Understanding PID Control, Part 7: Important PID Concepts

From the series: Understanding PID Control

With the overview of PID tuning techniques behind us, we can now move on to discussing two very important concepts with PID control: cascaded loops and discrete systems. Both of these concepts are fundamental to most practical control systems and they each change the way you approach and think about your problem. Therefore, I think they are worth spending some time to discuss in a bit more detail. I’m Brian, and welcome to a MATLAB Tech Talk.

What is cascade control? Well, to answer that, let’s go back to an example we used in the second video of this series: altitude control for a quadcopter drone. Recall the block diagram for this system. We start with a reference altitude. This is the altitude we want the drone to hover at. This is goes through a comparator to generate an error term, which then is fed into a PID controller that generates a propeller speed command.  This is commanded to the propeller, which begins to spin up and the output for this block is the true propeller speed. The propeller speed generates a force that acts on the drone adjusting the altitude. Finally the altitude is measured by a sensor and fed back into the comparator at the beginning.

This is our classic feedback loop and it has a single PID controller like we’re used to. But now let’s focus in on the propeller itself. Again, the propeller command is the input and the resulting speed is the output. But if you recall, videos 5 and 6 in this series talked about motor control. And what is a propeller other than a motor with a fancy bit on one end? So, instead of a single block labeled propeller, this is actually a small feedback loop itself. There is the command, which goes through its own comparator and the resulting error is fed into a PID controller. The output of the controller is voltage, which is applied to the motor, causing the motor to spin. The motor speed is measured by a sensor and fed back into the comparator.

So now we have two feedback loops in our system: one that controls the motor speed, and one that controls the drone altitude. We can call these two loops the inner and outer loop so that we can distinguish between the two. These are cascaded loops or nested loops. The outer loop drives the set point of the inner loop, and the inner loop affects the feedback path of the outer loop, So they are intimately connected. You would probably expect that with a system with two PID controllers interacting with each other that this would require a slightly different approach to controlling it versus a system with just a single loop. And you would be right - but it may not be as different as you expect.

The first question might be, why design a system with cascaded loops in the first place? Why not stick with the single loop approach that we have so much experience with? For example, a single loop approach would be to design one PID controller that takes altitude error and outputs a motor voltage.  In this case, the altitude directly drives the motor voltage - there is no middle step of first commanding a motor speed.  Here’s two reasons: The first is that with a cascaded approach, it can be easier to isolate problems in the system. In our single loop example, if we have a problem with the motor, then it can be difficult to test just the motor separate from the rest of the altitude system. With the cascaded loop scenario, we could run the motor controller closed loop and determine if the problem exists with the motor or with the rest of the system interacting with the motor. The second reason for cascaded loops is that multiple groups can work separate parts of the problem. One team might build and design the motor controller while another team is responsible for the altitude controller. This is especially the case if you buy a motor with an integrated motor controller. The supplier that built the motor will have already designed the inner loop for you, and you just need to incorporate the outer loop around it.

But a more important reason to have cascaded loops is to run them at different speeds to address different problems and sources of error. The motor controller can respond quickly to local disturbances, whereas the altitude controller can be tuned conservatively to reject sensor noise and increase stability.

For example, in cascaded control, let’s say that the propeller motor takes 10 volts to run at 100 rpm, which we’ll say is the speed required to hover the drone. Now we add some disturbance in our loop, maybe the battery voltage drops because another high current device in the drone is turned on, or perhaps the lubricant in the motor bearings get hot and the resistance drops, requiring less voltage for the same speed.  In both of these cases, the inner loop motor controller can sense this errors and adjust the motor voltage in a fraction of a second so that the propeller speed is barely affected. If the inner loop is fast enough, then the motor disturbances wouldn’t even be seen by the outer loop because there wouldn’t be a noticeable change in altitude. This would allow the outer loop to be much slower and only respond to relatively slow disturbances like wind gusts.

If there was only a single loop, then the altitude controller would have to sense that tiny change in altitude and quickly adjust the motor voltage in that fraction of a second. If we tuned the outer loop to be that aggressive, then it would respond quicker to changes in altitude but it would also respond quickly to altitude sensor noise. Not ideal. So it’s a much better system to have these cascaded loops with each loop targeting a different set of reference points and disturbances.

OK, now that we have an understanding of why we need cascaded loops, let’s quickly discuss a few ways to tune them. First off, if the inner loop is much faster than the outer loop, that is, the bandwidth of the loop is at least about 5-7 times higher, then it is possible to tune them separately. For our drone example, we could tune the motor controller first and make sure that when we command a speed the motor spins up quickly with the performance that we want. Since the inner loop is so fast, when the outer loop requests a motor speed, the motor spins to that speed to quickly that to the outer loop, it might as well be instantaneous.  Therefore, to tune the outer loop, you can make the assumption that the inner loop doesn’t exist and the command goes straight through. In this way, you are essentially tuning two single loops exactly like you are used to.

If the two loops must operate at around the same bandwidth, tuning becomes a little more difficult because we can’t claim that the inner loop is fast enough to be instantaneous. Now the inner loop performance does affect the outer loop.  I’ll highlight 3 different ways to approach tuning here, but not go into detail on any of them. If you’d like more information, check out the links below. The first way is just an iterative approach.  This method requires you to tune the inner loop with a first guess, then tune the outer loop while the inner loop is running. If this isn’t sufficient for both loops when you’re done, then go back and tweak the inner loop again and iterate back and forth until you have the performance you are looking for.

An easier way, however, is to treat the system as a multi-input multi-output system rather than two cascaded single-input single-output systems. Using a state-space representation, you can treat the whole model as a single dynamical system and tune both loops simultaneously using a variety of state space tuning methods.  Finally, we can use software to auto-tune them. Just like I showed in the last video, the auto-tuning capabilities of MATLAB and Simulink can be used to tune the two loops at the same time.

Alright, there is obviously more I could cover on cascaded loops, but now I want to turn your attention to a topic that will affect almost every PID controller you create - and that is a discrete PID controller.

Similar to what I just did with cascaded loops, for discrete control I really just want to use this opportunity to introduce the topic to you and maybe give just enough information to explain why learning the topic is important and how a discrete PID controller differs from a continuous PID controller. As always, links to more information are below including a series of videos on discrete control that I made on my channel.

So why discrete control? Why not always treat time as a continuous variable like how we experience it in real life? The reason is because the controllers we design are usually run on digital computers. And digital computers don’t run continuously, but instead update at each sample time. For example, that means that if a computer process is running at 1 Hz, then once a second the controller reads the sensors, performs some calculations, and then commands the actuators. The actuators keep that command for the full second before receiving a new command at the next sample time. Contrast that with a continuous system, or analog computer, where the actuator command is changing continuously and smoothly. Digital systems also have other unique characteristics like quantization and transport delay, but to explain how digital computers impact a controller design, I’m going to just focus on sample time for the rest of this video.

If the sample time is short enough compared to the dynamics of your system, then it behaves very similarly to the continuous system. This is because the controller can read the sensors and update the actuators so fast that it appears practically continuous. However, as the sample time increases, the discrete result differs more and more from the continuous result. To understand why, let’s imagine this scenario. You want to walk from point A to point B and to do so you have to walk through a winding hallway. If the lights are on and your eyes are open, then this is analogous to the continuous time domain. You can see everything as it happens and constantly adjust your path. In fact you could probably run through the hallway. If we swap the continuous light for a discrete light like a strobe, then things change. If the strobe light is very fast, so fast that it looks like the lights are just on, then the situation looks the same to you and you can run through the hallway. However, if the flashing of the light slows down so that you can only see once a second, or once every few seconds, then you can no longer run through the hallway, at least not with as much confidence as before. 

This is the effect that discretizing has on our continuous PID controller. If the sample time is relatively fast, the PID controller can see everything and react just the same way as before. However, if the sample time is relatively slow (that is the flashing of the light is slow) then the PID controller will have to be slowed down as well to keep it from metaphorically running into the walls.

Now, the structure of the PID algorithm looks different between the continuous, s-domain, and discrete, z-domain representations. They both have a filtered derivative term and there are integrals on both sides, but as you might expect since they look different, the Kp, Ki, and Kd terms are not the same between the two for equivalent systems. However, there are several different ways to convert from the s-domain to the z-domain, and again I go through many of them in the series on discrete control. So a typical approach to tuning a discrete PID controller is to first tune it in the s-domain and then convert it to the z-domain, while ensuring that the sample time is fast enough to keep the same behavior. Then you can implement this discrete PID controller on your digital computer and be confident that it will behave the way you want.

I want to end this video by quickly explaining why we learn and talk about controllers in the continuous domain first. Why don’t we build and tune our PID controller in the discrete domain from the beginning? The answer is the same as for why we use linear models to approximate our nonlinear systems. It’s because solving problems in the s-domain is easier, and we have a lot more tools at our disposal for the continuous domain, just like we have a lot more tools for linear systems over nonlinear systems. In addition, often the plant being controlled is itself continuous, and so modeling both the plant and the controller in the same continuous domain makes the problem simpler. And as we’ve just seen, as long as certain conditions are met, like a fast enough sample time, then these imperfect representations are still good enough to get the job done. And it’s easier than the alternative.

So, I hope this quick discussion gave you a little insight into cascade control and discrete PID controllers. If you don’t want to miss the next tech talk video, don’t forget to subscribe to this channel. Also, if you want to check out my channel, control system lectures, I cover more control theory topics there as well. Thanks for watching. I’ll see you next time.

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