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For a simulation for testing on automatic cruise control, I came across the equation:

$$ v_{n+1} = (1 - k_1 / m) v_n + (1 - k_b) \begin{pmatrix} T_n \\ θ_n \\ \end{pmatrix} $$

where:

  • $T$ = throttle position
  • $k_1$ = viscous friction
  • $k_b = k_2 / m$
  • $k_2 = m g \sin(θ)$
  • $v$ = velocity

$k_b$ doesn't make sense. The matrix part doesn't make sense to me. Can anyone expound the equation?

Why isn't the angle put in sin or cos first?

Also, why is there viscous friction in solid physics? Isn't the angular component in $k_2$ enough?

SOURCE:

A Fuzzy Logic Book: "scribd.com/doc/105335356/124/INDUSTRIAL-APPLICATIONS"; Page 508. Number 13.2.

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