A Note on Predefined-Time Stability

This note presents a new characterization for predefined-time-stable systems based on Lyapunov stability. In contrast to the previous results in predefined-time stability, the proposed characterization allows the construction of predefined-time stabilizing controllers with polynomial terms instead of exponential ones, removing the exponential nature hypothesis of predefined-time-stable systems. Moreover, the existing Lyapunov characterization of predefined-time stability is shown to be a consequence of the new theorem presented in this paper. Finally, the proposed approach is used for the construction of robust predefined-time stabilizing controllers for first-order systems. A simulation example shows the feasibility of the proposed methods.

[1]  Alexander G. Loukianov,et al.  A discontinuous recurrent neural network with predefined time convergence for solution of linear programming , 2014, 2014 IEEE Symposium on Swarm Intelligence.

[2]  Andrey Polyakov,et al.  Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems , 2012, IEEE Transactions on Automatic Control.

[3]  Emilio Roxin,et al.  On finite stability in control systems , 1966 .

[4]  Wilfrid Perruquetti,et al.  Finite time stability conditions for non-autonomous continuous systems , 2008, Int. J. Control.

[5]  Alexander G. Loukianov,et al.  A second order predefined-time control algorithm , 2017, 2017 14th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE).

[6]  V. Haimo.  Finite time controllers , 1986 .

[7]  David Gómez-Gutiérrez,et al.  Variable Structure Predefined‐Time Stabilization of Second‐Order Systems , 2018, Asian Journal of Control.

[8]  Leonid Fridman,et al.  Uniform Second-Order Sliding Mode Observer for mechanical systems , 2010, 2010 11th International Workshop on Variable Structure Systems (VSS).

[9]  Alexander G. Loukianov,et al.  A class of predefined-time stable dynamical systems , 2018 .

[10]  Alexander G. Loukianov,et al.  Predefined-time stability of dynamical systems with sliding modes , 2015, 2015 American Control Conference (ACC).

[11]  Z. Zuo,et al.  Fixed-time stabilization and consensus of nonholonomic systems , 2016 .

[12]  Alessandro Astolfi,et al.  Homogeneous Approximation, Recursive Observer Design, and Output Feedback , 2008, SIAM J. Control. Optim..

[13]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[14]  Zhengtao Ding,et al.  Fixed-Time Consensus Tracking for Multiagent Systems With High-Order Integrator Dynamics , 2018, IEEE Transactions on Automatic Control.

[15]  Alexander G. Loukianov,et al.  On optimal predefined‐time stabilization , 2017 .

[16]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[17]  A. Bacciotti,et al.  Liapunov functions and stability in control theory , 2001 .

[18]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[19]  Alexander G. Loukianov,et al.  Semi-global predefined-time stable systems , 2017, 2017 14th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE).

[20]  Leonid M. Fridman,et al.  Stability notions and Lyapunov functions for sliding mode control systems , 2014, J. Frankl. Inst..

[21]  Leonid M. Fridman,et al.  Design of a prescribed convergence time uniform Robust Exact Observer in the presence of measurement noise , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

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