Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach

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Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach Yongming Lia,b,c , Shaocheng Tonga,∗ , Tieshan Lib , Xingjian Jingc a Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, Liaoning 121001, PR China b Navigation College, Dalian Maritime University, Dalian, Liaoning 116026, PR China c Department of Mechanical Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 20 August 2012; received in revised form 1 February 2013; accepted 3 February 2013

Abstract This paper considers the adaptive fuzzy robust control problem for a class of single-input and single-output (SISO) stochastic nonlinear systems in strict-feedback form. The systems under study possess unstructured uncertainties, unknown dead-zone, uncertain dynamics and unknown gain functions. In the controller design, fuzzy logic systems are adopted to approximate the unknown functions, and the uncertain nonlinear system is therefore transformed into an uncertain parameterized system with unmodeled dynamics. By combining the backstepping technique with the stochastic small-gain approach, a novel adaptive fuzzy robust control scheme is developed. It is shown that the proposed control approach can guarantee that the closed-loop system is input-statepractically stable (ISpS) in probability, and the output of the system converges to a small neighborhood of the origin by appropriately tuning several design parameters. Simulation results are provided to illustrate the effectiveness of the proposed control approach. © 2013 Elsevier B.V. All rights reserved. Keywords: Stochastic nonlinear system; Fuzzy logic systems; Fuzzy control; Dead-zone; Stochastic small gain approach

1. Introduction Since fuzzy set theory was founded by Zadeh in [1], a great progress has been achieved in fuzzy control. In the early stage, some pioneering results in fuzzy control were gained by Mamdani and his colleagues [2–5] via the linguistic approaches [6–8]. Subsequently, remarkable developments have been witnessed in the following decade in the applications of fuzzy control in various practical problems, such as water quality control [9,10], automatic train operation system [11,12], elevator control [13], and so on. Gradually, the fuzzy control has become one of the most versatile tools due to its nice performance in various applications. However, the main drawbacks of the aforementioned fuzzy control approaches did not provide the stability analysis for the control systems, which limits its potential applications in engineering practice. It is firstly shown in [14,15] that a fuzzy logic system is a universal approximator. That is, a fuzzy logic system can be used to approximate any nonlinear system with a required accuracy ∗ Corresponding author. Tel.: +86 416 4199101; fax: +86 416 4199415.

E-mail address: [email protected] (S. Tong). 0165-0114/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.02.002 Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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provided that enough rules are given. Based on the universal approximation theorem and by incorporating fuzzy logic systems into adaptive control schemes, the stable fuzzy adaptive controllers are first proposed in [16]. Afterwards, various adaptive fuzzy control approaches for nonlinear systems have been developed [17–20]. However, the control approaches mentioned above are restricted only to those nonlinear systems satisfying the matching conditions. That is, the unknown nonlinearities appear in the same equation as the control input in a state space representation [21]. If the nonlinear systems do not satisfy the matching conditions, the adaptive fuzzy control approaches mentioned above cannot be implemented. In the past decades, many fuzzy adaptive control schemes have been reported by combining the backstepping technique with the adaptive fuzzy logic systems [22–27]. Fuzzy adaptive backstepping control schemes provide a systematic framework for the design of tracking or regulation strategies, which importantly can deal with the uncertain nonlinear systems without satisfying the matching conditions. However, the existing adaptive fuzzy backstepping control approaches only consider deterministic nonlinear systems. It is well known that stochastic disturbance often exists in practical systems and it is usually a troublesome source resulting in instability of control systems. Therefore, investigation on stability analysis and control design for stochastic nonlinear systems has also received increasing attentions, for example [28–30] and the references therein. In recent years, several fuzzy adaptive control schemes have also been developed for stochastic nonlinear systems by using the backstepping technique and combining with adaptive fuzzy logic systems. Wang [31] proposed a fuzzy adaptive control scheme for a class of uncertain strictfeedback stochastic nonlinear systems with unknown virtual control gain function. Yu and Du [32] proposed a neural network (NN) adaptive state feedback control for a class of uncertain non-affine stochastic non-linear systems with time-varying delay. Chen, Zhou and Li [33–35] presented several adaptive neural network control schemes to handle the output feedback control for uncertain stochastic nonlinear systems. However, these results do not consider the problem of nonsmooth nonlinear inputs in the controlled systems. It is well known that nonsmooth nonlinear inputs such as dead zone, backlash, and hysteresis are common in industrial systems, for example, mechanical connections, hydraulic actuators, electric servomotors and other application systems. Especially, dead zone is one of the most important nonsmooth nonlinearities in many industrial processes (for example, some biomedical systems, mechanical and electrical components like valves, hydraulic servovalves and DC servo motors are all with dead-zone inputs). Dead zone can severely limit system control performance and even results in instability of the control system. Several stable adaptive fuzzy or neural networks control algorithms have been proposed by [36–45] for a class of nonlinear systems preceded by unknown dead-zones. Liu and Zhou [36] and Roopaei et al. [37] proposed two adaptive fuzzy control schemes for a class of SISO nonlinear systems with dead-zone input, respectively. Hsu [38] and Hsu et al. [39] considered the problems of sliding mode control for uncertain nonlinear systems with or without time delays, respectively, but the adaptive control technique was not used in [38,39]. Whereas, two fuzzy adaptive control schemes were proposed for a class of multi-input multi-output (MIMO) nonlinear systems with or without known and unknown control direction by using the Nussbaum function technique and adaptive variable-structure control technique in [40,41]. Recently, [42,43] presented two novel adaptive fuzzy control algorithms for a class of unknown time-varying delay multivariable nonlinear systems with both unknown actuator nonlinearities (dead-zone or backlash-like hysteresis) and unknown control direction. Two fuzzy adaptive observerbased approaches were presented to deal with the projective synchronization problem of a class of chaotic systems to involve both sector nonlinearity and dead-zone in [44,45], respectively. However, the main limitation in these results is that the considered nonlinear systems must also satisfy the matching conditions. More recently, a robust adaptive NN backstepping controller was proposed in [46] for a class of SISO nonlinear pure-feedback systems with dead-zone and without satisfying the matching condition, and the stability of the resulting closed-loop systems was proved by using integral-type Lyapunov functions. An adaptive neural controller was proposed for a class of SISO nonlinear systems with dead-zone and multiple time-delays based on dynamic surface control technique in [47]. Zhang and Ge [48,49] extended the results of [46,47] to the MIMO nonlinear strict-feedback systems with dead-zone and without satisfying the matching condition. The latest work in [50] investigated the adaptive fuzzy backstepping output feedback control for a class of nonlinear systems with the unmeasured states. However, the controlled systems considered in [46–50] are all deterministic nonlinear systems, and thus the results are not straightforward to be generalized for stochastic nonlinear systems. Yu and Du [51] proposed a neural network adaptive control design for a class of stochastic nonlinear systems with unknown dead-zone. Nevertheless, the stochastic nonlinear systems considered in [51] do not include the unmodeled dynamics. Therefore, the designed adaptive controller lacks the robustness to unmodeled dynamics or disturbances [52]. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Motivated by the discussions and observations above, in this paper, a new adaptive fuzzy robust backstepping control approach is proposed for a class of nonlinear strict-feedback systems with unknown dead zone and unmodeled dynamics. In the controller design, the fuzzy logic systems are utilized to approximate the unknown functions and to transform the considered uncertain nonlinear system into an uncertain parameterized system with unmodeled dynamics. By combining backstepping technique with stochastic small-gain approach, a new adaptive fuzzy robust control is therefore developed. It is shown that the proposed control approach can guarantee that the closed-loop system is inputstate practically stability in probability, and the output of the system converges to a small neighborhood of the origin. Compared with the existing results, the main contributions of this paper can be summarized as follows: (i) Based on the stochastic small-gain theorem and the backstepping design technique, the proposed adaptive control method not only ensure the stability of the control system, but also has the robustness to the unmodeled dynamics, which cannot be solved by the existing results such as those in [32–51]; (ii) since the developed design approach does not need to either construct the dead-zone inverse or require the knowledge of bounds of dead-zone slopes, the proposed control scheme can avoid the over-parameterization problem existing in [50]; and (iii) by estimating the norms of parameter vectors rather than each element of the parameter vectors, the proposed control scheme contains fewer adaptive parameters to be tuned on-line than the previous results in [33–35], which makes the computational burden significantly alleviated. 2. Problem statements and preliminaries 2.1. Problem statements and basic assumptions Consider the following uncertain stochastic nonlinear system with dead-zones and unmodeled dynamics: dz = q1 (x, z) dt + q2 (x, z) dw d xi = [gi (x i )xi+1 + f i (x i ) + i (x, z)] dt + iT (x i ) dw, i = 1, . . . , n − 1 d xn = [gi (x)u + f n (x) + n (x, z)] dt + nT (x) dw y = x1 with the dead-zones in the input described by ⎧ ⎪ ⎨ m r (v), v ≥ br bl < v < br u = D(v) = 0, ⎪ ⎩ m l (v), v ≤ bl

(1)

(2)

where x i = [x1 , x2 , . . . , xi ]T ∈ R i , i = 1, 2, . . . , n (x = x n ) are the states of the system and y ∈ R is the output of the system, respectively. w is r-dimensional standard Brownian motion defined on the complete probability space (, F, P) with  being a sample space, F being a -field, and P being the probability measure. z ∈ R n 0 are unmodeled dynamics and i (x, z) the dynamic disturbances. f i (·), gi (·) : R i → R, q1 (x, z), q2 (x, z), i (x, z) : R n+n 0 → R, iT (·) : R i → R r for i = 1, 2, . . . , n are unknown smooth nonlinear functions. u ∈ R is the output of the dead-zone, v ∈ R is the input of the dead-zone, bl < 0 and br > 0 are the unknown parameters of the unknown dead-zone. The control objective is to design an adaptive fuzzy state feedback control law v for system (1) such that for all initial conditions, all the signals in the closed-loop systems are input-state-practically stability (ISpS) in probability as t → ∞. Throughout the paper, the following assumptions are made on system (1). Assumption 1 (Yu and Du [51]). For i = 1, . . . , n, nonlinear functions gi (x i ) are unknown, but their signs are known, and there exist positive constants g and g¯ such that 0 < g ≤ |gi (x i )| ≤ g¯ < ∞, ∀x i ∈ R i . Without loss of generality, it is further assumed that 0 < g ≤ gi (x i ) ≤ g¯ < ∞, for i = 1, . . . , n. Assumption 2 (Yu and Du [51]). For 1 ≤ i ≤ n, there exist nonnegative unknown smooth functions i such that  i (x i ) ≤ ik=1 |xk |i (x i ). Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Assumption 3 (Zhang and Ge [49], Yu and Du [51]). Assume that the output of dead-zone is not measurable, m l (v) and m r (v) are uncertain smooth functions, and there exist unknown positive constants nl0 , nl1 , nr 0 and nr 1 such that 0 < nl0 ≤ m l (v) ≤ nl1 , ∀v ∈ (−∞, bl ] 0 < nr 0 ≤ m r (v) ≤ nr 1 , ∀v ∈ [br , +∞) and 0 ≤ min{nl0 , nr 0 } is a known positive constant, where m l (v) = dm l (s)/ds|s=v and m r (v) = dm r (s)/ds|s=v . Based on Assumption 3 the dead-zone (2) can be written as follows [49,51]: u = D(v) = T (t)(t)v + d(v)

(3)

where (t) = [ ¯ r (t), ¯ l (t)]T , (t) = [nr (v(t)), nl (v(t))]T ,   1, v(t) > bl , 1, v(t) < br ¯ l (t) = ¯ r (t) = 0, v(t) ≥ br 0, v(t) ≤ bl ,  nr (v) =

0,

v ≤ bl ,

m r ( r (v)), bl < v < +∞,

 nl (v) =

m l ( l (v)), −∞ < v < br 0,

v ≥ br

⎧

v ≥ br ⎪ ⎨ −m r ( r (v))br ,



−[m (

(v)) + m (

(v))]v, bl < v < br d(v) = r r l l ⎪ ⎩ −m l ( l (v))bl , v ≤ bl where l (v) ∈ (v, bl ), if v < bl ; l (v) ∈ (bl , v), if bl ≤ v < br ; r (v) ∈ (br , v), if br < v; r (v) ∈ (v, br ), if bl < v ≤ br , and |d(v)| ≤ ∗ , ∗ is an unknown positive constant with ∗ = (nr 1 + nl1 ) max{br , −bl }. Assumption 4 (Jiang [53], Wu et al. [54]). For each 1 ≤ i ≤ n, there exists an unknown positive constant pi∗ , such that |i | ≤ pi∗ i1 (x i ) + pi∗ i2 (|z|)

(4)

where i1 and i2 are known nonnegative smooth functions, with i2 (0) = 0. Assumption 5 (Jiang [53], Wu et al. [54]). The unmodeled dynamics is ISpS, i.e., for z-system in (1), there exists a Lyapunov function V0 (z) such that 0 (|z|) ≤ V0 (z) ≤ ¯ 0 (|z|)

(5)

V0 ≤ − 0 (|z|) + 0 (|y|) + d¯0

(6)

where 0 , ¯ 0 , 0 and 0 are known class ∞ -functions, and d¯0 ≥ 0 is a known constant. 2.2. Mathematical preliminaries Consider a stochastic nonlinear system d x = f (x, u) dt + (x, u) dw(t)

(7)

where x ∈ R n , u ∈ R m are the state and the input of the system, respectively. w is an r-dimensional independent standard Wiener process, and f (·) : R m+n → R n and (·) : R m+n → R n×r are locally Lipschitz and satisfy f (0, 0) = 0, (0, 0) = 0. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Let V (x) denote an infinitesimal generator of C 2 positive function V (x): R n → R along stochastic differential equation (7) defined by   2 1 *V (x) * V T f (x) + Tr  (x, u) 2 (x, u) V (x) = 2 *x *x where Tr(·) denotes the matrix trace. Definition 1. The system (7) is said to be input-to-state practically stable (ISpS) in probability if for any  > 0, there exist a class -function , and a class ∞ -function and a constant d ≥ 0 such that P{|x(t)| < (|x(0)|, t) + ( u t ) + d} ≥ 1 − , ∀t ≥ 0, x0 ∈ R n \{0}

(8)

When d = 0 in (8), the ISpS property becomes the input-to-state stability (ISS) property defined in [54]. Theorem 1 (Jiang [53], Wu et al. [54]). For system (7), if there exist C 2 function V (x), class ∞ -functions , ¯ ,  and -function , and a constant d ≥ 0 such that (|x|) ≤ V (x) ≤ ¯ (|x|) V (x) ≤ − (|x|) + (|u|) + d

(9) (10)

Then system (7) is ISpS in probability. Consider the following interconnected stochastic systems d x1 = f 1 (x1 , x2 , 1 ) dt + g1 (x1 , x2 , 1 ) dw1t d x2 = f 2 (x1 , x2 , 2 ) dt + g2 (x1 , x2 , 2 ) dw2t

(11)

where x = (x1T , x2T )T ∈ R n 1 +n 2 is the state of system, i (i = 1, 2) denotes exterior disturbance and/or interior uncertainty, w1t and w2t are independent standard Wiener processes. Theorem 2 (Wu et al. [54], Stochastic small-gain theorem). Suppose that both the x1 -system and x2 -system in (11) are ISpS in probability with (1 , x2 ) as input and x1 as state, and (2 , x1 ) as input and x2 as state, respectively, i.e., for any 1 and 2 > 0 such that P{|x1 (t)| < 1 (|x1 (0)|, t) + 1 ( x2t ) + w ( 1t ) + d1 } ≥ 1 − 1

(12)

P{|x2 (t)| < 2 (|x2 (0)|, t) + 2 ( x1t ) + w ( 2t ) + d2 } ≥ 1 − 2

(13)

where i are class -functions, i and w are class ∞ -functions, and di is a nonnegative constant, i=1,2. If there exist nonnegative parameters 1 , 2 and s0 such that the nonlinear gain functions 1 and 2 satisfy (1 + 1 ) 1 (s) ◦ (1 + 2 ) 2 (s) ≤ s, ∀s ≥ s0

(14)

then the interconnected system (11) is ISpS in probability with  = (1 , 2 ) as input and x = (x1 , x2 ) as state, i.e., for any  > 0, there exist a class -function , a class ∞ -function w (·) and a constant d ≥ 0 such that P{|x(t)| < (|x(0)|, t) + w ( t ) + d} ≥ 1 − 

(15)

2.3. 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: R l : If x1 is F1l and x2 is F2l and . . . and xn is Fnl , then y is G l , l = 1, 2, . . . , N Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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where x = (x1 , x2 , . . . , xn )T and y are FLS input and output, respectively. N is the number of If–then rules; F1l , . . . , Fnl and G l are fuzzy sets for linguistic variables x1, x2 , . . . , xn and y, respectively. Through singleton fuzzifier, center average defuzzification and product inference, the FLS can be expressed as [16] n N l=1 y¯l i=1  Fil (x i ) (16) y(x) =  N n l=1 [ i=1  F l (x i )] i

where  F l (xi ) and G l (y) are the membership functions of fuzzy sets Fil and G l , respectively, and y¯l = max y∈R G l (y). i Define the fuzzy basis functions as n i=1  F l (x i ) l =  N n i l=1 [ i=1  F l (x i )] i

Denoting = [ y¯1 , y¯2 , . . . , y¯ N ] = [W1 , W2 , . . . , W N ] and (x) = [ 1 (x), 2 (x), . . . , N (x)]T , then fuzzy logic system (16) can be rewritten as WT

y(x) = W T (x)

(17)

Lemma 1 (Wang [16]). For any continuous function f(x) defined over a compact set  and any given positive constant , there exists a fuzzy logic system (17) and an ideal parameter vector W ∗ such that sup | f (x) − W ∗T (x)| ≤ 

x∈

Remark 1. It is noticed that Lemma 1 is the universal approximation lemma, which is cited by most references about adaptive fuzzy control. Assumptions 1–5, and Theorems 1 and 2 come from Refs. [49,51,53,54]. This paper is different from Refs. [49,51,53,54]. Zhang and Ge [49] investigated the adaptive fuzzy control problem for a class of nonlinear systems with dead-zone and unknown control directions. The system considered in [49] is a deterministic nonlinear system, not a stochastic nonlinear system and it does not include the unmodeled dynamics. Yu and Du [51] investigated the adaptive fuzzy control problem for a class of stochastic nonlinear systems, but it did not consider the problem of unmodeled dynamics. Jiang [53] and Wu et al. [54] gave the definitions of small-gain theorem and stochastic small-gain theorem. The adaptive control approaches in [53,54] are only suitable for the unstructured uncertainties which are to be linear with the unknown parameters (linear-in-parameters), or the unstructured uncertainties which only contain y, rather than x1 , . . . , xn . The unstructured uncertainties in this paper are completely unknown nonlinear functions of x1 , . . . , xn . Moreover, the results [53,54] did not consider the dead-zone problem either. 3. Adaptive fuzzy backstepping controller design In this section, an adaptive fuzzy state feedback control design approach is developed by combining the backstepping technique with the small-gain theorem. To introduce the design of controller, we need to make the following coordinate transformation: 1 =  (y 2 )y i = xi − i−1 , i = 2, . . . , n

(18)

where (y 2 )>0 is sufficiently smooth function with (0) = 0, which will be chosen appropriately later such that  (y 2 ) is strictly positive over R+ , where  (y 2 ) is the value of the derivative of (y 2 ) at y 2 ; i−1 is the intermediate control function, which will be specified later. Step 1: Consider the following Lyapunov function candidate: g 2 1 1 2 ˜ + p˜ (19) V1 = (y 2 ) + 2 21 1 2¯ 1 1 where 1 > 0 and ¯ 1 > 0 are design parameters; pi ≥ max{ p ∗j , p ∗2 j ; 1 ≤ j ≤ i}, i = 1, . . . , n; pˆ i and i are the ∗ ∗ ˜ estimates of pi and  , respectively ; p˜ i = pˆi − pi and i = i − ; the definition of ∗ will be given later, i = 1, . . . , n. i

i

i

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For the given V1 , associated with the stochastic differential equation (1), we define the differential operator  as follows:   2 g ˙ * V *V1 1 1 1 T V1 = p˜ 1 p˙˜ 1 + Tr 1 2 1 (g1 (x1 )x2 + f 1 (x1 ) + 1 (x, z)) + ˜ 1 ˜ 1 + 1 2 *y *y ¯ 1 = y (g1 (x1 )x2 + f 1 (x1 ) + 1 (x, z)) +

g 1

1 1 ˜ 1 ˙ 1 + p˜ p˙ˆ + (2

y 2 +  )1T (x1 )1 (x1 ) ¯1 1 1 2

(20)

where Tr denotes the matrix trace. By using Assumptions 2, 4 and Young inequality, we have

2 1 2 ((2 y

+  )1T (x1 )1 (x1 )) ≤ 21 (2

y 2 +  (y 2 ))x12 21 (x1 )

(21)

y 1 (x, z) ≤ |y |( p1∗ 11 (x1 ) + p1∗ 12 (|z|)) ≤ p1 (y )2 (211 (x1 ) + 41 ) +

1 4

+ 212 (|z|)

Substituting (21) and (22) into (20), we have

 1 V1 ≤ y g1 (x1 )x2 + f 1 (x1 ) + (2

y 2 +  )y21 (x1 ) 2

g 1 1 1 2

2 ˙ ˜ ˙ p˜ 1 pˆ 1 + p1 (y ) 11 (x1 ) + + + 212 (|z|) + 1 1 + ¯ 1 4 4 1 g 1 1 ≤ y [g1 (x1 )x 2 + f¯1 (x1 )] + ˜ 1 ˙ 1 − (y )2 − g¯ (y )4 1 4 4

1 ˙ 1 1 2

2 + + 212 (|z|) + p˜ 1 pˆ − (y ) 11 (x1 ) + ¯1 1 4 4

(22)

(23)

where f¯1 (x1 ) = f 1 (x1 )+(1/2 )(2

y 2 + )y21 (x1 )+ pˆ 1 y (211 (x1 )+ 41 )+ 41 y + g¯ 41 (y )3 is an unknown nonlinear function. Since f 1 (x1 ) and 21 (x1 ) are unknown, f¯1 (x1 ) cannot be directly implemented to construct the virtual controller 1 . By employing a fuzzy logic system W1T 1 (x1 ) to model f¯1 (x1 ), f¯1 (x1 ) can be expressed as f¯1 (x1 ) = W1∗T 1 (x1 ) + 1 (x1 )

(24)

where 1 (x1 ) refers to the fuzzy minimum approximation error and satisfies |1 (x1 )| ≤ ∗1 , and ∗1 is a positive constant. Substituting (18) and (24) into (23), we have 1 1 V1 ≤ y (g1 (x1 )(2 + 1 ) + W1∗T 1 (x1 ) + 1 (x1 )) − g¯ (y )2 − (y )4 4 4

g 1 1 ˙ 1 + ˜ 1 ˙ 1 + p˜ 1 (25) + + 212 (|z|) pˆ 1 − (y )2 211 (x1 ) + 1 4 4 ¯ 1 It is noticed that, for any variable x ∈ R and constant  > 0, the following inequality holds: x  0 ≤ |x| − x tanh ≤ 0.2785  By using Young inequality and about inequality, we have y W1∗T 1 (x1 ) ≤ |y | W1∗ 1 (x1 )

y 1 + 0.2785g∗1 ≤ y g∗1 1 tanh  y 1 (x1 ) ≤ 41 (y )2 + ∗2 1

(26) (27)

Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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where ∗1 = W1∗ /g. Substituting (26) and (27) into (23), we have

g 1 1 ˙ V1 ≤ y [g1 (x1 )(2 + 1 )] + ˜ 1 ˙ 1 + p˜ 1 pˆ 1 − (y )2 211 (x1 ) + 1 4 ¯ 1



1 y 1 1

∗ ¯ )4 + + 212 (|z|) + ∗2 + 0.2785g∗1 − g(y 1 + y g1 1 tanh 4 4 

g 1 4 1 1 ˙ 2

2 ˜ ˙ ≤ 1 g1 2 + 1 g1 1 − g pˆ 1 − (y ) 11 (x1 ) + ¯ + 1 1 + p˜ 1 4 1 1 4 ¯ 1



y 1 + 212 (|z|) + d¯1 +y g∗1 1 tanh  where d¯1 =

+ 0.2785g∗1 + ∗2 1 . Now, choose intermediate control function 1 as

y 1 2 1 = −yv1 (y ) − 1 1 tanh 

(28)

1 4

where v1 (y 2 ) is a smooth non-decreasing function with v1 (0) > 0, then we obtain



y 1 1 g1 1 ≤ −1 yg1 v1 − 1 g1 1 tanh  where the property of 1 ≥ 0 has been used in (30) and will be explained in Remark 1. Substituting (29) and (30) into (28), we have



 1 y 1 1 ˙ 4

˜ 1 − y 1 tanh V1 ≤ 1 g1 2 − 1 yg1 v1 − b M 1 + g 1 4 1 

1 ˙ 1 + 212 (|z|) + d¯1 + p˜ 1 pˆ 1 − (y )2 211 (x1 ) + 4 ¯ 1 Furthermore, take the adaptation functions 1 and pˆ 1 as



y 1 − 1 1 ˙ 1 = 1 y 1 tanh 



1 p˙ˆ 1 = ¯ 1 (y )2 211 (x1 ) + − ¯ 1 pˆ 1 4

(29)

(30)

(31)

(32) (33)

where 1 and ¯ 1 are positive design parameters. Then, by the following inequalities: ¯ 41 + 41 g ¯ 42 + 21 g¯ 1 g1 2 ≤ 41 g −g1 ˜ 1 1 = −g1 ˜ 1 (˜ 1 + ∗1 ) ≤ − −¯ 1 p˜ 1 pˆ 1 ≤ −

(34) g1 2

2 ˜ 1 +

g1 2

∗2 1

¯ 1 2 ¯ 1 2 p˜ + p 2 1 2 1

(35) (36)

We can obtain g1 2 ¯ 1 2 1 4 V1 ≤ −1 yg1 v1 + g ¯ 2 + 212 (|z|) − ˜ − p˜ + d1 4 2 1 2 1

(37)

where d1 = 21 g¯ + (g1 /2)∗2 ¯ 1 /2) p12 + d¯1 . 1 + ( Remark 2. From (32), since



y 1 y 1 tanh ≥0 

(38)

We obtain ˙ 1 ≥ −1 1 1 . If the initial condition 1 (0) ≥ 0 holds, then 1 (t) ≥ 0 for all t ≥ 0. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Step i: (2 ≤ i ≤ n − 1). Based on the coordinate transformation i = xi − i−1 , and Itô formula, one has ⎛ ⎞ i−1  * i−1 di = ⎝gi (x i )xi+1 + f i (x i ) − L i−1 −  j (x, z) + i (x, z)⎠ dt *x j j=1 ⎛ + ⎝i −

i−1  * i−1 j=1

*x j

⎞T  j ⎠ dw

(39)

with L i−1 =

i−1  * i−1 j=1

*x j

(g j (x j )x j+1 + f j (x j )) +

i−1  * i−1 j=1

* j

i−1 2 1  * i−1 T ˙ j +   2 *x j *xk j k j,k=1

(40)

Consider the Lyapunov function candidate g 2 1 1 2 ˜ + p˜ i Vi = Vi−1 + i4 + 4 2i i 2¯ i

(41)

where i > 0 and ¯ i > 0 are design parameters. Using the same procedures as Step 1, it follows that ⎛ ⎞ i−1  * i−1  j (x, z) + i (x, z)⎠ Vi = Vi−1 + i3 ⎝gi (x i )xi+1 + f i (x i ) − L i−1 − *x j j=1 ⎛ ⎞T ⎛ ⎞ i−1 i−1   g * i−1 ⎠ ⎝ * i−1 ⎠ 1 3 2⎝ i − j  j + ˜ i ˙ i + p˜ i p˙ˆ i + i i − ¯ 2  *x *x i i j j j=1 j=1 It is noticed that  2  4     i−1 i−1      * * 3 2 3 3 i−1 i−1 4    j  ≤ i i − j +  i −  2 i 4 4 *x j *x j     j=1 j=1

(42)

(43)

i3 i (x, z) ≤ |i3 |( pi∗ i1 (x i ) + pi∗ i2 (|z|)) ≤ pi (i3 i1 (x i ))2 + −i3

i−1  * i−1 j=1

*x j

1 4

2 + 41 pi (i3 )2 + i2 (|z|)

(44)

 i−1    3 * i−1  ∗ ∗    j (x, z) ≤ i *x ( p j  j1 (x j ) + p j  j2 (|z|)) j

j=1

≤ pi

i−1  j=1

i3

* i−1  j1 (x j ) *x j

2 +

2  i−1 i−1 i − 1 1  3 * i−1 + 2j2 (|z|) i + pi 4 4 *x j j=1 j=1

(45)

Substituting (43)–(45) into (42) yields ⎛

⎞  4   i−1  * i−1  3  ⎜ ⎟ i − Vi ≤ Vi−1 + i3 ⎝gi (x i )xi+1 + f i (x i ) − L i−1 + i  j + pi i3 h i ⎠  4  *x j   j=1 +1 +

i−1  g i −1 1 2 + ˜ i ˙ i + p˜ i p˙ˆ i + 2j2 (|z|) + i2 (|z|) 4 i ¯ i j=1

Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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i−1  g i −1 ≤ Vi−1 + i3 (gi (x i )xi+1 + f¯i (x i )) + 1 + 2j2 (|z|) + ˜ i ˙ i + 4 i

2 +i2 (|z|) + p˜ i

1 ˙ pˆ − i6 h i ¯i i

j=1

(46)

where hi = and

2

2 i−1 i−1  * i−1 1 1  * i−1 + + (i1 (x i ))2 +  j1 (x j ) 4 4 *x j *x j j=1 j=1

 4   i−1    * 3 1 1 i−1  ¯ j + pˆ i i3 h i + g f i (x i ) = f i (x i ) − L i−1 + i i − ¯ i + i3  4  4 4 *x j 

(47)

j=1

is an unknown function, the fuzzy logic system WiT i (x i ) is used to approximate f¯i (x i ). Then f¯i (x i ) can be depicted as f¯i (x i ) = Wi∗T i (x i ) + i (x i )

(48)

where i (x i ) refers to the fuzzy minimum approximation error and satisfies |i (x i )| ≤ i∗ , and i∗ is a positive constant. Substituting (48) into (46), we have 1 4 1 6 ¯ − i Vi ≤ Vi−1 + i3 gi (x i )xi+1 + i3 Wi∗T i (x i ) + i3 i (x i ) − g 4 i 4

i−1  g i −1 1 ˙ 2 +1 + 2j2 (|z|) + i2 (|z|) + p˜ i (49) + ˜ i ˙ i + pˆ i − i6 h i 4 i ¯ i j=1

According to 0 ≤ |x| − x tanh(x/) ≤ 0.2785, we can obtain   i3 i 3 3 |i i | − i i tanh ≤ 0.2785  Multiply Wi∗ = gi∗ to the above inequality, we get   i3 i ∗ 3 3 ∗ ≤ 0.2785gi∗ Wi |i | i − i gi i tanh  and

 Wi∗ |i3 | i

≤

i3 gi∗ i tanh

i3 i 

 + 0.2785gi∗

Since the term i3 Wi∗T i (x i ) in (49) satisfies i3 Wi∗T i (x i ) ≤ Wi∗ |i3 i | we obtain i3 Wi∗T i (x i )

 ≤

i3 gi∗ i tanh

i3 i 

 + 0.2785gi∗

(50)

By using Young inequality, we have i3 i (x i ) ≤ 41 i6 + i∗2

(51)

Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Substituting (18), (50) and (51) into (49) and recursively using the process employed in the former steps, we have Vi ≤ −1 yg1 v1 −

i−1 

ck gk 4k +

k=2

−

i−1  ¯ k

2

k=1

i−1 

2k2 (|z|) +

k=1

i−2  k  k=1 j=1

p˜ k2 + di−1 + i3 gi (x i )xi+1 + 1 +

2 +i2 (|z|) + p˜ i

1 ˙ pˆ i − i6 h i ¯ i

2j2 (|z|) −

i−1  gk k=1

2

2 ˜ k

i−1  g i −1 + ˜ i ˙ i + 2j2 (|z|) 4 i

 + i3 gi∗ i tanh

j=1

i3 i 



+0.2785gi∗ + i∗2

(52)

Choose intermediate control function i as   i3 i 3 i = −ci i − i − i i tanh 4 

(53)

where ci is a positive design parameter, then we obtain   i3 i 3 4 3 4 3 i gi i ≤ −ci gi i − gi i − i gi i tanh 4 

(54)

From (54), (52) can be rewritten as Vi ≤ −1 yg1 v1 −

i 

ck gk 4k +

k=2

−

i−1  ¯ k

2

k=1



+ p˜ i

i 

2k2 (|z|) +

k=1

+ d¯i + i3 gi (x i )xi+1

1 ˙ pˆ − i6 h i ¯i i

2j2 (|z|) −

k=1 j=1

 p˜ k2

i−1  k 

+ g ˜ i

i−1  gk k=1

2

2 ˜ k

  i3 i 1˙ 3 i − i i tanh i 

(55)

where d¯i = di−1 + 1 + (i − 1)/4 + 0.2785gi∗ + i∗2 . Choose the adaptation functions i and pˆ i as     3  i i 3 ˙ i = i i i tanh − i i  p˙ˆ i = ¯ i [i6 h i − ¯ i pˆi ]

(56) (57)

where i and ¯ i are positive design parameters. Then, by the following inequalities: i3 gi i+1 ≤

3 4

gi i4 +

− gi ˜ i i ≤ − − ¯ i p˜ i pˆ i ≤ −

gi 2

1 4

4 g ¯ i+1

2 ˜ i +

gi 2

(58) i∗2

(59)

¯ i 2 ¯ i 2 p˜ + pi 2 i 2

(60)

We can obtain Vi ≤ −1 yg1 v1 −

i  k=2

k i i−1  i i    gk 2  ¯ k 2 1 4 ¯ i+1 + ˜ k − p˜ + di (61) ck gk 4k + g 2k2 (|z|) + 2j2 (|z|) − 4 2 2 k k=1

k=1 j=1

k=1

k=1

where di = d¯i + (gi /2)i∗2 + (¯ i /2) pi2 . Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Step n: This is the final step. The actual control signal v will be constructed. By means of (18) and Itô formula, we have ⎛ ⎞ n−1  * n−1 dn = ⎝gn (x)(T (t)(t)v + d(v)) + f n (x) − L n−1 −  j (x, z) + n (x, z)⎠ dt *x j j=1 ⎛ + ⎝n −

n−1  * n−1 j=1

*x j

⎞T  j ⎠ dw

(62)

where L n−1 is given in (40) with i = n. Consider the Lyapunov function candidate g0 2 1 1 2 Vn = Vn−1 + 4n + ˜ n + p˜ n 4 2n 2¯ n

(63)

Taking (46) with i = n, xn+1 = u. Using the fuzzy logic system WnT n (x) to approximate the unknown function f¯n (x), and repeating the similar methods in (48), (50) and (51), we have 3n f¯n (x)

≤

3n g0 ∗n n tanh



3n n 

1 + 0.2785g0 ∗n + 6n + ∗2 n 4

(64)

where i∗ = Wi∗ /(g0 ). Further, we have Vn ≤ −1 yg1 v1 −

n−1 

ck gk 4k +

k=2

−

n−1  ¯ k k=1

+

n−1 

2

n−1 

2k2 (|z|) +

k=1

n−2  k 

2j2 (|z|) −

k=1 j=1

p˜ k2 + dn−1 + 3n gn (x)(T (t)(t)v + d(v)) + 1 +

2j2 (|z|) + 2n2 (|z|) + p˜ n

j=1

n−1  gk k=1

2

2 ˜ k

n − 1 g0 ˜ ˙ + n n 4 n

3

 1 ˙ pˆ n − 6n h n + 3n g∗n n tanh n n  ¯ n

+0.2785g∗n + ∗2 n

(65)

where h n is given in h i with i = n. Choose actual control input v and the adaptation functions n and pˆ n as 3

3  v = −cn n − n − n n tanh n n 4 

(66)



3  − n n ˙ n = n 3n n tanh n n 

(67)

p˙ˆ n = ¯ n [6n h n − ¯ n pˆ n ]

(68)

where cn , n and ¯ n are positive design parameters. Since T (t)(t) ≥ 0 and |d(v)| ≤ ∗ , the following results hold: 3

3  3n gn (x)(T (t)(t)v ≤ −cn gn 0 4n − gn 0 4n − g0 3n n n tanh n n 4  3n gn (x)d(v) ≤

3 1 g ¯ ∗4 gn 0 4n + 4 40

(69)

(70)

Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Substituting (66)–(70) into (65) results in Vn ≤ − y 2 gv1 −

n−1 

ck gk 4k − cn gn 0 4n −

k=2

−

n  k=1

¯ k 2 p˜ + 2 k

n−1  gk k=1

n 

2k2 (|z|) +

k=1

k n−1  

2

2 ˜ k −

n−1  g0 k k=1

2

2 ˜ k

2j2 (|z|) + d

(71)

k=1 j=1

¯ ∗4 . If we choose  and v1 to satisfy where d = dn = d¯n + (g0 i /2)i∗2 + (¯ i /2) pi2 + (1/40 )g  (y 2 )y 2 gv1 (y 2 ) ≥ 21 (y 2 )

(72)

By substituting (72) into (71), one gets n−1 n−1 n−1   gk 2  g0 k 2 1 ˜ k − ˜ k Vn ≤ − (y 2 ) − ck gk 4k − cn gn 0 4n − 2 2 2 k=2

−

n  ¯ k k=1

2

p˜ k2 +

k=1

n 

2k2 (|z|) +

k=1

k n−1  

k=1

2j2 (|z|) + d

(73)

k=1 j=1

Letting c = min{4c2 g, . . . , 4cn−1 g, 4cn g0 , 1, 1 1 , . . . , n n , ¯ 1 ¯ 1 , . . . , ¯ n ¯ n }

(74)

Eq. (73) becomes Vn ≤ −cVn +

n 

2k2 (|z|) +

k=1

k n−1  

2j2 (|z|) + d

(75)

k=1 j=1

4. Small-gain design and stability analysis Since the function i2 (i = 1, . . . , n) is smooth and vanishes at the origin, according to [53,54], there exists a smooth k∞ -function  such that n 

2k2 (|z|) +

k=1

n−1  k 

2j2 (|z|) ≤ (V0 (z))

(76)

k=1 j=1

Substituting (76) into (75) results in Vn ≤ −cVn + (V0 (z)) + d

(77)

As stated in [53,54], there always exist a smooth function (y 2 ) and arbitrarily small parameters d and d¯ such that (y ) + d¯ ≥ 4 −1 ◦ 0 2



1 2 y ,  (y 2 ) > 0 d

(78)

where (0) = 0. According to Assumption 5, one gets |z| ≤ 0−1 (V0 (z)) and |z| ≥ ¯ −1 0 (V0 (z)) Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Because of the item of (y 2 )/2 is included in Vn , from (71), one has



1¯ 1 2 1 2 y ≤

0 (y ) + d d 4 4

1¯ 1 2 (y ) + d ≤ 2 2

1¯ d ≤ (Vn ) + 2

(79)

By substituting (79) into V0 , one gets V0 ≤ − 0 (|z|) + 0 (|y|) + d¯0

1 2 ≤ − 0 (|z|) + 0 y + 0 (d ) + d¯0 d ¯ ≤ − 0 ◦ ¯ −1 0 ◦ V0 (z) + (Vn ) + d

(80)

where d¯ = 0 ( 21 d¯ ) + 0 (d ) + d¯0 .



By Theorem 1, for any i > 0 (i = 1, 2), there exist class  function i (·, ·) and constant di > 0 such that 

P{|Vn (X )| < 1 (Vn (0), t) + 1 ( V0t ) + d 1 } ≥ 1 − 1 

P{|V0 (z)| < 2 (V0 (0), t) + 2 ( Vnt ) + d 2 } ≥ 1 − 2

(81) (82)

where X (t) = (y, 2 , . . . , n , ˜ 1 , . . . , ˜ n , p˜ 1 , p˜ 2 , . . . , p˜ n )T and

1 (s) = (4da1 (s)/c)v¯1

(83)

v¯2

2 (s) = 0−1 ◦ (¯ 0 ◦ 2¯ −1 0 ◦ 2da2 (s))

(84)

 d1

= (4d¯a1 d/c)v¯1 + (2db1 /c)v¯1

(85)

 d2

−1 v¯2 ¯ ¯ v¯2 = 0−1 ◦ (¯ 0 ◦ 2¯ −1 0 ◦ 2¯ −1 0 ◦ (2da2 d)) + 0 ◦ (¯ 0 ◦ (db2 ))

(86)

and dai , d¯ai and v¯i hold with dai ≥ d¯ai ≥ 1 and v¯i ≥ 1. To check the stochastic small-gain condition, we select class ∞ -function as 

1/v¯1 1/v¯2 d 1 1 −1 1 c

(s) = 0 ◦ ¯ 0 ◦ 0 ◦ −1 ◦ s 2da2 2 1 + 2 4da1 1 + 1

(87)

with design parameter 0 < d ≤ 1. From the smoothness (y 2 ) and (0) = 0, there exists a smooth function ¯ (y 2 ) such that (y 2 ) = y 2 ¯ (y 2 ). By selecting v1 (y 2 ) to satisfy v1 (y 2 ) ≥

¯ (y 2 )  (y 2 )

(88)

The aforementioned analysis and small-gain design are summarized in the following theorem. Theorem 3. For stochastic nonlinear system with dead-zones and unmodeled dynamics (1), under Assumptions 1–5, the controller (66), with the intermediate control functions (29) and (53) parameter laws (32), (33), (56), (57), (67) and (68) guarantee that the closed-loop system is ISpS in probability. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Proof. From the arguments above, it is clear that the choice of in (87) guarantees that (1 + 1 ) 1 ◦ (1 + 2 ) 2 (s) ≤ s, ∀s ≥ 0

(89)

From (81), (82) and (89), and according to Theorem 2, for any  > 0, by choosing v¯i and dbi large enough (i = 1, 2), there exists a class  function  such that for all t ∈ [0, ∞), P{|V¯ ((t))| < (V¯ ((0)), t) + dc } ≥ 1 − 

(90)

where (t) = (z, y, 2 , . . . , n , ˜ 1 , . . . , ˜ n , p˜ 1 , p˜ 2 , . . . , p˜ n )T , V¯ = (V0 , Vn )T , and the parameter dc can be chosen as 



−1 −1 −1 dc = (1 + −1 1 )(1 + 3 )(1 + 3 )[ d1 + 1 (1 + 2 )(1 + 3 )(1 + 3 ) d2 )] 



−1 −1 −1 +(1 + −1 2 )(1 + 3 )(1 + 3 )[ d2 + 2 (1 + 1 )(1 + 3 )(1 + 3 ) d1 )]

(91)

with 3 being any positive constant. The guidelines of the main parameters selections in the control system are summarized as follows: (1) The parameters ck in (53), i in (56), ¯ i in (57) (k = 2, . . . , n, i = 1, . . . , n) and  in (53) are chosen to satisfy ck > 0, i > 0, ¯ i > 0 and  > 0, respectively. From (75) and the definitions of c and d, we can find that increasing the value of ck , i , ¯ i and decreasing the value of  can help to reduce the value of |y|. However, if one increases the values of ck , i , ¯ i and decreases the value of , the control energy will become larger. Therefore, the trade-off between the better transient performance and control action should be considered in practical controller design. (2) Design parameters i in (56) and ¯ i in (57) (i = 1, . . . , n) are used in the -modification, the appropriate choice of i and ¯ i can prevent i and pˆi (i = 1, . . . , n) to drift. Similar to the remark in [46,48], i and ¯ i are chosen to satisfy i > 0 and ¯ i > 0, respectively. (3) Design parameters d in (78), dai and v¯i (i = 1, 2) in (83)–(86), and d in (87) are required to satisfy that d > 0, dai > 1, v¯i > 0 and 0 < d ≤ 1, respectively. Choosing dai and v¯i larger and d smaller can guarantee that |y| is smaller and the convergence rates of y is faster. However, both cases will lead to the increase in the control energy.  5. Simulation studies In this section, the effectiveness of the proposed method and the control performance is illustrated by the following examples. Example 1 (Numerical example). Consider the following nonlinear stochastic system: dz = q1 (x, z) dt + q2 (x, z) dw d x1 = [g1 (x1 )x2 + f 1 (x1 ) + 1 (x, z)] dt + 1 (x) dw d x2 = [g2 (x 2 )u + f 2 (x 2 ) + 2 (x, z)] dt + 2 (x) dw y = x1

(92) √ where q1 (x, z) = −2.5z + 0.25x12 , q2 (x, z) = 2x12 /8, g1 (x1 ) = 1 + x12 /(1 + x12 ), 1 (x1 ) = x12 , g2 (x 2 ) = 1+ x12 x22 /(1 + x12 x22 ), 2 (x 2 ) = 0.9x12 cos(x2 ), 1 = z sin x13 , f 1 (x1 ) = x12 , f 2 (x 2 ) = x12 sin x1 + 1.7x22 , 2 = 0.9z cos x1 , and ⎧ v ≥ 2.5 ⎪ ⎨ (1 − 0.3 sin(v))(v − 2.5), −1.5 < v < 2.5 u = D(v) = 0, ⎪ ⎩ (0.8 − 0.2 cos(v))(v + 1.5), v ≤ −1.5 It is easy to get g = 1, g¯ = 2. For z-system in (92), with the choice of Lyapunov function V0 (z) = 0 (|z|) = ¯ 0 (|z|) = z 2 , we can verify that V0 ≤ −4V0 + 18 y 4 , which implies that Assumption 2 is satisfied for c0 = 4, 0 = y 4 /8, d¯0 = 0. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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According to (76), it is easy to obtain that (s) = d s, d = 3

(93)

By setting 1 = 2 = 1, v¯1 = 1 and v¯2 = 1, from (87), one has

(s)−1 =

48da1 da2 s d c

(94)

where c = min{1, 4c2 g0 , 1 1 , 2 2 , ¯ 1 ¯ 1 , ¯ 2 ¯ 2 }. Substituting 0 = y 4 /8 and (94) into (78) gives (y 2 ) = c y 4

(95)

where c = 24da1 da2 /(d cd2 ). It is easy to verify that (y 2 ) satisfies (78) and  (y 2 ) > 0, and by choosing v1 (y 2 ) = 10, then (72) holds. Define fuzzy membership functions as  F l (x1 ) = exp[−(x1 − 4 + l)2 /6],  F l (x2 ) = exp[−(x2 − 24 + 6l)2 /6], l = 1, . . . , 7. 1

2

The FLSs W1T 1 (x1 ) and W2T 2 (x1 , x2 ) are used to approximate unknown functions f¯1 and f¯2 , respectively, where W1T = [W11 , W12 , W13 , W14 , W15 , W16 , W17 ], W2T = [W21 , W22 , W23 , W24 , W25 , W26 , W27 ], 1 (x1 ) = [ 11 , 12 , 13 , 14 , 15 , 16 , 17 ]T , 2 (x1 , x2 ) = [ 21 , 22 , 23 , 24 , 25 , 26 , 27 ]T , exp[−(x1 − 4 + j)2 /6] 1 j (x1 ) = 7 2 n=1 exp[−(x 1 − 4 + n) /6] and exp[−(x1 − 4 + j)2 /6] × exp[−(x2 − 24 + 6 j)2 /6] , 2 j (x1 , x2 ) = 7 2 2 n=1 exp[−(x 1 − 4 + n) /6] × exp[−(x 2 − 24 + 6n) /6]

j = 1, . . . , 7.

The intermediate control function 1 , actual control input v, and adaptation functions 1 , pˆ 1 , 2 and pˆ 2 are chosen as

y 1 2 1 = −yv1 (y ) − 1 1 tanh (96)    32 2 3 v = −c2 2 − 2 − 2 2 tanh (97) 4 



˙1 = 1 y tanh y 1 − 1 1 1 



1 p˙ˆ 1 = ¯ 1 (y )2 211 (x1 ) + − ¯ 1 pˆ 1 4     3  2 3 2 ˙ 2 = 2 2 2 tanh − 2 2  p˙ˆ 2 = ¯ 2 [62 h 2 − ¯ 2 pˆ 2 ]

(98) (99)

(100) (101)

where h 2 = 41 + 41 (* 1 /*x1 )2 + (21 (x 2 ))2 + ((* 1 /*x1 )11 (x1 ))2 with 11 = (sin x13 )2 and 21 = (cos x1 )2 . The design parameters are chosen as c2 = 1, 1 = 2 = ¯ 1 = ¯ 2 = 1,  = 0.01, 1 = ¯ 1 = 2 = ¯ 2 = 0.1, da1 = da2 = 1, d = 1 and d = 120. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Fig. 1. The trajectory of state x1 .

Fig. 2. The trajectory of state x2 .

The initial conditions are chosen as x1 (0) = 2.5, x2 (0) = 0.04, z(0) = 0.5, 1 (0) = 2 (0) = 0.01,

pˆ 1 (0) = pˆ 2 (0) = 0.

The simulation results are shown in Figs. 1–3, where the trajectories of x1 and x2 are shown by Figs. 1 and 2, respectively. The trajectories of the input of the dead-zone (actual control input) v and the output of the dead-zone u are shown in Fig. 3. Example 2 (Application example). We consider a one-link manipulator with the inclusion of motor dynamics and stochastic disturbances to illustrate the application of the control method proposed in this paper. The dynamics equation Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Fig. 3. The trajectory of v (red) and u (black). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

of such system is given by [34]  D q¨ + B q˙ + N sin(q) =  +  ˙ + Hm  = u − K m q˙ M

(102)

where q, q˙ and q¨ denote the link position, velocity, and acceleration, respectively.  is the torque produced by the electrical subsystem,  = 1 (q(t), q(t)) ˙ w, ˙ where 1 (q(t), q(t)) ˙ = q 2 (t) cos(q(t)) ˙ and w represents the torque stochastic disturbance. u is the control input used to represent the electromechanical torque. D = 1 kg m2 is the mechanical inertia, B=1 N m s/rad is a positive the coefficient of viscous friction at the joint, N=10 is a positive constant related to the mass of the load and the coefficient of gravity, M=0.1 H is the armature inductance, H = 1.0  is the armature resistance, and K m = 0.2 N m/A is the back electromotive force coefficient. Let x1 = q, x2 = q, ˙ and x3 = , and suppose that system (102) exists the unmodeled dynamics and disturbances, then the dynamics given by (102) can be written in the following form: dz = q1 (z, y) dt + q2 (z, y) dw d x1 = x2 dt d x2 = [g2 (x1 , x2 )x3 + f 2 (x1 , x2 ) + 2 (, x)] dt + 1 (x1 , x2 ) dw d x3 = [g3 (x)u + f 3 (x) + 3 (, x)] dt y = x1

(103)

where q1 (z, y) = −5z + 0.125y 2 , q2 (z, y) = y 2 /16, 1 (x1 , x2 ) = 0.9x12 cos(x2 ), 2 (z, x) = x12 z, 3 (z, x) = z sin(x1 ), f 2 (x1 , x2 ) = −x2 − 10 sin(x1 ), f 3 (x) = −2x2 − 10x3 , g2 (x1 , x2 ) = 1, g3 (x) = 10 and ⎧ ⎪ ⎨ 1.4(v − 1.5), v ≥ 1.5 −1.3 < v < 1.5 u = D(v) = 0, ⎪ ⎩ 0.6(v + 1.3), v ≤ −1.3 It is easy to get g = 1, g¯ = 2 and 0 = 0.6. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Fig. 4. The fuzzy membership functions.

For z-system, with the choice of Lyapunov function V0 () = 0 (||) = ¯ 0 (||) = 2 , we can verify V0 ≤ −4V0 + 18 y 4 , which implies that Assumption 3 is satisfied for c0 = 4, 0 = y 4 /8 and d0 = 0. In the simulation studies, the if–then rules are chosen as R 1 : If x1 is F11 , x2 is F21 and x3 is F31 , then y is G 1 ; R 2 : If x1 is F12 , x2 is F22 and x3 is F32 , then y is G 2 ; R 3 : If x1 is F13 , x2 is F23 and x3 is F33 , then y is G 3 ; R 4 : If x1 is F14 , x2 is F24 and x3 is F34 , then y is G 4 ; R 5 : If x1 is F15 , x2 is F25 and x3 is F35 , then y is G 5 ; where fuzzy sets are chosen as F11 = (N L), F21 = (N L), F31 = (N L), F12 = (N S), F22 = (N S), F32 = (N S), F13 = (Z O), F23 = (Z O), F33 = (Z O), F14 = (P S), F24 = (P S), F34 = (P S), F15 = (P L), F25 = (P L), F35 = (P L), which are defined over the interval [−6, 6] for variables x1 , x2 and x3 , respectively. NL, NS, ZO, PS and PL denote negative large, negative small, zero, positive small, positive large, respectively. Their center points are selected as −6, −3, 0, 3, 6, respectively. The corresponding fuzzy membership functions are given by

 (xi − 9 + 3l)2  F l (xi ) = exp − , i = 1, 2, 3, l = 1, . . . , 5 i 5 as shown in Fig. 4. Construct fuzzy basis functions exp[−(x1 − 9 + 3 j)2 /5] , 1 j (x1 ) = 5 2 n=1 exp[−(x 1 − 9 + 3n) /5] exp[−(x1 − 9 + 3 j)2 /5] × exp[−(x2 − 9 + 3 j)2 /5] 2 j (x1 , x2 ) = 5 , 2 2 n=1 exp[−(x 1 − 9 + 3n) /5] × exp[−(x 2 − 9 + 3n) /5]

j = 1, . . . , 5

Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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and exp[−(x1 − 9 + 3 j)2 /5] × exp[−(x2 − 9 + 3 j)2 /5] × exp[−(x3 − 9 + 3 j)2 /5] , 3 j (x1 , x2 , x3 ) = 5 2 2 2 n=1 exp[−(x 1 − 9 + 3n) /5] × exp[−(x 2 − 9 + 3n) /5] × exp[−(x 3 − 9 + 3n) /5] j = 1, . . . , 5 Define 1 (x1 ) = [ 11 , 12 , 13 , 14 , 15 , 16 , 17 ]T , 2 (x1 , x2 ) = [ 21 , 22 , 23 , 24 , 25 , 26 , 27 ]T , 3 (x1 , x2 , x3 ) = [ 31 , 32 , 33 , 34 , 35 ]T , W1T = [W11 , W12 , W13 , W14 , W15 ], W2T = [W21 , W22 , W23 , W24 , W25 ] and W3T = [W31 , W32 , W33 , W34 , W35 ], and obtain FLSs W1T 1 (x1 ), W2T 2 (x1 , x2 ) and W3T 3 (x1 , x2 , x3 ). W1T 1 (x1 ), W2T 2 (x1 , x2 ) and W3T 3 (x1 , x2 , x3 ) are used to approximate unknown functions f¯1 , f¯2 and f¯3 , respectively. The intermediate control functions 1 , 2 , actual control input v, and adaptation functions 1 , pˆ 1 , 2 , pˆ 2 , 3 and pˆ 3 are chosen as

y 1 2 1 = −yv1 (y ) − 1 1 tanh    32 2 3 2 = −c2 2 − 2 − 2 2 tanh 4    33 3 3 v = −c3 3 − 3 − 3 3 tanh 4 



˙1 = 1 y tanh y 1 − 1 1 1 



1 − ¯ 1 pˆ 1 p˙ˆ 1 = ¯ 1 (y )2 211 (x1 ) + 4     3 ˙2 = 2 3 tanh 2 2 − 2 2 2 2  p˙ˆ 2 = ¯ 2 [62 h 2 − ¯ 2 pˆ 2 ]     3  3 3 3 ˙ 3 = 3 3 3 tanh −  3 3  p˙ˆ 3 = ¯ 3 [63 h 2 − ¯ 3 pˆ 3 ] where 1 1 h2 = + 4 4 and 1 1 h3 = + 4 4



* 1 *x1



2

* 2 *x1

+ (21 (x 2 ))2

2

+

* 2 *x2

2 

+ (31 (x 3 )) + 2

* 2 21 (x 2 ) *x2

2

with 21 = x14 and 31 = (sin(x1 ))2 . The design parameters are chosen as c2 = 2, c3 = 3, i = ¯ i = 2,  = 0.01, i = ¯ i = 0.2, v¯1 = 1, v¯2 = 1, da1 = da2 = 2, d = 1 and d = 80 (i = 1, 2, 3). The initial conditions are chosen as x1 (0) = 0.6, x2 (0) = 0.2, x3 (0) = 0, z(0) = 0.3, i (0) = 0.02,

pˆ i (0) = 0 (i = 1, 2, 3).

Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Fig. 5. The trajectory of state x1 .

Fig. 6. The trajectory of state x2 .

The simulation results are shown in Figs. 5–8, where the trajectories of x1 , x2 and x3 are shown by Figs. 5, 6 and 7, respectively. The trajectories of the input of the dead-zone (actual control input) v and the output of the dead-zone u are shown in Fig. 8. Remark 3. Figs. 3 and 8 exhibit the trajectories of the input of the dead-zone v (red line) and the output of the dead-zone u (black line) of systems (92) and (102), respectively. In fact, we really need to find the actual control input (the input of the dead-zone) v by using the backstepping technique. Therefore, we draw the trajectory v in Figs. 3 and 8. However, we know u is a piecewise function of v from the definition of dead-zone of (2), so we also draw the trajectory of u in Figs. 3 and 8 in order to show the relation of u and v. From Examples 1 and 2, we can conclude that the proposed fuzzy control approach can guarantee that all the signals in the closed-loop system are bounded in probability, and the system output converges to a small neighborhood of the origin. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Fig. 7. The trajectory of state x3 .

Fig. 8. The trajectory of v (red) and u (black). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

6. Conclusions In this paper, a new adaptive fuzzy robust backstepping control approach has been proposed for a class of nonlinear strict-feedback systems with unknown dead zone and unmodeled dynamics. In controller design, the fuzzy logic systems are employed to model the uncertain nonlinear dynamics, and by using the backstepping technique and stochastic smallgain approach, a new adaptive fuzzy robust control method is developed. It has been proved that the proposed control approach can guarantee that: (a) the closed-loop system is input-state practically stability in probability, and (b) the output of the system converges to a small neighborhood of the origin. The main features of the proposed adaptive control approach are that it can not only ensure the stability of the control system, but also compensate the dead zone and the unmodeled dynamics. Moreover, the overparameterization problem existing in previous results can be effectively avoided. Please cite this article as: Y. Li, et al., Adaptive fuzzy control of uncertain stochastic nonlinear systems with unknown dead zone using small-gain approach, Fuzzy Sets and Systems (2013), http://dx.doi.org/10.1016/j.fss.2013.02.002

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Acknowledgment This work was supported in part by the National Natural Science Foundation of China (Nos. 61074014, 61203008, 51179019), the Program for Liaoning Innovative Research Team in University (No. LT2012013), the Program for Liaoning Excellent Talents in University (No. LR2012016), the Natural Science Foundation of Liaoning Province (No. 20102012), and competitive internal research grants of Hong Kong Polytechnic University (No. A-PL07). The authors would like to gratefully acknowledge the constructive comments and useful suggestions from the editor and anonymous reviewers for this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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