in modern control theory, often modeled mathematically using the standard bound function
to bloat deeply into the saturation zone, causing massive transient overshoots and prolonged settling times when the system attempts to reverse direction. In the presence of
sat(vk)=sign(vk)×min(|vk|,umax)sat open paren v sub k close paren equals sign open paren v sub k close paren cross min of open paren the absolute value of v sub k end-absolute-value comma u sub max of end-sub close paren
sat(vk)≈umax×tanh(vkumax)sat open paren v sub k close paren is approximately equal to u sub max of end-sub cross hyperbolic tangent open paren the fraction with numerator v sub k and denominator u sub max of end-sub end-fraction close paren
