Getting Started

Constructing 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.

You can specify 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 = [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. See Model Interconnection Functions in "Creating and Manipulating Models" online for more details on concatenation operations for LTI systems.

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 the Structures and Cell Arrays in the MATLAB online documentation for more details on cell arrays.

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

You can specify this MIMO transfer matrix 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)
  

The Control System Toolbox responds with

• 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 , the cell array entries N{i,j} and D{i,j} should be row-vector representations of the numerator and denominator of , the entry of the transfer matrix .

MIMO Example: Jet Transport Aircraft

Accessing I/O Pairs in MIMO Systems