Dynamical systems – an introduction

It is said that one picture is worth a thousand words – so let it be two thousand:

Fig. 1 A dynamical system – as seen by an engineer
Fig. 2 “Life is life” dynamical system

A dynamic system is one whose behavior, described by certain quantities, changes in time. We may be interested only in the end result (that is called a steady state), but if we are really to talk about the system dynamics, then the end result will be only one of the elements to be considered, and the main emphasis will be on how the steady state is reached (or why there is no steady state).

Let’s imagine that we are dealing with a simple lifting mechanism, which after switching on should push something to a given height (note: the example is simplified as much as possible, so that you can easily imagine what is shown in the diagram below). In the simplest model (not to scare you, there will be no diagram or equations for this example) the input (control) is the force to be used, and the output – the height to which we managed to lift our weight. Obviously, the greater the weight, the greater the force to be used. The device is constructed in such a way that after switching it on (giving a constant force at the input), after some (preferably short) time the feeder reaches some constant height and its position stops changing (it settles – hence the name “steady state”).

The first question to ask is, “How much force should we use?” Let us assume that we can answer this question. However, this is not enough for us to declare success. Depending on what components were used to construct the lifting mechanism, the position of the mechanism may change in time in different ways, always reaching the set height we require (see figure below).

Fig. 3 Three output signals , obtained for varius system parameters. All converge to the same steady state, but in different ways. The red line represents a waveform that could be considered unsatisfactory, due to the long time required to reach steady state. The blue line represents the response of the system, which is much faster, but there are oscillations that may be unacceptable. The black line could be considered a satisfactory system response

Here we come to the first reason of learning system dynamics – acquiring the ability to predict how a system will behave in response to a particular input and using this knowledge in system design (including the selection of components that make up the system). We look at various features of such systems – at first glance, let us focus on the (perhaps) most obvious:

  • How fast will the system reach steady state (this still needs to be defined – e.g. after how long will the output value stay within +/-5% of the steady state value)?
  • Will there be oscillations in the system, or will the system reach a steady state without oscillations (aperiodic behaviour)?
  • How large is the overshot of the response?
Fig.4 Aperiodic system response . Tss denotes the time needed to reach the steady state
Fig. 5 Oscillatory system response . Tss denotes the time needed to reach the steady state

Of course, you can find answers to these questions by trial and error – but it’s probably better to do some calculations and simulations first, and only then build a prototype of the system that you would like to sell for a lot of money… Especially that trial and error method may prove to be very expensive …

Once we know how the system responds to a particular input form, we can move on to the next step – how to shape the controls so that the system behaves exactly as we expect it to. For example, if we are dealing with a robot that has to move horizontally an element from one place to another, the question is: How to control the motors that move the individual parts of the robot? And if we want to charge the battery of an electric car – how to change the voltage (current) in the charging device to do it in the shortest time?

The problem is that even if we designed and constructed the system, we only know its parameters with some accuracy. Moreover, as I will show in the subsequent posts, any model that allows us to answer the questions asked above is always a simplified description of reality. In addition, the system behavior, may also be affected by some interference. Therefore, instead of an open loop structure (as in Fig. 1), a closed loop control system, using the information from the measuring devices to achieve the desired effect, is preferred:

Fig. 6 The simplest closed-loop control system structure

Automatic control – here we come (well, some day and post, at least). That’s life, as well:

Fig.7 Closed-loop for anyone

If you liked the post and think it might be worth your friends reading it – please recommend it :).

What is it about ?

Why have I created this blog ? I guess, mainly to fulfill my desire to teach everyone about everything… In lectures there is usually no time to talk about “the basics of the basics” – because I wouldn’t have time to get to the point before the end of the semester. Additionally, I would bore those students who know the basics. I hope that by reading my posts at least some of you will be motivated to learn and to get to know the mechanisms that affect us all.

Though engineering is displayed prominently at the head of the main page, I hope that people of different backgrounds might find something interesting in my posts. I’ve started with introduction to systems dynamics but I hope I’ll have time (and motivation) to write about optimization and decision making (prospective managers and decision makers are welcome to read and comment), control systems, elements of computer science, operations research and whatever else might be of interest to my students and others who want to gain some understanding of what is going on around us (not only in technical systems) and how to live with it 🙂

Well, not exactly – I will present (comprehensive, hopefully), introduction to various methods of decribing and analysis of simple (and then, not so simple) systems (technical, biological), their control, as well as optimization methods and their applications. That’s for a start – and then I will see if there’s any interest in my writing and either expand to new topics or bequeath the blog to my descendants (if I cannot leave a fortune to them…)

For some time now I’ve been observing that what’s obvious to me isn’t necessarily obvious to others, my students in particular (who could guess?). I will try to remember that – that’s why sometimes my posts may seem too basic. However, I would like to reach out not only to my own students, but also to others, who may have a different background, as well as to everyone who has discovered a hidden engineer at heart or who simply needs a little understanding of the blog topics for personal or professional development. You may even find that some of the entries will be of value to the absolutely-nothing-about-engineer-in-me people.

The first blog posts will be about dynamical systems, their description, different points of view to look at them. There will be concrete examples (initially very simple, not to say primitive – but only following simple examples, we can understand the more complex ones), so that every person reading them hopefully will be able to imagine a system I am writing about and relate formulae or diagrams to the behaviour of the system. I will strive to make the entries understandable to anyone who wants to comprehend what is going on, not just memorize a formula and after passing an exam (paper, colloquium, etc.) immediately forget it.

In posts devoted to systems dynamics, I will use simple mechanical, electrical and “hydraulic” systems as examples. I assume that everyone has ever poured something from one container to another (maybe even through a funnel), pushed or turned something, switched on a lightbulb – hence the examples. If it goes well, we will move from such lightbulb switching to salmon trapping in a sea farm in a Norwegian fjord.

Once we know how dynamical systems behave, we will start talking about how to control them. And then, optimization will also appear …

If you find my posts interesting, please put a word around…