On the ?2-stability of time-varying linear and nonlinear discrete-time MIMO systems

Y.V.VENKATESH 已出版文章查询
Y.V.VENKATESH
本平台内已出版文章查询
1

+ 作者地址

1Department of ECE, National University of Singapore, Singapore


0
  • 摘要
  • 参考文献
  • 相关文章
  • 统计
New conditions are derived for the 2-stability of time-varying linear and nonlinear discrete-time multiple-input multiple-output (MIMO) systems, having a linear time time-invariant block with the transfer function Γ(z), in negative feedback with a matrix of periodic/aperiodic gains A(k),k =0,1,2,. . . and a vector of certain classes of non-monotone/monotone nonlinearities?( · ), without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features: i) They involve the positive definiteness of the real part (as evaluated on |z| = 1) of the product of Γ(z) and a matrix multiplier function of z. ii) For periodic A(k), one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A(k), but for aperiodic A(k), which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of (A(k+1),A(k)),k=1,2,. . . . iii) They are distinct from and less restrictive than recent results in the literature.

[1] A. I. Lur’e;V. N. Postnikov .On the theory of stability of control systems[J].Prikladnaya Mathematika i Mekhanika,1944,8(03):246-248.

[2] Shorten R;Wirth F;Mason O;Wulff K;King C .Stability criteria for switched and hybrid systems[J].SIAM Review,2007(4):545-592.

[3] Hai Lin;Panos J. Antsaklis .Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results[J].IEEE Transactions on Automatic Control,2009(2):308-322.

[4] D. Liberzon.Switching in Systems and Control[M].Boston:Birkhauser,2003

[5] S. Sun;S. S. Ge.Switched Linear Systems:Control and Design[M].Berlin:Springer-Verlag,2005

[6] G. Szego .On the absolute stability of sampled-data control systems[J].Proceedings of the National Academy of Sciences(USA),1963,50(03):558-560.

[7] Premaratne K.;Jury E.I. .Discrete-time positive-real lemma revisited: the discrete-timecounterpart of the Kalman-Yakubovitch lemma[J].IEEE Transactions on Circuits and Systems.I.Fundamental Theory and Applications,1994(11):747-750.

[8] Yamamoto Y. .A function space approach to sampled data control systems andtracking problems[J].IEEE Transactions on Automatic Control,1994(4):703-713.

[9] Y. Z. Tsypkin .On the stability in the large of nonlinear sampled-data systems[J].Doklady Akademi Nauk SSSR(USSR),1962,145:52-55.

[10] G. Zames .On the input-output stability of time-varying nonlinear feedback systems-Parts 1 and 2[J].IEEE Transactions on Automatic Control,1966,11(02):228-238,465-476.

[11] V. M. Popov .Absolute stability of nonlinear systems of automatic control[J].Automation and Remote Control,1962,22(08):857-875.

[12] Y. Z. Tsypkin .A criterion for absolute stability of automatic pulse systems with monotonic characteristics of the nonlinear element[J].Doklady Akademi Nauk SSSR(USSR),1964,155:1029-1032.

[13] E. I. Jury;B. W. Lee .On the stability of a certain class of nonlinear sampled-data systems[J].IEEE Transactions on Automatic Control,1964,9(01):51-61.

[14] R. P. O’Shea;M. I. Younis .A frequency-time domain stability criterion for sampled data systems[J].IEEE Transactions on Automatic Control,1967,12(06):719-724.

[15] G. Zames;P. Falb .Stability conditions for systems with monotone and slope restricted nonlinearities[J].SIAM Journal on Control,1968,6(01):89-108.

[16] Y. V. Venkatesh .Riesz-Thorin theorem and p-stability of nonlinear time-varying discrete systems[J].Journal of Mathematical Analysis and Applications,1988,135(02):627-643.

[17] Oliveira, M.Z.;Gomes Da Silva Jr., J.M.;Coutinho, D. .Stability analysis for a class of nonlinear discrete-time control systems subject to disturbances and to actuator saturation[J].International Journal of Control,2013(4/6):869-882.

[18] P. A. Bliman;A. M. Krasnosel’ski .Popov absolute stability crite-rion for time-varying multivariable nonlinear systems[OL].http://citeseerx.ist.psu.edu/viewdoc/summary doi=10.1.1.27.9354,1999.

[19] Altshuller, D. A. .Delay-Integral-Quadratic Constraints and Stability Multipliers for Systems With MIMO Nonlinearities[J].IEEE Transactions on Automatic Control,2011(4):738-747.

[20] Z. Huang;Y. V. Venkatesh;C. Xiang.On frequency-domain L2-stability conditions for a class of switched linear and nonlinear systems-Part II:MIMO[M].Singapore:National University of Singapore,2011:1-30.

[21] Z. Huang.Stability Analysis of Switched Systems[M].Singapore:National University of Singapore,2011

[22] Zhihong HUANG,Y. V. VENKATESH,Cheng XIANG,Tong Heng LEE.Frequency-domain L2-stability conditions for time-varying linear and nonlinear MIMO systems[J].控制理论与应用(英文版),2014(01):13-34.

[23] M. S. Branicky .Multiple Lyapunov functions and other analysis tools for switched and hybrid systems[J].IEEE Transactions on Automatic Control,1998(4):475-482.

[24] Mehmet Akar;Kumpati S. Narendra .On the existence of common quadratic Lyapunov functions for second-order linear time-invariant discrete-time systems[J].International journal of adaptive control and signal processing,2002(10):729-751.

[25] Tingshu Hu;Zongli Lin .Absolute Stability Analysis of Discrete-Time Systems With Composite Quadratic Lyapunov Functions[J].IEEE Transactions on Automatic Control,2005(6):781-797.

[26] Kapila V.;Haddad W.M. .A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach[J].IEEE Transactions on Automatic Control,1996(1):149-152.

[27] W. M. Haddad;V. Kapila.Robust stabilization for discrete-time systems with slowly time-varying uncertainty[A].New Orleans:IEEE,1995:202-207.

[28] Mircea Lazar;W. P. Maurice H. Heemels;Andy R. Teel .Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems[J].IEEE Transactions on Automatic Control,2009(10):2421-2425.

[29] G. Zhai;B. Hu;K. Yasuda et al.Stability and 2-gain analysis of discrete-time switched systems[J].Transactions of the Institute of Systems Control and Information Engineers,2002,15(03):117-125.

[30] G. Zhai.Stability and 2-gain analysis of switched symmetric systems[A].Boston:Birkhauser,2003

[31] W. M. Haddad;D. S. Bernstein .Explicit construction of quadratic Lyapunov functions for the smal gain, positivity, circle, and Popov theorems and their application to robust stability-Part II:Discrete-time theory[J].International Journal of Robust and Nonlinear Control,1994,4(11):249-265.

[32] Ahmad, N. S.;Heath, W. P.;Li, G. .LMI-Based Stability Criteria for Discrete-Time Lur'e Systems With Monotonic, Sector- and Slope-Restricted Nonlinearities[J].IEEE Transactions on Automatic Control,2013(2):459-465.

[33] N. S. Ahmad;W. P. Heath;G Li.Lyapunov functions for generalized discrete-time multivariable Popov criterion[A].Milano:Elsevier Science,Ltd,2011:3392-3397.

[34] W. P. Heath;Guang Li .Lyapunov functions for the multivariable Popov criterion with indefinite multipliers[J].Automatica,2009(12):2977-2981.

[35] N. Ahmad;W. Heath;G. Li.Lyapunov functions for discrete-time multivariable Popov criterion with indefinite multipliers[A].Piscataway:IEEE,2010:1559-1564.

[36] L. Gurwitz .Stability of discrete linear inclusion[J].Linear Algebra and its Applications,1995,231(01):47-85.

[37] Daniel Liberzon;Joao P. Hespanha;A. Stephen Morse .Stability of switched systems: a Lie-algebraic condition[J].Systems and Control Letters,1999(3):117-122.

[38] Guisheng Zhai;Derong Liu;Imae J.;Kobayashi T. .Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems[J].IEEE Transactions on Circuits and Systems, II. Express briefs,2006(2):152-156.

[39] Tatsushi Ooba;Yasuyuki Funahashi .On the simultaneous diagonal stability of linear discrete-time systems[J].Systems and Control Letters,1999(3):175-180.

[40] Monovich, T;Margaliot, M .ANALYSIS OF DISCRETE-TIME LINEAR SWITCHED SYSTEMS: A VARIATIONAL APPROACH[J].SIAM Journal on Control and Optimization,2011(2):808-829.

[41] Margaliot M .Stability analysis of switched systems using variational principles: An introduction[J].Automatica,2006(12):2059-2077.

[42] Margaliot M.;Langholz G. .Necessary and sufficient conditions for absolute stability: the case of second-order systems[J].IEEE Transactions on Circuits and Systems.I.Fundamental Theory and Applications,2003(2):227-234.

[43] Molchanov A.P.;Derong Liu .Robust absolute stability of time-varying nonlinear discrete-time systems[J].IEEE Transactions on Circuits and Systems.I.Fundamental Theory and Applications,2002(8):1129-1137.

[44] J. H. Davis .Discrete systems with periodic feedback[J].SIAM Journal on Control,1972,10(01):1-13.

[45] A. S. Morse .Supervisory control of families of linear set-point control ers-Part I:exact matching[J].IEEE Transactions on Automatic Control,1996,41(10):1413-1431.

[46] Raymond A. Decarlo;Michael S. Branicky;Stefan Pettersson;Bengt Lennartson .Perspectives and results on the stability and stabilizability of hybrid systems[J].Proceedings of the IEEE,2000(7):1069-1082.

[47] J. C. Geromel;P. Colaner .H∞ and dwel time specifications of continuous-time switched linear systems[J].IEEE Transactions on Automatic Control,2010,55(01):207-212.

[48] J. P. Hespanha.Stability of switched systems with average dwel-time[A].Piscataway:IEEE,1999:2655-2660.

[49] L. Zhang;P. Shi .Stability, 2-gain and asynchronous H∞control of discrete-time switched systems with average dwel time[J].IEEE Transactions on Automatic Control,2009,54(09):2193-2200.

[50] X. Zhao;L. Zhang;P. Shi et al.Stability and stabilization of switched linear systems with mode-dependent average dwel time[J].IEEE Transactions on Automatic Control,2012,57(07):1809-1815.

[51] R. W. Brockett;J. L. Wil ems .Frequency-domain stability criteria-Parts 1 and 2[J].IEEE Transactions on Automatic Control,1965,10(7/10):255-261,407-413.

[52] M. A. L. Thathachar .Stability of systems with power-law nonlinearities[J].AUTOMATICA,1970,6(05):721-730.

[53] Z. Huang;Y. V. Venkatesh;C. Xiang et al.Frequency-domain 2-stability conditions for switched linear and nonlinear SISO systems[J].International Journal of Systems Science,2014,45(03):682-703.

[54] A. G. Dewey;E. I. Jury .A stability inequality for a classof nonlinear feedback systems[J].IEEE Transactions on Automatic Control,1966,11(01):54-63.

[55] Y. V. Venkatesh .Global variation criteria for the L2-stability of nonlinear time varying systems[J].SIAM Journal on Mathematical Analysis,1978,19(03):568-581.


DOI: http://dx.doi.org/10.1007/s11768-014-4045-7

语种: 英文   


期刊热词
  • + 更多
  • 字体大小