Finite-time stability of cascaded time-varying systems

The uniform global finite-time stability is discussed for a cascaded time-varying system consisting of two uniformly finite-time stable subsystems. It is shown that a forward completeness condition is enough to ensure the uniform global finite-time stability of the system. For ease of reference, a particular result with a growth rate condition is also deduced. These stability results are applied to the tracking control problem of a non-holonomic wheeled mobile robot in kinematic model. Two tracking control laws are developed respectively for two different cases of the desired rotate velocity. Both control laws are continuous and can control the mobile robot to track the desired trajectory in finite time. Simulation results are provided to show the effectiveness of the method.

[1]  Dennis S. Bernstein,et al.  Finite-Time Stability of Continuous Autonomous Systems , 2000, SIAM J. Control. Optim..

[2]  Luca Zaccarian,et al.  On the Literature's two different definitions of uniform global asymptotic stability for nonlinear systems , 2006 .

[3]  C. Samson.  Control of chained systems application to path following and time-varying point-stabilization of mobile robots , 1995, IEEE Trans. Autom. Control..

[4]  Zhong-Ping Jiang,et al.  A small-gain control method for nonlinear cascaded systems with dynamic uncertainties , 1997, IEEE Trans. Autom. Control..

[5]  Claude Samson,et al.  Velocity and torque feedback control of a nonholonomic cart , 1991 .

[6]  Henk Nijmeijer,et al.  Tracking Control of Mobile Robots: A Case Study in Backstepping , 1997, Autom..

[7]  M. Kawski.  Homogeneous Stabilizing Feedback Laws , 1990 .

[8]  P. Kokotovic,et al.  A positive real condition for global stabilization of nonlinear systems , 1989 .

[9]  M. Vidyasagar.  Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability , 1980 .

[10]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[11]  Jie Huang,et al.  On an output feedback finite-time stabilization problem , 2001, IEEE Trans. Autom. Control..

[12]  Guowu Yang,et al.  Global finite-time stabilization: from state feedback to output feedback , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[13]  E. Panteley,et al.  On global uniform asymptotic stability of nonlinear time-varying systems in cascade , 1998 .

[14]  S. Bhat,et al.  Finite-time stability of homogeneous systems , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[15]  R. Suárez,et al.  Global stabilization of nonlinear cascade systems , 1990 .

[16]  João Pedro Hespanha,et al.  Stabilization of nonholonomic integrators via logic-based switching , 1999, Autom..

[17]  Dennis S. Bernstein,et al.  Geometric homogeneity with applications to finite-time stability , 2005, Math. Control. Signals Syst..

[18]  Wei Lin,et al.  Global finite-time stabilization of a class of uncertain nonlinear systems , 2005, Autom..

[19]  Zhihong Man,et al.  Multi-input uncertain linear systems with terminal sliding-mode control , 1998, Autom..

[20]  Eduardo Sontag.  Remarks on stabilization and input-to-state stability , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[21]  Xinghuo Yu,et al.  Terminal sliding mode control design for uncertain dynamic systems , 1998 .

[22]  E. Moulay,et al.  Finite time stability of nonlinear systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[23]  Yu-Ping Tian,et al.  Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control , 2002, Autom..

[24]  Frédéric Mazenc,et al.  Asymptotic tracking of a reference state for systems with a feedforward structure , 2000, Autom..

[25]  L. Praly,et al.  Asymptotic tracking of a state reference for systems with a feedforward structure , 1997, 1997 European Control Conference (ECC).

[26]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[27]  David Angeli,et al.  A Unifying Integral ISS Framework for Stability of Nonlinear Cascades , 2001, SIAM J. Control. Optim..

[28]  A. Astolfi.  Discontinuous control of nonholonomic systems , 1996 .

[29]  Zhihong Man,et al.  Non-singular terminal sliding mode control of rigid manipulators , 2002, Autom..

[30]  Fumio Miyazaki,et al.  A stable tracking control method for an autonomous mobile robot , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[31]  Antonio Loría,et al.  Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems , 2001, Autom..

[32]  Zhong-Ping Jiang,et al.  A recursive technique for tracking control of nonholonomic systems in chained form , 1999, IEEE Trans. Autom. Control..

[33]  Warren E. Dixon,et al.  Nonlinear Control of Wheeled Mobile Robots , 2001 .

[34]  A. Fuller,et al.  Stability of Motion , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[35]  S. Shankar Sastry,et al.  Stabilization of trajectories for systems with nonholonomic constraints , 1992, Proceedings 1992 IEEE International Conference on Robotics and Automation.

[36]  Elena Panteley,et al.  Exponential tracking control of a mobile car using a cascaded approach , 1998 .

[37]  P. Hartman.  Ordinary Differential Equations , 1965 .

[38]  Hassan K. Khalil,et al.  Nonlinear Systems Third Edition , 2008 .

[39]  Claude Samson,et al.  Feedback control of a nonholonomic wheeled cart in Cartesian space , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

[40]  Frank L. Lewis,et al.  Control of a nonholonomic mobile robot: backstepping kinematics into dynamics , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[41]  Vimal Singh,et al.  Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[42]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[43]  Yiguang Hong,et al.  Finite-time stabilization and stabilizability of a class of controllable systems , 2002, Syst. Control. Lett..

[44]  N. Rouche,et al.  Stability Theory by Liapunov's Direct Method , 1977 .