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Robust adaptive decentralized fuzzy control for stochastic large-scale nonlinear systems with dynamical uncertainties
Neurocomputing 97 (2012) 33–43
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Robust adaptive decentralized fuzzy control for stochastic large-scale nonlinear systems with dynamical uncertainties$ Tong Wang, Shaocheng Tong n, Yongming Li Department of Mathematics, Liaoning University of Technology, Jinzhou, Liaoning 121000, China
a r t i c l e i n f o
abstract
Article history: Received 15 February 2012 Received in revised form 5 May 2012 Accepted 29 May 2012 Communicated by H. Zhang Available online 3 July 2012
In this paper, a robust adaptive fuzzy decentralized backstepping output feedback control approach is proposed for a class of uncertain large-scale stochastic nonlinear systems with unknown nonlinear functions, dynamical uncertainties and without the measurements of the states. The fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy state observer is designed for estimating the unmeasured states. To solve the problem of the dynamical uncertainties, the changing supply function technique is incorporated into the backstepping recursive design technique, and a new robust adaptive fuzzy decentralized output feedback control approach is constructed. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing design parameters appropriately. A simulation example is provided to show the effectiveness of the proposed approach. & 2012 Elsevier B.V. All rights reserved.
Keywords: Stochastic nonlinear large-scale systems Fuzzy adaptive decentralized control Backstepping technique Changing supply function
1. Introduction In the past decades, fuzzy logic systems (FLSs) and neural networks (NNs) as universal approximators have been extensively used for the modeling and controller design for uncertain nonlinear systems [1,2], and various adaptive fuzzy and NN decentralized control approaches have been developed for large-scale nonlinear systems, for example see [3–9]. In [1–6], adaptive NN and fuzzy decentralized control schemes were developed for some classes of uncertain nonlinear large-scale systems with measurable states. By designing the state observer, [7–9] proposed adaptive fuzzy and NN output feedback decentralized control approaches for the uncertain nonlinear large-scale systems with immeasurable states. Generally, these adaptive NN and fuzzy decentralized control approaches can deal with a larger class of nonlinear large-scale systems with unknown nonlinear functions and without a priori knowledge about the interconnections, and achieve satisfactory control performances. However, a major constraint is that the unknown functions and interconnections of the considered systems are in the range space of the input matrix, which is so-called to satisfy the matching condition in
$ This work was supported by the National Natural Science Foundation of China (no. 61074014), the Outstanding Youth Funds of Liaoning Province (no. 2005219001). n Corresponding author. E-mail address: [email protected] (S. Tong).
0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.05.017
[10,11], thus the above mentioned adaptive control approaches cannot be applied to those uncertain nonlinear large-scale systems without satisfying the matching condition. To handle the control problem of uncertain nonlinear largescale systems without satisfying the matching condition, several adaptive NN and fuzzy decentralized controllers have been developed by using the backstepping recursive technique [12–17]. In [12,13] adaptive NN decentralized backstepping state feedback was developed to control approaches for the uncertain large-scale systems without or with time delays. By designing the K-filters or state observers, [14,15] investigated adaptive NN decentralized backstepping output feedback controllers for a class of the uncertain large-scale systems with the immeasurable states, while [16,17] considered adaptive fuzzy decentralized output feedback controllers for a class of uncertain large-scale systems with immeasurable states and unknown control directions. More recently, the investigations on stochastic nonlinear systems have received considerable attention, and many results on adaptive NN or fuzzy backstepping technique obtained for deterministic nonlinear systems in the strict-feedback form have been successfully extended to the stochastic nonlinear systems [18–23]. Authors in [18,19] first proposed adaptive NN backstepping output feedback control design approaches for a class of SISO stochastic systems in the strict-feedback form by using a linear state observer. Authors in [20] first proposed adaptive NN backstepping output feedback control design approaches for a class of SISO stochastic systems in the strict-feedback form by
34
T. Wang et al. / Neurocomputing 97 (2012) 33–43
using a linear state observer. Authors in [21–23] investigated the adaptive NN and fuzzy decentralized output feedback stabilization problem for a class of stochastic large-scale nonlinear systems. However, the above control approaches in [18–23] do not consider the stochastic nonlinear systems containing the unmodeled dynamics and dynamical disturbances (i.e., dynamical uncertainties). As stated in [24,25], the unmodeled dynamics and dynamical disturbances often exist in many practical nonlinear systems, and they are also the major reason of resulting in the instability of the control systems. Therefore, to study the stochastic large-scale nonlinear systems with consideration of dynamical uncertainties is very important in control theory and applications. The purpose of this paper is to investigate the adaptive decentralized fuzzy control for a class of large-scale stochastic nonlinear systems with three uncertainties, i.e., unknown nonlinear functions, dynamical uncertainties and unmeasured states. In the control design, fuzzy logic systems are employed to approximate the unknown nonlinear functions, and a fuzzy state observer is designed to estimate the unmeasured states. To solve the problem of the dynamical uncertainties, the changing supply function technique is incorporated into the backstepping recursive design technique, a new robust adaptive decentralized fuzzy backstepping output feedback control scheme is constructed. The main advantages of the proposed adaptive fuzzy decentralized control approach are summarized as follows: (i) by designing a new fuzzy state observer, not a reduced-order state observer used in [21,22,27–29], the proposed state observer can obtain the better estimations of the unmeasured states; (ii) by incorporating the changing supply function technique into the backstepping recursive design technique, the proposed adaptive decentralized control approach can applied to a larger class of stochastic nonlinear systems and has the robustness to the dynamical uncertainties compared with the existing results in [21–23]; and (iii) it is mathematically proved that the resulting closed-loop system are SGUUB in probability and the output converges to a small neighborhood of the origin.
2. Some notations and preliminary results 2.1. Stability in probability
Lemma 1. ([28]) Consider the stochastic system (2) and assume that f(t,0), h(t,0) are bounded uniformly in t. If there exist functions V(t,x), m1(U), m2(U)AkN, constants c1 40,c2 Z0, and a nonnegative function W(t,x), such that
m1 ðjxjÞ rVðt,xÞ r m2 ðjxjÞ, ‘Vðt,xÞ rc1 Wðt,xÞ þ c2 , then (i) there exists an almost surely unique solution on (0,N) for the system (2). (ii) the solution process is bounded in probability, when W(t,x) ZcV(t,x) for a constant c 40. 2.2. Fuzzy logic systems A FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine, and the defuzzifier. The knowledge base is composed of a collection of fuzzy. If-then rules of the following form: Rl : If x1 is F l1 and x2 is F l2 and. . .and xn is F ln , Then y is Gl ,l ¼ 1,2,. . .,N where x ¼ ðx1 ,x2 . . .xn ÞT and y are FLS input and output, respectively, mF l ðxi Þ and mGl ðyÞare the membership function of fuzzy sets i F li and Gl, N is the number of inference rules. Through singleton fuzzifier, center average defuzzification and product inference, the FLS can be expressed as PN Qn l ¼ 1 yl i ¼ 1 mF li ðxi Þ yðxÞ ¼ PN Qn ð3Þ l ¼ 1 ½ i ¼ 1 mF l ðxi Þ i
where yl ¼ maxy A R mGl ðyÞ. Define the fuzzy basis functions as Qn i ¼ 1 mF l ðxi Þ fl ¼ PN Qn i l ¼ 1 ½ i ¼ 1 mF li ðxi Þ T Denoting y ¼ y1 ,y2 ,. . .yN ¼ y1 , y2 ,. . ., yN and jðxÞ ¼ j1 ðxÞ, j2 ðxÞ,. . ., jN ðxÞT , then fuzzy logic system (3) can be rewritten as T
yðxÞ ¼ y jðxÞ
ð4Þ
Consider the following time-varying stochastic system: dx ¼ ðf ðt,xÞ þgðt,xÞuÞ dt þ hðt,xÞ dw
ð1Þ
where w is an r-dimensional standard Brownian motion, xARn is the state, uAR is the control input, f,g:R þ Rn-Rn and h:R þ Rn-Rn r. Definition 1. ([26–28]) For any given V(t,x) associated with the stochastic differential Eq. (1), define the differential operator ‘ as follows: ( ) @V @V @V 1 @2 V T þ f ðt,xÞ þ gðt,xÞu þ Tr h ðt,xÞ 2 hðt,xÞ ‘Vðt,xÞ ¼ @t @x @x 2 @x For control-free stochastic nonlinear system of the form dx ¼ f ðt,xÞ dt þhðt,xÞ dw
ð2Þ
the following stability notions introduced will be used throughout the paper.
Lemma 2. For any continuous function f(x) defined over a compact set O and any given positive constant e, there exist a fuzzy logic system (4) and a ideal parameter vector yn such that [1] nT supf ðxÞy jðxÞ r e xAO
3. System descriptions and basic assumptions In this paper, we consider the following stochastic nonlinear large-scale systems: dzi ¼ f i0 ðt,zi ,yÞ dt þ g i0 ðt,zi ,yÞ dwi dxij ¼ ½xi,j þ 1 þ f ij ðx ij Þ þ Dij ðt,zi ,yÞ dt þg ij ðt,zi ,yÞ dwi j ¼ 1,. . .,ni 1, ^ dxi,ni ¼ ½ui þ f i,ni ðx i,n Þ þ Di,ni ðt,zi ,yÞ dt þ g i,ni ðt,zi ,yÞ dwi i
Definition 2. ([26–28]) The solution process {x(t),t Z0} of stochastic system (2) is said to be bounded in probability, if limc-1 sup0 r t o 1 PxðtÞ 4 c ¼ 0. Definition 3. ([28]) Consider the system (2) with f(t,0) 0, h(t,0) 0. The equilibrium f(t,0) 0 is globally stable in probability if for any e 40, there exists a class k function g(U) such that P{9x(t)9o g(9x09)} Z1 e, 8t Z0, x0ARn\{0}.
yi ¼ xi1
ð5Þ T
i
where x ij ¼ ½xi1 ,. . .,xij A R , i ¼ 1,. . .,N,j ¼ 1,. . .,ni are the states, uiAR, yiAR are the control input and output of the system, and zi A Rmi are unmodeled dynamics and Dij(t,zi,y) are the dynamical disturbances. fij(U), j ¼1,2,y,ni are unknown smooth nonlinear functions. fi0(U), gi0(U), Dij(U) and gij(U) are uncertain functions; wiAR is an independent standard Brownian motion defined on a
T. Wang et al. / Neurocomputing 97 (2012) 33–43
complete probability space. In this paper, only the output yi is assumed to be available for measurement. Assumption 1. ([28]) For each i ¼ 1,. . .,N, j ¼ 1,. . .,ni there are unknown constants pnij Z0, and known smooth functions jij0 Z0, jijl Z0, cij0 Z0 and cijl Z0 such that 8ðt,zi ,yÞ A R þ Rmi RN , N X Dij ðt,zi ,yÞ r pn j ðzi Þ þ pn jijl ðyl Þ ij ij0 ij l¼1
35
Assumption 5. ([23] )There are positive constants enij unknown n n and dij such as eij r enij and dij r dij . Denote oij ¼ eij dij, by Assumption 5, one has oij r n n n n eij þ dij ¼ oij , where oij is also an unknown constant. In this paper, the parameters adaptation laws will be designed later to estimate enij and onij , respectively. Rewrite (5) in the following form dzi ¼ f i0 ðt,zi ,yÞ dt þ g i0 ðt,zi ,yÞ dwi 2 3 ni X 4 dxi ¼ Ai xi þ K i yi þ Bij ðf ij ðx ij Þ þ Di ðt,zi ,yÞ þ Bi ui Þ5dt þ Gi ðt,zi ,yÞ dwi
N X g ðt,zi ,yÞ rpn c ðzi Þ þ pn cijl ðyl Þ ij ij ij0 ij l¼1
j¼1
Without loss of generality, this paper assumes that jij0(0) ¼0 and cij0(0) ¼0. Assumption 2. ([28]) For each zi-subsystem in (5), i ¼ 1,. . .,N, there exist function V zi ðzi Þ and kN functions a i ðzi Þ, ai ðzi Þ,
ai0(9zi9) and gil(9yl9), l ¼ 1,. . .,N such that 8ðt,zi ,yÞ A R þ Rmi RN , a i ðzi Þ rV zi ðzi Þ r ai ðzi Þ ð6Þ ‘V zi ðzi Þ r
N X
gil ðyl Þai0 ðzi Þ
ð7Þ
l¼1
Assumption 3. ([28]) There exist known smooth functions ciz and ci0 satisfying @V z =@zi r c ðzi Þ and :g i0 ðt,zi ,yÞ: r c ðzi Þ, i ¼ 1,. . .,N ð8Þ iz i0 i
Assumption 4. ([23]) There exist a set of known constants mij, j ¼1,2,y,ni for 8X1,X2ARi, the following inequality holds f ðX 1 Þf ðX 2 Þ r mij :X 1 X 2 :: ij
ij
where :X 1 X 2 : expresses the 2-norm of vector X1 X2. Control objective: The control task is to design an adaptive output feedback controller using the output yi and state estimations x^ ij so that the closed-loop system is bounded in probability and the outputs of the system can be regulated to a small neighborhood of the origin in probability.
4. Fuzzy state observer design
yi ¼ C i xi where 2
ð9Þ 2
3 2 3 ki1 0 6 7 6 7 ^ I K i ¼ 4 ^ 5, Bi ¼ 4 ^ 5, ki,ni 0 . . . 0 ki,ni 1 Di ¼ Di1 ðt,zi ,yÞ ^ Di,ni ðt,zi ,yÞ , T Bij ¼ 0 . . . 1 . . . 0 , h iT Gi ðt,zi ,yÞ ¼ g i1 . . . g i,ni : Ci ¼ 1 . . . 0 0 3
ki1
6 Ai ¼ 4
7 5,
Choose vector Ki such that matrix Ai is a strict Hurwitz, therefore, for any a given positive definite matrix Q i ¼ Q i T 4 0, there exists a positive definite matrix P i ¼ Pi T such that Ai T Pi þP i Ai ¼ Q i
ð10Þ
Design a fuzzy state observer as x_^ i ¼ Ai x^ i þK i yi þ
ni X
Bij f^ ij ðx^ ij 9yij Þ þBi ui
j¼1
y^ i ¼ C i x^ i
ð11Þ
Let the observer error vector as ei ¼
xi x^ i pni
ð12Þ
where pni ¼ maxf1,pnij ,pijn4 1 r j r ni g. From (9), (11) and (12), one has 0 ni D 1 X dei ¼ @Ai ei þ n Bij f ij ðx ij Þf ij ðx^ ij Þ þ f ij ðx^ ij Þf^ ij ðx^ ij yij Þ þ ni dt pi j ¼ 1 pi þ
Gi ðt,zi ,yÞ dwi pni
Note that in the system (5), the functions f ij ðx ij Þ, j ¼ 1,2,. . .,ni are unknown and the states xi2 ,. . .,xi,ni are not measured directly, thus a state observer should be established to obtain the estimates of xi2 ,. . .,xi,ni . By Lemma 2, it can be assumed that the unknown functions f ij ðx ij Þ, j ¼ 1,2,. . .,ni can be approximated by the following fuzzy logic systems
where F i ¼ ½F i1 ,. . .,F i,ni T ¼ ½ðf i1 ðxi1 Þf i1 ðx^ i1 ÞÞ,. . .,ðf i,ni ðx i,n Þ f i,ni ðx^ i,n ÞÞT i i and di ¼ ½di1 ,. . ., di,ni T . Choose the following Lyapunov candidate Vi0 as
T f^ ij ðx^ ij 9yij Þ ¼ yij jij ðx^ ij Þ,
V i0 ¼
1 rj r ni , T
where x^ ij ¼ ½x^ i1 , x^ i2 ,. . ., x^ ij are the estimates of x ij ¼ ½xi1 ,xi2 ,. . .,xij and denote xi ¼ x i,n and x^ i ¼ x^ i,n . i i n The optimal parameter vectors yij is defined as ynij ¼ argminyij A Oij ½supx^ A Uij f^ ij ðx^ ij 9yij Þf ij ðx^ ij Þ:
T
¼ ðAi ei þ
1 T ðe P e Þ2 2 i i i
ð13Þ
ð14Þ
From (12)–(14), one has 2 T e P ðF þ di þ Di Þ pni i i i h i 1 þ n2 Tr GTi ð2Pi ei eTi Pi þ eTi Pi ei P i ÞGi pi 4
‘V i0 rlmin ðQ i Þlmin ðPi Þ:ei : þ eTi P i ei
ij
where Oij and Uij are bounded compact sets for yij and x^ ij , respectively. The corresponding minimum approximation error eij and dij are defined by eij ¼ f ij ðx^ ij Þf^ ij ðx^ ij ynij Þ, dij ¼ f ij ðx^ ij Þf^ ij ðx^ ij yij Þ
1 G ðt,z ,yÞ ðF þ di þ Di ÞÞdt þ i n i dwi pni i pi
ð15Þ
Since jijl, cijl, i,l ¼ 1,2,. . .,N, j ¼ 1,2,. . .,ni , are smooth functions, ð1Þ ð2Þ there exist smooth nonnegative functions kijl ðyl Þ, kijl ðyl Þ such that !4 !4 N N N X X X ð1Þ j ðy Þ r k y 4 þ8 j ð0Þ ð16Þ ijl
l¼1
l
ijl
l¼1
l
ijl
l¼1
36
T. Wang et al. / Neurocomputing 97 (2012) 33–43
N X
!4
cijl ðyl Þ
N X
r
l¼1
!4
N X
ð2Þ
kijl yl 4 þ 8
l¼1
change of coordinates
cijl ð0Þ
ð17Þ
l¼1
^ i2 ,. . ., o ^ i,j1 ,yi , p^ i Þ wi1 ¼ yi , wij ¼ x^ ij ai,j1 ðx^ i,j1 , yi1 ,. . ., yi,j1 , e^ i1 , o
ð23Þ
By Assumptions 1, 3 and 4, and the fact that pni Z 1, one has the following inequalities
where ai,j 1(U)ðj ¼ 2,. . .,ni Þ is an intermediate control, yij,p^ i , e^ i1 and ^ i,j1 are the estimates of the parameters ynij , pi,eni1 and oni,j1 , o
2 T 3 1 n 4 8=3 4 e P e e T P d r :P : :ei : þ :di : pni i i i i i i 2 i 2
ð18Þ
respectively. Step 1: From the second equation in (5), and according to It o^ ’s differentiation rule, one has
0 1 ni X 2 T 3 1 8=3 4 4 e P e e T P F r :P : :ei : þ @ m A:ei : pni i i i i i i 2 i 2 j ¼ 1 ij
ð19Þ
dwi1 ¼ ½xi2 þ f i1 ðx i1 Þ þ Di1 ðt,zi ,yÞdt þ g i1 ðt,zi ,yÞ dwi nT ¼ ½wi2 þ ai1 þpni ei2 þ F i1 þ ei1 þ yi1 ji1 ðx^ i1 Þ þ Di1 ðt,zi ,yÞ dt
þ g i1 ðt,zi ,yÞ dwi ni X 2 T 3 2 4 2 e P e eT P D r :Pi : :ei : þ 4ni :P i : ½jij0 4 ðzi Þ n i i i i i i pi 2 j¼1 !4 3 N N X X ð1Þ 4 k y þ8 j ð0Þ 5 þ ijl
Choose the Lyapunov function candidate as V i1 ¼ V i0 þ ð20Þ
ijl
l
l¼1
where
l¼1
4
2
2
l¼1
ri1 4 0,
r i1 40
and
r i1 40
are
design
ð25Þ parameters.
n4=3
Define pi ¼ max1,pni ,pni 4 ,pi . For the convenience of the following derivations, we cite the following Lemma 3:
Lemma 3. ([23]) For any a variable x, and any a constant k40, 0 the inequality jxjx tanhðx=kÞ r0:2785k ¼ k is satisfied. From (23)–(25), the infinitesimal generator of Vi1 satisfies
2
r ð2:Pi : þ lmax ðPi ÞÞ:ei : þ 4ni ð2:Pi : þ lmax ðP i ÞÞ 2 !4 3 ni N N X X X 4 ð2Þ 4 4cij0 ðzi Þ þ kijl yl þ 8 cijl ð0Þ 5 j¼1
1 4 1 ~T ~ 1 2 1 w þ y y þ e~ þ p~ 2 4 i1 2r i1 i1 i1 2r i1 i1 2r i1 i
y~ i1 ¼ yni1 yi1 , e~ i1 ¼ eni1 e^ i1 and p~ i ¼ p^ i pi are the parameters errors.
h i 1 Tr Gi T ð2Pi ei ei T P i þ ei T Pi ei P i ÞGi pni 2 h i 1 1 ¼ n2 Tr Gi P i ei ðGi Pi ei ÞT þ n2 ðei T P i ei ÞTrðGi T Pi Gi Þ pi pi 1 2 4 4 2 r ð2:Pi : þ lmax ðPi ÞÞ :ei : þ :Gi =pi n : 2 2
ð24Þ
‘V i1 ¼ ‘V i0 þ w3i1 ðwi2 þ ai1 þpni ei2 þ F i1 þ ei1 þ yni1T ji1 ðx^ i1 Þ ð21Þ
l¼1
þ Di1 ðt,zi ,yÞÞ þ
3 2 1 T 1 1 w g g T þ y~ i1 y_~ i1 þ e~ i1 e_~ i1 þ p~ i p_~ i 2 i1 i1 i1 r i1 r i1 r i1 ð26Þ
Substituting (16)–(21) into (15) gives
4
2
‘V i0 rli0 :ei : þ Ci0 ðzi Þ þ4ni :Pi : 2
þ 4ni ð2:P i :
ni X
N X
j¼1
l¼1
ni X
N X
j¼1
l¼1
2 þ lmax ðP i ÞÞ
!
By Assumptions 1, 3 and 4, one has
ð1Þ
kijl yl 4
3 4
ð2Þ kijl yl 4
ð22Þ
þC i0
where
li0 ¼ lmin ðQ i Þlmin ðP i Þ3:P i :
2
Ci0 ðzi Þ ¼ 4ni :Pi :
ni X
8=3
j¼1
C i0 ¼
2
ni X
N X
j¼1
l¼1
N X
!4
jijl ð0Þ
3 4
1 4
w3i1 F i1 r mi1 w3i1 :ei : r w4i1 þ m4i1 :ei :4 3 4
c4ij0 ðzi Þ
j¼1
ni X 1 2 n 4 :d : þ 32ni :P i : 2 i j¼1
þ lmax ðP i ÞÞ
ni X
ð27Þ
ð28Þ
1 4
w3i1 pni ei2 r ðpni Þ4=3 w4i1 þ :ei :4
ni 7 1X 2 2 :Pi : lmax ðPi Þ m 2 2 j ¼ 1 ij
j4ij0 ðzi Þ þ4ni ð2:Pi :2 þ l2max ðPi ÞÞ
1 4
w3i1 wi2 r w4i1 þ w4i2
!
2
þ 32ni ð2:P i :
w3i1 Di1 r w3i1 pni1 ji10 ðzi Þ þ
N X
ð29Þ
!
ji1l ðyl Þ
l¼1 N N X X 3 ð1Þ r ðpni1 Þ4=3 w4i1 þ2ðj4i10 ðzi Þ þ ki1l y4l þ 8 ji1l ð0ÞÞ4 4 l¼1 l¼1
!
ð30Þ
l¼1
!4
cijl ð0Þ
5. Adaptive fuzzy decentralized controller design and stability analysis In this section, by using the designed state observer in the above section, an adaptive controller together with the parameter adaptive laws are to be constructed by using the backstepping design technique and the changing supply function technique, and the stability analysis of the closed-loop system will be given. 5.1. Adaptive fuzzy decentralized controller design The adaptive fuzzy decentralized output feedback control design consists of ni-steps, i ¼ 1, ,N, each step is based on the
N X 3 2 3 wi1 g i1 g Ti1 r w2i1 pni12 ci10 ðzi Þ þ ci1l ðyl Þ 2 2 l¼1
!2
N N X X 3 n4 4 ð2Þ r pi1 wi1 þ 6c4i10 ðzi Þ þ 6 ki1l y4l þ48 ci1l ð0Þ 4 l¼1 l¼1
!4
ð31Þ Substituting (27)–(31) into (26) and by using Lemma 3, gives 0 ni X 1 4 3 9 4 3@ ‘V i1 rli1 :ei : þ wi2 þ yi ai1 þ yi þ p^ i yi þ giji yi þ 2kð1Þ i1i yi 4 2 4 j¼1 ð2Þ T þ 6ki1i yi þ yi1 ji1 ðx^ i1 Þ þ e^ i1 tanhðy3i =kÞ þ Ci1 ðzi Þ þHi1 ðyÞ T 1 þ Pi1 ðyÞ þ C i1 þ y~ i1 y_ i1 þ ji1 ðx^ i1 Þy3i r i1 1 _ 1 _ 9 4 ð32Þ p^ i yi þ e~ i1 e^ i1 þ y3i tanhðy3i =kÞ þ p~ i 4 r i1 r i1
T. Wang et al. / Neurocomputing 97 (2012) 33–43
where li1 ¼ li0 ð1=4Þm4i1 ð1=4Þ, Hi1 ðyÞ ¼
Pni
j¼2
PN
l ¼ 1,l a i
0
1 @ a i,m1 Aðxi1 x^ i1 Þ þ @kim kij @x^ ij j¼1
gijl y4l ,
2 2 ð2Þ gijl ¼ 4ni :Pi :2 kð1Þ ijl þ 4ni ð2:P i : þ lmax ðP i ÞÞkijl ,
Ci1 ðzi Þ ¼ Ci0 ðzi Þ þ2j4i10 ðzi Þ þ 6c4i10 ðzi Þ, N X
Pi1 ðyÞ ¼ 2
l ¼ 1,l a i
C i1 ¼ C i0 þ16
N X
ð1Þ ki1l y4l þ 6
N X
V im ¼ V i,m1 þ
l ¼ 1,l a i
fi1l ð0Þ
þ48
l¼1
N X
!4
ci1l ð0Þ
0
þ ei1 k : n
l¼1
Design the intermediate control function ai1 and the adaptation functions yi1, e^ i1 and $i1 as 3 2
9 4
ai1 ¼ bi1 yi Fi1 ðy2i Þyi yi p^ i yi
ni X
giji yi 2kð1Þ i1i yi
j¼1
ð2Þ T 6ki1i yi yi1 ji1 ðx^ i1 Þe^ i1 tanhðy3i =kÞ
ð33Þ
y_ i1 ¼ ri1 ji1 ðx^ i1 Þy3i si1 yi1
ð34Þ
e_^ i1 ¼ ri1 y3i tanhðy3i =kÞsi1 e^ i1
ð35Þ
1 4
‘V i1 r li1 :ei : þ w4i2 y4i ðbi1 þ Fi1 ðy2i ÞÞ þ Ci1 ðzi Þ þHi1 ðyÞ þ Pi1 ðyÞ
si1 ~ T s 1 y i1 yi1 þ i1 e~ i1 e^ i1 þ p~ i ðp_^ i $i1 Þsi1 p~ i p^ i r i1
r i1
r i1
ð37Þ
Step m(2rmrni 1): From the mþ1 equation in (5), and similar to Step 1, one has dwim ¼ ðx^ i,m þ 1 þ f^ im ðx^ im 9yim Þ þ kim pni ei1 Þ dt
m1 X j¼1
@ai,m1 _ @a y ij dt i,m1 p_^ i dt @yij @p^ i
m1 m1 X @ai,m1 X @ai,m1 @ai,m1 _ @a _^ dt ^ i1 dt i,m1 dyi e dx^ ij o ij ^ ij @yi @e^ i1 @x^ ij @o j¼1
2 T
¼ 4aim þ wi,m þ 1 þ yim jim ðx^ im Þ þ ðkim
j¼2
m1 X j¼1
kij
@ai,m1 Þðxi1 x^ i1 Þ @x^ ij
m1 X @ai,m1 @ai,m1 _ _^ @ai,m1 p_^ @ai,m1 e_^ y ij o ij i1 i ^ ij @ y @o @p^ i @e^ i1 ij j¼1 j¼2 m1 X
@ai,m1 1 @2 ai,m1 ðxi2 þ f i1 ðx i1 Þ þ Di1 ðt,zi ,yÞÞ g i1 ðt,zi ,yÞ2 2 @y2i @yi # m1 X @ai,m1 @a ðx^ i,j þ 1 þ f^ ij ðx^ ij yij ÞÞ dt i,m1 g i1 ðt,zi ,yÞdwi ^ ij @yi @ x j¼1 T @a ¼ aim þ wi,m þ 1 þ y~ im jim ðx^ im Þ þ Oim þ oim i,m1 p_^ i @p^ i
@ai,m1 g i1 ðt,zi ,yÞ dwi @yi
where
m 1 X j¼1
m 1 X j¼2
þ
2 3 2 @ai,m1 1 ~ T _~ 1 wim g i1 ðt,zi ,yÞ þ y y þ o~ o_~ @yi 2 r im im im r im im im
m 1 X
@ai,m1 _ @a y ij i,m1 e_^ i1 @ y @e^ i1 ij j¼1
@ai,m1 T ðx^ i,j þ 1 þ yij jij ðx^ ij ÞÞ @x^ ij @ai,m1 _ @a ^ ij i,m1 ðx^ i2 þ yTi1 ji1 ðx^ i1 ÞÞ o ^ ij @yi @o
ð41Þ
By Assumptions 1, 3 and 4, one has 3 4
1 4
ð42Þ
w3im
@ai,m1 n 3 @ai,m1 4=3 1 4 pi ei2 r pi w4im þ :ei : 4 4 @yi @yi
ð43Þ
w3im
@ai,m1 3 @ai,m1 4=3 4 1 F i1 r wim þ m4i1 :ei :4 4 4 @yi @yi
ð44Þ
w3im
@ai,m1 3 @ai,m1 4=3 4 1 4 di1 r wim þ dni 4 4 @yi @yi
ð45Þ
! N X @ai,m1 @a Di1 rw3im i,m1 pni1 ji10 ðzi Þ þ ji1l ðyl Þ @yi @yi l¼1 ! 4=3 N N X X 3 @ai,m1 ð1Þ 4 4 4 4 r pi wim þ 2ðji10 ð zi Þ þ ki1l yl þ 8 ji1l ð0ÞÞ @yi 4 l¼1 l¼1
w3im
ð46Þ
l¼1
1 @2 ai,m1 @a g i1 ðt,zi ,yÞ2 i,m1 ðpni ei2 þ F i1 þ di1 þ Di1 ðt,zi ,yÞÞ dt 2 2 @yi @yi
Oim ¼ yTim jim ðx^ im Þ
ð40Þ
2 3 2 @ai,m1 1 @2 ai,m1 wim g i1 ðt,zi ,yÞ w3im g i1 ðt,zi ,yÞ2 2 2 @yi @y2i !2 ! N X 3 2 @ai,m1 2 1 3 @2 ai,m1 n2 zi Þ þ y Þ r wim wim c ð c ð p l i10 i1l i1 2 2 @yi @y2i l¼1 0 !2 1 9 @ai,m1 4 1 3 @2 ai,m1 A 3 r @ wim þ wim wim pi þ 6c4i10 ðzi Þ 4 4 @yi @y2i !4 N N X X ð2Þ ki1l y4l þ 32 ci1l ð0Þ ð47Þ þ6
1 4 1 ~T ~ 1 w þ y y þ o~ 2 4 im 2r im im im 2r im im
n where rim 40 and r im 4 0 are design parameters. y~ im ¼ yim yim n ~ im ¼ oim o ^ im are the parameters errors. yim and o ^ im are the and o n estimates of yim and onim , respectively. From Eqs. (37), (38) and (40), one can obtain T @a ‘V im r ‘V i,m1 þ w3im aim þ wi,m þ 1 þ y~ im jim ðx^ im Þþ Oim þ oim i,m1 p_^ i @p^ i ! @ai,m1 n 1 @2 ai,m1 2 ðpi ei2 þ F i1 þ di1 þ Di1 ðt,zi ,yÞÞ g i1 ðt,zi ,yÞ 2 @y2i @yi
ð36Þ
where si1, si1 and si1 40 are design parameters. From Eqs. (32)–(36), it follows that
þC i1 þ
ð39Þ
w3im wi,m þ 1 r w4im þ w4i,m þ 1
9 $i1 ¼ r i1 y4i r i1 si1 p^ i 4
4
m 1 X
Consider the following Lyapunov function candidate
ð2Þ ki1l y4l ,
!4
37
ð38Þ
l¼1
Substituting Eqs. (42)–(47) into (41) and by using Lemma 3, gives 3 @ai,m1 4=3 4 ‘V im r lim :ei : þ C im þ w3im ðaim þ Oim þ wim þ wim 4 @yi 3 @ai,m1 4=3 3 @ai,m1 4=3 þ p^ i wim þ p^ i wim 4 4 @yi @yi 0 !2 1 4 2 9 @ a 1 @ a i,m1 i,m1 3 Ap^ i þ @ wim þ wim 4 4 @yi @y2i @a 1 ^ im tanhðw3im =kÞ i,m1 $im Þ þ w4i,m þ 1 þo 4 @p^ i T 1 1 @a 3 þ y~ im wim jim ðx^ im Þ y_ im þ p~ i w3im i,m1 r im @p^ i r i1 4 2 _ ðp^ $ Þs p~ p^ y ðb þ F ðy ÞÞ þ C ðz Þ i
im
i1 i i
i
i1
i1
i
im
i
38
T. Wang et al. / Neurocomputing 97 (2012) 33–43
þHi1 ðyÞ þ Pim ðyÞ þ Lim ðyi Þ þ ~ im ðw3im tanhðw3im =kÞ þo
si1 ~ T s y i1 yi1 þ i1 e~ i1 e^ i1 r i1
where
r i1
1 _ ^ Þ o r im im
ð48Þ ð1Þ
where lim ¼ li,m1 ð1=4Þð1=4Þm4i1 , Lim ðyi Þ ¼ Li,m1 ðyi Þ þ 2ki1i y4i þ ð2Þ 6ki1i y4i ,
Pim ðyÞ ¼ Pi,m1 ðyÞ þ 2
N X
N X
ð1Þ
ki1l y4l þ 6
l ¼ 1,l a i
C im ¼ C i,m1 þ
N X 1 n 4 di þ 16 ji1l ð0Þ 4 l¼1
!4
þ 32
N X
!4
ci1l ð0Þ
0
þ onim k :
l¼1
Design the intermediate control function aim and the adapta^ im and $im as tion functions yim, o 3 @ai,m1 4=3 3 @ai,m1 4=3 aim ¼ bim wim Oim wim wim p^ i wim 4 2 @yi @yi 0 !2 1 9 @ai,m1 4 1 3 @2 ai,m1 A @ai,m1 @ þ p^ i w þ wim $im 4 im 4 @yi @p^ i @y2 i
3 im =kÞ
^ im tanhðw o
@ai,ni 1 _ @a y ij i,ni 1 e_^ i1 @ y @e^ i1 ij j¼1
@ai,ni 1 T ðx^ i,j þ 1 þ yij jij ðx^ ij ÞÞ @x^ ij
n i 1 X
@ai,ni 1 _ @a ^ ij i,ni 1 ðx^ i2 þ yTi1 ji1 ðx^ i1 ÞÞ o ^ ij @yi @ o j¼2 0 1 n i 1 X @ a i,n 1 i Aðxi1 x^ i1 Þ kij þ @ki,ni @x^ ij
ð2Þ
l ¼ 1,l a i
n i 1 X j¼1
ki1l y4l ,
Cim ðzi Þ ¼ Ci,m1 ðzi Þ þ 2j4i10 ðzi Þ þ 6c4i10 ðzi Þ,
n i 1 X
Oi,ni ¼ yTi,ni ji,ni ðx^ i,ni Þ
j¼1
Consider the following Lyapunov function candidate V i,ni ¼ V i,ni 1 þ
1 4 1 ~T ~ 1 w þ y y þ o~ 2 4 i,ni 2r i,ni i,ni i,ni 2r i,ni i,ni
n where r i,ni 40 and r i,ni 4 0 are design parameters. y~ i,ni ¼ yi,ni yi,ni n ~ i,ni ¼ oi,n o ^ i,ni are the parameters errors.yi,ni and o ^ i,ni are and o i
n
estimates of yi,ni and oni,ni , respectively. Similar to the derivations in Step m, one can obtain
ð49Þ 4
y_ im ¼ r im 3im
ð50Þ
_^ ¼ r w3 tanhðw3 =kÞs o o im im im im ^ im im
ð51Þ
w jim ðx^ im Þsim yim
$im ¼ $i,m1 þ ri1 w3im 0
‘V i,ni r li,ni :ei : þ C i,ni
ni X
bij w4ij y4i ðbi1 þ Fi1 ðy2i ÞÞ
j¼2 ni X sij ~ T y y þ Ci,ni ðzi Þ þ Hi1 ðyÞ þ Li,ni ðyi Þ þ Pi,ni ðyÞ þ r ij ij j ¼ 1 ij
3 @ai,m1 4=3 wim 2 @yi
þ
!2 11 9 @ai,m1 4 1 3 @2 ai,m1 AA @ þ w þ wim 4 im 4 @yi @y2i
ð58Þ
ð52Þ
ni X sij si1 ~ ^ ^ ij si1 p~ i p^ i e i1 ei1 þ o~ ij o
r i1
j¼2 4
rli,ni :ei :
ni X
r ij
bij w4ij y4i ðbi1 þ Fi1 ðy2i ÞÞ þ Ci,ni ðzi Þ þ Hi1 ðyÞ
j¼2
From (48)–(52), it follows that 4
‘V im rlim :ei : þ C im
m X
bij 4ij þ
w
j¼2
1 r i1
p~ i w3im
@ai,m1 _ ðp^ i $im Þ @p^ i
1 si1 p~ i p^ i y4i ðbi1 þ Fi1 ðy2i ÞÞ þ w4i,m þ 1 þ Cim ðzi Þ þ Hi1 ðyÞ 4 m m X X sij ~ T sij s ^ ij y ij yij þ i1 e~ i1 e^ i1 þ o~ ij o þ Pim ðyÞ þ Lim ðyi Þ þ r r r ij i1 j¼1 j ¼ 2 ij
~ ij , p~ i Þ þ C i þ C i,ni þ Li,ni ðyi Þ þ Pi,ni ðyÞ þ di ðy~ ij , e~ ij , o where ~ ij , p~ i Þ ¼ di ðy~ ij , e~ ij , o Ci ¼
ð53Þ Step ni: By setting ni ¼m, one has the control function uiand the ^ i,ni , and $i,ni as follows: adaptation functions yi,ni , o 1 3 @ai,ni 1 4=3 3 @ai,ni 1 4=3 ui ¼ bi,ni wi,ni Oi,ni wi,ni wi,ni p^ i wi,ni 4 4 2 @yi @yi 0 1 !2 @ai,ni 1 4 1 3 @2 ai,ni 1 A 9 ^ i,ni tanhðw3i,n =kÞ @ wi,ni þ wi,ni p^ i o i @yi 4 4 @y2i @ai,ni 1 þ $i,ni @p^ i
ð54Þ
y_ i,ni ¼ ri,ni w3i,ni fi,ni ðx^ i,ni Þsi,ni yi,ni
ð55Þ
3 3 _^ ^ i,ni o i,ni ¼ r i,ni wi,ni tanhðwi,ni =kÞsi,ni o
ð56Þ
p_^ i ¼ $i,ni ¼ $i,ni 1 þr i1 w3i,ni
3 @ai,ni 1 2 @yi
C i,ni
si1 2
p2i þ
ni X
ni ni sij ~ 2 si1 2 X sij 2 si1 ~ 2 X pi y ij e~ i1 o~ ij ,
2
sij
2r ij j¼1
yijn2 þ
j¼1
si1 2r i1
2r ij
n2 ei1 þ
2r i1
ni X
sij
2r ij j¼2
j¼2
wi,ni
0 !2 11 @ai,ni 1 4 1 3 @2 ai,ni 1 AA 9 @ w þ wi,ni þ 4 i,ni 4 @yi @y2i
2r ij
oijn2
!4 !4 N N X X 1 n 4 ¼ C i,ni 1 þ di þ 16 ji1l ð0Þ þ 32 ci1l ð0Þ þ oni,ni k0 4 l¼1 l¼1
5.2. Changing supply function design and stability analysis Next, by using the changing supply function technique proposed by [28], we will construct the function Fi1( ) introduced in the first step. From (59), first choose a smooth nonnegative function Fi1( ) such that
Fi1 ðy2i Þy4i Li,ni ðyi Þ Z Fi ðy2i Þy4i 4=3
ð59Þ
ð60Þ
with Fi( ) being a smooth nonnegative function to be designed. P i 4 4 Choose parameterli,ni 4 0, and let V 0i ¼ li,ni :ei : þ nj ¼ 1 bij wij , then it follows from (59) and (60) that
ð57Þ
‘V i,ni r V 0i þ Ci,ni ðzi Þ þ Hi1 ðyÞ þ Pi,ni ðyÞFi ðy2i Þy4i ~ ij , p~ i Þ þ C i þ C i,ni þdi ðy~ ij , e~ ij , o
ð61Þ
T. Wang et al. / Neurocomputing 97 (2012) 33–43
Take the following Lyapunov function candidate for the whole system N X
VN ¼
V i,ni
ð62Þ
i¼1
Then it follows from (62) that
‘V N r
N X
V 0i
i¼1
Fi ðy2i Þy4i þ
i¼1
N X
þ
N X
N X
i¼1
ð63Þ
P PN ~ ij , e~ ij , y~ ij , p~ i Þ ¼ N ~ ~ ~ ~ where dðo i ¼ 1 di ðo ij , e ij , y ij , p i Þ and C ¼ i ¼ 1 ðC i þ C i,ni Þ. Notice that by the definitions of Hi1(y) and Pi,ni ðyÞ, we have
N X
nl X
N X
2
i ¼ 1,i a l
l¼1j¼1
! kl1i y4i
i ¼ 1,i a l
i ¼ 1,i a l
Z Fi0 ðy2i Þy4i
ð64Þ
with Fi0( ) being a smooth non-decreasing function to be designed in the next subsection. From (63) and (64), we have
‘V N r
N X i¼1
V 0i
N X
Fi0 ðy2i Þy4i þ
i¼1
N X i¼1
Assumption 6. ([28])For the functions fij0( ),cij0( ), ciz( ) and ci0( ) and ai0( ) given by Assumptions 2–3, the following condition holds: limsup s-0 þ
j
ai0 ðsÞ
o1
ð65Þ
ð69Þ
P Case 2. If N l¼ 1 gil ð yl Þ Z ð1=2Þai0 ð zi Þ, then in this case, we have V zi ðzi Þ r ai ðzi Þ r Zi ðyl Þ, and " # N N X X r ðV z Þ g ðyl Þai0 ðzi Þ r r ðZ ðyl ÞÞg ðyl Þr ðV z Þai0 ðzi Þ: i
il
i
Lemma 4. Under Assumption 6, and if [28]
Z s Z 1 ½xi ða i 1 ðsÞÞ0 exp ½zi ða i 1 ðtÞÞ1 dt ds o1
il
i
i
i
l¼1
For these two cases, it follows that " # N N X X 1 ri ðV zi Þ gil ðyl Þai0 ðzi Þ r ri ðZi ðyl ÞÞgil ðyl Þ ri ðV zi Þai0 ðzi Þ: 2 l¼1 l¼1 Consider the Lyapunov function candidate for the entire system W ¼ VN þ
N X
Ui
where VN is defined by (62). Then, it follows from (63), (68) and (69) that
‘W r
N X
V 0i
i¼1
þ
ð67Þ
i¼1
r
N X 1
2 i¼1
N X
N X
Ci,ni ðzi Þ
i¼1
N X
Fi0 ðy2i Þy4i þ
i¼1
N X 1 i¼1
4
r0i ðV zi Þc2iz ðzi Þc2i0 ðzi Þ þdðo~ ij , e~ ij , y~ ij , p~ i Þ þC
V 0i
i¼1
holds. Then, under the controller (54), the control laws (35) and (55)–(57), the closed-loop system has an almost surely unique solution on (0,N) and the solution process is bounded in
Fi0 ðy2i Þy4i þ
N X 1 þ ri ðZi ðyl ÞÞgil ðyl Þ ri ðV zi Þai0 ðzi Þ 2 i¼1l¼1 i¼1
Then there exist a non-decreasing positive function ri( ) such that 8x A Rmi
Theorem 2. Consider the system (3) and (4). Under the Assumptions 1–6 and the conditions of lemma 4, if g ðsÞ ð68Þ limsup il 4 o 1, 1 ri,l r N, s s-0 þ
N X
N X N X
ð66Þ
0
1 1 r ðV z ðz ÞÞa ðz ÞCi,ni ðzi Þ Z r0ii ðV zi ðzi ÞÞc2iz ðzi Þc2i0 ðzi Þ 4 i i i i0 i 2
i
l¼1
i¼1
According to [28], from Assumptions 2–3 and 5, one can construct continuous increasing functions xi( ) and zi( ) satisfy2 2 ing xi ðsÞai0 ðsÞ Z4Ci,ni ðsÞ and zi ðsÞai0 ðsÞ Z2ciz ðsÞci0 ðsÞ.
0
1 2
To show the last step of (69), we consider the following two cases separately: P Case 1. If N l ¼ 1 gil ð yl Þ r ð1=2Þai0 ð zi Þ, then in this case, we have that " # N X 1 ri ðV zi Þ gil ðyl Þai0 ðzi Þ r ri ðV zi Þai0 ðzi Þ 2 l¼1
~ ij , e~ ij , y~ ij , p~ i Þ þ C Ci,ni ðzi Þ þ dðo
To construct the nonnegative functionFi0( ), the following Assumption 6 is introduced.
4 2 2 4 ij0 ðsÞ þ cij0 ðsÞ þ ciz ðsÞci0 ðsÞ
where Zi ¼ ai ða1 i0 ð2N gil ðUÞÞÞA k1 .
ð2Þ
i ¼ 1,i a l
j¼1
ri ðZi ðyl ÞÞgil ðyl Þ ri ðV zi Þai0 ðzi Þ
1 0 r ðV z Þc2 ðz Þc2i0 ðzi Þ 2 i i iz i
þ
In order to make ‘V N be non-positive, we first choose Fi( ) in (60) (1rirN) to ensure the non-positiveness of the sum of allyirelated terms of the right-hand of the above inequality (63). To this end, choose a smooth nonnegative function Fi( ) such that ! nl nl N N N X X X X X ð1Þ ð2Þ Fi ðy2i Þy4i glji y4i 2 kl1i þ6 kl1i y4i j ¼ 2 i ¼ 1,i a l
N X
r
glji y4i
N X
ð1Þ
kl1i þ 6
By Ito^ formula, Assumptions 2 and 3, we have @V T 2 1 0 zi ‘U i ðzi Þ ¼ ri ðV zi Þ‘V zi þ ri ðV zi Þ g i0 2 @zi
l¼1
N X
l ¼ 1 j ¼ 2 i ¼ 1,i a l
i¼1
þ
nl N X X
Proof. Suppose that ri( ) is the changing supply function defined in Lemma 4. Let Z V z ðzi Þ i ri ðtÞ dt U i ðzi Þ ¼ 0
~ ij , e~ ij , y~ ij , p~ i Þ þ C Ci,ni ðzi Þ þ dðo
ðHi1 ðyÞ þ Pi,ni ðyÞÞ ¼
probability. Moreover, the outputs y ¼ ½y1 ,. . .yN T can be regulated into a small neighborhood of the origin in probability.
ðHi1 ðyÞ þ Pi,ni ðyÞÞ
i¼1
N X
39
N X N X
ri ðZi ðyl ÞÞgil ðyl Þ
i¼1l¼1
ri ðV zi Þai0 ðzi Þ þdðo~ ij , e~ ij , y~ ij , p~ i Þ þC
ð70Þ
It is east to get that N X N X i¼1l¼1
ri ðZi ðyl ÞÞgil ðyl Þ ¼
N X N X i¼1l¼1
rl ðZl ðyi ÞÞgli ðyi Þ
ð71Þ
40
T. Wang et al. / Neurocomputing 97 (2012) 33–43
Then from (68), we can construct a smooth non-decreasing function Fi0(s2)AkN such that N X
Fi0 ðs2 Þ Z
ri ðZi ðsÞÞ sup
t A ð0,s
l¼1
Fi0 ðy2i Þy4i Z
N X
gli ðtÞ t4
ji1j ðx^ i1 Þ ¼ P5
rl ðZl ðyi ÞÞgli ðyi Þ
j12j ðx^ 11 , x^ 12 Þ ¼
Fi0 ðy2i Þy4i Z
i¼1
The fuzzy logic systems can be expressed in the form 5 X ^f ðx^ y Þ ¼ yT j ðx^ Þ ¼ yTi1j ji1j ðx^ i1 Þ, i ¼ 1,2: i1 i1 i1 i1 i1 i1 j¼1 5 X ^f ðx^ y Þ ¼ yT j ðx^ , x^ Þ ¼ yT12j j12j ðx^ 11 , x^ 12 Þ 12 12 12 12 12 11 12 j¼1
Thus, by (70) we have N X
V 0i
i¼1
N X 1
4 i¼1
exp½ðx^ 11 3 þ nÞ2 =4 exp½ðx^ 12 3 þ nÞ2 =4
j ¼ 1,. . .,5:
ri ðZi ðyl ÞÞgil ðyl Þ
i¼1l¼1
‘W r
exp½ðx^ 11 3 þjÞ2 =4 exp½ðx^ 12 3 þjÞ2 =4 5 P n¼1
This together with (71) ensures N X N X
We obtain fuzzy basis functions as follows: exp½ðx^ i1 3þ jÞ2 =16 ,j ¼ 1,. . .,5 2 ^ n ¼ 1 exp½ðxi1 3 þnÞ =16
, and hence
l¼1
N X
l ¼ 1,. . .,5
r0i ðV zi Þai0 ðzi Þ þ dðo~ ij , e~ ij , y~ ij , p~ i Þ þ C
where Define W1 ¼
N X
V 0i þ
i¼1
N X 1
4 i¼1
yTi1 ¼ ½yi11 , yi12 , yi13 , yi14 , yi15 , yT12 ¼ ½y121 , y122 , y123 , y124 , y125 ,
r0i ðV zi Þai0 ðzi Þdðo~ ij , e~ ij , y~ ij , p~ i Þ
ji1 ðx^ i1 Þ ¼ ½ji11 ðx^ i1 Þ, ji12 ðx^ i1 Þ, ji13 ðx^ i1 Þ, ji14 ðx^ i1 Þ, ji15 ðx^ i1 ÞT , i ¼ 1,2,
Then, it is easy to see that W1 is positive-definite and radically ^ , e^ , y, pÞ ^ and satisfies unbounded in its arguments ðe, w,z, o
‘W r W 1 þC e ¼ ½eT1 ,. . .,eTN T , ¼ ½ ^ 1 ,. . ., ^ N T , ^
ð72Þ ¼ ½ T1 ,. . ., TN T , z ¼ ½zT1 ,. . .,zTN T , ¼ ½ ^ 1 ,. . ., ^ N T , y ¼ ½y1 ,. . ., yN T .
where w w w p^ ¼ ½p^ 1 ,. . ., ^ o o e e e p^ N T ,o By theorem 1, the closed-loop system has an almost surely unique solution on (0,N), and moreover, the solution of the closed-loop system is bounded in probability, i.e., for any given e 40, there exist a k‘ function b and a k function g such that 8tZ0, one has
^ o ^ , e^ , yÞ o bð9ðeð0Þ, wð0Þ,zð0Þ, pð0Þ, ^ P ðe, w,z, p, ^ ð0Þ, e^ ð0Þ, yð0ÞÞ9,tÞ þ gðCÞ Z 1e o ^ ^ ð0Þ, e^ ð0Þ, yð0ÞÞ a 0. From the definition where ðeð0Þ, wð0Þ,zð0Þ, pð0Þ, o of C i,ni and in the similar argurments in [27–30], C can be made small if we choose the design parameters appropriately. 6. Simulation study In this section, the feasibility of the proposed method and the control performances are illustrated by the following example: Example. Consider the following stochastic nonlinear system 1 dz1 ¼ ð3z1 þ x211 sint þ x21 Þdt þ pffiffiffi z1 cosx11 dw1 2 dx11 ¼ ðx12 þ sinðx11 Þ þ 0:1z1 þ 0:5x21 Þ dt þ dw1 dx12 ¼ ðx12 þ x11 þ u1 Þ dt þdw1 y1 ¼ x11 1 1 1 dz2 ¼ 4z2 þ x221 þ x211 dt þ pffiffiffi z2 sinx21 dw2 4 4 2 dx21 ¼ ðu2 þ x221 þx221 x11 Þ dt þ z2 x21 dw2 y2 ¼ x21 Define fuzzy membership as follows:
j12 ðx^ 11 , x^ 12 Þ ¼ ½j121 ðx^ 11 , x^ 12 Þ, j122 ðx^ 11 , x^ 12 Þ, j123 ðx^ 11 , x^ 12 Þ, j124 ðx^ 11 , x^ 12 Þ, j125 ðx^ 11 , x^ 12 ÞT Then the state observer (13) is dx^ 11 ¼ ðx^ 12 þ f^ 1 ðx^ 11 y11 Þ þ 10ðx11 x^ 11 ÞÞdt dx^ 12 ¼ ðu1 þ f^ 2 ðx^ 12 y12 Þ þ12ðx11 x^ 11 ÞÞdt dx^ 21 ¼ ðu2 þ f^ 21 ðx^ 21 y21 Þ þ10ðx21 x^ 21 ÞÞdt Given a positive definite matrix Q¼I, solving (10) to obtain the positive definite matrix 0:6500 0:5000 P1 ¼ 0:5000 0:4708 With the notations of Assumption 1, we can take j110(z1)¼ 9z19, j111(9y19)¼0, j112(9y29)¼0.59y29, j120(z1)¼0, j121(9y19)¼0, j122(9y29)¼0, c110(z1)¼0, c111(9y19)¼ 0, c112 (9y 29)¼0, c120(z1)¼0, c121(9y19)¼0, c122(9y29)¼0, j210(z2)¼0, j211 ðy1Þ ¼ 0:5y21 , j212 ðy2 Þ ¼ 0:5y22 , c210 ðz2 Þ ¼ 0:5z22 , c211(9y19)¼0, c212 ðy2 Þ ¼ 0:5y22 , pnij ¼ 1. For convenience, choose fijl(0)¼0 and cijl(0) ¼0, respectively. ð1Þ ð2Þ Then from (16) and (17) we know that k111 ¼ 0, k111 ¼ 0 and ð1Þ ð2Þ 2 2 k212 ¼ ð1=4Þy2 , k212 ¼ ð1=4Þy2 . According to (32), we can get g111 ¼0 and g121 ¼0. By Assumption 2, we can get a 1 ðsÞ ¼ ð1=4Þs4 , a1 ðsÞ ¼ ð1=2Þs4 ,
g11 ðsÞ ¼ ð1=4Þs8 , g12 ðsÞ ¼ ð1=4Þs4 , a10 ðsÞ ¼ ð3=4Þs4 . a 2 ðsÞ ¼ ð1=2Þs2 , a2 ðsÞ ¼ s2 , g21 ðsÞ ¼ ð1=8Þs4 , g22 ¼ ð1=8Þs4 , a20 ðsÞ ¼ 7=2 s2 . According to Lemma 4 and (60) and (64), by choosing r1(s)¼80 and r2 ðsÞ ¼ ð4=3Þs þ 8, we can get F10 ðy21 Þ ¼ ð20 þ ð1=42ÞÞ y41 þ 1,F20 ðy22 Þ ¼ 21y22 þð1=42Þy62 ; F11 ðy21 Þ ¼ ð21 þ ð1=42ÞÞ y41 þ1, F21 ðy22 Þ ¼ 21y22 þ ð1=42Þy62 þ 1. The controller and parameter adaptive laws for the subsystem 1 are given as follows: 3 2
9 4
a11 ¼ b11 y1 F11 ðy21 Þy1 y1 p^ 1 y1 ð73Þ ð2Þ
T
2 X
g1j1 y1 2kð1Þ 111 y1
j¼1
6k111 y1 y11 j11 ðx^ 11 Þe^ 11 tanhðy31 =kÞ
ð74Þ
mF li1 ðx^ i1 Þ ¼ exp½ðx^ i1 3 þlÞ2 =16, l ¼ 1,. . .,5:
y_ 11 ¼ r 11 j11 ðx^ 11 Þy31 s11 y11
ð75Þ
mF l12 ðx^ 11 , x^ 12 Þ ¼ exp½ðx^ 11 3 þlÞ2 =4 exp½ðx^ 12 3 þlÞ2 =4,
e_^ 11 ¼ r11 y31 tanhðy31 =kÞs11 e^ 11
ð76Þ
T. Wang et al. / Neurocomputing 97 (2012) 33–43
3 4
$11 ¼ r 11 y41 r11 s11 p^ 1
0
ð77Þ
1 3 @a11 4=3 w12 u1 ¼ b12 w12 O12 w12 4 4 @y1 0 !2 1 9 @a11 4 1 3 @2 a11 A @ p^ 1 w þ w12 4 12 @y1 4 @y21
3 @a11 4=3 @a ^ 12 tanhðw312 =kÞ þ 11 $12 p^ 1 w12 o 2 @y1 @p^ 1
!2 11 4 2 9 @ a 1 @ a 11 11 3 AA þ @ w12 þ w12 4 4 @y1 @y21
1 ð1Þ u2 ¼ b21 y2 F21 ðy22 Þy2 p^ 2 y2 2k212 y2 2 ð2Þ
ð78Þ
ð79Þ
_^ ¼ r w3 tanhðw3 =kÞs o o 12 12 12 12 ^ 12 12
ð80Þ
T
6k212 y2 y21 f21 ðx^ 21 Þe^ 21 tanhðy2 =kÞ
ð82Þ
y_ 21 ¼ r 21 f21 ðx^ 21 Þy2 s21 y21
ð83Þ
e_^ 21 ¼ r21 y2 tanhðy2 =kÞs21 e^ 21 :
ð84Þ
1 p_^ 2 ¼ r 21 y22 r 21 s21 p^ 2 2
ð85Þ
Fig. 3. x21(solid) and x^ 21 (dotted). Fig. 1. x11(solid) and x^ 11 (dotted).
Fig. 2. x12(solid) and x^ 12 (dotted).
ð81Þ
The controller and parameter adaptive laws for the subsystem 2 are given as follows:
y_ 12 ¼ r12 w312 f12 ðx^ 12 Þs12 y12
3 @a11 4=3 w12 p_^ 1 ¼ $12 ¼ $11 þ r 11 w312 2 @y1
41
Fig. 4. z1(solid) and z2(dotted).
42
T. Wang et al. / Neurocomputing 97 (2012) 33–43
In this simulation, the design parameters in (74)–(85) are chosen as r11 ¼ r12 ¼r21 ¼0.1, r 11 ¼ r 12 ¼ r 21 ¼ 1, r 11 ¼ r 21 ¼ 0:01, s11 ¼ s12 ¼ s21 ¼1, s11 ¼ 0:8, s12 ¼ 0:3, s21 ¼ 12, s11 ¼ 1:4, s21 ¼ 10:5, b11 ¼0.25, b12 ¼0.25, b21 ¼0.25, k¼0.1 The initial conditions are chosen as x11 ð0Þ ¼ 0:05,x12 ð0Þ ¼ 0:5,x21 ð0Þ ¼ 0:05,z1 ð0Þ ¼ z2 ð0Þ ¼ 10, x^ 11 ð0Þ ¼ x^ 12 ð0Þ ¼ x^ 21 ¼ 0p^ 1 ð0Þ ¼ p^ 2 ð0Þ ¼ e^ 11 ¼ e^ 21 ¼ 0,
yT11 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9 yT12 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9 yT21 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9 The simulation results are shown by Figs. 1–6. From the simulation results in Figs. 1–6, it clearly shows that the proposed control approach can achieve the good control performances even if the controlled nonlinear stochastic system contains the unknown functions, the unmodeled dynamics and unmeasured states.
7. Conclusions In this paper, an observer-based adaptive fuzzy decentralized output feedback control approach has been proposed for a class of uncertain large-scale stochastic nonlinear systems with unknown functions, dynamic uncertainties and without the direct measurements of state variables. In the design, fuzzy logic systems are utilized to approximate the unknown functions and a fuzzy state observer is developed. By using the fuzzy state observer and based on the principle of the adaptive backstepping technique and the changing supply function technique, a new robust adaptive fuzzy decentralized output feedback control scheme is synthesized. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing appropriate design parameters. The future researches will mainly concentrate on the tracking output feedback control problem and the tolerant-control design problem for the large-scale stochastic nonlinear systems based on the result of this paper. References
Fig. 5. :yi1 : (solid) and .:yi2 :.(dotted).
Fig. 6. u1 (solid) and u2(dotted).
[1] L.X. Wang, Adaptive fuzzy systems and control, Prentice Hall, Englewood Cliffs, NJ, 1994. [2] H.G. Zhang, Modeling, identification, and control of a class of nonlinear systems, IEEE Trans. Fuzzy Syst. 9 (2) (2001) 349–354. [3] J.T. Spooner, K.M. Passino, Decentralized adaptive control of nonlinear systems using radial basis neural networks, IEEE Trans. Autom. Control 44 (11) (1999) 2050–2057. [4] C.C. Hua, X.P. Guan, P. Shi, Adaptive fuzzy control for uncertain interconnected time-delay systems, Fuzzy Sets Syst. 153 (3) (2005) 447–458. [5] S.N. Huang, K.K. Tan, T.H. Lee, Decentralized control of a class of large-scale nonlinear systems using neural networks, Automatica 41 (9) (2005) 1645–1649. [6] S.C. Tong, H.X. Li, G.R. Chen, Adaptive fuzzy decentralized control for a class of large-scale nonlinear systems, IEEE Trans. Syst., Man, Cybernatics B 34 (1) (2004) 770–774. [7] D. Lin, X.Y. Wang, Observer-based decentralized fuzzy neural sliding mode control for interconnected unknown chaotic systems via network structure adaptation, Fuzzy Sets Syst. 161 (15) (2010) 2066–2080. [8] Y.S. Huang, Robust decentralized adaptive fuzzy control for a class of largescale MIMO nonlinear systems, Int. J. Robust and Nonlinear Control (2011)htt p://dx.doi.org/10.1002/rnc.1780. [9] Y.S. Huang, M. Wu, Robust decentralized direct adaptive output feedback fuzzy control for a class of large-sale nonaffine nonlinear systems, Inf. Sci. 181 (1) (2011) 2392–2404. [10] S. Jain, F. Khorrami, Decentralized adaptive control of a class of large-scale interconnected nonlinear systems, IEEE Trans. Autom. Control 42 (2) (1997) 136–157. [11] S. Jain, F. Khorrami, Decentralized adaptive output feedback design for largescale nonlinear systems, IEEE Trans. Autom. Control 42 (5) (1997) 729–735. [12] W.S. Chen, J.M. Li, Globally decentralized adaptive backstepping neural network tracking output control for unknown nonlinear interconnectioned systems, Asian J. Control 12 (1) (2010) 96–102. [13] S.J. Yoo, J.B. Park, Neural-network-based decentralized adaptive control for a class of large-scale nonlinear systems with unknown time-varying delays, IEEE Trans. Syst., Man, Cybernetics B 39 (5) (2009) 1316–1323. [14] W.S. Chen, J.M. Li, Decentralized output-feedback neural control for systems with unknown interconnections, IEEE Trans. Syst., Man, Cybern. B 38 (1) (2008) 258–266. [15] C.C. Hua, X.P. Guan, Output feedback control for time delay nonlinear interconnected systems using neural networks, IEEE Trans. neural networks 18 (4) (2008) 673–688. [16] S.C. Tong, C.L. Liu, Y.M. Li, Fuzzy adaptive decentralized control for large-scale nonlinear systems with dynamical uncertainties,, IEEE Trans. Fuzzy Syst. 18 (5) (2010) 845–861. [17] S.C. Tong, C.L. Liu., Y.M. Li, Adaptive fuzzy decentralized control for largescale nonlinear systems with time varying delays and unknown highfrequency gain sign, IEEE Trans. Syst., Man, Cybernetics B 41 (2) (2011) 474–485. [18] W.S. Chen, L.C. Jiao, Adaptive NN backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays, IEEE Trans. Syst., Man, Cybernetics B 40 (3) (2010) 939–950. [19] W.S. Chen, L.C. Jiao, Z.B. Du, Output-feedback adaptive dynamic surface control of stochastic nonlinear systems using neural network, IET Control Theory Appl. 12 (2010) 3012–3021.
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[20] S.C. Tong, Y. Li, Y.M. Li, Y.J. Liu, Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems, IEEE Trans. Syst., Man, Cybernetics B 41 (6) (2011) 1693–1704. [21] J. Li, W.S. Chen, J.M. Li, Adaptive NN output-feedback decentralized stabilization for a class of large-scale stochastic nonlinear strict-feedback systems, Int. J. Robust Nonlinear Control 21 (3) (2011) 425–472. [22] W.S. Chen, L.C. Jiao, Decentralized backstepping output-feedback control for stochastic interconnected systems with time-varying delays using neural networks, Neural Comput. Appl. (2011)http://dx.doi.org/10.1007/s00521011-0590-x. [23] Y. Li, Y.M. Li, S.C. Tong, Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large-scale systems, Neurocomputing 83 (2012) 38–46. [24] Z.P. Jiang, L. Praly, Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties, Automatica 34 (7) (1998) 825–840. [25] Z.J. Wu, X.J. Xie, P. Shi, Robust adaptive output-feedback control for nonlinear systems with output unmodeled dynamics, Int. J. Robust Nonlinear Control 18 (11) (2008) 1162–1187. [26] H. Deng, M. Krstic, Output-feedback stochastic nonlinear stabilization, IEEE Trans. Autom. Control 44 (2) (1999) 328–333. [27] Z.J. Wu, X.J. Xie, S.Y. Zhang, Adaptive backstepping controller design using stochastic small-gain theorem, Automatica 43 (4) (2007) 608–620. [28] S.J. Liu, J.F. Zhang, Z.P. Jiang, Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems, Automatica 43 (2) (2007) 238–251. [29] W. Chen, L. Jiao, R. Li, J. Li, Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances, IEEE Trans. Fuzzy Syst. 18 (4) (2010) 674–685. [30] W. Chen, Z. Zhang, Globally stable adaptive backstepping fuzzy control for output-feedback systems with unknown high-frequency gain sign, IEEE Fuzzy Sets and Syst. 161 (6) (2010) 821–836.
Tong Wang received the B.S. degree from the Department of Mathematics, Shanxi University, Taiyuan, China, in 2010. He is now working towards the M.E. degree in control theory and control engineering from Liaoning University of Technology, Jinzhou, China. His current research interests include fuzzy control, adaptive control, and stochastic control.
43 Shaocheng Tong received the B.A. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.A. degree in fuzzy mathematics from Dalian Marine University, PRC, and the Ph.D. degree in fuzzy control from Northeastern University, PRC, in 1982, 1988, and 1997, respectively. Currently, he is a Professor in the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, PRC. His research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control.
Yongming Li received the B.S. degree and the M.S. degree in applied mathematics from Liaoning University of Technology, Jinzhou, China, in 2004 and 2007, respectively. He is currently a lecturer in the Department of Basic Mathematics, Liaoning University of Technology. Now, he is pursuing his Ph.D. degree in transportation information engineering and control in Dalian Maritime University. His current research interests include fuzzy and neural networks control, and nonlinear adaptive control.
Contents lists available at SciVerse ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Robust adaptive decentralized fuzzy control for stochastic large-scale nonlinear systems with dynamical uncertainties$ Tong Wang, Shaocheng Tong n, Yongming Li Department of Mathematics, Liaoning University of Technology, Jinzhou, Liaoning 121000, China
a r t i c l e i n f o
abstract
Article history: Received 15 February 2012 Received in revised form 5 May 2012 Accepted 29 May 2012 Communicated by H. Zhang Available online 3 July 2012
In this paper, a robust adaptive fuzzy decentralized backstepping output feedback control approach is proposed for a class of uncertain large-scale stochastic nonlinear systems with unknown nonlinear functions, dynamical uncertainties and without the measurements of the states. The fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy state observer is designed for estimating the unmeasured states. To solve the problem of the dynamical uncertainties, the changing supply function technique is incorporated into the backstepping recursive design technique, and a new robust adaptive fuzzy decentralized output feedback control approach is constructed. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing design parameters appropriately. A simulation example is provided to show the effectiveness of the proposed approach. & 2012 Elsevier B.V. All rights reserved.
Keywords: Stochastic nonlinear large-scale systems Fuzzy adaptive decentralized control Backstepping technique Changing supply function
1. Introduction In the past decades, fuzzy logic systems (FLSs) and neural networks (NNs) as universal approximators have been extensively used for the modeling and controller design for uncertain nonlinear systems [1,2], and various adaptive fuzzy and NN decentralized control approaches have been developed for large-scale nonlinear systems, for example see [3–9]. In [1–6], adaptive NN and fuzzy decentralized control schemes were developed for some classes of uncertain nonlinear large-scale systems with measurable states. By designing the state observer, [7–9] proposed adaptive fuzzy and NN output feedback decentralized control approaches for the uncertain nonlinear large-scale systems with immeasurable states. Generally, these adaptive NN and fuzzy decentralized control approaches can deal with a larger class of nonlinear large-scale systems with unknown nonlinear functions and without a priori knowledge about the interconnections, and achieve satisfactory control performances. However, a major constraint is that the unknown functions and interconnections of the considered systems are in the range space of the input matrix, which is so-called to satisfy the matching condition in
$ This work was supported by the National Natural Science Foundation of China (no. 61074014), the Outstanding Youth Funds of Liaoning Province (no. 2005219001). n Corresponding author. E-mail address: [email protected] (S. Tong).
0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.05.017
[10,11], thus the above mentioned adaptive control approaches cannot be applied to those uncertain nonlinear large-scale systems without satisfying the matching condition. To handle the control problem of uncertain nonlinear largescale systems without satisfying the matching condition, several adaptive NN and fuzzy decentralized controllers have been developed by using the backstepping recursive technique [12–17]. In [12,13] adaptive NN decentralized backstepping state feedback was developed to control approaches for the uncertain large-scale systems without or with time delays. By designing the K-filters or state observers, [14,15] investigated adaptive NN decentralized backstepping output feedback controllers for a class of the uncertain large-scale systems with the immeasurable states, while [16,17] considered adaptive fuzzy decentralized output feedback controllers for a class of uncertain large-scale systems with immeasurable states and unknown control directions. More recently, the investigations on stochastic nonlinear systems have received considerable attention, and many results on adaptive NN or fuzzy backstepping technique obtained for deterministic nonlinear systems in the strict-feedback form have been successfully extended to the stochastic nonlinear systems [18–23]. Authors in [18,19] first proposed adaptive NN backstepping output feedback control design approaches for a class of SISO stochastic systems in the strict-feedback form by using a linear state observer. Authors in [20] first proposed adaptive NN backstepping output feedback control design approaches for a class of SISO stochastic systems in the strict-feedback form by
34
T. Wang et al. / Neurocomputing 97 (2012) 33–43
using a linear state observer. Authors in [21–23] investigated the adaptive NN and fuzzy decentralized output feedback stabilization problem for a class of stochastic large-scale nonlinear systems. However, the above control approaches in [18–23] do not consider the stochastic nonlinear systems containing the unmodeled dynamics and dynamical disturbances (i.e., dynamical uncertainties). As stated in [24,25], the unmodeled dynamics and dynamical disturbances often exist in many practical nonlinear systems, and they are also the major reason of resulting in the instability of the control systems. Therefore, to study the stochastic large-scale nonlinear systems with consideration of dynamical uncertainties is very important in control theory and applications. The purpose of this paper is to investigate the adaptive decentralized fuzzy control for a class of large-scale stochastic nonlinear systems with three uncertainties, i.e., unknown nonlinear functions, dynamical uncertainties and unmeasured states. In the control design, fuzzy logic systems are employed to approximate the unknown nonlinear functions, and a fuzzy state observer is designed to estimate the unmeasured states. To solve the problem of the dynamical uncertainties, the changing supply function technique is incorporated into the backstepping recursive design technique, a new robust adaptive decentralized fuzzy backstepping output feedback control scheme is constructed. The main advantages of the proposed adaptive fuzzy decentralized control approach are summarized as follows: (i) by designing a new fuzzy state observer, not a reduced-order state observer used in [21,22,27–29], the proposed state observer can obtain the better estimations of the unmeasured states; (ii) by incorporating the changing supply function technique into the backstepping recursive design technique, the proposed adaptive decentralized control approach can applied to a larger class of stochastic nonlinear systems and has the robustness to the dynamical uncertainties compared with the existing results in [21–23]; and (iii) it is mathematically proved that the resulting closed-loop system are SGUUB in probability and the output converges to a small neighborhood of the origin.
2. Some notations and preliminary results 2.1. Stability in probability
Lemma 1. ([28]) Consider the stochastic system (2) and assume that f(t,0), h(t,0) are bounded uniformly in t. If there exist functions V(t,x), m1(U), m2(U)AkN, constants c1 40,c2 Z0, and a nonnegative function W(t,x), such that
m1 ðjxjÞ rVðt,xÞ r m2 ðjxjÞ, ‘Vðt,xÞ rc1 Wðt,xÞ þ c2 , then (i) there exists an almost surely unique solution on (0,N) for the system (2). (ii) the solution process is bounded in probability, when W(t,x) ZcV(t,x) for a constant c 40. 2.2. Fuzzy logic systems A FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine, and the defuzzifier. The knowledge base is composed of a collection of fuzzy. If-then rules of the following form: Rl : If x1 is F l1 and x2 is F l2 and. . .and xn is F ln , Then y is Gl ,l ¼ 1,2,. . .,N where x ¼ ðx1 ,x2 . . .xn ÞT and y are FLS input and output, respectively, mF l ðxi Þ and mGl ðyÞare the membership function of fuzzy sets i F li and Gl, N is the number of inference rules. Through singleton fuzzifier, center average defuzzification and product inference, the FLS can be expressed as PN Qn l ¼ 1 yl i ¼ 1 mF li ðxi Þ yðxÞ ¼ PN Qn ð3Þ l ¼ 1 ½ i ¼ 1 mF l ðxi Þ i
where yl ¼ maxy A R mGl ðyÞ. Define the fuzzy basis functions as Qn i ¼ 1 mF l ðxi Þ fl ¼ PN Qn i l ¼ 1 ½ i ¼ 1 mF li ðxi Þ T Denoting y ¼ y1 ,y2 ,. . .yN ¼ y1 , y2 ,. . ., yN and jðxÞ ¼ j1 ðxÞ, j2 ðxÞ,. . ., jN ðxÞT , then fuzzy logic system (3) can be rewritten as T
yðxÞ ¼ y jðxÞ
ð4Þ
Consider the following time-varying stochastic system: dx ¼ ðf ðt,xÞ þgðt,xÞuÞ dt þ hðt,xÞ dw
ð1Þ
where w is an r-dimensional standard Brownian motion, xARn is the state, uAR is the control input, f,g:R þ Rn-Rn and h:R þ Rn-Rn r. Definition 1. ([26–28]) For any given V(t,x) associated with the stochastic differential Eq. (1), define the differential operator ‘ as follows: ( ) @V @V @V 1 @2 V T þ f ðt,xÞ þ gðt,xÞu þ Tr h ðt,xÞ 2 hðt,xÞ ‘Vðt,xÞ ¼ @t @x @x 2 @x For control-free stochastic nonlinear system of the form dx ¼ f ðt,xÞ dt þhðt,xÞ dw
ð2Þ
the following stability notions introduced will be used throughout the paper.
Lemma 2. For any continuous function f(x) defined over a compact set O and any given positive constant e, there exist a fuzzy logic system (4) and a ideal parameter vector yn such that [1] nT supf ðxÞy jðxÞ r e xAO
3. System descriptions and basic assumptions In this paper, we consider the following stochastic nonlinear large-scale systems: dzi ¼ f i0 ðt,zi ,yÞ dt þ g i0 ðt,zi ,yÞ dwi dxij ¼ ½xi,j þ 1 þ f ij ðx ij Þ þ Dij ðt,zi ,yÞ dt þg ij ðt,zi ,yÞ dwi j ¼ 1,. . .,ni 1, ^ dxi,ni ¼ ½ui þ f i,ni ðx i,n Þ þ Di,ni ðt,zi ,yÞ dt þ g i,ni ðt,zi ,yÞ dwi i
Definition 2. ([26–28]) The solution process {x(t),t Z0} of stochastic system (2) is said to be bounded in probability, if limc-1 sup0 r t o 1 PxðtÞ 4 c ¼ 0. Definition 3. ([28]) Consider the system (2) with f(t,0) 0, h(t,0) 0. The equilibrium f(t,0) 0 is globally stable in probability if for any e 40, there exists a class k function g(U) such that P{9x(t)9o g(9x09)} Z1 e, 8t Z0, x0ARn\{0}.
yi ¼ xi1
ð5Þ T
i
where x ij ¼ ½xi1 ,. . .,xij A R , i ¼ 1,. . .,N,j ¼ 1,. . .,ni are the states, uiAR, yiAR are the control input and output of the system, and zi A Rmi are unmodeled dynamics and Dij(t,zi,y) are the dynamical disturbances. fij(U), j ¼1,2,y,ni are unknown smooth nonlinear functions. fi0(U), gi0(U), Dij(U) and gij(U) are uncertain functions; wiAR is an independent standard Brownian motion defined on a
T. Wang et al. / Neurocomputing 97 (2012) 33–43
complete probability space. In this paper, only the output yi is assumed to be available for measurement. Assumption 1. ([28]) For each i ¼ 1,. . .,N, j ¼ 1,. . .,ni there are unknown constants pnij Z0, and known smooth functions jij0 Z0, jijl Z0, cij0 Z0 and cijl Z0 such that 8ðt,zi ,yÞ A R þ Rmi RN , N X Dij ðt,zi ,yÞ r pn j ðzi Þ þ pn jijl ðyl Þ ij ij0 ij l¼1
35
Assumption 5. ([23] )There are positive constants enij unknown n n and dij such as eij r enij and dij r dij . Denote oij ¼ eij dij, by Assumption 5, one has oij r n n n n eij þ dij ¼ oij , where oij is also an unknown constant. In this paper, the parameters adaptation laws will be designed later to estimate enij and onij , respectively. Rewrite (5) in the following form dzi ¼ f i0 ðt,zi ,yÞ dt þ g i0 ðt,zi ,yÞ dwi 2 3 ni X 4 dxi ¼ Ai xi þ K i yi þ Bij ðf ij ðx ij Þ þ Di ðt,zi ,yÞ þ Bi ui Þ5dt þ Gi ðt,zi ,yÞ dwi
N X g ðt,zi ,yÞ rpn c ðzi Þ þ pn cijl ðyl Þ ij ij ij0 ij l¼1
j¼1
Without loss of generality, this paper assumes that jij0(0) ¼0 and cij0(0) ¼0. Assumption 2. ([28]) For each zi-subsystem in (5), i ¼ 1,. . .,N, there exist function V zi ðzi Þ and kN functions a i ðzi Þ, ai ðzi Þ,
ai0(9zi9) and gil(9yl9), l ¼ 1,. . .,N such that 8ðt,zi ,yÞ A R þ Rmi RN , a i ðzi Þ rV zi ðzi Þ r ai ðzi Þ ð6Þ ‘V zi ðzi Þ r
N X
gil ðyl Þai0 ðzi Þ
ð7Þ
l¼1
Assumption 3. ([28]) There exist known smooth functions ciz and ci0 satisfying @V z =@zi r c ðzi Þ and :g i0 ðt,zi ,yÞ: r c ðzi Þ, i ¼ 1,. . .,N ð8Þ iz i0 i
Assumption 4. ([23]) There exist a set of known constants mij, j ¼1,2,y,ni for 8X1,X2ARi, the following inequality holds f ðX 1 Þf ðX 2 Þ r mij :X 1 X 2 :: ij
ij
where :X 1 X 2 : expresses the 2-norm of vector X1 X2. Control objective: The control task is to design an adaptive output feedback controller using the output yi and state estimations x^ ij so that the closed-loop system is bounded in probability and the outputs of the system can be regulated to a small neighborhood of the origin in probability.
4. Fuzzy state observer design
yi ¼ C i xi where 2
ð9Þ 2
3 2 3 ki1 0 6 7 6 7 ^ I K i ¼ 4 ^ 5, Bi ¼ 4 ^ 5, ki,ni 0 . . . 0 ki,ni 1 Di ¼ Di1 ðt,zi ,yÞ ^ Di,ni ðt,zi ,yÞ , T Bij ¼ 0 . . . 1 . . . 0 , h iT Gi ðt,zi ,yÞ ¼ g i1 . . . g i,ni : Ci ¼ 1 . . . 0 0 3
ki1
6 Ai ¼ 4
7 5,
Choose vector Ki such that matrix Ai is a strict Hurwitz, therefore, for any a given positive definite matrix Q i ¼ Q i T 4 0, there exists a positive definite matrix P i ¼ Pi T such that Ai T Pi þP i Ai ¼ Q i
ð10Þ
Design a fuzzy state observer as x_^ i ¼ Ai x^ i þK i yi þ
ni X
Bij f^ ij ðx^ ij 9yij Þ þBi ui
j¼1
y^ i ¼ C i x^ i
ð11Þ
Let the observer error vector as ei ¼
xi x^ i pni
ð12Þ
where pni ¼ maxf1,pnij ,pijn4 1 r j r ni g. From (9), (11) and (12), one has 0 ni D 1 X dei ¼ @Ai ei þ n Bij f ij ðx ij Þf ij ðx^ ij Þ þ f ij ðx^ ij Þf^ ij ðx^ ij yij Þ þ ni dt pi j ¼ 1 pi þ
Gi ðt,zi ,yÞ dwi pni
Note that in the system (5), the functions f ij ðx ij Þ, j ¼ 1,2,. . .,ni are unknown and the states xi2 ,. . .,xi,ni are not measured directly, thus a state observer should be established to obtain the estimates of xi2 ,. . .,xi,ni . By Lemma 2, it can be assumed that the unknown functions f ij ðx ij Þ, j ¼ 1,2,. . .,ni can be approximated by the following fuzzy logic systems
where F i ¼ ½F i1 ,. . .,F i,ni T ¼ ½ðf i1 ðxi1 Þf i1 ðx^ i1 ÞÞ,. . .,ðf i,ni ðx i,n Þ f i,ni ðx^ i,n ÞÞT i i and di ¼ ½di1 ,. . ., di,ni T . Choose the following Lyapunov candidate Vi0 as
T f^ ij ðx^ ij 9yij Þ ¼ yij jij ðx^ ij Þ,
V i0 ¼
1 rj r ni , T
where x^ ij ¼ ½x^ i1 , x^ i2 ,. . ., x^ ij are the estimates of x ij ¼ ½xi1 ,xi2 ,. . .,xij and denote xi ¼ x i,n and x^ i ¼ x^ i,n . i i n The optimal parameter vectors yij is defined as ynij ¼ argminyij A Oij ½supx^ A Uij f^ ij ðx^ ij 9yij Þf ij ðx^ ij Þ:
T
¼ ðAi ei þ
1 T ðe P e Þ2 2 i i i
ð13Þ
ð14Þ
From (12)–(14), one has 2 T e P ðF þ di þ Di Þ pni i i i h i 1 þ n2 Tr GTi ð2Pi ei eTi Pi þ eTi Pi ei P i ÞGi pi 4
‘V i0 rlmin ðQ i Þlmin ðPi Þ:ei : þ eTi P i ei
ij
where Oij and Uij are bounded compact sets for yij and x^ ij , respectively. The corresponding minimum approximation error eij and dij are defined by eij ¼ f ij ðx^ ij Þf^ ij ðx^ ij ynij Þ, dij ¼ f ij ðx^ ij Þf^ ij ðx^ ij yij Þ
1 G ðt,z ,yÞ ðF þ di þ Di ÞÞdt þ i n i dwi pni i pi
ð15Þ
Since jijl, cijl, i,l ¼ 1,2,. . .,N, j ¼ 1,2,. . .,ni , are smooth functions, ð1Þ ð2Þ there exist smooth nonnegative functions kijl ðyl Þ, kijl ðyl Þ such that !4 !4 N N N X X X ð1Þ j ðy Þ r k y 4 þ8 j ð0Þ ð16Þ ijl
l¼1
l
ijl
l¼1
l
ijl
l¼1
36
T. Wang et al. / Neurocomputing 97 (2012) 33–43
N X
!4
cijl ðyl Þ
N X
r
l¼1
!4
N X
ð2Þ
kijl yl 4 þ 8
l¼1
change of coordinates
cijl ð0Þ
ð17Þ
l¼1
^ i2 ,. . ., o ^ i,j1 ,yi , p^ i Þ wi1 ¼ yi , wij ¼ x^ ij ai,j1 ðx^ i,j1 , yi1 ,. . ., yi,j1 , e^ i1 , o
ð23Þ
By Assumptions 1, 3 and 4, and the fact that pni Z 1, one has the following inequalities
where ai,j 1(U)ðj ¼ 2,. . .,ni Þ is an intermediate control, yij,p^ i , e^ i1 and ^ i,j1 are the estimates of the parameters ynij , pi,eni1 and oni,j1 , o
2 T 3 1 n 4 8=3 4 e P e e T P d r :P : :ei : þ :di : pni i i i i i i 2 i 2
ð18Þ
respectively. Step 1: From the second equation in (5), and according to It o^ ’s differentiation rule, one has
0 1 ni X 2 T 3 1 8=3 4 4 e P e e T P F r :P : :ei : þ @ m A:ei : pni i i i i i i 2 i 2 j ¼ 1 ij
ð19Þ
dwi1 ¼ ½xi2 þ f i1 ðx i1 Þ þ Di1 ðt,zi ,yÞdt þ g i1 ðt,zi ,yÞ dwi nT ¼ ½wi2 þ ai1 þpni ei2 þ F i1 þ ei1 þ yi1 ji1 ðx^ i1 Þ þ Di1 ðt,zi ,yÞ dt
þ g i1 ðt,zi ,yÞ dwi ni X 2 T 3 2 4 2 e P e eT P D r :Pi : :ei : þ 4ni :P i : ½jij0 4 ðzi Þ n i i i i i i pi 2 j¼1 !4 3 N N X X ð1Þ 4 k y þ8 j ð0Þ 5 þ ijl
Choose the Lyapunov function candidate as V i1 ¼ V i0 þ ð20Þ
ijl
l
l¼1
where
l¼1
4
2
2
l¼1
ri1 4 0,
r i1 40
and
r i1 40
are
design
ð25Þ parameters.
n4=3
Define pi ¼ max1,pni ,pni 4 ,pi . For the convenience of the following derivations, we cite the following Lemma 3:
Lemma 3. ([23]) For any a variable x, and any a constant k40, 0 the inequality jxjx tanhðx=kÞ r0:2785k ¼ k is satisfied. From (23)–(25), the infinitesimal generator of Vi1 satisfies
2
r ð2:Pi : þ lmax ðPi ÞÞ:ei : þ 4ni ð2:Pi : þ lmax ðP i ÞÞ 2 !4 3 ni N N X X X 4 ð2Þ 4 4cij0 ðzi Þ þ kijl yl þ 8 cijl ð0Þ 5 j¼1
1 4 1 ~T ~ 1 2 1 w þ y y þ e~ þ p~ 2 4 i1 2r i1 i1 i1 2r i1 i1 2r i1 i
y~ i1 ¼ yni1 yi1 , e~ i1 ¼ eni1 e^ i1 and p~ i ¼ p^ i pi are the parameters errors.
h i 1 Tr Gi T ð2Pi ei ei T P i þ ei T Pi ei P i ÞGi pni 2 h i 1 1 ¼ n2 Tr Gi P i ei ðGi Pi ei ÞT þ n2 ðei T P i ei ÞTrðGi T Pi Gi Þ pi pi 1 2 4 4 2 r ð2:Pi : þ lmax ðPi ÞÞ :ei : þ :Gi =pi n : 2 2
ð24Þ
‘V i1 ¼ ‘V i0 þ w3i1 ðwi2 þ ai1 þpni ei2 þ F i1 þ ei1 þ yni1T ji1 ðx^ i1 Þ ð21Þ
l¼1
þ Di1 ðt,zi ,yÞÞ þ
3 2 1 T 1 1 w g g T þ y~ i1 y_~ i1 þ e~ i1 e_~ i1 þ p~ i p_~ i 2 i1 i1 i1 r i1 r i1 r i1 ð26Þ
Substituting (16)–(21) into (15) gives
4
2
‘V i0 rli0 :ei : þ Ci0 ðzi Þ þ4ni :Pi : 2
þ 4ni ð2:P i :
ni X
N X
j¼1
l¼1
ni X
N X
j¼1
l¼1
2 þ lmax ðP i ÞÞ
!
By Assumptions 1, 3 and 4, one has
ð1Þ
kijl yl 4
3 4
ð2Þ kijl yl 4
ð22Þ
þC i0
where
li0 ¼ lmin ðQ i Þlmin ðP i Þ3:P i :
2
Ci0 ðzi Þ ¼ 4ni :Pi :
ni X
8=3
j¼1
C i0 ¼
2
ni X
N X
j¼1
l¼1
N X
!4
jijl ð0Þ
3 4
1 4
w3i1 F i1 r mi1 w3i1 :ei : r w4i1 þ m4i1 :ei :4 3 4
c4ij0 ðzi Þ
j¼1
ni X 1 2 n 4 :d : þ 32ni :P i : 2 i j¼1
þ lmax ðP i ÞÞ
ni X
ð27Þ
ð28Þ
1 4
w3i1 pni ei2 r ðpni Þ4=3 w4i1 þ :ei :4
ni 7 1X 2 2 :Pi : lmax ðPi Þ m 2 2 j ¼ 1 ij
j4ij0 ðzi Þ þ4ni ð2:Pi :2 þ l2max ðPi ÞÞ
1 4
w3i1 wi2 r w4i1 þ w4i2
!
2
þ 32ni ð2:P i :
w3i1 Di1 r w3i1 pni1 ji10 ðzi Þ þ
N X
ð29Þ
!
ji1l ðyl Þ
l¼1 N N X X 3 ð1Þ r ðpni1 Þ4=3 w4i1 þ2ðj4i10 ðzi Þ þ ki1l y4l þ 8 ji1l ð0ÞÞ4 4 l¼1 l¼1
!
ð30Þ
l¼1
!4
cijl ð0Þ
5. Adaptive fuzzy decentralized controller design and stability analysis In this section, by using the designed state observer in the above section, an adaptive controller together with the parameter adaptive laws are to be constructed by using the backstepping design technique and the changing supply function technique, and the stability analysis of the closed-loop system will be given. 5.1. Adaptive fuzzy decentralized controller design The adaptive fuzzy decentralized output feedback control design consists of ni-steps, i ¼ 1, ,N, each step is based on the
N X 3 2 3 wi1 g i1 g Ti1 r w2i1 pni12 ci10 ðzi Þ þ ci1l ðyl Þ 2 2 l¼1
!2
N N X X 3 n4 4 ð2Þ r pi1 wi1 þ 6c4i10 ðzi Þ þ 6 ki1l y4l þ48 ci1l ð0Þ 4 l¼1 l¼1
!4
ð31Þ Substituting (27)–(31) into (26) and by using Lemma 3, gives 0 ni X 1 4 3 9 4 3@ ‘V i1 rli1 :ei : þ wi2 þ yi ai1 þ yi þ p^ i yi þ giji yi þ 2kð1Þ i1i yi 4 2 4 j¼1 ð2Þ T þ 6ki1i yi þ yi1 ji1 ðx^ i1 Þ þ e^ i1 tanhðy3i =kÞ þ Ci1 ðzi Þ þHi1 ðyÞ T 1 þ Pi1 ðyÞ þ C i1 þ y~ i1 y_ i1 þ ji1 ðx^ i1 Þy3i r i1 1 _ 1 _ 9 4 ð32Þ p^ i yi þ e~ i1 e^ i1 þ y3i tanhðy3i =kÞ þ p~ i 4 r i1 r i1
T. Wang et al. / Neurocomputing 97 (2012) 33–43
where li1 ¼ li0 ð1=4Þm4i1 ð1=4Þ, Hi1 ðyÞ ¼
Pni
j¼2
PN
l ¼ 1,l a i
0
1 @ a i,m1 Aðxi1 x^ i1 Þ þ @kim kij @x^ ij j¼1
gijl y4l ,
2 2 ð2Þ gijl ¼ 4ni :Pi :2 kð1Þ ijl þ 4ni ð2:P i : þ lmax ðP i ÞÞkijl ,
Ci1 ðzi Þ ¼ Ci0 ðzi Þ þ2j4i10 ðzi Þ þ 6c4i10 ðzi Þ, N X
Pi1 ðyÞ ¼ 2
l ¼ 1,l a i
C i1 ¼ C i0 þ16
N X
ð1Þ ki1l y4l þ 6
N X
V im ¼ V i,m1 þ
l ¼ 1,l a i
fi1l ð0Þ
þ48
l¼1
N X
!4
ci1l ð0Þ
0
þ ei1 k : n
l¼1
Design the intermediate control function ai1 and the adaptation functions yi1, e^ i1 and $i1 as 3 2
9 4
ai1 ¼ bi1 yi Fi1 ðy2i Þyi yi p^ i yi
ni X
giji yi 2kð1Þ i1i yi
j¼1
ð2Þ T 6ki1i yi yi1 ji1 ðx^ i1 Þe^ i1 tanhðy3i =kÞ
ð33Þ
y_ i1 ¼ ri1 ji1 ðx^ i1 Þy3i si1 yi1
ð34Þ
e_^ i1 ¼ ri1 y3i tanhðy3i =kÞsi1 e^ i1
ð35Þ
1 4
‘V i1 r li1 :ei : þ w4i2 y4i ðbi1 þ Fi1 ðy2i ÞÞ þ Ci1 ðzi Þ þHi1 ðyÞ þ Pi1 ðyÞ
si1 ~ T s 1 y i1 yi1 þ i1 e~ i1 e^ i1 þ p~ i ðp_^ i $i1 Þsi1 p~ i p^ i r i1
r i1
r i1
ð37Þ
Step m(2rmrni 1): From the mþ1 equation in (5), and similar to Step 1, one has dwim ¼ ðx^ i,m þ 1 þ f^ im ðx^ im 9yim Þ þ kim pni ei1 Þ dt
m1 X j¼1
@ai,m1 _ @a y ij dt i,m1 p_^ i dt @yij @p^ i
m1 m1 X @ai,m1 X @ai,m1 @ai,m1 _ @a _^ dt ^ i1 dt i,m1 dyi e dx^ ij o ij ^ ij @yi @e^ i1 @x^ ij @o j¼1
2 T
¼ 4aim þ wi,m þ 1 þ yim jim ðx^ im Þ þ ðkim
j¼2
m1 X j¼1
kij
@ai,m1 Þðxi1 x^ i1 Þ @x^ ij
m1 X @ai,m1 @ai,m1 _ _^ @ai,m1 p_^ @ai,m1 e_^ y ij o ij i1 i ^ ij @ y @o @p^ i @e^ i1 ij j¼1 j¼2 m1 X
@ai,m1 1 @2 ai,m1 ðxi2 þ f i1 ðx i1 Þ þ Di1 ðt,zi ,yÞÞ g i1 ðt,zi ,yÞ2 2 @y2i @yi # m1 X @ai,m1 @a ðx^ i,j þ 1 þ f^ ij ðx^ ij yij ÞÞ dt i,m1 g i1 ðt,zi ,yÞdwi ^ ij @yi @ x j¼1 T @a ¼ aim þ wi,m þ 1 þ y~ im jim ðx^ im Þ þ Oim þ oim i,m1 p_^ i @p^ i
@ai,m1 g i1 ðt,zi ,yÞ dwi @yi
where
m 1 X j¼1
m 1 X j¼2
þ
2 3 2 @ai,m1 1 ~ T _~ 1 wim g i1 ðt,zi ,yÞ þ y y þ o~ o_~ @yi 2 r im im im r im im im
m 1 X
@ai,m1 _ @a y ij i,m1 e_^ i1 @ y @e^ i1 ij j¼1
@ai,m1 T ðx^ i,j þ 1 þ yij jij ðx^ ij ÞÞ @x^ ij @ai,m1 _ @a ^ ij i,m1 ðx^ i2 þ yTi1 ji1 ðx^ i1 ÞÞ o ^ ij @yi @o
ð41Þ
By Assumptions 1, 3 and 4, one has 3 4
1 4
ð42Þ
w3im
@ai,m1 n 3 @ai,m1 4=3 1 4 pi ei2 r pi w4im þ :ei : 4 4 @yi @yi
ð43Þ
w3im
@ai,m1 3 @ai,m1 4=3 4 1 F i1 r wim þ m4i1 :ei :4 4 4 @yi @yi
ð44Þ
w3im
@ai,m1 3 @ai,m1 4=3 4 1 4 di1 r wim þ dni 4 4 @yi @yi
ð45Þ
! N X @ai,m1 @a Di1 rw3im i,m1 pni1 ji10 ðzi Þ þ ji1l ðyl Þ @yi @yi l¼1 ! 4=3 N N X X 3 @ai,m1 ð1Þ 4 4 4 4 r pi wim þ 2ðji10 ð zi Þ þ ki1l yl þ 8 ji1l ð0ÞÞ @yi 4 l¼1 l¼1
w3im
ð46Þ
l¼1
1 @2 ai,m1 @a g i1 ðt,zi ,yÞ2 i,m1 ðpni ei2 þ F i1 þ di1 þ Di1 ðt,zi ,yÞÞ dt 2 2 @yi @yi
Oim ¼ yTim jim ðx^ im Þ
ð40Þ
2 3 2 @ai,m1 1 @2 ai,m1 wim g i1 ðt,zi ,yÞ w3im g i1 ðt,zi ,yÞ2 2 2 @yi @y2i !2 ! N X 3 2 @ai,m1 2 1 3 @2 ai,m1 n2 zi Þ þ y Þ r wim wim c ð c ð p l i10 i1l i1 2 2 @yi @y2i l¼1 0 !2 1 9 @ai,m1 4 1 3 @2 ai,m1 A 3 r @ wim þ wim wim pi þ 6c4i10 ðzi Þ 4 4 @yi @y2i !4 N N X X ð2Þ ki1l y4l þ 32 ci1l ð0Þ ð47Þ þ6
1 4 1 ~T ~ 1 w þ y y þ o~ 2 4 im 2r im im im 2r im im
n where rim 40 and r im 4 0 are design parameters. y~ im ¼ yim yim n ~ im ¼ oim o ^ im are the parameters errors. yim and o ^ im are the and o n estimates of yim and onim , respectively. From Eqs. (37), (38) and (40), one can obtain T @a ‘V im r ‘V i,m1 þ w3im aim þ wi,m þ 1 þ y~ im jim ðx^ im Þþ Oim þ oim i,m1 p_^ i @p^ i ! @ai,m1 n 1 @2 ai,m1 2 ðpi ei2 þ F i1 þ di1 þ Di1 ðt,zi ,yÞÞ g i1 ðt,zi ,yÞ 2 @y2i @yi
ð36Þ
where si1, si1 and si1 40 are design parameters. From Eqs. (32)–(36), it follows that
þC i1 þ
ð39Þ
w3im wi,m þ 1 r w4im þ w4i,m þ 1
9 $i1 ¼ r i1 y4i r i1 si1 p^ i 4
4
m 1 X
Consider the following Lyapunov function candidate
ð2Þ ki1l y4l ,
!4
37
ð38Þ
l¼1
Substituting Eqs. (42)–(47) into (41) and by using Lemma 3, gives 3 @ai,m1 4=3 4 ‘V im r lim :ei : þ C im þ w3im ðaim þ Oim þ wim þ wim 4 @yi 3 @ai,m1 4=3 3 @ai,m1 4=3 þ p^ i wim þ p^ i wim 4 4 @yi @yi 0 !2 1 4 2 9 @ a 1 @ a i,m1 i,m1 3 Ap^ i þ @ wim þ wim 4 4 @yi @y2i @a 1 ^ im tanhðw3im =kÞ i,m1 $im Þ þ w4i,m þ 1 þo 4 @p^ i T 1 1 @a 3 þ y~ im wim jim ðx^ im Þ y_ im þ p~ i w3im i,m1 r im @p^ i r i1 4 2 _ ðp^ $ Þs p~ p^ y ðb þ F ðy ÞÞ þ C ðz Þ i
im
i1 i i
i
i1
i1
i
im
i
38
T. Wang et al. / Neurocomputing 97 (2012) 33–43
þHi1 ðyÞ þ Pim ðyÞ þ Lim ðyi Þ þ ~ im ðw3im tanhðw3im =kÞ þo
si1 ~ T s y i1 yi1 þ i1 e~ i1 e^ i1 r i1
where
r i1
1 _ ^ Þ o r im im
ð48Þ ð1Þ
where lim ¼ li,m1 ð1=4Þð1=4Þm4i1 , Lim ðyi Þ ¼ Li,m1 ðyi Þ þ 2ki1i y4i þ ð2Þ 6ki1i y4i ,
Pim ðyÞ ¼ Pi,m1 ðyÞ þ 2
N X
N X
ð1Þ
ki1l y4l þ 6
l ¼ 1,l a i
C im ¼ C i,m1 þ
N X 1 n 4 di þ 16 ji1l ð0Þ 4 l¼1
!4
þ 32
N X
!4
ci1l ð0Þ
0
þ onim k :
l¼1
Design the intermediate control function aim and the adapta^ im and $im as tion functions yim, o 3 @ai,m1 4=3 3 @ai,m1 4=3 aim ¼ bim wim Oim wim wim p^ i wim 4 2 @yi @yi 0 !2 1 9 @ai,m1 4 1 3 @2 ai,m1 A @ai,m1 @ þ p^ i w þ wim $im 4 im 4 @yi @p^ i @y2 i
3 im =kÞ
^ im tanhðw o
@ai,ni 1 _ @a y ij i,ni 1 e_^ i1 @ y @e^ i1 ij j¼1
@ai,ni 1 T ðx^ i,j þ 1 þ yij jij ðx^ ij ÞÞ @x^ ij
n i 1 X
@ai,ni 1 _ @a ^ ij i,ni 1 ðx^ i2 þ yTi1 ji1 ðx^ i1 ÞÞ o ^ ij @yi @ o j¼2 0 1 n i 1 X @ a i,n 1 i Aðxi1 x^ i1 Þ kij þ @ki,ni @x^ ij
ð2Þ
l ¼ 1,l a i
n i 1 X j¼1
ki1l y4l ,
Cim ðzi Þ ¼ Ci,m1 ðzi Þ þ 2j4i10 ðzi Þ þ 6c4i10 ðzi Þ,
n i 1 X
Oi,ni ¼ yTi,ni ji,ni ðx^ i,ni Þ
j¼1
Consider the following Lyapunov function candidate V i,ni ¼ V i,ni 1 þ
1 4 1 ~T ~ 1 w þ y y þ o~ 2 4 i,ni 2r i,ni i,ni i,ni 2r i,ni i,ni
n where r i,ni 40 and r i,ni 4 0 are design parameters. y~ i,ni ¼ yi,ni yi,ni n ~ i,ni ¼ oi,n o ^ i,ni are the parameters errors.yi,ni and o ^ i,ni are and o i
n
estimates of yi,ni and oni,ni , respectively. Similar to the derivations in Step m, one can obtain
ð49Þ 4
y_ im ¼ r im 3im
ð50Þ
_^ ¼ r w3 tanhðw3 =kÞs o o im im im im ^ im im
ð51Þ
w jim ðx^ im Þsim yim
$im ¼ $i,m1 þ ri1 w3im 0
‘V i,ni r li,ni :ei : þ C i,ni
ni X
bij w4ij y4i ðbi1 þ Fi1 ðy2i ÞÞ
j¼2 ni X sij ~ T y y þ Ci,ni ðzi Þ þ Hi1 ðyÞ þ Li,ni ðyi Þ þ Pi,ni ðyÞ þ r ij ij j ¼ 1 ij
3 @ai,m1 4=3 wim 2 @yi
þ
!2 11 9 @ai,m1 4 1 3 @2 ai,m1 AA @ þ w þ wim 4 im 4 @yi @y2i
ð58Þ
ð52Þ
ni X sij si1 ~ ^ ^ ij si1 p~ i p^ i e i1 ei1 þ o~ ij o
r i1
j¼2 4
rli,ni :ei :
ni X
r ij
bij w4ij y4i ðbi1 þ Fi1 ðy2i ÞÞ þ Ci,ni ðzi Þ þ Hi1 ðyÞ
j¼2
From (48)–(52), it follows that 4
‘V im rlim :ei : þ C im
m X
bij 4ij þ
w
j¼2
1 r i1
p~ i w3im
@ai,m1 _ ðp^ i $im Þ @p^ i
1 si1 p~ i p^ i y4i ðbi1 þ Fi1 ðy2i ÞÞ þ w4i,m þ 1 þ Cim ðzi Þ þ Hi1 ðyÞ 4 m m X X sij ~ T sij s ^ ij y ij yij þ i1 e~ i1 e^ i1 þ o~ ij o þ Pim ðyÞ þ Lim ðyi Þ þ r r r ij i1 j¼1 j ¼ 2 ij
~ ij , p~ i Þ þ C i þ C i,ni þ Li,ni ðyi Þ þ Pi,ni ðyÞ þ di ðy~ ij , e~ ij , o where ~ ij , p~ i Þ ¼ di ðy~ ij , e~ ij , o Ci ¼
ð53Þ Step ni: By setting ni ¼m, one has the control function uiand the ^ i,ni , and $i,ni as follows: adaptation functions yi,ni , o 1 3 @ai,ni 1 4=3 3 @ai,ni 1 4=3 ui ¼ bi,ni wi,ni Oi,ni wi,ni wi,ni p^ i wi,ni 4 4 2 @yi @yi 0 1 !2 @ai,ni 1 4 1 3 @2 ai,ni 1 A 9 ^ i,ni tanhðw3i,n =kÞ @ wi,ni þ wi,ni p^ i o i @yi 4 4 @y2i @ai,ni 1 þ $i,ni @p^ i
ð54Þ
y_ i,ni ¼ ri,ni w3i,ni fi,ni ðx^ i,ni Þsi,ni yi,ni
ð55Þ
3 3 _^ ^ i,ni o i,ni ¼ r i,ni wi,ni tanhðwi,ni =kÞsi,ni o
ð56Þ
p_^ i ¼ $i,ni ¼ $i,ni 1 þr i1 w3i,ni
3 @ai,ni 1 2 @yi
C i,ni
si1 2
p2i þ
ni X
ni ni sij ~ 2 si1 2 X sij 2 si1 ~ 2 X pi y ij e~ i1 o~ ij ,
2
sij
2r ij j¼1
yijn2 þ
j¼1
si1 2r i1
2r ij
n2 ei1 þ
2r i1
ni X
sij
2r ij j¼2
j¼2
wi,ni
0 !2 11 @ai,ni 1 4 1 3 @2 ai,ni 1 AA 9 @ w þ wi,ni þ 4 i,ni 4 @yi @y2i
2r ij
oijn2
!4 !4 N N X X 1 n 4 ¼ C i,ni 1 þ di þ 16 ji1l ð0Þ þ 32 ci1l ð0Þ þ oni,ni k0 4 l¼1 l¼1
5.2. Changing supply function design and stability analysis Next, by using the changing supply function technique proposed by [28], we will construct the function Fi1( ) introduced in the first step. From (59), first choose a smooth nonnegative function Fi1( ) such that
Fi1 ðy2i Þy4i Li,ni ðyi Þ Z Fi ðy2i Þy4i 4=3
ð59Þ
ð60Þ
with Fi( ) being a smooth nonnegative function to be designed. P i 4 4 Choose parameterli,ni 4 0, and let V 0i ¼ li,ni :ei : þ nj ¼ 1 bij wij , then it follows from (59) and (60) that
ð57Þ
‘V i,ni r V 0i þ Ci,ni ðzi Þ þ Hi1 ðyÞ þ Pi,ni ðyÞFi ðy2i Þy4i ~ ij , p~ i Þ þ C i þ C i,ni þdi ðy~ ij , e~ ij , o
ð61Þ
T. Wang et al. / Neurocomputing 97 (2012) 33–43
Take the following Lyapunov function candidate for the whole system N X
VN ¼
V i,ni
ð62Þ
i¼1
Then it follows from (62) that
‘V N r
N X
V 0i
i¼1
Fi ðy2i Þy4i þ
i¼1
N X
þ
N X
N X
i¼1
ð63Þ
P PN ~ ij , e~ ij , y~ ij , p~ i Þ ¼ N ~ ~ ~ ~ where dðo i ¼ 1 di ðo ij , e ij , y ij , p i Þ and C ¼ i ¼ 1 ðC i þ C i,ni Þ. Notice that by the definitions of Hi1(y) and Pi,ni ðyÞ, we have
N X
nl X
N X
2
i ¼ 1,i a l
l¼1j¼1
! kl1i y4i
i ¼ 1,i a l
i ¼ 1,i a l
Z Fi0 ðy2i Þy4i
ð64Þ
with Fi0( ) being a smooth non-decreasing function to be designed in the next subsection. From (63) and (64), we have
‘V N r
N X i¼1
V 0i
N X
Fi0 ðy2i Þy4i þ
i¼1
N X i¼1
Assumption 6. ([28])For the functions fij0( ),cij0( ), ciz( ) and ci0( ) and ai0( ) given by Assumptions 2–3, the following condition holds: limsup s-0 þ
j
ai0 ðsÞ
o1
ð65Þ
ð69Þ
P Case 2. If N l¼ 1 gil ð yl Þ Z ð1=2Þai0 ð zi Þ, then in this case, we have V zi ðzi Þ r ai ðzi Þ r Zi ðyl Þ, and " # N N X X r ðV z Þ g ðyl Þai0 ðzi Þ r r ðZ ðyl ÞÞg ðyl Þr ðV z Þai0 ðzi Þ: i
il
i
Lemma 4. Under Assumption 6, and if [28]
Z s Z 1 ½xi ða i 1 ðsÞÞ0 exp ½zi ða i 1 ðtÞÞ1 dt ds o1
il
i
i
i
l¼1
For these two cases, it follows that " # N N X X 1 ri ðV zi Þ gil ðyl Þai0 ðzi Þ r ri ðZi ðyl ÞÞgil ðyl Þ ri ðV zi Þai0 ðzi Þ: 2 l¼1 l¼1 Consider the Lyapunov function candidate for the entire system W ¼ VN þ
N X
Ui
where VN is defined by (62). Then, it follows from (63), (68) and (69) that
‘W r
N X
V 0i
i¼1
þ
ð67Þ
i¼1
r
N X 1
2 i¼1
N X
N X
Ci,ni ðzi Þ
i¼1
N X
Fi0 ðy2i Þy4i þ
i¼1
N X 1 i¼1
4
r0i ðV zi Þc2iz ðzi Þc2i0 ðzi Þ þdðo~ ij , e~ ij , y~ ij , p~ i Þ þC
V 0i
i¼1
holds. Then, under the controller (54), the control laws (35) and (55)–(57), the closed-loop system has an almost surely unique solution on (0,N) and the solution process is bounded in
Fi0 ðy2i Þy4i þ
N X 1 þ ri ðZi ðyl ÞÞgil ðyl Þ ri ðV zi Þai0 ðzi Þ 2 i¼1l¼1 i¼1
Then there exist a non-decreasing positive function ri( ) such that 8x A Rmi
Theorem 2. Consider the system (3) and (4). Under the Assumptions 1–6 and the conditions of lemma 4, if g ðsÞ ð68Þ limsup il 4 o 1, 1 ri,l r N, s s-0 þ
N X
N X N X
ð66Þ
0
1 1 r ðV z ðz ÞÞa ðz ÞCi,ni ðzi Þ Z r0ii ðV zi ðzi ÞÞc2iz ðzi Þc2i0 ðzi Þ 4 i i i i0 i 2
i
l¼1
i¼1
According to [28], from Assumptions 2–3 and 5, one can construct continuous increasing functions xi( ) and zi( ) satisfy2 2 ing xi ðsÞai0 ðsÞ Z4Ci,ni ðsÞ and zi ðsÞai0 ðsÞ Z2ciz ðsÞci0 ðsÞ.
0
1 2
To show the last step of (69), we consider the following two cases separately: P Case 1. If N l ¼ 1 gil ð yl Þ r ð1=2Þai0 ð zi Þ, then in this case, we have that " # N X 1 ri ðV zi Þ gil ðyl Þai0 ðzi Þ r ri ðV zi Þai0 ðzi Þ 2 l¼1
~ ij , e~ ij , y~ ij , p~ i Þ þ C Ci,ni ðzi Þ þ dðo
To construct the nonnegative functionFi0( ), the following Assumption 6 is introduced.
4 2 2 4 ij0 ðsÞ þ cij0 ðsÞ þ ciz ðsÞci0 ðsÞ
where Zi ¼ ai ða1 i0 ð2N gil ðUÞÞÞA k1 .
ð2Þ
i ¼ 1,i a l
j¼1
ri ðZi ðyl ÞÞgil ðyl Þ ri ðV zi Þai0 ðzi Þ
1 0 r ðV z Þc2 ðz Þc2i0 ðzi Þ 2 i i iz i
þ
In order to make ‘V N be non-positive, we first choose Fi( ) in (60) (1rirN) to ensure the non-positiveness of the sum of allyirelated terms of the right-hand of the above inequality (63). To this end, choose a smooth nonnegative function Fi( ) such that ! nl nl N N N X X X X X ð1Þ ð2Þ Fi ðy2i Þy4i glji y4i 2 kl1i þ6 kl1i y4i j ¼ 2 i ¼ 1,i a l
N X
r
glji y4i
N X
ð1Þ
kl1i þ 6
By Ito^ formula, Assumptions 2 and 3, we have @V T 2 1 0 zi ‘U i ðzi Þ ¼ ri ðV zi Þ‘V zi þ ri ðV zi Þ g i0 2 @zi
l¼1
N X
l ¼ 1 j ¼ 2 i ¼ 1,i a l
i¼1
þ
nl N X X
Proof. Suppose that ri( ) is the changing supply function defined in Lemma 4. Let Z V z ðzi Þ i ri ðtÞ dt U i ðzi Þ ¼ 0
~ ij , e~ ij , y~ ij , p~ i Þ þ C Ci,ni ðzi Þ þ dðo
ðHi1 ðyÞ þ Pi,ni ðyÞÞ ¼
probability. Moreover, the outputs y ¼ ½y1 ,. . .yN T can be regulated into a small neighborhood of the origin in probability.
ðHi1 ðyÞ þ Pi,ni ðyÞÞ
i¼1
N X
39
N X N X
ri ðZi ðyl ÞÞgil ðyl Þ
i¼1l¼1
ri ðV zi Þai0 ðzi Þ þdðo~ ij , e~ ij , y~ ij , p~ i Þ þC
ð70Þ
It is east to get that N X N X i¼1l¼1
ri ðZi ðyl ÞÞgil ðyl Þ ¼
N X N X i¼1l¼1
rl ðZl ðyi ÞÞgli ðyi Þ
ð71Þ
40
T. Wang et al. / Neurocomputing 97 (2012) 33–43
Then from (68), we can construct a smooth non-decreasing function Fi0(s2)AkN such that N X
Fi0 ðs2 Þ Z
ri ðZi ðsÞÞ sup
t A ð0,s
l¼1
Fi0 ðy2i Þy4i Z
N X
gli ðtÞ t4
ji1j ðx^ i1 Þ ¼ P5
rl ðZl ðyi ÞÞgli ðyi Þ
j12j ðx^ 11 , x^ 12 Þ ¼
Fi0 ðy2i Þy4i Z
i¼1
The fuzzy logic systems can be expressed in the form 5 X ^f ðx^ y Þ ¼ yT j ðx^ Þ ¼ yTi1j ji1j ðx^ i1 Þ, i ¼ 1,2: i1 i1 i1 i1 i1 i1 j¼1 5 X ^f ðx^ y Þ ¼ yT j ðx^ , x^ Þ ¼ yT12j j12j ðx^ 11 , x^ 12 Þ 12 12 12 12 12 11 12 j¼1
Thus, by (70) we have N X
V 0i
i¼1
N X 1
4 i¼1
exp½ðx^ 11 3 þ nÞ2 =4 exp½ðx^ 12 3 þ nÞ2 =4
j ¼ 1,. . .,5:
ri ðZi ðyl ÞÞgil ðyl Þ
i¼1l¼1
‘W r
exp½ðx^ 11 3 þjÞ2 =4 exp½ðx^ 12 3 þjÞ2 =4 5 P n¼1
This together with (71) ensures N X N X
We obtain fuzzy basis functions as follows: exp½ðx^ i1 3þ jÞ2 =16 ,j ¼ 1,. . .,5 2 ^ n ¼ 1 exp½ðxi1 3 þnÞ =16
, and hence
l¼1
N X
l ¼ 1,. . .,5
r0i ðV zi Þai0 ðzi Þ þ dðo~ ij , e~ ij , y~ ij , p~ i Þ þ C
where Define W1 ¼
N X
V 0i þ
i¼1
N X 1
4 i¼1
yTi1 ¼ ½yi11 , yi12 , yi13 , yi14 , yi15 , yT12 ¼ ½y121 , y122 , y123 , y124 , y125 ,
r0i ðV zi Þai0 ðzi Þdðo~ ij , e~ ij , y~ ij , p~ i Þ
ji1 ðx^ i1 Þ ¼ ½ji11 ðx^ i1 Þ, ji12 ðx^ i1 Þ, ji13 ðx^ i1 Þ, ji14 ðx^ i1 Þ, ji15 ðx^ i1 ÞT , i ¼ 1,2,
Then, it is easy to see that W1 is positive-definite and radically ^ , e^ , y, pÞ ^ and satisfies unbounded in its arguments ðe, w,z, o
‘W r W 1 þC e ¼ ½eT1 ,. . .,eTN T , ¼ ½ ^ 1 ,. . ., ^ N T , ^
ð72Þ ¼ ½ T1 ,. . ., TN T , z ¼ ½zT1 ,. . .,zTN T , ¼ ½ ^ 1 ,. . ., ^ N T , y ¼ ½y1 ,. . ., yN T .
where w w w p^ ¼ ½p^ 1 ,. . ., ^ o o e e e p^ N T ,o By theorem 1, the closed-loop system has an almost surely unique solution on (0,N), and moreover, the solution of the closed-loop system is bounded in probability, i.e., for any given e 40, there exist a k‘ function b and a k function g such that 8tZ0, one has
^ o ^ , e^ , yÞ o bð9ðeð0Þ, wð0Þ,zð0Þ, pð0Þ, ^ P ðe, w,z, p, ^ ð0Þ, e^ ð0Þ, yð0ÞÞ9,tÞ þ gðCÞ Z 1e o ^ ^ ð0Þ, e^ ð0Þ, yð0ÞÞ a 0. From the definition where ðeð0Þ, wð0Þ,zð0Þ, pð0Þ, o of C i,ni and in the similar argurments in [27–30], C can be made small if we choose the design parameters appropriately. 6. Simulation study In this section, the feasibility of the proposed method and the control performances are illustrated by the following example: Example. Consider the following stochastic nonlinear system 1 dz1 ¼ ð3z1 þ x211 sint þ x21 Þdt þ pffiffiffi z1 cosx11 dw1 2 dx11 ¼ ðx12 þ sinðx11 Þ þ 0:1z1 þ 0:5x21 Þ dt þ dw1 dx12 ¼ ðx12 þ x11 þ u1 Þ dt þdw1 y1 ¼ x11 1 1 1 dz2 ¼ 4z2 þ x221 þ x211 dt þ pffiffiffi z2 sinx21 dw2 4 4 2 dx21 ¼ ðu2 þ x221 þx221 x11 Þ dt þ z2 x21 dw2 y2 ¼ x21 Define fuzzy membership as follows:
j12 ðx^ 11 , x^ 12 Þ ¼ ½j121 ðx^ 11 , x^ 12 Þ, j122 ðx^ 11 , x^ 12 Þ, j123 ðx^ 11 , x^ 12 Þ, j124 ðx^ 11 , x^ 12 Þ, j125 ðx^ 11 , x^ 12 ÞT Then the state observer (13) is dx^ 11 ¼ ðx^ 12 þ f^ 1 ðx^ 11 y11 Þ þ 10ðx11 x^ 11 ÞÞdt dx^ 12 ¼ ðu1 þ f^ 2 ðx^ 12 y12 Þ þ12ðx11 x^ 11 ÞÞdt dx^ 21 ¼ ðu2 þ f^ 21 ðx^ 21 y21 Þ þ10ðx21 x^ 21 ÞÞdt Given a positive definite matrix Q¼I, solving (10) to obtain the positive definite matrix 0:6500 0:5000 P1 ¼ 0:5000 0:4708 With the notations of Assumption 1, we can take j110(z1)¼ 9z19, j111(9y19)¼0, j112(9y29)¼0.59y29, j120(z1)¼0, j121(9y19)¼0, j122(9y29)¼0, c110(z1)¼0, c111(9y19)¼ 0, c112 (9y 29)¼0, c120(z1)¼0, c121(9y19)¼0, c122(9y29)¼0, j210(z2)¼0, j211 ðy1Þ ¼ 0:5y21 , j212 ðy2 Þ ¼ 0:5y22 , c210 ðz2 Þ ¼ 0:5z22 , c211(9y19)¼0, c212 ðy2 Þ ¼ 0:5y22 , pnij ¼ 1. For convenience, choose fijl(0)¼0 and cijl(0) ¼0, respectively. ð1Þ ð2Þ Then from (16) and (17) we know that k111 ¼ 0, k111 ¼ 0 and ð1Þ ð2Þ 2 2 k212 ¼ ð1=4Þy2 , k212 ¼ ð1=4Þy2 . According to (32), we can get g111 ¼0 and g121 ¼0. By Assumption 2, we can get a 1 ðsÞ ¼ ð1=4Þs4 , a1 ðsÞ ¼ ð1=2Þs4 ,
g11 ðsÞ ¼ ð1=4Þs8 , g12 ðsÞ ¼ ð1=4Þs4 , a10 ðsÞ ¼ ð3=4Þs4 . a 2 ðsÞ ¼ ð1=2Þs2 , a2 ðsÞ ¼ s2 , g21 ðsÞ ¼ ð1=8Þs4 , g22 ¼ ð1=8Þs4 , a20 ðsÞ ¼ 7=2 s2 . According to Lemma 4 and (60) and (64), by choosing r1(s)¼80 and r2 ðsÞ ¼ ð4=3Þs þ 8, we can get F10 ðy21 Þ ¼ ð20 þ ð1=42ÞÞ y41 þ 1,F20 ðy22 Þ ¼ 21y22 þð1=42Þy62 ; F11 ðy21 Þ ¼ ð21 þ ð1=42ÞÞ y41 þ1, F21 ðy22 Þ ¼ 21y22 þ ð1=42Þy62 þ 1. The controller and parameter adaptive laws for the subsystem 1 are given as follows: 3 2
9 4
a11 ¼ b11 y1 F11 ðy21 Þy1 y1 p^ 1 y1 ð73Þ ð2Þ
T
2 X
g1j1 y1 2kð1Þ 111 y1
j¼1
6k111 y1 y11 j11 ðx^ 11 Þe^ 11 tanhðy31 =kÞ
ð74Þ
mF li1 ðx^ i1 Þ ¼ exp½ðx^ i1 3 þlÞ2 =16, l ¼ 1,. . .,5:
y_ 11 ¼ r 11 j11 ðx^ 11 Þy31 s11 y11
ð75Þ
mF l12 ðx^ 11 , x^ 12 Þ ¼ exp½ðx^ 11 3 þlÞ2 =4 exp½ðx^ 12 3 þlÞ2 =4,
e_^ 11 ¼ r11 y31 tanhðy31 =kÞs11 e^ 11
ð76Þ
T. Wang et al. / Neurocomputing 97 (2012) 33–43
3 4
$11 ¼ r 11 y41 r11 s11 p^ 1
0
ð77Þ
1 3 @a11 4=3 w12 u1 ¼ b12 w12 O12 w12 4 4 @y1 0 !2 1 9 @a11 4 1 3 @2 a11 A @ p^ 1 w þ w12 4 12 @y1 4 @y21
3 @a11 4=3 @a ^ 12 tanhðw312 =kÞ þ 11 $12 p^ 1 w12 o 2 @y1 @p^ 1
!2 11 4 2 9 @ a 1 @ a 11 11 3 AA þ @ w12 þ w12 4 4 @y1 @y21
1 ð1Þ u2 ¼ b21 y2 F21 ðy22 Þy2 p^ 2 y2 2k212 y2 2 ð2Þ
ð78Þ
ð79Þ
_^ ¼ r w3 tanhðw3 =kÞs o o 12 12 12 12 ^ 12 12
ð80Þ
T
6k212 y2 y21 f21 ðx^ 21 Þe^ 21 tanhðy2 =kÞ
ð82Þ
y_ 21 ¼ r 21 f21 ðx^ 21 Þy2 s21 y21
ð83Þ
e_^ 21 ¼ r21 y2 tanhðy2 =kÞs21 e^ 21 :
ð84Þ
1 p_^ 2 ¼ r 21 y22 r 21 s21 p^ 2 2
ð85Þ
Fig. 3. x21(solid) and x^ 21 (dotted). Fig. 1. x11(solid) and x^ 11 (dotted).
Fig. 2. x12(solid) and x^ 12 (dotted).
ð81Þ
The controller and parameter adaptive laws for the subsystem 2 are given as follows:
y_ 12 ¼ r12 w312 f12 ðx^ 12 Þs12 y12
3 @a11 4=3 w12 p_^ 1 ¼ $12 ¼ $11 þ r 11 w312 2 @y1
41
Fig. 4. z1(solid) and z2(dotted).
42
T. Wang et al. / Neurocomputing 97 (2012) 33–43
In this simulation, the design parameters in (74)–(85) are chosen as r11 ¼ r12 ¼r21 ¼0.1, r 11 ¼ r 12 ¼ r 21 ¼ 1, r 11 ¼ r 21 ¼ 0:01, s11 ¼ s12 ¼ s21 ¼1, s11 ¼ 0:8, s12 ¼ 0:3, s21 ¼ 12, s11 ¼ 1:4, s21 ¼ 10:5, b11 ¼0.25, b12 ¼0.25, b21 ¼0.25, k¼0.1 The initial conditions are chosen as x11 ð0Þ ¼ 0:05,x12 ð0Þ ¼ 0:5,x21 ð0Þ ¼ 0:05,z1 ð0Þ ¼ z2 ð0Þ ¼ 10, x^ 11 ð0Þ ¼ x^ 12 ð0Þ ¼ x^ 21 ¼ 0p^ 1 ð0Þ ¼ p^ 2 ð0Þ ¼ e^ 11 ¼ e^ 21 ¼ 0,
yT11 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9 yT12 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9 yT21 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9 The simulation results are shown by Figs. 1–6. From the simulation results in Figs. 1–6, it clearly shows that the proposed control approach can achieve the good control performances even if the controlled nonlinear stochastic system contains the unknown functions, the unmodeled dynamics and unmeasured states.
7. Conclusions In this paper, an observer-based adaptive fuzzy decentralized output feedback control approach has been proposed for a class of uncertain large-scale stochastic nonlinear systems with unknown functions, dynamic uncertainties and without the direct measurements of state variables. In the design, fuzzy logic systems are utilized to approximate the unknown functions and a fuzzy state observer is developed. By using the fuzzy state observer and based on the principle of the adaptive backstepping technique and the changing supply function technique, a new robust adaptive fuzzy decentralized output feedback control scheme is synthesized. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by choosing appropriate design parameters. The future researches will mainly concentrate on the tracking output feedback control problem and the tolerant-control design problem for the large-scale stochastic nonlinear systems based on the result of this paper. References
Fig. 5. :yi1 : (solid) and .:yi2 :.(dotted).
Fig. 6. u1 (solid) and u2(dotted).
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Tong Wang received the B.S. degree from the Department of Mathematics, Shanxi University, Taiyuan, China, in 2010. He is now working towards the M.E. degree in control theory and control engineering from Liaoning University of Technology, Jinzhou, China. His current research interests include fuzzy control, adaptive control, and stochastic control.
43 Shaocheng Tong received the B.A. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.A. degree in fuzzy mathematics from Dalian Marine University, PRC, and the Ph.D. degree in fuzzy control from Northeastern University, PRC, in 1982, 1988, and 1997, respectively. Currently, he is a Professor in the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, PRC. His research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control.
Yongming Li received the B.S. degree and the M.S. degree in applied mathematics from Liaoning University of Technology, Jinzhou, China, in 2004 and 2007, respectively. He is currently a lecturer in the Department of Basic Mathematics, Liaoning University of Technology. Now, he is pursuing his Ph.D. degree in transportation information engineering and control in Dalian Maritime University. His current research interests include fuzzy and neural networks control, and nonlinear adaptive control.