Lets consider a signal. its mean over a period of time T shuld be:

So if a given value of V is needed you only have to keep this intrgal unchanged (it is not necessary for it to be a voltage, it can also be an intensity or whatever).
The first way to achieve thtat cuold be keepeng the value of
as close as possible to de value of the mean. This also keeps all the moments of the signal close to 0 for example the variance.This parameter very important as far as noise is concernd. This way is almost perffect but for is very inefficient. alwys less than the 25%, so it should not be used for hight power devices unless there is no choice.
Everythng that keeps the integral unchanged coud be used. Since it is a linear system on f(t), it remains unchanged if we change the complete set to a different one. it is also linear in dt (we could even choose the absolute value and consider it a scalar product).
But as far as this topic is concernd we could take its definition instead of the integral.

Then, a good way to do it could be splitting every intervel in two different intervals, so as to

Then ist just switching on and off. Then if we choose a signal bigger than the maximum of F(t) and take the product with (actually convolution) a fucntion as the following

which switch on in a and off in b.
Hera comes the problem of choosing the time T in order to tihnk of the mean.
If there ire periods, its obvious that should be the shortest (of them).
If ther is no period, the time shuld be related the most relevant for the system, thus the problem is the the same as the one we find when passing from the micorscopical to macroscopical scales, as happens in electormognatism or thermodynamics. Then a time longer tha the one needed for switching (each individual switch should be negligible to the time the mean is computed) and much faster than every relevant phenomenon in the controlled sysytem.
So it can be different in different systems for example a oven (or e furnance, for what matters ist the same), this used to be done with 5- 15 secondes on and the mean, then culd be computed over 10 minutes or so. This is possible because the system has a huge heat capacity.
The oven is a very special case as a computer power supply also is. this reqeuires a really fast answer and has other constraints, mainly a voltage limit that shuld not be surpassed and a lot of noises coming from higer fourier components that should be filtered.
Those contraints (like for axample an additional current limitation) and filters for such a things shuld be taken into account as we will do in a further post.
Conmutation
The switching times could be choosen in different ways, form pre-computed values to all sorts of feedback systems (including the simple and obsolete bimetallic thermostat)
Some of those thigs should be treated in a further post but not every possible system. some of them will be treated if and only if i were asked for.
The mean is a function
We could think it in meny ways. but it implys that for every time, the mean shouldd follow a function. Then if those derivatives are very differnt (one to each other.) then is a case of the Schwartz thoerm for oscillating integrals.
So:
.
This needs the norm of the derivative of the menan [latex} V(t) [/latex] should always be lesser than
or if there are discontinuitues in
there should be less than in f(t)
This is not very clear and should be prooved in casewise . anyway it is necessary to proove it in a epsilon-delta that for the averge case might resemble the Schwatrz theorm or just be a simple manipulation of the definition (Stieltjes, Lebesgue or even Riemann definitions).
A simple way for the typical case can be treated calculating the means locally (it it has less discontinuities than V(t) and then comparing each other and with the times (frequencies) that will be present on the system. then could be considered as continuous. (if we consider for the slower signal a interval (ball) much bigger than the period of the fastst one its quit likely for us to proove it)
¿how to get it?
Let’s consider the follwing integral:

So for changing the mean it’s only necessary to change $t_0$
It can be achheved in a dynamical way, for example comparing the desired signal with a sawtooth. then the conmutation elment (MOSFET) will close the circuit (conduct) if and only if the sawtooth is smoller than the signal it is compared with. Later, an example will be shown.
how can we filter it?
What we have is a signal that con be written as a Fourier series. The N first components should be the same as the ones for the desired function (this lastone should be used as a control signal to the actual one). Then the differences should be smaller than the radius or ball ke spoke abaut. Then in the superior order compnents we will find the effects of conmutation. this idea is the same we spoke abaut befor but told by its effects (with fourier series).
Then we can filer it with a lowwpass filter (able to absorb the power needed…). that will cut off voltage peaks and higer fourier compnents.
PHSVM
(phase space vector modulation)
It is exactly the same idea but taking into account n independent integrals over the same ball.