBH) test to prove that parallel interconnection of two linear controllable single-input single-output systems is controllable if and only if they have no common eigenvalue?!

Answer 1

A parallel interconnection of two systems has the following block diagram

Here $H_1$ and $H_2$ can be written as the following two state space models

$$ \left\{\begin{matrix} \dot{x}_1 = A_1\,x_1 + B_1\,u_1 \\ y_1 = C_1\,x_1 + D_1\,u_1 \end{matrix}\right. $$ $$ \left\{\begin{matrix} \dot{x}_2 = A_2\,x_2 + B_2\,u_2 \\ y_2 = C_2\,x_2 + D_2\,u_2 \end{matrix}\right. $$

When considering the parallel interconnection then $u=u_1=u_2$ and $y=y_1+y_2$ so the combined state space model gives

$$ \begin{matrix} \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} A_1 & 0 \\ 0 & A_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} B_1 \\ B_2 \end{bmatrix} u \\ y = \begin{bmatrix} C_1 & C_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} D_1 & D_2 \end{bmatrix} u \end{matrix} $$

Now you have to consider that $\left[A_i - \lambda\,I \quad B_i\right]$ is full rank for all $\lambda \in \mathbb{C}$, so eigenvalues of $A_i$.

Now by looking at this same criteria of the parallel interconnection you can say something about its controllability. However I believe that the statement that the parallel interconnection is controllable if and only if they have no common eigenvalue, only holds for the single input case. For the multiple input case this condition becomes a sufficient but not necessary condition. For example when $\begin{bmatrix}B_1^\top & B_2^\top\end{bmatrix}^\top$ is already full rank, then it does not matter what $A_1$ and $A_2$ are.