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On strong stabilizability of MIMO infinite-dimensional systems. (English) Zbl 1448.93254

Summary: The strong stabilization problem, i.e., the problem of designing a real stable stabilizing controller, is considered for multi-input multi-output infinite-dimensional real linear systems. The considered systems may have infinitely many poles and zeros in the open right-half-plane, as well as on the imaginary axis. It is shown that the well-known parity interlacing property (pip) for real-rational systems is also a necessary condition in the most general case as long as the plant has coprime factorizations over \(\mathcal{H}^\infty \), which is a necessary condition for stabilizability. Furthermore, it is shown that pip is also sufficient under some additional mild assumptions.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C35 Multivariable systems, multidimensional control systems

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References:

[1] Abedor, J. L., & Poolla, K. (1989). On the strong stabilization of delay systems. In Proceedings of the IEEE conference on decision and control. (pp. 2317-2318). Tampa, FL, U.S.A.
[2] Churchill, R. V.; Brown, J. W., Complex variables and applications (1984), McGraw-Hill: McGraw-Hill New York · Zbl 0546.30003
[3] Curtain, R.; Weiss, G.; Weiss, M., Coprime factorizations for regular linear systems, Automatica, 32, 1519-1531 (1996) · Zbl 0870.93025
[4] Curtain, R. F.; Zwart, H., An introduction to infinite-dimensional linear systems theory (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0839.93001
[5] Davison, E. J.; Wang, S. H., Properties of linear time-invariant multivariable systems subject to arbitrary output and state feedback, IEEE Transactions on Automatic Control, AC-18, 24-32 (1973) · Zbl 0262.93018
[6] Duren, P. L., Theory of \(H^P\) spaces (1970), Academic Press: Academic Press New York · Zbl 0215.20203
[7] Gümüşsoy, S., Coprime-inner/outer factorization of SISO time-delay systems and FIR structure of their optimal H-infinity controllers, International Journal of Robust and Nonlinear Control, 22, 981-998 (2012) · Zbl 1273.93063
[8] Gümüşsoy, S.; Özbay, H., Stable \(\mathcal{H}^\infty\) controller design for time-delay systems, International Journal of Control, 81, 546-556 (2008) · Zbl 1152.93361
[9] Gümüşsoy, S.; Özbay, H., Sensitivity minimization of strongly stabilizing controllers for a class of unstable time-delay systems, IEEE Transactions on Automatic Control, 54, 590-595 (2009) · Zbl 1367.93162
[10] Hoffman, K., Banach spaces of analytic functions (2007), Dover · Zbl 0117.34001
[11] Krstic, M.; Smyshlyaev, A., Boudary control of PDEs: A course on backstepping designs (2008), SIAM: SIAM Philadelphia · Zbl 1149.93004
[12] Loreto, M. D.; Bonnet, C.; Loiseau, J. J., Stabilization of neutral time-delay systems, (Loiseau, J. J.; Michiels, W.; Niculescu, S.-I.; Sipahi, R., Topics in time delay systems. Topics in time delay systems, Lecture notes in control and information sciences, no. 388 (2009), Springer-Verlag: Springer-Verlag Berlin), 209-219
[13] Michiels, W.; Niculescu, S. I., Stability and stabilization of time-delay systems (2007), SIAM: SIAM Philadelphia · Zbl 1140.93026
[14] Mikkola, K. M., Real solutions to control, approximation, and factorization problems, SIAM Journal on Control and Optimization, 50, 1071-1086 (2012) · Zbl 1263.93203
[15] Miller, G., Numerical analysis for engineers and scientists (2014), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 1294.00002
[16] Mortini, R., Reducibility of function pairs in \(H_{\mathbb{R}}^\infty \), St. Petersburg Mathematical Journal, 23, 1013-1022 (2012) · Zbl 1277.30041
[17] Nguyen, L. H. V., & Bonnet, C. (2012). Coprime factorizations of MISO fractional time-delay systems. In Proceedings of the international symposium on mathematical theory of networks and systems. Melbourne, Australia.
[18] Özbay, H., Introduction to feedback control theory (1999), CRC Press: CRC Press Boca Raton · Zbl 0980.93001
[19] Özbay, H. (2008). On strongly stabilizing controller synthesis for time delay systems, In Proceedings of the 17th IFAC world congress. (pp. 6342-6346). Seoul, Korea.
[20] Quadrat, A., On a generalization of the Youla-Kučere parameterization. Part I: The fractional ideal approach to SISO systems, Systems & Control Letters, 50, 135-148 (2003) · Zbl 1157.93348
[21] Quadrat, A., On a general structure of the stabilizing controllers based on stable range, SIAM Journal on Control and Optimization, 42, 6, 2264-2285 (2004) · Zbl 1069.93032
[22] Smith, M. C., On stabilization and the existence of coprime factorizations, IEEE Transactions on Automatic Control, 34, 1005-1007 (1989) · Zbl 0693.93057
[23] Staffans, O., Feedback stabilization of a scalar functional differential equation, Journal of Integral Equations, 10, 319-342 (1985) · Zbl 0592.93046
[24] Staffans, O., Stabilization of a distributed system with a stable compensator, Mathematics of Control, Signals, and Systems, 5, 1-22 (1992) · Zbl 0745.93066
[25] Staffans, O., Coprime factorizations and well-posed linear systems, SIAM Journal on Control and Optimization, 36, 1268-1292 (1998) · Zbl 0919.93040
[26] Suyama, K. (1991). Strong stabilization of systems with time-delays. In Proceedings of the 1991 international conference on industrial electronics, control and instrumentation. (pp. 1758-1763). Kobe, Japan.
[27] Toker, O.; Özbay, H., On the rational \(\mathcal{H}^\infty\) controller design for infinite dimensional plants, International Journal of Robust and Nonlinear Control, 6, 383-397 (1996) · Zbl 0866.93032
[28] Treil, S., The stable rank of the algebra \(H^\infty\) equals 1, Journal of Functional Analysis, 109, 130-154 (1992) · Zbl 0784.46037
[29] Ünal, H. U., & Iftar, A. (2008). Stable \(\mathcal{H}^\infty\) controller design for systems with multiple time-delays: The case of data-communication networks. In Proceedings of the 17th IFAC world congress. (pp. 13348-13354). Seoul, Korea.
[30] Ünal, H. U.; Iftar, A., Stable \(\mathcal{H}^\infty\) controller design for systems with multiple input/output time-delays, Automatica, 48, 563-568 (2012) · Zbl 1244.93046
[31] Ünal, H. U.; Iftar, A., Stable \(\mathcal{H}^\infty\) flow controller design using approximation of FIR filters, Transactions of the Institute of Measurement and Control, 34, 3-25 (2012)
[32] Vidyasagar, M., Control system synthesis: A factorization approach (1985), M.I.T. Press: M.I.T. Press Cambridge, MA · Zbl 0655.93001
[33] Vyhlidal, T.; Zitek, P., QPmR - Quasi-Polynomial root-finder: Algorithm update and examples, (Vyhlidal, T.; Lafay, J.-F.; Sipahi, R., Delay systems: from theory to numerics and applications (2014), Springer: Springer New York), 299-312 · Zbl 1275.93033
[34] Wakaiki, M.; Yamamoto, Y., Stable controller design for mixed sensitivity reduction of infinite-dimensional systems, Systems & Control Letters, 72, 80-85 (2014) · Zbl 1297.93141
[35] Wakaiki, M.; Yamamoto, Y.; Özbay, H., Sensitivity reduction by strongly stabilizing controllers for MIMO distributed parameter systems, IEEE Transactions on Automatic Control, 57, 2089-2094 (2012) · Zbl 1369.93171
[36] Wakaiki, M.; Yamamoto, Y.; Özbay, H., Stable controllers for robust stabilization of systems with infinitely many unstable poles, Systems & Control Letters, 62, 511-516 (2013) · Zbl 1279.93093
[37] Wakaiki, M.; Yamamoto, Y.; Özbay, H., Sensitivity reduction by stable controllers for MIMO infinite dimensional systems via the tangential Nevanlinna-Pick interpolation, IEEE Transactions on Automatic Control, 59, 1099-1105 (2014) · Zbl 1360.93231
[38] Watanabe, K., Finite spectrum assignment and observer for multivariable systems with commensurate delays, IEEE Transactions on Automatic Control, AC-31, 543-549 (1986) · Zbl 0596.93009
[39] Wick, B. D., Stabilization in \(H_{\mathbb{R}}^\infty ( \mathbb{D} )\), Publicacions Matematiques, 54, 25-52 (2010), Also see: Corrigenda. 55 (2011) 251-260 · Zbl 1193.46033
[40] Wu, Z.; Michiels, W., Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method, Journal of Computational and Applied Mathematics, 236, 2499-2514 (2012) · Zbl 1237.65065
[41] Zeren, M., & Özbay, H. (1997). On stable \(\mathcal{H}^\infty\) controller design. In Proceedings of the American control conference. (pp. 1302-1306). Albuquerque, New Mexico, U.S.A.
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