# What is the “waterbed effect” in control system design?

I recently stumbled across some notes on the "Waterbed effect" in some notes by A. Megretski for an MIT course on "multivariate control systems". Here's an excerpt:

A common effect, usually associated with unstable zeroes and poles of the open loop plant, makes it theoretically impossible to make certain closed loop transfer functions “small” simultaneously at all frequencies: if amplitude of the frequency response is reduced in one part of the spectrum, it may have to get larger in the other part. This effect, sometimes called the waterbed effect, can be explained mathematically in terms of integral inequalities imposed on the closed loop transfer functions. In the basis of such results is the affine characterization of all possible closed loop responses, as well as the Cauchy integral relation for analytical functions.

I don't think I've ever heard of this before. Could someone explain the effect in more practical terms? When am I likely to encounter this effect in practice?

## 1 Answer

If I am understanding this paper please correct me if I am wrong:

A common effect, usually associated with unstable zeroes and poles of the open
loop plant, makes it theoretically impossible to make certain closed loop transfer
functions “small” simultaneously at all frequencies:


This is talking about Pole Zero Cancellation in realizable control systems. Essentially:

$$\frac{1}{s-\alpha}$$

is unstable for a step response however:

$$\frac{s-\alpha_1}{s-\alpha_2} = 1$$ where $$\alpha_1 = \alpha_2$$

which is stable; however, due to paramter variation (resistor/capacitor tolerance) it is impossible to cancle out an unstable pole. alpha_1 and alpha_2 may never perfectly align to cancel each other out. (maybe through digital controls)

if amplitude of the frequency
response is reduced in one part of the spectrum, it may have to get larger in the other
part. This effect, sometimes called the waterbed effect, can be explained mathematically
in terms of integral inequalities imposed on the closed loop transfer functions.


Basically, if alpha_1 increses then this "water bed effect" is caused by alpha_2 draging down the frequency response longer becofre alpha_1 zero kicks in.

essentially the Frequency repsonse would look like this if they werent matched:

--------\
\
\-------------


instead of this when they are matched exactly which looks like this:

----------------------------------


(That is, a flat response)

If the opposit happens (alpha_2 is made larger, you should see the opposit effect of this response)

             -----------------
/
/
-----/


.

In the basis of such results is the affine characterization of all possible
closed loop responses, as well as the Cauchy integral relation for analytical
functions.


Is answered by this paper: