Homogeneous discrete differentiation of functions with unbounded higher derivatives

Homogeneous sliding-mode-based differentiators (HD) are known for their high asymptotic accuracy. Their practical realization is computer-based and requires discretization. The corresponding combination of a discrete system with a continuous-time input signal produces hybrid dynamics. In the case of the most usual one-step Euler discretization that hybrid system lacks the homogeneity of its predecessor and loses its ultimate accuracy. Nevertheless, the discrete differentiator can be modified, restoring the homogeneity and the accuracy of HD. Similarly to HD, the proposed homogeneous discrete differentiator can also be used to differentiate signals with a variable upper bound of the highest derivative. Simulation results confirm the theoretical results.

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

[2]  Leonid M. Fridman,et al.  High order sliding mode observer for linear systems with unbounded unknown inputs , 2010, Int. J. Control.

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

[4]  Arie Levant,et al.  Quasi-continuous high-order sliding-mode controllers , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

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

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

[7]  A. F. Filippov.  Equations with the Right-Hand Side Continuous in x and Discontinuous in t , 1988 .

[8]  Peter C. Müller,et al.  A Simple Nonlinear Observer for a Class of Uncertain Mechanical Systems , 2007, IEEE Transactions on Automatic Control.

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

[10]  Leonid Fridman.  Chattering analysis in sliding mode systems with inertial sensors , 2003 .

[11]  Abdelhamid Rabhi,et al.  Estimation of contact forces and tire road friction , 2007, 2007 Mediterranean Conference on Control & Automation.

[12]  G. Bartolini.  Chattering phenomena in discontinuous control systems , 1989 .

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

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

[15]  Leonid M. Fridman,et al.  An exact and uniformly convergent arbitrary order differentiator , 2011, IEEE Conference on Decision and Control and European Control Conference.

[16]  Joseph Z. Ben-Asher,et al.  Aircraft Pitch Control via Second-Order Sliding Technique , 2000 .

[17]  Arie Levant,et al.  Discretization issues of high-order sliding modes , 2011 .

[18]  Giorgio Bartolini,et al.  First and second derivative estimation by sliding mode technique , 2000 .

[19]  Katsuhisa Furuta,et al.  Frequency characteristics of Levant's differentiator and adaptive sliding mode differentiator , 2007 .

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

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

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

[23]  J. Barbot,et al.  Nonlinear Observer for Autonomous Switching Systems with Jumps , 2007 .

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

[25]  Michael Defoort,et al.  A novel higher order sliding mode control scheme , 2009, Syst. Control. Lett..

[26]  Qudrat Khan,et al.  Robust Parameter Estimation of Nonlinear Systems Using Sliding-Mode Differentiator Observer , 2011, IEEE Transactions on Industrial Electronics.

[27]  Christopher Edwards,et al.  Sliding mode control : theory and applications , 1998 .

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

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

[30]  A. Levant.  Sliding order and sliding accuracy in sliding mode control , 1993 .

[31]  Giorgio Bartolini,et al.  Robust speed and torque estimation in electrical drives by second-order sliding modes , 2003, IEEE Trans. Control. Syst. Technol..

[32]  A. Levant,et al.  Discrete-Time Sliding-Mode-Based Differentiation , 2013 .

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

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

[35]  Arie Levant,et al.  Chattering Analysis , 2007, IEEE Transactions on Automatic Control.