NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

Figure 2.2: Mechanical Model of Flexible Beam System,
The exible beam is treated as capable only of motion in the plane of the diagram.
If perturbed the free end of the beam will oscillate according to the classical beam
equation:
A(x) @2 w(x t)
@t 2
+ @2
@x 2 [EI(x)@2 w(x t)
@x 2 ]=f(x t) (2.1)
where f(x t) is the applied force on the beam in the y direction,and w(x t) is the
de ection of a point a distance x from the xed end at time = t. Unfortunately, the
partial derivatives in this equation are not suitable for the state-space model that we
seek, something of the form:
where x 2 IR n .
_x = f(x)+g(x)u(x) (2.2)
A simpler idea would be to model the exible beam as a rigid beam with a
single torsional spring at the xed end (see Figure 2.3). A state-space equation can
be obtained for this model using the Euler-Lagrange (EL) equations:
d
dt (@Lp(qp qp) _
) ;
@qp _
@Lp(qp qp) _
= Mup
@qp
9
(2.3)