Laplace Transform of Sign() or sgn() functions

[macardoso]

TL;DR Summary

Trying to model friction of a linear motor. Need help with state space model. What is the Laplace Transform of sign(x_dot)? I think this is the right sub-forum?

Trying to model friction of a linear motor in the process of creating a state space model of my system. I've found it easy to model friction solely as viscous friction in the form b * x_dot, where b is the coefficient of viscous friction (N/m/s) and x_dot represents the motor linear velocity.
However, this viscous friction only model fails to accurately represent the system since the Coulomb friction in my motor is roughly 400x that of the viscous friction for my operational speed range. I would like to use the friction model b_c * sign(x_dot) + b_v * x_dot, where b_c represents the coefficient of sliding (Coulomb) friction, b_v is the coefficient of viscous friction, and x_dot is the motor linear velocity. The sign() function is needed since the direction of the friction force opposes the direction of motion.
There is also a static friction component that I'm ignoring for now.
My issue: I can't seem to find a Laplace transform for sign(x_dot) to allow me convert my equations of motion into a state-space model. The simple model b * x_dot becomes b * X(s) * s. What would b_c * sign(x_dot) + b_v * x_dot become??

Thanks,
An engineer trying to relearn control theory 10 years after college...

 

[anuttarasammyak]

$$sgn(x-c)=2 H(x-c)-1$$
where H is Heaviside step function.
From Laplace tansform table
$$L\{sgn(x-c)\}(s)=2 \frac{e^{-cs}}{s}-\frac{1}{s}$$