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Frequency-domain L 2-stability conditions for time-varying linear and nonlinear MIMO systems

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Abstract

The paper deals with the L 2-stability analysis of multi-input-multi-output (MIMO) systems, governed by integral equations, with a matrix of periodic/aperiodic time-varying gains and a vector of monotone, non-monotone and quasi-monotone nonlinearities. For nonlinear MIMO systems that are described by differential equations, most of the literature on stability is based on an application of quadratic forms as Lyapunov-function candidates. In contrast, a non-Lyapunov framework is employed here to derive new and more general L 2-stability conditions in the frequency domain. These conditions have the following features: i) They are expressed in terms of the positive definiteness of the real part of matrices involving the transfer function of the linear time-invariant block and a matrix multiplier function that incorporates the minimax properties of the time-varying linear/nonlinear block. ii) For certain cases of the periodic time-varying gain, they contain, depending on the multiplier function chosen, no restrictions on the normalized rate of variation of the time-varying gain, but, for other periodic/aperiodic time-varying gains, they do. Overall, even when specialized to periodic-coefficient linear and nonlinear MIMO systems, the stability conditions are distinct from and less restrictive than recent results in the literature. No comparable results exist in the literature for aperiodic time-varying gains. Furthermore, some new stability results concerning the dwell-time problem and time-varying gain switching in linear and nonlinear MIMO systems with periodic/aperiodic matrix gains are also presented. Examples are given to illustrate a few of the stability theorems.

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Authors and Affiliations

  1. Department of Electrical & Computer Engineering, National University of Singapore, Singapore, Singapore

    Zhihong Huang, Y. V. Venkatesh, Cheng Xiang & Tong Heng Lee

Authors
  1. Zhihong Huang
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  2. Y. V. Venkatesh
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  3. Cheng Xiang
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  4. Tong Heng Lee
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Correspondence to Cheng Xiang.

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Zhihong HUANG received his B.S. degree in Electrical and Electronic Engineering from Zhejiang University, China in 2002, and Ph.D. degree in Electrical and Computer Engineering from National University of Singapore (NUS) in 2011. He joined the Standard Chartered Bank, Singapore where he is a quantitative analyst with the modeling and analytical group. His research interests are switched systems and stochastic models.

Y. V. VENKATESH (SM-IEEE’91) received his Ph.D. degree from the Indian Institute of Science (IIS), Bangalore. He was an Alexander von Humboldt fellow at the Universities of Karlsruhe, Freiburg, and Erlangen, Germany; a national research council fellow (USA) at the Goddard Space Flight Center, Greenbelt, MD; and a visiting professor at the Institut fuer Mathematik, Technische Universitat Graz, Austria, Institut fuer Signalverarbeitung, Kaiserslautern, Germany, National University of Singapore, Singapore and others. Apart from stability theory, his research interests include 3D computer and robotic vision, signal processing, pattern recognition, biometrics, hyperspectral image analysis, and neural networks. As a professor at IIS, he was also the Dean of Engineering Faculty and, earlier, the Chairman of the Electrical Sciences Division. He is a fellow of the Indian Academy of Sciences, the Indian National Science Academy, and the Indian Academy of Engineering. He is on the editorial board of the International Journal of Information Fusion.

Cheng XIANG received his B.S. degree in Mechanical Engineering from Fudan University, China in 1991, M.S. degree in Mechanical Engineering from the Institute of Mechanics, Chinese Academy of Sciences in 1994, and Ph.D. degree in Electrical Engineering from Yale University in 2000. From 2000 to 2001, he was a financial engineer at Fannie Mae, Washington D.C. He has been with the National University of Singapore since 2001. At present, he is an associate professor with the Department of Electrical and Computer Engineering, the National University of Singapore. His research interests include pattern recognition, intelligent control and systems biology.

Tong Heng LEE received his B.A. degree with First Class Honors in the Engineering Tripos from Cambridge University, England, in 1980, M.E. degree from NUS in 1985, and Ph.D. degree from Yale University in 1987. He is a professor in the Department of Electrical and Computer Engineering at the National University of Singapore (NUS) and also a professor in the NUS Graduate School, NUS NGS. He was a past vice-president (research) of NUS. Dr. Lee’s research interests are in the areas of adaptive systems, knowledge-based control, intelligent mechatronics and computational intelligence. He currently holds associate editor appointments in the IEEE Transactions in Systems, Man and Cybernetics; IEEE Transactions in Industrial Electronics; Control Engineering Practice (an IFAC journal); and the International Journal of Systems Science (Taylor and Francis, London). In addition, he is the deputy editor-in-chief of IFAC Mechatronics journal. Dr. Lee was a recipient of the Cambridge University Charles Baker Prize in Engineering; the 2004 ASCC (Melbourne) Best Industrial Control Application Paper Prize; the 2009 IEEE ICMA Best Paper in Automation Prize; and the 2009 ASCC Best Application Paper Prize. He has also co-authored five research monographs (books) and holds four patents (two of which are in the technology area of adaptive systems, and the other two are in the area of intelligent mechatronics). He is the recipient of the 2013 ACA Wook Hyun Kwon Education Prize.

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Huang, Z., Venkatesh, Y.V., Xiang, C. et al. Frequency-domain L 2-stability conditions for time-varying linear and nonlinear MIMO systems. Control Theory Technol. 12, 13–34 (2014). https://doi.org/10.1007/s11768-014-0182-2

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