NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

Response in cm
4
3
2
1
0
−1
−2
−3
SGA: Nonlinear Optimal
Open Loop Response
−4
0 1 2 3 4 5
Time
6 7 8 9 10
Figure 4.13: SGA: Nonlinear H 2 vs. Open Loop Response
linearized H 2 (Figure 4.14 controllers in damping the vibrating beam. It did not use
as much control e ort as either of these controllers, however, so it did use control
e ort very e ciently.
Since this control was fully nonlinear, in the sense that it was a nonlinear
function of the state of the system, it was hoped that it could outperform the standard
linearized approach. The fact that the LQR control provided better performance
could be attributed to the fact that not enough basis functions were used, or perhaps
the wrong basis functions were used. Only polynomials were used for basis functions,
but logarithms or exponential functions might have performed better.
4.2.6 SGA: Nonlinear H1 Control
The performance of the SGA to the HJI equation is comparable to that of
the SGA to the HJB equation (see Figure 4.23). It seems that in both the H 2 and
the H1 cases, the linear LQR design yields better overall performance, as shown in
Figure 4.19. It was hoped though, that the nonlinear solutions would perform better
in hardware { the added uncertainties should demand more selective controllers.
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