# How to add a constant to a control system block diagram

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

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?

• @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? Jun 22 at 14:48

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. $$H(s) = \begin{bmatrix}\frac{V(s)}{U(s)} & \frac{V(s)}{C(s)}\end{bmatrix}.$$

• 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. Jun 22 at 15:18
• 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. Jun 22 at 17:19