How to create a transfer function system model from code with given parameters?

I want to replicate a transfer function system model I created with tfest in code.

Here is the model I want to replicate.

>> tf21 = tfest(data_est,2,1,'Ts',1)
tf21 =
 
  From input "u1" to output "y1":
          0.9614 z^-1
  ----------------------------
  1 - 1.663 z^-1 + 0.8313 z^-2
 
Name: tf21
Sample time: 1 seconds
Discrete-time identified transfer function.

Parameterization:
   Number of poles: 2   Number of zeros: 1
   Number of free coefficients: 3
   Use "tfdata", "getpvec", "getcov" for parameters and their uncertainties.

Status:                                              
Estimated using TFEST on time domain data "data_est".
Fit to estimation data: 84.81%                       
FPE: 1.025, MSE: 1.013                               
>> tf21.Numerator

ans =

         0    0.9614

>> tf21.Denominator

ans =

    1.0000   -1.6634    0.8313
    

Here's what I tried:

>> numerator = [0    0.9614];
>> denominator = [1.0000   -1.6634    0.8313];
>> tf_si = tf(numerator,denominator,'ts',1)

tfsi =
 
          0.9614
  ----------------------
  z^2 - 1.663 z + 0.8313
 
Sample time: 1 seconds
Discrete-time transfer function.

This doesn't look right. I need the same function that maps inputs to outputs. How do I set the coefficients for z^-2 and z^-1 on the denominator so they are the same as in the model I am trying to replicate?

Answer 1

[Star Strider]

Try this:

numerator = [0 0.9614 0];
denominator = [1.0000   -1.6634    0.8313 0];
tf_si = tf(numerator,denominator,'ts',1, 'Variable','z^-1')

producing:

tf_si =
 
          0.9614 z^-1
  ----------------------------
  1 - 1.663 z^-1 + 0.8313 z^-2
 
Sample time: 1 seconds
Discrete-time transfer function.

All the relevant vector elements need to be specified.