Time scale state feedback h-stabilisation of linear systems under Lipschitz-type disturbances

This paper studies the h-stabilisation problem of certain classes of perturbed systems on time scales. Sufficient conditions for the control law design are proposed to ensure the h-stability of the closed-loop dynamical system under Lipschitz-type disturbances. Using the Gronwall inequality approach with time scale theory, the h-stability of the closed-loop system is investigated in non-uniform time domains with bounded graininess. Some numerical examples are provided to show the feasibility of the obtained results using the proposed approach for systems evolving on some arbitrary time scales.

[1]  Z. Bartosiewicz,et al.  Stability and stabilizability of linear time-delay systems on homogeneous time scales , 2017 .

[2]  Alaa E. Hamza,et al.  Stability of abstract dynamic equations on time scales , 2012 .

[3]  M. Pinto,et al.  Variationally stable difference equations , 1997 .

[4]  An LMI approach to stability for linear time-varying system with nonlinear perturbation on time scales , 2011 .

[5]  Robert J. Marks,et al.  Linear state feedback stabilisation on time scales , 2009, 0910.3034.

[6]  Corentin Briat,et al.  Dynamic equations on time-scale : Application to stability analysis and stabilization of aperiodic sampled-data systems , 2011 .

[7]  V. Lakshmikantham,et al.  Stability of moving invariant sets and uncertain dynamic systems on time scales , 1998 .

[8]  T. Laleg‐Kirati,et al.  Sufficient conditions for uniform exponential stability and h-stability of some classes of dynamic equations on arbitrary time scales , 2019, Nonlinear Analysis: Hybrid Systems.

[9]  Jianfu Yang,et al.  MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS , 2009 .

[10]  L. Fausett,et al.  Controllability, Observability and Realizability Criteria on Time Scale Dynamical Systems , 2004 .

[11]  Do Duc Thuan,et al.  Stabilizability and robust stabilizability of implicit dynamic equations with constant coefficients on time scales , 2016, IMA J. Math. Control. Inf..

[13]  Zbigniew Bartosiewicz,et al.  Stability, stabilization and observers of linear control systems on time scales , 2007, 2007 46th IEEE Conference on Decision and Control.

[14]  Fabian R. Wirth,et al.  A spectral characterization of exponential stability for linear time-invariant systems on time scales , 2003 .

[15]  A. Peterson,et al.  Dynamic Equations on Time Scales: An Introduction with Applications , 2001 .

[16]  Michael Defoort,et al.  On the consensus tracking investigation for multi-agent systems on time scale via matrix-valued Lyapunov functions , 2018, Autom..

[17]  Harry L. Trentelman,et al.  Proceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems (MTNS) , 2004 .

[18]  P. Kloeden,et al.  Lyapunov functions for linear nonautonomous dynamical equations on time scales , 2006 .

[19]  A. Peterson,et al.  Dynamic Equations on Time Scales , 2001 .

[20]  h-Stability for linear dynamic equations on time scales , 2006 .

[21]  Mohammad Hossein Shafiei,et al.  Partial Finite-Time Stabilization of Perturbed Nonlinear Systems Based on the Novel Concept of Nonsingular Terminal Sliding Mode Method , 2020 .

[22]  Variationally Stable Difference Systems , 2001 .

[23]  Zbigniew Bartosiewicz,et al.  Unification of continuous-time and discrete-time systems: the linear case∗ , 2004 .

[24]  Ravi P. Agarwal,et al.  Difference equations and inequalities , 1992 .

[25]  Nguyen Huu Du,et al.  On the exponential stability of dynamic equations on time scales , 2007 .

[26]  Tuba Gulsen,et al.  A certain class of surfaces on product time scales with interpretations from economics , 2018 .

[27]  Zhenkun Huang,et al.  Consensus of multi-agent system with distributed control on time scales , 2016, Appl. Math. Comput..

[28]  Jeffrey J. DaCunha,et al.  Transition matrix and generalized matrix exponential via the Peano-Baker series , 2005 .

[29]  Mohamed Ali Hammami,et al.  State feedback stabilization of a class of uncertain nonlinear systems on non-uniform time domains , 2016, Syst. Control. Lett..

[30]  E. Pawłuszewicz.  Null-Controllability of Linear Systems on Time Scales , 2012 .

[31]  M. Bohner,et al.  Controllability and observability of time-invariant linear dynamic systems , 2012 .

[32]  M. Pinto.  PERTURBATIONS OF ASYMPTOTICALLY STABLE DIFFERENTIAL SYSTEMS , 1984 .

[33]  R. M. C. Mestrom,et al.  Design and Numerical Analysis of an Electrostatic Energy Harvester With Impact for Frequency Up-Conversion , 2020, Journal of Computational and Nonlinear Dynamics.

[34]  J. J. Dacunha.  Stability for time varying linear dynamic systems on time scales , 2005 .

[35]  S. Choi,et al.  Variationally stable dynamic systems on time scales , 2012, Advances in Difference Equations.

[36]  Li Yang,et al.  Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales , 2014, Appl. Math. Comput..

[37]  H. Leiva,et al.  Approximate Controllabilty of Semilinear Dynamic Equations on Time Scale , 2018, Asian Journal of Control.

[38]  Robert J. Marks,et al.  CONTROLLABILITY, OBSERVABILITY, REALIZABILITY, AND STABILITY OF DYNAMIC LINEAR SYSTEMS , 2009, 0901.3764.

[39]  F. V. Vleck,et al.  Stability and Asymptotic Behavior of Differential Equations , 1965 .

[40]  Zbigniew Bartosiewicz,et al.  On stabilisability of nonlinear systems on time scales , 2013, Int. J. Control.

[41]  Yi Zhang,et al.  Lie symmetry analysis on time scales and its application on mechanical systems , 2019 .

[42]  Tingwen Huang,et al.  Global exponential synchronization of nonautonomous recurrent neural networks with time delays on time scales , 2018, Appl. Math. Comput..

[43]  Stability of nonlinear differential systems , 1992 .

[44]  M. Bohner,et al.  Linear Programming Problems on Time Scales , 2017 .

[45]  Robert J. Marks,et al.  State feedback stabilization of linear time-varying systems on time scales , 2010, 2010 42nd Southeastern Symposium on System Theory (SSST).