A Supertwisting Algorithm for Systems of Dimension More Than One

6472 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 11, NOVEMBER 2014

A Supertwisting Algorithm for Systems ofDimension More Than One

Michael V. Basin, Senior Member, IEEE, and Pablo Cesar Rodríguez Ramírez

Abstract—This paper presents a data-driven homogeneous con-tinuous supertwisting algorithm for systems of dimension morethan one. The conditions of finite-time convergence to the originare obtained, and the robustness of the designed algorithm isdiscussed. This paper concludes with numerical simulations illus-trating performance of the designed algorithms.

Index Terms—Sliding mode control, supertwisting, systems ofdimension more than one.

I. INTRODUCTION

I T IS well known that the classical discontinuous slidingmode control provides finite-time convergence for a sys-

tem of dimension one [1]. A finite-time stabilizing controlfor a system of dimension two is realized using the twistingalgorithm [2], where the second-order sliding mode control isalso discontinuous. Both algorithms are robust with respect tobounded disturbances. On the other hand, using a continuoussecond-order sliding mode supertwisting algorithm [3], a stateof a dimension one system can be stabilized along with its firstderivative. The supertwisting algorithm is robust with respectto unbounded disturbances satisfying a Lipschitz condition.The finite-time convergence of the designed algorithms is con-ventionally established using geometrical techniques [2], [3],direct Lyapunov method [4]–[6], or homogeneity approach [7],[8]. The explicit Lyapunov functions for their second-ordersupertwisting algorithms can be found in [6]. The homogene-ity approach, mentioned even in the classical book [9], wasconsistently developed in the mentioned papers and applied tothe observer design in [10]. High-gain observer-based slidingmode control schemes were designed in ([11]–[16]). Variousmodifications of the sliding mode technique have always beenactively used in industrial applications ([17]–[23]), includingdata-driven ones ([24]–[27]). More recent results on data-drivencontrol and monitoring can be found in [28] and [29]. A recentpaper [30] has presented a modified supertwisting algorithm forsystems of dimension more than one, which however fails to

Manuscript received August 10, 2013; revised October 26, 2013 andDecember 24, 2013; accepted December 25, 2013. Date of publicationJanuary 21, 2014; date of current version June 6, 2014. This work wassupported in part by the French Centre National de la Recherche Scientifique(CNRS) and in part by the Mexican National Council of Science and Technol-ogy (CONACyT) under Grant PICS 25232, Grant 170660, and Grant 129081.

The authors are with the Department of Physical and Mathematical Sciences,Autonomous University of Nuevo León, San Nicolás de Los Garza 66451-144, Mexico (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2014.2301932

provide finite-time convergence to the origin for some systemstates.

This paper corrects the indicated flaw and presents a data-driven homogeneous continuous supertwisting algorithm forsystems of dimension more than one, which assures finite-timeconvergence to the origin for all system states. First, the caseof dimension two is addressed. The conditions of finite-timeconvergence to the origin equilibrium are obtained, and the ro-bustness of the designed algorithm is discussed. Similar resultsare then obtained for systems of dimension more than two.This paper concludes with numerical simulations illustratingperformance of the designed algorithms.

This paper is organized as follows. The problem statement isgiven in Section II. Some preliminary results are presented inSection III. A supertwisting-like control algorithm for systemsof dimension two is designed in Section IV. The correspond-ing examples are provided in Section V. A supertwisting-like control algorithm for systems of dimension more thantwo is presented in Section VI and illustrated by examples inSection VII. Section VIII concludes this paper. The proofs ofall theorems and lemmas are given in the Appendix. A briefconference version of this paper was presented in [30].

II. CONTROL PROBLEM STATEMENT

Consider a double-integrator dynamic system of dimen-sion two

x1(t) = x2(t) x1(t0) = x10

x2(t) = u(t) x1(t0) = x20 (1)

where x(t) = [x1(t), x2(t)] ∈ R2 is the system state, andu(t) ∈ R is the control input.

In the classical second-order sliding mode control theory, afinite-time stabilizing control for system (1) is designed usingthe twisting algorithm [2] in the form

u(t) = −k1sgn (x1(t))− k2sgn (x2(t)) (2)

where k1 and k2 > 0 are certain positive constants, and thesignum function of a scalar x is defined as sgn(x) = 1, if x > 0;sgn(x) = 0, if x = 0; and sgn(x) = −1, if x < 0 ([9]).

On the other hand, if a scalar dynamic system is of dimen-sion one

x(t) = u(t) x(t0) = x0 (3)

a continuous finite-time stabilizing control for system (3) canbe designed using the supertwisting algorithm [3] as follows:

u(t) = −λ |x(t)|1/2 sgn (x(t))− α

∫ t

t0

sgn (x(s)) ds (4)

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BASIN AND RODRÍGUEZ RAMÍREZ: SUPERTWISTING ALGORITHM FOR SYSTEMS OF DIMENSION MORE THAN ONE 6473

where λ > 0 and α > 0 are certain positive constants. Notethat applying the continuous control (4) to system (3) resultsin a second-order sliding mode, i.e., both x(t) and x(t) con-verge to zero for a finite time. In other words, the continu-ous control (4) yields finite-time convergence similar to thatproduced by a classical discontinuous sliding mode controlu(t) = −Ksgn(x(t)), where K > 0 is sufficiently large, forsystem (3).

In this paper, we propose a homogeneous supertwisting-likecontinuous modification of the twisting control algorithm (2) asfollows:

u(t) = −λ0

∣∣∣∣∣∣t∫

t0

x1(s)ds

∣∣∣∣∣∣1/4

sgn

⎛⎝

t∫t0

x1(s)ds

⎞⎠

− λ1 |x1(t)|1/3 sgn (x1(t))− λ2 |x2(t)|1/2 sgn (x2(t))

− α

t∫t0

sgn (x2(s)) ds (5)

where λ0, λ1, λ2 > 0, and α > 0 are certain positive constants.It would be demonstrated that the designed continuous control(5) works similarly to the twisting control (2), i.e., results infinite-time convergence of both states x1(t) and x2(t) of system(1) to the origin. The announced result is formalized in the nextsections and then proved in the Appendix. Note that the controllaw (5) is a state feedback one since the system states (1) areassumed available for control design.

III. PRELIMINARY RESULTS

Hereinafter in this paper, we say that the system state (tra-jectory) x(t) ∈ Rn is finite-time (globally) convergent to apoint xf ∈ Rn, if for any initial condition x(t0) = x0 ∈ Rn

there exists a time moment T (x0) < ∞ such that x(t) = xf forall t ≥ T .

Before proceeding to the main result for dimension twosystems, let us present the following lemmas.

Lemma 1: Consider a dynamic system (1) of dimension two.Then, the following modified supertwisting control law:

u(t) = −λ1 |x1(t)|1/3 sgn (x1(t))

−λ2 |x2(t)|1/2 sgn (x2(t))− α

t∫t0

sgn (x2(s)) ds (6)

yields finite-time convergence of both states x1(t) and x2(t) toa point [x1f , 0].

Proofs of all the lemmas are given in the Appendix.Consider now a system (1) in the presence of a disturbance

x1(t) = x2(t) x1(t0) = x10

x2(t) = u(t) + ξ(t) x1(t0) = x20 (7)

where ξ(t) satisfies the Lipschitz condition with constant L.System (7) can be still stabilized at a point [x1f , 0] in view ofthe following theorem.

Lemma 2: Consider a dynamic system (7) of dimension twoin the presence of a disturbance ξ(t) satisfying the Lipschitzcondition with constant L. Then, the modified supertwistingcontrol law (5) yields finite-time convergence of both statesx1(t) and x2(t) to a point [x1f , 0], provided that the follow-ing conditions hold for control gains: α > L and λ2

2 > 2(α+L)2/(α− L).

The preceding results can be generalized as follows. Con-sider a dynamic system of dimension n > 2

x1(t) =x2(t) x1(t0) = x10

x2(t) =x3(t) x2(t0) = x20

· · ·xn(t) =u(t) xn(t0) = xn0 (8)

using the notation for system (1). We propose a generaliza-tion the supertwisting-like continuous control algorithm (5) asfollows:

u(t) = −v1(t)− v2(t)− · · · − vn(t) + vn+1(t) (9)

where vi(t) = λi|xi(t)|γisgn(xi(t)), i = 1, . . . , n

vn+1(t) = |(s(t))|γ/(1−γ) sgn (s(t))

s(t) = − α

t∫t0

sgn (xn(s)) ds

and certain positive constants λ1, . . . , λn > 0, α > 0, and ex-ponents γi, i = 1, . . . , n, are assigned according to [7] to yieldhomogeneous finite-time convergence of all the states of theclosed-loop system (8), (9). Namely, γi ∈ (0, 1), i = 1, . . . , n,satisfy the recurrent relations γi−1 = γiγi+1/(2γi+1 − γi), i =2, . . . , n, γn+1 = 1, and γn = γ, where γ belongs to an interval(1− ε, 1), ε > 0. Reference [7] establishes that there existssuch an ε > 0 that the homogeneous (in view of definition of γi)closed-loop system (8), (9) without the term vn+1(t) is globallyfinite-time convergent to the origin.

Lemma 3: Consider a dynamic system (8) of dimension n >2. Then, the modified supertwisting control law (9) yields finite-time convergence of the states x1(t), . . . , xn−1(t), xn(t) to apoint [x1f , . . . , x(n−1)f , 0].

Remark 1: Theorem 4 ensures finite-time stability of system(8) with control law (9) with respect to an equilibrium point[x1f , . . . , x(n−1)f , 0] located in the manifold xn = 0, whichis however different from the origin. Then, it follows from(8) that x2f = x3f = · · · = x(n−1)f = 0. Thus, the equilibriumpoint is given by [x1f , 0 . . . , 0, 0] and located in the manifoldx2 = x3 = · · · = xn = 0.

Consider now a system (8) in the presence of a disturbance

x1(t) =x2(t) x1(t0) = x10

x2(t) =x3(t) x2(t0) = x20

· · ·xn(t) =u(t) + ξ(t) xn(t0) = xn0 (10)


Fig. 1. Graphs of the system states (1) upon applying the control law (5) with initial conditions x10 = 1000 and x20 = 1000.

Fig. 2. Graphs of the system states (1) upon applying the control law (5) with initial conditions x10 = 1000 and x20 = −1000.

where ξ(t) satisfies the Lipschitz condition with constantL. System (10) can be still stabilized at a point [x1f , . . . ,x(n−1)f , 0] in view of the following theorem.

Lemma 4: Consider a dynamic system (10) of dimension n>2 in the presence of a disturbance ξ(t) satisfying the Lipschitzcondition with constant L. Then, the modified supertwistingcontrol law (9) yields finite-time convergence of the statesx1(t), . . . , xn−1(t), xn(t) to a point [x1f , . . . , x(n−1)f , 0], pro-vided that the following conditions hold for control gains: α >L and λ2

n > 2(α+ L)2/(α− L).

IV. SUPERTWISTING ALGORITHM FOR

DIMENSION TWO SYSTEMS

The results for the control law (5) are given as follows.Theorem 1: Consider a dynamic system (1) of dimension

two. Then, the modified supertwisting control law (5) yieldsfinite-time convergence of both states x1(t) and x2(t) to theorigin.

Proofs of all the theorems are given in the Appendix.Theorem 2: Consider a dynamic system (7) of dimension two

in the presence of a disturbance ξ(t) satisfying the Lipschitzcondition with constant L. Then, the modified supertwistingcontrol law (5) yields finite-time convergence of both statesx1(t) and x2(t) to the origin, provided that the following condi-tions hold for control gains: α>L and λ2

2>2(α+L)2/(α−L).

V. EXAMPLES: I. DIMENSION TWO

This section presents examples of designing a finite-time sta-bilizing regulator for a dynamic system (1) of dimension two,based on the modified supertwisting regulator (5) in Theorems 1and 2. The given examples correspond to benchmark problemsof stabilizing an electric circuit without active resistance, withand without current disturbances, for a finite time.

1) Consider a linear system (1). The modified supertwistingregulator (5) is applied with the control gains selectedas λ0 = 0.325, λ1 = 10, λ2 = 5, and α = 1. The initialconditions are assigned as x10 = 1000 and x20 = 1000.The obtained results are shown in Fig. 1. Fig. 2 shows theresults for the initial conditions x10 = 1000 and x20 =−1000.

2) Consider a linear system (8) with disturbance ξ(t) =sin(1000t). Again, the modified supertwisting regulator(5) is applied with the control gains selected as λ0 =0.325, λ1 = 10, λ2 = 5, and α = 1. The initial conditionsare assigned as x10 = 1000 and x20 = 1000. The ob-tained results are shown in Fig. 3. Fig. 4 shows the resultsfor the initial conditions x10 = 1000 and x20 = −1000.

This example clearly demonstrates that the sufficient condi-tions for the control gains in Theorem 2 are too conservative,and the finite-time convergence takes place with much relaxed


Fig. 3. Graphs of the system states (7) with disturbance ξ(t) = sin(1000t) upon applying the control law (5) with initial conditions x10 = 1000 andx20 = 1000.

Fig. 4. Graphs of the system states (7) with disturbance ξ(t) = sin(1000t) upon applying the control law (5) with initial conditions x10 = 1000 andx20 = −1000.

values. In particular, the value of constant L in this exampleis equal to 1000 due to high-frequency sinusoidal oscillationssin(1000t).

VI. SUPERTWISTING ALGORITHM FOR DIMENSION

MORE THAN TWO SYSTEMS

The main result can be generalized as follows. Consider adynamic system (8) of dimension n > 2. We propose a gen-eralization the supertwisting-like continuous control algorithm(8) as follows:

u(t) = −v0(t)− v1(t)− v2(t)− · · · − vn(t) + vn+1(t)(11)

where

v0(t) = λ0

∣∣∣∣∣∣t∫

t0

x1(s)ds

∣∣∣∣∣∣γ0

sgn

⎛⎝

t∫t0

x1(s)ds

⎞⎠

vi(t) = λi |xi(t)|γi sgn (xi(t)) , i = 1, . . . , n

vn+1(t) = |(s(t))|γ/(1−γ) sgn (s(t))

s(t) = −α

t∫t0

sgn (xn(s)) ds

and certain positive constants λ0, λ1, . . . , λn > 0, α > 0, andexponents γi, i = 0, . . . , n, are assigned according to [7] toyield homogeneous finite-time convergence of all the states ofthe closed-loop system (8), (11). Namely, γi ∈ (0, 1) and i=0,. . . , n satisfy the recurrent relations γi−1 = γiγi+1/(2γi+1 −γi), i = 2, . . . , n, γn+1 = 1, and γn = γ, where γ belongs toan interval (1− ε, 1), ε > 0. Reference [7] establishes that thereexists such an ε > 0 that the homogeneous (in view of the def-inition ofγi) closed-loop system (8), (11) without the terms v0(t)and vn+1(t) is globally finite-time convergent to the origin.

It would be demonstrated that the designed continuouscontrol (11) works similarly to the supertwisting control (5),i.e., results in finite-time convergence of the states x1(t),x2(t), . . . , xn(t) of the system (8) to the origin. The announcedresult is formalized in the next theorem and then proved in theAppendix.

Theorem 3: Consider a dynamic system (8) of dimensionn > 2. Then, the modified supertwisting control law (11) yieldsfinite-time convergence of the states x1(t), . . . , xn−1(t), xn(t)to the origin.

Consider now a system (10) in the presence of a disturbance,where ξ(t) satisfies the Lipschitz condition with constant L.System (10) can be still stabilized at the origin in view of thefollowing theorem.


Fig. 5. Graphs of the system states (8) upon applying the control law (11) with initial conditions x10 = x20 = x30 = 1000.

Fig. 6. Graphs of the system states (8) upon applying the control law (11) with initial conditions x10 = x30 = 1000 and x20 = −1000.

Theorem 4: Consider a dynamic system (10) of dimensionn > 2 in the presence of a disturbance ξ(t) satisfying theLipschitz condition with a constant L. Then, the modifiedsupertwisting control law (11) yields finite-time convergenceof the states x1(t), . . . , xn−1(t), xn(t) to the origin, providedthat the following conditions hold for control gains: α>L andλ2n>2(α+L)2/(α−L).

VII. EXAMPLES: II. DIMENSION MORE THAN TWO

This section presents examples of designing a finite-timestabilizing regulator for a dynamic system (8) of a dimensionmore than two, based on the modified supertwisting regulator(11) in Theorems 3 and 4. The given examples correspond tobenchmark problems of stabilizing an electric circuit withoutactive resistance, with and without current disturbances, for afinite time, where the control law is generated by an actuator,which integrates a primary control input.

1) Consider a linear 3-D system

x1(t) = x2(t) x1(t0) = x10

x2(t) = x3(t) x2(t0) = x20

x3(t) = u(t) x3(t0) = x30. (12)

The modified supertwisting regulator (11)

u(t) = −λ0

∣∣∣∣∣∣t∫

t0

x1(s)ds

∣∣∣∣∣∣1/5

sgn

⎛⎝

t∫t0

x1(s)ds

⎞⎠

− λ1 |x1(t)|1/4 sgn (x1(t))−λ2 |x2(t)|1/3sgn(x2(t))

− λ3 |x3(t)|1/2 sgn (x3(t))−α

t∫t0

sgn(x3(s)) ds

is applied with the control gains selected as λ0 = λ1 =λ2 = 20, λ3 = 10, and α = 1. The initial conditions areassigned as x10 = x20 = x30 = 1000. The obtained re-sults are shown in Fig. 5. Fig. 6 shows the results forinitial conditions x10 = x30 = 1000 and x20 = −1000.

2) Consider a linear system (12) with disturbance ξ(t) =sin(1000t). Again, the modified supertwisting regulator(8) is applied with the control gains selected as λ0 =λ1 = λ2 = 20, λ3 = 10, and α = 1. The initial condi-tions are assigned as x10 = x20 = x30 = 1000. The ob-tained results are shown in Fig. 7. The correspondingcontinuous control law is presented in Fig. 8. Fig. 9 showsthe results for the initial conditions x10 = x30 = 1000


Fig. 7. Graphs of the system states (10) with disturbance ξ(t) = sin(1000t) upon applying the control law (11) with initial conditions x10 = x20 =x30 = 1000.

Fig. 8. Graphs of the control law (11) applied to the system (10) with disturbance ξ(t) = sin(1000t) and initial conditions x10 = x20 = x30 = 1000.

Fig. 9. Graphs of the system states (10) with disturbance ξ(t) = sin(1000t) upon applying the control law (11) with initial conditions x10 = x30 = 1000 andx20 = −1000.

and x20 = −1000. The corresponding continuous controllaw is given in Fig. 10.

This example again demonstrates that the sufficient condi-tions for the control gains in Theorem 4 are too conservative,and the finite-time convergence takes place with much relaxedvalues. In particular, the value of constant L in this example

is equal to 1000 due to high-frequency sinusoidal oscillationssin(1000t).

VIII. CONCLUSION

This paper has presented a data-driven homogeneous contin-uous supertwisting algorithm for systems of dimension more


Fig. 10. Graphs of the control law (11) applied to the system (10) with disturbance ξ(t) = sin(1000t) and initial conditions x10 = x30 = 1000 and x20 =−1000.

than one, which is globally convergent to the origin for a finitetime for any initial condition and also robust with respect todisturbances with a bounded changing rate. The designed tech-nique generalizes the seminal continuous supertwisting algo-rithm, which was proven to be highly effective for stabilizationof both system state and its derivative, to systems of dimensionmore than one. This advance leads to a possibility of applyinga continuous finite-time stabilization control law to technicalplants, where a conventional sliding mode control cannot bereliably employed due to effects pertinent to its discontinuousnature, such as short circuiting. Typical examples of industrialelectronics devices where the designed technique could be usedinclude induction motors, antilock braking systems, vibrationattenuators, and many others.

APPENDIX

A) Proof of Lemma 1: The system (1), (5) can be recast inthe time-invariant form

x1(t) =x2(t) x1(t0) = x10

x2(t) = −λ1 |x1(t)|1/3 sgn (x1(t))

− λ2 |x2(t)|1/2 sgn (x2(t)) + x3(t) x2(t0) = x20

x3(t) = −αsgn (x2(t)) x3(t0) = 0.

(13)

The vector field f on the right-hand side of (13) can berepresented as the sum of two homogeneous vector fields,i.e., f = g1 + g2, where g1 = [x2,−λ1|x1(t)|1/3sgn(x1(t))−ρλ2|x2(t)|1/2sgn(x2(t)), 0], ρ ∈ (0, 1), and g2 = [0,−(1−ρ)λ2|x2(t)|1/2sgn(x2(t)) + x3(t),−αsgn(x2(t))] are of ho-mogeneity degrees m1 = m2 = −1. The field g1 providesthe finite-time stability at a point [0, 0, x3(t0)] in view ofits homogeneity and Lyapunov function V (x1, x2) = λ1(3/4)|x1(t)|4/3 + (1/2)|x2(t)|2. The field g2 corresponds to a su-pertwisting algorithm [3], which converges to a point [x1f , 0, 0]for a finite time. The theorem assertion now follows from [7],applying inequality (35) in that theorem with U(x) = −c2,

c2 = const > 0, and inequality (36) there with the right-handside coefficient −(c1 + c2) and taking into account that theLyapunov function for supertwisting has a continuous totalderivative in time along the trajectory; hence, the results ofTheorem 6.2, Lemma 4.2, and [7] hold. �

B) Proof of Lemma 2: System (7) can be recast in the time-invariant form

x1(t) =x2(t) x1(t0) = x10

x2(t) =−λ1 |x1(t)|1/3 sgn (x1(t))

− λ2 |x2(t)|1/2 sgn (x2(t)) + x3(t) x2(t0) = x20

x3(t) =−αsgn (x2(t)) + ξ(t) x3(t0) = 0

(14)

where ξ(t) exists and is bounded for almost all t ≥ t0. Thetheorem assertion follows from Lemma 1 and the convergenceconditions for a supertwisting algorithm [3]. �

C) Proof of Lemma 3: The system (8), (9) can be recast inthe time-invariant form

x1(t) =x2(t) x1(t0) = x10

x2(t) =x3(t) x2(t0) = x20

· · ·xn−1(t) =xn(t) xn−1(t0) = x(n−1)0

xn(t) = −v1(t)− v2(t)− · · · − vn(t) + |xn+1(t)|γ/(1−γ)

× sgn (xn+1(t)) xn(t0) = xn0

xn+1(t) = −αsgn (xn(t)) xn+1(t0) = 0. (15)

Similarly to (13), the vector field f on the right-hand side of(13) can be represented as the sum of two homogeneous vectorfields, i.e., f = g1 + g2, where g1 = [x2, x3, . . . , xn,−v1(t)−v2(t)− . . . vn−1(t)− ρvn(t), 0], ρ ∈ (0, 1), and g2 = [0, 0,. . . , 0,−(1− ρ)vn(t) + vn+1(t),−αsgn(xn(t))] are of homo-geneity degrees m1 = m2 = (γ − 1)/γ < 0. The homogeneitydegree m2 for g2 can always be selected as m2 = m1 = (γ −1)/γ, by virtue of exponent γ/(1− γ) for vn+1, making


the entire system (15) homogeneous. The field g1 provides thefinite-time stability at a point [0, . . . , 0, xn+1(t0)] in view of[7]. The field g2 introduces a modification of a supertwisting al-gorithm [3], which converges to a point [x1f , . . . , x(n−1)f , 0, 0]for a finite time, in view of Lyapunov function V (xn, xn+1) =α|xn(t)|+ (1− γ)|xn+1(t)|1/(1−γ). The theorem assertionnow follows from [7], applying inequality (35) in that theoremwith U(x) = −c2, c2 = const > 0, and inequality (36) therewith the right-hand side coefficient −(c1 + c2) and taking intoaccount that V (xn, xn+1) has a continuous total derivative intime along the trajectory; hence, the results of Theorem 6.2,Lemma 4.2, and [7] hold. �

D) Proof of Lemma 4: System (10) can be recast in the time-invariant form

x1(t)=x2(t) x1(t0)= x10

x2(t)=x3(t) x2(t0)= x20

· · ·

xn−1(t)=xn(t) xn−1(t0)= x(n−1)0

xn(t)= −v1(t)− v2(t)− · · · − vn(t) + |xn+1(t)|γ/(1−γ)

× sgn (xn+1(t)) xn(t0)= xn0

xn+1(t)= −αsgn(xn(t)) + ξ(t) xn+1(t0)= 0

where ξ(t) exists and is bounded for almost all t ≥ t0. Thetheorem assertion follows from Lemma 3 and the convergenceconditions for a supertwisting algorithm [3]. �

E) Proof of Theorem 1: The system (1), (5) can be recast inthe time-invariant form, introducing a new fictitious variable x0

x0(t) =x1(t) x0(t0) = 0

x1(t) =x2(t) x1(t0) = x10

x2(t) =−λ0 |x0(t)|1/4sgn(x0(t))−λ1 |x1(t)|1/3sgn(x1(t))

− λ2 |x2(t)|1/2sgn(x2(t))+ x3(t) x2(t0)=x20

x3(t) =−αsgn (x2(t)) x3(t0) = 0.

The theorem assertion follows from Lemma 3 and Remark 1 ofthis paper. �

F) Proof of Theorem 2: System (7) can be recast in the time-invariant form, introducing a new fictitious variable x0

x0(t) =x1(t) x0(t0) = 0

x1(t) =x2(t) x1(t0) = x10

x2(t) =−λ0 |x0(t)|1/4sgn(x0(t))−λ1 |x1(t)|1/3sgn(x1(t))

− λ2 |x2(t)|1/2sgn(x2(t))+ x3(t) x2(t0) = x20

x3(t) =−αsgn (x2(t)) + ξ(t) x3(t0) = 0

where ξ(t) exists and is bounded for almost all t ≥ t0. The theo-rem assertion follows from Theorem 1 of this paper and theconvergence conditions for a supertwisting algorithm [3]. �

G) Proof of Theorem 3: The system (8), (11) can be recastin the time-invariant form

x0(t)=x1(t) x0(t0)= 0

x1(t)=x2(t) x1(t0)= x10

x2(t)=x3(t) x2(t0)= x20

· · ·xn−1(t)=xn(t) xn−1(t0)= x(n−1)0

xn(t)=−v0(t)−v1(t)−v2(t)−· · ·−vn(t)

+ |xn+1(t)|γ/(1−γ)sgn (xn+1(t)) xn(t0)=xn0

xn+1(t)=−αsgn (xn(t)) xn+1(t0)= 0.

The theorem assertion follows from Lemma 4 and Remark 1 ofthis paper. �

H) Proof of Theorem 4: The system (10) can be recast in thetime-invariant form

x0(t)=x1(t) x0(t0)= 0

x1(t)=x2(t) x1(t0)= x10

x2(t)=x3(t) x2(t0)= x20

· · ·xn−1(t)=xn(t) xn−1(t0)= x(n−1)0

xn(t)=−v0(t)−v1(t)−v2(t)−· · ·−vn(t)

+ |xn+1(t)|γ/(1−γ) sgn(xn+1(t)) xn(t0)= xn0

xn+1(t)= −αsgn (xn(t)) + ξ(t) xn+1(t0)= 0

where ξ(t) exists and is bounded for almost all t ≥ t0. The theo-rem assertion follows from Theorem 3 of this paper and theconvergence conditions for a supertwisting algorithm [3]. �

ACKNOWLEDGMENT

The authors would like to thank Prof. W. Perruquetti,Dr. D. Efimov, Dr. A. Polyakov, E. Bernuau, and other mem-bers of the Lille University–Institut National de Recherche enInformatique et en Automatique (INRIA) team involved in theNon-A Project for their useful comments, suggestions, andcriticisms.

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Michael V. Basin (M’95–SM’07) received the Ph.D.degree in physical and mathematical sciences witha major in automatic control and system analysisfrom Moscow Aviation Institute, Moscow, Russia,in 1992.

He was a Senior Scientist with the Institute ofControl Sciences, Russian Academy of Sciences,Moscow, in 1992–1996 and a Visiting Professorwith the University of Nevada, Reno, NV, USA, in1996–1997. Since 1998, he has been a Full Professorwith the Autonomous University of Nuevo León, San

Nicolás de Los Garza, Mexico. Since 1992, he has published more than 100research papers in international refereed journals and more than 150 papers inproceedings of leading IEEE and International Federation of Automatic Controlconferences and symposiums. He is the author of the monograph New Trendsin Optimal Filtering and Control for Polynomial and Time-Delay Systems(Springer, 2008). His works are cited more than 1000 times. His researchinterests include optimal filtering and control problems, stochastic systems,time-delay systems, identification, sliding mode control, and variable structuresystems.

Dr. Basin is a member of the IEEE Industrial Electronics Society TechnicalCommittee on Data-Driven Control and Monitoring and the IEEE ControlSystems Society Technical Committees on Variable Structure Systems/SlidingMode Control and Intelligent Control. He served as the Editor-in-Chief of theJournal of The Franklin Institute (2010–2013) and as an Associate Editor ofAutomatica, IET Control Theory and Applications, and the International Jour-nal of Systems Science. He was awarded the title of Highly Cited Researcher byThomson Reuters (International Science Institute), the publisher of the ScienceCitation Index, in 2009.

Pablo Cesar Rodríguez Ramírez received thePh.D. degree in industrial physics engineering fromthe Autonomous University of Nuevo León, SanNicolás de Los Garza, Mexico, in 2013.

He currently holds a postdoctoral scholarship withthe Center for Research and Advanced Studies ofthe National Polytechnic Institute (CINVESTAV),Monterrey, Mexico. His research interests includeoptimal filtering and control problems, stochasticsystems, identification, and sliding mode control.

A Supertwisting Algorithm for Systems of Dimension More Than One

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