Understanding the ss() and tf2ss() functions and their differences

Question

Following this question, I am trying to comprehend the functionality of MATLAB's ss() and tf2ss() functions and their differences. Consider the code below:

Kf = 0.2;   
J = 0.02; 

h2 = tf(1, [J Kf]);

dcm1 = ss(h2);
dcm2 = tf2ss(h2.Numerator{1}, h2.Denominator{1});

partially copied from here. Now I expect the dcm1 and dcm2 to be identical but here is what I get:

>> dcm1

dcm1 =
 
  A = 
        x1
   x1  -10
 
  B = 
       u1
   x1   8
 
  C = 
         x1
   y1  6.25
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

>> dcm2

dcm2 =

   -10

I would appreciate it if you could help me understand why am I getting two different results? how I can have identical results using the tf2ss() function? In other words, I want to create a dcm2 which is identical to dcm1 but using the tf2ss() function.

StaticBeagle · Accepted Answer · 2022-02-18T01:17:38.960

1

ss returns a state-space system, similar to tf they both return continuous systems in ss and tf representation respectively.

ss2tf returns the A, B, C, D matrices which are the result of converting the transfer function tf into its state-space representation.

There's one caveat though, the state-space representation of a transfer function is not unique. It seems like Matlab uses two different algorithms for ss and tf2ss. You can verify that both systems return the same transfer function:

Kf = 0.2;   
J = 0.02; 

h2 = tf(1, [J Kf]);

dcm1 = ss(h2);
[A, B, C, D] = tf2ss(h2.Numerator{1}, h2.Denominator{1});

dcm1 =

[A, B, C, D]

A = -10
B = 1
C = 50
D = 0

Now let's compare transfer functions:

[b, a] = ss2tf(A, B, C, D)
% yields
b = [0 50]
a = [1 10]

[b, a] = ss2tf(dcm1.a, dcm1.b, dcm1.c, dcm1.d)
% yields
b = [0 50]
a = [1 10]

edited Feb 18 '22 at 01:17

answered Feb 17 '22 at 23:21

StaticBeagle

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have you tried the suggested line above? The result is not identical! – Foad S. Farimani Feb 17 '22 at 23:28
@FoadS.Farimani the matrices will contain the same values, but the result won't be wrapped in a continuous time system representation. If you try printing `dcm1.a` and `A` to the console you should see `-10`. – StaticBeagle Feb 17 '22 at 23:32
It appears that the `ss()` function, returns `ss` class, but the `tf2ss()` returns `double`. – Foad S. Farimani Feb 17 '22 at 23:32
@FoadS.Farimani yep that is correct. – StaticBeagle Feb 17 '22 at 23:33
nope, `dcm1 --> A = -10 , B = 8, C = 6.25, D = 0` but `dcm2 --> -10 1 50 0` – Foad S. Farimani Feb 17 '22 at 23:34
1

@FoadS.Farimani I've updated my answer. The reason `ss` and `tf2ss` are returning different results is because the state-space representation of a transfer function is not unique, and it seems like each method use a different algorithm. – StaticBeagle Feb 18 '22 at 01:20
This is strange! Thanks for the elaboration. I will do more research and come back. – Foad S. Farimani Feb 18 '22 at 07:16
Thanks for your kind support. [This follow-up Q/A](https://stackoverflow.com/q/71213364/4999991) might also be interesting for you. – Foad S. Farimani Feb 23 '22 at 09:15
1

@FoadS.Farimani thanks for sharing the info. These were a good set of questions! – StaticBeagle Feb 23 '22 at 14:00

Understanding the ss() and tf2ss() functions and their differences

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