A novel adaptive gain super twisting sliding mode controller.pdf

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Automatica 48 (2012) 759–769 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica A novel adaptive-gain supertwisting sliding mode controller: Methodology and application Yuri Shtessel a , Mohammed Taleb b,c , Franck Plestan b a The University of Alabama in Huntsville, Huntsville, USA b LUNAM Université, Ecole Centrale de Nantes, IRCCyN UMR CNRS 6597, Nantes, France c Moulay Ismail University, Meknes, Morocco article info Article history: Received 20 January 2011 Received in revised form 12 July 2011 Accepted 28 September 2011 Available online 20 March 2012 Keywords: Second order sliding mode Adaptive control Electropneumatic actuator abstract A novel super-twisting adaptive sliding mode control law is proposed for the control of an electropneumatic actuator. The key-point of the paper is to consider that the bounds of uncertainties and perturbations are not known. Then, the proposed control approach consists in using dynamically adapted control gains that ensure the establishment, in a finite time, of a real second order sliding mode. The important feature of the adaptation algorithm is in non-overestimating the values of the control gains. A formal proof of the finite time convergence of the closed-loop system is derived using the Lyapunov function technique. The efficiency of the controller is evaluated on an experimental set-up. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction During the past two decades, the development of high- performance linear/nonlinear controllers (Brun, Sesmat, Thomas- set, & Scavarda, 1999; Brun, Thomasset, & Bideaux, 2002; Chiang, Chen, & Tsou, 2005; Edge, 1997; Hamiti, Voda-Besançon, & Roux- Buisson, 1996; Kimura, Hara, Fujita, & Kagawa, 1997; Kyoungk- wan & Shinichi, 2005; Ming-Chang & Shy-I, 1995; Miyajima, Fujita, Sakaki, Kawashima, & Kagawa, 2007; Rao & Bone, 2006; Richard & Scavarda, 1996; Schultea & Hahn, 2004; Smaoui, Brun, & Thomas- set, 2006) yields the possibility of reaching high accuracy posi- tioning for a pneumatic actuator. However, due to uncertainties, robust controllers are necessary to ensure positioning with high precision. In this respect, sliding mode controllers are used for elec- tropneumatic actuators (Bouri & Thomasset, 2001; Paul, Mishra, & Radke, 1994; Smaoui, Brun, & Thomasset, 2005; Yang & Lilly, 2003). Sliding mode control is one of the best choices for control- ling perturbed systems with matched disturbances/uncertainties (Edwards & Spurgeon, 1998; Utkin, Guldner, & Shi, 1999). The price This work has been partially supported by CNRS through ‘‘Musclair’’ PEPS project. The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Warren E. Dixon under the direction of Editor Andrew R. Teel. E-mail addresses: [email protected] (Y. Shtessel), [email protected] (M. Taleb), [email protected] (F. Plestan). for achieving the robustness/insensitivity to these disturbances is control chattering (Boiko, 2008; Edwards & Spurgeon, 1998; Frid- man, 2001, 2002; Utkin et al., 1999). The traditional ways for re- ducing chattering are as follows: (a) Replacing the discontinuous control function by ‘‘saturation’’ or ‘‘sigmoid ones’’ (Burton & Zinober, 1986; Slotine & Li, 1991). This approach yields continuous control and chattering elimination. However, it constrains the sliding system’s trajectories not to the sliding surface but to its vicinity losing the robustness to the disturbances. (b) Using the higher order sliding mode control techniques (Djemaï, Barbot, & Busawon, 2008; Laghrouche, Plestan, & Glu- mineau, 2007; Laghrouche, Smaoui, Plestan, & Brun, 2006; Levant, 2003, 2005; Plestan, Glumineau, & Laghrouche, 2008; Shtessel, Shkolnikov, & Levant, 2007). This approach allows driving to zero the sliding variable and its consecutive deriva- tives in the presence of the disturbances/uncertainties increas- ing the accuracy of the sliding variable stabilization, and has still been successfully applied for the control of electropneu- matic actuators (Girin & Plestan, 2009; Laghrouche et al., 2006). However, the main challenge of high order sliding mode con- trollers is the use of high order time derivatives of the slid- ing variable. It is worth noting that some second order sliding mode control, the popular super-twisting algorithm (Levant, 1993) and gain-commuted controller (Plestan, Moulay, & Glu- mineau, 2010a) only require measurement of the sliding vari- able whereas the other second order sliding mode controllers also need the time derivative of the sliding variable. 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.02.024

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adaptive gain super twisting

Transcript of A novel adaptive gain super twisting sliding mode controller.pdf

Page 1: A novel adaptive gain super twisting sliding mode controller.pdf

Automatica 48 (2012) 759–769

Contents lists available at SciVerse ScienceDirect

Automatica

journal homepage: www.elsevier.com/locate/automatica

A novel adaptive-gain supertwisting sliding mode controller: Methodologyand application✩

Yuri Shtessel a, Mohammed Taleb b,c, Franck Plestan b

a The University of Alabama in Huntsville, Huntsville, USAb LUNAM Université, Ecole Centrale de Nantes, IRCCyN UMR CNRS 6597, Nantes, Francec Moulay Ismail University, Meknes, Morocco

a r t i c l e i n f o

Article history:Received 20 January 2011Received in revised form12 July 2011Accepted 28 September 2011Available online 20 March 2012

Keywords:Second order sliding modeAdaptive controlElectropneumatic actuator

a b s t r a c t

A novel super-twisting adaptive sliding mode control law is proposed for the control of anelectropneumatic actuator. The key-point of the paper is to consider that the bounds of uncertainties andperturbations are not known. Then, the proposed control approach consists in using dynamically adaptedcontrol gains that ensure the establishment, in a finite time, of a real second order sliding mode. Theimportant feature of the adaptation algorithm is in non-overestimating the values of the control gains.A formal proof of the finite time convergence of the closed-loop system is derived using the Lyapunovfunction technique. The efficiency of the controller is evaluated on an experimental set-up.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

During the past two decades, the development of high-performance linear/nonlinear controllers (Brun, Sesmat, Thomas-set, & Scavarda, 1999; Brun, Thomasset, & Bideaux, 2002; Chiang,Chen, & Tsou, 2005; Edge, 1997; Hamiti, Voda-Besançon, & Roux-Buisson, 1996; Kimura, Hara, Fujita, & Kagawa, 1997; Kyoungk-wan & Shinichi, 2005; Ming-Chang & Shy-I, 1995; Miyajima, Fujita,Sakaki, Kawashima, & Kagawa, 2007; Rao & Bone, 2006; Richard &Scavarda, 1996; Schultea & Hahn, 2004; Smaoui, Brun, & Thomas-set, 2006) yields the possibility of reaching high accuracy posi-tioning for a pneumatic actuator. However, due to uncertainties,robust controllers are necessary to ensure positioning with highprecision. In this respect, slidingmode controllers are used for elec-tropneumatic actuators (Bouri & Thomasset, 2001; Paul, Mishra,& Radke, 1994; Smaoui, Brun, & Thomasset, 2005; Yang & Lilly,2003). Sliding mode control is one of the best choices for control-ling perturbed systems with matched disturbances/uncertainties(Edwards & Spurgeon, 1998; Utkin, Guldner, & Shi, 1999). The price

✩ This work has been partially supported by CNRS through ‘‘Musclair’’ PEPSproject. The material in this paper was partially presented at the 18th IFACWorld Congress, August 28–September 2, 2011, Milano, Italy. This paper wasrecommended for publication in revised form by Associate Editor Warren E. Dixonunder the direction of Editor Andrew R. Teel.

E-mail addresses: [email protected] (Y. Shtessel),[email protected] (M. Taleb),[email protected] (F. Plestan).

0005-1098/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2012.02.024

for achieving the robustness/insensitivity to these disturbances iscontrol chattering (Boiko, 2008; Edwards & Spurgeon, 1998; Frid-man, 2001, 2002; Utkin et al., 1999). The traditional ways for re-ducing chattering are as follows:

(a) Replacing the discontinuous control function by ‘‘saturation’’or ‘‘sigmoid ones’’ (Burton & Zinober, 1986; Slotine & Li,1991). This approach yields continuous control and chatteringelimination. However, it constrains the sliding system’strajectories not to the sliding surface but to its vicinity losingthe robustness to the disturbances.

(b) Using the higher order sliding mode control techniques(Djemaï, Barbot, & Busawon, 2008; Laghrouche, Plestan, & Glu-mineau, 2007; Laghrouche, Smaoui, Plestan, & Brun, 2006;Levant, 2003, 2005; Plestan, Glumineau, & Laghrouche, 2008;Shtessel, Shkolnikov, & Levant, 2007). This approach allowsdriving to zero the sliding variable and its consecutive deriva-tives in the presence of the disturbances/uncertainties increas-ing the accuracy of the sliding variable stabilization, and hasstill been successfully applied for the control of electropneu-matic actuators (Girin & Plestan, 2009; Laghrouche et al., 2006).However, the main challenge of high order sliding mode con-trollers is the use of high order time derivatives of the slid-ing variable. It is worth noting that some second order slidingmode control, the popular super-twisting algorithm (Levant,1993) and gain-commuted controller (Plestan, Moulay, & Glu-mineau, 2010a) only require measurement of the sliding vari-able whereas the other second order sliding mode controllersalso need the time derivative of the sliding variable.

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(c) Using controllers with dynamical gains. Recently, adaptivesliding mode controllers have been proposed, the interestbeing the adaptation of the gain magnitude with respect touncertainty/perturbation effects. Then, a reduced gain induceslower chattering. In Plestan, Shtessel, Brégeault, and Poznyak(2010b), an adaptive (first order) sliding mode controller hasbeen proposed and has been evaluated for the control of anelectropneumatic actuator.

The main objective of this paper is to bring together two of theprevious chattering reduction approaches, gain adaptation andhigh order sliding mode control. The obtained controller, basedon the well-known super-twisting (Levant, 1993) second ordersliding mode algorithm, does not require any information on theboundaries of the disturbance and its gradient except for theirexistence. It will yield to the very first application to a real systemof such method.

The super-twisting control law (STW) is one of the mostpowerful second order continuous slidingmode control algorithmsthat handles a relative degree equal to one. It generates thecontinuous control function that drives the sliding variable andits derivative to zero in finite time in the presence of thesmooth matched disturbances with bounded gradient, when thisboundary is known. Since STW algorithm contains a discontinuousfunction under the integral, chattering is not eliminated butattenuated. The main disadvantage of STW control algorithmis that it requires the knowledge of the boundaries of thedisturbance gradient. In many practical cases this boundarycannot be easily estimated. The overestimating of the disturbanceboundary yields to larger than necessary control gains, whiledesigning the STW control law. The adaptive-gain STW (ASTW)control law, which handles the perturbed plant dynamics withthe additive disturbance/uncertainty of certain class with theunknown boundary, was proposed in Shtessel, Moreno, Plestan,Fridman, and Poznyak (2010). In this latter paper, a novel adaptiveSTW control law that continuously drives the sliding variable andits derivative to zero in the presence of the bounded disturbancewith the unknown boundary, has been proposed. The finiteconvergence time is estimated. The proof is based on recentlyproposed Lyapunov function (Moreno & Osorio, 2008; Polyakov& Poznyak, 2009). The current paper is extending the result ofShtessel et al. (2010) to a larger class of nonlinear uncertainsystems.

This paper is organized as follows. Section 2 gives themain methodological results concerning the novel adaptivesuper-twisting algorithm. Section 3 describes the experimentalset-up composed by two electropneumatic actuators. Then,Section 4 described the position controller of one of the bothelectropneumatic actuators (the second being controlled in forceby an another way), and displays obtained experimental results.

2. Novel adaptive supertwisting controller design

2.1. Problem statement

Consider a single-input uncertain nonlinear system

x = f (x)+ g(x)u (1)

where x ∈ X ⊂ Rn is a state vector (X is a compact set), u ∈ Ris a control function, f (x) ∈ Rn is a differentiable, partially knownvector-field. Assume that

• A1. A sliding variable σ = σ(x, t) ∈ R is designed so that thesystem’s (1) desirable compensated dynamics are achieved inthe sliding mode σ = σ(x, t) = 0.

• A2. The relative degree of system (1) with the sliding variableσ(x, t)with respect to u equals one, and the internal dynamicsare stable.

Therefore, the input–output dynamics can be presented

σ =∂σ

∂t+∂σ

∂xf (x)

a(x,t)

+∂σ

∂xg(x)u

b(x,t)

= a(x, t)+ b(x, t)u. (2)

Also, it is assumed that

• A3. The function b(x, t) ∈ R is uncertain and can be presentedas

b(x, t) = b0(x, t)+∆b(x, t) (3)

where b0(x, t) > 0 is a known function and ∆b(x, t) is abounded perturbation so that|∆b(x, t)|b0(x, t)

= γ (x, t) ≤ γ1 < 1

∀x ∈ Rn and t ∈ [0,∞)with an unknown boundary γ1.• A4. The function a(x, t) ∈ R is presented as

a(x, t) = a1(x, t)+ a2(x, t) (4)

with the bounded terms

|a1(x, t)| ≤ δ1|σ |1/2

|a2(x, t)| ≤ δ2(5)

where the finite boundaries δ1, δ2 > 0 exist but are not known.

Finally, one gets

σ = a(x, t)+

1 +

∆b(x, t)b0(x, t)

b1(x,t)

ω (6)

where ω = b0(x, t)u. From A3, one gets

• A5.

1 − γ1 ≤ b1(x, t) ≤ 1 + γ1. (7)

The problem is to drive the sliding variable σ and its derivativeσ to zero in finite time in the presence of the bounded additive (5)andmultiplicative (3) perturbationswith the unknown boundariesδ1, δ2, γ1 > 0 by means of continuous control without the controlgain overestimation.

The classical SMC and the second order sliding (2-sliding)mode controllers, including the continuous STWcontrol algorithm,can robustly handle such problem if the boundaries of theperturbations are known. The main disadvantage of the classicalSMC is in introducing control chattering, while SOSM controllersare able to attenuate it. In this work we are looking for anadaptive-gain STW (ASTW) algorithm that is able to address thisproblem via generating continuous control function (chatteringattenuation) so that its gains are adapted to the unknown additiveand multiplicative perturbations with the unknown boundariesand without the control gain overestimation.

2.2. Control structure

The following STW control is considered (Levant, 1993)

ω = −α|σ |1/2sign(σ )+ v

v = −β

2sign(σ )

(8)

where the adaptive gains

α = α(σ , σ , t)β = β(σ , σ , t)

(9)

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are to be defined. The solution of system (6) is understood in thesense of Filippov (Filippov, 1988). Assuming the uncertain functionb1(x, t) to be an uncertain piece-wise constant, control system (6)and (8) can be rewritten as

σ = −αb1(x, t)|σ |1/2sign(σ )+ ω∗ + a1(x, t)

ω∗ = −βb1(x, t)

2sign(σ )+ a2(x, t)+ b1(x, t)v (10)

ω∗(0) = 0

where ω∗ = a2 + b1(x, t)v. Next, for brevity, the notations b1 anda2 will be used for the terms b1(x, t) and a2(x, t). Let us assumethat the term b1v is bounded with unknown boundary δ3 > 0, i.e.

|b1v| ≤|b1|2

t

0βdτ ≤ δ3. (11)

It is demonstrated below that the adaptive gain β = β(σ , σ , t)is bounded with uncertain boundary β∗ > 0, i.e. |β| ≤ β∗. ThenEq. (11) becomes

|b1v| ≤12|b1|β∗t ≤ δ3. (12)

It is worth noting that Eq. (12) is valid on any finite time interval. Itis shown below that the dynamics of (10) are considered on a finitetime interval only. Finally, the boundary of the uncertain functionχ(x, t) = a2(x, t)+ b1v exists, but is unknown. This is

|χ(x, t)| ≤ δ2 + δ3 = δ4. (13)

The control design problem is reduced to designing ASTW control(8)–(9) that drives σ , σ → 0 given by (10) in finite time in thepresence of the bounded additive (5), (13) and multiplicative (3)perturbations with the unknown boundaries δ1, δ4, γ1 > 0.

The idea of designing ASTW is to dynamically increase the con-trol gains α(t) and β(t) until the 2-sliding mode establishes. Thenthe gains shall start reducing. This gain reduction shall be reversedas soon as the sliding variable or its derivative start deviating fromthe equilibrium point σ = σ = 0 in 2-sliding mode. Therefore, arule (a detector) that detects the beginning of a destruction of thesliding mode shall be constructed and incorporated in the ASTWcontrol law that allows not-overestimating the control gains α(t)and β(t). This ‘‘detector’’ is proposed to design by introducing adomain |σ | ≤ µ that is used as follows: as soon as this domain isreached the gains α(t) and β(t) start dynamically reducing untilthe system trajectories leave the domain. Then the gains start dy-namically increasing in order to force the trajectories back to thedomain in finite time.

Remark 1. A selection of the mentioned ‘‘detector’’ is not unique.In particular, since the super-twisting algorithm is a 2-SMCcontroller, the value of σ can be also taken into account, whiledynamically adapting the gains of the controller. For instance, thegains α(t) and β(t) can start decreasing if the condition |σ | +

c|σ | ≤ µ1 or σ 2+ cσ 2

≤ µ2, c > 0, µ1 > 0, µ2 > 0, is satisfied.Such choice of the ‘‘detector’’ could yield a smaller domain of theconvergence and will be analyzed in a future work. �

2.3. Main results

The main result of the paper is formulated in the followingtheorem.

Theorem 1. Consider system (10). Suppose that the functionsa1(x, t), a2(x, t) and b1(x, t) satisfy Assumptions A3 and A5 forsome unknown gains δ1, δ2, γ1 > 0. Then, for any initial conditions

x(0), σ (0), there exist a finite time 0 < tF and a parameter µ (as soonas the condition, δ4 being defined as δ4 = δ2 + δ3,

α >δ1

λ+ 4ε2

− ε (4δ4 + 1)

λ (1 − γ1)+

2εδ1 − 2δ4 − λ− 4ε2

212ελ (1 − γ1)

holds, if |σ(0)| > µ) so that a real 2-sliding mode, i.e.|σ | ≤ η1and |σ | ≤ η2, is established ∀t ≥ tF via ASTW control (8) with theadaptive gains (α(0) > αm)

α =

ω1

γ1

2sign(|σ | − µ), if α > αm

η, if α ≤ αm

β = 2εα

(14)

where ε, λ, γ1, ω1, η are arbitrary positive constants, and η1 ≥ µ,η2 > 0. The parameter αm is an arbitrary small positive constant. �

Proof. The proof is split into two steps. In the first step, wewill present system (10) in a form convenient for the Lyapunovanalysis. In order to do this a new state vector is introduced

z = [z1 z2]T =|σ |

1/2sign(σ ) ω∗

T(15)

and system (10) can be rewritten as

z1 =1

2|z1|(−αb1z1 + z2 + a1(x, t))

z2 = −βb12|z1|

z1 + χ(x, t)(16)

where χ(x, t) = a2(x, t)+ b1(x, t)v. Eq. (16) can be rewritten in avector-matrix formatz1z2

=

12|z1|

−αb1 1−βb1 0

A(z1)

z1z2

+1

2|z1|

1 00 2|z1|

G(z1)

a1(x, t)χ(x, t)

. (17)

Due to Assumption A4 and (13) we can write

a1(x, t) = ρ1(x, t)|σ |1/2sign(σ ) = ρ1(x, t)z1

χ(x, t) =ρ2(x, t)

2sign(σ ) =

ρ2(x, t)2

z1|z1|

(18)

where ρ1(x, y), ρ2(x, t) are some bounded functions so that

0 < ρ1(x, t) ≤ δ1, 0 < ρ2(x, t) ≤ 2δ4.

Eq. (17) can be rewritten in view of Eq. (18)z1z2

= A(z1)

z1z2

(19)

where

A(z1) =1

2|z1|

− (αb1 − ρ1(x, t)) 1− (βb1 − ρ2(x, t)) 0

.

It can be observed that

(a) if z1, z2 → 0 in finite time then σ , σ → 0 in finite time;(b) |z1| = |σ |

1/2 and sign(z1) = sign(σ ).

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In the second step of the proof, the stability analysis of system(17) is performed. In order to do it the following Lyapunov functioncandidate is introduced

V (z1, z2, α, β) = V0 +1

2γ1

α − α∗

2+

12γ2

β − β∗

2 (20)

where (with λ > 0, ε > 0)

V0(z) =λ+ 4ε2

z21 + z22 − 4εz1z2 = zTPz

P =

λ+ 4ε2 −2ε

−2ε 1

(21)

and α∗ > 0, β∗ > 0 are some constants. It is worth noting that thematrix P is positive definite if λ > 0 and ε are any real number. Thederivative of the Lyapunov function candidate (20) is presented

V (z, α, β) = zTAT (z1)P + PA(z1)

z +

1γ1

α − α∗

α

+1γ2

β − β∗

β. (22)

The first term of (22) is computed taking into account (17) and (19)

V0 = zTAT (z1)P + PA(z1)

z ≤ −

12|z1|

zT Q z. (23)

The symmetric matrix Q is computed taking into account (5) and(18)

Q =

Q11 Q12

Q21 4ε

(24)

with

Q11 = 2λαb1 + 4εb1(2εα − β)

− 2(λ+ 4ε2)ρ1(x, t)+ 4ερ2(x, t)

Q12 = Q21 =βb1 − 2εαb1 − λ− 4ε2

+ 2ερ1(x, t)− ρ2(x, t).

In order to guarantee the positive definiteness of the matrix Q , weenforce

β = 2εα. (25)

The matrix Q will be positive definite with a minimal eigenvalueλmin(Q ) ≥ 2ε if

α >δ1

λ+ 4ε2

− ε (4δ4 + 1)

λ (1 − γ1)+

2εδ1 − 2δ4 − λ− 4ε2

212ελ (1 − γ1)

.

(26)

In view of (23) and assuming that Eqs. (25)–(26) hold, it is easy toshow that

V0 ≤ −rV 1/20 (27)

where

r =ελ

1/2min(P)

λmax(P). (28)

Indeed, since

V0(z) ≤ −1

2|z1|zT Q z ≤ −

2ε2|z1|

zT z = −ε

|z1|∥z∥2 (29)

and

λmin(P)∥z∥2≤ zTPz ≤ λmax(P)∥z∥2 (30)

where ∥z∥2= z21 + z22 = |σ | + z22 and

|z1| = |σ |1/2

≤ ∥z∥ ≤V 1/20 (z)

λ1/2min{P}

(31)

then

V0(z) ≤ −rV 1/20 , r =

ελ1/2min(P)

λmax(P). (32)

Now, in view of Eqs. (22) and (27) can be rewritten (with εα =

α − α∗ and εβ = β − β∗)

V (z, α, β) = zTPz + zTPz +1γ1εαα +

1γ2εβ β

≤ −1

|z1|zT Q z +

1γ1εαα +

1γ2εβ β

≤ −rV 1/20 +

1γ1εαα +

1γ2εβ β

= −rV 1/20 −

ω1√2γ1

|εα| −ω2

√2γ2

|εβ |

+1γ1εαα +

1γ2εβ β +

ω1√2γ1

|εα| +ω2

√2γ2

|εβ |. (33)

Taking into account a well-known inequalityx2 + y2 + z2

1/2≤ |x| + |y| + |z|

and in view of Eq. (20), we can derive

− rV 1/20 −

ω1√2γ1

|εα| −ω2

√2γ2

|εβ | ≤ −η0V (z, α, β) (34)

with η0 = min(r, ω1, ω2). Taking into account Eq. (34), we canrewrite Eq. (33) as

V (z1, z2, α, β) ≤ −η0V (z1, z2, α, β)+

1γ1εαα

+1γ2εβ β +

ω1√2γ1

|εα| +ω2

√2γ2

|εβ |. (35)

Now, we assume that the adaptation law (14) makes the adaptivegainsα(t) andβ(t)bounded (this assumptionwill be proven later).Then there exist positive constants α∗, β∗ such that α(t)−α∗ < 0and β(t) − β∗ < 0, ∀t ≥ 0. In view of the above assumption,Eq. (35) can be reduced to the following

V (z1, z2, α, β) ≤ −η0 [V (z1, z2, α, β)]1/2

− |εα|

1γ1α −

ω1√2γ1

εβ 1γ2β −

ω2√2γ2

. (36)

It gives

V (z1, z2, α, β) ≤ −η0 [V (z1, z2, α, β)]1/2 + ξ (37)

with

ξ = − |εα|

1γ1α −

ω1√2γ1

εβ 1γ2β −

ω2√2γ2

. (38)

C1. Suppose that |σ | > µ and α(t) > αm for all t ≥ 0. Then, inview of (14)

α = ω1

γ1

2(39)

and

ξ = −|εβ |

1γ2β −

ω2√2γ2

. (40)

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Y. Shtessel et al. / Automatica 48 (2012) 759–769 763

After selecting ε =ω22ω1

γ2γ1

and differentiating (25) we obtain

β = 2εα → β = 2εα → β

= εω12γ1 → β = ω2

γ2

2. (41)

In view of Eq. (41) the term ξ in (40) becomes ξ = 0 and

V (z1, z2, α, β) ≤ −η0 [V (z1, z2, α, β)]1/2 . (42)

It is worth noting that for the finite time convergence α(t)must satisfy inequality (26). It means that α(t) shall increasein accordance with (39) until (26) is met that guarantees thepositive definiteness of the matrix Q and validity of (42). Afterthat the finite time tF convergence to the domain |σ | ≤ µ isguaranteed according to (42).

C2. Suppose that |σ | < µ thenα(t) is reducing in accordancewith(14) that takes a form

α =

−ω1

γ1

2, if α > αm

η, if α ≤ αm

(43)

and the term

ξ =

2|α − α∗

|ω1

√2γ1

, if α > αm

−|αm − α∗+ η · t|

η

γ1−

ω1√2γ1

, if α ≤ αm

(44)

becomes (or can be) positive (one recalls that αm is a smallparameter). It is worth noting that second equation in (44) isvalid only for finite time, since as soon as α becomes less orequal to αm, its value immediately starts increasing such thatα = αm + η · t . Then, the first equation in (44) becomes valid.

In view of (44) the derivative of the Lyapunov function (37)becomes sign indefinite and |σ |may become larger thanµ dueto decrease of the control gains α(t) and β(t).

As soon as |σ | becomes greater than µ the condition thatdefines C1 case holds so that σ reaches the domain |σ | ≤ µ againin finite time, and so on. Therefore, during the adaptation processthe sliding variable σ reaches the domain |σ | ≤ µ in finite timethen may leave this domain for a finite time, and it is guaranteedthat it always stays in a larger domain |σ | ≤ η1, η1 > µ in a realsliding mode.

Inside the domain |σ | ≤ µ, the value |σ | can be estimated inaccordance with (10), (14) and (25)

|σ | ≤ ((1 − γ1)α(t1)+ δ1) µ1/2

+ [ε(1 − γ1)α(t1)+ δ4] (t2 − t1)= η2 (45)

where t1 and t2 are the time instants when σ(t) enters the domain|σ | ≤ µ and leaves this domain respectfully.When |σ(t)| becomesµ < |σ | ≤ η1 then

|σ | ≤ (1 + γ1)η1/21 + ε

α(t2)+ ω1

η1γ1

2

(t3 − t2)

+ δ1η1/21 + δ4(t3 − t2) = η2 (46)

where t2 and t3 (t3 > t2) are the time instantswhen σ(t) leaves thedomain |σ | ≤ µ and enters this domain afterwards respectfully.Combining the conditions (45) and (46) we obtain

|σ | ≤ max(η2, η2) = η2. (47)

Note that Eqs. (45) and (46) prove only the existence of the realsliding mode domain

W = {σ , σ : |σ | ≤ η1, |σ | ≤ η2, η1 > µ} (48)

since it is practically impossible to calculate the values η1, η2.Theorem 1 is proven. �

Now we can prove the assumption about boundedness of α(t)and β(t).

Proposition 1. The adaptive gains α(t) and β(t) are bounded. �

Proof. In the domain µ < |σ | ≤ η1, a solution to (14) can beconstructed as

α = α(0)+ ω1

γ1

2· t, 0 ≤ t ≤ tF . (49)

Therefore α(t) is bounded. The adaptive gain β(t) is also bounded,since β(t) = 2εα(t). Inside the domain |σ | ≤ µ the control gainsα(t) andβ(t) are decreasing. Therefore, the gainsα(t) andβ(t) arebounded in the real 2-sliding mode. Proposition 1 is proven. �

Now we can easily estimate finite reaching time.

Proposition 2. As soon as inequality (26) is fulfilled in finite time t0,the adaptive-gain STW control law (14) drives the sliding variable σfrom the initial condition |σ(t0)| ≥ µ and its derivative to the domainW = {σ , σ : |σ | ≤ η1, |σ | ≤ η2, η1 > µ} in finite time that isestimated as

tF ≤2V 1/2(t0)η0

(50)

where η0 = min(r, ω1, ω2). �

Proof. Inequality (26) is fulfilled in finite time, since its right handside is bounded and the adaptive gain α(t) is increasing linearlywith respect to time in accordance with (14). Assuming µ = 0implies σ , σ → 0 in finite time tr that can be estimated by tr ≤

2V1/2(t0)η0

, which is obtained by a direct integration of inequality (37)with ξ = 0. For µ > 0, σ , σ → W (σ , σ ) in finite time tF ≤ tr .Proposition 2 is proven. �

It is worth noting that if the ‘‘detector’’ for adaptive gainreduction, the term sign(|σ | − µ) in the gain adaptation law (14),is eliminated (by making µ = 0), then the adaptive gain law (14)shall be changed to (with α(0) > 0)

α =

ω1

γ1

2if σ = 0

0, if σ = 0

β = 2εα.

(51)

In this case the ASTW control law will drive the system’s (1)trajectory to the ideal 2-sliding mode, i.e. σ = σ = 0 in finitetime. However, the adaptive control gains α(t) and β(t) can beoverestimated. This result is formulated in the following corollary.

Corollary 1. Consider system (10). Suppose that the functionsa1(x, t), a2(x, t) and b1(x, t) satisfy Assumptions A3 and A5 forsome unknown gains δ1, δ2, γ1 > 0. Then for any initial conditionsx(0), σ (0), the second order sliding mode, i.e.σ = σ = 0, willbe established in finite time via STW control (8) with the adaptivegains (51). �

The Propositions 1 and 2 are obviously valid for the case ofASTW control in (8) and (51).

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764 Y. Shtessel et al. / Automatica 48 (2012) 759–769

Fig. 1. Photo of the electropneumatic system—On the left hand side is the‘‘main’’ actuator whose its position is controlled. On the right hand side is the‘‘perturbation’’ actuator whose the load force is controlled.

3. Electropneumatic system

3.1. Description

The electropneumatic system (see Figs. 1–2) is composed bytwo actuators. The first one, named the ‘‘main’’ one (left handside), is a double acting electropneumatic actuator controlled bytwo servodistributors (Fig. 2) and is composed by two chambersdenoted P and N . Piston diameter is 80 mm and rod diameter25 mm. With a source pressure equal to 7 bar, the maximumforce developed by the actuator is 2720 N . The air mass flowrates qm entering in the chambers are modulated by two three-way servodistributors. The pneumatic jack horizontally moves aload carriage of mass M . This carriage is coupled to the secondelectropneumatic actuator, the so-called ‘‘perturbation’’ one. Aspreviously mentioned, the goal of this latter is to produce adynamical load force on the main actuator. The actuator has thesame mechanical characteristics than the main one, but the airmass flow rate is modulated by a single five-way servodistributor.In the sequel of the paper, only the control of the ‘‘main’’actuator position is considered; note that the force control ofthe ‘‘perturbation’’ actuator is currently made by an analogic PIDcontroller developed by the test bench constructor. In conclusion,the aim of this test bench is to evaluate performances of positioncontroller with respect to unknown dynamical perturbation force.

The experimental test bench is simulated with a fluid powersystems dedicated software AMESim (LMS SA Co.), and thecontrol law is developed under Matlab/Simulink (The MathworksCo.). It implies a cosimulation program (Figs. 3 and 4) throughlinks between AMESim and Matlab/Simulink. In Fig. 4, the block‘‘AMESim Model’’ makes the link between Matlab simulationand Amesim simulator described by Fig. 3. A consequence ofthe cosimulation is that two models are used: a ‘‘simulation’’model simulated by AMESim, and a ‘‘control’’ model simulated byMatlab/Simulink. It can be summarized as follows

• the ‘‘simulation’’ model takes into account physical phenomenaas temperature variations, experimental values of mass flowrate delivered by each servodistributors, dynamics of servodis-tributors, dry friction..., and is developed under Amesim. Theperturbation force is viewed as an input.

• the ‘‘control’’ model is simpler than the previous (for example,mass flow rates models are written as pressures polynomials(Sesmat & Scavarda, 1996)) and issued in order to designthe nonlinear position controller under Matlab/Simulink. Theperturbation force is supposed unknown.

Matlab/Simulink allows to use a DS1104 board (dSpace Co.) onwhich the control law is implemented. In the sequel, the experi-mental results have been obtained with a 1 ms sample time.

3.2. Simulation model

A standard pneumatic actuator is equipped by a pneumaticdamper in order to protect the piston: this protection avoidshigh clashes between the piston and the external structure of theactuator. The damper is composed by a restrictionwhich limits theexhaust mass flow rate. In order to obtain maximum performance,this restriction has been deleted. It implies that, in a first step, thecontrol law has to be evaluated on cosimulation. The cosimulationis using the ‘‘simulation’’ model developedwith AMESim software,thismodel trying to be as close as possible to the physical behavior.

Servodistributor model. The servodistributor model is composedin two parts, a dynamic part and a static one:

• Dynamic part is modelized by a second order transfer functionidentified from experimental measure

F(s) =ω2

ns

s2 + 2 · ζs · ωnss + ω2ns

(52)

with ωns = 246 rad s−1 and ζs = 0.707.• Static part is modelized by an experimental table in which

mass flow rate is given in function of ratio pressure (upstream/downstream) and control voltage (Sesmat & Scavarda, 1996).

Pneumatic chamber variable volume model. Each chamber ofthe pneumatic actuator is considered as a variable volume, inwhich the air mass evolves with time. State the classical followingassumptions (Shearer, 1956):

• H1. Air is perfect gas and its kinetic inconsequential.• H2. The pressure and the temperature are homogeneous in each

chamber.• H3. The mass flow is pseudo-stationary.

The first dynamic principle applied to the air mass and thethermodynamic evolution of air in each chamber read as (with X =

P or N) (Shearer, 1956)

dpXdt

= −γpXVX

dVX

dt+γ rTrVX

qmXin −γ rTXVX

qmXout

+(γ − 1)

VX

dQX

dtdTXdt

= −(γ − 1)TXVX

dVX

dt+

rTXpXVX

(γ Tr − TX )qmXin

−rT 2

X

pXVX(γ − 1)qmXout + (γ − 1)

TXpXVX

dQX

dt

(53)

with γ the adiabatic constant, Tr the temperature inside theupstream tank, qmXin the mass flow rate brought inside the Xchamber, and qmXout the mass flow rate brought outside the Xchamber. QX , the thermal exchange with the X chamber wall, isdescribed by Assumption A4.

• H4. The thermal exchange is due only by conduction describedby

dQX

dt= λScX (TcX − TX ) (54)

with λ the thermal exchange coefficient by conduction, ScX thetotal area inside a X chamber, and TcX the temperature of the Xchamber wall.

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Y. Shtessel et al. / Automatica 48 (2012) 759–769 765

Fig. 2. Scheme of electropneumatic system—This figure displays the mechanical and software structures. The software structure is based on a dSpace board on which theposition controller of the ‘‘main’’ actuator is implemented. The mechanical structure is composed by two actuators, the ‘‘main’’ one (left hand side) and the ‘‘perturbation’’one (right hand side).

Fig. 3. AMESim model for cosimulation.

Fig. 4. MATLAB/Simulink control law for cosimulation.

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766 Y. Shtessel et al. / Automatica 48 (2012) 759–769

Mechanical model. The second Newton law gives

dvdt

=1M

S (pP − pN)− Ff − bvv − F

dydt

= v

(55)

with friction force Ff including stiction, Coulomb and Stribeckphenomena.

Samplers and saturation. Samplers are added in AMESim’smodelin order to take into account samplers of acquisition card; sampletime is 1 ms which is very smaller than the natural frequency ofthis electropneumatic system. So it is not necessary to discretizethe model all the control law are synthesize in continuous time.Saturation signal control are added, i.e. |usat | = 10 V.

3.3. Control model

This model is developed in order to design the control lawin order to obtain a simplest version which allows the design ofcontrol law. The following hypotheses are added

• H5. The process is polytropic and characterized by coefficient k(with 1 < k < γ ).

• H6. The leakage between system and atmosphere are neglected.• H7. The temperature variations in each chamber are incon-

sequential with respect to the supply temperature, i.e. TP =

TN = T .

Therefore, pressures dynamics reads as

dpXdt

= −kpXVX

dVX

dt+

krTVX

qmXin − qmXout

. (56)

• H8. The leakages between the two chambered, and between theservodistributor and the jack are negligible.

• H9. Supply and exhaust pressure are supposed to be constant.

By defining qm(uX , pX ) := qmXin − qmXout , one gets

dpPdt

= −kpP

VP(y)dVP(y)

dt+

krTVP

qm(uP , pP)

dpNdt

= −kpN

VN(y)dVN(y)

dt+

krTVN

qm(uN , pN).(57)

• H10. All dry frictions forces are neglected.1

• H11. There is no control signal saturation.• H12. Dynamic part of servodistributor is neglected.2.• H13. Static part of servodistributor depends on pressures and

control value

qm(uX , pX ) = ϕ (pX )+ ψ (pX , sign (uX )) uX

with ϕ andψ5th-order polynomials with respect to pX (Sesmat& Scavarda, 1996) and issued from experimental measures.

• H14. Only the position of the actuator is controlled, whichmeans that the problem is a single input–single output (SISO).It implies that uP = −uN = u.

1 The viscous friction forces have been identified on real system: it has beenestablished that the carriage presents such frictions bvv with bv = 30.2 This hypothesis has been made given that the servodistributor dynamics are

much faster than the mechanical part of the system and are considered as singularperturbed unmodeled dynamics (Fridman, 2003; Soto-Cota, Fridman, Loukianov, &Canedo, 2006)

With VP(y) = V0 + S · y and VN(y) = V0 − S · y (V0 being equalto the half of the cylinder volume), the model used for the designof controller is a nonlinear system and reads as

pP =krTVP(y)

ϕP + ψP · u −

SrT

pPv

pN =krT

VN(y)

ϕN − ψN · u +

SrT

pNv

v =1M

[S (pP − pN)− bvv − F ]

y = v

(58)

with F the unknown perturbation force, ϕP = ϕ(pP), ϕN = ϕ(pN),

ψP = ψ (pP , sign (u)) ,ψN = ψ (pN , sign (−u)) .

It is obvious that the system (58) reads as the nonlinear system(1) with x = [pP pN v y]T ,

f (x) =

krTVP(y)

ϕP −

SrT

pPv

krTVN(y)

ϕN +

SrT

pNv

1M

[S (pP − pN)− bvv − F ]

v

,

g(x) =

krTVP(y)

ψPkrT

VN(y)ψN 0 0

T

.

3.4. Control problem formulation

In system (58), the vector-fields f (x) and g(x) are partiallyknown. Furthermore, the perturbation terms F and bvv areunknown and bounded with unknown bounds, whereas thefunctions ϕP , ϕN , ψP and ψN give only an estimation of the massflow rate, the ‘‘quality’’ of this estimation being evaluated withdifficulty. The lack of knowledge of the perturbation/uncertaintiesbounds makes it difficult designing a robust sliding modecontroller without overestimating the control gains. The problemis then to design a robust continuous adaptive-gain super-twistingsliding mode controller u that drives the output (the position y)of the electropneumatic actuator to follow a prescribed profile(see Fig. 5—dotted line) in the presence of bounded externalperturbations (see external forces F— Fig. 9) with unknownbounds.

4. ASTW slidingmode control of an electropneumatic actuator:design and experimental results

The aim of the control law is to get a good accuracy in termof position tracking for the desired trajectory displayed in Fig. 5(dotted line) in spite of the parametric uncertainties given above;furthermore, the dry frictions have not been taken into account inthe controller design. Following the previous section, consider thesliding variable

σ = ω2n(y − yd(t))+ 2ζωn(y − yd(t))+ (y − yd(t)) (59)

with ζ = 0.7 and ωn = 50 rad s−1. As system (58)–(59) admits arelative degree equal to 1 for σ with respect to u, one gets

σ = a(·)+ b(·) · u. (60)

From Laghrouche, Smaoui, Brun, and Plestan (2004), functions aand b are bounded for x ∈ X. Furthermore, the function b fulfills

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Y. Shtessel et al. / Automatica 48 (2012) 759–769 767

Fig. 5. External force—1000 N. Measured (solid line) and desired (dotted line)actuator position y (m) versus time (s).

Assumption A3 for x ∈ X. Furthermore, the function a can bewritten as a = a0 +∆a. It yields that the control law reads as

u =1b0(−a0 + ω) (61)

with control input ω defined by (8)

ω = −α|σ |1/2sign(σ )+ v

v = −β

2sign(σ ),

(62)

the gains α and β being defined by (14)

α =

ω1

γ1

2sign(|σ | − µ), if α > αm

η, if α ≤ αm

β = 2εα.

(63)

Parameters of the controller have been tuned as (this tuninghas been made in order to get the best behavior and highperformances)

ε = 1, γ1 = 2, ω1 = 200, µ = 0.7,αm = 0.01, η = αm.

Several experimental tests have beenmade in order to highlightthe properties of this class of controllers: then, experimentationshave beenwith two differentmagnitudes of perturbations in orderto show the adaptation of the gain with respect to it. Secondly,experimental tests have beenwith a constant gain,whose the valuehas been sufficiently large in order to counteract the worst case ofuncertainties/perturbations: it yields to an overestimation of thegain.

Figs. 5–6 display desired and current positions, and the positiontracking error for an external 1000 N force, respectively, and showthe effectiveness of the adaptive-gain STW controller given thatthe trajectory tracking is accurate in spite of bounded additive andmultiplicative uncertainties with unknown bounds.

The time history of the adaptive gain α(t) is shown in Fig. 7,where the reduction of the gain during is demonstrated whereasthe tracking is accurate in spite of the external force F (see Fig. 9).With a lower magnitude of the perturbation force (500 N), Fig. 11shows that the gain is lower: there is really an adaptation of thegainmagnitudewith respect to the perturbationmagnitude. Fig. 10also shows that the tracking error is less and has smaller transients.

Fig. 8 displays the control input which is far from saturation.Note that, in case of non adaptive gain, the input is larger: in fact,if all the uncertainties/perturbations are taken into account in the‘‘worst’’ case, the gain has to be strongly over-estimated and hasbeen tuned at 4000. Fig. 12 displays the control input: it is clearthat the control magnitude is larger.

Fig. 6. External force—1000 N. Position tracking error (m) versus time (s).

Fig. 7. External force—1000 N. Gain α(t) versus time (s).

Fig. 8. External force—1000 N. Control input u(t) (V ) versus time (s).

Fig. 9. External perturbation force (N) versus time (s).

5. Conclusions

A novel adaptive-gain real super-twisting (ASTW) slidingmode controller is proposed. The both drift uncertain termand multiplicative perturbation are assumed to be boundedwith unknown boundaries. The proposed Lyapunov-based ASTWcontroller design dynamically adapted control gain that ensures

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768 Y. Shtessel et al. / Automatica 48 (2012) 759–769

Fig. 10. External force—500 N. Position tracking error (m) versus time (s).

Fig. 11. External force—500 N. Gain α(t) versus time (s).

Fig. 12. External force—1000 N. Constant gain—α = 4000. Control input u(t) (V )versus time (s).

the establishment, in a finite time, of a real second order sliding(2-sliding) mode. The proposed ASTW sliding mode controldoes not overestimate the values of the control gains. Finiteconvergence time is estimated. The efficacy of the proposed ASTWsliding mode control algorithm is confirmed via its application toposition control of an electropneumatic actuator.

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Yuri Shtessel received the M.S. and Ph.D. degrees in Elec-trical Engineering with focus on Automatic Control fromthe Chelyabinsk State Technical University, Chelyabinsk,Russia in 1971 and 1978, respectively. Since 1993, hehas been with the Electrical and Computer Engineer-ing Department, The University of Alabama in Huntsville,where his present position is Professor. His research in-terests include sliding mode control and observation withapplications to electric power system, aerospace vehiclecontrol and blood glucose regulation. He published morethan 300 technical papers. Dr. Shtessel is a recipient of the

IEEE Third Millennium Medal for the outstanding contributions to control systemsengineering, 2000. He holds the ranks of Associate Fellow of AIAA and Senior Mem-ber of IEEE.

Mohammed Taleb received his Diploma in Engineeringin Electromechanical Engineering from ENSAM, MoulayIsmail University, Meknes, Morocco in 2003, and hisMaster’s degree in automatic control from the EcoleCentrale deNantes, Nantes, France, in 2010. He is currentlyworking toward the Ph.D. degree in the Institut deRecherche en Communication et CybernŽtique de Nantes(IRCCyN), Ecole Centrale de Nantes, France. His researchinterests include robust nonlinear control and adaptivehigher order sliding mode and their applications toelectropneumatic actuators.

Franck Plestan received the Ph.D. in Automatic Controlfrom the Ecole Centrale de Nantes, France, in 1995. FromSeptember 1996 to August 2000, he was with LouisPasteur University, Strasbourg, France. In September 2000,he joined the Ecole Centrale de Nantes, Nantes, Francewhere he is currently Professor. His research interestsinclude robust nonlinear control (adaptive/higher ordersliding mode), theoretical aspects of nonlinear observerdesign and control of electromechanical and mechanicalsystems (pneumatic actuators, biped robots, electricalmotors).