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Nonlinear Multi-Mode Robust Control for Small Telescopes

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Nonlinear Multi-Mode Robust Control for Small Telescopes NONLINEAR MULTI-MODE ROBUST CONTORL FOR SMALL TELESCOPES By WILLIAM LOUNSBURY Submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical Engineering and Computer Science CASE WESTERN RESERVE UNIVERSITY January, 2015 1 Case Western Reserve University School of Graduate Studies We hereby approve the thesis of William Lounsbury, candidate for the degree of Master of Science*. Committee Chair Mario Garcia-Sanz Committee Members Marc Buchner Francis Merat Date of Defense: November 12, 2014 * We also certify that written approval has been obtained for any proprietary material contained therein 2 Contents List of Figures ……………….. 3 List of Tables ……………….. 5 Nomenclature ………………..5 Acknowledgments ……………….. 6 Abstract ……………….. 7 1. Background and Literature Review ……………….. 8 2. Telescope Model and Performance Goals ……………….. 9 2.1 The Telescope Model ……………….. 9 2.2 Tracking Precision ……………….. 14 2.3 Motion Specifications ……………….. 15 2.4 Switching Stability and Bumpiness ……………….. 15 2.5 Saturation ……………….. 16 2.6 Backlash ……………….. 16 2.7 Static Friction ……………….. 17 3. Linear Robust Control ……………….. 18 3.1 Unknown Coefficients……………….. 18 3.2 QFT Controller Design ……………….. 18 4. Regional Control ……………….. 23 4.1 Bumpless Switching ……………….. 23 4.2 Two Mode Simulations ……………….. 25 5. Limit Cycle Control ……………….. 38 5.1 Limit Cycles ……………….. 38 5.2 Three Mode Control ……………….. 38 5.3 Three Mode Simulations ……………….. 51 6. Future Work ……………….. 55 6.1 Collaboration with the Pontifical Catholic University of Chile ……………….. 55 6.2 Senior Project ……………….. 56 Bibliography ……………….. 58 3 List of Figures Figure 1: TRAPPIST Photo ……………….. 9 Figure 2: Control Design – Azimuth Aggressive ……………….. 20 Figure 3: Control Design – Azimuth Moderate ……………….. 21 Figure 4: Control Design – Azimuth Aggressive ……………….. 21 Figure 5: Control Design – Altitude Moderate ……………….. 22 Figure 6: System Diagram ……………….. 25 Figure 7: Two-mode altitude controller comparison ……………….. 26 Figure 8: Two-mode azimuth controller comparison ……………….. 27 Figure 9: Two-mode altitude controller torque ……………….. 28 Figure 10: Two-mode azimuth controller torque ……………….. 29 Figure 11: Two-mode altitude bumpless vs. bumpy ……………….. 30 Figure 12: Two-mode azimuth bumpless vs. bumpy ……………….. 31 Figure 13: Two-mode altitude bumpless torque ……………….. 31 Figure 14: Two-mode azimuth bumpless torque ……………….. 32 Figure 15: Two-mode reference tracking robustness ……………….. 33 Figure 16: Two-mode torque robustness ……………….. 33 Figure 17: Two-mode reference tracking disturbance rejection ……………….. 35 4 Figure 18: Two-mode reference tracking robustness ……………….. 35 Figure 19: Two-mode difficult unsticking ……………….. 37 Figure 20: Two-mode easy sticking ……………….. 37 Figure 21: DFE algorithm ……………….. 40 Figure 22: DFE filter design ……………….. 42 Figure 23: DFE example output ……………….. 44 Figure 24: Arbitrary nonlinearity input/output diagram ……………….. 45 Figure 25: Arbitrary nonlinearity describing function ……………….. 46 Figure 26: DFE limit cycle prediction ……………….. 47 Figure 27: Limit cycle simulation ……………….. 48 Figure 28: DF limit cycle control design ……………….. 49 Figure 29: Bang-bang control example ……………….. 50 Figure 30: Three-mode step ……………….. 51 Figure 31: Three-mode step and ramp ……………….. 52 Figure 32: Three-mode disturbance rjection ……………….. 53 Figure 33: Three-mode robustness ……………….. 55 5 List of Tables Table 1: Telescope properties affecting the theoretical model ……………….. 13 Table 2: Telescope properties affecting desired resolution ……………….. 14 Table 3: Motion Specifications ……………….. 15 Table 4: Table of tables that do not list themselves ……………….. 59 Nomenclature DF – Describing Function DFE – Describing Function Estimator QFT – Quantitative Feedback Theory Alt-Azimuth mount – A telescope mount that allows for rotation normal to the Earth’s surface (azimuth) and a perpendicular axis (altitude) Arc Second – A unit of angular measurement equal to 1/3600th of 1 degree 6 Acknowledgments I would like to extend my thanks to Dr. Mario Garcia-Sanz for his help, encouragement, and guidance throughout my telescope research. Thanks are due as well to Dr. Cenk Cavusoglu for his help through my undergraduate and early graduate career at CWRU. Finally, I would like to thank my family for their ongoing care and support. 7 Nonlinear Multi-Mode Robust Control For Small Telescopes Abstract By William Lounsbury This paper introduces an innovative robust and nonlinear control design methodology for high-performance motor control in optical telescopes less than one meter in diameter. The dynamics of optical telescopes typically vary according to azimuth and altitude angles, temperature, friction, speed, and acceleration leading to nonlinearities and plant parameter uncertainty. The methodology proposed in this paper combines robust Quantitative Feedback Theory (QFT) techniques, the describing function method, and optimal control with nonlinear switching strategies that achieve simultaneously the best characteristics of a set of very active (fast) robust QFT controllers, very stable (slow) robust QFT controllers, and a pair of controllers designed around system limit cycles for high precision. A general dynamic model and a variety of specifications from several different commercially available amateur Newtonian telescopes are used for the controller design as well as the simulation and validation. It is also proven that the nonlinear/switching controller is stable for any switching strategy and switching velocity, according to described frequency conditions based on common quadratic Lyapunov functions and the circle criterion. 8 1. Literature Review Modern astronomy is mostly performed by very large telescope installations and satellites, with the realm of small telescopes confined to amateur astronomy. For this reason, most of the research performed on telescope control systems refers to the construction and maintenance of large telescope control systems. However, with a reasonably large industry for amateur astronomy and many universities installing custom small-scale telescopes for student use, the high-precision control techniques used for these systems is well worth investigating. The industry surrounding amateur astronomy consists of a few telescope manufacturers as well as a large number of custom systems designed by hobbyist engineers. These engineers do not publish much on the way that they do things, but the current state of the art within the companies’ telescopes can be found within the patents filed on their control systems. Stepper motor control is a very popular solution due to the high position precision that these motors can achieve. For instance, a telescope control system patented by Krewalk and Silverburg describes the usage of this technique which does not involve any particularly advanced control algorithms, opting instead for a much more complicated mechanical system to deal with the nonlinear control issues [1]. They claim to solve the problem of backlash by using a mechanism to continuously apply tension to each of the gears within the system to maintain contact. While this solution can be effective, the hardware used is very sensitive to damage and will fatigue with usage. A software solution that can eliminate backlash is desirable. The stepper motor solution also does not 9 take into account the friction and inertias of the system, and thus it cannot be optimized to a particular range of telescopes, and is not a robust solution. Even with stepper motor systems, modern control telescopes still suffer from large motion blur. The 60 cm TRAPPIST telescope, located in Chile, is such a modern telescope, as seen by a photo that it took in figure 1 [2]. Figure 1 The stars in the above image are clearly blurred out of their normal circular shape by either the motion of the Earth that the telescope was unable to compensate for, or incidental movements introduced to the telescope by its control system. In order to eliminate all of these issues, another solution must be found. 2. Telescope Model and Performance Goals 2.1 The Telescope model The telescope modeled in this paper is based on a simple 24cm reflecting telescope on an altitude-azimuth mount. This consists of a hollow cylinder for the tube and a disc for the mirror at the end. In the linear case, the frictional force is assumed to 10 always be proportional to the velocity with no sticking effects. Also, note that both axes of rotation intersect at the center of mass. This gives the telescope the property of constant potential energy. Therefore, potential energy is not taken into account in the model derivation. Realistically, a system can never be perfectly modeled. The linear model of the telescope relies heavily on two different factors, the inertia and the friction coefficient, neither of which is easy to measure while the telescope is being operated and both of which change with the usage of the telescope. Specifically, friction will change with temperature, humidity, and the usage of the telescope. The moment of inertia depends on the mass and positioning of various components in the telescope, and the moment of inertia about the azimuthal axis will change with altitudinal position. A vertically oriented telescope is easier to rotate than a horizontally oriented one. A key part of understanding this problem is being able to quantify not only a typical value for inertia or friction, but to calculate a range of values that these coefficients could take on. Solutions to this “range of coefficients”
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