PID Controllers

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.

An integral controller gives zero SSE for a step input.
A derivative control terms often produces faster response.

A PID controller operates on the error in a feedback system and does the following:

A PID controller calculates a term proportional to the error - the P term.
A PID controller calculates a term proportional to the integral of the error - the I term.
A PID controller calculates a term proportional to the derivative of the error - the D term.
The three terms - the P, I and D terms, are added together to produce a control signal that is applied to the system being controlled.

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.

A proportional controller may not give SSE performance needed in a system.
An integral controller may give SSE performance, but slow a system down.
Adding a derivative term may help cure both of those problems.

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.

There are three controllers acting in concert.
The three controllers are:

A proportional (top),
An Integral (center),
A Derivative (bottom).

The proportional, integral and derivative outputs are added together.

The PID controller can be thought of as having a transfer function.

The PID controller transfer function can be obtained by adding the three terms.

PID(s) = K_p + K_i/s+ sK_d

The transfer function can be combined into a pole-zero form.

PID(s) = [sK_p + K_i+ s²K_d]/s
Since there is a quadratic in the numerator, there are two zeroes in this transfer function as well as the obvious pole at the origin, s = 0.

Now, here's a good way to think about the effect of using a PID controller.

The PID controller transfer function really adds a pole at the origin, and two zeroes that can be anywhere in the s-plane that the designer wants, depending upon the designer's choice of the three gains.
PID(s) = [sK_p + K_i+ s²K_d]/s
This gives the designer an incredibly large number of possibilities.

Examples

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

Example

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.

For the engine + load:

G_L = 1, t_L = 2 sec.

For the controller:

G_C = K_I (variable), t_C = 1 sec.

For the sensor:

G_S = 1, t_S = .2 sec.

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.

The problem with this system is the two branches that start at s = -0.5 (There is a 2 second time constant in the plant that produces this pole.) and at s = -1 (And, this is the pole from the controller.)
Those two branches are the cause of the slow response and the general limitations on response time in this system.
Those branches come together and eventually move into the right half of the s-plane - where the system becomes unstable.
Those two branches are the cause of the slow response and the general limitations on response time in this system.

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.

The zeroes could be real, in which case they would be in positions like those seen below.

The zeroes could be complex, in which case they would be in positions like those seen below.

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.

We'll give you two real zeroes at -2 and -8.
We'll give you two complex zeroes at -2 + j3 and -2 - j3.
These are obviously not the only choices there are, but we'll start with one of these.
And, don't forget when we do this we will add a pole at the origin as well - in both cases.

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.

Adding the two zeroes does produce a change in the locus - not all for the better, although now only two branches go to infinity.
The two zeroes in the LHP move the centroid (Center of Gravity) to the right, and while the closed loop poles do not move as quickly into the RHP, they get there. And, we should have thought of that before we tried it, but now we know.

Make a mental note that adding poles and zeroes will always have an effect on the center of gravity (centroid).

The plot does raise some questions.

Would we be better off with zeroes further to the right? If the zeroes are further to the right, the center of gravity will be changed - and for the better since it will move to the left!

This root locus plot shows the same system with one change. Instead of a zero at s = -8, that zero is now at s = -1.5.

Now the center of gravity is in the LHP, so when the two poles go to infinity they head off in a direction that keeps them in the LHP.
When the two poles come together, they do so at about s = -0.5.
As the poles become complex, they move gradually to the left producing a system that is probably faster.
As the poles move to the left, the damping ratio deteriorates.

That's another thing to keep in mind. When we make changes we need to examine what possibilities the system allows. In the system above, either the speed of response is poorer (when there are closed loop poles on the segment between 0 and -0.5) or the damping ratio is poor (when the poles do move a little further into the left half of the s-plane.) Got to remember to check the damping ratio - especially if there is a requirement on damping or overshoot.

All of the above raises another question:

Should we consider complex zeroes? We need to consider how using complex zeroes might affect the issues we are concerned about above.

This root locus plot shows the system with two complex zeroes at -2 + j3 and at -2 - j3.

This looks pretty disastrous.
The two poles closest to the RHP move to the RHP fairly quickly.

Time response is slow to begin with, and it gets worse as you increase the gain.
Instability is even worse than a low damping ratio. Actually, you can view instability as a negative damping ratio, so possibly this can be considered a damping ratio problem also.

At higher gains those poles get back into the LHP, but it doesn't look like there is any gain that produces faster, better damped responses.

It's time to evaluate where we are. We have tried the following:

Zeroes at -2 and -8.
Zeroes at -2 and -1.5,
Two complex zeroes at -2 + j3 and at -2 - j3.

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.

There is a problem because it seems to be difficult to move the closed loop poles very far into the left half of the complex plane - in order to produce a fast system.
In all of these cases there is always a pole that is slow, or when the poles become complex, it is not possible to get much damping.
However, we can see that we have a substantial amount of control over where we can get the closed loop poles to locate. It might be that we just haven't explored this completely enough.

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.

Now, the zeroes "capture" the poles closest to the RHP.

The first time we used complex zeroes, the root locus branches captured by the zeroes were the branches further to the left.

The capture takes place only after a little excursion into the RHP.
If we built this system, we would need a high enough gain to ensure those two poles were in the LHP.
We would always have the problem that a lower gain would make the system unstable.

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.

This choice looks half-way promising.

The closed loop poles are as far into the left half plane as we have had them.
The damping ratio is decent also. It looks like we might be able to get a damping ratio of 0.6 or better in both pairs of complex closed-loop poles. That's an indicator of a system without too much overshoot.

We need to simulate the system to get a better idea of behavior.
We might explore zeroes located differently yet. Here is an example that shows some other possibilities.
After looking at these systems, look again at using two real zeroes.

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.

If we expand the zeroes, we get a factor [s² + 2s + 2].
We will need to remember that the root locus gain is 20 (note the black dots on the root locus plot above.

The root locus gain is K_rl= 2.5 K_i, so K_i = 8.

We need to adjust for the gains of the three blocks when they are expressed in root locus form.
To get a transfer function of 8 [s² + 2s + 2]/s, we need to have:

8[s² + 2s + 2]/s = [sK_p + K_i+ s²K_d]/s, or
K_d= 8,
K_p= 16,
K_i= 16,

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.

Using a general PID controller can give you performance that is better than using proportional or integral alone, or even a PI or PD.
The improvement comes because you can locate two zeroes where you want them.

In the example getting two complex zeroes at a good location was critical. It might not be possible to get the kind of improvement we got using a PI which would give you a real zero in either case.

Using a PID gives you the zero error behavior you get from integral control.
PID controllers are no panacea, but used wisely can help you design control systems that work well.

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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