Why does the step response to my discrete system not match between MATLAB and Simulink?

Hello, I am trying to figure out why I get different step responses to the same discrete system when I model the system in Simulink and MATLAB. Below is an example of what I am experiencing. The Following MATLAB code

G = tf(24.3,[3746 137.4 1]) % My original continuous time system.
Gz = c2d(G,1) % Discretization of the system with Ts = 1sec sample time.

step(Gz) %Step response to Gz

Produces

>> StepResponse_Comparison

G =

             24.3
    ----------------------
    3746 s^2 + 137.4 s + 1

Continuous-time transfer function.

Gz = 
  0.003204 z + 0.003165
  ---------------------
  z^2 - 1.964 z + 0.964

Sample time: 1 seconds
Discrete-time transfer function.

The model of the same system in Simulink (At least I think its the same). Sample time is set to 1 in the step block and inherited by the transfer function block. Initial conditions are all 0.

Produces

From what I understand these two plots should match. Any explanation for what is going on here would be much appreciated. Thank you.

Answer 1

[Steve Lee]

I had same problem.

Answer 2

[Luca Ferro]

Had the same problem earlier, it's late but it can be helpful for others i guess:

instead of using the discrete transfer function block use the LTI system block.

For PID blocks use the PID controller block as usual.

There is no need for ZOH or Rate transition blocks.

Also try to include the delays inside of the transfer functions, avoiding the use of the external Delay block.

Answer 3

[Paul]

The explanation for the Simulink resutls is that the coefficients entered in the Discrete Transfer Function block don't represent the actual transfer function G(z).

Here is how Gz is developed

G = tf(24.3,[3746 137.4 1]) % My original continuous time system.

G =
 
           24.3
  ----------------------
  3746 s^2 + 137.4 s + 1
 
Continuous-time transfer function.

Gz = c2d(G,1) % Discretization of the system with Ts = 1sec sample time.

Gz =
 
  0.003204 z + 0.003165
  ---------------------
  z^2 - 1.964 z + 0.964
 
Sample time: 1 seconds
Discrete-time transfer function.

Based on what's shown in the display, the transfer function entered into Simulink was

Gsimulink = tf([0.003204 0.003165],[1 -1.964 0.964],1)

Gsimulink =
 
  0.003204 z + 0.003165
  ---------------------
  z^2 - 1.964 z + 0.964
 
Sample time: 1 seconds
Discrete-time transfer function.

But Gz and Gsimulink result in different responses, with Gsimulink matching the Simulink solution.

step(Gz,Gsimulink,800)

The actual numerator and denominator of Gz are

format long e
Gz.num{:}

ans = 1×3

0 3.204093815520879e-03 3.165157912686405e-03

Gz.den{:}

ans = 1×3

1.000000000000000e+00 -1.963723295999098e+00 9.639854051237154e-01

Apparently all (well, maybe not all, but certainly more than three) of those significant digits matter.

The solution is to either enter Gz.num{:} and Gz.den{:} as the Numerator and Denominator block parameters in the Discrete Transfer Function block, or enter Gz as the LTI System Variable parameter in the LTI system block, as suggested by @Luca Ferro

[Luca Ferro]

i already tried that road and it doesn's seem like a perfect match:

When doing a set_param function it requires a string and during the conversion (red rectangle), it loses precision.

While by loading from workspace it actually work as intended:

As @Paul said they are equivalent, i was just missing the idea of working with the workspace instead of setting the parameters directly.

Do you know if Discrete Transfer Fcn block offer some advantages compared to LTI System block?

[Paul]

Use num2str to not lose that precision if needing to set the parameter as a literal vector

dcTf = tf(1,[0.2228665 0.770675 1]);
format long e
disDcTf = c2d(dcTf,0.01)

disDcTf =
 
  0.0002218 z + 0.0002192
  -----------------------
   z^2 - 1.966 z + 0.966
 
Sample time: 0.01 seconds
Discrete-time transfer function.

disDcTf.den{:}

ans = 1×3

1.000000000000000e+00 -1.965569927908843e+00 9.660109411503381e-01

"[" + string(num2str(disDcTf.den{:},16)) + "]"

ans = '[1     -1.965569927908843     0.9660109411503381]'

Don't know if there are any corner cases where the string literal will be different in the last bit or something like that. IMO, It's best to avoid this conversion and set the parameters with Gz.num{:} and Gz.den{:} or use the LTI System block with Gz.

The DTTF block offers some additional features, like setting initial states, external reset, and frame based processing. I don't think the LTI System block has those.