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Fuzzy Adaptive Quantized Control for a Class of Stochastic Nonlinear Uncertain Systems Zhi Liu, Fang Wang, Yun Zhang, and C. L. Philip Chen, Fellow, IEEE

Abstract—In this paper, a fuzzy adaptive approach for stochastic strict-feedback nonlinear systems with quantized input signal is developed. Compared with the existing research on quantized input problem, the existing works focus on quantized stabilization, while this paper considers the quantized tracking problem, which recovers stabilization as a special case. In addition, uncertain nonlinearity and the unknown stochastic disturbances are simultaneously considered in the quantized feedback control systems. By putting forward a new nonlinear decomposition of the quantized input, the relationship between the control signal and the quantized signal is established, as a result, the major technique difficulty arising from the piece-wise quantized input is overcome. Based on fuzzy logic systems’ universal approximation capability, a novel fuzzy adaptive tracking controller is constructed via backstepping technique. The proposed controller guarantees that the tracking error converges to a neighborhood of the origin in the sense of probability and all the signals in the closed-loop system remain bounded in probability. Finally, an example illustrates the effectiveness of the proposed control approach. Index Terms—Adaptive quantized control, backstepping technique, hysteretic quantizer, stochastic nonlinear systems.

I. I NTRODUCTION ECENTLY, quantized feedback control has received much attention, due to its wide application in digital control, networked systems, hybrid systems, and unmanned (aerial, ground, and underwater) vehicles (see [1]–[6]). The primary feature of quantized feedback control is that the signals between the controller and the plant are remotely implemented via communication channels with limited bandwidth. In such systems, the control signal being transmitted to

R

Manuscript received October 19, 2014; revised February 7, 2015; accepted February 16, 2015. This work was supported in part by the National Natural Science Foundation of China under Project U1134004, in part by the Science Fund for Distinguished Young Scholars under Grant S20120011437, in part by 2011 Zhujiang New Star, in part by the Ministry of education of New Century Excellent Talent under Grant NCET-12-0637, in part by the 973 Program of China under Grant 2011CB013104, and in part by the Doctoral Fund of Ministry of Education of China under Grant 20124420130001. This paper was recommended by Associate Editor H.-X. Li. Z. Liu and Y. Zhang are with the Faculty of Automation, Guangdong University of Technology, Guangzhou 510006, China (e-mail: [email protected]). F. Wang is with the Faculty of Automation, Guangdong University of Technology, Guangzhou 510006, China, and also with the College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China. C. L. P. Chen is with the Faculty of Science and Technology, University of Macau, Macau 999078, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2405616

the plant is a piece-wise constant function of time. An important issue of quantized control is to guarantee a certain closed-loop performance while requiring low communication rate. In recent years, the quantized control problem for linear and nonlinear plants has been an active topic, many interesting results have been reported in [4]–[10]. Among them, [4]–[6] are for linear quantized systems, [7]–[11] are for nonlinear quantized systems. However, such works in above references require the controlled systems are completely known. In practical engineering applications, as the plant is often subject to uncertain factors, it is hard to model it accurately. Therefore, the controller should be provided with adaptive capabilities to deal with these uncertainties. However, there are very few adaptive control schemes for uncertain system with quantized input. In [12] and [13], a Lyapunov-based adaptive quantized control strategy was proposed for linear uncertain systems. In [14], by proposing a new hysteretic quantizer, an adaptive control scheme was developed for nonlinear uncertain systems with quantized input. The hysteretic quantizer introduced in [14] can avoid the oscillation caused by logarithmic quantizer. However, in [14], it is hard to obtain the stability condition of the system in advance. To overcome this limitation, a backstepping-based adaptive stabilization approach was developed for parametric strict-feedback systems with hysteretic quantized input in [15]. It is worth noting that the results in [14] and [15] were based on the assumption that the uncertain nonlinearities are known functions whose parameters are unknown and linear with respect to those known functions. In addition, parameters are bounded as a prior knowledge and nonlinear functions require to satisfy global Lipschitz continuity condition (see [14], [15]). If such a prior knowledge of the structure is not available, the approach becomes infeasible. By using the fuzzy logic systems or neural networks’ universal approximation ability, such restrictions have been removed in conventional fuzzy adaptive design [16]–[36]. In these developed control schemes, the neural networks or fuzzy logic systems were used to approximate unknown nonlinearities, and adaptive controllers were constructed recursively based on the backstepping technique. However, the quantized control is quite different from the classical control theory. In the former, the control signal is transmitted to the plant after being quantized, the data are only available with finite precision and the system input is a piece-wise function. In the latter, it is assumed that any data of the system can be used with arbitrary precision, may be invalid in the presence of capacity-limited or signal quantization feedback. Therefore,

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it is difficult to extend the control approaches in [16]–[36] to quantized control problem. If the unknown stochastic disturbances and nonlinear functions are considered simultaneously in strict-feedback systems preceded by quantized input signal, it will be more challenging. On the other hand, the existing works concentrate on quantized stabilization, while the quantized feedback tracking problem has more practical meaning and recovers stabilization as a special case. Instead of forcing the output to the origin or a set point of interest, the aim of quantized tracking problem is to construct a quantized feedback controller which makes the output can follow a reference signal. However, according to [37], this problem has not drawn any attention in the present literature. Based on the above discussion, in this paper, we will consider the tracking problem of a class of stochastic nonlinear systems in strict-feedback form preceded by quantized input. To the best of authors’ knowledge, it is the first time, the tracking control problem of stochastic strict-feedback nonlinear systems with quantized input signal is investigated. Compared with the existing results, the main contributions of this paper are highlighted as follows. 1) From the viewpoint of design approach, a new nonlinear decomposition of the quantized input is introduced by using sector bound property of hysteretic quantizer. Based on the new nonlinear decomposition, the design difficulty for the boundedness of “disturbances-like” term in traditional linear decomposition is dealt with, as a result, the restrictive assumptions in [15] for uncertain functions of the system are removed. 2) Compared with the existing works on quantized control nonlinear systems, where the control schemes in [14] and [15] are designed only for systems in which nonlinear functions are known or linear parameterizations, this paper will investigate an adaptive control for a more general class of stochastic strict-feedback systems in which nonlinear functions and stochastic disturbances terms are completely unknown. By employing fuzzy logic systems’ universal approximation property, the technical difficulty of the unknown nonlinear functions and the Itˆo stochastic differentiation is resolved. Therefore, the systems in this paper are more general and more interesting. In addition, the norm of weight vector of the fuzzy logic system rather than the weight vector elements themselves is used as the estimated parameter, which significantly reduces the number of adaptive parameters. Therefore, the developed control scheme is more suitable for practical applications. 3) Compared with the existing works on quantized control problem, where the quantized stabilization is considered in [1]–[15] and [37], this paper will study the quantized feedback tracking problem. The quantized stabilization problem can be seen as a special case of the quantized feedback tracking problem. The latter is more interesting and complicated than the former. The rest of this paper is organized as follows. The preliminaries and problem formulation are presented in Section II.

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A novel fuzzy adaptive control scheme is proposed in Section III. A simulation example is provided to illustrate the effectiveness of the proposed control scheme in Section IV. Finally, Section V concludes this paper. II. P RELIMINARIES AND P ROBLEM F ORMULATION A. Stochastic Stability To introduce some definitions and lemmas on stochastic stability, consider the following stochastic nonlinear system: dx = f (x, t)dt + h(x, t)dw

(1) R+

stands for the state variable, f : × → where x ∈ Rn , h : Rn × R+ → Rn×r are locally Lipschitz functions in x and satisfy f (0, t) = h(0, t) = 0 for ∀t ≥ 0; w indicates an independent r-dimension standard Brownian motion defined on the complete probability space (, F, {Ft }t≥0 , P) with  being a sample space, F being a σ −field, {Ft }t≥0 being a filtration, and P being a probability measure. Definition 1 [38]: For twice continuously differentiable function V(x, t), define a differential operator L as follows:   2 ∂V 1 ∂V T∂ V + f + Tr h LV = h (2) ∂t ∂x 2 ∂x2 where Tr represents the matrix trace. Remark 1: As stated in [38], the second-order differential ∂ 2 V/∂x2 in Itˆo correction term 1/2Tr{hT ∂ 2 V/∂x2 h} will make the controller design much more difficult than that of the deterministic system. Lemma 1 [26]: Suppose that there exist a function V(x, t) ∈ C2,1 , two positive constants c and b, κ∞ −functions α1 and α2 , such that  α1 (x) ≤ V(x, t) ≤ α2 (x) (3) LV ≤ −cV(x, t) + b Rn

Rn

for ∀x ∈ Rn and ∀t > 0. Then, there exists an unique strong solution of (1) for each x0 ∈ Rn and the system is bounded in probability. Lemma 2 (Young’s Inequality [39]): For ∀(x, y) ∈ R2 , the following inequality holds: εp 1 xy ≤ |x|p + q |y|q p qε where ε > 0, p > 1, q > 1, and (p − 1)(q − 1) = 1. Lemma 3 [40]: Consider the dynamic system of the form θ˙ˆ (t) = −γ θˆ (t) + κρ(t) (4) where γ and κ are positive constants and ρ(t) is a positive function, then θ (t) ≥ 0 for ∀t ≥ t0 for any given bounded initial condition θˆ (t0 ) ≥ 0. B. Problem Formulation In this paper, we consider a tracking control problem for a class of stochastic strict-feedback systems with input quantization. Its dynamics can be written in the following form: ⎧ ⎨ dxi = (xi+1 + fi (¯xi ))dt + ψiT (¯xi )dw, 1 ≤ i ≤ n − 1 (5) dx = (q(u(t)) + fn (¯xn ))dt + ψnT (¯xn )dw ⎩ n y = x1

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Remark 3: From Fig. 1 and (6), it can be seen that quantizer parameters δ ∈ (0, 1) determines the communication rate, the larger δ is, the lower the communication rate is. Therefore, how to ensure the control performance with a low frequency, it is a interesting work. As shown later, by proposing a novel design scheme, the restrictive conditions posed on the quantizer parameter δ and system uncertainties in [15] are removed. C. Fuzzy Logic Systems Fig. 1.

Map of q(u(t)) for u > 0.

where x¯ i = [x1 , x2 , . . . , xi ]T ∈ Ri (i = 1, 2, . . . , n) is the state vector, y ∈ R stands for system output, and w is defined in (1). fi (.) : Ri → R and ψi (.) : Ri → Rr (i = 1, 2, . . . , n) are unknown smooth nonlinear functions. u(t) ∈ R is the control signal to be quantized, q(u(t)) is an input of the system and takes the quantized values. To avoid chattering, in this paper, a hysteretic quantizer, which is introduced in [14] and [15], is employed. The hysteretic quantizer in [14] and [15] can be regarded as a switched combination of two asymmetric logarithmic quantizers, the map of q(u(t)) for u > 0 is displayed in Fig. 1. According to [15], the hysteretic quantizer q(u(t)) is defined as follows: ⎧ ui ui sgn(u), ˙ < 0, or ⎪ 1+δ < |u| ≤ ui , u ⎪ ui ⎪ ⎪ ui < |u| ≤ 1−δ , u˙ > 0 ⎪ ⎪ ui ⎪ ⎪ ui (1 + δ)sgn(u), ui < |u| ≤ 1−δ , u˙ < 0, or ⎨ ui (1+δ) ui q(u(t)) = ˙ >0 1−δ < |u| ≤ 1−δ , u ⎪ umin ⎪ ⎪ , u ˙ < 0, or 0, 0 ≤ |u| < ⎪ 1+δ ⎪ umin ⎪ ⎪ ˙ >0 ⎪ 1+δ ≤ |u| ≤ umin , u ⎩ othercases q(u(t− )), (6) ρ 1−i umin (i

where ui = = 1, 2, . . .) and δ = (1 − ρ)/(1 + ρ) with parameters umin > 0 and 0 < ρ < 1. Then, q(u(t)) ∈ U = {0, ±ui , ±ui (1 + δ), i = 1, 2, . . .}. In the hysteresis quantizer (6), the range of the dead-zone for q(u(t)) is determined by the positive parameter umin , the positive parameter ρ can be viewed as a measure of quantization density. Remark 2: It should be noted that the hysteresis quantized model has been investigated for deterministic parametric strictfeedback systems in [14] and [15]. However, in [14] and [15], nonlinear functions fi (¯xi ) are expressed as the product of unknown parameters and known functions, i.e., fi (¯xi ) = φiT (¯xi )ηi with ηi being unknown parameters and φiT (¯xi ) being known functions. In (5), the restrictive conditions posed on fi (¯xi ) can be removed. In addition, unknown stochastic disturbances ψi (¯xi ) are neglected in [14] and [15]. In fact, in many physical systems, the structure of nonlinear functions fi (¯xi ) is often unknown and stochastic disturbances generally exist. Therefore, in contrast with the existing quantized control system, (5) is more general and interesting. But so far, there are no researches for the control of general stochastic strictfeedback nonlinear systems with input quantization due to its complexity. The challenge directly motivates the investigation of this paper.

In this paper, a fuzzy logic system will be used to approximate a continuous function f (x) defined on some compact set . Adopt the singleton fuzzifier, the product inference, and the center-average defuzzifier to deduce the following fuzzy rules: Rl : If x1 is F1l and . . . and xn is Fnl Then y is Gl , l = 1, 2, . . . , N where x = [x1 , x2 , . . . , xn ]T ∈ Rn and y ∈ R are the input and the output of the fuzzy system, respectively, Fil and Gl are fuzzy sets in R, μFl (xi ) represents the membership function i of fuzzy set Fil , and N is the number of the rules. By employing the singleton function, the center average defuzzification and the product inference [41], the output of the fuzzy system is n N l=1 l i=1 μFil (xi ) y(x) =  (7) N n l=1 i=1 μF l (xi ) i

where l = max μGl (y),  = (1 , 2 , . . . , N )T . y∈R

Let

n

i=1 μF l (xi ) ξl (x) =  i N n l=1 i=1 μF l (xi ) i

(x))T .

and ξ(x) = (ξ1 (x), ξ2 (x), . . . , ξN logic system can be rewritten as

Furthermore, the fuzzy

y(x) = T ξ(x).

(8)

Lemma 4 [41]: Let f (x) be a continuous function defined on a compact set . Then, for ∀ε > 0, there exists a fuzzy logic system (8) such that (9) sup f (x) − T ξ(x) ≤ ε. x∈

The objective of this paper is to design an adaptive fuzzy control law for u(t) to guarantee that all closed-loop signals are bounded in probability and the system output y follows a desired reference signal yd in the sense of mean quartic value. To develop the control design, the following assumption is required. Assumption 1: The reference signal yd (t) and its time derivatives up to the nth order are continuous and bounded. Remark 4: Compared with the existing quantized stabilization works, this paper focus on the quantized feedback tracking problem. Note that if yd (t) = 0, then the quantized tracking

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problem is equivalent to the quantized stabilization problem. Therefore, the quantized tracking problem is more general and interesting than the quantized stabilization, it is a challenging work. III. A DAPTIVE F UZZY T RACKING C ONTROL D ESIGN In this section, an adaptive fuzzy control scheme via backstepping technique is proposed for (5). The backstepping design procedure contains n steps and is developed based on the following coordinate transformation: z1 = x1 − yd , zi = xi − αi−1 , i = 2, . . . , n

(10)

where αi−1 is an intermediate control function, which will (i) be specified later. Define a vector function as y¯ d = (1) (i) T (i) [yd , yd , . . . , yd ] , i = 1, 2, . . . , n, where yd denotes the ith time derivative of yd . In each step of the backstepping design procedure, a fuzzy logic system i (Xi ) is employed to approximate an unknown function f¯i . To this end, define a constant as follows. θi = i 2 , i = 1, 2, . . . , n. Remark 5: In the most existing backstepping design, all the elements of weighting vectors in fuzzy logic systems need to be estimated online. For any nth order nonlinear system, if N fuzzy sets for all the variables in the fuzzy controller are used, there will be a total of nN parameters to be estimated online in their adaptive fuzzy control schemes. That means there will be many parameters need to be estimated online. As a result, the on-line learning time will become prohibitively large. To alleviate the online computation burden, by estimating the Euclidean norm of each weight vector, the number of the adaptive parameters is reduced to n in this paper. Therefore, this algorithm can reduce the computation burden significantly. Note that q(u(t)) is a quantized value, which makes the controller design difficult. For convenience of the controller design, we first give the following theorem on the relationship between q(u(t)) and u(t). Theorem 1: There exist functions G(u) and D(t) such that the quantized value q(u(t)) satisfies the following equality: q(u(t)) = G(u)u(t) + D(t)

(11)

where G(u) and D(t) satisfy the following equalities: 1 − δ ≤ G(u) ≤ 1 + δ, |D(t)| ≤ umin .

(12)

Proof: 1) For |u| > umin , from Fig. 1 and using sector bound property, it can be seen that the following inequality holds: q(u(t)) ≤ 1 + δ. (13) 1−δ ≤ u(t) Define g(u) = q(u(t))/u(t), then q(u(t)) = g(u)u(t)

(14)

where d(t) = −u(t), |d(t)| ≤ umin .

 G(u) =

and

 D(t) =

g(u), |u| > umin 1, |u| ≤ umin

(16)

0, |u| > umin −u(t), |u| ≤ umin

(17)

then q(u(t)) = G(u)u(t) + D(t) holds, G(u) and D(t) satisfy (12). The proof is completed. Remark 6: In the traditional linear decomposition q(u(t)) = u(t) + d(t), since the disturbances-like term d(t) satisfies |d(t)| ≤ |δu(t)|, it is very difficult to show the boundedness of d(t) for control design. Different from the existing decomposition approach, a new nonlinear decomposition (11) of the quantized input is established. Based on the proposed nonlinear decomposition of the quantized input, the design difficulty for the boundedness of disturbances-like term D(t) is resolved, and then the conventional analysis tools such as fuzzy control theory, can be applied to study the quantization effect. The restrictive conditions posed on quantized parameter and uncertain functions in [15] can be removed. So, Theorem 1 becomes a key to develop the following backstepping design scheme. Step 1: Consider stochastic system (5), noting that z1 = x1 − yd , the error dynamic is dz1 = (x2 + f1 (x1 ) − y˙ d )dt + ψ1T dw.

(18)

Choose a stochastic Lyapunov function candidate as V1 =

θ˜ 2 z41 + 1 4 2λ1

(19)

where λ1 > 0 is a design constant and θ˜1 = θ1 − θˆ1 is the parameter error. By (2), (10), and (18), one has 3 LV1 = z31 (z2 + α1 + f1 (x1 ) − y˙ d ) + z21 ψ1T ψ1 2 ˜θ1 θ˙ˆ1 − . λ1

(20)

Applying Lemma 2, the following inequalities hold: 3 2 T 3 3 z ψ ψ1 ≤ l1−2 z41 ψ1 4 + l12 2 1 1 4 4 z4 3z4 z31 z2 ≤ 1 + 2 4 4 where l1 is a positive design constant. Substituting and (22) into (20) yields

 3z1 3 + α1 − y˙ d + l1−2 z1 ψ1 4 LV1 ≤ z31 f1 + 4 4 4 z θ˜1 θ˙ˆ1 3 . + 2 + l12 − 4 4 λ1

(21) (22) (21)

(23)

Define a new function f¯1 = f1 + (3/4)z1 + (3/4)l1−2 z1 ψ1 4 − y˙ d , then (23) can be rewritten as

where 1 − δ ≤ g(u) ≤ 1 + δ. 2) For |u| ≤ umin , from Fig. 1, it is clear that q(u(t)) = 0 = u(t) + d(t)

Let

(15)

z4 θ˜1 θ˙ˆ1 3 LV1 ≤ z31 α1 + z31 f¯1 + l12 + 2 − . 4 4 λ1

(24)

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Since f¯1 contains the unknown function f1 and ψ1 , f¯1 cannot be implemented in practice. According to Lemma 4, for any given constant ε1 > 0, there exists a fuzzy logic system T1 ξ1 (X1 ) such that f¯1 = T1 ξ1 (X1 ) + δ1 (X1 ), |δ1 (X1 )| ≤ ε1

(25)

where X1 = (x1 , yd , y˙ d ). According to Young’s inequality, it follows: z31 f¯1 ≤

1 6 T 1 3 1 z1 θ1 ξ1 ξ1 + a21 + z41 + ε14 . 2 2 4 4 2a1

The virtual control signal is chosen as

 3 1 z1 − 2 θˆ1 z31 ξ1T ξ1 α1 = − k1 + 4 2a1

(26)

(27)

where γ1 is a positive design constant. Remark 7: It is noticed that (28) meets the conditions of Lemma 3. Therefore, if the given initial condition θˆ1 (t0 ) ≥ 0, then θˆ1 (t) ≥ 0, for ∀t > t0 . In fact, it is always reasonable to choose θˆ1 (t0 ) ≥ 0 in a practical situation, as θˆ1 is an estimation of θ1 . This property will be used in each design step. Substituting (26)–(28) into (24), we have LV1 ≤ −k1 z41 +

ε14

3 γ1 + l12 + + + θ˜1 θˆ1 . 4 4 4 4 λ1 a21

It is noted that γ1 γ1 2 γ1 2 θ˜1 θˆ1 ≤ − θ˜ + θ . λ1 2λ1 1 2λ1 1 z4 γ1 2 θ˜1 + 1 + 2 2λ1 4

(30)

(31)

(32) where i−1 i−1    ∂αi−1    ∂αi−1 ˙ θˆj fj x¯ j + xj+1 + ∂xj ∂ θˆj j=1

+

i−1  ∂αi−1 ( j)

j=0

∂yd

( j+1)

yd

+

i−1 1  ∂ 2 αi−1 T ψ ψq . 2 ∂xp ∂xq p p,q=1

j=1

3 + li2 (35) 4 z4 3 (36) z3i zi+1 ≤ z4i + i+1 4 4 where li is a design constant. Substituting the above inequalities into (34) and straightforward derivation gives the following result:   i−1 i−1  γj 2 3 kj z4j + θ˜j + j + li2 + z3i f¯i LVi ≤ − 2λj 4 j=1

j=1

+

z3i αi

1 1 + z4i+1 − θ˜i θ˙ˆi 4 λi

(37)

 4   i−1   ∂α 3 i−1   ψ f¯i = fi (¯xi ) − Lαi−1 + li−2 zi  − ψ j  + zi  i 4 ∂xj  

with ki being a positive design parameter. Similarly, a fuzzy logic system Ti ξi (Xi ) is utilized to T (i)T approximate f¯i , where Xi = [¯xiT , θ¯ˆi−1 , y¯ d ]T ∈ Zi ⊂ R3i with θ¯ˆ = [θˆ , θˆ , . . . , θˆ ]T . According to Lemma 4, f¯ i−1

1

2

i−1

(33)

i

can be expressed as f¯i = Ti ξi (Xi ) + δi (Xi ), |δi (Xi )| ≤ εi

(38)

where εi is any given positive constant. Furthermore, repeating the same method used in (26), the following inequality can be obtained: 1 1 3 1 z3i f¯i ≤ 2 z6i θi ξiT ξi + a2i + z4i + εi4 . (39) 2 4 4 2ai The virtual control signal is chosen as 

3 1 zi − 2 θˆi z3i ξiT ξi αi = − ki + 4 2ai

(40)

where ki > 0 and ai > 0 are design constants. The adaptation law is taken as λi θ˙ˆi = 2 z6i ξiT ξi − γi θˆi , θˆi (0) ≥ 0 2ai

Choose a stochastic Lyapunov function as 1 1 2 θ˜ Vi = Vi−1 + z4i + 4 2λi i

j=1

j=1

j=1

j=1

j=1

1 − θ˜i θ˙ˆi . (34) λi By the completion of squares and Young’s inequality, the following inequalities hold:  2  4     i−1 i−1      3 2 ∂α ∂α 3 i−1 i−1 −2 4    zi ψi − ψj  ≤ li zi ψi − ψj   2  ∂xj 4 ∂xj   

where f¯i is defined as

where 1 = 3l12 /4 + a21 /4 + ε14 /4 + (γ1 /2λ1 )θ12 . The term z42 /4 will be dealt with in the next step. Step i(2 ≤ i ≤ n − 1): Based on the coordinate transformation zi = xi − αi−1 and Itˆo formula, we have ⎞T ⎛ i−1  ∂αi−1 ⎠ dzi = ( fi (¯xi ) + xi+1 − Lαi−1 )dt + ⎝ψi − ψj dw ∂xj

Lαi−1 =

LVi = LVi−1 + z3i (zi+1 + αi + fi (¯xi ) − Lαi−1 ) ⎛ ⎞T ⎛ ⎞ i−1 i−1   3 ∂αi−1 ⎠ ⎝ ∂αi−1 ⎠ + z2i ⎝ψi − ψj ψj ψi − 2 ∂xj ∂xj

(29)

Equation (29) can be rewritten in the form LV1 ≤ −k1 z41 −

where λi is a positive design constant, and θ˜i = θi − θˆi refers to the parameter error. Using the similar procedure as step 1, it follows:

j=1

where k1 > 0 and a1 > 0 are design constants. The adaptation law is taken as λ1 (28) θ˙ˆ1 = 2 z61 ξ1T ξ1 − γ1 θˆ1 , θˆ1 (0) ≥ 0 2a1

z42

5

where γi is a positive design constant.

(41)

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Similar to (30), the following inequality holds: γi 2 γi 2 γi θ˜i θˆi ≤ − θ˜ + θ . λi 2λi i 2λi i Substituting (39)–(42) into (37) yields   i i  γj 2 1 θ˜j + kj z4j + j + z4i+1 LVi ≤ − 2λj 4 j=1

where f¯n is defined as (42)

 4   n−1    ∂α 3 7 n−1 ¯fn = fn (¯xn ) − Lαn−1 + ln−2 zn ψn − ψj    + 4 zn . 4 ∂x j   j=1

(50)

(43)

j=1

where j = (γj /2λj )θj2 + (1/2)a2j + (3/4)lj2 + (1/4)εj4 , j = 1, 2, . . . , i. Step n: According to Itˆo formula and (10), we can get ⎞T ⎛ n−1  ∂α n−1 dzn = ( fn (¯xn ) + u(t) − Lαn−1 )dt + ⎝ψn − ψj ⎠ dw ∂xj j=1

(44)

Similarly, for any given positive constant εn , the fuzzy logic system Tn ξn (Xn ) is utilized to approximate the unknown function f¯n . According to Lemma 4, f¯n can be expressed as f¯n = Tn ξn (Xn ) + δn (Xn ), |δn (Xn )| ≤ εn .

(51)

Using Young’s inequality, one has z3n f¯n ≤

1 6 T 1 3 1 z θn ξ ξn + a2n + z4n + εn4 . 2 4 4 2a2n n n

(52)

The control signal is chosen as

where Lαn−1 =

n−1 n−1    ∂αn−1    ∂αn−1 ˙ θˆj fj x¯ j + xj+1 + ∂xj ∂ θˆj j=1

j=1

+

n−1  ∂αn−1 ( j)

j=0

∂yd

( j+1)

yd

+

n−1 1  ∂ 2 αn−1 T ψ ψq . 2 ∂xp ∂xq p p,q=1

Consider the following stochastic Lyapunov function: 1 1 2 θ˜ (45) Vn = Vn−1 + z4n + 4 2λn n where λn is a positive design constant, and θ˜n = θn − θˆn refers to the parameter error. From (2) and (11), we have LVn = LVn−1 + z3n ( fn (¯xn ) + G(u)u(t) + D(t) − Lαn−1 ) ⎛ ⎞T ⎛ ⎞ n−1 n−1   3 ∂αn−1 ⎠ ⎝ ∂αn−1 ⎠ + z2n ⎝ψn − ψn − ψj ψj 2 ∂xj ∂xj j=1

j=1

θ˜n ˙ (46) θˆn λn where G(u) and D(t) satisfy inequality (12). Similar to (35), by the completion of squares, the following inequality holds:  2     n−1    3 2 ∂α 3 3 n−1  2 −2 4  ψ zn  l l − ψ ≤ + z n j n n n ψn  2  ∂xj 4 4   j=1 4  n−1  ∂αn−1  − ψj  (47)  ∂xj  −

j=1

where ln is a design constant. On the other hand, the following equality holds: 3 1 (48) z3n D(t) ≤ z4n + u4min . 4 4 From the inequalities (43) with (i = n − 1), (46) and (47) can be rewritten in the following form:   n−1 n−1  γj 2 3 LVn ≤ − kj z4j + θ˜j + j + z3n f¯n − z4n 2λj 4 j=1

j=1

+

z3n G(u)u(t) +

32 1 4 (49) l + u 4 n 4 min

u = −kn zn −

z3n θˆn ξnT ξn 2a2n (1 − δ)

(53)

where kn > 0 and an > 0 are design constants. The adaptation law is taken as λn θ˙ˆn = 2 z6n ξnT ξn − γn θˆn , θˆn (t0 ) ≥ 0 2an

(54)

where γn is a positive design constant. According to Lemma 3, if the given initial condition θˆn (t0 ) ≥ 0, then θˆn (t) ≥ 0 for ∀t > t0 . Therefore, from (52) and (12), the following equality holds: z3n G(u)u(t) ≤ −(1 − δ)kn z4n −

1 6 T z θˆn ξ ξn . 2a2n n n

(55)

Substituting (52)–(55) into (49), we can get   n−1 n−1  γj 2 3 1 4 ˜ kj zj + θ + j + ln2 + a2n LVn ≤ − 2λj j 4 2 j=1

j=1

1 1 γn + εn4 + u4min − (1 − δ)kn z4n + θ˜n θˆn . 4 4 λn Furthermore, based on the inequality γn γn 2 γn 2 θ˜n θˆn ≤ − θ˜n + θ λn 2λn 2λn n the following result holds:   n n  γj 2 1 cj z4j + θ˜j + LVn ≤ − j + u4min 2λj 4 j=1

(56)

(57)

j=1

where cj = kj (1 ≤ j ≤ n − 1), cn = (1 − δ)kn , and j = (γj /2λj )θj2 + (1/2)a2j + (3/4)lj2 + (1/4)εj4 (1 ≤ j ≤ n). So far, by adopting backstepping technique, the fuzzy adaptive control design has been completed. The main result will be summarized by the following theorem. Theorem 2: Consider the strict-feedback stochastic nonlinear system (5) with Assumption 1, preceded by hysteresis quantizer (6). For bounded initial conditions, the control (53), together with the intermediate virtual control signals (40) and adaptive laws (41) guarantee the following. 1) All the signals in the closed-loop system remain bounded in probability.

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7

2) The variables zj converge to a compact set Z , which is defined by ⎧ ⎫ n ⎨  4d ⎬

Z = zj E |zj |4 ≤ . (58) ⎩ c ⎭ j=1

Proof: 1) Choose the stochastic Lyapunov function as V = Vn . Define c = min{4cj , γj , j = 1, 2, . . . , n} and d = n  + (1/4)u4min , from (57), it follows: j j=1 LV ≤ −cV + d, t ≥ 0.

(60)

where E(·) indicates an expectation operator. From (60), we can get 

d −ct d e + (61) 0 ≤ E[V(t)] ≤ V(0) − c c which means that E[V(t)] ≤

d , t → ∞. c

From (61) and (62), we can obtain ⎛ ⎞ n  4d , t → ∞. E⎝ z4j ⎠ ≤ 4E[V(t)] ≤ c

Fuzzy adaptive control design process.

(59)

From the definition of V and Lemma 1, zj and θ˜j are bounded in probability. Since θj is a constant, θˆj is also bounded. Since z1 = x1 − yd and yd is bounded, it can be seen that x1 is bounded in probability. Considering that α1 is the function of z1 and θˆ1 , therefore α1 is bounded in probability, thus x2 = z2 + α1 is also bounded in probability. Similarly, it can be seen that αj and xj ( j = 3, . . . , n) are bounded in probability. From (53), it can be seen that u is also bounded. Thus, all the signals in the closed-loop systems remain bounded in probability. 2) Furthermore, according to [42, Th. 4.1], the following inequality holds: dE(V(t)) ≤ −cE(V(t) + d, t ≥ 0 dt

Fig. 2.

(62)

(63)

j=1

Accordingly, the error signal zj eventually converges to the compact set Z defined in (58). Remark 8: From (63) and the definition of c and d, it can be seen that tracking error z1 depends on design parameters kj , γj , λj , aj , and lj and the quantized parameters δ and umin . By increasing kj , γj , meanwhile reducing λj , aj , lj , and δ, umin , the tracking error will diminish. Remark 9: Different from the research result in [15], even though the quantized parameter δ takes any value in (0, 1) without the need for any additional conditions, the proposed scheme can still guarantee the desired performance and the boundedness of the closed-loop system.

Finally, the following block diagram Fig. 2 can describe clearly the fuzzy control design process of this section.

IV. S IMULATION E XAMPLE In this section, a numerical example is presented to demonstrate the utility of the proposed quantized adaptive fuzzy scheme. Example: Consider the following second-order strictfeedback stochastic nonlinear system with a hysteretic quantized input: dx1 = (x2 + f1 (¯x1 ))dt + ψ1 (¯x1 )dw dx2 = (q(u) + f2 (¯x2 ))dt + ψ2 (¯x2 )dw y = x1

(64)

where x1 and x2 denote the state variables, y is the system output, and q(u) denotes hysteretic quantized input defined in (6). The control objective is to design a quantized adaptive fuzzy controller such that all the signals remain bounded in probability and the system output y follows the desired reference signal yd = sin(t) + sin(0.5t). As described in [4], there are many advantages in introducing networks into industrial control systems such as reduced weight, low cost, simple installation, and maintenance. Industrial applications include process control systems and automated manufacturing systems. In the networked industrial control systems, the devices are mutually connected via communication cables which are of limited capacity. First of all, the quantization effect issue should be taken into account. On the other hand, it is inevitable that the practical control engineering systems are subjected to the effects of stochastic disturbance. Therefore, the simulation example (63) is important not only to the development of control theory itself but also to the synthesis of practical networked industrial control systems. Remark 10: It should be pointed out that the quantized control schemes proposed in [14] and [15] are only suitable for a deterministic system. But (63) contains the stochastic disturbance terms ψ1 (¯x1 ) and ψ2 (¯x2 ), therefore the approaches proposed in [14] and [15] cannot control (63).

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Fig. 3.

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Tracking error under cases 1 and 2.

To construct the adaptive fuzzy controller, the fuzzy membership functions are chosen as follows:  

−(x + 2)2 −(x + 1.5)2 , μF2 (x) = exp μF1 (x) = exp i i 2 2  

2 −(x + 1) −(x + 0.5)2 μF3 (x) = exp , μF4 (x) = exp i i 2 2 

2

−x −(x − 0.5)2 , μF6 (x) = exp μF5 (x) = exp i i 2 2  

2 −(x − 1) −(x − 1.5)2 μF7 (x) = exp , μF8 (x) = exp i i 2 2 

−(x − 2)2 μF9 (x) = exp . i 2 By Theorem 2, the virtual control signal, the actual controller, and the adaptive laws are designed as 

3 1 z1 − 2 z31 θˆ1 ξ1T (X1 )ξ1 (X1 ) α1 = − k1 + 4 2a1 z3 θˆ2 ξ T ξ2 u = −k2 z2 − 22 2 2a2 (1 − δ) λ i θˆ˙i = 2 z6i ξiT (Xi )ξi (Xi ) − γi θˆi , i = 1, 2 2ai where z1 = x1 − yd , z2 = x2 − α1 , X1 = [x1 , yd , y˙ d ]T , and X2 = [¯x2T , θˆ1 , y¯ d(2)T ]T . In simulation, fi (¯xi ) and ψi (¯xi ) are chosen as f1 (¯x1 ) = (1 − sin2 (x1 ))x1 , ψ1 (¯x1 ) = 0.5 cos(x1 ), f2 (¯x2 ) = −3.5x2 + x1 x22 , and ψ2 (¯x2 ) = 0.1x1 sin(2x1 x2 ). To show the effects of the quantized parameters δ and μmin on the control performance, the quantized parameters and the design parameters are chosen as the following two cases. Case 1: δ = 0.7, μmin = 0.2, k1 = k2 = 10, a1 = 0.8, a2 = 1, λ1 = 20, λ2 = 25, γ1 = 1, and γ2 = 2. Case 2: δ = 0.15, μmin = 0.1, and the other design parameters are the same as the parameters in case 1. The initial conditions are all chosen as [x1 (0), x2 (0)]T = [0.05, −0.5]T , and [θˆ1 (0), θˆ2 (0)]T = [0.1, 0.2]T . The simulation results are shown in Figs. 3–8. The simulation results are shown in Figs. 3–8. Fig. 3 displays the tracking error under two cases. Figs. 4–6 show that the state variable x2 and the adaptive parameters θˆ1 and θˆ2 are all bounded under two cases. Figs. 7 and 8 show the control signal u and the quantized signal q(u).

Fig. 4.

State variable x2 under cases 1 and 2.

Fig. 5.

Adaptive parameter θˆ1 under cases 1 and 2.

Fig. 6.

Adaptive parameter θˆ2 under cases 1 and 2.

Fig. 7.

Control input u under cases 1 and 2.

On the other hand, we compare the control performances under the two cases by a quantitative evaluation. To this end, we define the index of the tracking

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Fig. 8.

Quantized signal q(u) under cases 1 and 2.

Fig. 10.

9

Adaptive parameters θˆ1 and θˆ2 under three random situations.

TABLE I P ERFORMANCE C OMPARISON U NDER THE T WO C ASES

Fig. 11. Control input u and the quantized signal q(u) under three random situations.

Fig. 9.

Tracking error y − yd and state x2 under three random situations.

M 2 error k=1 [y(k) − yd (k)] , define the control index as M as 2 k=1 [u(k)] , where M denotes the number of sampling data. The output and the control indexes are calculated from 0 to 20 s with a sampling period of 0.01 s. The control performance comparison results are shown in Table I. From Figs. 3 and 8 and Table I, it can be seen that, the smaller the quantized parameters δ and μmin are, the smaller the tracking error. However, the smaller δ implies the demand for bandwidth is more high. Therefore, in practice, a tradeoff between the control performance and the bandwidth should be made. To show the effectiveness of the proposed scheme under different uncertain conditions, for random variable with normal distribution dw ∼ N(0, dt), we make simulation under three random disturbance situations, where w indicates an independent standard Brownian motion. In the simulation, the same design parameters and the same initial conditions as the ones in case 2 are used. The simulation results are displayed by Figs. 9–11. From Figs. 9–11, it can be seen that the desired

Fig. 12.

Trajectory of the system output y.

control performance can be guaranteed even if the system is affected by random disturbances. To further explain that the restrictive condition for quantized parameter in [15] can be removed, we consider the control performance for ∀δ ∈ [0.1, 0.8] and μmin = 0.5. The design parameters are chosen as k1 = k2 = 8, a1 = 0.8, a2 = 1, λ1 = 20, λ2 = 25, γ1 = 1, and γ2 = 2. The simulation is run with the initial conditions [x1 (0), x2 (0)]T = [1, 0.5]T , and [θˆ1 (0), θˆ2 (0)]T = [0.1, 0.2]T . The simulation results are displayed in Figs. 12–15. Fig. 12 displays the trajectory of the system output y. It can be seen that, for ∀δ ∈ [0.1, 0.8], the system output can track the reference signal yd to a bounded compact set. Figs. 13 and 14 show that the state variables x2 and the adaptive parameters θˆ1 and θˆ2 are bounded. Fig. 15 shows the control signal u and the quantized input q(u). The simulation

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bound property of hysteretic type quantizer, a new nonlinear decomposition of the quantized input is introduced. Based on the new nonlinear decomposition, the design difficulty for the boundedness of disturbances-like term in traditional linear decomposition is dealt with, as a result, the restrictive conditions for quantized parameters and system uncertainties in [15] are removed. In addition, the norm of weight vector of the fuzzy logic system rather than the weight vector elements themselves is used as the estimated parameter, which significantly reduces the number of adaptive parameters. Finally, simulation results show the effectiveness of the main result. Fig. 13.

State variable x2 .

ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments and suggestions that have improved the presentation of this paper. R EFERENCES

Fig. 14.

Adaptive parameters θˆ1 and θˆ2 .

Fig. 15.

True control input u and quantized input q(u).

results show that although the system is subjected to random disturbances and fi (¯xi ) are not linear parametric, the system output still follows the given reference signal closely and the closed-loop signals remain bounded for ∀δ ∈ [0.1, 0.8]. In addition, the proposed control scheme contains only two adaptive parameters needed to be estimated online, which reduces the on-line computing burden obviously. V. C ONCLUSION Based on the backstepping technique, an adaptive fuzzy tracking scheme has been developed for a class of stochastic strict feedback nonlinear systems preceded by hysteretic quantized input. In this paper, nonlinear functions and stochastic disturbance terms are completely unknown. By employing fuzzy logic systems’ universal approximation property, the technical difficulty of the unknown nonlinear functions and the Itˆo stochastic differentiation is resolved. By using sector

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Zhi Liu received the B.S. degree from the Huazhong University of Science and Technology, Wuhan, China, the M.S. degree from Hunan University, Changsha, China, and the Ph.D. degree from Tsinghua University, Beijing, China, in 1997, 2000, and 2004, respectively, all in electrical engineering. He is currently a Professor with the Department of Automation, Guangdong University of Technology, Guangzhou, China. His current research interests include fuzzy logic systems, neural networks, robotics, and robust control.

Fang Wang received the B.S. degree from Qufu Normal University, Qufu, China, and the M.S. degree from Shandong Normal University, Jinan, China, in 1997 and 2004, respectively. She is currently pursuing the Ph.D. degree with the Department of Automation, Guangdong University of Technology, Guangzhou, China. Since 2005, she has been at the Shandong University of Science and Technology, Qingdao, China. Her current research interests include fuzzy control, neural network control, backstepping control, and adaptive control.

Yun Zhang received the B.S. and M.S. degrees from Hunan University, Changsha, China, in 1982 and 1986, respectively, and the Ph.D. degree from the South China University of Science and Technology, Guangzhou, China, in 1998, all in automatic engineering. He is currently a Professor with the Department of Automation, Guangdong University of Technology, Guangzhou. His current research interests include intelligent control systems, network systems, and signal processing.

C. L. Philip Chen (S’88–M’88–SM’94–F’07) received the M.S. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, and the Ph.D. degree in electrical engineering from Purdue University, West Lafayette, IN, USA, in 1985 and 1988, respectively. He was a Tenured Professor, a Department Head and an Associate Dean in two different universities in the U.S. for 23 years. He is currently the Dean of the Faculty of Science and Technology, and a Chair Professor with the Department of Computer and Information Science, University of Macau, Macau, China. His current research interests include systems, cybernetics, and computational intelligence. Dr. Chen has been an Editor-in-Chief of the IEEE T RANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS : S YSTEMS since 2014. He is currently an Associate Editor of several IEEE transactions. He is also the Chair of Technology Education in Computer Engineering 9.1 Economic and Business Systems of International Federation of Automatic Control. He is a fellow of the American Association for the Advancement of Science.

Fuzzy Adaptive Quantized Control for a Class of Stochastic Nonlinear Uncertain Systems.

In this paper, a fuzzy adaptive approach for stochastic strict-feedback nonlinear systems with quantized input signal is developed. Compared with the ...
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