Control Engineering conver transfer function MIMO to state space

I have transfer function 2x2 MIMO with 2 inputs 2 outputs and 3 states.

How can i convert the transfer function to state space form using Matlab?

Answer 1

[Shashi Kiran]

Hi @meng fowler,

To convert a 2x2 MIMO transfer function system with two inputs, two outputs, and three states into a state-space representation in MATLAB, you can follow these steps:

• To create the transfer function matrix for a MIMO system (given the individual transfer functions between the inputs and outputs).

s = tf('s');  % Define the Laplace variable

% Transfer functions from input 1 to outputs
TF11 = (s + 2)/(s^3 + 2*s^2 + 3*s + 4);  % Output 1, Input 1
TF21 = (2*s + 1)/(s^3 + s^2 + 2*s + 3);  % Output 2, Input 1

% Transfer functions from input 2 to outputs
TF12 = (s + 3)/(s^3 + s^2 + 3*s + 2);    % Output 1, Input 2
TF22 = (s + 1)/(s^3 + 2*s^2 + s + 1);    % Output 2, Input 2

% Create the MIMO transfer function matrix (2x2 system)
sys_tf = [TF11 TF12; TF21 TF22];

• Use MATLAB’s 'ss' function to convert the transfer function model to a state-space representation.

% Convert to state-space representation
sys_ss = ss(sys_tf);

% Display the state-space matrices
A = sys_ss.A % Represents the system dynamics.

A = 12×12

-2.0000 -1.5000 -2.0000 0 0 0 0 0 0 0 0 0 2.0000 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.0000 -1.0000 -1.5000 0 0 0 0 0 0 0 0 0 2.0000 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.0000 -1.5000 -1.0000 0 0 0 0 0 0 0 0 0 2.0000 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 -2.0000 -1.0000 -1.0000

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B = sys_ss.B % Maps the inputs to the states.

B = 12×2

1 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 2

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C = sys_ss.C % Maps the states to the outputs.

C = 2×12

0 0.5000 1.0000 0 0 0 0 0.2500 0.7500 0 0 0 0 0 0 0 1.0000 0.5000 0 0 0 0 0.5000 0.5000

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D = sys_ss.D % Direct input-output relationship

D = 2×2

0 0 0 0

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Please refer to the following documentations for more details about the functions:

1. ss:https://www.mathworks.com/help/control/ref/ss.html
2. tf: https://www.mathworks.com/help/control/ref/tf.html

Hope this helps.