Differential Equations

A differential equation system is given by: for all t ∈ Reals₊

z'(t) = g(z(t), v(t)).

Here

z'(t) is the derivative of z with respect to t, evaluated at t ;

t ∈ Reals₊ stands for continuous time;

z: Reals₊ → Reals ^N is the state response; and

v: Reals₊→ Reals ^N is the input signal.

Here, the state is updated continuously rather than discretely, but otherwise, this is similar to a state machine with an infinite state space. In fact, we can approximate the derivative by the difference

z(t + δ ) - z(t) ≈ δ g(z(t), v(t)).

If we sample z and v at times 0, δ , 2δ , ..., nδ, (n+1)δ , ... and denote the nth sample value by

s(n) = z(nδ ), x(n) = v(nδ ),

we get the difference equation

s(n + 1) - s(n) = δ g(s(n), x(n))

This is called a difference equation because the difference on the left is analogous to a differential in a differential equation. This can be rewritten as a state update equation,

s(n + 1) = s(n) + δ g(s(n), x(n))

Thus, a discrete approximation to a differential equation is a state machine with an infinite state space.

Week 6

Differential Equations