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NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

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Response in m<br />

0.05<br />

0.04<br />

0.03<br />

0.02<br />

0.01<br />

0<br />

−0.01<br />

−0.02<br />

Linear Robust<br />

Linear Optimal<br />

−0.03<br />

0 1 2 3 4 5 6 7<br />

Time<br />

Figure 5.4: Linear Robust vs. Linear Optimal<br />

though Figures 5.6 and 5.7 show that the linearized controllers' attenuation was more<br />

uniform.<br />

Since its performance increases with respect to the linearized designs in going<br />

from simulation to the hardware testbed, the passivity based control design is justi ed.<br />

Its robustness is also apparent from its portability between simulation and hardware.<br />

5.4 Successive Galerkin Approximations<br />

The successive Galerkin approximations yields the best results in hardware.<br />

Both the HJB solution and the HJI solution outperform the linearized controlsaswell<br />

as the passivity based control, as shown in Figures 5.9, 5.10, and 5.11. Figure 5.16<br />

shows that the nonlinear robust approximation slightly outperforms the HJB solution,<br />

perhaps because its design emphasizes robustness with respect to the unmodelled<br />

e ects of the exible beam.<br />

The SGA method succeeds in outperforming standard linear approaches as<br />

well as a passivity based design. Its implementation is straightforward, and it is eas-<br />

ily tuned to optimize its performance. Its excellent performance in hardware speaks<br />

52

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