# What is the physical intuition behind an "integrator" in control theory

I am watching some videos on control theory by John Rossiter, and they are very good. So far I have watched most of the videos on classical control, including transfer function, feedback, etc.

One topic that keep arising is the importance of integrators to help reduce the offset of the system to zero. That is, in order to reduce the difference between the target level and the actual level of the system to zero, an integrator is required in the simple first and second order systems provided.

But I am not clear on what is meant by an integrator. The presentation in the videos is about the Laplace transform structure of an integrator as a pole located at zero. That is, the inclusion of a $$\frac{1}{s}$$ in the transfer function. The integrator can be included in the controller or the plant itself. I understand the math easy enough.

However, I am not clear on what is the physical intuition behind an integrator? Like is the integrator some sort of mechanical component of a control system, or some sort of design aspect. I suppose I am trying to find some examples of systems that contain integrators, so that I can understand what action the integrator is performing.

An integrator is something that "remembers" how long a steady-state error has persisted and ramps the gain in proportion to how long the error has existed.

So, imagine that we have a chart with error on the y-axis and time on the x-axis. the cumulative "persistence" of an error will scale as the area under that curve, which is an integral over time.

So, you integrate the magnitude of the steady-state error over time to get a system response that will, over time, progressively try harder and harder to force the error to zero.

• thanks for the response. I think I have a better understanding now. The integrator is like a computer thing or an aspect of the measurement system. In the videos, the author makes a big deal about the inclusion of the integrator in the transfer function, so I had the mistaken impression that the integrator was an aspect of the physical system or "plant." But the more I read, etc., seems like the integrator is just a way to track the cumulative error between the current measurement and the target level over the recent history of the system. Thanks again for helpng me understand this. Commented Oct 11, 2023 at 6:10

I'm an electronic engineer, so to me, an integrator is a capacitor. Capacitors integrate charge current to get Q, total charge, and capacitor voltage is proportional to total charge.

The most obvious place where a capacitor is used as part of an integrator is in an Analog Computer, used for simulation of control circuits that contain integrators. Of course, now we just use digital simulations instead :).

Anyway, integrators can be used in any kind of process control. You might need to force the number of matches in a matchbox to '48': for regulatory reasons, your long term error in describing content should trend to zero.

But the concept of integral control comes from navigation, where you want your long term error to trend to zero so that you don't miss your target, or hit a rock instead of a channel. Control circuits for ships and rockets control the speed and direction of the unit, and you need to integrate that to get location and destination.

Hopefully it will help – the integrator performs mathematical integration (value accumulation). Similarly, the derivative performs derivation (direction and steepness of the change in the accumulated value).

Both operations require, as described above, the "retention" of the previous value. They represent a way of reacting to a change in value.

Yes, this can be a property of both the regulated system and the controller - a system (its description) that is integrative, derivative, or a combination of both (physical systems that are capable of storing energy). See https://en.wikipedia.org/wiki/Accumulator_(energy) They form (and describe) so-called dynamical systems. See https://en.wikipedia.org/wiki/Dynamical_system the subject of dynamical systems theory https://en.wikipedia.org/wiki/Dynamical_systems_theory

Electronically, they can be implemented by an analog circuit using a capacitor or inductor, often using an operational amplifier, but also digitally, using a memory component. See https://en.wikipedia.org/wiki/Integrator For serial circuit with passive components see https://en.wikipedia.org/wiki/RC_circuit#Series_circuit

For a mechanical or hydraulic system, represented by a mass and a spring, a damper or a reservoir and a fluid, its inertia, see https://www.ecs.csun.edu/~nhuttho/me584/Chapter%203%20Mechanical%20Systems_part1_forclass.pdf

Mechanical (analog) computers were also constructed. See "The Mechanical Integrator - a machine that does calculus" https://www.youtube.com/watch?v=s-y_lnzWQjk Resp. https://en.wikipedia.org/wiki/Analog_computer

In the physical sense, an integrator adds up your input over time.

Electrical Current * time -> (change in) Electrical Charge

Fluid Flow * time -> (change in) Volume

Heat Flow * time -> (change in) Temperature (for a given thermal mass)

Object Velocity * time -> (change in) Object Position

Force * time -> (change in) Momentum

You might see this in the "plant" of a control loop. For example if you're controlling the fluid volume in a tank (system output), there might be an intermediate input that is the fluid flow. In practice you might have another thing like a valve which controls your flow, so this is a little bit of an example.

You might also see an integrator in the "compensator" of the control loop. In that case, it's purely mathematical, and if we want to have intuition, we should think about the control loop in a purely mathematical sense. The integrator gives you "lots of response at low frequency and no response at high frequency". The bode plot representation is a nice way to approach this.