# Block Diagrams with offset in control theory

I have an equation in the form $$y = mx+b$$. In block diagram form, I could draw this with a proportional gain block and a summer. I was wondering if there are some clever tricks where I can make this equation with just a proportional block, as it is often times much simpler to deal with a bunch of series connected blocks rather than including summing junctions and what not. For the sake of argument, let's say for example the equation I have is $$y = 5x+2$$.

Obviously no. You specify that 2 needs to be added, but you want to realize this without adding the 2. You can't have it both ways. Either you need the constant added or you don't. You have to decide. You can't add 2 without adding 2. I'm not sure where the confusion is coming from.

By the way, with the constant added it is no longer a linear system, so some linear system analisys techniques don't apply.

• Huh? Adding a constant doesn't affect the linearity of a system! – Dave Tweed Sep 12 '12 at 12:21
• @DaveTweed, Olin's correct. For linearity, you must only have $f(x+y) = f(x) + f(y)$ (superposition), you must also have $f(ax) = af(x)$ (homogeneity). But $5ax + 2 \neq a(5x + 2)$. Remember, the order in which you place your linear sub blocks should not matter. But clearly, placing an integrator before this block would give a different system than placing an integrator after it. – Alfred Centauri Sep 12 '12 at 13:05
• Adding a constant makes a system non-linear. Linearity requires that a linear combination of inputs gives the same linear combination of outputs. If your system adds a constant, linearity doesn't hold. – Juancho Sep 12 '12 at 13:16
• When we talk about "linear systems" we're usually just talking about a nonlinear system that is operating with approximately linear behavior about some operating point. In this case, for example, you could redefine your output variable as y' = y-2, et voila, a linear system! – The Photon Sep 12 '12 at 16:44
• It's an affine system. – Ben Voigt Sep 12 '12 at 18:11

If I understand you right, you want to avoid this:

I cant think of a way to get rid of the sum block.

But if it just because you want to draw less, I suggest to draw somthing like this:

It might look lame and non-creative, but for non-linear function (like sinus, etc.) it is common ( at least I have seen it alot) to write them in their own block.