Let's say my setup is something simple: I'm trying to heat some liquid to a temperature $T$. I can influence the temperature of the liquid by applying some potential difference $V$ across the liquid. This means that if I were to try to hold the temperature at some setpoint $T_{set}$ using a PID controller, then I would output some control variable $u(t)$.

Clearly all the terms used in calculating $u(t)$ would be in terms of the measured temperature $T$. The question is, what exactly is the output $u(t)$? Is this in units of temperature or voltage?

I don't think this is in temperature because I actually control the voltage. In that case, should I be setting the new voltage to u(t)? Or is u(t) more like a feedback such that I actually set $V_{new} \leftarrow V_{curr} + u(t)$?

  • 1
    $\begingroup$ The output of your PID control is the drive signal for your plant. $\endgroup$
    – DKNguyen
    Sep 28, 2022 at 4:44
  • $\begingroup$ You must complete the feedback loop. Your temperature based PID output must be connected to some way to generate heat that will be measured. $\endgroup$ Sep 29, 2022 at 14:49

2 Answers 2


Think of yourself as a PID controller controlling the speed of your car. The PID output is your foot! Is this in units of velocity? No. It's just % pedal push or foot angle. In practice it's just "apply more pedal" or "apply less pedal".

In many temperature control applications the output is used to adjust the duty cycle of the heater. For example, I work with industrial plastic film heat sealing systems which have a response time of about 10 s. We typically run the heaters on a 2 s timer and the PID output of 0 to 100% adjusts the heater on-time from 0 to 2 s. So, in this example the PID output is time, not temperature.


The following graph might be a helpful perspective to your problem (this is one of the perspectives).

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Figure: source:parallax)

The temperature of the Liquid is the sensor measurement that comes as an output from the System.

The error e(t) is the difference between set point and the Liquid's Temperature.

As you can see this is provided to the different block (Proportional, Integrative, Differential). In each block you see a constant and the formula.

For example the proportional control output $u_p(t)$ is:

$$u_p(t) = K_P \cdot e(t)$$

Which means that the $K_P$ "has units" of $\frac{V}{^oC}$ (if temperature is measured in Celcius).


  • $K_i$ has units $\frac{V}{^oC\cdot s}$
  • $K_d$ has units $\frac{V\cdot s}{^oC}$

All this ensure that the different control outputs $u_P, u_I, u_D$ all have units that are accepted by the actuator (Voltage in your case).


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