A few basic principles and some experimentation can get you a long way in
mechatronics, but the time comes when they must be backed up by mathematical
theory. That is not to deny that the empirical approach will get you
much further than the theory on its own.

I have tried to feature theory that will actually be useful, rather than
mental party tricks that are only really relevant to answering examination
questions. We cannot avoid differential and difference equations. They are
the essential substance of the things that we are trying to control. We must
be able to spy out the state variables that define the behavior of the system
and then derive state equations to describe them.

One vital use for the differential equations is to enable us to simulate the
systems in software and try out the algorithms, whether based on theory or
pragmatism, to obtain an early verdict on the likelihood of the success of the
outcome. But unless the algorithm is tried out on a real system, simulation is
a mere mental pastime.

We have to include methods of testing for stability, for despite the legion
of thermostats happy in their limit cycles, there are many systems where true
stability is essential. Transfer functions certainly have their uses, but they are
just one piece of the jigsaw puzzle and not an end in themselves. When we
use mathematical shorthand to put our state equations into matrix form,
we find ourselves led into the world of transformations and eigenvalues and
we need to brush up on a mathematical toolkit.

When we step back to the mechanical reality of moving parts, accelerations
and forces, we again find that the coordinate geometry is leading us down the
matrix path. Rotations become tensors, and the articulation of a robot arm
involves a chain of affine transformations, performed very neatly by a few
matrix software operations.