# How do I find specific transfer functions from a plant of a MIMO system?

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?

For any linear time-invariant system the transfer function will be

$$G(s) = YU^{-1} = C (sI-A)^{-1}B + D,$$

and the blank matrix in you question will be $$\G(s) \in \mathcal{R}^{4 \times 6} \$$, with the top element of the output matrix $$\ z\$$

$$z_{1} = ([C_{1i}] (sI-A)^{-1}B + D)U,$$

where $$\ [C_{1i}] \$$ is the row vector with the elements in the first row of $$\C\$$. Do notice that $$\z_1(u_1,u_2,\dots,n_2) \$$.

If you permute the inputs in $$\U\$$ it is the same as applying an invertible matrix $$\ P\$$ to have a $$\ \hat{U} = PU\$$. So, as long as $$\ P\$$ is just a permutation (a single 1 per line and no columns has more than a single 1), if $$\G(1,1)\$$ gives you a TF you will be able to find that same TF somewhere in $$\ \hat{G} \$$.

Moreover, if you have

$$\begin{bmatrix} d_1\\ d_2\\ u_1\\ u_2\\ n_1\\ n_2 \end{bmatrix} \rightarrow G,$$

$$\begin{bmatrix} u_1\\ u_2\\ d_1\\ d_2\\ n_1\\ n_2 \end{bmatrix} \rightarrow \hat{G},$$

The permutation is

$$P = \begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}$$

$$G(1,1) = \hat{G}(1,3).$$
• Thanks for answering, so if I have understood, if I select $G(1,1)$, doesn' t matter if I do a permutation of the inputs or of the outputs, I will alway get the same result, so always the same transfer function for $G(1,1)$. But how do I know which transfer function is in position $G(1,1)$? For example I know that the transfer function from the disturbance to the output is the sensitivity function, how do I understand if it is in $G(1,1)$? Thank you again.
• "if $G(1,1)$ gives you a TF you will be able to find that same TF somewhere in $\hat{G}$." but in many situations $G(1,1) \neq \hat{ G}(1,1)$.
• Not sure what you mean, but if you want to find how $u_4 \rightarrow y_1$ look at $z_{1,u_4} = z_{y_1,u_4} = ([C_{1,i}] (sI-A)^{-1}[B_{j,4}] + [D_{j,4}])u_4,$