## Happenings – 2013 Feb 15

Well…. All week long I had expected to say that the close approach of an asteroid was a non-event… I expected to say that it shot thru our ring of communication satellites without hitting anything, and just kept going.

And do it did. (Hmm. Would the NSA actually tell us if it lost a secret satellite?)

What none of us were expecting, of course, was the meteorite that crashed into Russia. Since it came from an entirely different direction, we believe that it was not associated with the asteroid. It broke a lot of glass, and injured a lot of people, but it wasn’t exactly devastating. I understand that people are already selling alleged pieces of it.

I’m really beginning to feel overwhelmed by my classes. I doubt that a technical post will go out this Monday, but we’ll see. At the moment, however, I don’t plan on trying to assemble one.

## introduction

I want to see exactly how we get offset under P-only control, and how PI control eliminates the offset. I’m going to use Example 3 again… I’m going to look at P and PI control… and I’m going to use the Tyreus-Luyben (“T-L”) tuning rules, which we’ve seen before.

When I started this, I was wondering about two things:

• Is the control effort nonzero when we have offset?
• We don’t always have offset under P-only control, do we?

I can’t say I’ve become an expert in offset, but I’m a little more comfortable with it. Now what I’d like to know is why the control effort goes to zero when we do not have offset – but that’s a question still looking for an answer.

Let me tell you up front what we will find:

• With P-only control, we may have offset: the output will not tend to the set point.
• In that case, the use of PI-control will eliminate offset.
• But if our plant (G) has a pole of any order at the origin, then P-only control does not lead to offset.
• In that case, you could imagine that the plant itself includes integral control.

## Happenings – 2013 Feb 9

There have been no major changes in the past week.

Oh, I did have to report for jury duty twice in the past week. The first day, I filled out a 15-page questionnaire… the second day, having read my answers, they dismissed me from further consideration.

In the meantime, I have made a little progress on MATLAB, but I still have a lot to learn. Sometimes I feel completely overwhelmed by 3 courses and MATLAB and a blog and work – and I vow not to start the fourth class… but the actual requirements for the courses aren’t very demanding, so I’m keeping up, although I’m not doing anywhere near what I had hoped to do.

An oft-repeated question in the discussion fora for the “Control of Mobile Robots” class is: how can we have offset under proportional-only control? I think it would be worthwhile to look at that, so I expect to have a short technical post ready for this coming Monday.

## Happenings – 2013 Feb 2

I think I’m going to do something different this weekend… and I don’t know that it will lead to a technical post on Monday… but it seems to be what I need to do.

Play with MATLAB. We’re using it in two of the courses I’m taking – Image Processing, and Control of Mobile Robots. So I’ve got a backlog of work in the Image Processing class, and I ought to get moving in the Controls class. On top of it all, as I’ve said before, I’ve got a hefty pile of books that use MATLAB.

Now, the student version prohibits me from using it for commercial or professional purposes. I don’t care to argue with them over whether this blog is “professional”, so unless I buy my own full-fledged version, I believe that it is simpler to not publish any of my MATLAB exercises out here – not unless I move up from the student license.

So I’ve got MATLAB itself to play with, and Simulink… and toolkits for image processing, controls, and symbolic calculations, and wavelets… and more, but that’s a heck of a lot to look at already.

Unless something quick and easy pops into my head over the weekend, I expect no technical post to go out this Monday. I expect to be doing plenty of applied mathematics, I just don’t expect to publish any of it.

## The Economic Order Quantity – a simple calculus application

The EOQ (Economic Order Quantity) formula is a deceptively simple model. It comes from Zipkin’s “Foundations of Inventory Management” (Irwin/McGraw-Hill, 2000, 0-256-11379-3) and it is the very first model in the book. It was first published 100 years ago, in 1913 – the model, not the book!.

When all is said and done, it’s a simple application of freshman calculus.

Imagine that we sell or use up one product, at a known constant rate $\lambda\$. Periodically, we order more of this product, to replenish our inventory I(t). Further, there is a known constant lead time L – between when we place an order and when we receive it (actually, when we can sell or use it, so this includes unloading and storing). If our inventory will go to zero at t = T, then, at the very latest, we must place an order at T – L:

## Happenings – 2013 Jan 26

Since putting out the latest technical post, I haven’t done a whole lot. I’m falling a little behind in my two Coursera classes (Intro Computational Finance, and Image Processing)… I have not been playing with Matlab, not much anyway… I have an idea for the next control theory post, but I don’t know about getting it done this weekend.

Still, I have a couple of ideas other than control theory for a short technical post this weekend, and I hope to put some time into Matlab. And to catch up on the assignments and the lectures for my classes – a third one (Control of Mobile Robots) starts this Monday… the fourth one (Signal Processing) starts on Feb 18, a week before the first one ends – so I’ll be taking 4 classes for only one week. Whew!

We’ll see how it goes.

## Control Theory – Example 3: PI, PD, and PID

There’s something I forgot to do when I looked at PID tuning: I meant to look at the Bode plots for the controllers, literally to see what they do individually. In addition, I’m going to mention real derivative control; I’ve long known that the simple PID includes “ideal” derivative control, but I only just realized just how unrealistic it is.

It’s probably just as well that this end up in a separate post.

Let’s just dive in to Bode plots.

## PID with Ziegler-Nichols rules

Recall our plant… first the definition, then the resulting transfer function.

We found, by looking at the gain margin for the open loop Bode plot of the plant that the ultimate gain and ultimate period were 10 and $2 \pi\$: