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Non-linear adaptive sliding mode switching controlwith average dwell-timeLei Yu a , Maoqing Zhang a & Shumin Fei ba School of Mechanical and Electrical Engineering, Soochow University, Suzhou 215021,Chinab School of Automation, Southeast University, Nanjing 210096, China

Version of record first published: 28 Jul 2011.

To cite this article: Lei Yu, Maoqing Zhang & Shumin Fei (2011): Non-linear adaptive sliding mode switching control withaverage dwell-time, International Journal of Systems Science, DOI:10.1080/00207721.2011.604739

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International Journal of Systems Science2011, 1–8, iFirst

Non-linear adaptive sliding mode switching control with average dwell-time

Lei Yua*, Maoqing Zhanga and Shumin Feib

aSchool of Mechanical and Electrical Engineering, Soochow University, Suzhou 215021, China;bSchool of Automation, Southeast University, Nanjing 210096, China

(Received 17 September 2010; final version received 1 July 2011)

In this article, an adaptive integral sliding mode control scheme is addressed for switched non-linear systems inthe presence of model uncertainties and external disturbances. The control law includes two parts: a slide modecontroller for the reduced model of the plant and a compensation controller to deal with the non-linear systemswith parameter uncertainties. The adaptive updated laws have been derived from the switched multiple Lyapunovfunction method, also an admissible switching signal with average dwell-time technique is given. The simplicity ofthe proposed control scheme facilitates its implementation and the overall control scheme guarantees the globalasymptotic stability in the Lyapunov sense such that the sliding surface of the control system is well reached.Simulation results are presented to demonstrate the effectiveness and the feasibility of the proposed approach.

Keywords: adaptive integral sliding mode control; switched multiple Lyapunov function; average dwell-time

1. Introduction

The control of switched non-linear systems which arise

in many practical processes is an important topic in the

field of control (Branicky 1998; Liberzon 2003; Han,

Ge, and Lee 2009; Wu 2009; Yang, Cocquempot, and

Jiang 2009; Yu, Fei, and Li 2010; Zhang and Shi 2011).

Typically, a switched system consists of a number of

subsystems and a switching law (or signal), which

defines a specific subsystem being activated during a

certain interval of time. Many significant approaches

have been developed to deal with this control problem

in the past few years.On the other hand, sliding mode control is a well-

known robust control technique against parameter

uncertainties, external disturbances and unmodelled

dynamics (Roh and Oh 2000; Chen 2006; Guan and

Pan 2008). However, it has some disadvantages such as

chattering or high frequency oscillation in practical

applications. Adaptive control is an effective approach

to handle parameter variations (Ruan, Yang, Wang,

and Li 2006; Labiod and Guerra 2007; Liu, Tong, and

Li 2010). Known as a valid method to overcome

system uncertainties, especially uncertainties derived

from uncertain parameters, adaptive control schemes

are used to automatically adjust the response of the

controller to compensate for changes in the response of

the plant. Therefore, adaptive sliding mode control has

the advantages of combining the robustness of variable

structure methods with the control capability ofadaptive control. Recently, the adaptive sliding modecontroller designed for linear and non-linear systemshas been studied by some researchers (Chen 2006;Guan and Pan 2008). Chen (2006) has addressed anadaptive sliding controller for an electro-hydraulicsystem driven by a double-rod actuator with non-linearuncertain parameters. Also, an adaptive sliding modetracking controller is designed for a general class ofdiscrete-time MIMO systems with unknown parame-ters and disturbances (Guan and Pan 2008).

However, the adaptive sliding mode controlmethod applying to switched non-linear systems isreally very few. In this article, we will develop theadaptive integral sliding mode control scheme for aclass of switched non-linear systems with modeluncertainties and external disturbance. The mainfeatures of this study include: (1) we have designedthe adaptive sliding mode control scheme for simplicityusing some concise definitions, which is suitable forpractical implementation; (2) by designing an adaptiveintegral sliding mode compensation controller, a novel-type multiple switched Lyapunov function is devel-oped; (3) the update laws derived from the switchedmultiple Lyapunov function method can ensure theboundedness of parameter estimates. The asymptoti-cally stable adaptive controller together with aneffective adaptation law designed in this article cancompensate for the external disturbance of switched

*Corresponding author. Email: slender2008@gmail.com

ISSN 0020–7721 print/ISSN 1464–5319 online

� 2011 Taylor & Francis

DOI: 10.1080/00207721.2011.604739

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non-linear systems, and especially for the non-linearuncertain parameters.

The outline of this article is as follows. In Section 2,problem formulation and the detailed model arepresented. In Section 3, the proposed adaptive integralsliding mode compensation control strategy is givenfor the uncertain non-linear switched systems. InSection 4, the experiment is set up and results arediscussed using two examples. Finally, a conclusion isthen followed in Section 5.

2. Problem formulation

Consider a class of uncertain switched non-linearsystems as follows:

x1�¼ x2 þ !�ðtÞ,1ðxÞ � �

x2�¼ x3 þ !�ðtÞ,2ðxÞ � �

� � �

x�

n�1¼ xn þ !�ðtÞ,n�1ðxÞ � �

xn�¼ f�ðtÞðxÞ þ g�ðtÞðxÞ � uþ !�ðtÞ,nðxÞ � �þ d�ðtÞðtÞ

8>>>>>>>><>>>>>>>>:

ð1Þ

where t � 0, x ¼ ðx1, x2, . . . , xnÞT2 Rn denotes the

state vector of the systems, which is available; u 2 Rand y 2 R are input and output, respectively; �ðtÞ :½0,þ1Þ is the piecewise constant switching signaltaking value from the finite index set � ¼

deff1, 2, . . . , pg.

If �ðtÞ ¼ i, then we say that the ith subsystem is activeand the remaining subsystems are inactive.d ðtÞ 2 L2½0,1Þ is the external disturbance; � ¼ð�1, �2, . . . , �nÞ

T2 Rn is the given uncertain constant

parameter and fiðxÞ, giðxÞ,!iðxÞ ði 2 �Þ are allsmooth functions.

The control problem is to guarantee that theresulting closed-loop system is asymptotically stablein the Lyapunov sense and the sliding surface of thecontrol system is reached. The following assumptionsare made on system (1).

Assumption 2.1: The plant order n, fiðxÞ, giðxÞ and!iðxÞ are known.

Assumption 2.2: The external disturbance d(t) isbounded, assuming that the upper bound of the distur-bance d(t) is D, i.e. jd ðtÞj � D, where D4 0 is given.

3. Design of adaptive integral sliding mode

compensation controller

To ensure the stability of the closed-loop system andthe reachability of the sliding mode surface, we willdesign the control law which is composed of two

parts: one slide mode controller us for the reduced

model of the plant and the other controller, as a

compensator, uc to deal with the non-linear systems

with parameter uncertainties. So we define the control

law as follows:

u ¼ us þ uc ð2Þ

Substituting (2) into (1) we have:

xn�¼ f�ðtÞðxÞ þ g�ðtÞðxÞ � us þ g�ðtÞðxÞ � uc

þ !�ðtÞ,nðxÞ � �þ d�ðtÞðtÞ ð3Þ

In order to illustrate the proposed control scheme

in practical implementation, we take the notations in

the following form:

Fi ¼ ½x2,x3, . . . , xn, fiðxÞ þ giðxÞus þ diðtÞ�Tð4Þ

Wi ¼ ½!1ðxÞ,!2ðxÞ, . . . ,!nðxÞ�T

ð5Þ

Gi ¼ ½0, 0, . . . , giðxÞ�T

ð6Þ

From (4)–(6), system (1) can be expressed as:

x�¼ Fi þWi � �þ Gi � uc ð7Þ

Let the sliding mode surface be defined by:

s ¼ Kc xðtÞ � xðt0Þ �

Z t

t0

Fixð�Þ d�

� �ð8Þ

As we know, when x�¼ Fi, the system is global

exponential asymptotical stability such that

limt!1ðWi � �þ Gi � ucÞ ¼ 0 ð9Þ

Then, the dynamics of sliding mode surface can be

given by:

s�¼ Kc½WiðxÞ�þ GiðxÞuc� ð10Þ

According to feedback linearisable techniques and

the control theory of sliding model control

(Kanellakopoulos, Kokotovic, and Morse 1991;

Yoon, Park, and Yoon 2008), the control law of

system (1) is chosen as:

us ¼1

giðxÞð�Ksx� fiðxÞÞ

uc ¼1

Kc � Gi�� sgnðsÞ � KcFi�� �

8>><>>: ð11Þ

where

sgnðsÞ ¼

1 for s4 0

0 for s ¼ 0

�1 for s5 0

8><>:

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then the following equation (i.e. the control law) can be

written easily from Equations (9) and (10):

u ¼1

giðxÞ�Ksx� fiðxÞð Þ þ

1

Kc � Gi�� sgnðsÞ � KcFi�� �

ð12Þ

where � is the known positive parameter; Ks ¼

ðks1, ks2, . . . , ksnÞ and Kc ¼ ðkc1, kc2, . . . , kcnÞ are,

respectively, n-dimensional constant vectors; � is the

estimate value of �, and the estimate error ~� is

described by

~� ¼ �� � ð13Þ

The estimate � for the sliding model control system

is updated by the following normalised adaptive law:

��

¼ ½sKcWiðxÞ�T

ð14Þ

For the switching signal �, a switching sequence is

given by:X:¼ ði0, t0Þ, ði1, t1Þ, . . . , ðik, tkÞ, . . . , jik 2 �, k 2 N� �

ð15Þ

where ðik, tkÞ denotes that the ikth subsystem is

switched on at tk, and the ikþ1th subsystem is switched

off at tkþ1. Where t0 is the initial time and tk 4 0 is the

kth switching time. When t 2 ½tk, tkþ1Þ, the trajectory of

the switched non-linear system (1) is produced by the

ikþ1th subsystem, defining �P ¼ tk � tk�1 as dwell-time

of the ikth subsystem.

Assumption 3.1: For t 2 ðtk�1, tk� 2 �m ðm 2 � ¼def

f1, 2, . . . , ngÞ and t 2 ðtk, tkþ1� 2 �mþ1, there is a con-

stant 0 � � � 1 such that

sðtkþ1Þ�� �� � � � sðtkÞ�� �� ð16Þ

In this article, it is assumed that � ¼ 1 (Colaneri,

Geromel, and Astolfi 2008; Mohamad and Liu 2008).

Definition 3.1: Given some family of switching signals

�, for each � and t � t0 � 0, let N�ðt0, tÞ denote the

number of discontinuities of �in the interval ½t0, tÞ

(Kanellakopoulos, Kokotovic, and Morse 1991;

Hespanha and Morse 1999; Persis, Santis, and

Morse 2003). For given �D, N0 4 0, we denote by

�ave½�D,N0� the set of all switching signals for which

N�ðt0, tÞ � N0 þt� t0�D

ð17Þ

The constant �D is called the average dwell-time and

the chatter bound is N0. So the switching signal

�ave½�D,N0� � � consisting of all switching signals has

the same persistent dwell-time �D 4 0 and the same

persistent chatter bound N0 4 0.

Assumption 3.2: For each piecewise constant switchingsignal �, the switched non-linear system can be definedas follows:

x�¼ f�ðx, u, d Þ ð18Þ

For each � 2 �, it is shown that (18) is uniformlyasymptotically stable over � such that

xk k � ðc xðt0Þ�� ��Þ1=$e�ðt�t0Þ $

þ �0

Z t

t0

uð�Þ�� ��2 d� þ ðtÞ 8t � t0 4 0 ð19Þ

where c, $, �0 are suitable positive real numbers and ðtÞ the continuous non-negative function.

Lemma 3.1: Given a set �f¼Df fiðx, u, d Þ : i 2 �g of

non-linear maps from Rn to itself for which Assumption3.2 holds (Hespanha and Morse 1999; Persis, Santis,and Morse 2003). For any average dwell-time �D � �

D,

which �D is a finite constant and any chatter boundN0 4 0, the switching system (18) is uniformly asymp-totically stable over �ave½�D,N0� and there exist contin-uously differentiable functions Vi : R! Rn, i 2 �,positive constants �1, �2 and functions 1, 2 of classK1 satisfying that

�1 xk kð Þ � ViðxÞ � �2 xk kð Þ ð20Þ

@Vi

@x� fiðx, u, d Þ � �21 � ViðxÞ ð21Þ

ViðxÞ � 2 � Vj ðxÞ ð22Þ

for each x 2 Rn and i, j 2 �.

Remark 1: The idea of Assumption 3.1 is that even ifthere exists a Lyapunov function Vi for each sub-system, individually, restrictions must be placed on theswitching scheme to guarantee stability of the overallswitched system. Also, as we know, the initial value ofthe sliding mode surface in this study is sð0Þ4 0.According to the theory of sliding mode control, theresulting closed-loop control system in this article isstable in Lyapunov sense such that the function ofsliding mode surface sðtÞ is monotonically non-increasing.

Remark 2: From the definition of a Lyapunovfunction, Vi is monotonically non-increasing on everytime interval where the ith subsystem is active, and theset �m represents the part of the state space where

V�

i � 0. The idea of Assumption 3.1 is that even if thereexist the sliding mode surface and the Lyapunovfunction for each subsystem, restrictions must beplaced on the switching scheme to guarantee stabilityof the overall switched system. In fact, it is possible toconstruct examples of globally asymptotically stable

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systems and a switching rule that sends all trajectories

to infinity (Branicky 1998). A sufficient condition to

guarantee Lyapunov stability is to require, as in (16),

that for every mode i, the value of Vi at the beginning

of each interval on which the ith subsystem is active

not exceeding the value at the beginning of such

previous interval.

Based on the above analysis, for the uncertain

switched non-linear system (1), we have the following

result.

Theorem 3.1: Consider system (1) satisfying

Assumptions (2.1, 2.2, 3.1 and 3.2) and Lemma 3.1

controlled by the adaptive controller (12). With the

proposed adaptation law and the switching signal with

average dwell-time method, it can be guaranteed that the

resulting closed-loop switched system is asymptotically

stable in the Lyapunov sense and uniformly ultimately

bounded while the actual output follows the desired

output signal and the sliding surface of the control

system is reached.

Proof: Define a switched multiple Lyapunov function

candidate to analyse the stability of system (1) as:

V ¼1

2s2 �

Xni¼1

iðtÞ þ ~�T ~�

!ð23Þ

with the characteristic function:

iðtÞ :¼1 t 2�i

0 t =2 �i

�,

�i ¼ ft j the ith subsystem is active at time instant tg

For t 2 ðtk�1, tk� 2 �m and t 2 ðtk, tkþ1� 2 �mþ1, from

(16) and (23), we have:

DVðtÞ ¼ Vðtkþ1Þ � VðtkÞ

¼1

2sðtkþ1Þ

2�1

2sðtkÞ

2

¼1

2sðtkþ1Þ�� ��2� 1

2sðtkÞ�� ��2 5 0 ð24Þ

From the switching signal (17), for 8t 2 ½tk, tkþ1� 2 �m,

taking the time derivative of V, we get:

V�

¼ s s�þ ~�T ~�

�

¼ sKc½WiðxÞ�þ GiðxÞuc� þ ~�T ~��

¼ s½KcWiðxÞ�� � sgnðsÞ � KcWiðxÞ�� þ ~�T ~��

¼ �� sj j þ sKcWiðxÞð�� �Þ þ ~�T ~��

ð25Þ

Then, according to the update law (12) and ~��

¼ ���

, we

obtain:

V�

¼ �� sj j þ sKcWiðxÞ ~�� ~�T ��

¼ �� sj j þ ~�T½sKcWiðxÞ�T� ~�T �

�

¼ �� sj j � 0 ð26Þ

So the stability of closed-loop system can be guaran-

teed. In what follows, it is verified that the sliding

surface of the control system can be reached, i.e.

limt!1 sðtÞ ¼ 0.The equality V

�

¼ ��jsj implies that s is

integrable as

limt!1

Z t

0

sj j dt ¼1

�Vð0Þ � VðtÞ½ �:

As we know, since Vð0Þ is bounded and VðtÞ is mono-

tonically non-increasing and bounded, it has been

deduced that limt!1

R t0 sj j dt and s

�are both bounded.

According to Barbalat’s Lemma (Slotine and Li 1991;

Yu, Fei, and Li 2010), all the variables of the closed-

loop system are bounded in the presence of model

uncertainties and external disturbances. Also, we can

conclude that sðtÞ will asymptotically converge to zero

by choosing the design parameters appropriately, i.e.

limt!1

sðtÞ ¼ 0:

So the sliding surface of the control system has been

reached.So far, the proof of Theorem 1 and the

controller design with adaptive integral sliding mode

method have been completed.

Remark 3: In the adaptive slide mode controller

designed by this article, uc has to be discontinuous, so

this will lead to chattering. Generally, chattering must

be eliminated for the controller to perform properly. It

can be achieved by smoothing out the control discon-

tinuity for uc as follows (Roh and Oh 2000; Ho, Ho,

and Rad 2009):

uc ¼1

Kc � Gi�� sgn

s

"

� �� KcFi�

� �ð27Þ

where "4 0 is a design parameter.

Remark 4: The ultimate bound for the system track-

ing error can be made smaller by increasing the given

uncertain constant parameter. This will result in

increased interaction with unknown or unmodelled

plant dynamics.

Remark 5: Though the update laws using tuning

functions (14) without projection always ensure the

boundedness of parameter estimates, the correspond-

ing parameter bounds are dependent of

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design parameters. As will be seen in Section 3, theboundedness of parameter estimates is desired indealing with the control problem for the uncertainswitched non-linear systems.

Remark 6: The control law u in (12) is only true overa compact set Uc � Rn on which giðxÞ 6¼ 0, also Gi 6¼ 0.

4. Illustrative example

In this section, we apply our proposed adaptiveintegral sliding mode controller for two cases. Thefirst example is to let the model of switched non-linearsystems including non-linear uncertainties and externaldisturbance. The second practical example for electro-hydraulic servo system is choosing the model ofswitched non-linear systems without uncertainties. Sothe Example 2 is the special case of the Example 1.

Example 1: Consider the switched systems withuncertainties as follows:

X21

:

x1�¼ x2 þ !11 � �1

x2�¼ x3 þ !12 � �2

x3�¼ x31 þ x22 þ 5uþ !13 � �3 þ d1

8><>: ð28Þ

X22

:

x1�¼ x2 þ !21 � �1

x2�¼ x3 þ !22 � �2

x3�¼ x21 þ 2uþ !23 � �3 þ d2

8><>: ð29Þ

where d ¼ ½d1 d2� ¼ ½0:5 sinðtÞ 0:6 sinðtÞ�, !1 ¼

½!11 !12 !13� ¼ ½2x21 � x2 x1 � x

22 sinx1 � x2�, !2 ¼

½!21 !22 !23� ¼ ½x1 � sin x2 2x2 � x3 x1 � x3�. Thecontrol parameters for the controller (12) were chosenas ks ¼ ½1 2 1�, kc ¼ ½1 1 1�, � ¼ ½2 2 2�, � ¼ 3, �D ¼ 3and the initial values of state vectors are xð0Þ ¼½�1 � 2 3�T. Simulation results are shown inFigures 1–3. From Figure 1, which denotes the curve

of the system state vector, the system vector can

asymptotically converge to zero. In addition, the

sliding surface which converges to 0 is well achieved

in Figure 2. Respectively, the curve of the adaptive law

and the control signal are shown in Figures 3 and 4.

Also, Figure 5 shows the switching signal. From the

simulation results, it can be concluded that the

obtained theoretic results are feasible and efficient for

the given switched non-linear systems.

Example 2: In this example, we verify the validity of

the design approach for the plant (1) with n¼ 3. When

� ¼ 0, the model of an electro-hydraulic servo system is

given by (Mihajlov, Nikolic, and Antic 2002):

X11

:

x1�¼ x2

x2�¼ x3

x3�¼ 11x1 þ 21x2 þ 31x3 þ b1uþ d1

8><>: ð30Þ

0 2 4 6 8 10 12 14 16 18 20–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

3

Time (s)

x1x2x3

Figure 1. Curve of the system state vector for Example 1.

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

q<

Figure 3. The curve of the adaptive law for Example 1.

0 2 4 6 8 10 12 14 16 18 20–0.5

0

0.5

1

1.5

2s

Time (s)

Slid

e su

rfac

e

Figure 2. Sliding surface performance for Example 1.

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X12

:

x1�¼ x2

x2�¼ x3

x3�¼ 12x1 þ 22x2 þ 32x3 þ b2uþ d2

8><>: ð31Þ

where

1i ¼4CtpeK

miVt, 2i ¼

K

miþ4A2emiVt

, 3i ¼4eCtp

miVt,

bi ¼4AemiVt

kvCdw

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPs � sgnðxvÞPL

�

s,

di ¼ �4eCtpFe

Vt�

Fe

�

mi:

The physical parameters of this electro-hydraulic servo

system are given in Table 1. In addition, the mass of

actuator and load is chosen as m1 ¼ 20 kg, m2 ¼ 24 kg.

Using the method of the Example 2 designed in

this article, Figures 6 and 7 show the simulation

0 2 4 6 8 10 12 14 16 18 20–0.5

0

0.5

1

1.5

2

Time (s)

Slid

e su

rfac

e

s

Figure 7. Sliding surface performance for Example 2.

0 2 4 6 8 10 12 14 16 18 20–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

3

Time (s)

x1

x2x3

Figure 6. Curve of the system state vector for Example 2.

Table 1. Physical parameters of the electro-hydraulic servosystem.

Name Symbol Unit

Supply pressure Ps PaTotal actuator volume Vt M3

Effective bulk modulus e PaActuator ram area A m2

Total leakage coefficient Ctp M3/(s �Pa)Discharge coefficient CdSpool valve area gradient W MFluid mass density � Kg/m3

Mass of actuator and load mi KgSpring constant K N/mLoad pressure PL PaSpool displacement xv MCoulomb friction Fe N

0 1 2 3 4 5 6 7 8 9 10111213141516171819200

0.5

1

1.5

2

2.5

3

Time (s)

s(t

)

Figure 5. The switching signal.

0 2 4 6 8 10 12 14 16 18 20–1.5

–1

–0.5

0

0.5

1

Time (s)

Con

trol

inpu

t

u

Figure 4. The curve of the control signal for Example 1.

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results for the proposed adaptive sliding mode controlhaving satisfactory control performance. So theexperimental results point out that the stable andwell-damped response of the control system isobtained.

Remark 7: From the comparison and analysis of thetwo examples, we can see that: (1) the controller ofthe system proposed in this article can both guaranteethe global stability of the closed-loop system and theultimate convergence of all the states in the closed-loop system; (2) the proposed control scheme hassatisfactory performance and robustness even in thepresence of unmodelled dynamics and disturbances.Therefore, such an adaptive integral sliding modecompensation controller would be inadequate toaddress the complex control systems where thereexist appreciable non-parametric uncertainties whichinclude unmodelled dynamics, external disturbancesand other imperfections in the estimates of systemparameters.

Remark 8: Even though the switching conditionsrequire knowledge of the temporal evolution of theclosed-loop state, xðtÞ, the a priori knowledge of thesolution of the constrained closed-loop non-linearsystem (which is difficult to obtain in general) is notneeded for the practical implementation of theproposed approach. Instead, the supervisor canmonitor (on-line) how x evolves in time to determineif and when the switching conditions are satisfied. Ifthe conditions are satisfied for the desired targetmode at some time, then switching can take placesafely.

Remark 9: In this section, the proposed adaptivesliding mode controller incorporates the capability tomaintain stable performance for switched non-linearsystems. From the control point of view, designing asatisfactory robust controller to enhance robustnessand maintain boundedness is very important for thecontrol performance. H-infinity control method as apopular robust strategy may be another choice. Fromthe application point of view, the control schemeproposed is suitable for practical implementation.Also, from the modelling point of view, we haveconsidered the uncertainty and external disturbances.So we can use neural networks or fuzzy systems as atool for modelling non-linear functions up to a smallerror tolerance for future work.

5. Conclusion

In this article, we have investigated the adaptiveintegral sliding mode control method with averagedwell-time technique for a class of uncertain switched

non-linear systems. It is proved that closed-loop systemis asymptotically stable in the Lyapunov sense and thesliding surface of the control system is obtained.Simulation results show that the proposed controlmethod has satisfactory performance and robustness inthe presence of model uncertainties and externaldisturbance. The control scheme is simple and practicalto improve the dynamic system’s transient responseand robustness, also has a broad future in engineeringapplications. Therefore, from the practical point ofview, the technique can successfully be applied topractical control systems and this can be extended tothe other non-affine non-linear systems.

Acknowledgements

The authors gratefully acknowledge the anonymousreviewers for their constructive and insightful comments forfurther improving the quality of this study. This study issupported by the National Natural Science Foundation ofChina, nos 60835001, 60804017 and 60764001.

Notes on contributors

Lei Yu was born in Xuancheng, P.R.China, 1983. He received his MSdegree in Control Theory andControl Engineering from HefeiUniversity of Technology, China, in2008 and the PhD in AutomaticControl Theory and Applicationsfrom Southeast University in 2011.He is now a Lecturer in the College of

Mechanical and Electrical Engineering, Soochow University.His research interests include switched nonlinear systems,robust adaptive control, neural network control, etc.

Maoqing Zhang was born in 1954.He received his MS degree fromYanshan University, Qinghuangdao,P.R. China, 1983. He is now aProfessor in the Research Institute ofMechanical and ElectricalEngineering at Soochow University,Suzhou, P.R. China. His researchinterests include control theory and

control engineering, and fuzzy control, etc.

Shumin Fei was born in 1961. Hereceived his PhD from BeihangUniversity, Beijing, P.R. China,1995. From 1995 to 1997, he didPostdoctoral Research in theResearch Institute of Automation atSoutheast University. He is now aProfessor in the Research Institute ofAutomation at Southeast University,

Nanjing, P.R. China. His research interests include theanalysis and synthesis of nonlinear systems, robust control,adaptive control and analysis and synthesis of time-delaysystems, etc.

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