Criterion of Approximation for Designing 2 × 2 Feedback Systems with Inputs Satisfying Bounding Conditions
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Published:
Nov 11, 2014
Keywords:
Feedback systems Rational approximants Two-input two-output systems Non-rational systems Peak outputs Method of inequalities
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Abstract
A common practice in designing a feedback system with a non-rational transfer matrix is to replace the non-rational matrix with an appropriate rational approximant during the design process so that reliable and effcient computational tools for rational systems can be utilized. Consequently, a criterion of approximation is required to ensure that the controller obtained from the approximant still provides satisfactory results for the original system. This paper derives such a criterion for the case of two-input two-output feedback systems in which the design objective is to ensure that the errors and the controller outputs always stay within prescribed bounds whenever the inputs satisfy certain bounding conditions. For a given rational approximant matrix, the criterion is expressed as a set of inequalities that can be solved in practice. It will be seen that the criterion generalizes a known result for single-input single-output systems. Finally, a controller for a binary distillation column is designed by using the criterion in conjunction with the method of inequalities.
Article Details
How to Cite
Chuman, T., & Arunsawatwong, S. (2014). Criterion of Approximation for Designing 2 × 2 Feedback Systems with Inputs Satisfying Bounding Conditions. ECTI Transactions on Electrical Engineering, Electronics, and Communications, 13(1), 1–10. https://doi.org/10.37936/ecti-eec.2015131.170968
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Circuits and Systems
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References
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[29] R. K. Wood and M. W. Berry, "Terminal composition control of binary distillation column," Chemical Engineering Sci., vol. 28, no. 9, pp. 1707-1717, 1973.
[30] G. A. Baker Jr. and P. Graves-Morris, Padé Approximants, Cambridge University Press, New York, USA, 1996.
[31] R. Curtain and K. Morris, "Transfer functions of distributed parameter systems: A tutorial," Automatica, vol. 45, no. 5, pp. 1101-1116, 2009.
[2] V. Zakian, "Simplication of linear timeinvariant systems by moment approximants," Int. J. Control, vol. 18, no. 3, pp. 455-460, 1973.
[3] J. Lam, "Model reduction of delay systems using Padé approximants," Int. J. Control, vol. 57, no. 2, pp. 377-391, 1993.
[4] G. Gu, P. P. Khargonekar and E. B. Lee, "Approximation of infinite-dimensional systems," IEEE Trans. Automat. Control, vol 34, no. AC-6, pp. 610-618, 1989.
[5] B. J. Birch and R. Jackson, "The behaviour of linear systems with inputs satisfying certain bounding conditions," J. Electron. Control, vol. 6, no. 4, pp. 366-375, 1959.
[6] P. G. Lane, "The principle of matching: a necessary and suffcient condition for inputs restricted in magnitude and rate of change," Int. J. Control, vol. 62, no. 4, pp. 893-915, 1995.
[7] W. Silpsrikul and S. Arunsawatwong, "Computation of peak output for inputs restricted in L2 and L1 norms using nite dierence schemes and convex optimization," Int. J. Automation Computing, vol. 6, no. 1, pp. 7-13, 2009.
[8] W. Silpsrikul and S. Arunsawatwong, "Computation of peak output for inputs satisfying many bounding conditions on magnitude and slope," Int. J. Control, vol. 83, no. 1, pp. 49-65, 2010.
[9] V. Zakian, "New formulation for the method of inequalities," Proc. Institution Elect. Engineers, vol. 126, no. 6, pp. 579-584, 1979.
[10] V. Zakian, "A criterion of approximation for the method of inequalities," Int. J. Control, vol. 37, no. 5, pp. 1103-1111, 1983.
[11] V. Zakian, A framework for design: theory of majorants, Control Systems Centre Report No. 604, University of Manchester Institute of Science and Technology, Manchester, UK, 1984.
[12] V. Zakian, "A performance criterion," Int. J. Control, vol. 43, no. 3, pp. 921-931, 1986.
[13] V. Zakian, "Perspectives of the principle of matching and the method of inequalities," Int. J. Control, vol. 65, no. 1, pp. 147-175, 1996.
[14] V. Zakian, Control Systems Design: A New Framework, Springer-Verlag, London, 2005.
[15] S. Arunsawatwong, "Stability of retarded delay differential systems," Int. J. Control, vol. 65, no. 2, pp. 347-364, 1996.
[16] S. Arunsawatwong, "Stability of Zakian IMN recursions for linear delay dierential equations," BIT Numerical Math., vol. 38, no. 2, pp. 219-233, 1998.
[17] S. Arunsawatwong and V. Q. Nguyen, "Design of retarded fractional delay dierential systems using the method of inequalities," Int. J. Automation Computing, vol. 6, no. 1, pp. 22-28, 2009.
[18] W. Silpsrikul and S. Arunsawatwong, "Design of fractional PI controllers for a binary distillation column with disturbances restricted in magnitude and slope," In Proc. 18th IFAC World Congr., Milan, Italy, pp. 7702-7707, 2011.
[19] A. T. Bada, "Design of delayed control systems with Zakian's method," Int. J. Control, vol. 40, pp. 773-781, 1984.
[20] A. T. Bada, "Design of delayed control systems using Zakian's framework," Proc. Institution Elect. Engineers, vol. 132, no. 6, pp. 251-256, 1985.
[21] A. T. Bada, "Robust brake control for a heavyduty truck," Proc. Institution Elect. Engineers, vol. 134, no. 1, pp. 1-8, 1987.
[22] O. Taiwo, "The design of robust control systems for plants with recycle," Int. J. Control, vol. 43, pp. 671-678, 1986.
[23] V. S. Mai, S. Arunsawatwong, and E. H. Abed, "Design of uncertain nonlinear feedback systems with inputs and outputs satisfying bounding conditions," In Proc. 18th IFAC World Congr., Milan, Italy, pp. 10970-10975, 2011.
[24] H. H. Nguyen, Numerical Design of Feedback Systems with Backlash for Inputs Restricted in Magnitude and Slope, Master's thesis, Chulalongkorn University, Bangkok, Thailand, 2014.
[25] P. G. Lane, Design of Control Systems with Input and Output Satisfying Certain Bounding Conditions, PhD thesis, University of Manchester Institute of Science and Technology, Manchester, UK, 1992.
[26] S. Arunsawatwong, "Critical control of building under seismic disturbance," in Control Systems Design: A New Framework, V. Zakian, editor, Springer-Verlag, London, pp. 339-353, 2005.
[27] V. Zakian and U. Al-Naib, "Design of dynamical and control systems by the method of inequalities, " Proc. Institution Elect. Engineers, vol. 120, no. 11, pp. 1421-1427, 1973.
[28] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, New York, USA, 1975.
[29] R. K. Wood and M. W. Berry, "Terminal composition control of binary distillation column," Chemical Engineering Sci., vol. 28, no. 9, pp. 1707-1717, 1973.
[30] G. A. Baker Jr. and P. Graves-Morris, Padé Approximants, Cambridge University Press, New York, USA, 1996.
[31] R. Curtain and K. Morris, "Transfer functions of distributed parameter systems: A tutorial," Automatica, vol. 45, no. 5, pp. 1101-1116, 2009.