Re: C2D Function

[Lukas Reichlin]

On 27.11.2011, at 06:29, Lukas Reichlin wrote:

> On 26.11.2011, at 21:02, Luke Trowbridge wrote:
> 
>> Hello:
>> 
>> When I execute the following code:
>> 
>> 
>> num = [5.0576e20, 0, 0, 0];
>> den = [1, 2.5296e4, 1.002247e9, 1.4011535e13, 2.1985832e17, 1.417505e21, 
>> 7.4138682e24, 1.588969e28, 1.6328178e31];
>> atf = tf(num, den);
>> dtf = c2d(atf, 1e-5);
>> 
>> 
>> Octave 3.2.4 and control package 1.0.11 return a sensible result for the 
>> variable dtf.  However, when the same code is executed with Octave 3.4.3 and 
>> control package 2.2.1, the value of the variable dtf is given as follows:
>> 
>> Transfer function "dtf" from input "u1" to output ...
>> y1:  0
>> Continuous-time model.
>> 
>> 
>> The C2D function seems to work fine in octave 3.4.3 if I use transfer 
>> functions with numerator/denominator coefficients that are a little close 
>> together (within 10^15).  How should I go about this continuous to digital 
>> conversion in octave 3.4.3?
>> 
>> Thanks for any help,
>> Luke Trowbridge
> 
> Hi Luke
> 
> The problem is not c2d. The problem is the transfer function to state-space 
> conversion (internally, c2d is computed in state-space). You can try
> 
> num = [5.0576e20, 0, 0, 0];
> den = [1, 2.5296e4, 1.002247e9, 1.4011535e13, 2.1985832e17, 1.417505e21, 
> 7.4138682e24, 1.588969e28, 1.6328178e31];
> atf = tf(num, den);
> dtf = c2d(atf, 1e-5);
> ss (atf)
> 
> ans.d =
>       u1
>   y1   0
> 
> Continuous-time model.
> 
> The problem is caused by the very large num and den coefficients which lead 
> to very small numbers in the observer canonical form (Wolovich's observable 
> structure theorem) produced by Slicot TD04AD:
> 
> ss (atf)
> 
> ans.a =
>               x1          x2          x3          x4          x5          x6  
>         x7          x8
>   x1   3.411e-13  -1.598e-13    2.14e-14           0           0           0  
>          0      -163.3
>   x2       -1000   -2.97e-15    1.65e-13           0           0           0  
>          0       158.9
>   x3           0       -1000  -6.094e-14           0           0           0  
>          0      -74.14
>   x4           0           0        1000           0           0           0  
>          0      -14.18
>   x5           0           0           0      -1e+04           0           0  
>          0       21.99
>   x6           0           0           0           0      -1e+05           0  
>          0      -140.1
>   x7           0           0           0           0           0      -1e+06  
>          0   1.002e+04
>   x8           0           0           0           0           0           0  
>     -1e+05   -2.53e+04
> 
> ans.b =
>               u1
>   x1           0
>   x2           0
>   x3           0
>   x4  -5.058e+04
>   x5           0
>   x6           0
>   x7           0
>   x8           0
> 
> ans.c =
>            x1       x2       x3       x4       x5       x6       x7       x8
>   y1        0        0        0        0        0        0        0  -0.0001
> 
> ans.d =
>       u1
>   y1   0
> 
> Continuous-time model.
> 
> With the chosen default tolerance in control/@tf/__sys2ss__.m, these are 
> considered to be not of full rank and are cancelled. Result is a model with 
> no remaining states and feedthrough matrice 0.
> 
> You can change line 118 in control/@tf/__sys2ss__.m from
>  [a, b, c, d] = sltd04ad (ucoeff, dcoeff, index, sqrt (eps));
> to
>  [a, b, c, d] = sltd04ad (ucoeff, dcoeff, index, 0);
> to get the result above and the discrete system dtf:
> 
> 
> Transfer function "dtf" from input "u1" to output ...
> 
>      4.034e-07 z^7 + 8.808e-06 z^6 - 4.499e-06 z^5 - 3.418e-05 z^4 + 
> 3.573e-05 z^3 + 2.257e-06 z^2 - 8.186e-06 z - 3.408e-07
> y1:  
> -----------------------------------------------------------------------------------------------------------------------
>                   z^8 - 7.682 z^7 + 25.88 z^6 - 49.96 z^5 + 60.42 z^4 - 46.89 
> z^3 + 22.8 z^2 - 6.353 z + 0.7765             
> 
> Sampling time: 1e-05 s
> Discrete-time model.
> 
> I don't have a quick solution in general. But my advice is to properly scale 
> input-output channels of your models as we do *numerical* computations after 
> all where scaling is an important thing.
> 
> Best regards,
> Lukas

I revised the default tolerance for rank determination in TD04AD. Try replacing 
the code from control/@tf/__sys2ss__.m with the code from this link:

http://octave.svn.sourceforge.net/viewvc/octave/trunk/octave-forge/main/control/inst/%40tf/__sys2ss__.m?revision=9203&view=markup

My advice on scaling is still very important. If num and den coefficients get 
even larger than they already are, they will get cancelled.

Lukas