Last Updated on October 9, 2021

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Complexity theorists often quote the non-linearity of dynamical complex systems as a reason why we should not bother even try to predict their future states and just “go with the flow”. I find all such suggestions ill-advised.

Change Coordinate System

Before we give up all attempts to simplify an apparently complex situation we are trying to figure out, I would like to share my experience from dealing with a complex problem of “aerial target tracking“, and stress the importance of a point of view and the selection of an appropriate coordinate system (framework) in which to do the tracking.

Sensors (visual, IR radar, laser) and weapons, all work in the spherical coordinate system using the measured radial distance r (distance from the target to the origin point of the laser or radar sensor or the weapon), polar angle (or elevation) θ (vertical angle of the sensor or weapon platform with respect to the “polar” axis or the orthogonal complement elevation angle from the horizontal surface), and azimuth φ (angle of rotation from some referential “meridian” plane or center-line).

Source: Wikipedia

This coordinate system is very useful when collecting (measuring) positional data about aerial targets or for pointing weapons that will engage that target. However, this spherical coordinate system is less useful for target tracking, because the collected data measured by the sensors describing a target trajectory in the polar coordinate system is non-linear and looks something like this for a target moving in a straight line:

However, when the above (measured) data is transformed from spherical to cartesian coordinates (x, y and z), the situation is quite different:

Just by changing the coordinate system, the trajectory data becomes linear, so predicting the future position of an aerial target is now much easier then if the tracking (prediction) was done in the original (spherical) coordinate system.

So, the lesson to take home here is that just because something appears to be complex, it may not be so after giving it another look from a different perspective, because from that point of view it may present itself as a much simpler problem to solve.

Untangle the Threads

Another example I’d like to share is connected with the origins of my Model for a Dynamical System with Memory (the DSM). In the late 80’s while attending some advanced courses in Control Engineering I was introduced to the following representation of a non-linear dynamical system with memory (scanned from my old notebook):

The context was about dividing the system in two parts, the linear components (functions g and f) and the non-linear (memory) part, function F. While trying to wrap my head around this diagram I draw two versions of it, the original and the “untangled” one where the original connectivity was preserved but no connections (arrows) were crossing each other:

The moment I saw the “untangled” version of this block diagram I knew I had found a new framework that could remove (explain) a lot of complex issues in the future.

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