NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

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4.19 SGA: Nonlinear H1 vs. Linear Optimal Control . . . . . . . . . . . . 42 4.20 SGA: Nonlinear H1 vs. Linear Robust Control . . . . . . . . . . . . 43 4.21 SGA: Nonlinear H1 vs. Passivity Based Control . . . . . . . . . . . . 43 4.22 SGA: Nonlinear H1 vs. Backstepping Control . . . . . . . . . . . . . 44 4.23 SGA: Nonlinear H1 vs. SGA: Nonlinear H 2 . . . . . . . . . . . . . . 44 5.1 Open Loop Response of the FBS . . . . . . . . . . . . . . . . . . . . 50 5.2 Linear Optimal vs. Open Loop . . . . . . . . . . . . . . . . . . . . . 51 5.3 Linear Robust vs. Open Loop . . . . . . . . . . . . . . . . . . . . . . 51 5.4 Linear Robust vs. Linear Optimal . . . . . . . . . . . . . . . . . . . . 52 5.5 Passivity Based vs. Open Loop . . . . . . . . . . . . . . . . . . . . . 53 5.6 Passivity Based vs. Linear Optimal . . . . . . . . . . . . . . . . . . . 53 5.7 Passivity Based vs. Linear Robust . . . . . . . . . . . . . . . . . . . . 54 5.8 SGA: Nonlinear H 2 vs. Open Loop . . . . . . . . . . . . . . . . . . . 54 5.9 SGA: Nonlinear H 2 vs. Linear Optimal . . . . . . . . . . . . . . . . . 55 5.10 SGA: Nonlinear H 2 vs. Linear Robust . . . . . . . . . . . . . . . . . 55 5.11 SGA: Nonlinear H 2 vs. Passivity Based . . . . . . . . . . . . . . . . . 56 5.12 SGA: Nonlinear H1 vs. Open Loop . . . . . . . . . . . . . . . . . . . 56 5.13 SGA: Nonlinear H1 vs. Linear Optimal . . . . . . . . . . . . . . . . 57 5.14 SGA: Nonlinear H1 vs. Linear Robust . . . . . . . . . . . . . . . . . 57 5.15 SGA: Nonlinear H1 vs. Passivity Based . . . . . . . . . . . . . . . . 58 5.16 SGA: Nonlinear H1 vs. SGA: Nonlinear H 2 . . . . . . . . . . . . . . 58 xii
Chapter 1 Introduction 1.1 Motivation and Problem Description The reliability and e ectiveness of feedback control systems has been recog- nized for centuries: they provide a simple and robust way of regulating a large class of physical systems, and there exists a rich body of mathematical results that enable the systematic design of feedback control laws. A signi cant limitation of most of the classic results is that they apply to a rather narrow class of systems { those which can be modeled by systems of linear di erential equations. Despite the fact that many physical plants can be linearized, and thus an adequate small-signal control can be realized for the linearized system, the fact remains that almost all physical systems in- volve dynamics that cannot be modeled completely by a linear system. Such nonlinear systems have been the subject of much research in the past two decades researchers have sought to extend the results obtained for linear plants to non-linear plants. A good example of this is the search for the nonlinear optimal feedback controller. It is well known that the solution of the optimal linear control problem depends upon the solution of an algebraic Riccati equation. For the non-linear case, this result generalizes nicely: the optimal non-linear controller requires the solution of the Hamilton-Jacobi-Bellman (HJB) equation (which was actually discovered before the Riccati equations). This partial di erential equation is very di cult to solve, however, and the quest for a reliable and accurate approximation of its solution is an open problem. The Successive Galerkin Approximation (SGA) is one such method of approximation, and the details of its implementation will be described in Chapter 3. 1
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 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 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:
Page 70 and 71:
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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