Abstract
A comparison between an \(\mathcal{H}_\infty\) and \(\mathcal{H}_2\) multivariable controller designed for and implemented on the MIT SERC Interferometer testbed is presented. The testbed is a modally rich, lightly damped structure that exhibits non-minimum phase behavior and has a disturbance rejection performance metric. Both controllers were 38 state two-input two-output digital compensators implemented at a sampling frequency of 2 kHz, and both were designed using a frequency weighted loop shaping approach. The details of how the various design variables and their values were chosen for both methods are compared. Comparing the methods and the resulting control designs illustrates that the \(\mathcal{H}_\infty\) and \(\mathcal{H}_2\) controllers are quite similar. The similarity is in part due to the fact that both controllers nearly cancel all the minimum phase transmission zeros of the plant. We also analyze performance robustness properties using structured singular value (μ) methods together with experimental frequency response data.
This work was carried out at the MIT Space Engineering Research Center with support provided by NASA grant NAGW-1335.
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References
Hyland, D. C., Junkins, J. L., and Longman, R., “Active Control Technology for Large Space Structures,” Journal of Guidance, Control, and Dynamics, Vol. 16, Sept-Oct 1993, pp. 801–821.
Sparks, D. and Juang, J.-N., “Survey of Experiments and Experimental Facilities for Control of Flexible Structures,” Journal of Guidance, Control, and Dynamics, Vol. 15, Jul.-Aug. 1992, pp. 801–816.
Anderson, B. D. O. and Mingori, D. L., “Use of Frequency Dependence in Linear Quadratic Control Problems to Frequency-Shape Robustness,” Journal of Guidance, Control, and Dynamics, Vol. 8, No. 3, 1985, pp. 397–401.
Safonov, M. and Chiang, R., “CACSD Using the State Space L ∞ Theory — A design Example,” IEEE Transactions on Automatic Control, Vol. 33, May 1988, pp. 477–479.
Stein, G. and Athans, M., “The LQG/LTR Procedure for Multivariable Feedback Control Design,” IEEE Transactions on Automatic Control, Vol. 32, Feb. 1987, pp. 105–114.
McFarlane, D. and Glover, K., “A Loop Shaping Design Procedure Using \(\mathcal{H}_\infty\) Synthesis,” IEEE Transactions on Automatic Control, Vol. 37, June 1992, pp. 759–769.
Sparks, A., Banda, S., and Yeh, H.-H., “A Comparison of Loop Shaping Techniques,” in Proceedings of the 30th Conference on Decision and Control, (Brighton, England), Dec. 1991.
Lim, K., Maghami, P., and Joshi, S., “A Comparison of Controller Designs for an Experimental Flexible Structure,” in Proceedings of the American Control Conference, (Boston, MA), June 1991.
Zhou, K., “Comparison Between \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) Controllers,” IEEE Transactions on Automatic Control, Vol. 37, Aug. 1992, pp. 1261–1265.
Blackwood, G. H., Jacques, R. N., and Miller, D. W., “The MIT multipoint alignment testbed: Technology development for optical interferometry,” in Proceedings of the SPIE Conference on Active and Adaptive Optical Systems, (San Diego, CA), July 1991.
Anderson, E., Blackwood, G., and How, J., “Passive Damping in the MIT SERC Controlled Structures Testbed,” in Proceedings of the International Symposium on Active Materials and Adaptive Structures, Nov. 1991.
Anderson, E., Moore, D., and Fanson, J., “Development of an Active Member Using Piezoelectric and Electrostrictive Actuators for Control of Precision Structures,” in Proceedings of the 31st AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, 1990.
Balmès, E. and Crawley, E., “Designing and Modeling Structural Dynamics for Control. Applications to the MIT/SERC Interferometer Testbed,” in 1st SMAC Conference, (Nice, France), Dec. 1992.
Jacques, R., On-Line System Identification and Control Design for Flexible Structures, PhD thesis, Massachusetts Institute of Technology, 1994. SERC report #7-94.
Liu, K., Jacques, R., and Miller, D., “Frequency Domain Structural System Identification by Observability Range Space Extraction,” in Proceedings of the American Control Conference, (Baltimore, MDS), June 1994.
Jacques, R. and Miller, D., “Multivariable Model Identification from Frequency Response Data,” in Proceedings of the 32nd Conference on Decision and Control, 1993.
Lehtomaki, N., Castanon, D., Levy, B., Stein, G., Sandell, Jr., N., and Athans, M., “Robustness and Modeling Error Characterization,” IEEE Transactions on Automatic Control, Vol. AC-29, Mar. 1984, pp. 212–220.
Fleming, F., “The Effect of Structure, Actuator, and Sensor on the Zeros of Controlled Structures,” Master's thesis, Massachusetts Institute of Technology, 1991. MIT SERC report # 18–90.
Lublin, L. and Athans, M., “Application of \(\mathcal{H}_2\) Control to the MIT SERC Interferometer Testbed,” in Proceedings of the American Control Conference, (San Francisco, CA), June 1993.
Doyle, J., Glover, K., Khargonekar, P., and Francis, B., “State Space Solutions to Standard \(\mathcal{H}_2\)/\(\mathcal{H}_\infty\) Control Problems,” IEEE Transactions on Automatic Control, Vol. 34, Aug. 1989, pp. 831–847.
Doyle, J., Francis, B., and Tannenbaum, A., Feedback Control Theory, New York: Macmillan Publishing Company, 1992.
Packard, A. and Doyle, J., “The Complex Structured Singular Value,” Automatica, Vol. 29, No. 1, 1993, pp. 71–109.
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© 1995 Springer-Verlag London Limited
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Lublin, L., Athans, M. (1995). An experimental comparison of \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) designs for an interferometer testbed. In: Francis, B.A., Tannenbaum, A.R. (eds) Feedback Control, Nonlinear Systems, and Complexity. Lecture Notes in Control and Information Sciences, vol 202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0027676
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DOI: https://doi.org/10.1007/BFb0027676
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