NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ... NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
Response in m 0.05 0.04 0.03 0.02 0.01 0 −0.01 −0.02 Linear Robust Linear Optimal −0.03 0 1 2 3 4 5 6 7 Time Figure 5.4: Linear Robust vs. Linear Optimal though Figures 5.6 and 5.7 show that the linearized controllers' attenuation was more uniform. Since its performance increases with respect to the linearized designs in going from simulation to the hardware testbed, the passivity based control design is justi ed. Its robustness is also apparent from its portability between simulation and hardware. 5.4 Successive Galerkin Approximations The successive Galerkin approximations yields the best results in hardware. Both the HJB solution and the HJI solution outperform the linearized controlsaswell as the passivity based control, as shown in Figures 5.9, 5.10, and 5.11. Figure 5.16 shows that the nonlinear robust approximation slightly outperforms the HJB solution, perhaps because its design emphasizes robustness with respect to the unmodelled e ects of the exible beam. The SGA method succeeds in outperforming standard linear approaches as well as a passivity based design. Its implementation is straightforward, and it is eas- ily tuned to optimize its performance. Its excellent performance in hardware speaks 52
Response in m Response in m 0.05 0.04 0.03 0.02 0.01 0 −0.01 −0.02 Passivity Based Open Loop −0.03 0 1 2 3 4 5 6 7 Time 0.05 0.04 0.03 0.02 0.01 0 −0.01 −0.02 Figure 5.5: Passivity Based vs. Open Loop Passivity Based Linear Optimal −0.03 0 1 2 3 4 5 6 7 Time Figure 5.6: Passivity Based vs. Linear Optimal 53
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Response in m<br />
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 />
Passivity Based<br />
Open Loop<br />
−0.03<br />
0 1 2 3 4 5 6 7<br />
Time<br />
0.05<br />
0.04<br />
0.03<br />
0.02<br />
0.01<br />
0<br />
−0.01<br />
−0.02<br />
Figure 5.5: Passivity Based vs. Open Loop<br />
Passivity Based<br />
Linear Optimal<br />
−0.03<br />
0 1 2 3 4 5 6 7<br />
Time<br />
Figure 5.6: Passivity Based vs. Linear Optimal<br />
53
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