Mathematica: Simplifying a feedback system

[divB]

Hi,

I have given a complex feedback system in z domain like:

Y = 2*W2 + E
W2 = z^-1/(1-z^-1) * (W1 + (1-z^-1)*W2 - 1/2*(1-z^-1)*Y)
W1 = V - Y + z^-1/(1-z^-1) * 1/2 * (V - Y)

where Y is the system output, V is the system input, W1 and W2 are intermediate nodes and E is an error input

I and want to get an expression like

Y(z) = E(z)*(...) + X(z)*(...)

i.e., the system output as a sum of the error input and the input. Alternatively: signal transfer function and error transfer function.

Just defining the terms as above and using Solve[] or so is not really successful.


Anyone a hint how to do that?

Thanks
divB

 

[Hepth]

Does this get what you want?:

Solve[{Y == 2*W2 + e, W2 == z^(-1)/(1 - z^(-1))*(W1 + (1 - z^(-1))*W2 - 1/2*(1 - z^(-1))*Y),
   W1 == V - Y + z^(-1)/(1 - z^(-1))*1/2*(V - Y)}, {Y, W1, W2}];

Collect[Expand[%], e, Simplify]

Gives : (where I use little "e" as E, since E is the Exp[1] exponential.

$$Y\to \frac{e (-1+z)^3}{z^3}+\frac{V (-1+2 z)}{z^2}$$

 

[divB]

Woow! Thanks a lot.
It doesn't match with my hand calculations :-( - buts that's the reason I wanted to check it.

So the key here is to supply Solve[] a list of the indendent set of equations and a list of all inter connected variables?

 

[Hepth]

You need to supply it with N equations and solve for N independent variables. I don't know if i put in enough assumptions to let MM know if z was compelx or not, of it it just assumes it.

 

[alphy]

I would suggest using a mathematical software such as Mathematica to simplify the feedback system in z domain. Mathematica has a variety of built-in functions and tools that can help you simplify complex equations and solve for specific variables.

One approach you could take is to use the Simplify[] function to simplify each equation individually, and then use the Eliminate[] function to eliminate the intermediate variables W1 and W2. This would leave you with an expression for Y in terms of V and E.

Another approach could be to use the TransferFunctionModel[] function to create a transfer function model for the entire system, and then use the TransferFunctionSimplify[] function to simplify it further.

I would also recommend consulting the Mathematica documentation or reaching out to the Mathematica community for additional tips and techniques on simplifying feedback systems.