Units associated to PID Terms I am trying to understand PID Controllers .. I don't understand very well what Physical Quantity is associated to each PID Term, 
The Block Diagram of PID is this:

the PID Equation is this:
$$ u(t)= K_P e(t)+ K_I \int_0^t e( \gamma) d \gamma  +K_D \frac{d e(t)}{dt}   $$
where:
$$ u(t) = Controller Output $$
$$ e(t) = r(t) - y(t) = Error Term $$
$$ y(t) = Controlled Variable $$
$$ r(t) = Commander Variable $$
$$ K_P = Proportional Constant  $$
$$ K_I = Integral Constant  $$
$$ K_D = Derivative Constant  $$
The Commander Variable normally is a constant that is is known as the Set Point (SP)
So, I have heard some people saying that:


*

*Proportional Term is related with the Present, 

*Integral Term is related with the Past 

*Derivative Term is related with the Future
But I don't know, I am not completely comfortable with the idea, the Present of What?, The Future and Past of What? The Controlled Variable? the Controller Output? the System? the Error? .. All of them?
For example, let's take a Drone 

so we have that the Controller Output must be the speed of the propellers:
$$ u(t) = v(t) $$
and the controlled variable is going to be the position:
$$ y(t) = p(t) $$
Since y(t) has position as unit, and the Setpoint too, the error term is going to have position as unit as well:
$$ e(t) = SP - p(t) $$
So, in order to match the units for the controller output we have that each constant must have certain units and each term in going to contribute to the controller output depending on certain physical quantity of the system at that moment:


*

*Proportional Term Units:
$$ ( \frac{1}{s} ) (m) = \frac{m}{s}   $$

*Integral Term Units:
$$ ( \frac{1}{s^2} ) (m s) = \frac{m}{s}   $$

*Derivative Term Units:
$$ ( 1 ) ( \frac{m}{s}) = \frac{m}{s}   $$
Where: m=meters, s=seconds
What I don't understand is What does it mean the position multiplied by time?
I have seen that lastly there has been some research about Integral Kinematics, I recently discovered the word Absement, this is the Integral of the Position, and the News say that it was even measured for the first time by Maya Burhanpurkar, a 14 years old girl, which for me it is actually kind of surprisingly, I don't even have an idea of how you could measure that, also, I am an Native Spanish Speaker and I don't know how to translate that word, it is supposed to be a portmanteau of the words "Absence" and "Displacement", so, in Spanish absence is translated as "Ausencia", and displacement is translated as "Desplazamiento", so, following the same logic, the portmanteau should be: "Ausemiento" ? .. but I am not sure, I guess that is topic for another question, anyway, the thing is that I feel that this absement is associated to the Integral Part somehow because they have the same units, is this correct?
Another example, a Heater System that heats due the Joule Heating Effect Produced by a Nichrome Wire, so we have that the controller output must have Voltage as unit:
$$ u(t) = V_{rms}(t) $$
and the controlled variable is going to have temperature as unit:
$$ y(t) = T(t) $$
Since y(t) has temperature as unit, and the Setpoint too, so the error term must have temperature unit too:
$$ e(t) = SP - T(t) $$
So, in order to match the units of the controller output we have that each constant must have certain units and each term in going to contribute to the controller output depending on certain physical quantity of the system at that moment:


*

*Proportional Term Units: 
$$ ( \frac{V}{K} ) (K) = V   $$

*Integral Term Units: 
$$ ( \frac{V}{ Ks} ) (K s) = V   $$

*Derivative Term Units: 
$$ ( \frac{Vs}{ K} ) ( \frac{K}{s}) = V   $$
where: K=Kelvin, V=Voltage, s=Seconds
I interpret it the next way: 


*

*The Proportional Term is associated with the error due to the mere Temperature that the system has at that moment, so, depending on the temperature of the System, the proportional Term is going to change

*The Derivative Term is associated with the error due to the Rate at what the Temperature is changing, so depending on the Velocity at what the Temperature is changing, the Derivative Term is going change .. umm .. the Future of the Temperature? I guess somehow you could say the Velocity of the Temperature Transfer is related with its future, but I don't know .. 

*and the Integral Term is associated with the error due .. umm .. what is it Temperature Multiplied by Time?, What unit is that? What does it mean? The Past of the Temperature? The Absement of Temperature?
I think it would be interesting to add other terms associated to the error due other Physical Quantities of the System, although I know that there's no need to, with just PID you can control the system .. I even have heard that most of the time you only need the Proportional and the Integral Part, so a PI Controller .. but I don't know, is it possible to add this:
$$ K_{D^2} \frac{d^2 e(t)}{dt^2}   $$
to make it a $PID D^2 $ controller ?  
So, I guess a more General Equation could be: 
$$ u(t)= K_P e(t)+ \sum_{a=1}^b \int \!\!\!\!\underbrace{\cdots}_{b \text{ times}\ \ \, }\!\!\!\!\int K_{I^a} e( \gamma) \,\mathrm d\gamma^a  + \sum_n^m K_{D^n} \frac{d ^ne(t)}{dt^n}   $$ 
where a is how much accuracy you would like to achieve in the Integral Term and m is how much accuracy you would like to achieve in the Derivative Term
Maybe I am understanding all of this wrong, and the terms are not even associated with a Physical Quantity, but then, Why do they have the units that they have? 
Another thing I don't understand is how do you derivate or integrate e(t) = SP - y(t) if you don't know how is it the function .. in the real world you don't know what is it y(t) .. How do you Integrate/Derivate that ... I have read that in order to implement that in code you have to use finite differences:
$$ K_i \int_0^t e(t')dt' = \sum_{i=1}^k e(t_i) \Delta t  $$
$$ \frac{d e(t)}{ dt} = \frac{ e(t_k)-e(t_{k-1})}{\Delta t}   $$
is this true? I am not sure about how finite differences work .. Do you have any suggestions about a book on the Topic or something like that? I don't remember any of that in my calculus books ..
But, the important thing is that e(t) is never going to have an explicit function that describes it, right? How do you derivate/integrate something that you don't know what is it?
These are some of the concerns I have about PID Controllers, I hope you could help me to understand it better, thanks
 A: A PID controller actually controls error, as shown in your drawing.  The error signal is considered to be dimensionless, and that signal goes through the controller transfer function and through the process transfer function.
Units enter the picture when specifying the sensor that sends a feedback signal to the controller.  As an example of how this is done, the following steps are followed:
1) A control systems engineer notes the type of instrument that is reading the process (e.g., a flow element, a thermocouple, a pressure sensor, etc.).  This engineer may have to consult with a process engineer to determine the minimum and maximum normal operating conditions for that sensor, but these values determine the range that the instrument will be calibrated for.
2) With calibration data, an instrument technician will go to the instrument in the field and make adjustments on it.  He knows that the range of the electrical signal from the instrument will vary between 4 milliamps and 20 milliamps (aka mA).  With this knowledge, he adjusts that instrument such that it sends a 4 mA signal to the controller when it sees the minimum process reading and a 20 mA signal to the controller when it sees a maximum process reading.
3) The mA signal going to the control room is an analog signal, and that signal goes to an analog to digital converter at the control room.
4) The PID controller actually exists as computer code in the control computer. That computer sees and displays a 0% signal on a terminal when a 4 mA signal comes from the field instrument and it displays a 100% signal when a 20 mA signal comes from the field instrument.
5) There is some sort of data table in the control computer for the particular instrument that is being read, and that data table contains information for the actual process value and units that correspond to a 0% signal.  The data table also contains process data for the 100% signal.  The control computer uses this information to display the true value of what the process is doing, along with the appropriate units that exist in the data table, which allows operators to know exactly what the process is doing. 
6) When the controller is initially commissioned, the control engineer tunes it by adding proportional and integral to it until it briskly responds to setpoint changes and eliminates deviations from setpoint.  Note that derivative is normally not used in a PID controller because derivative is very sensitive to process noise, and it tends to cause the PID controller to over-react.  
7) The controller's output is a digital value corresponding to some percent of full range, and the digital value is converted back to a 4-20 mA signal by a digital to analog converter.
8) The 4-20 mA output of the controller goes to a control valve in the field that also has to be calibrated by the instrument technician.  That control valve manipulates a flow rate to control a temperature, pressure, level, etc.
Admittedly, this is a long explanation, but note that the actual units involved in a measurement are displayed to the operator, but the controller has no knowledge of what those units are or what they mean.  This enables a generic piece of code in the control computer to be used to control temperature, flow rate, pressure, level, etc., because all instruments in the field are sending a 4-20 mA signal to the control room, and every PID controller in the control computer is comparing its own 4-20 mA feedback signal with its setpoint to determine a percent error.
A: 
this is a block diagram for a  controller
the input signal  to the PID block is the error $e$ (desired value minus measurement value)
and the output signal is the actuator $u$.
thus the equation for a PID block  unit is:
$$u=P\,e+I\,\int\,e\,dt+D\,\frac{d\,e}{dt}\tag 1$$
where $PID$ are constant gain parameters 
the units of $PID$ are depended on the units of $e$  and $u$
$$[u]=[P][e]+[I][e]s+[D]\frac{[e]}{s}\tag 2$$ 
with the  "units" equation (2) you can obtain the units of $P$ $I$ and $D$
example I: mechanical system with hydraulic actuator
$[e]=[m]$
$[u]=[N]=\frac{kg\,m}{s^2}$
$\Rightarrow$ equation (2)
$$\left[\frac{kg\,m}{s^2}\right]=[P]m+[I]m\,s+[D]\frac{m}{s}$$
thus:
$[P]=\frac{kg}{s^2}$
$[I]=\frac{kg}{s^3}$
$[D]=\frac{kg}{s}$ 
example I: Rotation speed control
$[e]=[RPM]=\frac{\pi}{30}\frac{1}{s}$
$[u]=[electricity]=[A]$
$P=[?]$
$I=[?]$
$D=[?]$
