NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ... NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...
Response in cm 4 3 2 1 0 −1 −2 −3 Linearized Optimal Open Loop Response −4 0 1 2 3 4 5 Time 6 7 8 9 10 Figure 4.3: Linearized Optimal vs. Open Loop Response solutions to plants that are di cult to regulate using the standard linear approaches. Figure 4.3 shows the LQR design's performance plotted against the open loop re- sponse. This linearized H 2 approach gave almost the best performance in simulations. This is reasonable since the initial disturbance was small, and the system stayed within a region where linearization yields a good estimate of the true system dynamics. It should be said, though, that this control relied on relatively high gains (Kc = [;658 69:8 26 3:7]), and it was later discovered that such gains resulted in actuator dysfunction and instability on the hardware system. This really highlights the goal of nonlinear control design research: to nd control laws that will e ectively regulate systems when they operate in the presence of real-world nonlinearities or when they operate well beyond their linear region. 32
Response in cm 4 3 2 1 0 −1 −2 −3 Linearized Robust Open Loop Response −4 0 1 2 3 4 5 Time 6 7 8 9 10 4.2.2 Linearized H1 Figure 4.4: Linear H1 vs. Open Loop Response The linearized H1 control also gave good performance, though it could not match the performance of the linearized H 2 controller. In fact, this linearized H1 control could match neither the H 2 SGA nor the Backstepping control's performance. Figure 4.5 shows that the price of a more robustness is less performance. The gains produced by the linearized H1 design (Kc = [;62 16:7 1:5:46]) were muchlower than those of the LQR design. These gains were, in fact, more robust at least in the sense that they produced a control that succeeded in stabilizing the actual exible beam system unlike the gains produced by the linearized H 2 approach. 4.2.3 Passivity Based Control As seen in in Figure 4.6, the passivity based control exhibited some strange behavior: after suddenly attenuating the vibration at about t = 6 the oscillations then increase a little before dying out. Though this control is still a dramatic improvement on the open loop response, both of the linearized controllers provided better damping 33
- Page 1 and 2: NONLINEAR CONTROLLER COMPARISON ON
- Page 3 and 4: BRIGHAM YOUNG UNIVERSITY GRADUATE C
- Page 5 and 6: ABSTRACT NONLINEAR CONTROLLER COMPA
- Page 7 and 8: Contents Acknowledgments vi List of
- Page 9: List of Tables 4.1 Tabular Comparis
- Page 12 and 13: 4.19 SGA: Nonlinear H1 vs. Linear O
- Page 14 and 15: Figure 1.1: TORA System One of the
- Page 16 and 17: 1.3 Literature Review In surveying
- Page 19 and 20: Chapter 2 Plant Speci cations and M
- Page 21 and 22: Figure 2.2: Mechanical Model of Fle
- Page 23 and 24: Figure 2.4: Translational Oscillati
- Page 25 and 26: 2.3 Software Set-up All of the cont
- Page 27 and 28: Chapter 3 Overview of Control Strat
- Page 29 and 30: The optimal gain matrix was given b
- Page 31 and 32: 3.3 Passivity-Based Control Certain
- Page 33 and 34: limitations, k 1 and k 2 are tuning
- Page 35 and 36: Now we regard as the control variab
- Page 37 and 38: where V satis es the well known Ham
- Page 39 and 40: ecause one can quickly adjust the Q
- Page 41 and 42: Chapter 4 Simulation Results 4.1 Th
- Page 43: disturbances, whereas the hardware
- Page 47 and 48: Response in cm Response in cm 4 3 2
- Page 49 and 50: Response in cm Response in cm 5 4 3
- Page 51 and 52: Response in cm 4 3 2 1 0 −1 −2
- Page 53 and 54: Response in cm Response in cm 4 3 2
- Page 55 and 56: Response in cm Response in cm 4 3 2
- Page 57 and 58: It is seen in Figure 4.23 that the
- Page 59: all of the physical parameters exce
- Page 62 and 63: Response in m 0.05 0.04 0.03 0.02 0
- Page 64 and 65: Response in m 0.05 0.04 0.03 0.02 0
- Page 66 and 67: Response in m Response in m 0.05 0.
- Page 68 and 69: Response in m Response in m 0.05 0.
- Page 70 and 71: Response in m Response in m 0.05 0.
- Page 72 and 73: control was that it requires only t
- Page 74 and 75: to the Galerkin approximation techn
- Page 77 and 78: Bibliography [1] P. Kokotovic M. Ja
Response in cm<br />
4<br />
3<br />
2<br />
1<br />
0<br />
−1<br />
−2<br />
−3<br />
Linearized Robust<br />
Open Loop Response<br />
−4<br />
0 1 2 3 4 5<br />
Time<br />
6 7 8 9 10<br />
4.2.2 Linearized H1<br />
Figure 4.4: Linear H1 vs. Open Loop Response<br />
The linearized H1 control also gave good performance, though it could not<br />
match the performance of the linearized H 2 controller. In fact, this linearized H1<br />
control could match neither the H 2 SGA nor the Backstepping control's performance.<br />
Figure 4.5 shows that the price of a more robustness is less performance.<br />
The gains produced by the linearized H1 design (Kc = [;62 16:7 1:5:46])<br />
were muchlower than those of the LQR design. These gains were, in fact, more robust<br />
at least in the sense that they produced a control that succeeded in stabilizing the<br />
actual exible beam system unlike the gains produced by the linearized H 2 approach.<br />
4.2.3 Passivity Based Control<br />
As seen in in Figure 4.6, the passivity based control exhibited some strange<br />
behavior: after suddenly attenuating the vibration at about t = 6 the oscillations then<br />
increase a little before dying out. Though this control is still a dramatic improvement<br />
on the open loop response, both of the linearized controllers provided better damping<br />
33
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