NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

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to the Galerkin approximation technique, where again an optimal set of coe cients is sought that will regulate the system optimally in some region of the state space, . The di erence is that the linearized controls are only guaranteed to locally stabi- lize the system, whereas the Galerkin approximations take into account more global information. Also, the SGA technique creates a truly nonlinear control signal, and this translates into better and more robust performance when implemented on the exible beam set-up. On the other hand, the backstepping approach uses a Lyapunov approach { building a control Lyapunov function step by step, then implementing a very complex and non-linear control that is a composite of the control signals at the various steps. The passivity-based strategy seeks to exploit the structure and ow of the energy in the system by adding a dynamic in the feedback loop.This control dynamic is chosen to add damping arti cially to the closed-loop system and to shape the potential energy function for closed-loop stability andperformance. 6.2 Extensions to this Research Some interesting things have come to light throughout the testing of these controls: there does not seem to be any advantage of adding higher order terms to the Galerkin approximations. As higher power terms are added to the basis functions there was a slight degradation in performance. This would be an excellent extension to this research: to discover why adding higher power terms to the approximation does not improve the control. Also, it would be valuable to experiment with non- polynomial basis functions, to see if other functions might provide better control signals perhaps exponential, logarithmic, Bessel, or other functions would generate useful nonlinearities in the feedback signal. The selection of an appropriate and appropriate basis functions is also an area where more research should be conducted. It was observed that changing the region of stability, , could signi cantly improve or degrade the control. Future research might explore ways of mathematically determining the optimal size of for a given demand of robustness and performance within a given region of the state 62
space. As the number of states in the system increases, the number of basis functions required to implement a given order of approximation grows exponentially. This means that for systems with a large number of states, a technique must be found for selecting only the higher order basis functions that provide useful information and that will translate into e ective elements of the control signal. Future research could be done to determine how to automate such a procedure, and this might alsoleadto an understanding of why some higher order basis functions degrade performance. 63
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NONLINEAR CONTROLLER COMPARISON ON
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BRIGHAM YOUNG UNIVERSITY GRADUATE C
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ABSTRACT NONLINEAR CONTROLLER COMPA
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Contents Acknowledgments vi List of
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List of Tables 4.1 Tabular Comparis
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4.19 SGA: Nonlinear H1 vs. Linear O
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Figure 1.1: TORA System One of the
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1.3 Literature Review In surveying
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Chapter 2 Plant Speci cations and M
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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 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 72 and 73: control was that it requires only t
Page 77 and 78: Bibliography [1] P. Kokotovic M. Ja
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NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

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