NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

higher order dynamics of the beam should manifest themselves as forces acting on
the beam, while the desired output is the regulation of the states subject to limited
actuator power. Thus the linearized H1 control should, even under worst-case e ects
from modeling and linearization errors, generate an adequate, stabilizing control. Like
the LQR design strategy, linearH1 is a solved problem, the solution of which can be
obtained by solving the following modi ed algebraic Riccati equation:
0=PA+ A T P ; P (B 2B T
2 ; ;2 B 1B T
1 )P + C T C: (3.8)
The valid solutions are ones in which P 0 and are suchthatA;(B 2B 0 2; 2 B 1B 0 1)P is
asymptotically stable. Asearch is then performed to nd the smallest possible that
satis es these conditions, and the resulting solution (a function of this optimal ) is
guaranteed to minimize the L 2 gain from disturbance to output while simultaneously
stabilizing the system. The optimal feedback gain matrix K is next computed from
this solution as follows: K = ;B 0 2P . In these equations, B 1 is the matrix resulting
from a linearization of k(x), where
0
B
k(x) = B
@
0
KmKg(M +m)
Rm[(I+mc2 )(M +m);m2c2 cos2 ( )]
0
;mcKmKg cos( )KmKg
Rm[(I+mc 2 )(M +m);m 2 c 2 cos 2 ( )]
1
C (3.9)
C
A
and B 2 is the linearization of g(x) about the equilibrium point x =0. For the exible
beam system, B2 was found to be:
0 1
0
B C
B C
B;:69C
B2 = B C
B C : (3.10)
B 0 C
@ A
31:7
The parameters used to compute this control were Q = 2000 100 70 0 and
R = :01. The optimal gain matrix was computed to be K = ;62 17 1:5 :46
and the optimal gamma was was found to be = 302.
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