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Neural-Based Adaptive Output-Feedback Control for a Class of Nonstrict-Feedback Stochastic Nonlinear Systems Huanqing Wang, Kefu Liu, Xiaoping Liu, Bing Chen, and Chong Lin

Abstract—In this paper, we consider the problem of observerbased adaptive neural output-feedback control for a class of stochastic nonlinear systems with nonstrict-feedback structure. To overcome the design difficulty from the nonstrict-feedback structure, a variable separation approach is introduced by using the monotonically increasing property of system bounding functions. On the basis of the state observer, and by combining the adaptive backstepping technique with radial basis function neural networks’ universal approximation capability, an adaptive neural output feedback control algorithm is presented. It is shown that the proposed controller can guarantee that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in the sense of mean quartic value. Simulation results are provided to show the effectiveness of the proposed control scheme. Index Terms—Adaptive neural output-feedback control, backstepping, nonstrict-feedback structure, stochastic nonlinear systems.

I. I NTRODUCTION URING the past decades, approximation-based adaptive neural or fuzzy backstepping control for nonlinear systems with low-triangular structure has been extensively investigated, and some interesting results have been reported in the literature, for example, see [1]–[13] and the references therein. The main idea of the adaptive neural or fuzzy control methodology is that neural networks or fuzzy-logic systems are applied to approximate the unknown nonlinearities in system dynamics, and the classical adaptive technique is used to construct controllers via backstepping. By combining adaptive neural control with backstepping technique, in [1]–[10], many

D

Manuscript received March 30, 2014; revised July 7, 2014 and September 28, 2014; accepted October 7, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61304002, Grant 61473160, Grant 61333012, and Grant 61304003, in part by the Education Department of Liaoning Province under the General Project Research under Grant L2013424, in part by the Program for New Century Excellent Talents in University under Grant NECT-13-0696, and in part by the Program for Liaoning Innovative Research Team in University under Grant LT2013023. This paper was recommended by Associate Editor D. Wang. H. Wang is with the School of Mathematics and Physics, Bohai University, Jinzhou 121000, China (e-mail: [email protected]). K. Liu and X. Liu are with the Faculty of Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada (e-mail: [email protected]; [email protected]). B. Chen and C. Lin are with the Institute of Complexity Science, Qingdao University, Qingdao 266071, China (e-mail: [email protected]; [email protected]). 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.2014.2363073

adaptive neural control schemes are proposed for single-input and single-output (SISO) nonlinear systems and multiinput and multioutput (MIMO) nonlinear systems. Alternatively, several adaptive fuzzy backstepping control strategies [12]–[14] were developed to deal with the control problem of uncertain nonlinear systems with strict-feedback form. All the aforementioned control approaches, however, require that the states of controlled nonlinear systems are measurable. As described in [15] and [16], in practice, state variables are often unmeasurable for many nonlinear systems, which makes the aforementioned state feedback control algorithms invalid. In such cases, observer-based output feedback control schemes should be applied. By designing a neural-based observer, in [17], an adaptive observer backstepping control scheme was proposed for a class of nonlinear systems with unmeasured states, which ensures the boundedness of all variables in the overall control system. Meanwhile, in [18], a backstepping-based fuzzy adaptive variable structure output feedback controller was first developed for a class of uncertain nonlinear systems. Afterwards, some elegant approximation-based output feedback control approaches were presented for different kinds of nonlinear systems [19]–[28]. It is well known that stochastic disturbance often exists in practical systems and is a source of instability of control systems. Therefore, the investigation on stochastic nonlinear systems is a challenging and meaningful issue, and has received much attention in the control community in recent years. In particular, many results on backstepping technique obtained for deterministic nonlinear strict-feedback systems have been successfully extended to stochastic nonlinear systems. For example, Pan and Basar [29] solved the stabilization problem for a class of stochastic nonlinear strict-feedback systems based on a risk-sensitive cost criterion and using the classic quadratic Lyapunov function. The proposed result guarantees global asymptotic stability of the closed-loop systems in the sense of probability. As an alternative, by introducing quartic Lyapunov function, Deng et al. [30]–[32] proposed a backstepping design scheme for stochastic strict-feedback systems and then the results were extended to inverse optimal control of the stochastic case. Afterwards, this design idea formed a main method to solve some control problems for other types of stochastic nonlinear systems [33]–[38]. On the other hand, approximation-based adaptive backstepping control methods have also been extended to control stochastic systems with uncertain nonlinearities. In [39]–[41], several

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adaptive neural or fuzzy control approaches were presented to deal with the state feedback control of stochastic nonlinear systems. When the states of the controlled stochastic system were unmeasurable, Tong et al. [42], [43], Chen et al. [44], Li et al. [45], and Zhou et al. [46] proposed some interesting approximation-based output feedback control schemes for some classes of uncertain stochastic nonlinear systems. Although a large amount of work has been carried out on the construction of backstepping-based adaptive controllers for uncertain nonlinear systems, the systems under consideration were assumed to be in low-triangular structure, that is, the control systems were in strict-feedback form [1]–[3], [12]–[14] or in pure-feedback form [5]–[8], [40]. To make the backstepping-based adaptive control method applicable for the control of nonlinear systems in a more general form, some efforts have been made. In [47] and [48], adaptive backstepping control strategies were proposed for a class of semi-strict-feedback nonlinear systems, in which the unknown nonlinear function fi (·) in ith subsystem was the function of whole state variables, but its bounding function was required to be a function of the current state variables only. In order to further relax the restriction on the controlled systems, Chen et al. [49], [50] developed two approximate-based adaptive backstepping control schemes for a class of deterministic nonlinear systems with nonstrict-feedback structure. More recently, the problem of approximation-based adaptive state-feedback control was considered in [51] and [52] for a class of nonstrict-feedback stochastic nonlinear systems. The aforementioned results in [47]–[52], however, required that the states of nonlinear systems were available for measurement. When the controlled stochastic nonlinear systems are in nonstrict-feedback structure and the state variables are unavailable for measurement, the control strategies in [47]–[52] could be invalid. Therefore, how to design an adaptive neural output feedback controller for nonstrict-feedback stochastic nonlinear systems is a more challenging and meaningful work. Motivated by the above observations, this paper considers the problem of adaptive neural control via output feedback for a class of stochastic nonlinear systems with nonstrict-feedback structure. An state observer is constructed to estimate the unmeasurable state variables, and a variable separation method is developed to overcome the difficulty of nonstrict-feedback structure based on the monotonously increasing property of the bounding functions. By combining the adaptive backstepping technique with radial basis function (RBF) neural networks’ universal approximation capability, an adaptive neural output feedback control scheme is proposed. The proposed adaptive neural output-feedback controller can guarantee that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) in the sense of mean quartic value. The main contributions of this paper are summarized as follows. 1) A backstepping-based adaptive neural output-feedback control methodology is systematically developed for a class of nonstrict-feedback stochastic nonlinear systems with unmeasurable states, especially, the nonlinear functions and their bounding functions in both drift and diffusion terms are all related to the whole state variables.

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2) Only one adaptive parameter is involved in the proposed controller no matter how large the order of the nonlinear stochastic systems is. As a result, the computational burden is significantly alleviated. Finally, simulation results are provided to show the effectiveness of the proposed control scheme. The rest of this paper is organized as follows. The problem formulation and preliminaries are given in Section II. Approximation-based adaptive neural output feedback control design is presented in Section III. Two simulation examples were given in Section IV, followed by Section V which concludes this paper. II. P RELIMINARIES AND P ROBLEM F ORMULATION In this section, the definition of stochastic stability, some useful preliminaries and neural network approximation are first presented in Section II-A, and then the problem formulation is presented in Section II-B. To develop an output feedback control strategy, a state observer is constructed in Section II-C. A. Preliminaries Before presenting the main results, let us first introduce some necessary definitions and lemmas, consider the following stochastic system: (1) dx = f (x)dt + h(x)dw, ∀x ∈ Rn where w is an r-dimensional independent 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; x ∈ Rn is the state, f : Rn → Rn and h : Rn → Rn×r are locally Lipschitz functions in x and satisfy f (0) = 0, h(0) = 0. Definition 1 [34]: For any given V(x) ∈ C2 , associated with the stochastic differential equation (1), define the differential operator L as follows: 1 ∂ 2V ∂V f + Tr hT 2 h (2) LV = ∂x 2 ∂x where Tr(A) is the trace of A. Remark 1: As shown in [34], the term 1/2Tr{hT (∂ 2 V/∂x2 )h} in (2) is called Itô correction term where the second-order differential ∂ 2 V/∂x2 makes the controller design much more difficult than that of the deterministic system. Definition 2 [42]: The trajectory {x(t), t ≥ 0} of the stochastic system (1) is said to be semi-globally uniformly ultimately bounded in the pth moment, if for some compact set ∈ Rn and any initial state x0 = x(t0 ), there exist a constant ε > 0 and a time constant T = T(ε, x0 ) such that E(|x(t)|p ) < ε for all t > t0 + T. Especially, when p = 2, it is usually called semi-globally uniformly ultimately bounded in mean square. Lemma 1 [42]: Suppose that there exists a C2 function V(x) : Rn → R+ , two constants c1 > 0 and c2 > 0, class K∞ −functions α¯ 1 and α¯ 2 such that α¯ 1 (|x|) ≤ V(x) ≤ α¯ 2 (|x|) LV(x) ≤ −c1 V(x) + c2 for all x ∈ Rn and t > t0 . Then, there is an unique strong solution of system (1) for each x0 ∈ Rn and it satisfies c2 E[V(x)] ≤ V(x0 )e−c1 t + , ∀t > t0 . c1

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Lemma 2 (Young’s inequality [30]): For ∀(x, y) ∈ R2 , the following inequality holds: xy ≤

εp p 1 |x| + q |y|q p qε

where ε > 0, p > 1, q > 1, and (p − 1)(q − 1) = 1. In the developed control design procedure, RBF neural networks fnn (Z) will be used to approximate any continuous function f (Z) : Rn → R. The RBF neural networks are described in the following form: fnn (Z) = W T S(Z)

(3)

is the input vector with q being where Z ∈ Z ⊂ the neural networks input dimension, weight vector W = [w1 , w2 , . . . , wl ]T ∈ Rl , l > 1 is the neural networks node number, and S(Z) = [s1 (Z), s2 (Z), . . . , sl (Z)]T means the basis function vector with si (Z) being chosen as the commonly used Gaussian function of the form (Z − μi )T (Z − μi ) , i = 1, 2, . . . , l (4) si (Z) = exp − η2 Rq

where μi = [μi1 , μi2 , . . . , μiq ]T is the center of the receptive field and η is the width of the Gaussian function. In [55], it has been indicated that with sufficiently large node number l, the RBF neural networks (3) can approximate any continuous function f (Z) over a compact set Z ⊂ Rq to arbitrary any accuracy ε > 0 as f (Z) = W ∗ T S(Z) + δ(Z), ∀z ∈ z ∈ Rq where

(5)

W∗

is the ideal constant weight vector and defined as ∗ T W := arg min sup f (Z) − W S(Z) W∈R¯ l

Z∈Z

and δ(Z) is the approximation error and satisfies |δ(Z)| ≤ ε. The boundedness of basis function vector S(Z) in (3) is shown in the following lemma, which will be used to prove the result of Lemma 4. Lemma 3 [56]: Consider the Gaussian RBF networks (3) and (4). Let ρ := 1/2 mini=j μi − μj , then an upper bound of S(Z) is taken as S(Z) ≤

∞

3q(k + 2)q−1 e−2ρ

2 k2 /η2

:= s.

(6)

k=0

It has been shown in [6] that the constant s in Lemma 3 is a limited value and is independent of the variable Z and the dimension of neural weights l. B. Problem Formulation In this paper, we consider a class of SISO stochastic nonlinear system as follows: ⎧

T ⎨ dxi = xi+1 + fi (x) dt + Tψi (x)dw, 1 ≤ i ≤ n − 1 (7) dx = u + fn (x) dt + ψn (x)dw ⎩ n y = x1 where x = [x1 , x2 , . . . , xn ]T ∈ Rn , u ∈ R, and y ∈ R are the state vector, system input, and system output, respectively, w is an independent r-dimensional standard Brownian motion

defined on the complete probability space (, F, {Ft }t≥0 , P) with being a sample space, F a σ −field, {Ft }t≥0 a filtration and P a probability measure. fi (.) : Rn → R and ψi (.) : Rn → Rr , (i = 1, 2, . . . , n) are smooth nonlinear functions with fi (0) = 0 and ψi (0) = 0, which may not be linearly parameterized. It can be seen from (7) that fi (.) and ψi (.) are the functions of the whole state variables, so the system (7) is in nonstrictfeedback structure, which is different from the strict-feedback stochastic systems [41]–[44], where the system functions fi (.) and ψi (.) are independent of state variables xj , ( j = i + 1, . . . , n). Apparently, the system (7) is also different from pure-feedback stochastic nonlinear systems [40]. Remark 2: Many physical processes such as hyperchaotic inductor-capacitor oscillation circuit system [53] and ball and beam system [54] are in nonstrict-feedback structure. Generally, the existence of stochastic disturbance is inevitable in many practical systems. Therefore, many controlled practical physical systems can be expressed as the system (7). The objective of this paper is to design an adaptive neural output-feedback controller such that all the signals in the closed-loop system remain bounded in the sense of the four-moment. To facilitate the control design in Section III, the following assumption will be used in the subsequent developments. Assumption 1: For nonlinear functions fi (x) and ψi (x) in (7), there exist positive constants hi and κi , respectively, such that i = 1, 2, . . . , n

fi (x) − fi xˆ ≤ hi x − xˆ , ψi (x) − ψi xˆ ≤ κi x − xˆ . Remark 3: By choosing xˆ = 0 in Assumption 1, it follows | fi (x)| ≤ hi x and ψi (x) ≤ κi x . This implies that the monotonically increasing functions ρi (s) = hi s and i (s) = κi s with s ∈ R are the bounding functions of fi (.) and ψi (.). C. Observer Design To develop an output-feedback control strategy, an observerbased controller is designed to guarantee all the signals in the closed-loop system to be SGUUB. First, we design the following observer to estimate the unmeasured states:

xˆ˙ i = xˆ i+1 + li y − xˆ 1 , 1 ≤ i ≤ n − 1

(8) x˙ˆ n = u + ln y − xˆ 1 where xˆ i is the estimation of xi , 1 ≤ i ≤ n. The design parameters li are chosen such that the polynomial p(s) = sn + l1 sn−1 + · · · + ln1 s + ln is Hurwitz. Define the observer error e = x − xˆ , where xˆ = [ˆx1 , . . . , xˆ n ]T , and from (7) and (8), the observer error equation is given as follows:

(9) de = A0 e + F(x) dt + (x)dw where F(x) = [ f1 (x), f2 (x), . . . , fn (x)]T , (x) T T T [ψ1 (x), ψ2 (x), . . . , ψn (x)]T ⎤ ⎡ −l1 ⎢ −l2 In−1 ⎥ ⎥ ⎢ A0 = ⎢ . ⎥ ⎦ ⎣ .. −ln 0 · · · 0

=

(10)



and A0 is a Hurwitz matrix. So for a given matrix Q > 0, there exists a matrix P > 0 such that AT0 P + PA0 = −Q.

(11)

III. A DAPTIVE N EURAL C ONTROL D ESIGN AND S TABILITY A NALYSIS In this section, an adaptive backstepping-based neural output-feedback control scheme will be proposed. As usual, in the backstepping approach, the following change of coordinates is made: zi = xˆ i − αi−1 , i = 1, 2, . . . , n

(12)

where α0 = 0, αi is the virtual control signal and will be constructed during the controller design process in the following form: 1 αi = −ki zi − 2 z3i θˆ SiT (Zi )Si (Zi ), 1 ≤ i ≤ n (13) 2ai where ki and ai are positive design parameters. θˆ is the estimation of an unknown constant θ , which will be defined later. The adaptive law will be defined as θ˙ˆ =

n λ 6 T ˆ z S (Zi )Si (Zi ) − γ θ, 2 i i 2a i i=1

i = 1, 2, . . . ., n (14)

where λ and γ are positive design parameters, and Si (Zi ) is the basis function vector with Zi = [x¯ˆ i , θˆ ]T ∈ Ri+1 (i = 1, 2, . . . , n). Note that, when i = n, αn is the actual control input signal u(t). Remark 4: It is apparent that (14) means that for any initial condition θˆ (t0 ) ≥ 0, the solution θˆ (t) ≥ 0 holds for t ≥ t0 . ˆ ≥ 0. Thus, throughout this paper, it is assumed that θ(t) Lemma 4: For the coordinate transformations zi = xˆ i − αi−1 , for i = 1, 2, . . . , n, the following result holds: n xˆ ≤ |zi | χi zi , θˆ (15) i=1

where χi (zi , θˆ ) = (ki + 1) + 1/2a2i s2 z2i θˆ , for i = 1, 2, . . . , n − 1, and χn (.) = 1. Proof: Let α0 = 0, by using Lemma 3, (12) and (13), one has n n n

xˆ i = xˆ ≤ |zi + αi−1 | ≤ |zi | + |αi−1 | ≤ ≤

i=1 n i=1 n

|zi | +

i=1 n−1 i=1

i=1

1 2 2 ˆ ki + 2 zi s θ |zi | 2ai

χi zi , θˆ |zi |.

i=1

The Lemma 4 provides the relationship between ˆx and error signals zi (i = 1, 2, . . . , n), which, together with Assumption 1, plays a key role to deal with the whole state functions fi (.) and ψi (.) during the backstepping design procedure. Now, we summarize our main result in the following theorem.

Theorem 1: Consider the closed-loop stochastic nonlinear system consisting of plant (7), the state observer (8), the controller (13), and adaptive law (14) under Assumption 1. Suppose that for 1 ≤ i ≤ n, the packaged functions f¯i (Zi ) can be approximated by neural network WiT Si (Zi ) in the sense that the approximation error δi is bounded, then for bounded initial conditions, all the signals in the closed-loop system are fourmoment semi-globally uniformly ultimately bounded, and the error signals ei , zi , and θ˜ eventually converge to the compact set s defined by n μ 2 E e4i ≤ 2 s = ei , zi , θ˜ bλmin (P) ρ i=1 n μ 2λμ 4 ˜ E zi ≤ 4 , θ ≤ ,1 ≤ i ≤ n (16) ρ ρ i=1

2 where ρ = min{(2λ0 )/(bλ nmax (P)), 4ki , γ , i = 1, . . . , n}, n 4 μ = 1/4 ε + 1/2 a2 + (γ /2λ)θ 2 , and λmin (P) i=1 i i=1 i will be given later. Proof: To present the stability analysis, consider the Lyapunov function candidate as

(17) V = Ve + Vz + Vθ n z4 , Vθ = 1/2λθ˜ 2 , b, where Ve = b/2(eT Pe)2 , Vz = 1/4 i=1 i and λ are positive constants, and θ˜ = θ − θˆ is a parameter error with θˆ being the estimation of unknown constant θ . By using (2), (8), and (9), we can obtain

LV = b eT Pe eT AT0 P + PA0 e + 2b eT Pe eT PF(x)

+ 2bTr T (x) 2PeeT P + eT PeP (x) + z31 (ˆx2 + l1 e1 ) ⎛ n i−1 ∂αi−1 xˆ j+1 + lj e1 z3i ⎝xˆ i+1 + li e1 − + ∂ xˆ j i=2 j=1 ⎞ ∂αi−1 ˙ ⎠ 1 ˙ − θˆ − θ˜ θˆ λ ∂ θˆ

≤ −bλ e 4 + 2b eT Pe eT PF(x)

+ 2bTr T (x) 2PeeT P + eT PeP (x) + z31 (z2 + α1 + l1 e1 ) ⎛ n i−1 ∂αi−1 z3i ⎝zi+1 + αi + li e1 − lj e1 + ∂ xˆ j i=2 j=1 ⎞ i−1 n ∂αi−1 ∂αi−1 ˙ ⎠ − xˆ j+1 − z3i θˆ ∂ xˆ j ∂ θˆ j=1

1 − θ˜ θ˙ˆ λ

i=2

(18)

where λ = λmin (P)λmin (Q). λmin (P) and λmin (Q) are the smallest eigenvalues of matrixes P and Q, respectively, zn+1 = 0, and αn denotes the actual control signal u(t). Furthermore, by taking Assumption 1, Remark 3 and Young’s inequality into account, the following two inequalities


can be obtained:

2b eT Pe eT PF(x)

= 2b eT Pe eT P F(x) − F xˆ + F xˆ

≤ 2b e 3 P 2 F(x) − F xˆ + 2b e 3 P 2 F xˆ n 2 8 3 43 1 4 2 ≤ bη0 P 3 e + 4 b hi e 4 2 2η0 i=1 n 2 4 4 8 3 3 1 xˆ + bη0 P 3 e 4 + 4 b h2i 2 2η0 i=1 n 2 4 8 1 ≤ 3bη03 P 3 e 4 + 4 b h2i e 4 2η0 i=1 n 2 4 1 xˆ + b h2i 4 2η0 i=1 1 bc0 e 4 2η04 n 1 3 4 ˆ z + bc n χ , θ z4i (19) 0 i i 2η04 i=1

2bTr T (x) 2PeeT P + eT PeP (x) √ ≤ 6bn n (x) 2 P 2 e 2 √ √ 3bn n ≤ 3bn nη12 P 4 e 4 + (x) 4 η12 √

√ 2 24bn n 4 4 (x) − xˆ 4 ≤ 3bn nη1 P e + 2 η1 √

4 24bn n xˆ + η12 2 √ n √ 2 24bn n 2 4 4 ≤ 3bn nη1 P e + κi e 4 η12 i=1 2 √ n 24bn n 4 + κi2 xˆ η12 i=1 √ √ 2 24bn n 4 ≤ 3bn nη1 P e 4 + c1 e 4 η12 √ n 24bn n 3 4 + c n χi (zi , θˆ )z4i (20) 1 η12 i=1 4

5

n ˆ θ˙ˆ in (18), by using z3 (∂αi−1 /∂ θ) Then, for the term − i=2 i adaptive law in (14) and rearranging terms in the summation, we have −

n i=2

z3i

∂αi−1 ˙ θˆ ∂ θˆ

⎞ ⎛ n ∂α λ i−1 ⎝ z3i z6 ST S − γ θˆ ⎠ =− 2 j j j ˆ 2a ∂ θ j i=2 j=1 ⎛ ⎞ n i−1 ∂αi−1 λ 6 T ⎠ ∂αi−1 = z3i ⎝ zj Sj Sj γ θˆ − 2a2j ∂ θˆ ∂ θˆ n

i=2

−

i=2

8

3 34 4 1 2 4 l z1 + β1 e1 (21) 4 4β12 1 ⎛ ⎞ ⎛ ⎞4 3 i−1 i−1 ∂αi−1 ⎠ ∂αi−1 ⎠ 3 4⎝ lj e1 ≤ z − l l z3i ⎝li − i j i ∂ xˆ j ∂ xˆ j 4βi2 j=1 j=1 z31 l1 e1 ≤

+ where βi is a positive constant.

1 2 4 β e , i = 2, . . . , n 4 i 1

(22)

j=1

z3i ⎛

∂αi−1 ∂ θˆ

n j=i

λ 6 T zj Sj Sj 2a2j

⎞ i−1 ∂α λ ∂α i−1 i−1 ≤ z3i ⎝ z6 S T S ⎠ γ θˆ − 2 j j j ˆ ˆ 2a ∂ θ ∂ θ j i=2 j=1 i n λ ∂αj−1 6 T 3 . + (23) zi Si Si z j 2a2i ∂ θˆ n

≤ 3bη03 P 3 e 4 +

# # where c0 = ( ni=1 h2i )2 and c1 = ( ni=1 κi2 )2 . With the help of Young’s inequality, one has

n

i=2

j=2

Substituting the inequalities (19)–(23) into (18) results in $

4 √ 8 LV ≤ − bλ − 3bη03 P 3 − 3bn nη12 P 4 √ 1 24bn n − bc0 − c1 e 4 2η04 η12 n 1 2 4 3 43 3 + βi e1 + z1 z2 + α1 + 2 l1 z1 4 4β1 i=1 ⎧ ⎛ ⎞4 3 ⎪ n i−1 ⎨ ∂α 3 i−1 3 ⎝ ⎠ + zi zi+1 + αi + 2 zi li − lj ⎪ ∂ xˆ j 4βi ⎩

i=2

j=1

−

i−1 ∂αi−1 j=1

−

∂ xˆ j

xˆ j+1 +

∂αi−1 γ θˆ ∂ θˆ

i−1 ∂αi−1 λ 6 T zj Sj Sj ∂ θˆ j=1 2a2j

⎫ i λ 3 T 3 ∂αj−1 ⎬ + 2 zi Si Si z j 2ai ∂ θˆ ⎭ j=2 √ n 1 24bn n 1 3 4 ˆ z4i − θ˜ θ˙ˆ + z bc + c χ , θ n 0 1 i i 4 2 λ 2η0 η1 i=1 4 √ 8 1 ≤ − bλ − 3bη03 P 3 − 3bn nη12 P 4 − 4 bc0 2η0 √ n 24bn n 1 2 − c1 − βi e 4 2 4 η1 +

n i=1

3 z3i αi + f¯i (Zi ) − 4

i=1 n i=1

z4i −

1 ˙ θ˜ θˆ λ

(24)



where the result z3i zi+1 ≤ (3/4)z4i + (1/4)z4i+1 has been used in the above equation, and the functions f¯i (Zi ) are defined as 3 43 3 l1 z 1 + z 1 f¯1 (Z1 ) = 2 2 4β1 √ 1 24bn n 3 4 ˆ z1 (25) z + bc + c χ , θ n 0 1 1 1 2η04 η12 ⎛ ⎞4 3 i−1 ∂α 3 i−1 ¯fi (Zi ) = ⎝ ⎠ z i li − lj ∂ xˆ j 4βi2 j=1

Next, by constructing virtual control signal αi and actual control signal u in (13) with i = n, respectively, we have 1 6 T zi θˆ Si Si 2a2i 1 z3n u ≤ −kn z4n − 2 z6n θˆ SnT Sn . 2an

z3i αi ≤ −ki z4i −

√ n 1 24bn n 1 2 − bc0 − c1 − βi e 4 4 2η04 η12 i=1

∂αi−1 7 ∂αi−1 xˆ j+1 + γ θˆ zi − ˆ 4 ∂ xˆ j ∂ θ j=1

i−1 ∂αi−1 λ 6 T zj Sj Sj ∂ θˆ j=1 2a2j i 3 ∂αj−1 λ z + 2 z3i SiT Si j 2ai ∂ θˆ j=2 √ 1 24bn n 3 4 ˆ + z bc + c χ , θ zi n 0 1 i i 2η04 η12 i = 2, . . . , n − 1 ⎛ ⎞4 3 n−1 ∂α 3 n−1 ⎠ ¯fn (Zn ) = ⎝ z n ln − lj + zn ∂ xˆ j 4βn2

1 4 1 2 εi + ai 4 2 i=1 i=1 i=1 n λ 1 ˙ + θ˜ z6 ST S − θˆ . 2 i i i λ 2a i i=1

−

−

−

j=1

−

n

ki z4i +

i=1

n−1 ∂αn−1 λ 6 T − zj Sj Sj ∂ θˆ j=1 2a2j n 3 ∂αj−1 λ z + 2 z3n SnT Sn j 2an ∂ θˆ j=2 √ 1 24bn n 3 4 ˆ + z bc + c χ , θ zn . (26) n 0 1 n n 2η04 η12

Then, neural network WiT Si (Zi ) is used to compensate for the nonlinear function f¯i (Zi ) such that for any given εi > 0 f¯i (Zi ) = WiT Si (Zi ) + δi (Zi )

n

(30)

1 4 1 2 γ εi + ai + θ˜ θˆ . 4 2 λ n

n

i=1

i=1

(31)

By using (γ /λ)θ˜ θˆ ≤ −(γ /2λ)θ˜ 2 + (γ /2λ)θ 2 , we have 4 √ 8 LV ≤ − bλ − 3bη03 P 3 − 3bn nη12 P 4 √ n 1 24bn n 1 2 − bc0 − c1 − βi e 4 4 2η04 η12 i=1 −

n

γ 2 1 4 θ˜ + − εi 2λ 4 n

ki z4i

i=1 n

i=1

1 2 γ 2 + ai + θ . 2 2λ

(32)

i=1

where |δi (Zi )| ≤ εi and δi denotes the approximation error. Applying Young’s inequality results in z3i f¯i (Zi ) = z3i

where θ = max{ Wi 2 , i = 1, 2, . . . , n}.

n

ki z4i +

√ n 1 24bn n 1 2 − bc0 − c1 − βi e 4 4 2η04 η12 i=1

∂αn−1 ∂αn−1 xˆ j+1 + γ θˆ ∂ xˆ j ∂ θˆ

WiT Si Wi + z3i δi (Zi ) Wi 1 1 3 1 ≤ 2 z6i Wi 2 SiT Si + a2i + z4i + εi4 2 4 4 2ai 1 1 3 1 ≤ 2 z6i θ SiT Si + a2i + z4i + εi4 2 4 4 2ai

n

Now, taking the adaptive law θ˙ˆ in (14) into account results in 4 √ 8 LV ≤ − bλ − 3bη03 P 3 − 3bn nη12 P 4

j=1

n−1

(29)

Furthermore, by combining (24) with the equations from (27) to (29), we can obtain 4 √ 8 LV ≤ − bλ − 3bη03 P 3 − 3bn nη12 P 4

i−1

+

(28)

Choose the parameters η0 , η1 , and βi (i = 1, 2, . . . , n) such that 4 √ 8 λ0 = bλ − 3bη03 P 3 − 3bn nη12 P 4 √ n 1 24bn n 1 2 − bc − c − βi > 0 (33) 0 1 4 2η04 η12 i=1 then the following result holds: LV ≤ −λ0 e 4 −

(27)

n

ki z4i −

i=1

+

γ 2 θ˜ 2λ

1 4 1 2 γ 2 εi + ai + θ . (34) 4 2 2λ n

n

i=1

i=1


Define

2λ0 , 4ki , γ , i = 1, . . . , n ρ = min bλ2max (P) n n γ 2 1 4 1 2 θ εi + ai + μ= 4 2 2λ

7

i=1

(35) (36)

i=1

where λmax (P) denotes the maximal eigenvalue of matrix P. Next, we can further rewrite (34) as LV ≤ −ρV + μ.

(37)

Furthermore, based on (37), one has dE[V(t)] ≤ −ρE[V(t)] + μ dt

(38)

then, (38) satisfies

$ ) μ −ρt μ 0 ≤ E[V(t)] ≤ V(0) − e + ρ ρ

(39) Fig. 1.

which implies that 0 ≤ E[V(t)] ≤ V(0) +

μ , ∀t > 0. ρ

(40)

Therefore, based on the inequality (40) and the definition of V, all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in the sense of the four-moment, and θ˜ is semi-globally uniformly ultimately bounded in mean square. Next, it can be shown from (39) and Lemma 1 that 0 ≤ E[V(t)] ≤ V(0)e−ρt +

μ , ∀t > 0 ρ

(41)

which means that E[V(t)] ≤

μ , t → ∞. ρ

(42)

Thus, based on (17), all the error signals ei , zi , and θ˜ eventually converge to the compact set s specified in (16). The proof is thus completed. Remark 5: Note that several results on nonstrictfeedback systems via backstepping have been obtained in [49], [51], and [52]. The main differences between this result and the ones in [49], [51], and [52] are summarized as follows. 1) The state variables of the controlled systems in [49], [51], and [52] are assumed available for measurement, and in this paper, the system states are unmeasurable. 2) θ = max{ Wi 2 , i = 1, 2, . . . , n} is used as the estimated parameter in this paper, so only one adaptive law needs to be updated online for nth order systems, which is more suitable for higher order nonlinear systems. The design procedure of the controller can be visualized from the block diagram shown in Fig. 1. IV. S IMULATION E XAMPLES In this section, two simulation examples are used to illustrate the effectiveness of the approach proposed in this paper.

Block diagram of control system.

Example 1: Consider the following second-order stochastic nonlinear system:

⎧ 2 + 0.5x2 cos(x1 )dw ⎨ dx1 = x2 + x1 x2 dt 2 2 (43) dx = u + x1 sin x2 dt + x2 e−x1 dw ⎩ 2 y = x1 where x1 and x2 denote the state variables and u is the control input. It is apparent that this system is in nonstrictfeedback form because of the existence of the terms x1 x22 and 0.5x2 cos(x1 ) in the first subsystem. In addition, the system (43) can also be seen as a pure-feedback system by choosing f1 = x2 + x1 x22 . However, since ∂f1 /∂x2 > 0 cannot be guaranteed for all x2 . Therefore, the existing control schemes for pure-feedback systems are not suitable. For the system (43), the observer is design as

x˙ˆ 1 = xˆ 2 + l1 y − xˆ 1

(44) x˙ˆ 2 = u + l2 y − xˆ 1 . According to Theorem 1, the virtual control input signal and the actual control input are defined as 1 3 T z1 θˆ S1 (Z1 )S1 (Z1 ) 2a21 1 u = −k2 z2 − 2 z32 θˆ S2T (Z2 )S2 (Z2 ) 2a2

αi = −k1 z1 −

with the adaption laws being defined as λ ˆ θ˙ˆ = z6 ST (Zi )Si (Zi ) − γ θ. 2 i i 2a i i=1 2

The simulation is run under the initial conditions x(0) = [0.3, 0.3]T , xˆ (0) = [0.1, 0.1]T , and θˆ (0) = 0. In the simulation, the design parameters are taken as follows: l1 = l2 = 5, k1 = k2 = 5, a1 = a2 = 2, γ = 1, and λ = 2. In addition, the RBF neural networks are chosen in the following way. Neural network W1T S1 (Z1 ) contains 49 nodes



Fig. 2.

State variables x1 and xˆ 1 of Example 1.

Fig. 3.


with centers spaced evenly in the interval [ − 5, 5] × [ − 5, 5] with widths being equal to two. Neural network W2T S2 (Z2 ) contains 343 nodes with centers spaced evenly in the interval [ − 5, 5] × [ − 5, 5] × [ − 5, 5] with widths being equal to two. The simulation results are shown by Figs. 2–5. Fig. 2 shows the state x1 and its estimation xˆ 1 . Fig. 3 shows the state variable x2 and its estimation xˆ 2 . From Figs. 2 and 3, it can be seen that the trajectories of the estimation states xˆ 1 and xˆ 2 follow the actual states x1 and x2 well, respectively; and they converge to zero fast. Fig. 4 shows the control input signal u. Fig. 5 displays the adaptive parameter θˆ . Apparently, the simulation results show that good convergence performances are achieved and all the signals in the closed-loop system are bounded. Example 2: To show the applicability of the proposed control scheme, Consider a one-link manipulator with the inclusion of motor dynamics and stochastic disturbances [44]. The dynamic equation of the system is described by D¨q + B˙q + N sin(q) = τ + τd (45) Mm τ˙ + Hm τ = u − Km q˙ where q, q˙ , and q¨ are the link position, velocity, and acceleration, respectively, τ denotes the torque produced by the ˙ ψ2 (q, q˙ , τ ) = q2 cos(˙qτ ), w is motor, τd = ψ2 (q, q˙ , τ )w, the torque stochastic disturbance defined in the system (7), and u is the voltage applied to the motor or control input. D = 1 kg m2 denotes the mechanical inertia, B = 1 Nm s/rad is the coefficient of viscous friction at the joint, N = 10 Nm

Fig. 4.

Actual control input signal u of Example 1.

Fig. 5.

Adaptive parameter θˆ of Example 1.

represents a positive constant that is related to the mass of the load and the coefficient of gravity, Mm = 0.1H is the armature inductance, Hm = 1.0 is the armature resistance, and Km = 0.2 Nm/A is the back electromotive force coefficient. Let x1 = q, x2 = q˙ , and x3 = τ , then the nonlinear dynamics in (45) can be given by ⎧ dx1 = x2 dt ⎪ ⎪ ⎨ dx2 = (−x2 + x3 − 10 sin x1 )dt + ψ2 (q, q˙ , τ )dw (46) dx3 = (−2x2 − 10x3 + 10u)dt ⎪ ⎪ ⎩ y = x1 . For the system (46), the observer is defined as that in the form (8), and the controller u, and adaptive law θ˙ˆ are constructed in (13) with i = 3, and (14), respectively. The simulation is carried out with the initial condition [x1 (0), x2 (0), x3 (0)]T = [0.2, 0.2, 0.2]T , xˆ (0) = [0.1, 0.1, 0.1]T , θˆ (0) = 0, and the design parameters l1 = l2 = l3 = 6, k1 = k2 = 2, k3 = 20 a1 = a2 = a3 = 1, γ = 1, and λ = 2. Moreover, the RBF neural networks W1T S1 (Z1 ) and W2T S2 (Z2 ) are chosen the same as Example 1, and W3T S3 (Z3 ) contains 34 nodes with centers spaced evenly in the interval [−3, 3] × [−3, 3] × [−3, 3] × [−3, 3] with widths being equal to two. Figs. 6–10 show the simulation results. Once again, the simulation results show that the control can quickly stabilize the system. Remark 6: It should be mentioned that though several approximation-based adaptive output-feedback control


9

Fig. 6.


Fig. 9.

Fig. 7.


Fig. 10.

Actual control input signal u of Example 2.

Adaptive parameter θˆ of Example 2.

one adaptive parameter is required for an n-order nonlinear systems. Therefore, the control schemes in [42]–[44] cannot be directly applied to Examples 1 and 2, which shows the advantages of the proposed approach. V. C ONCLUSION

Fig. 8.


schemes have been developed in [42]–[44] for stochastic nonlinear systems. The main differences between our result and the ones in [42]–[44] lie in that: 1) from the controlled systems, the results are obtained in [42]–[44] for strict-feedback stochastic nonlinear systems, i.e., the nonlinear functions of the ith subsystem are independent of the latter state variables xj (i + 1 ≤ j ≤ n), and in this paper, we considered a class of nonstrict-feedback stochastic nonlinear systems, which is in a more general form and 2) the control schemes in [42]–[44] require to estimate each element of the neural weight vectors, and in this paper, we only need to estimate the maximum of the norms of neural weight vectors which means that only

In this research, an adaptive neural output feedback control algorithm has been proposed for a class of nonstrict-feedback stochastic nonlinear systems with unmeasurable states. The proposed control approach guarantees that all the signals in the closed-loop system remain semi-globally uniformly ultimately bounded in the sense of the four moment, and the observer errors and the system output converge to a small neighborhood around the origin. The main contributions of this paper are twofold: the proposed control scheme provides a method to control a class of nonstrict-feedback stochastic nonlinear systems with unmeasurable states, and the developed control law only requires one adaptive parameter for an n-order nonlinear system. Simulation results have been provided to show the effectiveness of our results. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments and suggestions, which helped to improve the presentation of this paper greatly.



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Huanqing Wang received the B.Sc. degree in mathematics from Bohai University, Jinzhou, China, the M.Sc. degree in mathematics from Inner Mongolia University, Huhhot, China, and the Ph.D. degree from the Institute of Complexity Science, Qingdao University, Qingdao, China, in 2003, 2006, and 2013, respectively. He is currently an Associate Professor with the School of Mathematics and Physics, Bohai University. His current research interests include nonlinear control, adaptive fuzzy control, and stochastic nonlinear systems.

Kefu Liu received the B.Eng. and M.Sc. degrees in mechanical engineering from the Central South University of Technology, Changsha, China, in 1981 and 1984, respectively, and the Ph.D. degree in mechanical engineering from the Technical University of Nova Scotia, Halifax, NS, Canada, in 1992. He joined Lakehead University, Thunder Bay, ON, Canada, in 1998, where he was a Full Professor with the Department of Mechanical Engineering. He was an Assistant Professor with St. Mary’s University, San Antonio, TX, USA, from 1993 to 1995, and with Dalhousie University, Halifax, NS, Canada, from 1995 to 1998. His current research interests include vibration control, control of nonlinear systems, and mechatronics. Dr. Liu is a member of Professional Engineers Ontario.

11

Xiaoping Liu received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the Northeastern University, Shenyang, China, in 1984, 1987, and 1989, respectively. He was with the School of Information Science and Engineering, Northeastern University, for over ten years. In 2001, he joined the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON, Canada. His current research interests include nonlinear control systems, singular systems, and robust control. Prof. Liu is a member of the Professional Engineers of Ontario.

Bing Chen received the B.A. degree in mathematics from Liaoning University, Shenyang, China, the M.A. degree in mathematics from the Harbin Institute of Technology, Harbin, China, and the Ph.D. degree in electrical engineering from Northeastern University, Shenyang, China, in 1982, 1991, and 1998, respectively. He is currently a Professor with the Institute of Complexity Science, Qingdao University, Qingdao, China. His current research interests include nonlinear control systems, robust control, and adaptive fuzzy control.

Chong Lin received the B.Sc. and M.Sc. degrees in applied mathematics from Northeastern University, Shenyang, China, in 1989 and 1992, respectively, and the Ph.D. degree in electrical and electronic engineering from Nanyang Technological University, Singapore, in 1999. He was a Research Associate with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, in 1999. From 2000 to 2006, he was a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. Since 2006, he has been a Professor with the Institute of Complexity Science, Qingdao University, Qingdao, China. He has published over 60 research papers and co-authored two monographs. His current research interests include systems analysis and control, robust control, and fuzzy control.