In their paper (1), the authors consider a nonlinear mechanical system represented by
$mẍ+(Kd+b)ẋ+Kpx=Kdu̇+Kpu−sign(ẋ)μkN$
1
In Eq. 1, $m$ is the mass of the system, $Kd$ is the derivative gain, $Kp$ is the proportional gain, $b$ is the viscous friction, $μk$ is the kinetic coefficient of friction, $N$ is the normal force, $x$ is the system displacement, and $u$ is the applied control input. The nonlinearity in the system is the Coulomb friction, which is represented by $sign(ẋ)μkN$.

The following comments are made regarding the approach and results in Ref. (1):

• 1
The authors rely on the exact cancellation of the nonlinear term. More precisely, they propose the following control law:
$u=1Kpsign(ẋ)μkN+v$
2
This control law obviously cancels the nonlinearity in the mathematical model given in Eq. 1 and results in a linear system.

Theory and good practice of control engineering strongly discourage the design of control laws that rely on exact cancellation of linear or nonlinear terms. This is due to the fact that inevitable uncertainties make such cancellation impossible.

After Eq. (11) of Ref. 1, the authors write that “The $μkN$ estimates were found for each pair of adjacent peaks and troughs and then averaged.” From this statement, it is inferred that there must be uncertainty in the nonlinear term $μkN$, and hence it cannot be canceled entirely by the control law in Eq. 2.

• 2

Assuming that the nonlinearity is canceled, the authors design an input shaper to generate a sequence of two step inputs of amplitudes $A1$ and $A2$ to be applied exactly at time instances $t1$ and $t2$, respectively. The authors apply the designed control law to a solder cell machine and report satisfactory results. They, however, do not evaluate the performance of their system in the presence of uncertainties either analytically or experimentally. It is not clear how the system would behave when (i) the nonlinearity is not exactly canceled; (ii) the system parameters are different from the nominal values; and (iii) the step inputs generated by the input shaper are not applied exactly at $t1$ and $t2$.

1.
Lawrence
,
J.
,
Singhose
,
W.
, and
Hekman
,
K.
, 2005, “
Friction-Compensating Command Shaping for Vibration Reduction
,”
ASME J. Vibr. Acoust.
0739-3717,
127
, pp.
307
314
.