NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

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Figure 1.1: TORA System One of the challenges of evaluating the non-linear control design techniques that are being developed is to nd a reasonable method of gauging their performance. There exists a large variation in the complexity, behavior, and dynamics within the set of non-linear systems, and a given control design might work very well on one speci c type of system while providing poor performance on another. Thus the need for a benchmark non-linear problem { a mathematical system that contains signi cant non-linearities, has a tractable mathematical model, and is fairly easy to implement in hardware. In recent years, the translational oscillator rotational actuator (TORA) [1] system (also referred to as the Non-linear Benchmark Problem or NLBP) has been proposed as just such a benchmark system. In this system a mass is restricted to a linear oscillatory motion by ideal springs, and the system is actuated by arotational proof mass. The coupling between the rotational and linear motion provides the non- linearity. Many di erent researchers have used this system as a testbed to evaluate various non-linear control strategies, so this system was selected as a way to evaluate and compare the performance of the successive Galerkin technique with other robust, non-linear control laws as well as with standard optimal linearized methods. Speci cally, six unique control laws are derived and implemented on a hardware implementation of the TORA system. A non-linear Galerkin approximation to the H 2 2
problem and a Galerkin approximation to the non-linear H1 problem are studied and compared with the results of a linearized optimal control, a linearized H1 control, a passivity-based control, and a control law based on integrator backstepping. These control strategies are chosen because they are current topics of research, and most are proposed as control laws that will be provide robust performance. The speci c hardware system, upon which the tests were performed, is de- scribed in Chapter 2 along with a derivation of the equations of motion. Chapter 3 presents an overview of the control strategies tested, and it explains the details of the implementations of each algorithm. Chapter 3 also explains the successive Galerkin approximation technique and shows how it is implemented on the hardware system. Chapter 4 explains the mechanics of the testbed and the methods of comparing performance, and a comparison of the six controls in simulation is presented. The actual results of the tests as performed in hardware are compared and discussed in Chap- ter 5, and an analysis of the robustness of the control laws is presented in Chapter 6. Finally Chapter 7 sums up the results of the experiment the conclusions of the research are presented along with recommendations for future research. 1.2 Contributions There is often a wide gap between mathematical systems and the physical plants these systems attempt to model. The TORA system is proposed in [2] as a benchmark problem so that nonlinear control algorithms can be compared on a common system { a system that can be easily implemented in hardware. While many control designs perform well in the idealized environment of mathematics, the true test of a nonlinear algorithm must lie in its regulation of a physical plant. The SGA technique is applied to the nonlinear benchmark problem, and it performs better than other well known nonlinear techniques when applied to a physical implementation of this TORA system. This result is the main contribution of this thesis: it o ers experimental evidence that the SGA method produces excellent results when applied to real-life problems and systems. This thesis also justi es further research into understanding, clarifying, and improving the SGA technique. 3
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 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.
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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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