04.3 Transfer Function Matrix - A Misconception

From 03.1 Creating a LTI Object in MATLAB - Part 1, there might be a misconception that transfer function, $\mathbf{H}$ is a single polynomial. Something like the following:

And that is true- but for a SISO (Single Input-Single Output) system.

If you think of a MIMO system, then transfer function, $H(s)$ is not a single polynomial alone. This is a matrix, $\mathbf{H(s)}$. For instance, look up the following MIMO system from 05.1 Controllability 1:

MIMO System

\(\begin{bmatrix} \dot{x_1}\\ \dot{x_2}\\ \dot{x_3} \end{bmatrix}= \begin{bmatrix} -1 & 0 & 0 \\ 0 & -2 & 1\\ 0 & 2 & -2 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} + \begin{bmatrix} 0&1\\ 1&0\\ 1&0 \end{bmatrix} \begin{bmatrix} u_1\\ u_2 \end{bmatrix}\) \([y] =\begin{bmatrix}1&0&-1 \end{bmatrix} \begin{bmatrix} x_1\\x_2\\x_3 \end{bmatrix}+\begin{bmatrix} 0&0\end{bmatrix}\begin{bmatrix}u_1\\u2 \end{bmatrix}\)

Explanation

In this system, we have $M = 2$ meaning, 2 inputs, right? So, if you think in terms of a SISO system, you will be tripped up, like, there should be 2 $u$’s, right? \(\frac{y(s)}{u(s)} = \frac{52(s+9)}{s^2+10s+26}\) Therefore, the only explanation is the input-output relationship is like the following: \(y = \begin{bmatrix}H_{11}(s) & H_{12}(s) \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}\) So there are two transfer functions! You interpret it as: $H_{11}$ is transfer path from $u_1$ to $y$ and $H_{12}$ is transfer path from $u_2$ to 1 only.

Solution to the State-Space System Above

\(H_{11}(s) = \frac{2}{s^3+8s^2+19s+12}\) \(H_{12}(s)=\frac{2s^2+10s+10}{s^3+8s^2+19s+12}\) Now, typically they share the same poles but that’s another discussion (TODO)

MATLAB Technique

Problem Statement: If we have a MIMO system, and we would like to extract the sub-system only for the first input

RC = 1;
A = (1/RC)*[-3 1 0; 1 -2 1; 0 1 -3];
B = (1/RC)*[2 0; 0 0; 0 2];
C = [0 0 1];
D = [0 0];
ss_sys = ss(A,B,C,D);


%% Now this ss_sys has two transfer function, right? H_11 and H_12
%% So to access this TF you need to index 

ss_sys11 = ss_sys(1,1);
ss_sys12 = ss_sys(1,2);