Systems Theory... well, Systems Theory Light.

Jul 17 2012 Published by under Uncategorized

When I was in grad school, all of my course work was focused on dynamical systems. The simplest, canonical example of this is a mass on a spring. Hang a spring from a hook, hang a weight from the spring. The gravity exerts a downward force on the mass, and the spring has a constant which acts opposite the direction of extension or compression. Thus, when you pull a spring, it pulls back, but when you  compress a spring, it pushes back (as opposed to, say, a slinky, which will collapse all the way).

Force, you may recall, acts to accelerate masses. Acceleration is the second derivative of position. So, if we want to write an equation which describes where the mass is at any given point in time, we use Newtonian motion, which is good enough for 99.9999% of the engineering systems in the world. And while Leibnitz came up with better notation than Newton did, it's not as amenable to typing things out in text, so I'll use Newtonian notation as well.

So, Newton tells us that Force is equal to Mass times Acceleration. Also, the force exerted by a spring is equal to a negative constant k times the distance the spring is stretched. Or F=ma=-kx. Well, using ' to indicate derivatives, this means that F=m*x''(t)=-k*x(t). Or, the force exerted on an object is equal to the spring compression constant times the excursion of the spring. When we solve this equation for x(t) (which takes a little bit of doing, but not too much, and which I won't go into here) we get the equation of the harmonic oscillator, or x(t)=A*cos((2*pi/T)*t+phi), where A is the amplitude, T is the period, and phi is the angle of offset, or phase shift. A and phi are functions of the initial conditions (i.e.,how far was the spring extended in the first place, and did you just let go, or give it a push?), and T is a function of the spring constant and the mass.

Now, this all works for magic springs. Meaning, they don't lose any energy to friction to materials effects. In the real world, we need to include these, which we model as a damping function (like a shock absorber). For this, we add an additional force term, which is proportional not to the position, but to the velocity of the mass. Velocity is the first derivative of position. This looks like: F=-k*x(t)-c*x'(t)=m*x''(t). where c is a constant which indicates how much the system resists motion, called the damping constant.

We rearrange this to x''(t)+(c/m)*x'(t)+(k/m)*x(t)=0, and solve. There are various ways of rearranging the constants in order to derive specific meanings from them, like the damping ratio or the angular frequency (because mathematically, a pendulum is really the same thing as a top; that is, oscillating is essentially the same thing as spinning modulo a factor of i ~ very roughly speaking).

Frequently, we want to drive these systems to behave in a certain way, and then we'll end up with a system that looks like: x''(t)+a*x'(t)+b*x(t)=u(t), where u(t) is some function that we'd like to try to force the system to follow. This type of system is often called a "canonical second order system", because it's well understood and is useful for modeling everything from masses on springs to radio communication.  The math gets a little ugly if you don't do differential equations, and I'm not going to plough through it here. Suffice to say, that playing around with systems like this is how I got started in systems theory.

Everything I've done above is basically at the sophomore-or-junior-in-college level of engineering, or at least it was eighteen years ago. As we go in systems theory, we learn to do larger order systems (meaning derivatives greater than second), but to do them by decomposing the system into matrices and vectors, so that we replace higher derivatives with vector algebra. Turning differential equations into algebra is essentially how dynamical systems get solved, by using Laplace transforms, or in the discrete case, z-transforms.

I don't do any of that anymore, and I never did any professionally. The reason I write about it is to point out some of the basic concepts of systems theory that I do use today, and those basic concepts, which I'll try to go into more tomorrow, are the interconnectedness of things, and the utility of rates and relationships in modeling things. At no time, in explaining the above example, did it matter how big the mass was, or how strong the spring or the damping agent. Those are just numbers, and numbers are the least interesting thing about the problem.

The thing that matters is the relationships between the things. The equation of motion, F=ma, which is a straightforward, linear equation which at first blush would seem to have very little to do with waves, and less to do with spinning. All it means is that if I push something, it'll go a little faster. But when we look more deeply into the arrangements of objects, and how they interact, we end up with behaviors which are very interesting: vibration, rotation. Which lead to music and electricity and other things which, from humble beginnings reveal fascinating phenomena.

That's systems theory: the combination of small objects and simple rules to model and understand and predict very complex behaviors. Tomorrow I'll get in to how we use this to model more exciting systems, like a medical clinic or something.

 

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