PID Controllers - General PIDs
Why Do You Need PID Controllers?
What is a PID Controller?
Properties Of PID Controllers
Using PID Controllers - An Example
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Why Do You Need PID Controllers?  What is a PID Controller?

        PID controllers are a family of controllers.  PID controllers are sold in large quantities and are often the solution of choice when a controller is needed to close the loop.  The reason PID controllers are so popular is that using PID gives the designer a larger number of options and those options mean that there are more possibilities for changing the dynamics of the system in a way that helps the designer.  If the designer works it right s/he can get the advantages of several effects.  In particular, starting with a proportional controller, and adding integral and derivative terms to the control the designer can take advantage of the following effects.

        A PID controller operates on the error in a feedback system and does the following:
Goals Of This Lesson

        PID controllers are widely used for control of many kinds of systems.  Your goals for this lesson are as follows:

  Given a system you want to control with a PID controller,
  Be able to use common methods of analysis for a system with a PID controller in order to predict the behavior of the system + controller, and to be able to choose PID parameters.

Why Use PID Controllers?  What Are They?

        You may need a PID controller in many situations, particularly in the following cases.

Properties Of PID Controllers

 PID controllers can be viewed as three terms - a proportional term, and integral term and a derivative term - added together.  PID controllers are also known as three-term controllers and three-mode controllers.  Here's a block diagram representation of the PID.

        The PID controller can be thought of as having a transfer function.         Now, here's a good way to think about the effect of using a PID controller.

        Here are some of the possible pole-zero configurations the designer can add.

E1  Two imaginary zeroes with the pole at the origin.

E1  Two real zeroes with the pole at the origin.

        The designer's challenge is to choose a set of controller gains that produces a system response within specifications.  Above we can see that this is equivalent to choosing one gain and a pole-zero pattern - staying within the allowable configurations.  The pole-zero pattern is really a zero pattern you can control and a pole that is constrained to be at the origin.

Using PID Controllers


E2   Let us look at some examples in which you might want to use PID controllers.  In the lessons on proportional control and integral control, we considered this system with these numbers.  Here is a link to the problem and system description.

        There were several conclusions that we drew from investigations of proportional control (Click here for the specifications.) (Click here for the conclusions.)and integral control used with this system.  (Click on the hot words to go to the sections where this system was investigated.)

 We found, generally, that proportional control would not work, but integral control could be used to meet the SSE specifications.  However, integral control had response times on the order of ten to twenty seconds.  The question we want to look at here is "Can we get faster response with a general PID?".

        Let's review the root locus for this system with an integral controller.  Here is the root locus for the system with a proportional controller.

        With a PID controller we get a pole at s = 0 and two zeroes anywhere we want them.  Basically, we can add zeroes two different ways.

Which situation would you like to try?  We will arbitrarily try a few ideas so that we can see what happens in the different situations that are possible.

        The original root locus looks like this.  This root locus has three black dots marking the pole positions when the root locus gain is 50.

       Now, add the two zeroes (at -2 and -8) and the pole at the origin.  Here is the resulting root locus.

        The plot does raise some questions.
        This root locus plot shows the system with two complex zeroes at -2 + j3 and at -2 - j3.

        It's time to evaluate where we are.  We have tried the following:         These pairs of zeroes were all added in conjunction with a pole at the origin, using a general PID approach.  In all cases we see the following.         Before we give up on this, let's try some other zero locations.  We are going to choose some complex zero locations that are a little further to the right.  This root locus plot shows the system with two complex zeroes at -1 + j3 and at -1 - j3.  Paradoxically, moving the zero location to the right permits the poles emanating from s = 0 and s = -0.5 to move further to the left.
        Overall, the idea of two complex zeroes looks like it can yield some interesting results.  It's the best we've had to this point.  We can "tweak" the zero location just a bit more.  Here is the root locus with zeroes at -1 + j and -1 - j.  The black dots are located at a root locus gain of 20.

        We really do need to do simulations to get a feel for how the system behaves.  To do a simulation we need to complete the PID design.         This system was simulated using Simulink/Matlab using the system below.

Here is the computed response.  (We did this using two different integration algorithms.  Using the 4th order Runge-Kutta produces an unstable calculation unless the user coerces the time interval small enough.  The time interval here was 0.05 sec.)

Questions And Conclusions/Comments on the extended example

        What can we conclude about general PID controllers, drawing especially on the extended example above?  Here are some conclusions we think are appropriate.

Links To Related Lessons

Other Lessons On PID Controllers

Material That You Might Want To Review Moving Along - More Advanced Material
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