The controller you are describing where you keep adding the error times a constant to the current output value is an Integrator. You are clearly integrating the error. A proportional controller would be setting the output to P times the error. It also matters what the output controls, e.g. whether it’s torque, or position, or velocity for a motor control system. (something proportional in velocity is integral in torque…)

The reason for a steady state error with P only is that as your system approaches the set-point the error signal gets smaller and smaller. Your control is Kp times that error signal and eventually the error will be small enough that Kp times the error won’t be enough to force it all the way to zero.

An Integrator “saves the day” by accumulating the error over time and therefore even the tiniest error will eventually accumulate to something large enough to force the controller to correct for it.

As an example let’s look at a P temperature controller for an oven. We have a heating element and we control the current to that element. Our set-point is 100 degrees. The oven is constantly losing heat to the outside environment (which is lets say at 0 degrees). We start up at an error of 100 and let’s say our P is set such that this is 10A. If P is set too large the system is going to be unstable but if we assume stability at some point the error will be so small such that P*error isn’t going to be enough to overcome the heat loss of the oven to the environment. You will reach equilibrium away from your set-point.

Consider the following system: (source: prenhall.com)

This is a proportional system with a plant G(s), gain K, input R(s) and output Y(s). The system has no disturbance modeled and has unit feedback, but using this we can show how the proportional controller (even in this simplified, ideal state) inherently produces an offset at the output.

Let’s express our output Y(s) in terms of everything else.

Y(s) = G(s) * K * (X(s) – Y(s))

Doing a few manipulations, you can get this:

Y(s) = X(s) * [G(s)*K/(1+G(s)*K)]

What this means is Y(s) is always proportional to the input signal X(s) multiplied by the system G(s)*K/(1+G(s)*K), which means there is always a biased output (unless K or G(s) go to infinity).

You have to remember, you’re dealing with real devices. In a perfect world, a control signal will have an effect, no matter how small. In the real world, control signals smaller than some threshold value don’t do anything at all. (Imagine a chihuahua trying to pull a Mack truck.)

In a proportional control system, the control signal is the error multiplied by the control gain. To prevent overshoot, instability, and other Bad Things, you make the gain small.

As your plant (controlled system) gets closer and closer to the desired value, the error gets smaller and smaller. When you multiply a smaller and smaller error by a small gain, eventually the product gets too small to have any effect, and is truncated to zero.

That’s your steady state error, the region around the desired output where the product of the error and the gain is too small to register.

After whatsisname’s downvote, I thought back. He’s right. I was confusing “dead band” with “steady state error”.

If you have a constant perturbing force (for lack of a better term), and you are using a PI controller, there will be some point at which the control signal (product of error and gain) exactly balances the perturbing force. At this point, the plant (system being controlled) experiences zero net force: the perturbing force and the correction cancel each other, and the plant just sits there, with a constant error, the error generating the signal to cancel out the perturbing force.

The first-level fix is to go to a PI controller, one that not only produces a correction based on the error, but also based in the integral over time of the error. The integral term is what “notices” the steady state error and moves to correct it.

The catch is that a PI controller can have overshoot problems. There are no silver bullets.