NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

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1.3 Literature Review In surveying the technical literature relevant to this thesis, there are two pri- mary topics: research on the topic of the successive Galerkin approximation method, and research devoted to the topic of the NLBP system. Additionally a short review of the literature pertaining to the various control strategies used to highlight the SGA method will be presented. The SGA method was rst published in [3] as a method for iteratively improv- ing a nonlinear feedback control. This publication included a proof of the algorithm's convergence and the stability of the resulting control law at each iteration. Addi- tionally, each iteration brings the control design closer to solving the HJB equation, and thus the method of successive approximation eventually yields an approximation of the nonlinear optimal solution with any desired accuracy, as the order of the Galerkin approximation increases and as the number of iterations goes to in nity. In [4] the algorithm was successfully applied to the inverted pendulum problem as well as several one-dimensional examples. In [5] the SGA technique was extended to the optimal robust nonlinear control problem. In the robust problem, a solution of the Hamilton-Jacobi-Isaacs equation is required with the minimum possible L 2 gain from disturbance to output. In [5] a proof is presented of the convergence and stability of the algorithm at each successive iteration, and su cient conditions for convergence and stability are presented for both the nonlinear optimal and robust algorithms. Additionally, [5] successfully applies the algorithm to a hydraulic actuation system, a missile autopilot design, and the control of an underwater vehicle system. The control of the TORA system was originally proposed as a model of a dual-spin spacecraft. The goal was to study how to circumvent the resonance capture e ect in [6], but in [7], [8] the system was studied to analyze the usefulness of a rotational actuator in damping out linear oscillations and vibrations. It was rst proposed as a benchmark problem for nonlinear control systems in [2]. A physical testbed is described consisting of a motor driven proof mass that was mounted on a linear air track. In [2], the three-fold objective of the Nonlinear Benchmark Problem is set forth: to stabilize the closed loop system, to exhibit good disturbance rejection 4
compared to the uncontrolled oscillator, and to require limited control e ort. A passivity-based control design is also described that uses the control to simulate a damped pendulum absorber. The International Journal of Robust and Nonlinear Control published a special issue devoted exclusively to the NLBP (volume 8, 1998) and the 1995 American Control Conference contained an invited session in which six papers were submitted on the topic of the Nonlinear Benchmark Problem. The literature devoted to the TORA system was used as the basis for the designs presented in this thesis. In particular, the passivity-based control strategy comes mainly from the implementation described in [9], while a more complete ex- planation of passivity-based control of Euler-Lagrange systems can be found in [10] or in [11]. The integrator backstepping controller was mainly derived from [1] where they supply the necessary variable substitutions and transformations to derive acas- cade controller. It should be noted that the cascade controller presented in [1] is almost identical to the one derived in this thesis, and this controller is a special case of the full controller one obtains using integrator backstepping as described in [12]. The linearized optimal H 2 controller was derived by the well known solution to the Riccati equation. A concise overview of the theory and application of optimal linear control can be found in [13]. The linearized H1 controller was also designed by the well known robust control technique as described in Chapter 6 of [14]. 5
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 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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