Integral Control Systems
What Is An Integral Control System?
Background
Using Integral Control & Steady State Analysis

What Is An Integral Control System?

Often control systems are designed using Integral Control.  In this control method, the control systems acts in a way that the control effort is proportional to the integral of the error.  You should have studied proportional control before tackling this lesson.  A proportional control system is shown in the block diagram below.

The proportional controller amplifies the error and applies a control effort to the system that is proportional to the error.  In integral control, the control effort is proportional to the integral so the controller now needs to be an integrator, and it will have a transfer function of Ki/s - not just a gain, Kp.

What Do You Need To Get From This Lesson?

This is a short lesson.  The goals are simple.

Given a closed loop, integral control system,
Know that the SSE is zero - exactly!
Be able to explain why the SSE can be zero even though there is no input to the integrator.

What Is Integral Control? - Some Background

Integral control is what you have when the signal driving the controlled system is derived by integrating the error in the system.  The transfer function of the controller is Kp/s, if you think in terms of transfer functions and Laplace transforms.  That is what is shown in the diagram below.

That's the general outline, but to understand how integral control really works, it helps to understand exactly what an integral is.  Let's consider that a while.

To use integral control you really need to understand what an integrator is and what an integral is.  Let's get back to basics.  An integral is really the area under a curve.  Let's assume that the independent variable is time, t.  Then as time goes on the area accumulates.  In math courses when they talk about integration, they picture it as the limit of a process of taking small incremental areas - shown below - and letting the interval, T, shrink to zero.  In digital integration, that visualization process is important.  Here is an approximation to an integral that is a sum of areas under the curve of the function being integrated.

If the integral starts at zero, then the integral is just the area under the curve.  Let's look at some implications of that.

• If the input goes to zero, then the integral stops changing and just has whatever value it had just before the input became zero.
• The integral can change in either direction as the signal goes positive and negative.  Negative area can subtract from positive area, lowering the value of an integral.
The first point here is very important because it has implications for the way that the error in the system behaves.  The second point has strong implications for overall system behavior, particularly for understanding overshoot in the output of an integral control system.

Using Integral Control

Let's look at the structure of a system to control liquid level in a tank.  The input is some desired level.  The output level is measured and fed back to be compared to the input, generating an error signal.  We integrate that error signal to get the voltage to be applied to the pump.

Consider this question.  What happens when the error signal is zero?

The answer to this question is that maybe nothing happens.  If the error signal is zero, then the output of the integrator stays constant!  That means that the voltage applied to the pump stays constant, and if everything is at steady state, the liquid level in the tank stays constant.

If the output level is the desired level, this is a desirable steady state.  Let's review this situation.

• If output level matches the desired level, the error is zero.
• Because the error is zero, the integrator output does not change.
• Because the integrator output doesn't change, if the rest of the system is at steady state nothing else changes.  All is copasetic!
This sounds too good to be true.  What could possibly go wrong?  Well there could be at least two problems
• The system has to reach steady state.  You'll need to learn something about system dynamics to ensure stability.  If the system starts to oscillate wildly, then it may not reach a steady state, so the zero state state behavior is never really seen.
• Although the error goes to zero, no guarantees about speed of response are given.
This has been a very cursory look at integral control.  You'll need to get into the details to really make it work.  The important point you need to take from this lesson is this.
• If you can design a stable integral control system, the steady state error (SSE) will be zero - exactly!
• The guarantee of zero steady state error may be important in a system.

You are a long way from a complete understanding of integral control - but you've made a good start.  There are lots of unanswered questions about integral control, and we'll give you some questions and links here and on the next page.

• Many of the questions revolve around system dynamics.  You need to be sure that you can predict how a system behaves.
• Does the system oscillate a lot before it gets to steady state?
• How long does the system take to get to steady state?  What do we mean by "How long?"?
Other questions revolve around how you can implement integral control.  There are two basic ways to implement integral control, and both are widely used.

The analog integrator can be used when the rest of the control system is implemented with analog components.

Another option is:

• Use digital integration.  You may want to check the introductory lesson on digital integration and implementation of integral control.
• If you want to use digital integration, you'll need to learn about digital integration algorithms, and you'll need to be conversant with sampled systems, and particularly Z-transform methods.

Summary

The main thing to take from this lesson is that integral control will produce zero SSE.  There's not if's, and's or but's about it.  It will be zero.  When you think about that, you may wonder why it isn't always used.  As with most things in life, there are advantages and disadvantages.  The disadvantage is that integral control might produce a closed loop system with significantly slower response times.  That's a subject that will take some knowledge about system response times and how they are related to the system you are controlling.  You'll need some knowledge about root locus.  Using the root locus you can get a handle on response times and how they are related to the parameters of the controlled system, and to the gain you choose for the integral controller.

Problems