Delay-Free Loops

Backward-Euler numerical integrator:

$$v_n \;=\;v_{n-1} + T\,\hat{\dot v}_n$$

Corresponding BE digital mass model:

$$\hat{v}_n \;=\;\hat{v}_{n-1} + \frac{T}{m} f_n$$

where $\hat{v}_n$ is the $n$ th sample of the estimated velocity, $f_n$ is the driving force at sample $n$ , $m$ is the mass, and $T$ is the sampling interval

• Note that a delay-free loop appears if $f_n$ depends on $v_n$ (e.g., due to friction):
  $$\hat{v}_n \;=\;\hat{v}_{n-1} + \frac{T}{m} f_n(\hat{v}_n)$$

• In such a case, the difference equation is not computable in this form

• Non-computable finite-differences schemes such as this are said to be implicit

• We can address this by using a forward-difference (``Forward Euler'')in place of a backward difference