Discrete differentiators based on sliding modes

Abstract Sliding-mode-based differentiation of the input f ( t ) yields exact estimations of the derivatives f , … , f ( n ) , provided an upper bound L ( t ) of | f ( n + 1 ) ( t ) | is available in real-time. In practice it involves discrete sampling and numerical integration of the internal variables between the measurements. Accuracy asymptotics of different discretization schemes are calculated for discrete noisy sampling, whereas sampling and integration steps are independently variable or constant. Proposed discrete differentiators restore the optimal accuracy asymptotics of their continuous-time counterparts. Event-triggered sampling is considered. Extensive numeric experiments are presented and analyzed.

[1]  Xinghuo Yu,et al.  Discrete-Time Terminal Sliding Mode Control Systems Based on Euler's Discretization , 2014, IEEE Transactions on Automatic Control.

[2]  Andrey Polyakov,et al.  Finite-time and fixed-time stabilization: Implicit Lyapunov function approach , 2015, Autom..

[3]  Driss Boutat,et al.  An observation algorithm for nonlinear systems with unknown inputs , 2009, Autom..

[4]  Leonid M. Fridman,et al.  Design of Continuous Twisting Algorithm , 2017, Autom..

[5]  Wilfrid Perruquetti,et al.  Finite-Time Observers: Application to Secure Communication , 2008, IEEE Transactions on Automatic Control.

[6]  Arie Levant,et al.  Homogeneity approach to high-order sliding mode design , 2005, Autom..

[7]  Arie Levant,et al.  Exact Differentiation of Signals with Unbounded Higher Derivatives , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[8]  Leonid M. Fridman,et al.  Super-Twisting Algorithm in presence of time and state dependent perturbations , 2018, Int. J. Control.

[9]  Cédric Join,et al.  Numerical differentiation with annihilators in noisy environment , 2009, Numerical Algorithms.

[10]  Arie Levant,et al.  On discretization of high-order sliding modes , 2016 .

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

[12]  Antonella Ferrara,et al.  Higher Order Sliding Mode Controllers With Optimal Reaching , 2009, IEEE Transactions on Automatic Control.

[13]  Yuri B. Shtessel,et al.  A novel adaptive-gain supertwisting sliding mode controller: Methodology and application , 2012, Autom..

[14]  Giorgio Bartolini,et al.  Approximability Properties for Second-Order Sliding Mode Control Systems , 2007, IEEE Transactions on Automatic Control.

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

[16]  Arie Levant,et al.  Globally convergent differentiators with variable gains , 2018, Int. J. Control.

[17]  L. Fridman,et al.  Higher‐order sliding‐mode observer for state estimation and input reconstruction in nonlinear systems , 2008 .

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

[19]  A. N. Atassi,et al.  Separation results for the stabilization of nonlinear systems using different high-gain observer designs ☆ , 2000 .

[20]  Xinghuo Yu,et al.  Nonlinear derivative estimator , 1996 .

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

[22]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[23]  Arie Levant,et al.  Weighted homogeneity and robustness of sliding mode control , 2016, Autom..

[24]  A.R. Teel,et al.  Homogeneous hybrid systems and a converse Lyapunov theorem , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[25]  A. Levant.  Robust exact differentiation via sliding mode technique , 1998 .

[26]  Yuri B. Shtessel,et al.  Aeronautical and space vehicle control in dynamic sliding manifolds , 2003 .

[27]  Sarah K. Spurgeon,et al.  Sliding mode observers: a survey , 2008, Int. J. Syst. Sci..

[28]  Jaime A. Moreno,et al.  Strict Lyapunov Functions for the Super-Twisting Algorithm , 2012, IEEE Transactions on Automatic Control.

[29]  Sarah K. Spurgeon,et al.  A Robust Exact Differentiator Toolbox for Matlab®/Simulink® , 2017 .

[30]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[31]  Alessandro Pisano,et al.  Sliding mode control: A survey with applications in math , 2011, Math. Comput. Simul..

[32]  Andrey Polyakov,et al.  On homogeneity and its application in sliding mode control , 2014, Journal of the Franklin Institute.

[33]  Arie Levant,et al.  Higher-order sliding modes, differentiation and output-feedback control , 2003 .

[34]  Chee Pin Tan,et al.  Sliding mode observers for fault detection and isolation , 2002 .

[35]  Xinghuo Yu,et al.  Sliding-Mode-Based Differentiation and Its Application , 2017 .

[36]  Franck Plestan,et al.  A new algorithm for high‐order sliding mode control , 2008 .

[37]  Martin Horn,et al.  Discrete-Time Implementation of Homogeneous Differentiators , 2020, IEEE Transactions on Automatic Control.

[38]  Giorgio Bartolini,et al.  A survey of applications of second-order sliding mode control to mechanical systems , 2003 .

[39]  Leonid M. Fridman,et al.  Robust exact uniformly convergent arbitrary order differentiator , 2013, Autom..

[40]  Arie Levant,et al.  Robust exact filtering differentiators , 2020, Eur. J. Control.

[41]  Arie Levant,et al.  Proper discretization of homogeneous differentiators , 2014, Autom..

[42]  Jaime A. Moreno,et al.  On strict Lyapunov functions for some non-homogeneous super-twisting algorithms , 2014, J. Frankl. Inst..

[43]  Xinghuo Yu,et al.  Sliding-Mode-Based Differentiation and Filtering , 2018, IEEE Transactions on Automatic Control.

[44]  Arie Levant,et al.  Discrete sliding-mode-based differentiators , 2016, 2016 14th International Workshop on Variable Structure Systems (VSS).

[45]  Arie Levant,et al.  Stability and robustness of homogeneous differential inclusions , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).