Home Page by Arie Levant

Levant, Arie ( formerly Levantovsky, Lev )


Professor of the Applied Mathematics Dept.   

School of Mathematical Sciences,
Tel-Aviv University,
Tel-Aviv 69978

E-mail: levant@tauex.tau.ac.il
Tel.: +972-3-6408812 (office), Fax: +972-3-6407543



"Real-time" differentiation of the noisy signal f = f0 +h,
representing the "unknown" basic signal f0(t) = sin(0.5 t) + 0.5 cos(t)
corrupted by the noise
h = 107 cos(104 t) - 2 107 sin(1.7 105 t) + 5 107 cos(1.33 105 t)

and sampled with the time period 10-5.

Filtering differentiator of the order 3 and the filtering order 7 is applied

Teaching        My papers on Google Scholar

Fall 2023. Reception Hours: Sunday, 15:10 - 16:00, Room 123b, Schreiber building.    ZOOM is possible on request      


1987 Ph.D. Thesis "Higher-order sliding modes and their application in control of uncertain processes". Adviser: Prof S.V. Emelyanov. Opponent (secondary adviser): Prof M.A. Krasnoselsky.

1983-1987 Postgraduate studies in mathematical control theory, Institute for System Studies of the USSR Academy of Sciences, Moscow.

1980 B.Sc. and M.Sc. from the  Moscow State University, Faculty of Mechanics and Mathematics, majored in the Theory of Differential Equations.

1980 M.Sc. Thesis "Singularities of Stability Region Boundary." Scientific Adviser: Prof V.I. Arnold.

Research Interests:

Nonlinear Control Theory, Sliding Mode Control, Homogeneous Discontinuous Control, Nonlinear Finite-Time-Exact Tracking and Observation, Robust Exact High-Order Differentiation.

Main scientific results:

Establishment of new branches of the Sliding-Mode Control (SMC) Theory - High-Order SMC and Homogeneous SMC; Real-Time Robust Exact Differentiation

Some papers in the PDF format:

Introduction of High-Order SMs:

A.Levant-Sliding order and sliding accuracy in sliding mode control, International Journal of Control, 58(6), 1247-1263, 1993

S.V.Emelyanov,S.K.Korovin,A.Levant-High order sliding modes in control systems, Computational Mathematics and Modeling, 7(3), 294-318, 1996

A.Levant-Robust exact differentiation via sliding mode technique, Automatica, 34(3), 379-384, 1998

A.Levant,A.Pridor,R.Gitizadeh, I.Yaesh, J. Z.Ben-Asher-Aircraft Pitch Control via Second Order Sliding Technique, AIAA Journal of Guidance, Control and Dynamics, 23(4), 586-594, 2000

Proofs for the paper IEEE Transactions on Automatic Control, 46(9), 2001, pp. 1447 - 1451 - see the paper in Int. J. of Control, 76 (9/10), 924-941, 2003

A.Levant-Universal SISO sliding-mode controllers with finite-time convergence, IEEE Transactions on Automatic Control, 46(9), 2001, pp. 1447 - 1451

L.Fridman,A.Levant-Higher order sliding modes.- in W. Perruquetti, J. P. Barbot, eds. "Sliding Mode Control in Engineering", Marcel Dekker, Inc., 53-101, 2002

Introduction of homogeneous SMC:

A..Levant-Construction principles of output-feedback 2-SMC design, in Proc. of the 41th IEEE CDC, Las-Vegas, Dec.10-13, 2002

A.Levant-Higher-order sliding modes, differentiation and output-feedback control, Int. J. of Control, 76 (9/10), 924-941, Special issue on Sliding-Mode Control, 2003

A.Levant-Homogeneity approach to high-order sliding mode design. - Automatica, 41(5), 823-830, 2005

A.Levant-Quasi-Continuous High-Order Sliding-Mode Controllers , IEEE Transactions on Automatic Control, 50(11), 1812-1816, 2005

A.Levant-Homogeneous quasi-continuous sliding-mode control. Lecture Notes in Control and Information Sciences, 334, 143--168, "Advances in Variable Structure and Sliding Mode Control", 2006

A.Levant-Construction principles of 2-sliding mode design. Automatica, 43(4), 576-586, 2007

A.Levant-Finite Differences in Homogeneous Discontinuous Control, IEEE Transactions on Automatic Control, 52(7), 1208 - 1217, 2007

A.Levant,L.Alelishvili-Integral High-Order Sliding Modes, IEEE Transactions on Automatic Control, 52(7), 1278 - 1282, 2007
Proofs for High-Order Integral Sliding Modes - see here and here

Artificial increase of the output relative degree:

L.Dorel(Alelishvili),A.Levant-On Chattering-Free Sliding-Mode Control, in Proc. of the 45th IEEE CDC, Mexico, Cancun, Dec.9-11, 2008

A.Levant,Y.Pavlov-Generalized homogeneous quasi-continuous controllers, Int.Journal of Robust and Nonlinear Control, 18(4-5), 385 - 398, 2008 - Special Issue: Advances in Higher Order Sliding Mode Control

A.Levant-Keynote "Homogeneous High-Order Sliding Modes", 17th IFAC World Congress, Seoul, July 6-11, 2008

A.Levant,A.Michael-Adjustment of High-Order Sliding-Modes (with ), Int.Journal of Robust and Nonlinear Control, 19(15), 1657 - 1672, 2009

A.Levant,L.Fridman-Homogeneous Sliding Modes in the Presence of Fast Actuators, IEEE Transactions on Automatic Control, 55(3), 810 - 814, 2010

Full version of the paper "Accuracy of Homogeneous Sliding Modes in the Presence of Fast Actuators" (with proofs)

Chattering Analysis:

A.Levant-Chattering Analysis, IEEE Transactions on Automatic Control, 55(6), 1380 - 1389, 2010

A.Levant-Finite-time stability and high relative degrees in sliding-mode control. Plenary Lecture at the 11th International Workshop VSS'2010, Mexico-City, Mexico, June 26, 2010, published in Lecture Notes in Control and Information Sciences, 412, 59-92, 2012

Introduction of Practical Relative Degree:

A.Levant-Practical Relative Degree Approach in Sliding-Mode Control. Plenary Lecture at the 12th Scientific Workshop VSS'11, Mumbay (India), January 12-14, 2012, published in Lecture Notes in Control and Information Sciences, 440, 97-115, 2013

Feasibility of fixed-time convergence:

A.Levant-On Fixed and Finite Time Stability in Sliding Mode Control. Corrected version. In_Proc._of_the 52th IEEE Conference on Decision and Control, CDC 2013, Florence, Italy, December 10-13, 2013

Arbitrary convergence-time function realization, including acceleration of MIMO finite-time convergence up to fixed-time convergence

Y.Dvir, D.Efimov, A.Levant, A.Polyakov, W.Perruquetti-Acceleration of finite-time-stable homogeneous systems, Int. Journal of Robust and Nonlinear Control, 28(5), 1757 - 1777, 2018

Discretization of homogeneous SMC:

A.Levant,M.Livne-Uncertain disturbances' attenuation by homogeneous MIMO sliding mode control and its discretization, IET Control Theory & Applications, 9(4), 515-525, 2015

S.Ding, A.Levant, S.Li - Simple homogeneous sliding-mode controller, Automatica, 67(5),22-32,2016

A.Levant, M.Livne - Weighted homogeneity and robustness of sliding mode control, Automatica, 72(10),186-193,2016

A.Levant, M.Livne, D.Lunz-On Discretization of High Order Sliding Modes, in Barbot,Fridman,Plestan (eds) "Recent Trends in Sliding Mode Control", 177-202,IET,2016

General weighted homogeneity of differerntial inclusions:

A.Levant, D.Efimov, A.Polyakov, W.Perruquetti-Stability and Robustness of Homogeneous Differential Inclusions,CDC 2016, Las-Vegas, December 12-15, 2016

Asymptotic optimality of SM-based differentiators:

A.Levant, M.Livne, X.Yu-Sliding-Mode-Based Differentiation and Its Application, 20th IFAC World Congress, Toulouse, July 9-14, France, 2017. A typo has been corrected in the simulation section.

Universal homogeneous control templates:

A.Levant-Non-Lyapunov SISO homogeneous control design, IEEE CDC2017

Fast-convergent SM-based differentiators with variable gains:

A.Levant,M.Livne-Globally convergent differentiators with variable gains, Int.J. of Control, published online March 2018

A.Levant-Filtering Differentiators and Observers, VSS, Graz, Austria, July 9-11, 2018

Lecture "Black-Box Control in Theory and Applications" at the Dalian Maritime University, Dalian, China, August 29, 2018

Lecture "Robust Exact Real-Time Differentiation" at the Dalian Maritime University, Dalian, China, August 31, 2018

Filtering unbounded noises in differentiation (still exact without noises!):

A.Levant,X.Yu-Sliding Mode Based Differentiation and Filtering, IEEE TAC, 63(9), 3061-3067, 2018

Filtering theory:

A.Levant,M.Livne-Robust exact filtering differentiators,European Journal of Control, available online, 2019

A.Levant-Homogeneous filtering and differentiation based on sliding modes, IEEE CDC'2019

Presentation of Filtering Differentiators (December, 2019)

Discretization of differentiators, event-triggered sampling:

M.Livne,A.Levant-Proper discretization of homogeneous differentiators, Automatica, 50(8), pp. 2007-2014, 2014

J.-P.Barbot,A.Levant,M.Livne,D.Lunz-Discrete differentiators based on sliding modes,Automatica, 112, available online, 2020


SMC for students: survey-tutorial, 2019

Lecture "Homogeneous Control and Observation of Uncertain SISO Systems", Dalian Maritime University, China, July 27, 2020, Mexico, January 13, 2021 ZOOM recording of July 27, 2020

A.Hanan,A.Levant,A.Jbara-Low-chattering discretization of homogeneous differentiators, IEEE TAC, 2022

A.Hanan, A.Jbara,A.Levant-Homogeneous sliding modes in noisy environments, in "Emerging Trends in Sliding Mode Control Theory and Application", Springer, 2021

A.Hanan, A.Jbara,A.Levant-Low-chattering discretization of sliding mode control, presented at IEEE CDC'2021, December 17

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