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Coprime factor model reduction for discrete-time uncertain systems. (English) Zbl 1300.93048
Summary: This paper presents a contractive coprime factor model reduction approach for discrete-time uncertain systems of LFT form with norm bounded structured uncertainty. A systematic approach is proposed for coprime factorization and contractive coprime factorization of the underlying uncertain systems. The proposed coprime factor approach overcomes the robust stability restriction on the underlying systems which is required in the balanced truncation approach. Our method is based on the use of linear matrix inequalities to construct the desired reduced dimension uncertain system model. Closed-loop robustness is discussed under additive coprime factor perturbations.

MSC:
93B11 System structure simplification
93C55 Discrete-time control/observation systems
93C41 Control/observation systems with incomplete information
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[1] J. Doyle, A. Packard, K. Zhou, Review of LFTs, LMIs, and \(\mu\), in: Proceedings of the 30th IEEE Conference on Decision and Control, Vol. 2, 1991, pp. 1227-1232.
[2] Moore, B., Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26, 17-32, (1981) · Zbl 0464.93022
[3] D. Enns, Model reduction with balanced realizations: an error bound and frequency weighted generalization, in: Proceedings of the 23rd IEEE Conference on Decision and Control, 1984, pp. 127-132.
[4] Glover, K., All optimal Hankel-norm approximations of linear multivariable systems and their \(L_\infty\) error bounds, Internat. J. Control, 39, 1115-1193, (1984) · Zbl 0543.93036
[5] Meyer, D., Fractional balanced reduction: model reduction via fractional representation, IEEE Trans. Automat. Control, 35, 12, 1341-1345, (1990) · Zbl 0723.93011
[6] Hinrichsen, D.; Pritchard, A. J., An improved error estimate for reduced-order models of discrete-time systems, IEEE Trans. Automat. Control, 35, 3, 317-320, (1990) · Zbl 0702.93020
[7] Zhou, K.; Doyle, J.; Glover, K., Robust and optimal control, (1996), Prentice-Hall Upper Saddle River, NJ
[8] W. Wang, J. Doyle, C. Beck, K. Glover, Model reduction of LFT systems, in: Proceedings of IEEE Conference on Decision and Control, Vol. 2, 1991, pp. 1233-1238.
[9] Beck, C.; Doyle, J.; Glover, K., Model reduction of multidimensional and uncertain systems, IEEE Trans. Automat. Control, 41, 10, 1466-1477, (1996) · Zbl 0862.93009
[10] Li, L.; Petersen, I. R., A Gramian-based approach to model reduction for uncertain systems, IEEE Trans. Automat. Control, 55, 2, 508-514, (2010) · Zbl 1368.93074
[11] C. Beck, P. Bendottii, Model reduction methods for unstable uncertain systems, in: Proceedings of the 36th IEEE Conference on Decision and Control, Vol. 4, 1997, pp. 3298-3303.
[12] Beck, C., Coprime factors reduction methods for linear parameter varying and uncertain systems, Systems Control Lett., 55, 3, 199-213, (2006) · Zbl 1129.93352
[13] L. Li, Coprime factor model reduction for continuous-time uncertain systems, in: Proceedings of the 46th IEEE Conference on Decision and Control, Cancun, Mexico, 2008, pp. 4227-4232.
[14] Apkarian, P.; Gahinet, P., A convex characterization of gain-scheduled \(H_\infty\) controllers, IEEE Trans. Automat. Control, 40, 5, 853-864, (1995) · Zbl 0826.93028
[15] Scorletti, G.; Ghaoui, L. E., Improved LMI conditions for gain scheduling and related control problems, Internat. J. Robust Nonlinear Control, 8, 10, 845-877, (1998) · Zbl 0935.93050
[16] Goddard, P. J., Performance-preserving controller approximation, (1995), Cambridge University, (Ph.D. thesis)
[17] G. Wood, P. Goddard, K. Glover, Approximation of linear parameter-varying systems, in: Proceedings of the 35th IEEE Conference on Decision and Control, Vol. 1, 1996, pp. 406-411.
[18] Wu, F., Induced \(\mathcal{L}_2\) norm model reduction of polytopic uncertain linear systems, Automatica, 32, 10, 1417-1426, (1996) · Zbl 0873.93018
[19] Haddad, W.; Kapila, V., Robust, reduced-order modeling for state-space systems via parameter-dependent bounding functions, IEEE Trans. Automat. Control, 42, 2, 248-253, (1997) · Zbl 0876.93019
[20] Lall, S.; Beck, C., Error-bounds for balanced model-reduction of linear time-varying systems, IEEE Trans. Automat. Control, 48, 6, 946-956, (2003) · Zbl 1364.93115
[21] Sandberg, H.; Rantzer, A., Balanced truncation of linear time-varying systems, IEEE Trans. Automat. Control, 49, 2, 217-229, (2004) · Zbl 1365.93062
[22] Chu, Y.-C.; Glover, K., Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities, IEEE Trans. Automat. Control, 44, 3, 471-483, (1999) · Zbl 0958.93059
[23] L. Li, Model reduction for linear parameter-dependent systems, in: Proceedings of the 17th IFAC World Congress, Seoul, Korea, 2008, pp. 4048-4053.
[24] Andersson, L.; Rantzer, A.; Beck, C., Model comparison and simplification, Internat. J. Robust Nonlinear Control, 9, 3, 157-181, (1999) · Zbl 0935.93020
[25] Andersson, L.; Rantzer, A., Frequency-dependent error bounds for uncertain linear models, IEEE Trans. Automat. Control, 44, 11, 2094-2098, (1999) · Zbl 1136.93366
[26] L. Li, I.R. Petersen, A Gramian-based approach to model reduction for uncertain systems, in: Proceedings of the 46th IEEE Conference on Decision and Control, 2007, pp. 4373-4378.
[27] L. Li, Coprime factor model reduction for discrete-time uncertain systems, in: Proceedings of the 48th IEEE Conference on Decision and Control, 2010, pp. 6213-6218.
[28] Georgiou, T.; Smith, M., Optimal robustness in the gap metric, IEEE Trans. Automat. Control, 35, 673-686, (1990) · Zbl 0800.93289
[29] Glover, K.; McFarlane, D., Robust stabilization of normalized coprime factor plant descriptions with \(H_\infty\)-bounded uncertainty, IEEE Trans. Automat. Control, 34, 8, 821-830, (1989) · Zbl 0698.93063
[30] Youla, D. C.; Jabr, H.; Bongiorno, J. J., Modern Wiener-Hopf design of optimal controllers-part II: the multivariable case, IEEE Trans. Automat. Control, 21, 3, 319-338, (1976) · Zbl 0339.93035
[31] E. Prempain, On coprime factors for parameter-dependent systems, in: Proceedings of 45th IEEE Conference on Decision and Control, 2006, pp. 5796-5800.
[32] Ober, R.; McFarlane, D., Balanced canonical forms for minimal systems: a normalized coprime factor approach, Linear Algebra Appl., 122-124, 23-64, (1989)
[33] Li, L., Structured model reduction and control for interconnected systems, (2005), University of California Los Angeles, (Ph.D. thesis)
[34] Ionescu, V.; Oara, C., A discrete-time reduced-order controller for robust stabilization of plant in the normalized coprime-factor description, IMA J. Math. Control Inform., 11, 3, 231-252, (1994) · Zbl 0811.93023
[35] Hoffmann, J. W., Normalized coprime factorizations in continuous and discrete time—a joint state-space approach, IMA J. Math. Control Inform., 13, 4, 359-384, (1996) · Zbl 0863.93024
[36] G.A. Murad, I. Postlethwaite, G. Da-Wei, Model reduction for non-minimal state-space systems, in: Proceedings of the 36th IEEE Conference on Decision and Control, Vol. 4, 1997, pp. 3304-3310.
[37] Li, L.; Paganini, F., Structured coprime factor model reduction based on lmis, Automatica, 41, 1, 145-151, (2005) · Zbl 1067.93010
[38] Doyle, J. C.; Glover, K.; Khargonekar, P. P.; Francis, B. A., State-space solutions to standard \(H_2\) and \(H_\infty\) control problems, IEEE Trans. Automat. Control, 34, 8, 831-847, (1989) · Zbl 0698.93031
[39] Vinnicombe, G., Frequency domain uncertainty and the graph topology, IEEE Trans. Automat. Control, 38, 1371-1383, (1993) · Zbl 0787.93076
[40] C. Beck, J. Doyle, Model reduction of behavioural systems, in: Proceedings of the 32nd IEEE Conference on Decision and Control, Vol. 4, 1993, pp. 3652-3657.
[41] Dullerud, G. E.; Paganini, F., A course in robust control theory: A convex approach, (2000), Springer New York · Zbl 0939.93001
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