NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

control Lyapunov function globally asymptotically stable. Backstepping essentially
tries to reduce the system to a number of subsystems in series. Then the stabilizing
inputs for the subsystem closest to the output are computed, and these inputs are
then treated as the outputs of the previous system, and a new set of inputs to this
next subsystem are computed to guarantee stability. This process is repeated until
the original control input is computed. Control laws generated by this methodol-
ogy, as shown in [1], are proven to stabilize the system globally and asymptotically.
However, this design procedure is the most di cult to compute and implement, and
the resulting control law is both complicated and cumbersome. The rst step is
to apply a variable transformation to the exible beam's non-dimensionalized state
equations (2.5) to obtain:
z 1 = x 1 + sin( )
z 2 = x 2 + _ cos( )
y 1 = x 3
y 2 = x 4
v =
1
1 ; 2 cos 2 ( )
cos( )(z1 ; (1 + _2 ) sin( )) + k2 mk2 g
Rmmc (z2 ; _ cos( )) + kmkgu
R ; m
(3.18)
This variable substitution and the feedback transformation from u to v (which is
a non-singular transformation because the parameter is always smaller than 1)
simpli ed the system equations to the following form:
z_ 1 = z2 z_ 2 = ;z1 + sin( )
y_ 3 = _
y_ 4 = = v
22
(3.19)