## MIMO Transfer Functions

MIMO transfer functions are two-dimensional arrays of elementary SISO transfer functions. There are two ways to specify MIMO transfer function models:

• Concatenation of SISO transfer function models

• Using `tf` with cell array arguments

### Concatenation of SISO Models

Consider the following single-input, two-output transfer function.

`$H\left(s\right)=\left[\begin{array}{c}\frac{s-1}{s+1}\\ \frac{s+2}{{s}^{2}+4s+5}\end{array}\right].$`

You can specify H(s) by concatenation of its SISO entries. For instance,

```h11 = tf([1 -1],[1 1]); h21 = tf([1 2],[1 4 5]); ```

or, equivalently,

```s = tf('s') h11 = (s-1)/(s+1); h21 = (s+2)/(s^2+4*s+5); ```

can be concatenated to form H(s).

```H = [h11; h21] ```

This syntax mimics standard matrix concatenation and tends to be easier and more readable for MIMO systems with many inputs and/or outputs.

Tip

Use `zpk` instead of `tf` to create MIMO transfer functions in factorized form.

### Using the tf Function with Cell Arrays

Alternatively, to define MIMO transfer functions using `tf`, you need two cell arrays (say, `N` and `D`) to represent the sets of numerator and denominator polynomials, respectively. See What Is a Cell Array? for more details on cell arrays.

For example, for the rational transfer matrix H(s), the two cell arrays `N` and `D` should contain the row-vector representations of the polynomial entries of

`$N\left(s\right)=\left[\frac{s-1}{s+2}\right],\text{ }D\left(s\right)=\left[\frac{s+1}{{s}^{2}+4s+5}\right].$`

You can specify this MIMO transfer matrix H(s) by typing

```N = {[1 -1];[1 2]}; % Cell array for N(s) D = {[1 1];[1 4 5]}; % Cell array for D(s) H = tf(N,D) ```
```Transfer function from input to output... s - 1 #1: ----- s + 1 s + 2 #2: ------------- s^2 + 4 s + 5 ```

Notice that both `N` and `D` have the same dimensions as H. For a general MIMO transfer matrix H(s), the cell array entries `N{i,j}` and `D{i,j}` should be row-vector representations of the numerator and denominator of Hij(s), the ijth entry of the transfer matrix H(s).