On stable \(\mathcal{H}^\infty\) controller design for plants with infinitely many unstable zeros and poles. (English) Zbl 1489.93025
Summary: In this work, stable \(\mathcal{H}^\infty\) controller design problem for linear time-invariant single-input single-output plants, which have infinitely many open-right-half-plane zeros and possibly infinitely many right-half-plane poles, is considered. Interpolation-based approach is developed to solve the problem. The approach requires to construct a unit in \(\mathcal{H}^\infty\) satisfying certain interpolation conditions at the open-right-half-plane zeros of the plant and \(\mathcal{H}^\infty\) norm constraint. In order to construct such a unit function, first, construction of an \(\mathcal{H}^\infty\) function satisfying interpolation conditions at given countably many distinct points in the open-right-half plane is presented. Then, by using the upper bound on the \(\mathcal{H}^\infty\) norm of the constructed function and the small-gain theorem, first, sufficient condition is presented for the solution of the sensitivity minimization problem by a stable controller and design methodology for such a controller is presented. Then, stable \(\mathcal{H}^\infty\) controller design approach for the considered class of plants is presented under certain assumptions. A numerical example is given to verify the presented results.
MSC:
| 93B36 | \(H^\infty\)-control |
| 93B55 | Pole and zero placement problems |
| 93C35 | Multivariable systems, multidimensional control systems |
Keywords:
stable controllers; interpolation; infinite-dimensional systems; mixed sensitivity minimizationReferences:
| [1] | Ahsen, M. E.; Özbay, H.; Niculescu, S.-I., Analysis of deterministic cyclic gene regulatory network models with delays (2015), Birkhäuser · Zbl 1315.92003 |
| [2] | Brown, J. W.; Churchill, R. V., Complex variables and applications (1984), McGraw-Hill Book Company: McGraw-Hill Book Company USA · Zbl 0546.30003 |
| [3] | Curtain, R.; Morris, K., Transfer functions of distributed parameter systems: A tutorial, Automatica, 45, 5, 1101-1116 (2009) · Zbl 1162.93300 |
| [4] | Curtain, R. F.; Zwart, H., An introduction to infinite-dimensional linear systems theory (1995), Springer-Verlag: Springer-Verlag New York, U.S.A · Zbl 0839.93001 |
| [5] | Duren, P. L., Theory of \(H^P\) spaces (1970), Academic Press: Academic Press New York, USA · Zbl 0215.20203 |
| [6] | Foias, C.; Özbay, H.; Tannenbaum, A., (Robust control of infinite dimensional systems: Frequency domain methods. Robust control of infinite dimensional systems: Frequency domain methods, Lecture notes in control and information sciences, No. 209 (1996), Springer-Verlag: Springer-Verlag London) · Zbl 0839.93003 |
| [7] | Gahinet, P.; Apkarian, P., An linear matrix inequality approach to \(H_\infty\) control, International Journal of Robust and Nonlinear Control, 4, 4, 421-448 (1994) · Zbl 0808.93024 |
| [8] | Gümüşsoy, S., Coprime-inner/outer factorization of SISO time-delay systems and FIR structure of their optimal \(H_\infty\) controllers, International Journal of Robust and Nonlinear Control, 50, 981-998 (2012) · Zbl 1273.93063 |
| [9] | Gümüşsoy, S.; Özbay, H., Remarks on strong stabilization and stable \(H^\infty\) controller design, IEEE Transactions on Automatic Control, 50, 2083-2087 (2005) · Zbl 1365.93398 |
| [10] | Gümüşsoy, S.; Özbay, H., Stable \(H^\infty\) controller design for time-delay systems, International Journal of Control, 81, 546-556 (2008) · Zbl 1152.93361 |
| [11] | 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 |
| [12] | Gümüşsoy, S.; Özbay, H., On feedback stabilization of neutral time delay systems with infinitely many unstable poles, IFAC-PapersOnLine, 51, 14, 118-123 (2018) |
| [13] | Hara, S.; Yamamoto, Y.; Omata, T.; Nakano, M., Repetitive control system: A new type servo system for periodic exogenous signals, IEEE Transactions on Automatic Control, 33, 7, 659-668 (1988) · Zbl 0662.93027 |
| [14] | Hoffman, K., Banach spaces of analytic functions (2007), Dover · Zbl 0117.34001 |
| [15] | Ito, H.; Ohmori, H.; Sano, A., Design of stable controllers attaining low \(H^\infty\) weighted sensitivity, IEEE Transactions on Automatic Control, 38, 485-488 (1993) · Zbl 0789.93090 |
| [16] | Jacob, B.; Morris, K.; Trunk, C., Minimum-phase infinite-dimensional second-order systems, IEEE Transactions on Automatic Control, 52, 9, 1654-1665 (2007) · Zbl 1366.93277 |
| [17] | Kline, M., Euler and infinite series, Mathematics Magazine, 56, 5, 307-314 (1983) · Zbl 0526.01015 |
| [18] | Koosis, P., Introduction to \(H_p\) spaces (1980), Cambridge University Press: Cambridge University Press Cambridge, U.K · Zbl 0435.30001 |
| [19] | Krstic, M., Delay compensation for nonlinear, adaptive, and PDE systems (2009), Birkhauser · Zbl 1181.93003 |
| [20] | Kuo, C.-F. J.; Shu-Chyuarn, L., Modal analysis and control of a rotating Euler-Bernoulli beam part I: control system analysis and controller design, Mathematical and Computer Modelling, 27, 5, 75-92 (1998) · Zbl 1185.93052 |
| [21] | Lee, P. H.; Soh, Y. C., Synthesis of simultaneous stabilizing \(H^\infty\) controller, International Journal of Control, 78, 1437-1446 (2005) · Zbl 1122.93331 |
| [22] | Lorenzini, C.; Flores, J. V.; Pereira, L. F.A.; Pereira, L. A., Resonant-repetitive controller with phase correction applied to uninterruptible power supplies, Control Engineering Practice, 77, 118-126 (2018) |
| [23] | Mikkola, K., Real solutions to control, approximation, and factorization problems, SIAM Journal on Control and Optimization, 50, 3, 1071-1086 (2012) · Zbl 1263.93203 |
| [24] | Morgans, A. S.; Annaswamy, A. M., Adaptive control of combustion instabilities for combustion systems with right-half plane zeros, Combustion Science and Technology, 180, 9, 1549-1571 (2008) |
| [25] | Mortini, R., Reducibility of function pairs in \(h_R^\infty \), St. Petersburg Mathematical Journal, 23, 6, 1013-1022 (2012) · Zbl 1277.30041 |
| [26] | Niculescu, S. I., (Delay effects on stability: A robust control approach. Delay effects on stability: A robust control approach, Lecture notes in control and information sciences, No. 269 (2001), Springer-Verlag: Springer-Verlag London) · Zbl 0997.93001 |
| [27] | Özbay, H., Stable \(H^\infty\) controller design for systems with time-delays, (Willems, J. C.; Hara, S.; Ohta, Y.; Fujioka, H., Perspectives in mathematical system theory, control, and signal processing. Perspectives in mathematical system theory, control, and signal processing, Lecture notes in control and information sciences, vol. 398 (2010), Springer-Verlag: Springer-Verlag Berlin), 105-113 · Zbl 1198.93074 |
| [28] | Özbay, H., An operator theoretic approach to robust control of infinite dimensional systems, TWMS Journal of Pure and Applied Mathematics, 4, 1, 3-19 (2013) · Zbl 1297.93089 |
| [29] | Özbay, H.; Gümüşsoy, S.; Kashima, K.; Yamamoto, Y., Frequency domain techniques for \(H^\infty\) control of distributed parameter systems (2018), SIAM · Zbl 1517.93001 |
| [30] | Shneiderman, D.; Palmor, Z., Properties and control of the quadruple-tank process with multivariable dead-times, Journal of Process Control, 20, 1, 18-28 (2010) |
| [31] | Smith, M. C., On stabilization and the existence of coprime factorizations, IEEE Transactions on Automatic Control, 34, 9, 1005-1007 (1989) · Zbl 0693.93057 |
| [32] | Toker, O.; Özbay, H., \( H_\infty\) optimal and suboptimal controllers for infinite dimensional SISO plants, IEEE Transactions on Automatic Control, 40, 751-755 (1995) · Zbl 0826.93026 |
| [33] | Ünal, H. U.; İftar, A., Stable \(H^\infty\) controller design for systems with multiple time-delays, Automatica, 48, 3, 563-568 (2012) · Zbl 1244.93046 |
| [34] | Ünal, H. U.; İftar, A., Stable \(H^\infty\) flow controller design using approximation of FIR filters, Transactions of the Institute of Measurement and Control, 34, 3-25 (2012) |
| [35] | Vidyasagar, M., Control system synthesis: A factorization approach (1985), M.I.T. Press: M.I.T. Press Cambridge, MA, U.S.A · Zbl 0655.93001 |
| [36] | 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 |
| [37] | Wakaiki, M.; Yamamoto, Y.; Özbay, H., Sensitivity reduction by strongly stabilizing controllers for MIMO distributed parameter systems, IEEE Transactions on Automatic Control, 57, 8, 2089-2094 (2012) · Zbl 1369.93171 |
| [38] | Wakaiki, M.; Yamamoto, Y.; Özbay, H., Stable controllers for robust stabilization of systems with infinitely many unstable poles, Systems & Control Letters, 62, 6, 511-516 (2013) · Zbl 1279.93093 |
| [39] | 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, 4, 1099-1105 (2014) · Zbl 1360.93231 |
| [40] | Wick, B. D., Corrigenda: “Stabilization in \(H_R^\infty ( D )\)”, Mathematical Publications, 55, 1, 251-260 (2011) · Zbl 1221.46054 |
| [41] | Yao, W.-S.; Tsai, M.-C.; Yamamoto, Y., Implementation of repetitive controller for rejection of position-based periodic disturbances, Control Engineering Practice, 21, 9, 1226-1237 (2013) |
| [42] | Yücesoy, V.; Özbay, H., Stable and robust controller synthesis for unstable time delay systems via interpolation and approximation, IFAC-PapersOnLine, 51, 14, 230-235 (2018) |
| [43] | Zeren, M., & Özbay, H. (1997). On stable \(H^\infty\) controller design. In Proceedings of the American control conference (pp. 1302-1306). |
| [44] | Zeren, M.; Özbay, H., On the synthesis of stable \(H^\infty\) controllers, IEEE Transactions on Automatic Control, 44, 431-435 (1999) · Zbl 1056.93620 |
| [45] | Zhou, K.; Doyle, J. C.; Glover, K., Robust and optimal control (1996), Prentice Hall: Prentice Hall Englewood Cliffs, U.S.A · Zbl 0999.49500 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
