Suppose I have a physical system, for example a mass-spring-damper system, which has been written in state space. Now, suppose that the matrices describing the system are \$A,B,C,D\$.
Moreover, let' s say it is a MIMO system, and I have \$6\$ inputs, which are \$u_{1},u_{2},d_{1},d_{2},n_{1},n_{2}\$, where the first two are the outputs of the controller, the second pair are the disturbances and the last ones are the noises.
Ans there are \$4\$ outputs, which are the displacement of a mass, defined as \$z_{1},z_{2}\$ each of which has two components.
So, I define the plant as:
G = ss(A,B,C,D)
now, I have my plant defined. Suppose now I want to find here the transfer functions of the systems, such as the sensitivity function, the complementary sensitivity function and the control sensitivity function, how could I do?
I know that I will have smething like:
\$\begin{bmatrix} z_{11}\\ z_{12}\\ z_{21}\\ z_{2,2} \end{bmatrix} = \begin{bmatrix} & & & \\ & & & \\ & & & \\ & & & \\ & & & \\ & & & \end{bmatrix} \cdot \begin{bmatrix} d_1\\ d_2\\ u_1\\ u_2\\ n_1\\ n_2 \end{bmatrix}\$
and to know the transfer function I need to look at the appropriate entry in the transfer matrix, but how do I know that the order of the inputs and of the outputs is this?
So, what I mean is that I could also have
\$\begin{bmatrix} z_{21}\\ z_{22}\\ z_{11}\\ z_{12} \end{bmatrix} = \begin{bmatrix} & & & \\ & & & \\ & & & \\ & & & \\ & & & \\ & & & \end{bmatrix} \cdot \begin{bmatrix} u_1\\ u_2\\ d_1\\ d_2\\ n_1\\ n_2 \end{bmatrix}\$
or other combinations, so if I do \$G(1,1)\$, in the first case I obtain a transfer function, and in the second case I obtain a different transfer function.
So, how can I do?