3-D Visualization and Optimization for SISO Linear Systems Using Parametrization of Two-Stage Compensator Design

Authors: Kazuyoshi Mori, Keisuke Hashimoto

Abstract:

In this paper, we consider the two-stage compensator designs of SISO plants. As an investigation of the characteristics of the two-stage compensator designs, which is not well investigated yet, of SISO plants, we implement three dimensional visualization systems of output signals and optimization system for SISO plants by the parametrization of stabilizing controllers based on the two-stage compensator design. The system runs on Mathematica by using “Three Dimensional Surface Plots,” so that the visualization can be interactively manipulated by users. In this paper, we use the discrete-time LTI system model. Even so, our approach is the factorization approach, so that the result can be applied to many linear models.

Keywords: Linear systems, visualization, optimization, two-Stage compensator design, Mathematica.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1124907

References:

[1] C. A. Desoer, R. W. Liu, J. Murray, and R. Saeks, “Feedback system design: The fractional representation approach to analysis and synthesis,” IEEE Trans. Automat.Contr., vol. AC-25, pp. 399-412, 1980.
[2] M. Vidyasagar, H. Schneider, and B. A. Francis, “Algebraic and topological aspects of feedback stabilization,” IEEE Trans. Automat. Contr., vol. AC-27, pp. 880-894, 1982.
[3] K.Mori, “Parametrization of all strictly causal stabilizing controllers,” IEEE Trans. Automat. Contr., vol. AC-54, pp.2211-2215, 2009.
[4] M. Vidyasagar, Control System Synthesis: A Factorization Approach, Cambridge, MA:MIT Press, 1985.
[5] K. Mori, “Multi-stage compensator design -a factorization approach-,” in Proceedings of The 8th Asian Control Conference(ASCC 2011), 2011, 1012-1017.
[6] V. R .Sule, “Feedback stabilization over commutative rings: The matrix case,” SIAM J. Control and Optim., vol. 32, no. 6, pp. 1675-1695, 1944.
[7] K. Mori and K. Abe, “Feedback stabilization over commutative rings: Further study of coordinate-free approach, ” SIAM J. Control and Optim., vol. 39, no. 6, pp. 1952-1973, 2001.
[8] S. Shankar and V. R.Sule, “Algebraic geometric aspects of feedback stabilization,” SIAM J. Control and Optim., vol. 30, no. 1, pp. 11-30, 1992.
[9] P. Wellin, Programming with Mathematica:An Introduction, Cambridge University Press, 2013.
[10] ThreeDimensionalSurfacePlots: URL: http://reference.wolfram.com/ mathematica/tutorial/ThreeDimensionalSurfacePlots.en.html
[11] S. Wolfram, “The Mathematica Book Forth Edition,”Wolfram Research, Inc.
[12] F. A. Aliev and V. B. Schneider, “Comments on Optimizing Simultaneously Over the Numerator and Denominator Polynomials in the Youla-Kucera Parameterization” IEEE Trans. Automat. Contr., vol. 52, no. 4 pp. 763, 2007.
[13] D. C. Youla, H. A. Jabr. and J. J. Bongiorno, “Modern Wiener-Hopf design of optimal controllers Part II: The multivariable case,” IEEE Trans. Automat. Contr., vol. AC-21, no. 3 pp. 319-338, Jun. 1976.
[14] V. Kuˇcela, “Stability of discrete linear control systems,” Preprintsc 6th IFAC World Congr., Boston, MA, 1975.
[15] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes Third Edition: Golden Section Search in One Dimension,” Cambridge university, pp. 492-495, 2007.
[16] K. Hashimoto and K. Mori, “Visualization Input-Output Relation with Parametrization of Two Stage Compensator Design”, The Society of Instrument and Control Engineers Tohoku Chapter., Japan, Fukushima, no. 284-7, Nov. 2013.
[17] K. Hashimoto and K. Mori, “Visualization of The Input-Output Relation of Multi-Input Multi-Output System Using Parametrization of Two-Stage Compensator Design”, Tohoku-Section Joint Convention of Institutes of Electrical and information Engineers, Japan, Yamagata, no. 2F01, Aug. 2013.
[18] K. Hashimoto and K. Mori, “Visualization Of The Input-output Relation Of Single-input Single-output System And Multi-input Multi-output System Using Parametrization Of Two-stage Compensator Design”, 34th International Association of Science and Technology for Development Conference on Modelling, Identification and Control MIC 2015, Austria, Innsbruck, no. 826-032, Feb. 2015.