For this example, consider the following SISO transfer function model:

Specify the numerator and denominator coefficients ordered in descending powers of s, and create the transfer function model.

numerator = 1;
denominator = [2,3,4];
sys = tf(numerator,denominator)

sys =
 
         1
  ---------------
  2 s^2 + 3 s + 4
 
Continuous-time transfer function.

For this example, consider the following discrete-time SISO transfer function model:

Specify the numerator and denominator coefficients ordered in descending powers of z and the sample time of 0.1 seconds. Create the discrete-time transfer function model.

numerator = [2,0];
denominator = [4,0,3,-1];
ts = 0.1;
sys = tf(numerator,denominator,ts)

sys =
 
        2 z
  ---------------
  4 z^3 + 3 z - 1
 
Sample time: 0.1 seconds
Discrete-time transfer function.

For this example, consider a transfer function model that represents a second-order system with known natural frequency and damping ratio.

The transfer function of a second-order system, expressed in terms of its damping ratio $$\zeta$$ and natural frequency $${\omega }_0$$, is:

Assuming a damping ratio, $$\zeta$$ = 0.25 and natural frequency, $${\omega }_0$$ = 3 rad/s, create the second order transfer function.

zeta = 0.25;
w0 = 3;
numerator = w0^2;
denominator = [1,2*zeta*w0,w0^2];
sys = tf(numerator,denominator)

sys =
 
         9
  ---------------
  s^2 + 1.5 s + 9
 
Continuous-time transfer function.

Examine the response of this transfer function to a step input.

stepplot(sys)

The plot shows the ringdown expected of a second-order system with a low damping ratio.

Create a transfer function for the discrete-time, multi-input, multi-output model:

with sample time ts = 0.2 seconds.

Specify the numerator coefficients as a 2-by-2 matrix.

numerators = {1 [1 0];[-1 2] 3};

Specify the coefficients of the common denominator as a row vector.

denominator = [1 0.3];

Create the discrete-time MIMO transfer function model.

ts = 0.2;
sys = tf(numerators,denominator,ts)

sys =
 
  From input 1 to output...
          1
   1:  -------
       z + 0.3
 
       -z + 2
   2:  -------
       z + 0.3
 
  From input 2 to output...
          z
   1:  -------
       z + 0.3
 
          3
   2:  -------
       z + 0.3
 
Sample time: 0.2 seconds
Discrete-time transfer function.

For more information on creating MIMO transfer functions, see MIMO Transfer Functions.

In this example, you create a MIMO transfer function model by concatenating SISO transfer function models. Consider the following single-input, two-output transfer function:

Specify the MIMO transfer function model by concatenating the SISO entries.

sys1 = tf([1 -1],[1 1]);		
sys2 = tf([1 2],[1 4 5]);
sys = [sys1;sys2]

sys =
 
  From input to output...
       s - 1
   1:  -----
       s + 1
 
           s + 2
   2:  -------------
       s^2 + 4 s + 5
 
Continuous-time transfer function.

For more information on creating MIMO transfer functions, see MIMO Transfer Functions.

For this example, create a continuous-time transfer function model using rational expressions. Using a rational expression can sometimes be easier and more intuitive than specifying polynomial coefficients of the numerator and denominator.

Consider the following system:

To create the transfer function model, first specify s as a tf object.

s = tf('s')

s =
 
  s
 
Continuous-time transfer function.

Create the transfer function model using s in the rational expression.

sys = s/(s^2 + 2*s + 10)

sys =
 
        s
  --------------
  s^2 + 2 s + 10
 
Continuous-time transfer function.

For this example, create a discrete-time transfer function model using a rational expression. Using a rational expression can sometimes be easier and more intuitive than specifying polynomial coefficients.

Consider the following system:

To create the transfer function model, first specify z as a tf object and the sample time Ts.

ts = 0.1;
z = tf('z',ts)

z =
 
  z
 
Sample time: 0.1 seconds
Discrete-time transfer function.

Create the transfer function model using z in the rational expression.

sys = (z - 1) / (z^2 - 1.85*z + 0.9)

sys =
 
        z - 1
  ------------------
  z^2 - 1.85 z + 0.9
 
Sample time: 0.1 seconds
Discrete-time transfer function.

For this example, create a transfer function model with properties inherited from another transfer function model. Consider the following two transfer functions:

For this example, create sys1 with the TimeUnit and InputDelay property set to 'minutes'.

numerator1 = [2,0];
denominator1 = [1,8,0];
sys1 = tf(numerator1,denominator1,'TimeUnit','minutes','InputUnit','minutes')

sys1 =
 
     2 s
  ---------
  s^2 + 8 s
 
Continuous-time transfer function.

propValues1 = [sys1.TimeUnit,sys1.InputUnit]

propValues1 = 1x2 cell
    {'minutes'}    {'minutes'}

Create the second transfer function model with properties inherited from sys1.

numerator2 = [1,-1];
denominator2 = [7,2,0,0,9];
sys2 = tf(numerator2,denominator2,sys1)

sys2 =
 
        s - 1
  -----------------
  7 s^4 + 2 s^3 + 9
 
Continuous-time transfer function.

propValues2 = [sys2.TimeUnit,sys2.InputUnit]

propValues2 = 1x2 cell
    {'minutes'}    {'minutes'}

Observe that the transfer function model sys2 has that same properties as sys1.

You can use a for loop to specify an array of transfer function models.

First, pre-allocate the transfer function array with zeros.

sys = tf(zeros(1,1,3));

The first two indices represent the number of outputs and inputs for the models, while the third index is the number of models in the array.

Create the transfer function model array using a rational expression in the for loop.

s = tf('s');                                                  
for k = 1:3                                                             
    sys(:,:,k) = k/(s^2+s+k);                                          
end
sys

sys(:,:,1,1) =
 
       1
  -----------
  s^2 + s + 1
 

sys(:,:,2,1) =
 
       2
  -----------
  s^2 + s + 2
 

sys(:,:,3,1) =
 
       3
  -----------
  s^2 + s + 3
 
3x1 array of continuous-time transfer functions.

For this example, compute the transfer function of the following state-space model:

Create the state-space model using the state-space matrices.

A = [-2 -1;1 -2];
B = [1 1;2 -1];
C = [1 0];
D = [0 1];
ltiSys = ss(A,B,C,D);

Convert the state-space model ltiSys to a transfer function.

sys = tf(ltiSys)

sys =
 
  From input 1 to output:
  s + 2.22e-16
  -------------
  s^2 + 4 s + 5
 
  From input 2 to output:
  s^2 + 5 s + 8
  -------------
  s^2 + 4 s + 5
 
Continuous-time transfer function.

For this example, extract the measured and noise components of an identified polynomial model into two separate transfer functions.

Load the Box-Jenkins polynomial model ltiSys in identifiedModel.mat.

load('identifiedModel.mat','ltiSys');

ltiSys is an identified discrete-time model of the form: $$y(t)=\frac{B}{F}u(t)+\frac{C}{D}e(t)$$, where $$\frac{B}{F}$$ represents the measured component and $$\frac{C}{D}$$ the noise component.

Extract the measured and noise components as transfer functions.

sysMeas = tf(ltiSys,'measured') 

sysMeas =
 
  From input "u1" to output "y1":
            -0.1426 z^-1 + 0.1958 z^-2
  z^(-2) * ----------------------------
           1 - 1.575 z^-1 + 0.6115 z^-2
 
Sample time: 0.04 seconds
Discrete-time transfer function.

sysNoise = tf(ltiSys,'noise')

sysNoise =
 
  From input "v@y1" to output "y1":
           0.04556 + 0.03301 z^-1
  ----------------------------------------
  1 - 1.026 z^-1 + 0.26 z^-2 - 0.1949 z^-3
 
Input groups:        
    Name     Channels
    Noise       1    
                     
Sample time: 0.04 seconds
Discrete-time transfer function.

The measured component can serve as a plant model, while the noise component can be used as a disturbance model for control system design.

Transfer function model objects include model data that helps you keep track of what the model represents. For instance, you can assign names to the inputs and outputs of your model.

Consider the following continuous-time MIMO transfer function model:

The model has one input $$-$$ Current, and two outputs $$-$$ Torque and Angular velocity.

First, specify the numerator and denominator coefficients of the model.

numerators = {[1 1] ; 1};
denominators = {[1 2 2] ; [1 0]};

Create the transfer function model, specifying the input name and output names.

sys = tf(numerators,denominators,'InputName','Current',...
        'OutputName',{'Torque' 'Angular Velocity'})

sys =
 
  From input "Current" to output...
                s + 1
   Torque:  -------------
            s^2 + 2 s + 2
 
                      1
   Angular Velocity:  -
                      s
 
Continuous-time transfer function.

For this example, specify polynomial ordering in discrete-time transfer function models using the 'Variable' property.

Consider the following discrete-time transfer functions with sample time 0.1 seconds:

Create the first discrete-time transfer function by specifying the z coefficients.

numerator = [1,0,0];
denominator = [1,2,3];
ts = 0.1;
sys1 = tf(numerator,denominator,ts)

sys1 =
 
       z^2
  -------------
  z^2 + 2 z + 3
 
Sample time: 0.1 seconds
Discrete-time transfer function.

The coefficients of sys1 are ordered in descending powers of z.

tf switches convention based on the value of the 'Variable' property. Since sys2 is the inverse transfer function model of sys1, specify 'Variable' as 'z^-1' and use the same numerator and denominator coefficients.

sys2 = tf(numerator,denominator,ts,'Variable','z^-1')

sys2 =
 
           1
  -------------------
  1 + 2 z^-1 + 3 z^-2
 
Sample time: 0.1 seconds
Discrete-time transfer function.

The coefficients of sys2 are now ordered in ascending powers of z^-1.

Based on different conventions, you can specify polynomial ordering in transfer function models using the 'Variable' property.

In this example, you will create a low-pass filter with one tunable parameter a:

Since the numerator and denominator coefficients of a tunableTF block are independent, you cannot use tunableTF to represent F. Instead, construct F using the tunable real parameter object realp.

Create a real tunable parameter with an initial value of 10.

a = realp('a',10)

a = 
       Name: 'a'
      Value: 10
    Minimum: -Inf
    Maximum: Inf
       Free: 1

Real scalar parameter.

Use tf to create the tunable low-pass filter F.

numerator = a;
denominator = [1,a];
F = tf(numerator,denominator)

F =

  Generalized continuous-time state-space model with 1 outputs, 1 inputs, 1 states, and the following blocks:
    a: Scalar parameter, 2 occurrences.

Type "ss(F)" to see the current value, "get(F)" to see all properties, and "F.Blocks" to interact with the blocks.

F is a genss object which has the tunable parameter a in its Blocks property. You can connect F with other tunable or numeric models to create more complex control system models. For an example, see Control System with Tunable Components.

In this example, you will create a static gain MIMO transfer function model.

Consider the following two-input, two-output static gain matrix m:

Specify the gain matrix and create the static gain transfer function model.

m = [2,4;...
    3,5];
sys1 = tf(m)

sys1 =
 
  From input 1 to output...
   1:  2
 
   2:  3
 
  From input 2 to output...
   1:  4
 
   2:  5
 
Static gain.

You can use static gain transfer function model sys1 obtained above to cascade it with another transfer function model.

For this example, create another two-input, two-output discrete transfer function model and use the series function to connect the two models.

numerators = {1,[1,0];[-1,2],3};
denominator = [1,0.3];
ts = 0.2;
sys2 = tf(numerators,denominator,ts)

sys2 =
 
  From input 1 to output...
          1
   1:  -------
       z + 0.3
 
       -z + 2
   2:  -------
       z + 0.3
 
  From input 2 to output...
          z
   1:  -------
       z + 0.3
 
          3
   2:  -------
       z + 0.3
 
Sample time: 0.2 seconds
Discrete-time transfer function.

sys = series(sys1,sys2)

sys =
 
  From input 1 to output...
       3 z^2 + 2.9 z + 0.6
   1:  -------------------
       z^2 + 0.6 z + 0.09
 
       -2 z^2 + 12.4 z + 3.9
   2:  ---------------------
        z^2 + 0.6 z + 0.09
 
  From input 2 to output...
       5 z^2 + 5.5 z + 1.2
   1:  -------------------
       z^2 + 0.6 z + 0.09
 
       -4 z^2 + 21.8 z + 6.9
   2:  ---------------------
        z^2 + 0.6 z + 0.09
 
Sample time: 0.2 seconds
Discrete-time transfer function.