### State feedback controller design for fractional order system in the triangular form

ZHAO Yige, WANG Yuzhen

1. School of Control Science and Engineering, Shandong University, Jinan 250061, Shandong, China
• Received:2015-06-26 Revised:2015-11-02 Online:2015-12-20 Published:2015-06-26

Abstract: Some new properties for Caputo fractional derivative were presented, and a sufficient condition of asymptotical stability for fractional order nonlinear systems was obtained based on the new properties. By using the Backstepping technique, the state feedback control design problem for fractional order nonlinear systems in the triangular form was investigated. Two illustrative examples were provided to illustrate the effectiveness of the main results.

CLC Number:

• TP273
 [1] PODLUBNY I. Fractional differential equations[M]. New York:Academic Press, 1999.[2] KILBAS A A, SRIVASTAVA H H, TRUJILLO J J. Theory and applications of fractional differential equations[M]. Elsevier Science B V:Amsterdam, 2006.[3] MILLER K S, ROSS B. An introduction to the fractional calculus and fractional differential equation[M]. New York:John Wiley, 1993.[4] MANABE S. The non-integer integral and its application to control systems[J]. Electrotechnical Journal of Japan, 1961, 6(3-4):83-87.[5] TAVAZOEI M S, HAERI M. A note on the stability of fractional order systems[J]. Mathematics and Computers in Simulation, 2009, 79(5):1566-1576.[6] CHEN Y Q, MOORE K L. Analytical stability bound for a class of delayed fractional-order dynamic systems[J]. Nonlinear Dynamics, 2002, 29(1):191-200.[7] AHN H S, CHEN Y Q, PODLUBNY I. Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality[J]. Applied Mathematics and Computation, 2007, 187(1):27-34.[8] MOORNANI K A, HAERI M. On robust stability of linear time invariant fractional-order systems with real parametric uncertainties[J]. ISA Transactions, 2009, 48(4):484-490.[9] CHEN Y Q, AHN H S, XUE D Y. Robust controllability of interval fractional order linear time invariant systems[J]. Signal Processing, 2006, 86(10):2794-2802.[10] MOORNANI K A, HAERI M. Necessary and sufficient conditions for BIBO-stability of some fractional delay systems of neutral type[J]. IEEE Transactions on Automatic Control, 2011, 56(1):125-128.[11] ABUSAKSAKA A B, PARTINGTON J R. BIBO stability of some classes of delay systems and fractional systems[J]. Systems & Control Letters, 2014, 64:43-46.[12] LAZAREVIĆ M P. Finite time stability analysis of PDα fractional control of robotic time-delay systems[J]. Mechanics Research Communications, 2006, 33(2):269-279.[13] LAZAREVIĆ M P, SPASIĆ A M. Finite-time stability analysis of fractional order time-delay systems:Gronwall's approach[J]. Mathematical and Computer Modelling, 2009, 49(3):475-481.[14] KAMAL S, RAMAN A, BANDYOPADHYAY B. Finite-time stabilization of fractional order uncertain chain of integrator:an integral sliding mode approach[J]. IEEE Transactions on Automatic Control, 2013, 58(6):1597-1602.[15] SHEN J, LAM J. Non-existence of finite-time stable equilibria in fractional-order nonlinear systems[J]. Automatica, 2014, 50(2):547-551.[16] LAKSHMIKANTHAM V, LEELA S, SAMBANDHAM M. Lyapunov theory for fractional differential equations[J]. Communications in Applied Analysis, 2008, 12(4):365-376.[17] BURTON T A. Fractional differential equations and Lyapunov functionals[J]. Nonlinear Analysis:Theory, Methods & Applications, 2011, 74(16):5648-5662.[18] LI Y, CHEN Y Q, PODLUBNY I. Mittag-Leffler stability of fractional order nonlinear dynamic systems[J]. Automatica, 2009, 45(8):1965-1969.[19] LI Y, CHEN Y Q, PODLUBNY I. Stability of fractional-order nonlinear dynamic systems:Lyapunov direct method and generalized Mittag-Leffler stability[J]. Computers and Mathematics with Applications, 2010, 59(5):1810-1821.[20] SADATI S J, BALEANU D, RANJBAR A, et al. Mittag-Leffler stability theorem for fractional nonlinear systems with delay[J]. Abstract and Applied Analysis, 2010, 2010:1-7.[21] AGUILA-CAMACHO N, DUARTE-MERMOUD M A, GALLEGOS J A. Lyapunov functions for fractional order systems[J]. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(9):2951-2957.[22] 韩正之, 陈鹏年, 陈树中. 自适应控制[M]. 北京:清华大学出版社, 2011.[23] XIE X J, DUAN N, YU X. State-feedback control of high-order stochastic nonlinear systems with SiISS inverse dynamics[J]. IEEE Transactions on Automatic Control, 2011, 56(8):1921-1926.[24] XIE X J, LIU L. A homogeneous domination approach to state feedback of stochastic high-order nonlinear systems with time-varying delay[J]. IEEE Transactions on Automatic Control, 2013, 58(2):494-499.
 [1] CHENG Daizhan, LI Zhiqiang. A survey on linearization of nonlinear systems [J]. JOURNAL OF SHANDONG UNIVERSITY (ENGINEERING SCIENCE), 2009, 39(2): 26-36.
Viewed
Full text

Abstract

Cited

Shared
Discussed