NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

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Response in cm Response in cm 5 4 3 2 1 0 −1 −2 −3 SGA: Nonlinear Robust Backstepping −4 0 1 2 3 4 5 Time 6 7 8 9 10 4 3 2 1 0 −1 −2 −3 Figure 4.22: SGA: Nonlinear H1 vs. Backstepping Control SGA: Nonlinear Robust SGA: Nonlinear Optimal −4 0 1 2 3 4 5 Time 6 7 8 9 10 Figure 4.23: SGA: Nonlinear H1 vs. SGA: Nonlinear H 2 44
It is seen in Figure 4.23 that the performance of the nonlinear H1 solution and the nonlinear H 2 solution provide similar performance, with the robust design being slightly inferior, similar to the linearized case. The question becomes now, what exactly is gained by using such 'robust' design methodologies? As is shown in Chapter 5, the H1 designs do provide better performance on the hardware, a vindication of robust design methodology. 4.3 Tabulated Results Table 4.1 shows how the numbers compare for the various control strate- gies. The linearized H 2 was easily the best performer, while the passivity-based control yielded the worst results. The backstepping algorithm was the best oscillation damper, despite its high rst-state energy, but it also used a lot more control e ort than the other designs. Disappointingly, the nonlinear control laws do not particu- larly stand out as notably superior to the linearized ones: perhaps the TORA system is too easily stabilized and not appropriate for a nonlinear benchmark problem. Table 4.1: Tabular Comparison of Simulated Results LQR Lin. H1 PBC Backstep HJB HJI R y(t) 2 dt(cm 2 ) 3.06 5.72 11.2 6.03 4.12 5.28 R u(t) 2 dt(V 2 ) 4.05 4.39 .945 21.12 4.5 1.366 4.4 Tuning and Ease of Implementation The easiest controls to tune are the linearized optimal control laws. Both LQR and the optimal H1 require only the adjustment of the weighting matrix Q, and to rapidly test the new values is as simple as loading the new feedback gain vector, Kc, onto the software workspace. These controls are also the easiest to implement, as they require only the linearization of the state-space model, and the solution of a Riccati 45
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 and 44: disturbances, whereas the hardware
Page 45 and 46: Response in cm 4 3 2 1 0 −1 −2
Page 47 and 48: Response in cm Response in cm 4 3 2
Page 51 and 52: Response in cm 4 3 2 1 0 −1 −2
Page 55: Response in cm Response in cm 4 3 2
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 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
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