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I have an equation of the system as let's say \$v=fu+c\$. I have the block diagram as enter image description here

Only problem is how do I get a Laplace domain transfer function out of this diagram for the output \$\large v\$ and input \$\large u\$. I could treat the constant \$\large c\$ like another input and the whole system as a MISO to get 2 transfer functions but that would mean in the transfer function of \$\Large\frac{v}{u}\$, the constant \$\large c\$ does not matter nor has any effect as it would not show up. Is it possible to get a SISO transfer function of a system with two inputs?

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  • \$\begingroup\$ @Hearth: in this context, the OP means a Laplace-domain transfer function. Adding a constant is a nonlinear operation (it's affine, but still nonlinear). So -- how can that be, when Laplace-domain analysis doesn't admit to nonlinear analysis? \$\endgroup\$
    – TimWescott
    Jun 22 at 14:48

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You cannot get a single-input, single-output transfer function of this system that takes both \$u\$ and \$c\$ into account.

Laplace-domain analysis only works for linear systems, i.e. ones that obey superposition. Strictly speaking, adding an independent constant makes the system nonlinear (you can check me on that: \$y = a x\$ is linear, and obeys superposition, but \$y = a x + b\$ does not). The resulting system is affine, which is the world's easiest nonlinearity to resolve, but it's still nonlinear.

You must do one of two things: do your linear analysis while ignoring \$c\$ then patch up the results, or treat the system as a multi-input, single-output system.

To "ignore" \$c\$, set \$c = 0\$, do your analysis, then add it back to the output.

There's a number of ways to treat the system as MISO. The easiest is to find two transfer functions: \$\frac{V(s)}{U(s)}\$ and \$\frac{V(s)}{C(s)}\$. Then -- because you've modeled a system that's linear but has two inputs -- you can use superposition to find \$v\$ by adding the system response to \$u\$ and the system response to \$c\$.

Alternately, you could model your system in state space, or you could model it with a vector-valued transfer function; i.e. $$H(s) = \begin{bmatrix}\frac{V(s)}{U(s)} & \frac{V(s)}{C(s)}\end{bmatrix}.$$

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  • \$\begingroup\$ Ok, I get that I could get the value of \$v\$ but I need a transfer function at the end as I need to calculate the H infinity norm and response to step input among other things. I am using Matlab for calculating this stuff but the transfer function thing has got me pinned down. \$\endgroup\$ Jun 22 at 15:18
  • \$\begingroup\$ Please edit your question with this new information, so that it is complete (this is a Stackexchange thing -- we're not like usual forums). In anticipating of you editing your question, I've updated my answer. The short answer -- no, you cannot have a SISO transfer function with two inputs, nor can you have any Laplace-domain transfer function for a nonlinear system. You can model the system as a MISO system and use that in your analysis. \$\endgroup\$
    – TimWescott
    Jun 22 at 17:19

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