“…According to equation (3), f ( x 1 , x 2 , t ) is the function of inner disturbance, and
Tf′ is external disturbance. The classical ESO considers f ( x 1 , x 2 , t ) and
Tf′ as an extended state together, and equation (3) can be rewritten as follows (Mei and Yu, 2021): where
x3=f(x1,x2,t)+Tf′ is the total disturbances, and w 1 ( t ) is the derivative of x 3 . By establishing an ESO for equation (7) through a nonlinear function, the classical ESO can be written as follows: where
fal(e,α,σ)=true{lefteσ1−α,true|etrue|≤σ|e|αsign(e),true|etrue|>σ as in Han(2009), σ is the length of the linear segment interval, e is the error between the estimated state and the actual output, and α is a constant between 0 and 1. z 1 , z 2 , z 3 are the estimated values of x 1 , x 2 , x 3, respectively, and β 01 , β 02 , β 03 are the parameter values of the ESO.…”