How to convert symbolic expressions to transfer functions

I am encountering the problem of converting a symbolic expression to become a transfer function. Specifically, the linear system I am dealing with contains a non-constant distributed delay term which requires performing an integration to obtain the corresponding transfer function. However, it seems that the integration operator int cannot be applied with tf variables directly.

On the other hand, if there is a way to convert symbolic expressions to transfer functions, then this problem can be easily handled in symbolic setting first.

Thanks a lot

[Karan Gill]

Could you provide your code? I didn't understand " int cannot be applied with tf variables directly". Seeing your code would help understand the problem better. Also, I see a file exchange link in the right column of this page that seems to be relevant: Syms to TF Conversion

Karan (Symbolic Math documentation)

[Star Strider]

I suggested a similar approach yesterday. It’s apparently not a polynomial.

[Qian Feng]

Here is the code,

r = 1; 
s = tf('s');
syms x 
A4 = [-1 x; -1-x^3   -1];
Ap = int(A4*exp(x*s),x, -r, 0);

The reason why we have an integration there is because I am dealing with a distributed delay term in the time domain.

The problem is that it seems we cannot mix a tf variable with a symbolic variable here.

However, the aforementioned integration can be easily handle if s is a symbolic variable, which is the reason why I asked about how to transfer a symbolic entity into a tf one.

Answer 1

[Walter Roberson]

https://www.mathworks.com/matlabcentral/fileexchange/10816-mimo-toolbox/content/Mimotools/tf2sym.m

[Qian Feng]

Thanks a lot, the problem is that I am looking for a solution to transfer symbolic entities into transfer functions not vice versa.

[Walter Roberson]

No, your reported error was about not being able to int() a tf. You therefore need to transform your tf into a sym so that you can int()

Karan already gave a link for transforming sym into tf: http://www.mathworks.com/matlabcentral/fileexchange/27302-syms-to-tf-conversion

Note: transforming an symbolic expression into a transfer function is only possible if there is at most one symbolic variable. The MATLAB control system facilities cannot work with coefficients that are symbolic. For more on what can be done with alterable control system coefficients, please see http://www.mathworks.com/matlabcentral/answers/305339-how-to-create-a-transfer-function-with-gain-k#comment_395202

[Qian Feng]

Thanks a lot. As a matter of fact, if I directly apply symbolic variable that is

r = 1; 
syms x s
A4 = [-1 x; -1-x^3   -1];
Ap = int(A4*exp(x*s),x, -r, 0);

then Ap can be easily obtained. The problem here is that I need Ap to be a transfer function to calculate the peak value of the frequency response. So that is why I asked a way to transfer symbolic variables to transfer functions.

[Walter Roberson]

Your Ap is a 2 x 2 array; how do you wish to convert to a single transfer function?

[Qian Feng]

A transfer matrix then. Ap is part of the operation of the transfer function I considered. Namely, G = C*inv(sI - A)*B + D, and the final G is a transfer function of SISO system.

[Saumik Kumar Dey]

A way to convert symbolic expressions in 's' in transfer function:

syms s

% Symbolic expression in 's'
symExp = (s+2)/(s^2+5*s+9);    

% Convert to a string
TFstr = char(symExp);

% Define 's' as transfer function variable
s = tf('s');

% Evaluate the expression: "TF = (s+2)/(s^2+5*s+9)"
eval(['TF = ',TFstr])

Run the above lines of code and you will get one Continuous-time transfer function varible "TF" in your workspace.

[Bill Tubbs]

Would be nice if this worked:

syms s
G = 5/(10*s+1);
subs(G,s,tf('s'))

but it raises

Error using sym/subs>normalize (line 217)

Substitution expression X must be a symbolic, cell, or numeric array.

[Walter Roberson]

A more robust formulation than using eval():

syms s
symExp = (s+2)/(s^2+5*s+9);

ExpFun = matlabFunction(symExp);
ExpFun = str2func(regexprep(func2str(ExpFun), '\.([/^\\*])', '$1'));

TF = ExpFun(tf('s'));

The regexp() is there to change ./ and .^ and .\ and .* and to / ^ \ * respectively, as the dotted form of the operations are not defined for tf .

The .' operator does not need to be changed as .' is defined for tf, and ' is a different operator for tf . (Also, it is difficult to get a .' or ' remaining in a symbolic expression, as they would normally have gotten expanded out.)

[leonard-riccardo wecke]

not working

[Walter Roberson]

syms s
symExp = (s+2)/(s^2+5*s+9);

ExpFun = matlabFunction(symExp);
ExpFun = str2func(regexprep(func2str(ExpFun), '\.([/^\\*])', '$1'));

TF = ExpFun(tf('s'));
TF

TF =
 
      s + 2
  -------------
  s^2 + 5 s + 9
 
Continuous-time transfer function.

Seems to be working to me? Do you have an example of it failing?

[Paul]

@Walter Roberson

Cool code. Siight mod to also handle the case when symExp is a constant.

syms s
symExp(s) = 5;
ExpFun = matlabFunction(symExp);
ExpFun = str2func(regexprep(func2str(ExpFun), '\.([/^\\*])', '$1'));
TF = tf(ExpFun(tf('s')));
TF

TF =
 
  5
 
Static gain.

[Sanjeet Kumar]

Great, works well, Thanks!

Answer 2

[HyunSang Park]

If you're just trying to find the peak value of the bode magnitude plot, might I suggest avoid using tf altogether? the peak value is when d(G(jw))/dw = 0. You can easily find the derivative with syms, and the plug in the w to the original tf.

Answer 3

[Murugan venkatesan]

In order to analyze the bifurcation, the input impedance expression how to plot the bode graph..

[Walter Roberson]

@Murugan venkatesan

I do not understand how people can use your answer to convert symbolic expressions to transfer functions? Could you show how your solution could be used for the example symExp = (s+2)/(s^2+5*s+9); ?