Abstract
A judicious choice of the state-space realization is required in order to account for the assumed smoothness of the state-space matrices with respect to the design parameters. The direct parameterization of poles and residues may be not appropriate, due to their possible non-smooth behavior with respect to design parameters. This is avoided in the proposed technique, by converting the pole-residue description to a Sylvester description which is computed for each root macromodel. This technique is used in combination with suitable parameterizing schemes for interpolating a set of state-space matrices, and hence the poles and residues indirectly, in order to build accurate parametric macromodels. The key features of the present approach are first the choice of a proper pivot matrix and second, finding a well-conditioned solution of a Sylvester equation. Stability and passivity are guaranteed by construction over the design space of interest. Pertinent numerical examples validate the proposed Sylvester technique for parametric macromodeling.
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Notes
- 1.
The exact generic realization \({\mathcal {S}} ({\mathbf {g}}) \) is analytically unknown in the sense that for each new value of \({\mathbf {g}}\) an oracle (or black-box function) has to be consulted.
- 2.
Note that multilinear interpolation satisfies both positivity and partition of unity.
References
Ferranti, F., Knockaert, L,. Dhaene, T.: Guaranteed passive parameterized admittance-based macromodeling, IEEE Trans. Adv. Pack., 33(3), 623–629 (2010)
Ferranti, F., Knockaert, L., Dhaene, T.: Parameterized S-parameter based macromodeling with guaranteed passivity. IEEE Microw. Wireless Compon Lett, 19(10), 608610 (2009)
Ferranti, F., Knockaert, L., Dhaene, T., Antonini, G.: Passivity preserving parametric macromodeling for highly dynamic tabulated data based on Lure equations. IEEE Trans. Micro. Theor. Tech. 58(12), 3688–3696 (2010)
Triverio, P., Nakhla, M., Grivet-Talocia, S.: Passive parametric modeling of interconnects and packaging components from sampled impedance, admittance or scattering data. Electron. Syst. Integr. Technol. Conf. pp. 16, Sept. (2010)
De Caigny, J., Camino, J.F., Swevers, J.: Interpolating model identication for siso linear parameter-varying systems. Mech. Syst. Signal Proc. 23(8), 23952417 (2009)
Samuel, E.R., Knockaert, L., Ferranti, F., Dhaene, T.: Guaranteed passive parameterized macromodeling by using sylvester state-space realizations. IEEE Trans. Microwave Theor. Tech. 61(4), 1444–1454 (April 2013)
Gilbert, E.G.: Controllability and observability in multi-variable control systems. SIAM J. Control 1(2), 128151 (1963)
Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 1731 (Feb. 1981)
Lovera, M., Mercere, G.: Identification for gain-scheduling: a balanced subspace approach. Am. Control Conf. 2007, pp. 858–863, (July 2007)
Anderson, B.D.O., Vongpanitlerd, S.: Network Analysis and Synthesis. NJ, Prentice-Hall, Englewood Cliffs (1973)
Ferranti, F., Knockaert, L., Dhaene, T., Antonini, G.: Parametric macromodeling for S-parameter data based on internal nonexpansivity, Int. J. Num. Model. Electron. Netw. Devices Fields 26(1), 1527 (2013)
Ferranti, F., Knockaert, L., Dhaene, T.: Passivity-preserving parametric macromodeling by means of scaled and shifted state-space systems, IEEE Trans. Microwave Theor. Tech. 59(10), 2394–2403, (Oct 2011)
De Souza, E., Bhattacharyya, S.P.: Controllability, observability and the solution of AX-XB = C. Lin. Alg. Appl. 39, 167–188 (1981)
Varga, A.: Robust pole assignment via sylvester equation based state feedback parametrization. Proceedings IEEE International Symposium on Computer-Aided Control System Design, pp. 13–18, 2000
Gustavsen, B., Semlyen, A.: Rational approximation of frequency domain responses by vector fitting. IEEE Trans. Power Delivery 14(3), 1052–1061 (July 1999)
Carvalho, J., Datta, K., Hong, Y.: A new block algorithm for full-rank solution of the Sylvester-observer equation. IEEE Trans. Autom. Control 48(12), 2223–2228 (Dec. 2003)
Boyd, S., Vandenberghe, L.: Convex Optimization, Cambridge University Press, Cambridge, U.K., 2004. Available at http://www.math.nus.edu.sg/ mattohkc/sdpt3.html
Löfberg, J.: " YALMIP: a toolbox for modeling and optimization in MATLAB. Proceedings of CACSD Conference, Taipei, Taiwan, 2004. Available: http://control.ee.ethz.ch/joloef/yalmip.php
Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming (web page and software), July 2008. Available: http://www.stanford.edu/boyd/cvx/
Mattingley, J.E., Boyd, S.: "CVXMOD: convex optimization software in Python (web page and software), Aug. 2008. Available: http://cvxmod.net/
Benavides, N.L., Carr, R.D., Hart, W.E.: Python optimization modeling objects (Pyomo). In: Proceedings of INFORMS Computing Society Conference, 2009. Available: https://software.sandia.gov/trac/pyutilib/export/30/trunk/doc/ pyomo.pdf
Grant, M., Boyd, S., Ye, Y.: Disciplined convex programming. In: Global optimization: from theory to implementation (Nonconvex Optimization and Its Applications), L. Liberti and N. Maculan, Eds. New York: Springer Science and Business Media, pp. 155–210, 2006
Mattingley, J., Boyd, S.: Real-time convex optimization in signal processing. IEEE Signal Proc. Mag. 27(3), 50–61 (2010)
Curtain, R.F.: Old and new perspectives on the positive-real lemma in systems and control theory. Z. Angew. Math. Mech. 79(9), 579–590 (1999)
Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear matrix inequalities in system and control theory. SIAM Studies in Applied Mathematics, 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994
Weiss, H., Wang, Q., Speyer, J.L.: System characterization of positive real conditions. IEEE. Trans. Autom. Contr. 39(3), 540–544 (1994)
Knockaert, L., Dhaene, T., Ferranti, F., De Zutter, D.: Model order reduction with preservation of passivity, non-expansivity and Markov moments. Syst Control Lett. 60(1), 53–61 (Jan. 2011)
Acknowledgments
This research has been funded by the Research Foundation Flanders (FWO) and the Interuniversity Attraction Poles Programme BESTCOM initiated by the Belgian Science Policy Office.
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Samuel, E.R., Knockaert, L., Dhaene, T. (2015). Passive Parametric Macromodeling by Using Sylvester State-Space Realizations. In: Ferrier, JL., Gusikhin, O., Madani, K., Sasiadek, J. (eds) Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-319-10891-9_18
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