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Adaptive neural tracking control for stochastic nonlinear strict-feedback systems with unknown input saturation
Information Sciences 269 (2014) 300–315
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Adaptive neural tracking control for stochastic nonlinear strict-feedback systems with unknown input saturation Huanqing Wang a,c, Bing Chen a,⇑, Xiaoping Liu b, Kefu Liu b, Chong Lin a a
Institute of Complexity Science, Qingdao University, Qingdao, 266071 Shandong, PR China Faculty of Engineering, Lakehead University, Orillia, ON P7A 5E1, Canada c School of Mathematics and Physics, Bohai University, Jinzhou, 121000 Liaoning, PR China b
a r t i c l e
i n f o
Article history: Received 11 January 2013 Received in revised form 4 June 2013 Accepted 22 September 2013 Available online 2 October 2013 Keywords: Adaptive neural tracking control Stochastic nonlinear system Input saturation Backstepping technique
a b s t r a c t In this paper, the problem of adaptive neural tracking control is considered for a class of single-input/single-output (SISO) strict-feedback stochastic nonlinear systems with input saturation. To deal with the non-smooth input saturation nonlinearity, a smooth nonaffine function of the control input signal is used to approximate the input saturation function. Classical adaptive technique and backstepping are used for control synthesis. Based on the mean-value theorem, a novel adaptive neural control scheme is systematically derived without requiring the prior knowledge of bound of input saturation. It is shown that under the action of the proposed adaptive controller all the signals of the closed-loop system remain bounded in probability and the tracking error converges to a small neighborhood around the origin in the sense of mean quartic value. Two simulation examples are provided to demonstrate the effectiveness of the presented results. 2013 Elsevier Inc. All rights reserved.
1. Introduction It is well known that stochastic disturbance, which is usually a source of instability of control systems, often exists in practical systems. Therefore, the control design of nonlinear stochastic systems has attracted increasing attention in recent years [9,10,16,27,29,30,36–38,49–54]. Many control design approaches for deterministic nonlinear systems have been successfully extended to stochastic nonlinear systems. Especially, backstepping technique [18] has been a popular tool for control design of stochastic nonlinear systems, see, e.g., [9,10,16,27,29,30,49–52] and the reference therein. In [30], the quadratic Lyapunov function is used to solve the stabilization problem for stochastic nonlinear strict-feedback systems based on a risk-sensitive cost criterion, and the proposed controller guarantees globally asymptotic stability in probability. In [9,10], a quartic Lyapunov function is applied for control design and stability analysis of stochastic nonlinear strict-feedback and output-feedback systems. Compared with the quadratic Lyapunov function, the quartic Lyapunov function can be used to easily deal with the high-order Hessian term. Since then, the quartic Lyapunov function has been widely applied for control design of stochastic nonlinear systems [16,29,49–52]. However, the aforementioned control schemes maybe invalid to control stochastic systems with unknown nonlinear function, because they require that the nonlinear dynamics models are known precisely or the unknown parameters appear linearly with respect to known nonlinear functions. During the past decades, many approximation-based adaptive neural (or fuzzy) control approaches have been developed to control uncertain lower-triangular nonlinear systems, and lots of significant results have been reported, for example, see [2–5,12–14,19,22,23,25,26,28,35,39–42,44,46,55–58] for deterministic nonlinear systems and [8,21,33,43,47] for stochastic ⇑ Corresponding author. Tel.: +86 0532 85953607. E-mail address: [email protected] (B. Chen). 0020-0255/$ - see front matter 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.09.043
H. Wang et al. / Information Sciences 269 (2014) 300–315
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nonlinear systems. In these proposed control schemes, radial basis function (RBF) neural networks (or fuzzy logic systems) are used to approximate uncertain smooth nonlinear functions, and then adaptive backstepping technique is applied to design controllers. For the deterministic systems, Ge et al. [12–14] develop several adaptive neural control schemes for SISO nonlinear systems and multi-input and multi-output (MIMO) nonlinear systems. In [57,58], the problem of adaptive neural tracking control is considered for MIMO nonlinear systems with dead-zone. Then, for stochastic systems, Psillakis and Alexandridis [33] proposes an adaptive neural network control scheme to solve the problem of output tracking control for uncertain stochastic nonlinear strict-feedback systems with unknown covariance noise. Alternatively, in [47], a fuzzy-based adaptive control scheme is presented for a class of uncertain strict-feedback stochastic nonlinear systems with unknown virtual control gain function. The proposed controller guarantees that all the signals in the closed-loop systems are semi-globally uniformly bounded in probability. Recently, in [8,21,24,43], several approximation-based adaptive control approaches are proposed for some classes of stochastic nonlinear strict-feedback time-delay (or delay-free) systems. In many practical systems, input saturation is one of the most important non-smooth nonlinearities. It often severely limits the system performance, gives rise to undesirable inaccuracy or leads to instability [32]. Therefore, the phenomenon of input saturation has to be considered when the controller is designed in practical industrial process control field. So far, many significant results on control design of the systems with input saturation have been obtained, for example, see [6,7,11,48,59]. In [59], a globally stable adaptive control approach is presented for minimum phase SISO systems with input saturation. Chen et al. [6] proposes a robust adaptive neural control for a class of MIMO nonlinear systems with input nonlinearities. By introducing auxiliary design systems to analyze the effect of input constraints, in [7], an adaptive tracking control is proposed for a class of uncertain nonlinear systems with non-symmetric input constraints, and the derived controller guarantees that the closed-loop system is semi-globally uniformly ultimately bounded stability. Wen et al. [48] considers the problem of adaptive control for a class of uncertain nonlinear systems in the presence of input saturation and external disturbance, in which two new schemes are developed to compensate for the effects of the saturation nonlinearity and disturbances. Though the aforementioned results take input saturation nonlinearity into account, the effect of stochastic disturbance is ignored. Note that stochastic disturbance and input constraint could be existed in many practical systems. Motivated by the above observations, this paper considers the problem of adaptive neural tracking control for the case of nonlinear strict-feedback systems with stochastic disturbance and input saturation simultaneously. The proposed adaptive neural control scheme guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. Compared with the existing results, the main idea of control design in this paper is that a smooth non-affine function of the control input signal is firstly used to approximate the saturation function, and furthermore, the mean-value theorem is used to transform the non-affine function into affine form, i.e., gðv Þ ¼ g v l v . Then, the classical adaptive technique and backstepping are used to design controller. The proposed design approach does not require the prior knowledge of the bound of input saturation. In addition, the number of adaptive parameters just depends on the order of the considered systems. So, it is reduced considerably. In this way, the computational burden is significantly alleviated. This paper is organized as follows. The preliminaries and problem formulation are given in Section 2. A novel adaptive neural control scheme is presented in Section 3. Section 4 gives two simulation examples to illustrate the effectiveness of our results, and Section 5 concludes the work. 2. Preliminaries and problem formulation The following notations are used throughout this paper. R denotes the set of all real numbers; Rn indicates the real n-dimensional space. For a given vector or matrix X, XT denotes its transpose; Tr{X} is its trace when X is a square matrix; and kXk denotes the Euclidean norm of a vector X. Ci denotes the set of all functions with continuous ith partial derivative. Consider the following strict-feedback stochastic nonlinear system given by:
8 T > < dxi ¼ ðg i ðxi Þxiþ1 þ fi ðxi Þ þ di ðt; xÞÞdt þ wi ðxi Þdw; 1 6 i 6 n 1; dxn ¼ ðg n ðxn Þuðv Þ þ fn ðxn Þ þ dn ðt; xÞÞdt þ wTn ðxn Þdw; > : y ¼ x1 ;
ð1Þ
where xi ¼ ½x1 ; x2 ; . . . ; xi T 2 Ri , x = [x1, x2, . . . , xn]T 2 Rn and y 2 R are the state variables and the system output, respectively; w denotes an r-dimensional standard Brownian motion defined on the complete probability space (X, F, P) with X being a sample space, F being a r-field, and P being a probability measure; fi(), gi(): Ri ? R, wi (): Ri ? Rr, (i = 1, 2, . . . , n) stand for the unknown smooth nonlinear functions with fi(0) = 0 and wi (0) = 0 (1 6 i 6 n), di(), i = 1, 2, . . . , n are the external disturbance uncertainties of the system. v is the control signal to be designed, and u(v) denotes the plant input subject to saturation nonlinearity described by
uðv Þ ¼ satðv Þ ¼
signðv Þumax ; jv j P umax ;
v;
jv j < umax ;
where umax is a unknown parameter of input saturation.
ð2Þ
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Remark 1. There exist many practical systems which are described by strict-feedback form, such as One-Link Robot system, Pendulum System With Motor, Single-Link Manipulator system [55], and Brusselator model [45]. Meanwhile, stochastic disturbance and input saturation are inevitable in practical process. Therefore, the aforementioned systems can be governed by nonlinear differential equations of the form (1). The control objective is to design an adaptive neural controller for system (1) such that the system output y follows the specified desired trajectory yd and all the signals in the closed-loop systems remain bounded in probability. From (2), it can be seen that there exists a sharp corner when jvj = umax. So backstepping technique cannot be directly applied to construct control input signal. To solve this problem, the method proposed in [48] will be implemented. By this method, a smooth function is used to approximate the saturation function and defined as
ev =umax ev =umax gðv Þ ¼ umax tanhðv =umax Þ ¼ umax v =umax : e þ ev =umax
ð3Þ
Then, sat(v) in (2) can be expressed in the following form:
satðv Þ ¼ gðv Þ þ dðv Þ;
ð4Þ
where d(v) = sat(v) g(v) is a bounded function and its bound can be obtained as
jdðv Þj ¼ jsatðv Þ gðv Þj 6 umax ð1 tanhð1ÞÞ ¼ D:
ð5Þ
Fig. 1 shows the saturation nonlinearity in (2) and its approximation function in (3). According to the mean-value theorem [1], there exists a constant l with 0 < l < 1, such that
gðv Þ ¼ gðv 0 Þ þ g v l ðv v 0 Þ; vÞ where g v l ¼ @gð j ¼ @ v v ¼v l
ð6Þ
4 2 ðev =umax þev =umax Þ
jv ¼v l , vl = lv + (1 l)v0. By choosing
v0 = 0, (6) can be written as
gðv Þ ¼ g v l v ;
ð7Þ
Substituting (4) into (1) and using (7) gives
8 T > < dxi ¼ ðg i ðxi Þxiþ1 þ fi ðxi Þ þ di ðt; xÞÞdt þ wi ðxi Þdw; 1 6 i 6 n 1; dxn ¼ ðg n ðxn Þðg v l v þ dðv ÞÞ þ fn ðxn Þ þ dn ðt; xÞÞdt þ wTn ðxn Þdw; > : y ¼ x1 :
ð8Þ
To facilitate control system design, the following assumptions and lemmas are presented and will be used in the subsequent developments. xi Þ is unknown, but the sign of g i ð xi Þ does not change, and there exist Assumption 1 ([3,14]). For 1 6 i 6 n, the function g i ð unknown constants bm and bM, such that
0 < bm 6 jg i ðxi Þj 6 bM < 1;
8xi 2 Ri :
ð9Þ
xi Þ is strictly either positive or negative. Without loss of generality, it is further assumed that Apparently, (9) implies that g i ð
0 < bm 6 g i ðxi Þ 6 bM ; Assumption
2 [45]. For
8xi 2 Ri : 1 6 i 6 n,
ð10Þ there
exist
unknown
smooth
positive
functions
8ðt; xÞ 2 Rþ X; jdi ðt; xÞj 6 hi ðxi Þ.
10 8 6
u(v)
4 2 0 −2 −4 −6 −8 −10
−20
−15
−10
−5
0
5
10
15
20
v Fig. 1. Saturation (solid-line: sat (v); Dot-line smooth function umax⁄ tanh (v/umax)).
hi ð xi Þ
such
that
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Assumption 3 [3]. The desired trajectory yd(t) and its nth order time derivatives are continuous and bounded. To introduce some useful conceptions and lemmas, consider the following stochastic system:
dx ¼ f ðxÞdt þ hðxÞdw;
ð11Þ
where x and w are defined in (1), and f() and h() are locally Lipschitz functions in x and satisfy f(0) = 0 and h(0) = 0. Definition 1. For any given V(x) 2 C2, associated with the stochastic differential Eq. (11), define the differential operator L as follows:
( ) 2 @V 1 T @ V f þ Tr h LV ¼ h ; @x 2 @x2
ð12Þ
where Tr(A) is the trace of A. n o T 2 ^ correction term or high-order Hessian term, in which the Remark 2. As stated in [29], the term 12 Tr h @@xV2 h is called Ito 2 second-order differential @@xV2 makes the controller design much more difficult than that of the deterministic system. Definition 2 [17]. The solution process {x(t), t P 0} of stochastic system (11) is said to be bounded in probability, if limc?1 sup06t<1P{kx(t)k > c} = 0, where P{B} denotes the probability of event B. Lemma 1 [33]. Consider the stochastic system (11). If there exists a positive definite, radially unbounded, twice continuously differentiable Lyapunov function V : Rn ! R, and constants a0 > 0, b0 P 0 such that
LVðxÞ 6 a0 VðxÞ þ b0 ; then (i) the system has a unique solution almost surely and (ii) the system is bounded in probability. Lemma 2 (Young’s inequality [9]). For "(x, y) 2 R2, the following inequality holds:
xy 6
ep p
jxjp þ
1 jyjq ; qeq
where e > 0, p > 1, q > 1, and (p 1)(q 1) = 1. Lemma 3 [31]. For any variable g 2 R and constant
0 6 jgj g tanh
g
6 d;
> 0, the following inequality holds.
d ¼ 0:2785:
ð13Þ
In this note, the following RBF neural networks will be used to approximate any continuous function f(Z): Rn ? R,
fnn ðZÞ ¼ W T SðZÞ;
ð14Þ
q
where Z 2 XZ R is the input vector with q being the neural networks input dimension, weight vector W = [w1, w2, . . . , wl]2 Rl, l > 1 is the neural networks node number, and S(Z) = [s1(Z), s2(Z), . . . , sl(Z)]T means the basis function vector with si(Z) being chosen as the commonly used Gaussian function of the form
T
" si ðZÞ ¼ exp
# ðZ li ÞT ðZ li Þ ; r2
i ¼ 1; 2; . . . ; l;
ð15Þ
where li = [li1, li2, . . . , liq]T is the center of the receptive field and r is the width of the Gaussian function. In [34], it has been indicated that with sufficiently large node number l, the RBF neural networks (14) can approximate any continuous function f(Z) over a compact set XZ Rq to arbitrary any accuracy e > 0 as
f ðZÞ ¼ W T SðZÞ þ dðZÞ;
8z 2 Xz 2 Rq ;
⁄
where W is the ideal constant weight vector and defined as
(
)
W :¼ arg min supjf ðZÞ W T SðZÞj ; W2Rl
Z2XZ
and d(Z) denotes the approximation error and satisfies j d(Z)j 6 e.
ð16Þ
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Lemma 4 [20]. Consider the Gaussian RBF networks (14) and (15). Let q :¼ taken as
kSðZÞk 6
1 2 mini–j k
li lj k, then an upper bound of kS(Z)k is
1 X 2 2 2 q1 3qðk þ 2Þ e2q k =r :¼ s:
ð17Þ
k¼0
It has been shown in [44] that the constant s in Lemma 3 is a limited value and is independent of the variable Z and the dimension of neural weights l.
3. Adaptive neural control design In this section, a backstepping-based design procedure will be proposed to construct the adaptive neural tracking controller for the original systems (1) with input saturation nonlinearity (2). The design procedure contains n steps and involves the following coordinate transformation:
z1 ¼ x1 yd ;
zi ¼ xi ai1 ;
i ¼ 2; . . . ; n;
ð18Þ
where ai is a virtual control signal to be designed for the corresponding i-subsystem based on an appropriate Lyapunov function Vi. During the design procedure, the virtual control signal and adaptive law will be constructed in the following form:
z3i kSi ðZ i Þk ; ai 3 ^h_i ¼ c h^i þ ki z3 kSi ðZ i Þk tanh zi kSi ðZ i Þk ; i i ai
ai ðZ i Þ ¼ ki zi h^i kSi ðZ i Þk tanh
ð19Þ ð20Þ
where 1 6 i 6 n, ki, ai, ci and kihare positiveiTdesign contants, Si(Zi) is the RBF neural network basis function vector with T T ðiÞT T ðiÞ Z 1 ¼ ½x1 ; yd ; y_ d 2 XZ1 R3 ; Z i ¼ xTi ; ^ 2 XZi R2iþ2 ði ¼ 2; . . . ; nÞ; ^ hi1 ; yd hi ¼ ½h^1 ; h^2 ; . . . ; h^i . y d denotes the vector composed ^ of yd and up to its ith order time derivative, hi is the estimation of an unknown constant hi which will be given at the ith step, Specially, an denotes the actual control input v. ^i ðtÞ P 0 for all t P 0. Note that h ^i is an hi ð0Þ P 0, then h Remark 3. It is easy to prove from (20) that if initial condition ^ estimation of hi, and the initial condition of (20) can be given by designer. So, it is reasonable to choose ^ hi ð0Þ P 0. This property will be used in each step of control design. In the following, for simplicity, the time variable t and the state vector xi will be omitted from the corresponding functions and denote Si(Zi) by Si. Step 1: Since z1 = x1 yd, the first subsystem of (1) gives
dz1 ¼ ðg 1 x2 þ f1 þ d1 y_ d Þdt þ wT1 dw:
ð21Þ
Consider Lyapunov function candidate as
V1 ¼
1 4 bm ~2 h ; z þ 4 1 2k1 1
ð22Þ
where ~ h1 ¼ h1 ^ h1 is the parameter error. It can be verified easily from (12) along (21) and using the completion of squares that
3 2 3 2 bm ~ ^_ h1 h1 ; LV 1 6 z31 g 1 x2 þ f1 þ d1 y_ d þ l1 z1 kw1 k4 þ l1 4 4 k1
ð23Þ
where l1 is a design constant. By means of Assumption 3, the following inequality holds:
z31 d1 6 jz1 j3 h1 ðx1 Þ 6
1 6 2 1 z h ðx1 Þ þ g211 : 2 2g211 1 1
ð24Þ
Substituting (24) into (23) yields
3 3 3 2 1 bm ~ ^_ h1 h1 ; LV 1 6 z31 ðg 1 x2 þ f 1 ðZ 1 ÞÞ z41 g 1 z41 þ l1 þ g211 4 4 4 2 k1
ð25Þ
2 2 where f 1 ðZ 1 Þ ¼ f1 y_ d þ 2g12 z31 h1 ðx1 Þ þ 34 l1 z1 kw1 k4 þ 34 z1 þ 34 g 1 z1 . Since the smooth functions f1, g1, h1 and w1 are unknown, 11 f 1 ðZ 1 Þ cannot be directly used to construct virtual control signal a1. Thus, an RBF neural network W T1 S1 ðZ 1 Þ is employed to approximate the function f 1 ðZ 1 Þ such that, for any given e1 > 0,
f 1 ðZ 1 Þ ¼ W T S1 ðZ 1 Þ þ d1 ðZ 1 Þ; jd1 ðZ 1 Þj 6 e1 1
ð26Þ
H. Wang et al. / Information Sciences 269 (2014) 300–315
305
with d1(Z1) being the approximation error. Then, according to Lemma 3, one has
3 3 1 z kS1 k 3 1 þ dbm h1 a1 þ z41 þ e41 ; z31f 1 ðZ 1 Þ ¼ z31 W T1 S1 þ z31 d1 6 jz31 jkW 1 kkS1 k þ z41 þ e41 6 z31 bm h1 kS1 k tanh 1 4 4 a1 4 4
ð27Þ
1k where the unknown constant h1 ¼ kW . bm Substituting (26) into (25) and using (27) gives
LV 1 6 z31 g 1 z2 þ z31 g 1 a1 þ z31 bm h1 kS1 k tanh
3 z1 kS1 k 1 3 3 2 1 bm ~ ^_ h1 h1 ; þ dbm h1 a1 þ e41 g 1 z41 þ l1 þ g211 a1 4 4 4 2 k1
ð28Þ
where z2 = x2 a1. At the present stage, constructing the virtual control signal a1 as
a1 ¼ k1 z1 ^h1 kS1 k tanh
3 z1 kS1 k ; a1
ð29Þ
then using (10), we have
3 z kS1 k : z31 g 1 a1 6 k1 bm z41 z31 bm ^h1 kS1 k tanh 1 a1
ð30Þ
From (30), rewrite (28) as
3 3 1 3 2 1 bm ~ z kS1 k _ h1 k1 z31 kS1 k tanh 1 ^h1 : LV 1 6 k1 bm z41 þ z31 g 1 z2 g 1 z41 þ dbm h1 a1 þ e41 þ l1 þ g211 þ 4 4 4 2 a1 k1
ð31Þ
_ By choosing adaptive law ^ h1 in (20) with i = 1, it follows
LV 1 6 k1 bm z41 þ z31 g 1 z2 þ dbm h1 a1 þ
1 4 3 2 1 2 b c e þ l þ g þ m 1 ~h1 ^h1 : 4 1 4 1 2 11 k1
ð32Þ
Furthermore, applying Young’s inequality yields
3 1 g z4 þ g z4 ; 4 1 1 4 1 2 bm c1 ~ ^ bm c1 ~2 bm c1 ~ bm c1 ~2 bm c1 2 h1 h1 ¼ h þ h1 h1 6 h þ h : k1 k1 1 k1 2k1 1 2k1 1
z31 g 1 z2 6
ð33Þ ð34Þ
Using (33) and (34), we can further have
LV 1 6 k1 bm z41
bm c1 ~2 1 3 2 1 bm c1 2 1 bm c1 ~2 1 h þ dbm h1 a1 þ e41 þ l1 þ g211 þ h þ q1 þ g 1 z42 ; h þ g z4 6 c1 z41 4 4 2 4 2k1 1 2k1 1 4 1 2 2k1 1
ð35Þ
m c1 2 where c1 ¼ k1 bm ; q1 ¼ dbm h1 a1 þ b2k h1 þ 14 e41 þ 34 l1 þ 12 g211 . The term 14 g 1 z42 will be dealt with in the next step. 1 ^ formula, we have Step 2: From z2 = x2 a1 and It o
2
T @ a1 dz2 ¼ ðg 2 x3 þ f2 þ d2 ‘a1 Þdt þ w2 w1 dw; @x1
ð36Þ
where
‘a1 ¼
@ a1 ðg x2 þ f1 þ d1 Þ þ N1 @x1 1
ð37Þ
with
N1 ¼
1 X @ a1 ðjÞ j¼0 @yd
ðjþ1Þ
yd
þ
@ a1 ^_ 1 @ 2 a1 T h1 þ w w : ^ 2 @x21 1 1 @ h1
ð38Þ
Choose the Lyapunov function as
1 bm ~2 h : V 2 ¼ V 1 þ z42 þ 4 2k2 2
ð39Þ
Furthermore, by (12) it can be verified that
T 3 @ a1 @ a1 bm ~ ^_ h2 h2 : LV 2 ¼ LV 1 þ z32 ðg 2 x3 þ f2 þ d2 ‘a1 Þ þ z22 w2 w1 w2 w1 2 @x1 @x1 k2
ð40Þ
By substituting (31) and (37) into (40) and using the completion squares to the term next to the last one in (40), one has
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H. Wang et al. / Information Sciences 269 (2014) 300–315
LV 2 6 c1 z41
bm c1 ~2 1 @ a1 3 2 @ a1 h1 þ q1 þ g 1 z42 þ z32 g 2 x3 þ f2 þ d2 ðg 1 x2 þ f1 þ d1 Þ N1 þ l2 z2 kw2 w1 k4 4 4 2k1 @x1 @x1
3 2 bm ~ ^_ h2 h2 ; þ l2 4 k2
ð41Þ
where l2 is a positive design constant. Using the similar way to (24) yields
2 @ a1 @ a1 h1 6 1 z6 @ a1 h2 þ 1 g2 ; d1 6 jz32 j 2 1 2 21 @x1 @x1 @x1 2g221 1 1 2 z32 d2 6 2 z62 h2 þ g222 : 2 2g22 z32
ð42Þ ð43Þ
With the help of (42) and (43), (41) can be written as
LV 2 6 c1 z41
2 3 bm c1 ~2 3 3 2 1X b h1 þ q1 þ z32 g 2 x3 þ f 2 ðZ 2 Þ z42 g 2 z42 þ l2 þ g2 m ~h2 ^h_ 2 ; 4 4 4 2 j¼1 2j k2 2k1
ð44Þ
where
2 f 2 ðZ 2 Þ ¼ f2 @ a1 ðg x2 þ f1 Þ þ 1 g z2 N1 þ 3z2 kw @ a1 w k4 þ 1 z3 @ a1 h2 þ 1 z3 h2 þ 3 z2 þ 3 g z2 : 1 1 2 1 1 2 2 4 4 2 @x1 @x1 @x1 2g221 2g222 2 2 4 4l2
ð45Þ
Note that f 2 ðZ 2 Þ is an unknown smooth function. Therefore, an RBF neural network W T2 S2 ðZ 2 Þ is used to model the unknown f 2 ðZ 2 Þ such that
f 2 ðZ 2 Þ ¼ W T S2 ðZ 2 Þ þ d2 ðZ 2 Þ; 2
ð46Þ
where the approximate error d2(Z2) satisfies jd2(Z2)j 6 e2 with e2 being a given positive constant. Similar to (27), the following inequality holds.
z32 f 2 ðZ 2 Þ 6 z32 bm h2 kS2 k tanh
3 z2 kS2 k 3 1 þ dbm h2 a2 þ z42 þ e42 ; a2 4 4
ð47Þ
2k where the unknown constant h2 ¼ kW . Substituting (46) into (44) and using the inequality (47), we have bm
LV 2 6 c1 z41
3 2 bm c1 ~2 1 3 2 1X z kS k h1 þ q1 þ dbm h2 a2 þ e42 þ l2 þ g22j þ z32 g 2 z3 þ z32 g 2 a2 þ z32 bm h2 kS2 k tanh 2 2 4 4 2 j¼1 a2 2k1
3 bm ~ ^_ h2 h2 ; g 2 z42 4 k2
ð48Þ
where z3 = x3 a2. h2 in (20) into account with i = 2, the following inequalities can be obtained. Then, take a2 in (19) and ^
3 z kS2 k ; z32 g 2 a2 6 k2 bm z42 z32 bm ^h2 kS2 k tanh 2 a2 3 1 z32 g 2 z3 6 g 2 z42 þ g 2 z43 : 4 4
ð49Þ ð50Þ
By using the above inequalities, we can rewrite (48) as 2 2 X bm c1 ~2 1 3 2 1X b c 1 h1 þ q1 þ dbm h2 a2 þ e42 þ l2 þ LV 2 6 cj z4j g22j þ m 2 ~h2 ^h2 þ g 2 z43 4 4 2 4 2k k 1 2 j¼1 j¼1 2 2 2 X X bm cj 2 X 1 ~h þ 6 cj z4j qj þ g 2 z43 ; j 4 2k j j¼1 j¼1 j¼1
ð51Þ
P bm cj 2 2 h2 ^ h2 6 12 ~ h22 þ 12 h22 has been used. where cj ¼ kj bm ; qj ¼ dbm hj aj þ 2k hj þ 14 e4j þ 34 lj þ 12 jk¼1 g2jk ; j ¼ 1; 2, and the inequality ~ j ^ formula, one has Step i (3 6 i 6 n 1): By using (18) and Ito
dzi ¼ ðg i xiþ1 þ fi þ di ‘ai1 Þdt þ wi
i1 X @ ai1 j¼1
@xj
!T wj
dw;
ð52Þ
where
‘ai1 ¼
i1 X @ ai1 j¼1
@xj
ðg j xjþ1 þ fj þ dj Þ þ Ni1
ð53Þ
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H. Wang et al. / Information Sciences 269 (2014) 300–315
with Ni1 ¼
Pi1 @ ai1 ^_ Pi1 @ai1 ðjþ1Þ 1 Pi1 @ 2 ai1 T þ 2 p;q¼1 @xp @xq wp wq . hj þ j¼0 ðjÞ yd j¼1 @ ^ h @yd
j
Consider Lyapunov function as
1 bm ~2 h : V i ¼ V i1 þ z4i þ 4 2ki i
ð54Þ
It follows immediately from (12) that
LV i ¼ LV i1 þ
z3i ðg i xiþ1
i1 X 3 @ ai1 þ fi þ di ‘ai1 Þ þ z2i wi w 2 @xj j j¼1
!T wi
i1 X @ ai1 j¼1
@xj
! wj
bm ~ ^_ hi hi ; ki
ð55Þ
where the term LVi1 can be obtained by a straightforward calculation as former steps.
LV i1 6
i1 i1 i1 X X bm cj 2 X 1 ~h þ cj z4j qj þ g i1 z4i ; j 4 2k j j¼1 j¼1 j¼1
ð56Þ
P bm cj 2 2 where cj ¼ kj bm ; qj ¼ dbm hj aj þ 2k hj þ 14 e4j þ 34 lj þ 12 jk¼1 g2jk ; j ¼ 1; 2; . . . ; i 1. j By using the completion of squares, the following inequality holds:
2
4
i1 i1 X X 3 2
@ ai1
3 2 3 2 4
@ ai1
z w w 6 li þ li zi wi w ;
2 i i j¼1 @xj j
4 4 @xj j
j¼1
ð57Þ
where li is a positive design parameter. Next, by following a same line used in the procedures from (42) and (43), we have
z3i
i1 X @ ai1 j¼1
z3i di 6
@xj
dj 6
2 i1 i1 i1 X X X @ ai1 1 6 @ ai1 1 2 2 jzi j3 j jhj 6 z hj þ g; i 2 2 ij @x @x g 2 j j ij j¼1 j¼1 j¼1
1 6 2 1 2 z h þ g : 2g2ii i i 2 ii
ð58Þ ð59Þ
Further, substituting (53), (56) and (57) into (55) and using the formulas (58), (59) and (55) can be rewritten as i1 i1 i1 i X X bm cj 2 X 3 3 3 1X b ~h þ LV i 6 cj z4j qj þ z3i ðg i xiþ1 þ f i ðZ i ÞÞ z4i g i z4i þ l2i þ g2ij m ~hi ^h_i ; j 4 4 4 2 2k ki j j¼1 j¼1 j¼1 j¼1
ð60Þ
where f i ðZ i Þ is defined as
f ðZ Þ ¼ f i i i
2 i1 i1 X X 3 2 @ ai1 1 3 @ ai1 1 1 2 2 ðg j xjþ1 þ fj Þ Ni1 þ li zi kwi wj k4 þ z hj þ 2 z3i hi þ g i1 zi 2 i 4 4 @xj @x @x 2 2 g g j j ij ii j¼1 j¼1
i1 X @ ai1 j¼1
3 3 þ zi þ g i zi 4 4
ð61Þ
Currently, by employing a neural networks W Ti Si ðZ i Þ to approximate the unknown smooth function f i ðZ i Þ and constructing _ the virtual control law ai and adaptive law ^ hi defined respectively in (19) and (20), and then repeating the similar procedure from (27)–(35) in Step 1, the following result is true. i i i X X bm cj 2 X 1 ~h þ LV i 6 cj z4j qj þ g i z4iþ1 ; j 4 2k j j¼1 j¼1 j¼1
ð62Þ
P bm cj 2 2 where cj ¼ kj bm ; qj ¼ dbm hj aj þ 2k hj þ 14 e4j þ 34 lj þ 12 jk¼1 g2jk ; j ¼ 1; 2; . . . ; i. j ^ formula, we have Step n: This is the final step, and the actual control input v will be constructed. By (18) and Ito
dzn ¼ ðg n ðg v l v þ dðv ÞÞ þ fn þ dn ‘an1 Þdt þ wn
n1 X @ an1 j¼1
where ‘an1 is given in (53) with i = n. Choose the following Lyapunov function candidate:
1 g V n ¼ V n1 þ z4n þ 0 ~h2n ; 4 2kn
@xj
!T wj
dw;
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where g0 will be specified later. By applying (12) and taking (62) with k = n 1 into account, it follows n1 n1 n1 n1 X X X bm cj 2 X 1 @ an1 ~h þ LV n 6 cj z4j qj þ g n1 z4n þ z3n ðg n g v l v þ dðv Þ þ fn þ dn ðg j xjþ1 þ fj þ dj Þ j 4 2k @xj j j¼1 j¼1 j¼1 j¼1 !T ! n1 n1 X X 3 @ an1 @ an1 g _ Nn1 Þ þ z2n wn wn 0 ~hn ^hn : 2 @x @x k j j n j¼1 j¼1
ð63Þ
Similar to (58) and (59), the inequalities below hold.
z3n
n1 X @ an1 j¼1
z3n dn 6
@xj
dj 6
2 n1 n1 X X 1 6 @ an1 1 2 2 z hj þ g ; 2 n 2 nj @x g 2 j nj j¼1 j¼1
1 6 2 1 2 z h þ g : 2g2nn n n 2 nn
ð64Þ ð65Þ
Subsequently, substituting (64) and (65) into (63) and utilizing completion squares to the term next to the last one in (63), one has n1 n1 n1 n X X bm cj 2 X 3 3 3 1X g ~h þ LV n 6 cj z4j qj þ z3n g n dðv Þ þ z3n ðg n g v l v þ f n ðZ n ÞÞ z4n g n z4n þ l2n þ g2nj 0 ~hn ^h_ n : j 4 4 4 2 2k kn j j¼1 j¼1 j¼1 j¼1
ð66Þ
where f n ðZ n Þ is given in (61) with i = n. Again, an RBF neural network W Tn Sn ðZ n Þ is used to model the unknown function f n ðZ n Þ such that, for any given positive constant en,
f n ðZ n Þ ¼ W T Sn ðZ n Þ þ dn ðZ n Þ; n
jdn ðZ n Þj 6 en ;
where dn(Zn) is the approximation error. Furthermore, the following result is satisfied.
z3n f n ðZ n Þ 6 z3n g 0 hn kSn k tanh
z3n kSn k 3 1 þ dg 0 hn an þ z4n þ e4n an 4 4
ð67Þ
with hn ¼ kWg0n k. Now, constructing the actual control input signal v in (19) with i = n and noting that 0 < g m 6 g v l 6 1, then we obtain
3 z kSn k ; z3n g n g v l v 6 kn g 0 z4n z3n g 0 ^hn kSn k tanh n an
ð68Þ
where the fact of g n g v l P bm g m has been used, and g0 = bmgm. Remark 4. The property of 0 < g m 6 g v l 6 1 is introduced in [48], here, for the purpose of stability analysis, we still employ this property.
Next, by (5) and (9) and Lemma 2, one has
z3n g n dðv Þ 6
3 1 g z4 þ b M D 4 ; 4 n n 4
ð69Þ
_ Furthermore, combining (66) with the formulas (67)–(69) and choosing adaptive law ^ hn in (20) with i = n, we can obtain n n1 n1 n X X bm cj 2 X 1 1 3 1X g c ~h þ LV n 6 cj z4j qj þ dg 0 hn an þ e4n þ bM D4 þ l2n þ g2nj þ 0 n ~hn ^hn : j 4 4 4 2 2k kn j j¼1 j¼1 j¼1 j¼1
ð70Þ
In addition, note that
g 0 cn ~ ^ g c g c hn hn 6 0 n ~h2n þ 0 n h2n ; kn 2kn 2kn (70) can be rewritten as n n1 n X X bm cj 2 g 0 cn 2 X ~h ~h þ LV n 6 cj z4j qj j n 2kj 2kn j¼1 j¼1 j¼1
ð71Þ
P bm cj 2 2 where cj ¼ kj bm ; qj ¼ dbm hj aj þ 2k hj þ 14 e4j þ 34 lj þ 12 jk¼1 g2jk ; j ¼ 1; 2; . . . ; n 1. cn ¼ kn g 0 ; qn ¼ dg 0 hn an þ 14 e4n þ 14 bM D4 j P 2 n g c 2 0 n þ 34 ln þ 12 j¼1 g2nj þ 2k h . n n So far, the adaptive neural control design has been completed based on backstepping technique. The main result of this research will be summarized as the following theorem.
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Theorem 1. Consider the uncertain stochastic nonlinear system (1), preceded by unknown input saturation nonlinearity (2), the controller (19), and adaptive law (20) under Assumptions 1–3. Assume there exists sufficiently large compact set XZj ðj ¼ 1; 2; . . . ; nÞ, such that Z j 2 XZj for all t P 0, then for bounded initial conditions with ^hj ð0Þ P 0, h remain in a compact set XZ, (i) all the signals in the closed-loop system are bounded in probability and the error signals zj and ~ which defined by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n b0 ~ 2kj b0 4 ~ XZ ¼ zj ; hj E½jzj j 6 4Vð0Þ þ 4 ;hj j 6 Vð0Þ þ ; j¼1 a0 bm a0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi) 2kn b0 ~ : j ¼ 1; 2; . . . ; n 1; jhn j 6 Vð0Þ þ g0 a0 (
ð72Þ
(ii) there exists a finite time T1 such that the quartic mean square tracking error converges to the following compact set X1, for all t > T1,
X1 ¼ fyðtÞ 2 RjE½jy yd j4 6 8
b0 ; 8t > T 1 g a0
with the time T1 will be given later.
Proof. (i) Let V = Vn, and define a0 = min{4cj, cj, j = 1, 2, . . . , n.} and b0 ¼
LV 6 a0 V þ b0 ;
Pn
j¼1
qj , (71) implies that
t P 0:
ð73Þ
hj are bounded in probability. Since hjis a constant, ^ hj is Therefore, from Lemma 1, for j = 1, 2, . . . , n, the error signals zj and ~ bounded in probability. Consequently, aj is also bounded in probability due to kSik 6 s and j tanh ()j 6 1. Hence, we conclude that all the signals xj = zj + aj1 in the closed-loop system remain bounded in the sense of probability. Furthermore, from (73) the following inequality can be obtained directly by [10](Th. 4.1)
dE½VðtÞ 6 a0 E½VðtÞ þ b0 ; dt
ð74Þ
which implies that
E½VðtÞ 6 ea0 t ½Vð0Þ þ
b0 ; a0
8t > 0;
ð75Þ
then it follows
E½VðtÞ 6 Vð0Þ þ
b0 ; a0
8t > 0;
ð76Þ
Pn
P g 0 ~2 bm ~2 where Vð0Þ ¼ j¼1 14 z4j ð0Þ þ n1 j¼1 2kj hj ð0Þ þ 2kn hn ð0Þ. Next, by combining the definition of V and (76), we conclude that there hj remain nin it forall the exists a compact set XZ defined in (72) such that the error signals zj and ~ otime. such that (ii) From (75), it can be further verified that there exists a finite time T 1 ¼ max 0; a10 ln a0 bV0ð0Þ
E½jy yd j4 6 4E½VðtÞ 6
8b0 ; a0
8t > T 1 :
Thus, to achieve the desired tracking performance, we can properly adjust the design parameters a0 and b0.
h
4. Simulation example In order to test the effectiveness of the proposed method, two examples are given in this section. The first example is a numerical example, and the second one is a practical system model which is taken from [15]. Example 1. Consider the following third-order nonlinear stochastic system:
8 2 > dx1 ¼ ðð1 þ x21 Þx2 þ 0:5x1 sin x1 þ x21 cosð1:5tÞÞdt þ e0:2x1 x21 dw; > > > > < dx ¼ 6 þ x2 x2 x þ x x3 þ x x2 sin3 ðtÞ dt þ ln 10 þ x2 x2 dw; 2 3 1 2 1 2 1 2 1 2 > 2 2 2 x1 x2 x3 > 2 2 > ¼ ðð0:5 þ e Þuð v Þ þ x x x þ x x x sinð2tÞÞdt þ sinðx dx 3 1 2 3 1 2 3 > 1 Þx3 dw; > : y ¼ x1 ;
ð77Þ
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where x1, x2 and x3 denote the state variables, y is the system output and u(v) represents the output of the saturation nonlinearity defined in (2) and the input saturation limit is chosen as umax = 3. It is apparent that the system satisfies Assumptions 1–3. The control objective is to design an adaptive neural control scheme such that all the signals remain bounded and the system output y follows the desired reference signal yd = 0.5(sin(t) + sin(0.5t)). According to Theorem 1, the virtual control law, the actual control law, and the adaptive law are defined as
3 zi kSi ðZ i Þk ; i ¼ 1; 2; ai 3 z kS ðZ Þk v ¼ k3 z3 ^h3 kS3 ðZ3 Þk tanh 3 3 3 ; a3 3 z kS ðZ Þk _^ i i hi ¼ ki z3i kSi ðZ i Þk tanh i ci ^hi ; i ¼ 1; 2; 3; ai
ai ðZ i Þ ¼ ki zi ^hi kSi ðZ i Þk tanh
T
T ðiÞT T where z1 = x1 yd, z2 = x2 a1, z3 = x3 a2, Z 1 ¼ ½x1 ; yd ; y_ d ; Z i ¼ ½ xTi ; ^ hi1 ; yd ; ði ¼ 2; 3Þ and the design parameters are taken as follows: k1 = 6, k2 = 3, k2 = 15, a1 = 5, a2 = 3, a3 = 5, c1 = 0.01, c2 = c3 = 0.1, k1 = 15, k2 = 10 and k3 = 1. The simulation is run T with the initial conditions [x1(0), x2(0), x3(0)]T = [0.01, 0.2, 0.1]T, and ½^ h1 ð0Þ; ^ h2 ð0Þ; ^ h3 ð0Þ ¼ ½0; 0; 0T . The simulation results are shown in Figs. 2–6. Fig. 2 shows the system output y and the reference signal yd. From Fig. 2, it can be seen that the good tracking performance has been achieved. Fig. 3 shows that the state variable x2 and x3 are bounded. Figs. 4 and 5 display the signal u and the signal v, respectively. Fig. 6 shows that the adaptive parameters ^ h1 ; ^ h2 and ^ h3 are bounded. Example 2. To show the applicability of the proposed control scheme, consider the following Brusselator model in dimensionless form [15]:
8 2 > < x_ 1 ¼ C ðD þ 1Þx1 þ x1 x2 ; x_ 2 ¼ Dx1 þ ð2 þ cosðx1 ÞÞu x21 x2 ; > : y ¼ x1 ;
ð78Þ
where x1 and x2 denote the concentrations of the reaction intermediates, C and D are positive parameters which describe the supply of reservoir chemicals. The following statement was given in [15] ‘‘As a simplified model depicting chemical reactions, the Brusselator model is derived from partial differential equations (PDE) after a series of approximations. Thus, there must exist modeling errors and other types of unknown nonlinearities in the practical chemical reactions’’. In addition, as a practical reaction process, stochastic disturbance and input saturation are unavoidable in Brusselator model. Therefore, the controlled Brusselator model is assumed as
8 2 > < dx1 ¼ ðC ðD þ 1Þx1 þ x1 x2 þ d1 ðt; x2 ÞÞdt þ f1 ðx1 Þdw; dx2 ¼ ðDx1 þ ð2 þ cosðx1 ÞÞu x21 x2 þ d2 ðt; x2 ÞÞdt þ f2 ðx1 ; x2 Þdw; > : y ¼ x1 ;
ð79Þ
where the terms d1 ðt; x2 Þ and d2 ðt; x2 Þ are the external disturbance terms, which come from the modeling errors, w is stochastic disturbance, f1(x1) and f2(x1, x2) are uncertain nonlinear functions, and u is the output of the saturation nonlinearity defined in (2) and the saturation limits are chosen as umax = 4 and umin = 4, respectively. 1 y y
0.8
d
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
10
20
30
40
Time(Sec) Fig. 2. System output y(t) and reference signal yd(t) in Example 1.
50
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1 x2 x3
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6
0
10
20
30
40
50
Time(Sec) Fig. 3. State variables x2 and x3 in Example 1.
4 u
3 2 1 0 −1 −2 −3 −4
0
10
20
30
40
50
Time(Sec) Fig. 4. The output of the saturation nonlinearity u in Example 1.
10
v
5
0
−5
−10
−15
0
10
20
30
40
Time(Sec) Fig. 5. The designed control input signal
v in Example 1.
50
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Fig. 6. The adaptive parameters ^ h1 ; ^ h2 and ^ h3 in Example 1.
4.5 y yd
4 3.5 3 2.5 2 1.5 1 0
10
20
30
40
50
Time(Sec) Fig. 7. System output y(t) and reference signal yd in Example 2.
3 x2
2.5 2 1.5 1 0.5 0 0
10
20
30
Time(Sec) Fig. 8. State variables x2 in Example 2.
40
50
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5 u
4 3 2 1 0 −1 −2 −3 −4 −5 0
10
20
30
40
50
Time(Sec) Fig. 9. The output of the saturation nonlinearity u in Example 2.
10 v
5 0 −5 −10 −15 −20 −25 −30 −35 0
10
20
30
40
Time(Sec) Fig. 10. The designed control input signal
v in Example 2.
Fig. 11. The adaptive parameters ^ h1 and ^ h2 in Example 2.
50
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3 In the simulation, choose d1 ðt; x2 Þ ¼ 0:7x21 sinðx2 Þ cosð1:5tÞ; d2 ðt; x2 Þ ¼ 0:5 x21 þ x22 sin t; f 1 ðx1 Þ ¼ 0:1 sin x21 and f2(x1, x2) = 0.3x2 cosx1, and choose the fuzzy membership functions defined in Example 1 and the reference signal yd = 3 + sint + 0.5 sin(0.5t). It is assumed that x1 – 0 as in [15]. According to Theorem 1, the virtual control input signal a1 ^_ i ði ¼ 1; 2Þ are and the actual control input v are defined in (19) with i = 1 and i = 2, respectively; and the adaptive law h constructed in (20). The simulation is carried out with the system parameters C = 1, D = 3, the initial condition T
½x1 ð0Þ; x2 ð0Þ; ^ h1 ð0Þ; ^ h2 ð0Þ ¼ ½3:2; 1; 0; 0T and the design parameters k1 = k2 = 10, a1 = a2 = 5, c1 = c2 = 0.1 and k1 = k2 = 1. Figs. 7–11 show the simulation results. 5. Conclusion A novel adaptive neural control design scheme has been proposed for a class of stochastic nonlinear strict-feedback systems with unknown input saturation nonlinearity. The presented adaptive neural controller guarantees that all the signals in the closed-loop system are bounded in probability, and the tracking error converges into a small neighborhood of origin in the sense of mean quartic value. Simulation results further illustrate the feasibility and the effectiveness of the presented approach. Our future research will mainly focus on the MIMO stochastic nonlinear systems with input saturation or other input nonlinearities based on the result in this paper. 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Information Sciences journal homepage: www.elsevier.com/locate/ins
Adaptive neural tracking control for stochastic nonlinear strict-feedback systems with unknown input saturation Huanqing Wang a,c, Bing Chen a,⇑, Xiaoping Liu b, Kefu Liu b, Chong Lin a a
Institute of Complexity Science, Qingdao University, Qingdao, 266071 Shandong, PR China Faculty of Engineering, Lakehead University, Orillia, ON P7A 5E1, Canada c School of Mathematics and Physics, Bohai University, Jinzhou, 121000 Liaoning, PR China b
a r t i c l e
i n f o
Article history: Received 11 January 2013 Received in revised form 4 June 2013 Accepted 22 September 2013 Available online 2 October 2013 Keywords: Adaptive neural tracking control Stochastic nonlinear system Input saturation Backstepping technique
a b s t r a c t In this paper, the problem of adaptive neural tracking control is considered for a class of single-input/single-output (SISO) strict-feedback stochastic nonlinear systems with input saturation. To deal with the non-smooth input saturation nonlinearity, a smooth nonaffine function of the control input signal is used to approximate the input saturation function. Classical adaptive technique and backstepping are used for control synthesis. Based on the mean-value theorem, a novel adaptive neural control scheme is systematically derived without requiring the prior knowledge of bound of input saturation. It is shown that under the action of the proposed adaptive controller all the signals of the closed-loop system remain bounded in probability and the tracking error converges to a small neighborhood around the origin in the sense of mean quartic value. Two simulation examples are provided to demonstrate the effectiveness of the presented results. 2013 Elsevier Inc. All rights reserved.
1. Introduction It is well known that stochastic disturbance, which is usually a source of instability of control systems, often exists in practical systems. Therefore, the control design of nonlinear stochastic systems has attracted increasing attention in recent years [9,10,16,27,29,30,36–38,49–54]. Many control design approaches for deterministic nonlinear systems have been successfully extended to stochastic nonlinear systems. Especially, backstepping technique [18] has been a popular tool for control design of stochastic nonlinear systems, see, e.g., [9,10,16,27,29,30,49–52] and the reference therein. In [30], the quadratic Lyapunov function is used to solve the stabilization problem for stochastic nonlinear strict-feedback systems based on a risk-sensitive cost criterion, and the proposed controller guarantees globally asymptotic stability in probability. In [9,10], a quartic Lyapunov function is applied for control design and stability analysis of stochastic nonlinear strict-feedback and output-feedback systems. Compared with the quadratic Lyapunov function, the quartic Lyapunov function can be used to easily deal with the high-order Hessian term. Since then, the quartic Lyapunov function has been widely applied for control design of stochastic nonlinear systems [16,29,49–52]. However, the aforementioned control schemes maybe invalid to control stochastic systems with unknown nonlinear function, because they require that the nonlinear dynamics models are known precisely or the unknown parameters appear linearly with respect to known nonlinear functions. During the past decades, many approximation-based adaptive neural (or fuzzy) control approaches have been developed to control uncertain lower-triangular nonlinear systems, and lots of significant results have been reported, for example, see [2–5,12–14,19,22,23,25,26,28,35,39–42,44,46,55–58] for deterministic nonlinear systems and [8,21,33,43,47] for stochastic ⇑ Corresponding author. Tel.: +86 0532 85953607. E-mail address: [email protected] (B. Chen). 0020-0255/$ - see front matter 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ins.2013.09.043
H. Wang et al. / Information Sciences 269 (2014) 300–315
301
nonlinear systems. In these proposed control schemes, radial basis function (RBF) neural networks (or fuzzy logic systems) are used to approximate uncertain smooth nonlinear functions, and then adaptive backstepping technique is applied to design controllers. For the deterministic systems, Ge et al. [12–14] develop several adaptive neural control schemes for SISO nonlinear systems and multi-input and multi-output (MIMO) nonlinear systems. In [57,58], the problem of adaptive neural tracking control is considered for MIMO nonlinear systems with dead-zone. Then, for stochastic systems, Psillakis and Alexandridis [33] proposes an adaptive neural network control scheme to solve the problem of output tracking control for uncertain stochastic nonlinear strict-feedback systems with unknown covariance noise. Alternatively, in [47], a fuzzy-based adaptive control scheme is presented for a class of uncertain strict-feedback stochastic nonlinear systems with unknown virtual control gain function. The proposed controller guarantees that all the signals in the closed-loop systems are semi-globally uniformly bounded in probability. Recently, in [8,21,24,43], several approximation-based adaptive control approaches are proposed for some classes of stochastic nonlinear strict-feedback time-delay (or delay-free) systems. In many practical systems, input saturation is one of the most important non-smooth nonlinearities. It often severely limits the system performance, gives rise to undesirable inaccuracy or leads to instability [32]. Therefore, the phenomenon of input saturation has to be considered when the controller is designed in practical industrial process control field. So far, many significant results on control design of the systems with input saturation have been obtained, for example, see [6,7,11,48,59]. In [59], a globally stable adaptive control approach is presented for minimum phase SISO systems with input saturation. Chen et al. [6] proposes a robust adaptive neural control for a class of MIMO nonlinear systems with input nonlinearities. By introducing auxiliary design systems to analyze the effect of input constraints, in [7], an adaptive tracking control is proposed for a class of uncertain nonlinear systems with non-symmetric input constraints, and the derived controller guarantees that the closed-loop system is semi-globally uniformly ultimately bounded stability. Wen et al. [48] considers the problem of adaptive control for a class of uncertain nonlinear systems in the presence of input saturation and external disturbance, in which two new schemes are developed to compensate for the effects of the saturation nonlinearity and disturbances. Though the aforementioned results take input saturation nonlinearity into account, the effect of stochastic disturbance is ignored. Note that stochastic disturbance and input constraint could be existed in many practical systems. Motivated by the above observations, this paper considers the problem of adaptive neural tracking control for the case of nonlinear strict-feedback systems with stochastic disturbance and input saturation simultaneously. The proposed adaptive neural control scheme guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. Compared with the existing results, the main idea of control design in this paper is that a smooth non-affine function of the control input signal is firstly used to approximate the saturation function, and furthermore, the mean-value theorem is used to transform the non-affine function into affine form, i.e., gðv Þ ¼ g v l v . Then, the classical adaptive technique and backstepping are used to design controller. The proposed design approach does not require the prior knowledge of the bound of input saturation. In addition, the number of adaptive parameters just depends on the order of the considered systems. So, it is reduced considerably. In this way, the computational burden is significantly alleviated. This paper is organized as follows. The preliminaries and problem formulation are given in Section 2. A novel adaptive neural control scheme is presented in Section 3. Section 4 gives two simulation examples to illustrate the effectiveness of our results, and Section 5 concludes the work. 2. Preliminaries and problem formulation The following notations are used throughout this paper. R denotes the set of all real numbers; Rn indicates the real n-dimensional space. For a given vector or matrix X, XT denotes its transpose; Tr{X} is its trace when X is a square matrix; and kXk denotes the Euclidean norm of a vector X. Ci denotes the set of all functions with continuous ith partial derivative. Consider the following strict-feedback stochastic nonlinear system given by:
8 T > < dxi ¼ ðg i ðxi Þxiþ1 þ fi ðxi Þ þ di ðt; xÞÞdt þ wi ðxi Þdw; 1 6 i 6 n 1; dxn ¼ ðg n ðxn Þuðv Þ þ fn ðxn Þ þ dn ðt; xÞÞdt þ wTn ðxn Þdw; > : y ¼ x1 ;
ð1Þ
where xi ¼ ½x1 ; x2 ; . . . ; xi T 2 Ri , x = [x1, x2, . . . , xn]T 2 Rn and y 2 R are the state variables and the system output, respectively; w denotes an r-dimensional standard Brownian motion defined on the complete probability space (X, F, P) with X being a sample space, F being a r-field, and P being a probability measure; fi(), gi(): Ri ? R, wi (): Ri ? Rr, (i = 1, 2, . . . , n) stand for the unknown smooth nonlinear functions with fi(0) = 0 and wi (0) = 0 (1 6 i 6 n), di(), i = 1, 2, . . . , n are the external disturbance uncertainties of the system. v is the control signal to be designed, and u(v) denotes the plant input subject to saturation nonlinearity described by
uðv Þ ¼ satðv Þ ¼
signðv Þumax ; jv j P umax ;
v;
jv j < umax ;
where umax is a unknown parameter of input saturation.
ð2Þ
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Remark 1. There exist many practical systems which are described by strict-feedback form, such as One-Link Robot system, Pendulum System With Motor, Single-Link Manipulator system [55], and Brusselator model [45]. Meanwhile, stochastic disturbance and input saturation are inevitable in practical process. Therefore, the aforementioned systems can be governed by nonlinear differential equations of the form (1). The control objective is to design an adaptive neural controller for system (1) such that the system output y follows the specified desired trajectory yd and all the signals in the closed-loop systems remain bounded in probability. From (2), it can be seen that there exists a sharp corner when jvj = umax. So backstepping technique cannot be directly applied to construct control input signal. To solve this problem, the method proposed in [48] will be implemented. By this method, a smooth function is used to approximate the saturation function and defined as
ev =umax ev =umax gðv Þ ¼ umax tanhðv =umax Þ ¼ umax v =umax : e þ ev =umax
ð3Þ
Then, sat(v) in (2) can be expressed in the following form:
satðv Þ ¼ gðv Þ þ dðv Þ;
ð4Þ
where d(v) = sat(v) g(v) is a bounded function and its bound can be obtained as
jdðv Þj ¼ jsatðv Þ gðv Þj 6 umax ð1 tanhð1ÞÞ ¼ D:
ð5Þ
Fig. 1 shows the saturation nonlinearity in (2) and its approximation function in (3). According to the mean-value theorem [1], there exists a constant l with 0 < l < 1, such that
gðv Þ ¼ gðv 0 Þ þ g v l ðv v 0 Þ; vÞ where g v l ¼ @gð j ¼ @ v v ¼v l
ð6Þ
4 2 ðev =umax þev =umax Þ
jv ¼v l , vl = lv + (1 l)v0. By choosing
v0 = 0, (6) can be written as
gðv Þ ¼ g v l v ;
ð7Þ
Substituting (4) into (1) and using (7) gives
8 T > < dxi ¼ ðg i ðxi Þxiþ1 þ fi ðxi Þ þ di ðt; xÞÞdt þ wi ðxi Þdw; 1 6 i 6 n 1; dxn ¼ ðg n ðxn Þðg v l v þ dðv ÞÞ þ fn ðxn Þ þ dn ðt; xÞÞdt þ wTn ðxn Þdw; > : y ¼ x1 :
ð8Þ
To facilitate control system design, the following assumptions and lemmas are presented and will be used in the subsequent developments. xi Þ is unknown, but the sign of g i ð xi Þ does not change, and there exist Assumption 1 ([3,14]). For 1 6 i 6 n, the function g i ð unknown constants bm and bM, such that
0 < bm 6 jg i ðxi Þj 6 bM < 1;
8xi 2 Ri :
ð9Þ
xi Þ is strictly either positive or negative. Without loss of generality, it is further assumed that Apparently, (9) implies that g i ð
0 < bm 6 g i ðxi Þ 6 bM ; Assumption
2 [45]. For
8xi 2 Ri : 1 6 i 6 n,
ð10Þ there
exist
unknown
smooth
positive
functions
8ðt; xÞ 2 Rþ X; jdi ðt; xÞj 6 hi ðxi Þ.
10 8 6
u(v)
4 2 0 −2 −4 −6 −8 −10
−20
−15
−10
−5
0
5
10
15
20
v Fig. 1. Saturation (solid-line: sat (v); Dot-line smooth function umax⁄ tanh (v/umax)).
hi ð xi Þ
such
that
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Assumption 3 [3]. The desired trajectory yd(t) and its nth order time derivatives are continuous and bounded. To introduce some useful conceptions and lemmas, consider the following stochastic system:
dx ¼ f ðxÞdt þ hðxÞdw;
ð11Þ
where x and w are defined in (1), and f() and h() are locally Lipschitz functions in x and satisfy f(0) = 0 and h(0) = 0. Definition 1. For any given V(x) 2 C2, associated with the stochastic differential Eq. (11), define the differential operator L as follows:
( ) 2 @V 1 T @ V f þ Tr h LV ¼ h ; @x 2 @x2
ð12Þ
where Tr(A) is the trace of A. n o T 2 ^ correction term or high-order Hessian term, in which the Remark 2. As stated in [29], the term 12 Tr h @@xV2 h is called Ito 2 second-order differential @@xV2 makes the controller design much more difficult than that of the deterministic system. Definition 2 [17]. The solution process {x(t), t P 0} of stochastic system (11) is said to be bounded in probability, if limc?1 sup06t<1P{kx(t)k > c} = 0, where P{B} denotes the probability of event B. Lemma 1 [33]. Consider the stochastic system (11). If there exists a positive definite, radially unbounded, twice continuously differentiable Lyapunov function V : Rn ! R, and constants a0 > 0, b0 P 0 such that
LVðxÞ 6 a0 VðxÞ þ b0 ; then (i) the system has a unique solution almost surely and (ii) the system is bounded in probability. Lemma 2 (Young’s inequality [9]). For "(x, y) 2 R2, the following inequality holds:
xy 6
ep p
jxjp þ
1 jyjq ; qeq
where e > 0, p > 1, q > 1, and (p 1)(q 1) = 1. Lemma 3 [31]. For any variable g 2 R and constant
0 6 jgj g tanh
g
6 d;
> 0, the following inequality holds.
d ¼ 0:2785:
ð13Þ
In this note, the following RBF neural networks will be used to approximate any continuous function f(Z): Rn ? R,
fnn ðZÞ ¼ W T SðZÞ;
ð14Þ
q
where Z 2 XZ R is the input vector with q being the neural networks input dimension, weight vector W = [w1, w2, . . . , wl]2 Rl, l > 1 is the neural networks node number, and S(Z) = [s1(Z), s2(Z), . . . , sl(Z)]T means the basis function vector with si(Z) being chosen as the commonly used Gaussian function of the form
T
" si ðZÞ ¼ exp
# ðZ li ÞT ðZ li Þ ; r2
i ¼ 1; 2; . . . ; l;
ð15Þ
where li = [li1, li2, . . . , liq]T is the center of the receptive field and r is the width of the Gaussian function. In [34], it has been indicated that with sufficiently large node number l, the RBF neural networks (14) can approximate any continuous function f(Z) over a compact set XZ Rq to arbitrary any accuracy e > 0 as
f ðZÞ ¼ W T SðZÞ þ dðZÞ;
8z 2 Xz 2 Rq ;
⁄
where W is the ideal constant weight vector and defined as
(
)
W :¼ arg min supjf ðZÞ W T SðZÞj ; W2Rl
Z2XZ
and d(Z) denotes the approximation error and satisfies j d(Z)j 6 e.
ð16Þ
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Lemma 4 [20]. Consider the Gaussian RBF networks (14) and (15). Let q :¼ taken as
kSðZÞk 6
1 2 mini–j k
li lj k, then an upper bound of kS(Z)k is
1 X 2 2 2 q1 3qðk þ 2Þ e2q k =r :¼ s:
ð17Þ
k¼0
It has been shown in [44] that the constant s in Lemma 3 is a limited value and is independent of the variable Z and the dimension of neural weights l.
3. Adaptive neural control design In this section, a backstepping-based design procedure will be proposed to construct the adaptive neural tracking controller for the original systems (1) with input saturation nonlinearity (2). The design procedure contains n steps and involves the following coordinate transformation:
z1 ¼ x1 yd ;
zi ¼ xi ai1 ;
i ¼ 2; . . . ; n;
ð18Þ
where ai is a virtual control signal to be designed for the corresponding i-subsystem based on an appropriate Lyapunov function Vi. During the design procedure, the virtual control signal and adaptive law will be constructed in the following form:
z3i kSi ðZ i Þk ; ai 3 ^h_i ¼ c h^i þ ki z3 kSi ðZ i Þk tanh zi kSi ðZ i Þk ; i i ai
ai ðZ i Þ ¼ ki zi h^i kSi ðZ i Þk tanh
ð19Þ ð20Þ
where 1 6 i 6 n, ki, ai, ci and kihare positiveiTdesign contants, Si(Zi) is the RBF neural network basis function vector with T T ðiÞT T ðiÞ Z 1 ¼ ½x1 ; yd ; y_ d 2 XZ1 R3 ; Z i ¼ xTi ; ^ 2 XZi R2iþ2 ði ¼ 2; . . . ; nÞ; ^ hi1 ; yd hi ¼ ½h^1 ; h^2 ; . . . ; h^i . y d denotes the vector composed ^ of yd and up to its ith order time derivative, hi is the estimation of an unknown constant hi which will be given at the ith step, Specially, an denotes the actual control input v. ^i ðtÞ P 0 for all t P 0. Note that h ^i is an hi ð0Þ P 0, then h Remark 3. It is easy to prove from (20) that if initial condition ^ estimation of hi, and the initial condition of (20) can be given by designer. So, it is reasonable to choose ^ hi ð0Þ P 0. This property will be used in each step of control design. In the following, for simplicity, the time variable t and the state vector xi will be omitted from the corresponding functions and denote Si(Zi) by Si. Step 1: Since z1 = x1 yd, the first subsystem of (1) gives
dz1 ¼ ðg 1 x2 þ f1 þ d1 y_ d Þdt þ wT1 dw:
ð21Þ
Consider Lyapunov function candidate as
V1 ¼
1 4 bm ~2 h ; z þ 4 1 2k1 1
ð22Þ
where ~ h1 ¼ h1 ^ h1 is the parameter error. It can be verified easily from (12) along (21) and using the completion of squares that
3 2 3 2 bm ~ ^_ h1 h1 ; LV 1 6 z31 g 1 x2 þ f1 þ d1 y_ d þ l1 z1 kw1 k4 þ l1 4 4 k1
ð23Þ
where l1 is a design constant. By means of Assumption 3, the following inequality holds:
z31 d1 6 jz1 j3 h1 ðx1 Þ 6
1 6 2 1 z h ðx1 Þ þ g211 : 2 2g211 1 1
ð24Þ
Substituting (24) into (23) yields
3 3 3 2 1 bm ~ ^_ h1 h1 ; LV 1 6 z31 ðg 1 x2 þ f 1 ðZ 1 ÞÞ z41 g 1 z41 þ l1 þ g211 4 4 4 2 k1
ð25Þ
2 2 where f 1 ðZ 1 Þ ¼ f1 y_ d þ 2g12 z31 h1 ðx1 Þ þ 34 l1 z1 kw1 k4 þ 34 z1 þ 34 g 1 z1 . Since the smooth functions f1, g1, h1 and w1 are unknown, 11 f 1 ðZ 1 Þ cannot be directly used to construct virtual control signal a1. Thus, an RBF neural network W T1 S1 ðZ 1 Þ is employed to approximate the function f 1 ðZ 1 Þ such that, for any given e1 > 0,
f 1 ðZ 1 Þ ¼ W T S1 ðZ 1 Þ þ d1 ðZ 1 Þ; jd1 ðZ 1 Þj 6 e1 1
ð26Þ
H. Wang et al. / Information Sciences 269 (2014) 300–315
305
with d1(Z1) being the approximation error. Then, according to Lemma 3, one has
3 3 1 z kS1 k 3 1 þ dbm h1 a1 þ z41 þ e41 ; z31f 1 ðZ 1 Þ ¼ z31 W T1 S1 þ z31 d1 6 jz31 jkW 1 kkS1 k þ z41 þ e41 6 z31 bm h1 kS1 k tanh 1 4 4 a1 4 4
ð27Þ
1k where the unknown constant h1 ¼ kW . bm Substituting (26) into (25) and using (27) gives
LV 1 6 z31 g 1 z2 þ z31 g 1 a1 þ z31 bm h1 kS1 k tanh
3 z1 kS1 k 1 3 3 2 1 bm ~ ^_ h1 h1 ; þ dbm h1 a1 þ e41 g 1 z41 þ l1 þ g211 a1 4 4 4 2 k1
ð28Þ
where z2 = x2 a1. At the present stage, constructing the virtual control signal a1 as
a1 ¼ k1 z1 ^h1 kS1 k tanh
3 z1 kS1 k ; a1
ð29Þ
then using (10), we have
3 z kS1 k : z31 g 1 a1 6 k1 bm z41 z31 bm ^h1 kS1 k tanh 1 a1
ð30Þ
From (30), rewrite (28) as
3 3 1 3 2 1 bm ~ z kS1 k _ h1 k1 z31 kS1 k tanh 1 ^h1 : LV 1 6 k1 bm z41 þ z31 g 1 z2 g 1 z41 þ dbm h1 a1 þ e41 þ l1 þ g211 þ 4 4 4 2 a1 k1
ð31Þ
_ By choosing adaptive law ^ h1 in (20) with i = 1, it follows
LV 1 6 k1 bm z41 þ z31 g 1 z2 þ dbm h1 a1 þ
1 4 3 2 1 2 b c e þ l þ g þ m 1 ~h1 ^h1 : 4 1 4 1 2 11 k1
ð32Þ
Furthermore, applying Young’s inequality yields
3 1 g z4 þ g z4 ; 4 1 1 4 1 2 bm c1 ~ ^ bm c1 ~2 bm c1 ~ bm c1 ~2 bm c1 2 h1 h1 ¼ h þ h1 h1 6 h þ h : k1 k1 1 k1 2k1 1 2k1 1
z31 g 1 z2 6
ð33Þ ð34Þ
Using (33) and (34), we can further have
LV 1 6 k1 bm z41
bm c1 ~2 1 3 2 1 bm c1 2 1 bm c1 ~2 1 h þ dbm h1 a1 þ e41 þ l1 þ g211 þ h þ q1 þ g 1 z42 ; h þ g z4 6 c1 z41 4 4 2 4 2k1 1 2k1 1 4 1 2 2k1 1
ð35Þ
m c1 2 where c1 ¼ k1 bm ; q1 ¼ dbm h1 a1 þ b2k h1 þ 14 e41 þ 34 l1 þ 12 g211 . The term 14 g 1 z42 will be dealt with in the next step. 1 ^ formula, we have Step 2: From z2 = x2 a1 and It o
2
T @ a1 dz2 ¼ ðg 2 x3 þ f2 þ d2 ‘a1 Þdt þ w2 w1 dw; @x1
ð36Þ
where
‘a1 ¼
@ a1 ðg x2 þ f1 þ d1 Þ þ N1 @x1 1
ð37Þ
with
N1 ¼
1 X @ a1 ðjÞ j¼0 @yd
ðjþ1Þ
yd
þ
@ a1 ^_ 1 @ 2 a1 T h1 þ w w : ^ 2 @x21 1 1 @ h1
ð38Þ
Choose the Lyapunov function as
1 bm ~2 h : V 2 ¼ V 1 þ z42 þ 4 2k2 2
ð39Þ
Furthermore, by (12) it can be verified that
T 3 @ a1 @ a1 bm ~ ^_ h2 h2 : LV 2 ¼ LV 1 þ z32 ðg 2 x3 þ f2 þ d2 ‘a1 Þ þ z22 w2 w1 w2 w1 2 @x1 @x1 k2
ð40Þ
By substituting (31) and (37) into (40) and using the completion squares to the term next to the last one in (40), one has
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H. Wang et al. / Information Sciences 269 (2014) 300–315
LV 2 6 c1 z41
bm c1 ~2 1 @ a1 3 2 @ a1 h1 þ q1 þ g 1 z42 þ z32 g 2 x3 þ f2 þ d2 ðg 1 x2 þ f1 þ d1 Þ N1 þ l2 z2 kw2 w1 k4 4 4 2k1 @x1 @x1
3 2 bm ~ ^_ h2 h2 ; þ l2 4 k2
ð41Þ
where l2 is a positive design constant. Using the similar way to (24) yields
2 @ a1 @ a1 h1 6 1 z6 @ a1 h2 þ 1 g2 ; d1 6 jz32 j 2 1 2 21 @x1 @x1 @x1 2g221 1 1 2 z32 d2 6 2 z62 h2 þ g222 : 2 2g22 z32
ð42Þ ð43Þ
With the help of (42) and (43), (41) can be written as
LV 2 6 c1 z41
2 3 bm c1 ~2 3 3 2 1X b h1 þ q1 þ z32 g 2 x3 þ f 2 ðZ 2 Þ z42 g 2 z42 þ l2 þ g2 m ~h2 ^h_ 2 ; 4 4 4 2 j¼1 2j k2 2k1
ð44Þ
where
2 f 2 ðZ 2 Þ ¼ f2 @ a1 ðg x2 þ f1 Þ þ 1 g z2 N1 þ 3z2 kw @ a1 w k4 þ 1 z3 @ a1 h2 þ 1 z3 h2 þ 3 z2 þ 3 g z2 : 1 1 2 1 1 2 2 4 4 2 @x1 @x1 @x1 2g221 2g222 2 2 4 4l2
ð45Þ
Note that f 2 ðZ 2 Þ is an unknown smooth function. Therefore, an RBF neural network W T2 S2 ðZ 2 Þ is used to model the unknown f 2 ðZ 2 Þ such that
f 2 ðZ 2 Þ ¼ W T S2 ðZ 2 Þ þ d2 ðZ 2 Þ; 2
ð46Þ
where the approximate error d2(Z2) satisfies jd2(Z2)j 6 e2 with e2 being a given positive constant. Similar to (27), the following inequality holds.
z32 f 2 ðZ 2 Þ 6 z32 bm h2 kS2 k tanh
3 z2 kS2 k 3 1 þ dbm h2 a2 þ z42 þ e42 ; a2 4 4
ð47Þ
2k where the unknown constant h2 ¼ kW . Substituting (46) into (44) and using the inequality (47), we have bm
LV 2 6 c1 z41
3 2 bm c1 ~2 1 3 2 1X z kS k h1 þ q1 þ dbm h2 a2 þ e42 þ l2 þ g22j þ z32 g 2 z3 þ z32 g 2 a2 þ z32 bm h2 kS2 k tanh 2 2 4 4 2 j¼1 a2 2k1
3 bm ~ ^_ h2 h2 ; g 2 z42 4 k2
ð48Þ
where z3 = x3 a2. h2 in (20) into account with i = 2, the following inequalities can be obtained. Then, take a2 in (19) and ^
3 z kS2 k ; z32 g 2 a2 6 k2 bm z42 z32 bm ^h2 kS2 k tanh 2 a2 3 1 z32 g 2 z3 6 g 2 z42 þ g 2 z43 : 4 4
ð49Þ ð50Þ
By using the above inequalities, we can rewrite (48) as 2 2 X bm c1 ~2 1 3 2 1X b c 1 h1 þ q1 þ dbm h2 a2 þ e42 þ l2 þ LV 2 6 cj z4j g22j þ m 2 ~h2 ^h2 þ g 2 z43 4 4 2 4 2k k 1 2 j¼1 j¼1 2 2 2 X X bm cj 2 X 1 ~h þ 6 cj z4j qj þ g 2 z43 ; j 4 2k j j¼1 j¼1 j¼1
ð51Þ
P bm cj 2 2 h2 ^ h2 6 12 ~ h22 þ 12 h22 has been used. where cj ¼ kj bm ; qj ¼ dbm hj aj þ 2k hj þ 14 e4j þ 34 lj þ 12 jk¼1 g2jk ; j ¼ 1; 2, and the inequality ~ j ^ formula, one has Step i (3 6 i 6 n 1): By using (18) and Ito
dzi ¼ ðg i xiþ1 þ fi þ di ‘ai1 Þdt þ wi
i1 X @ ai1 j¼1
@xj
!T wj
dw;
ð52Þ
where
‘ai1 ¼
i1 X @ ai1 j¼1
@xj
ðg j xjþ1 þ fj þ dj Þ þ Ni1
ð53Þ
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with Ni1 ¼
Pi1 @ ai1 ^_ Pi1 @ai1 ðjþ1Þ 1 Pi1 @ 2 ai1 T þ 2 p;q¼1 @xp @xq wp wq . hj þ j¼0 ðjÞ yd j¼1 @ ^ h @yd
j
Consider Lyapunov function as
1 bm ~2 h : V i ¼ V i1 þ z4i þ 4 2ki i
ð54Þ
It follows immediately from (12) that
LV i ¼ LV i1 þ
z3i ðg i xiþ1
i1 X 3 @ ai1 þ fi þ di ‘ai1 Þ þ z2i wi w 2 @xj j j¼1
!T wi
i1 X @ ai1 j¼1
@xj
! wj
bm ~ ^_ hi hi ; ki
ð55Þ
where the term LVi1 can be obtained by a straightforward calculation as former steps.
LV i1 6
i1 i1 i1 X X bm cj 2 X 1 ~h þ cj z4j qj þ g i1 z4i ; j 4 2k j j¼1 j¼1 j¼1
ð56Þ
P bm cj 2 2 where cj ¼ kj bm ; qj ¼ dbm hj aj þ 2k hj þ 14 e4j þ 34 lj þ 12 jk¼1 g2jk ; j ¼ 1; 2; . . . ; i 1. j By using the completion of squares, the following inequality holds:
2
4
i1 i1 X X 3 2
@ ai1
3 2 3 2 4
@ ai1
z w w 6 li þ li zi wi w ;
2 i i j¼1 @xj j
4 4 @xj j
j¼1
ð57Þ
where li is a positive design parameter. Next, by following a same line used in the procedures from (42) and (43), we have
z3i
i1 X @ ai1 j¼1
z3i di 6
@xj
dj 6
2 i1 i1 i1 X X X @ ai1 1 6 @ ai1 1 2 2 jzi j3 j jhj 6 z hj þ g; i 2 2 ij @x @x g 2 j j ij j¼1 j¼1 j¼1
1 6 2 1 2 z h þ g : 2g2ii i i 2 ii
ð58Þ ð59Þ
Further, substituting (53), (56) and (57) into (55) and using the formulas (58), (59) and (55) can be rewritten as i1 i1 i1 i X X bm cj 2 X 3 3 3 1X b ~h þ LV i 6 cj z4j qj þ z3i ðg i xiþ1 þ f i ðZ i ÞÞ z4i g i z4i þ l2i þ g2ij m ~hi ^h_i ; j 4 4 4 2 2k ki j j¼1 j¼1 j¼1 j¼1
ð60Þ
where f i ðZ i Þ is defined as
f ðZ Þ ¼ f i i i
2 i1 i1 X X 3 2 @ ai1 1 3 @ ai1 1 1 2 2 ðg j xjþ1 þ fj Þ Ni1 þ li zi kwi wj k4 þ z hj þ 2 z3i hi þ g i1 zi 2 i 4 4 @xj @x @x 2 2 g g j j ij ii j¼1 j¼1
i1 X @ ai1 j¼1
3 3 þ zi þ g i zi 4 4
ð61Þ
Currently, by employing a neural networks W Ti Si ðZ i Þ to approximate the unknown smooth function f i ðZ i Þ and constructing _ the virtual control law ai and adaptive law ^ hi defined respectively in (19) and (20), and then repeating the similar procedure from (27)–(35) in Step 1, the following result is true. i i i X X bm cj 2 X 1 ~h þ LV i 6 cj z4j qj þ g i z4iþ1 ; j 4 2k j j¼1 j¼1 j¼1
ð62Þ
P bm cj 2 2 where cj ¼ kj bm ; qj ¼ dbm hj aj þ 2k hj þ 14 e4j þ 34 lj þ 12 jk¼1 g2jk ; j ¼ 1; 2; . . . ; i. j ^ formula, we have Step n: This is the final step, and the actual control input v will be constructed. By (18) and Ito
dzn ¼ ðg n ðg v l v þ dðv ÞÞ þ fn þ dn ‘an1 Þdt þ wn
n1 X @ an1 j¼1
where ‘an1 is given in (53) with i = n. Choose the following Lyapunov function candidate:
1 g V n ¼ V n1 þ z4n þ 0 ~h2n ; 4 2kn
@xj
!T wj
dw;
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where g0 will be specified later. By applying (12) and taking (62) with k = n 1 into account, it follows n1 n1 n1 n1 X X X bm cj 2 X 1 @ an1 ~h þ LV n 6 cj z4j qj þ g n1 z4n þ z3n ðg n g v l v þ dðv Þ þ fn þ dn ðg j xjþ1 þ fj þ dj Þ j 4 2k @xj j j¼1 j¼1 j¼1 j¼1 !T ! n1 n1 X X 3 @ an1 @ an1 g _ Nn1 Þ þ z2n wn wn 0 ~hn ^hn : 2 @x @x k j j n j¼1 j¼1
ð63Þ
Similar to (58) and (59), the inequalities below hold.
z3n
n1 X @ an1 j¼1
z3n dn 6
@xj
dj 6
2 n1 n1 X X 1 6 @ an1 1 2 2 z hj þ g ; 2 n 2 nj @x g 2 j nj j¼1 j¼1
1 6 2 1 2 z h þ g : 2g2nn n n 2 nn
ð64Þ ð65Þ
Subsequently, substituting (64) and (65) into (63) and utilizing completion squares to the term next to the last one in (63), one has n1 n1 n1 n X X bm cj 2 X 3 3 3 1X g ~h þ LV n 6 cj z4j qj þ z3n g n dðv Þ þ z3n ðg n g v l v þ f n ðZ n ÞÞ z4n g n z4n þ l2n þ g2nj 0 ~hn ^h_ n : j 4 4 4 2 2k kn j j¼1 j¼1 j¼1 j¼1
ð66Þ
where f n ðZ n Þ is given in (61) with i = n. Again, an RBF neural network W Tn Sn ðZ n Þ is used to model the unknown function f n ðZ n Þ such that, for any given positive constant en,
f n ðZ n Þ ¼ W T Sn ðZ n Þ þ dn ðZ n Þ; n
jdn ðZ n Þj 6 en ;
where dn(Zn) is the approximation error. Furthermore, the following result is satisfied.
z3n f n ðZ n Þ 6 z3n g 0 hn kSn k tanh
z3n kSn k 3 1 þ dg 0 hn an þ z4n þ e4n an 4 4
ð67Þ
with hn ¼ kWg0n k. Now, constructing the actual control input signal v in (19) with i = n and noting that 0 < g m 6 g v l 6 1, then we obtain
3 z kSn k ; z3n g n g v l v 6 kn g 0 z4n z3n g 0 ^hn kSn k tanh n an
ð68Þ
where the fact of g n g v l P bm g m has been used, and g0 = bmgm. Remark 4. The property of 0 < g m 6 g v l 6 1 is introduced in [48], here, for the purpose of stability analysis, we still employ this property.
Next, by (5) and (9) and Lemma 2, one has
z3n g n dðv Þ 6
3 1 g z4 þ b M D 4 ; 4 n n 4
ð69Þ
_ Furthermore, combining (66) with the formulas (67)–(69) and choosing adaptive law ^ hn in (20) with i = n, we can obtain n n1 n1 n X X bm cj 2 X 1 1 3 1X g c ~h þ LV n 6 cj z4j qj þ dg 0 hn an þ e4n þ bM D4 þ l2n þ g2nj þ 0 n ~hn ^hn : j 4 4 4 2 2k kn j j¼1 j¼1 j¼1 j¼1
ð70Þ
In addition, note that
g 0 cn ~ ^ g c g c hn hn 6 0 n ~h2n þ 0 n h2n ; kn 2kn 2kn (70) can be rewritten as n n1 n X X bm cj 2 g 0 cn 2 X ~h ~h þ LV n 6 cj z4j qj j n 2kj 2kn j¼1 j¼1 j¼1
ð71Þ
P bm cj 2 2 where cj ¼ kj bm ; qj ¼ dbm hj aj þ 2k hj þ 14 e4j þ 34 lj þ 12 jk¼1 g2jk ; j ¼ 1; 2; . . . ; n 1. cn ¼ kn g 0 ; qn ¼ dg 0 hn an þ 14 e4n þ 14 bM D4 j P 2 n g c 2 0 n þ 34 ln þ 12 j¼1 g2nj þ 2k h . n n So far, the adaptive neural control design has been completed based on backstepping technique. The main result of this research will be summarized as the following theorem.
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Theorem 1. Consider the uncertain stochastic nonlinear system (1), preceded by unknown input saturation nonlinearity (2), the controller (19), and adaptive law (20) under Assumptions 1–3. Assume there exists sufficiently large compact set XZj ðj ¼ 1; 2; . . . ; nÞ, such that Z j 2 XZj for all t P 0, then for bounded initial conditions with ^hj ð0Þ P 0, h remain in a compact set XZ, (i) all the signals in the closed-loop system are bounded in probability and the error signals zj and ~ which defined by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n b0 ~ 2kj b0 4 ~ XZ ¼ zj ; hj E½jzj j 6 4Vð0Þ þ 4 ;hj j 6 Vð0Þ þ ; j¼1 a0 bm a0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi) 2kn b0 ~ : j ¼ 1; 2; . . . ; n 1; jhn j 6 Vð0Þ þ g0 a0 (
ð72Þ
(ii) there exists a finite time T1 such that the quartic mean square tracking error converges to the following compact set X1, for all t > T1,
X1 ¼ fyðtÞ 2 RjE½jy yd j4 6 8
b0 ; 8t > T 1 g a0
with the time T1 will be given later.
Proof. (i) Let V = Vn, and define a0 = min{4cj, cj, j = 1, 2, . . . , n.} and b0 ¼
LV 6 a0 V þ b0 ;
Pn
j¼1
qj , (71) implies that
t P 0:
ð73Þ
hj are bounded in probability. Since hjis a constant, ^ hj is Therefore, from Lemma 1, for j = 1, 2, . . . , n, the error signals zj and ~ bounded in probability. Consequently, aj is also bounded in probability due to kSik 6 s and j tanh ()j 6 1. Hence, we conclude that all the signals xj = zj + aj1 in the closed-loop system remain bounded in the sense of probability. Furthermore, from (73) the following inequality can be obtained directly by [10](Th. 4.1)
dE½VðtÞ 6 a0 E½VðtÞ þ b0 ; dt
ð74Þ
which implies that
E½VðtÞ 6 ea0 t ½Vð0Þ þ
b0 ; a0
8t > 0;
ð75Þ
then it follows
E½VðtÞ 6 Vð0Þ þ
b0 ; a0
8t > 0;
ð76Þ
Pn
P g 0 ~2 bm ~2 where Vð0Þ ¼ j¼1 14 z4j ð0Þ þ n1 j¼1 2kj hj ð0Þ þ 2kn hn ð0Þ. Next, by combining the definition of V and (76), we conclude that there hj remain nin it forall the exists a compact set XZ defined in (72) such that the error signals zj and ~ otime. such that (ii) From (75), it can be further verified that there exists a finite time T 1 ¼ max 0; a10 ln a0 bV0ð0Þ
E½jy yd j4 6 4E½VðtÞ 6
8b0 ; a0
8t > T 1 :
Thus, to achieve the desired tracking performance, we can properly adjust the design parameters a0 and b0.
h
4. Simulation example In order to test the effectiveness of the proposed method, two examples are given in this section. The first example is a numerical example, and the second one is a practical system model which is taken from [15]. Example 1. Consider the following third-order nonlinear stochastic system:
8 2 > dx1 ¼ ðð1 þ x21 Þx2 þ 0:5x1 sin x1 þ x21 cosð1:5tÞÞdt þ e0:2x1 x21 dw; > > > > < dx ¼ 6 þ x2 x2 x þ x x3 þ x x2 sin3 ðtÞ dt þ ln 10 þ x2 x2 dw; 2 3 1 2 1 2 1 2 1 2 > 2 2 2 x1 x2 x3 > 2 2 > ¼ ðð0:5 þ e Þuð v Þ þ x x x þ x x x sinð2tÞÞdt þ sinðx dx 3 1 2 3 1 2 3 > 1 Þx3 dw; > : y ¼ x1 ;
ð77Þ
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where x1, x2 and x3 denote the state variables, y is the system output and u(v) represents the output of the saturation nonlinearity defined in (2) and the input saturation limit is chosen as umax = 3. It is apparent that the system satisfies Assumptions 1–3. The control objective is to design an adaptive neural control scheme such that all the signals remain bounded and the system output y follows the desired reference signal yd = 0.5(sin(t) + sin(0.5t)). According to Theorem 1, the virtual control law, the actual control law, and the adaptive law are defined as
3 zi kSi ðZ i Þk ; i ¼ 1; 2; ai 3 z kS ðZ Þk v ¼ k3 z3 ^h3 kS3 ðZ3 Þk tanh 3 3 3 ; a3 3 z kS ðZ Þk _^ i i hi ¼ ki z3i kSi ðZ i Þk tanh i ci ^hi ; i ¼ 1; 2; 3; ai
ai ðZ i Þ ¼ ki zi ^hi kSi ðZ i Þk tanh
T
T ðiÞT T where z1 = x1 yd, z2 = x2 a1, z3 = x3 a2, Z 1 ¼ ½x1 ; yd ; y_ d ; Z i ¼ ½ xTi ; ^ hi1 ; yd ; ði ¼ 2; 3Þ and the design parameters are taken as follows: k1 = 6, k2 = 3, k2 = 15, a1 = 5, a2 = 3, a3 = 5, c1 = 0.01, c2 = c3 = 0.1, k1 = 15, k2 = 10 and k3 = 1. The simulation is run T with the initial conditions [x1(0), x2(0), x3(0)]T = [0.01, 0.2, 0.1]T, and ½^ h1 ð0Þ; ^ h2 ð0Þ; ^ h3 ð0Þ ¼ ½0; 0; 0T . The simulation results are shown in Figs. 2–6. Fig. 2 shows the system output y and the reference signal yd. From Fig. 2, it can be seen that the good tracking performance has been achieved. Fig. 3 shows that the state variable x2 and x3 are bounded. Figs. 4 and 5 display the signal u and the signal v, respectively. Fig. 6 shows that the adaptive parameters ^ h1 ; ^ h2 and ^ h3 are bounded. Example 2. To show the applicability of the proposed control scheme, consider the following Brusselator model in dimensionless form [15]:
8 2 > < x_ 1 ¼ C ðD þ 1Þx1 þ x1 x2 ; x_ 2 ¼ Dx1 þ ð2 þ cosðx1 ÞÞu x21 x2 ; > : y ¼ x1 ;
ð78Þ
where x1 and x2 denote the concentrations of the reaction intermediates, C and D are positive parameters which describe the supply of reservoir chemicals. The following statement was given in [15] ‘‘As a simplified model depicting chemical reactions, the Brusselator model is derived from partial differential equations (PDE) after a series of approximations. Thus, there must exist modeling errors and other types of unknown nonlinearities in the practical chemical reactions’’. In addition, as a practical reaction process, stochastic disturbance and input saturation are unavoidable in Brusselator model. Therefore, the controlled Brusselator model is assumed as
8 2 > < dx1 ¼ ðC ðD þ 1Þx1 þ x1 x2 þ d1 ðt; x2 ÞÞdt þ f1 ðx1 Þdw; dx2 ¼ ðDx1 þ ð2 þ cosðx1 ÞÞu x21 x2 þ d2 ðt; x2 ÞÞdt þ f2 ðx1 ; x2 Þdw; > : y ¼ x1 ;
ð79Þ
where the terms d1 ðt; x2 Þ and d2 ðt; x2 Þ are the external disturbance terms, which come from the modeling errors, w is stochastic disturbance, f1(x1) and f2(x1, x2) are uncertain nonlinear functions, and u is the output of the saturation nonlinearity defined in (2) and the saturation limits are chosen as umax = 4 and umin = 4, respectively. 1 y y
0.8
d
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
10
20
30
40
Time(Sec) Fig. 2. System output y(t) and reference signal yd(t) in Example 1.
50
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1 x2 x3
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6
0
10
20
30
40
50
Time(Sec) Fig. 3. State variables x2 and x3 in Example 1.
4 u
3 2 1 0 −1 −2 −3 −4
0
10
20
30
40
50
Time(Sec) Fig. 4. The output of the saturation nonlinearity u in Example 1.
10
v
5
0
−5
−10
−15
0
10
20
30
40
Time(Sec) Fig. 5. The designed control input signal
v in Example 1.
50
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Fig. 6. The adaptive parameters ^ h1 ; ^ h2 and ^ h3 in Example 1.
4.5 y yd
4 3.5 3 2.5 2 1.5 1 0
10
20
30
40
50
Time(Sec) Fig. 7. System output y(t) and reference signal yd in Example 2.
3 x2
2.5 2 1.5 1 0.5 0 0
10
20
30
Time(Sec) Fig. 8. State variables x2 in Example 2.
40
50
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5 u
4 3 2 1 0 −1 −2 −3 −4 −5 0
10
20
30
40
50
Time(Sec) Fig. 9. The output of the saturation nonlinearity u in Example 2.
10 v
5 0 −5 −10 −15 −20 −25 −30 −35 0
10
20
30
40
Time(Sec) Fig. 10. The designed control input signal
v in Example 2.
Fig. 11. The adaptive parameters ^ h1 and ^ h2 in Example 2.
50
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3 In the simulation, choose d1 ðt; x2 Þ ¼ 0:7x21 sinðx2 Þ cosð1:5tÞ; d2 ðt; x2 Þ ¼ 0:5 x21 þ x22 sin t; f 1 ðx1 Þ ¼ 0:1 sin x21 and f2(x1, x2) = 0.3x2 cosx1, and choose the fuzzy membership functions defined in Example 1 and the reference signal yd = 3 + sint + 0.5 sin(0.5t). It is assumed that x1 – 0 as in [15]. According to Theorem 1, the virtual control input signal a1 ^_ i ði ¼ 1; 2Þ are and the actual control input v are defined in (19) with i = 1 and i = 2, respectively; and the adaptive law h constructed in (20). The simulation is carried out with the system parameters C = 1, D = 3, the initial condition T
½x1 ð0Þ; x2 ð0Þ; ^ h1 ð0Þ; ^ h2 ð0Þ ¼ ½3:2; 1; 0; 0T and the design parameters k1 = k2 = 10, a1 = a2 = 5, c1 = c2 = 0.1 and k1 = k2 = 1. Figs. 7–11 show the simulation results. 5. Conclusion A novel adaptive neural control design scheme has been proposed for a class of stochastic nonlinear strict-feedback systems with unknown input saturation nonlinearity. The presented adaptive neural controller guarantees that all the signals in the closed-loop system are bounded in probability, and the tracking error converges into a small neighborhood of origin in the sense of mean quartic value. Simulation results further illustrate the feasibility and the effectiveness of the presented approach. Our future research will mainly focus on the MIMO stochastic nonlinear systems with input saturation or other input nonlinearities based on the result in this paper. Acknowledgements This work is partially supported by the Natural Science Foundation of China (61074008, 61304002, 61333012 and 61174033), and by the Education Department of Liaoning Province under the general project research Grant No. L2013424. References [1] T.M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, MA, 1963. [2] Y.C. Chang, B.S. Chen, A nonlinear adaptive H1 tracking control design in robotic systems via neural networks, IEEE Transactions on Control Systems Technology 5 (1996) 13–29. [3] B. Chen, X.P. Liu, K.F. Liu, C. Lin, Direct adaptive fuzzy control of nonlinear strict-feedback systems, Automatica 45 (2009) 1530–1535. [4] B. Chen, X.P. Liu, K.F. Liu, P. Shi, C. Lin, Direct adaptive fuzzy control for nonlinear systems with time-varying delays, Information Sciences 180 (2010) 776–792. [5] B. Chen, X.P. Liu, K.F. Liu, C. Lin, Adaptive control for nonlinear MIMO time-delay systems based on fuzzy approximation, Information Sciences 222 (2013) 576–592. [6] M. 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