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I have a nonlinear system of

$f = m \ddot{x} + b \dot{x} + c \sin{x} , \tag{1}$

for which I want to design a model based PD controller or as I have seen on other references using computed torque control or feedback linearization.

I can define error as:

$e = x - x_d , \tag{2}$

where $x_d$ is the desired value. Following this presentation the computed torque control law is:

$f = m \left( \ddot{x}_d - u \right) + N , \tag{3}$

where $u$ is

$u = -k_d \dot{e} -k_p e , \tag{4}$

and $N$ is

$N = b \dot{x} + c \sin{x} . \tag{5}$

Combining the above equations I get

$\ddot{e} - k_d \dot{e} - k_p e = 0 , \tag{6}$

which in addition to contradicting the above reference, is very confusing. I would appreciate if you could help me know if my calculations are correct or I'm making a mistake here? Thanks for your support in advance.

P.S. reading this paper it seems that $u$ should actually be defined as

$u = k_d \dot{e} + k_p e , \tag{7}$

which leads to

$\ddot{e} + k_d \dot{e} + k_p e = 0 , \tag{6}$