0Article history:Received 22 October 2014Received in revised form9 June 2015Accepted 4 July 2015Available online 1 August 2015

Keywords:Switched systemsStochastic systemsAdaptive controlUnknown nonsymmetric actuatordead-zone

This paper is concerned with the problem of adaptive tracking control for a class of switched stochasticnonlinear systems in nonstrict-feedback form with unknown nonsymmetric actuator dead-zone andarbitrary switchings. A variable separation approach is used to overcome the difficulty in dealing with thenonstrict-feedback structure. Furthermore, by combining radial basis function neural networks universalapproximation ability and adaptive backstepping technique with common stochastic Lyapunov functionmethod, an adaptive control algorithm is proposed for the considered system. It is shown that the targetsignal can be almost surely tracked by the systemoutput, and the tracking error is semi-globally uniformlyultimately bounded in 4th moment. Finally, the simulation studies for a ship maneuvering system arepresented to show the effectiveness of the proposed approach.

2015 Elsevier Ltd. All rights reserved.

1. Introduction

As a typical class of hybrid systems, switched systems havedrawn considerable attention in the past decades sincemany phys-ical systems can be mathematically modeled by such multiple-mode systems. Given their widespread applications, the studies onswitched systems never cease and a number of excellent resultshave been reported, see for example, Briat (2014), Li, Wen, Soh,and Xie (2002), Vesely` and Rosinov (2014), Xiang and Xiao (2014),Xie,Wen, and Li (2001), Zhang and Gao (2010), Zhang, Zhuang, andShi (2015), Zhao, Yin, Li, and Niu (2015) and Zhao, Zhang, Shi, andLiu (2012) and the references therein. In Zhang and Gao (2010) andZhao et al. (2012), the control problems for switched linear systemsin both continuous-time and discrete-time contexts were solvedunder average dwell time switching and mode-dependent aver-age dwell time switching, respectively. In Zhang et al. (2015), the

The material in this paper was not presented at any conference. This paper wasrecommended for publication in revised form by Associate Editor Changyun Wenunder the direction of Editor Miroslav Krstic.

E-mail addresses: xdzhaohit@gmail.com (X. Zhao), peng.shi@adelaide.edu.au(P. Shi), zxldwlsj@gmail.com (X. Zheng), lixianzhang@hit.edu.cn (L. Zhang).

authors considered H filtering for a class of switched linear dis-crete systems under their proposed persistent dwell-time switch-ing. The problem of switching stabilization for slowly switchedlinear systems with mode-dependent average dwell time was in-vestigated in Zhao et al. (2015) by using invariant subspace the-ory. Recent advances for switched systems in linear setting can befound in Briat (2014), Vesely` and Rosinov (2014) and Xiang andXiao (2014). In addition, for switched nonlinear systems, some suf-ficient conditions were derived in Xie et al. (2001) to ensure thatthe whole switched nonlinear system is input-to-state stabilizable(ISS) when each mode is ISS. A concept of generalized matrix mea-sure for nonlinear systemswas proposed in Li et al. (2002) to studythe stability of switched nonlinear systems directly.

Note that during the most recent years, the adaptive back-stepping-based control problems have been investigated for a classof switched nonlinear systems (Han, Ge, & Lee, 2009; Long & Zhao,2015; Ma & Zhao, 2010; Wang, 2014). To list a few, the globalstabilization problem for switched nonlinear systems in lowertriangular form was investigated in both Long and Zhao (2015)and Ma and Zhao (2010) by using common Lyapunov functionand multiple Lyapunov functions, respectively; Han et al. (2009)presented an adaptive neural control design for a class of switchednonlinear systems with uncertain switching signals. However, the

http://dx.doi.org/10.1016/j.automatica.2015.07.022Automatica 60 (

Contents lists availa

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journal homepage: www.els

Brief paper

Adaptive tracking control for switched stwith unknown actuator dead-zone

Xudong Zhao a,b, Peng Shi c,d, Xiaolong Zheng a, Lixiana College of Engineering, Bohai University, Jinzhou 121013, Liaoning, Chinab Department of Mechanical Engineering, The University of Hong Kong, Hong Kongc School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5d College of Engineering and Science, Victoria University, Melbourne, Vic. 8001, Australiae School of Astronautics, Harbin Institute of Technology, Harbin, 150001, China

a r t i c l e i n f o a b s t r a c t0005-1098/ 2015 Elsevier Ltd. All rights reserved.2015) 193200

ble at ScienceDirect

atica

evier.com/locate/automatica

ochastic nonlinear systems

Zhang e

05, Australia

194 X. Zhao et al. / Automat

systems in these studies are of a strict-feedback formwhich greatlylimits the applicability in practice.

On the other hand, it is well known that stochastic disturbancefrequently exists in engineering applications. The systems subjectto the stochastic disturbance are usually termed as stochasticsystems that have also been widely probed, with switchingdynamics (Hou, Fu, & Duan, 2013; Wu, Cui, Shi, & Karimi, 2013;Wu, Yang, & Shi, 2010; Wu, Zheng, & Gao, 2013; Zhang, Wu,& Xia, 2014) or without (Wu, Xie, & Zhang, 2007; Xie & Duan,2010; Xie, Duan, & Yu, 2011; Xie, Duan, & Zhao, 2014; Xie & Liu,2012). To mention a few, the dissipativity-based sliding modecontrol was adopted in Wu, Zheng et al. (2013) for switchedstochastic linear systems. Stabilization problems for stochasticnonlinear systems with Markovian switching were studied inWu et al. (2010). By using a novel homogeneous dominationapproach, the output feedback stabilization problem was studiedin Xie and Liu (2012) for stochastic high-order nonlinear systemswith time-varying delay. The global stabilization was consideredin Xie et al. (2011) for high-order stochastic nonlinear systemswith stochastic integral input-to-state stability inverse dynamics.The concept of input-to-state practical stability is extended tostochastic nonlinear systems in Wu et al. (2007), upon which theyconsidered the control problem for a class of stochastic nonlinearsystems with unmodeled dynamics by using stochastic small-gaintheorem.However,most of these control strategies require that thenonlinear stochastic systems are known precisely or the unknownparameters appear linearly with respect to the known nonlinearfunctions. Furthermore, existing results on switched stochasticsystems also suppose that the systems are of a strict-feedbackform. Clearly, these ideal requirements cannot be satisfied inmanypractical situations.

Moreover, dead-zone characteristics are encountered in manyphysical components of control systems. They are particularlycommon in actuators, such as hydraulic servo-valve, electricservomotors, and biomedical systems. It is well known that asystem will become oscillating, and the regulation quality willdecline if some undesired dead-zone nonlinearities occur in thesystem. Thus, it is more realistic and reliable to design controllersbased on the system model where the dead-zone nonlinearitiesare taken into consideration. However, up to now, few resultson adaptive control have been developed for switched stochasticnonlinear systems with dead-zone characteristics.

It is seen from the above observations that it is of bothpractical and theoretical significance to investigate the problemof adaptive tracking control for nonstrict-feedback switchedstochastic systems with actuator dead-zone, which is howeverchallenging and has not been studied so far. This motivates usto carry out the present study. In this paper, a new approachof constructing common virtual control functions is proposed forthe studied system, and a backstepping-based adaptive controlmethodology is systematically developed with low computationburden since the controller only has two parameters need to bemodulated. The contributions of the paper lie in that: (i) ourconsidered switched systemmodel is of a nonstrict-feedback form;(ii) the uncertainty can be completely unknown; (iii) the unknownnonsymmetric actuator dead-zone is taken into account; (iv) thestochastic disturbance is considered in the switched systemmodel;and (v) fewer parameters need to be designed which is moreefficient in practice.

Notation Rn denotes the n-dimensional space, R+ is the set ofall nonnegative real numbers. C i stands for a set of functions withcontinuous ith partial derivatives. For a givenmatrixA (or vector v),AT (or vT ) denotes its transpose, and Tr{A} denotes its tracewhen Ais a square.K represents the set of functions: R+ R+, which arecontinuous, strictly increasing and vanishing at zero;K denotes

a set of functions which is of classK and unbounded. In addition, refers to the Euclidean vector norm.ica 60 (2015) 193200

2. Preliminaries and problem formulation

Consider the following switched stochastic nonlinear system innonstrict-feedback form:

dxi = (gi, (t)xi+1 + fi, (t)(x))dt + Ti, (t)(x)dw,1 i n 1,

dxn = (gn, (t)v(t) + fn, (t)(x))dt + Tn, (t)(x)dw,v(t) = D(t)(u(t)),y = x1, (1)where x = (x1, x2, . . . , xn)T Rn is the system state, w is an r-dimensional independent standard Brownian motion defined onthe complete probability space (,F , {Ft}t0 , P)with being asample space,F a -field, {Ft}t0 a filtration, P a probability mea-sure, and y is the system output; (t) : [0,) M = {1, 2, . . . ,m} represents the switching signal; v(t), u(t) R are the actua-tor output and input. For any i = 1, 2, . . . , n and k M, fi,k(x):Rn R, i,k : Rn Rr are locally Lipschitz unknown nonlinearfunctions and gi,k are positive known constants.

The nonsymmetric dead-zone nonlinearity is considered in thepaper, which is defined as the form in Tao and Kokotovi (1994):

vk = Dk(uk) =mrk(uk brk), uk brk0, blk < uk < brkmlk(uk + blk), uk blk .

(2)

Here, mrk > 0 and mlk > 0 stand for the right and the left slopesof the dead-zone characteristic, respectively. brk > 0 and blk > 0represent the breakpoints of the input nonlinearity. It should benoticed that, in this paper,mrk ,mlk , brk and blk are unknown.

In order to facilitate the analysis and design, it is assumed thatthe nonsymmetric dead-zone nonlinearity can be reformulated as:

vk = Dk(uk)+ k, (3)where Dk(uk) is an unknown smooth function, k is the errorbetween Dk(uk) and Dk(uk)with |k| k.

Moreover, we have

vk = uk + (Dk(uk) uk + k)= uk + k(uk)+ k, (4)

where k(uk) = Dk(uk)uk is an unknown function. The controllercan be designed as

uk = uck uk . (5)Then (4) can be rewritten as

vk = uck + k(uk) uk + k. (6)where uk is the compensator of dead-zone nonlinearity and uck isa main controller of system (1).

Our control objective is to design state-feedback controllerssuch that a given time-varying signal yd(t) can be tracked by theoutput of system (1) under arbitrary switching, while overcomingthe problem of actuator dead-zone. In this paper, we also assumethat the following assumptions hold:

Assumption 1. The tracking target yd(t) and its time derivativesup to nth order y(n)d (t) are continuous and bounded. Also, it isassumed that |yd(t)| d, where d > 0 is a constant knownapriori.Assumption 2. There exist strictly increasing smooth functionsi,k(), i,k() : R+ R+ with i,k(0) = i,k(0) = 0 such thatfor i = 1, 2, . . . , n and k M ,|fi,k(x)| i,k(x). (7)i,k(x) i,k(x). (8)

X. Zhao et al. / Automat

Remark 1. The increasing properties of i,k(), i,k() imply that ifai, bi 0, for i = 1, 2, . . . , n, then i,k(ni=1 ai) ni=1 i,k(nai),i,k(

ni=1 bi)

ni=1 i,k(nbi). Notice that i,k(s), i,k(s) are

smooth functions, and i,k(0) = i,k(0) = 0. Therefore, it followsthat there exist smooth functions hi,k(s), i,k(s) such that i,k(s) =shi,k(s), i,k(s) = si,k(s)which results in

i,k

n

j=1aj

nj=1

najhi,k(naj). (9)

i,k

n

j=1bj

nj=1

nbji,k(nbj). (10)

In Sanner and Slotine (1992), it has been proved that the radialbasis function (RBF) neural networks can approximate any contin-uous real function f (Z) over a compact set Z Rq. Specifically,for arbitrary > 0, there exists a neural networkW T S(Z) such that

f (Z) = W T S(Z)+ (Z), (Z) , (11)where Z Z Rq, W = [w1, w2, . . . , wl]T is the ideal constantweight vector, and S(Z) = [s1(Z), s2(Z), . . . , sl(Z)]T is the basisfunction vector, with l > 1 being the number of the neural net-work nodes and si(Z) being chosen as Gaussian functions, i.e., fori = 1, 2, . . . , l

si(Z) = exp(Z i)T (Z i)

2i

, (12)

where i = [i1, i2, . . . , iq]T is the center vector, and i is thewidth of the Gaussian function.

Definition 1. Consider the stochastic system dx = f (x, t)dt +h(x, t)dw. For any given V (x, t) C2,1, define the differential op-eratorL as follows:

LV = Vt

+ Vx

f (x, t)+ 12TrhT2Vx2

h. (13)

Definition 2. The trajectory {x(t), t 0} of switched stochasticsystem (1) is said to be semi-globally uniformly ultimatelybounded (SGUUB) in pth moment, if for any compact subset Rn and all x(t0) = x0 , there exist a constant > 0 and a timeconstant T = T (, x0) such that E(|x(t)|p) < , for all t > t0+T . Inparticular, when p = 2, it is usually called SGUUB in mean square.Lemma 1 (Krsti & Hua, 1998). Suppose that there exist a C2,1function V (x, t) : Rn R+ R+, two constants c1 > 0 and c2 > 0,classK functions 1 and 2 such that1(|x|) V (x, t) 2(|x|)LV c1V (x, t)+ c2

for all x Rn and t > t0. Then, there is an unique strong solution ofsystem (1) for each x0 Rn, which satisfiesE[V (x, t)] V (x0, t0)ec1t + c2c1 , t > t0.

Lemma 2 (Polycarpou & Ioannou, 1996). For any R and > 0,the following inequality holds:

0 | | tanh

, (14)with = 0.2785.ica 60 (2015) 193200 195

3. Main result

In this section a systemic control design and stability analysisprocedure will be presented by using adaptive backsteppingtechnique (the readers may refer to Krsti and Kokotovi (1995)and Krsti, Kokotovi, and Kanellakopoulos (1995) for more detailsabout adaptive backstepping technique). For i = 1, 2, . . . , n 1,let us define a common virtual control function i as

i = 1gi,mini + 34

zi 12a2i

z3i STi Si

, (15)

where i, ai > 0 are design parameters, gi,min = min{gi,k : k M}, zi represents the new state after the coordinate transforma-tion: zi = xi i1, 0 = yd. is an unknown constant that willbe specified later. Si = Si(Xi) is the basis function vector. Xi =[xTi ,

T

i , y(i)Td ]T with xi = [x1, x2, . . . , xi]T , i = [1, 2, . . . , i]T ,

y(i)d = [yd, yd, . . . , y(i)d ]T . The z-system can be obtained as follows:dzi = (gi,kxi+1 + fi,k Li1)dt

+i,k

i1j=0

i1xj

j,k

Tdw, 1 i n 1

dzn = (gn,kvk + fn,k Ln1)dt

+n,k

n1j=0

n1xj

j,k

Tdw, (16)

where the differential operator L is defined in Definition 1, thenLi1 is given by:

Li1 = i1

+

i1s=1

i1xs

(fs,k + gs,kxs+1)

+i1s=0

i1y(s)d

y(s+1)d +12

i1p,q=1

2i1xpxq

Tp,kq,k. (17)

Consider the following common stochastic Lyapunov functioncandidate

V =n

i=1

14z4i +

12r1

2 + 12r2

2, (18)

where r1, r2 > 0 are design parameters; = , = with and will be specified later; and represent the estima-tion of and respectively.

Lemma 3. From the coordinate transformations zi = xi i1, i =1, 2, . . . , n, 0 = yd, the following inequality holds

x n

i=1|zi|i(zi, )+ d, (19)

where i(zi, ) = 1gi,min [(i+ 34 )+ 12a2i z2i S

Ti Si]+1, for i = 1, 2, . . . ,

n 1, and n = 1.Proof. From Assumption 1 and (15), one gets

x n

i=1|xi|

ni=1

(|zi| + |i1|) n

i=1|zi| + yd

+n1i=1

1

gi,min

i + 34

+ 1

2a2iz2i S

Ti Si

|zi|

n

i=1|zi|i(zi, )+ d.The proof of Lemma 3 is completed here.

196 X. Zhao et al. / Automat

By using Definition 1, (16) and (17),LV can be given by

LV =n1i=1

z3ifi,k + gi,kxi+1

i1s=0

i1y(s)d

y(s+1)d

i1

i1s=1

i1xs

(fs,k + gs,kxs+1)

12

i1p,q=1

2i1xpxq

Tp,kq,k

+ 32z2i

i,k i1j=0

i1xj

j,k

2

+ z3nfn,k + gn,kvk

n1s=0

n1y(s)d

y(s+1)d

n1s=1

n1xs

(fs,k + gs,kxs+1) n1

12

n1p,q=1

2n1xpxq

Tp,kq,k

1

r1

+ 32z2n

n,k n1j=0

n1xj

n,k

2

1r2

=n

i=1

z3ifi,k

i1s=1

i1xs

fs,k

i1s=0

i1y(s)d

y(s+1)d i1s=1

i1xs

gs,kxs+1

i1

1

2

i1p,q=1

2i1xpxq

Tp,kq,k

+ 32z2i

i,k i1j=0

i1xj

j,k

2 1r1

1r2 +

n1i=1

z3i gi,kxi+1 + z3ngn,kvk. (20)By using Assumption 2 and Lemma 3, one has

z3i

fi,k

i1s=1

i1xs

fs,k(x)

= z3i

is=1

i1,sfs,k(x)

34nz4i

is=1

(i1,s)43 +

is=1

nl=1

z4l 4s,k(zl, )

+ z3i is=1

i1,ss,k((n+ 1)d), (21)where 4s,k(zl, ) = 14 (n+1)44l (zl, )h4s,k((n+1)|zl|l(zl, )), 0xs= 0, and i1,s is defined as i1,s = i1xs , s = 1, 2, . . . , i 1,i1,i = 1.

Then, the following inequality can be obtained

32z2i

i,k i1j=0

i1xj

j,k

2

9 n i1 n8i2(n+ 1)2nz4i +

l=1z4l

4i,k(zl, )+

j=1 l=1z4l

4j,k(zl, )ica 60 (2015) 193200

+ 98i2(n+ 1)2nz4i

i1j=1

i1xj

4

+ 98i2(n+ 1)2z4i l2ii 4i,k((n+ 1)d)+

ij=1

l2ij

+ 98i2(n+ 1)2z4i

i1j=1

i1xj

4l2ij

4j,k((n+ 1)d), (22)

where lij is a positive constant, and0xj

= 0 since 0 = yd, and

12z3i

i1p,q=1

2i1xpxq

Tp,kq,k (i 1)i1s=1

nl=1

z4l 4s,k(zl, )

+ 18(n+ 1)2nz6i

i1s=1

i1j=1

2i1xsxj

2

+ 12(n+ 1) z3i i1

s=1

i1j=1

2i1xsxj 2s,k((n+ 1)d), (23)

where 4s,k(zl, ) = 12 (n+1)44l (zl, )4s,k((n+1)|zl|l(zl, )), s =1, 2, . . . , i 1.

Substituting (21)(23) into (20) gives

LV n

i=1

34nz4i

is=1

(i1,s)43 +

ni=1

is=1

nl=1

z4l 4s,k(zl, )

+n

i=1

z3i is=1

i1,ss,k((n+ 1)d)+

ni=1

i1s=1

nl=1

(i 1)z4l 4s,k(zl, )

+n

i=1

i1s=1

i1j=1

18(n+ 1)2nz6i

2i1xsxj

2

+n

i=1

i1s=1

i1j=1

12(n+ 1) z3i 2s,k((n+ 1)d) 2i1xsxj

+

ni=1

98i2(n+ 1)2nz4i +

nl=1

z4l 4i,k(zl, )

+i1j=1

nl=1

z4l 4j,k(zl, )

+ 98i2(n+ 1)2nz4i

i1j=1

i1xj

4

+ 98i2(n+ 1)2z4i l2ii 4i,k((n+ 1)d)+

ij=1

l2ij

+ 98i2(n+ 1)2z4i

i1j=1

i1xj

4l2ij

4j,k((n+ 1)d)

+n

i=1z3i

i1s=0

i1y(s)d

y(s+1)d i1

i1s=1

i1xs

gs,kxs+1

+

n1i=1

z3i gi,kxi+1 + z3ngn,kvk 1r1 1

r2. (24)

X. Zhao et al. / Automat

Define Ui,k as

Ui,k =i

s=1|i1,s|s,k((n+ 1)d)

+ 12(n+ 1)

i1s=1

i1j=1

2i1xsxj 2s,k((n+ 1)d). (25)

By using Lemma 2 one hasz3i Ui,k z3i Ui,k tanh z3i Ui,ki,k+ i,k. (26)

It can be noticed in (24) that

n1i=1

z3i gi,kxi+1 =n1i=1

z3i gi,kzi+1 +n1i=1

gi,kz3i i, (27)

andn

i=1

is=1

nl=1

z4l 4s,k(zl, ) =

ni=1

z4in

s=1(n s+ 1)4s,k(zi, ),

ni=1

(i 1)i1s=1

nl=1

z4l 4s,k(zl, )=

ni=1

z4in1s=1

(ns)(i1)4s,k(zi, ),n

i=1

ij=1

nl=1

z4l 4j,k(zl, ) =

ni=1

z4in

j=1(n j+ 1)4j,k(zi, ).

For any i = 1, 2, . . . , n and k M , define fi,k as

fi,k = 34nzii

s=1(i1,s)

43 + zi

ns=1

(n s+ 1)4s,k(zi, )

+ zin1s=1

(n s)(i 1)4s,k(zi, )

+i1s=1

i1j=1

18(n+ 1)2nz3i

2i1xsxj

2

+ 98i2(n+ 1)2zi

i1j=1

i1xj

4l2ij

4j,k((n+ 1)d)

+ zin

j=1(n j+ 1)4j,k(zi, )+

98i2(n+ 1)2nzi

+ 98i2(n+ 1)2zil2ii 4i,k((n+ 1)d)

+ 98i2(n+ 1)2nzi

i1j=1

i1xj

4

i1s=0

i1y(s)d

y(s+1)d i1

i1s=1

i1xs

gs,kxs+1

+Ui,k tanhz3i Ui,ki,k

+ gi,kzi+1, (28)

with zn+1 = 0.Substituting (6) and (26)(28) into (24) yields

LV n1i=1

z3i (fi,k + gi,ki)+ z3n fn,k + z3ngn,k(uck + k uk + k)

1 1 n i r1

r2 +

i=1i,k +

j=1l2ij . (29)ica 60 (2015) 193200 197

By using neural networks approximation ability and Youngsinequality, the following inequalities can be obtained.

z3i fi,k = z3i W Ti,kSi,k + z3i i,k 1

2a2iz6iWi,k2 STi,kSi,k + a2i2 + 34 z4n +

4i,k

4,

12a2i

z6i iSTi Si +

a2i2+ 3

4z4i +

4i

4, (30)

z3n(k + k) = z3nW T,kS,k + z3n(,k + k)

12a2

z6nST S +

a22+ 3z

4n + 44

, (31)

where i,k =Wi,k2 , ,k = W,k2 , i = max{i,k : k M},

= max{,k : k M},i,k i, ,k + k .

Substituting (30) and (31) into (29) gives

LV n1i=1

z3i

z3i i2a2i

STi Si + gi,ki+ z3n

z3nn2a2n

STn Sn + gn,kuck

+ z3ngn,k

12a2

z3nST S uk

+ gn,k

a22+ 3

4z4n +

4

4

+n

i=1

2a2i + 3z4i + 4i

4

1

r1

1r2 +

ni=1

i +

ij=1

l2ij

, (32)

wherei = max{i,k, k M}.Design the virtual control function as

i = 1gi,mini + 34

zi 12a2i

z3i STi Si

, (33)

where = ni=1 i is the estimation of , and i > 0 is a designparameter.

The true actuator input is given as

uk = uck uk , (34)where

uck =1gn,k

n + 34

zn 12a2n

z3n STn Sn

, (35)

uk = + 34

zn + gn,max2a2gn,k

z3n ST S, (36)

with n, , an, a > 0 being the design parameters, gn,max = max{gn,k, k M}, gn,min = min{gn,k, k M}, and the estimation of . The adaptive laws can be designed as

=

ni=1

r12a2i,min

z6i STi Si 1 , (37)

= gn,maxr2

2a2,minz6nS

T S 2 . (38)

Then, one can get from (32)(38) that

LV n

i=1iz4i z4n + gn,k

a22+

4

4

+

ni=1

a2i2+

4i

4

1 2

n i

+

r1 +

r2 +

i=1i +

j=1l2ij . (39)

198 X. Zhao et al. / Automat

It is true that

= ( ) 122 + 1

22, (40)

= ( ) 122 + 1

22. (41)

Combining (39) with (40) and (41) gives

LV n

i=1iz4i

1

2r12 2

2r22

+ gn,ka22+

4

4

+

ni=1

a2i2+

4i

4

+

ni=1

i +

ij=1

l2ij

+ 1

2

2r1+ 2

2

2r2

p0V + q0, (42)where n := n + , p0 = min{4i, 1, 2 : 1 i n}, q0 =n

i=1(a2i2 +

4i4 )+

ni=1i +ij=1 l2ij+ 122r1 + 222r2 +gn,k( a22 +

44 ). By using Lemma 1, we have

dE[V (t)]dt

p0E[V (t)] + q0, (43)Then, the following inequality holds

0 E[V (t)] V (0)ep0t + q0p0, (44)

where V (0) = nj=1 z2j (0)4 + 12r1 (0)2 + 12r2 (0)2. Eq. (44) meansthat all the signals in the closed-loop system are bounded inprobability. It follows from (44) that

E[|zi|4] 4q0p0 , t . (45)Now, we are ready to present our main result in the following

theorem.

Theorem 1. Consider the closed-loop system (1)with unknown non-symmetric actuator dead-zone (2). Suppose that for 1 i n, k M, the unknown functions fi,k can be approximated by neural net-works in the sense that the approximation error i,k are bounded. Un-der the state feedback controller (34) and the adaptive laws (37), (38),the following statements hold:

(i) All the signals of the closed-loop z-system (16) under arbitraryswitching are SGUUB in 4th moment and

P

limt

ni=1

E[|zi|4] 4q0p0

= 1.

(ii) The output y of the closed-loop system (1) under arbitrary switch-ing can be almost surely regulated to a small neighborhood of thetarget signal.

Proof. It is not difficult to complete the proof by the abovederivations.

Remark 2. From (37) and (38), one can see that there are onlytwo adaptive parameters that need to be modulated in our results.Hence, the problemof over parameterization can be avoided by ourapproach, and the computation burden can be greatly reduced.

4. An illustrative exampleIn this section, the simulation studies for a ship maneuveringsystem are used to illustrate the effectiveness of our results.ica 60 (2015) 193200

The shipmaneuvering system can be described by the followingNorrbin nonlinear model (Lim & Forsythe, 1983).T(vs)h+ h+ (vs)h3 = K(vs) + T(vs)(, h, )w,where T(vs) is the time constant, h = denotes the yaw rate, stands for the heading angle, (vs) is Norrbin coefficient, K(vs)represents the rudder gain, is the rudder angle and w standsfor an r-dimensional independent standard Brownian motion,(vs)(, h, ) : R3 R3r is an unknown function, and (vs)is the switching signal which satisfies:

(vs) =1, 0 < vs vL2, vL < vs vM3, vM < vs vT

vL, vM , vT represent the value of low speed, medium speed andtop speed, respectively.

A simplified mathematical model of the rudder system can bedescribed as follows:TE, (vs) + = KE, (vs)E, (vs),where TE, (vs) represents the rudder time constant, stands for theactual rudder angle, KE, (vs) denotes the rudder control gain andE, (vs) is the rudder order.

Let x1 = , x2 = h, x3 = , v(vs) = E, (vs), we have thefollowing switched nonlinear system model with actuator dead-zone to describe the dynamic behavior of the ship with low speed,medium speed and high speed respectively.dx1 = x2dt,dx2 = (f(vs) + b(vs)x3)dt + T(vs)d,dx3 =

1TE, (vs)

x3 + KE, (t)TE, (vs)v(vs)

dt,

v(vs) = D(u(vs))where f(vs) = 1T(vs) x2

(vs)T(vs)

x32, b(vs) = K(vs)T(vs) .The vessel data comes from a ship which has a length overall

of 160.9 m. vL = 3.7m/s, K1 = 32 s1, T1 = 30 s, 1 =40 s2, TE,1 = 4 s, KE,1 = 2; vM = 7.5 m/s, K2 = 11.4 s1, T2 =63.69 s, 2 = 30 s2, TE,2 = 2.5 s, KE,2 = 1; vT = 15.3 m/s, K3 =5.1 s1, T3 = 80.47 s, 3 = 25 s2, TE,3 = 1 s, KE,3 = 0.72;The initial conditions are x1(0) = 2, x2(0) = 0.05, x3(0) =0.03, (0) = 10, (0) = 1. We construct the basis functionvectors S1, S2, S3 and S using 11, 15, 21 and 48 nodes, the centers1, 2, 3, evenly spaced on [1.5, 4.5][3, 4][10, 8],[5, 4] [30, 20] [0.5, 5.5], [5.5, 8] [12, 25] [0.1, 2] and [10, 2][60, 2][0.2, 10.5], thewidths 1 =1.2, 2 = 2.2, 3 = 2, = 1.8. The design parameters are a1 =a2 = a3 = a = 10, r1 = 2, r2 = 10, 1 = 0.5, 2 = 0.1, 1 =2 = 3 = 5, = 3. The desired trajectory is yd = 10 sin 0.08t .

According to Theorem 1, the adaptive laws , and the controllaws uck , uk are chosen, respectively, as

=

3i=1

0.01z6i STi Si 0.5 ,

= 0.036z63ST S 0.1,uck =

1g3,k

[5.75z3 0.005z33 ST3 S3],

uk = 3.75z3 +0.00057

g3,kz33 S

T S,

where uk = uck uk , z1 = x1 yd, z2 = x2 1, z3 = x3 2and 1, 2 are given by

= 5.75z 0.005z3ST S ,1 1 1 1 12 = 92z2 0.08z32 ST2 S2.

X. Zhao et al. / Automat

Fig. 1. Tracking performance.

Fig. 2. The responses of adaptive laws.

In order to give the simulation results, we assume that

vk = D(uk) =10(uk 50), uk 500, 60 < uk < 5020(uk + 60), uk 60

and 1 = 0.5x1 sin x2x3, 2 = 0.25x21x2 cos x2, 3 = 0.1x1x3. Thesimulation results are shown in Figs. 14, where Fig. 1 presents thesystem output and target signal yd, Fig. 2 shows the trajectoriesof adaptive laws, Fig. 3 demonstrates the responses of D(uck)(without dead-zone compensation controller) and D(uck uk)(with dead-zone compensation controller), and Fig. 4 illustratesthe evolution of switching signal. From Fig. 1, it can be seenthat the output can track the target signal yd within a smallbounded error. On the other hand, Fig. 3 verifies that the dead-zonenonlinearity can be compensated by uk .

5. Conclusions

This paper has investigated the tracking control problem fora class of switched stochastic nonlinear systems in nonstrict-feedback form under arbitrary switchings, where the unknownnonsymmetric actuator dead-zone is taken into account. Adaptivestate feedback controllers are designed for the considered systems.It is shown that the target signal can be almost surely tracked bythe system output within a small bounded error, and the trackingerror is SGUUB in 4th moment. In our future works, we will payattention to the control problems of more complicated systems

such as high-order switched stochastic systems by using efficientadaptive algorithms.ica 60 (2015) 193200 199

Fig. 3. The responses of D(uck uk ) and D(uck ).

Fig. 4. The response of switching signal.

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (61203123, 61322301, 61403041), theAustralian Research Council (DP140102180, LP140100471), the111 Project (B12018), the Liaoning Excellent Talents in University(LR2014035), the Natural Science Foundation of HeilongjiangProvince (F201417, JC2015015), and the Fundamental ResearchFunds for the Central Universities, China (HIT.BRETIII.201211,HIT.BRETIV.201306).

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Xudong Zhao was born in Harbin, China, on July. 7. 1982.He received the B.S. degree in Automation from HarbinInstitute of Technology in 2005 and the Ph.D. degree fromControl Science and Engineering from Space Control andInertial Technology Center, Harbin Institute of Technologyin 2010. Dr. Zhao was a lecturer and an associateprofessor at the China University of Petroleum, China.Since 2013, he joined Bohai University, China, where heis currently a professor. He is also a Postdoctoral Fellowin The University of Hong Kong from January 2014. Dr.Zhao serves as associate editor for Neurocomputing andica 60 (2015) 193200

International Journal of General Systems, and reviewer for various peer-reviewedjournals, including Automatica and IEEE Transactions on Automatic Control. Inaddition, he has been granted as the outstanding reviewer for Automatica, IETControl Theory & Applications and Journal of the Franklin Institute. His researchinterests include hybrid systems, positive systems, multi-agent systems, fuzzysystems, Hinf control, filtering and their applications. His works have been widelypublished in international journals and conferences.

Peng Shi (M95-SM98-F15) received the B.Sc. degreein Mathematics from Harbin Institute of Technology,China; the M.E. degree in Systems Engineering fromHarbin Engineering University, China; the Ph.D. degree inElectrical Engineering from the University of Newcastle,Australia; the Ph.D. degree in Mathematics from theUniversity of South Australia; and the D.Sc. degree fromthe University of Glamorgan, UK.

Dr. Shi was a post-doctorate and lecturer at theUniversity of South Australia; a senior scientist in theDefence Science and Technology Organisation, Australia;

and a professor at the University of Glamorgan, UK. Now, he is currently aprofessor at The University of Adelaide, and Victoria University, Australia. Hisresearch interests include system and control theory, computational intelligence,and operational research; and he has published widely in these areas.

Dr. Shi has actively served in the editorial board of a number of journals,including Automatica, IEEE Transactions on Automatic Control; IEEE Transactionson Fuzzy Systems; IEEE Transactions on Cybernetics; IEEE Transactions on Circuitsand Systems-I; and IEEE Access. He is a Fellow of the Institute of Electricaland Electronic Engineers, the Institution of Engineering and Technology, and theInstitute of Mathematics and its Applications. He is a Member of College of Expert,Australian Research Council, andwas the Chair of Chapter on Control and AerospaceElectronic Systems at the IEEE South Australia Section.

Xiaolong Zheng was born in Hubei Province, China onOctober 8, 1990. He received the B.S. degree in automationfrom Yangtze University College of Technology andEngineering, Jingzhou, China, in 2013. He is studying forthe M.S. degree in control theory and control engineeringin Bohai University, Jinzhou, China. His research interestsinclude adaptive control, switched nonlinear systems,intelligent control, stochastic nonlinear systems and theirapplications.

Lixian Zhang received the Ph.D. degree in control sci-ence and engineering fromHarbin Institute of Technology,China, in 2006. From Jan. 2007 to Sep. 2008, he worked asa postdoctoral fellow in the Dept. Mechanical Engineer-ing at Ecole Polytechnique de Montreal, Canada. He wasa visiting professor at Process Systems Engineering Labo-ratory,Massachusetts Institute of Technology (MIT) duringFeb. 2012 toMarch 2013. Since Jan. 2009, he has beenwiththe Harbin Institute of Technology, China, where he is cur-rently a Professor in the Research Institute of IntelligentControl and Systems.

Dr. Zhangs research interests include nondeterministic and stochastic switchedsystems, networked control systems, model predictive control and their applica-tions. He serves as Associated Editor for various peer-reviewed journals includingIEEE Transactions on Automatic Control, IEEE Transactions on Cybernetics, etc., andwas a leading Guest Editor for the Special Section of Advances in Theories and In-dustrial Applications of Networked Control Systems in IEEE Transactions on Indus-trial Informatics.

Adaptive tracking control for switched stochastic nonlinear systems with unknown actuator dead-zoneIntroductionPreliminaries and problem formulationMain resultAn illustrative exampleConclusionsAcknowledgmentsReferences