Adaptive neural output feedback control for stochastic nonlinear time-delay systems with input and output quantization

Communicated by Bo Shen

Accepted Manuscript

Adaptive Neural Output Feedback Control For Stochastic Nonlinear Time-Delay Systems With Input And Output Quantization Yekai Yang, Zhaoxu Yu, Shugang Li, Jitao Sun PII: DOI: Reference:

S0925-2312(17)31856-8 10.1016/j.neucom.2017.12.023 NEUCOM 19157

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

15 September 2017 24 November 2017 11 December 2017

Please cite this article as: Yekai Yang, Zhaoxu Yu, Shugang Li, Jitao Sun, Adaptive Neural Output Feedback Control For Stochastic Nonlinear Time-Delay Systems With Input And Output Quantization, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.12.023

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Adaptive Neural Output Feedback Control For Stochastic Nonlinear Time-Delay Systems With Input And Output Quantization Yekai Yang1 , Zhaoxu Yu1 , Shugang Li2 , Jitao Sun3

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1. Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, Department of Automation, East China University of Science and Technology, Shanghai China 200237; 2. Department of Information Management, Shanghai University, Shanghai, China,200444; 3. Institute for Intelligent Systems, University of Johannesburg, South Africa.

Abstract

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The problem of output feedback adaptive tracking control is studied for a class of stochastic nonlinear time-delay systems in which the measured output and input signals are quantized by two sector-bounded quantizers respectively. An observer including the quantized input and output signals is designed to estimate the unknown system states, and the unknown system functions with less restrictions are dealt with by using the neural network(NN)’s approximation. By combining the backstepping technique and the Lyapunov-Krasovskii method, an observer-based adaptive neural quantized tracking control scheme is presented for this class of systems. The stability analysis indicates that the tracking error can converge to a small neighborhood of the origin while all closed-loop signals are 4-moment(or 2moment) semi-globally uniformly ultimately bounded (SGUUB). Finally, two illustrative examples are provided to demonstrate the feasibility and effectiveness of the proposed design methodology.

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Keywords: Stochastic nonlinear systems, quantization, output feedback, time delay, neural network 1. Introduction With the wide application of networked control, quantization is often useful and inevitable in a lot of practical control systems [1]-[2]. The distinct Preprint submitted to Neurocomputing

December 14, 2017

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advantages of utilizing quantization schemes come from sufficient precision and low communication rate [3], such that quantized feedback control has become a focal topic in the control community. Usually, many quantized feedback control strategies (e.g., see in [4]-[10]) are presented in this situation that only the input signal between the plant and the controller is quantized. However, a variety of practical remotely controlled systems always include both the measurement quantization and input signal quantization [11]. Thus it is of greater practical significance to study the quantized feedback control for nonlinear systems with both input and output quantization. The stability problem has been firstly investigated for a class of linear systems with the quantized input and output in [12]. Thereafter, the study was extended to global output feedback stabilization for a class of nonlinear systems with quantized input and output in [13]-[15]. However, stochastic disturbance and time delay are not involved in the aforementioned results on input and output quantization. On the other hand, stochastic disturbance exists in the various practical systems, such that stability analysis and control synthesis of stochastic nonlinear systems have received much more attention in the past two decades (e.g., see in [16]-[31]). Especially, for a variety of non-networked stochastic nonlinear systems, many interesting adaptive control schemes have been proposed in [17]-[26]. However, these control schemes without involving quantization can not be applicable to network-based stochastic nonlinear systems. In order to solve the quantized feedback control problems for stochastic nonlinear systems, a few works are reported in [27]-[31]. In [27], an adaptive fuzzy quantized control strategy was developed for a class of stochastic strict-feedback nonlinear systems, and a new nonlinear decomposition of the Hysteretic quantized input was presented. Based on the decomposition of quantized input in [27], some other adaptive quantized control strategies have been proposed for various stochastic nonlinear systems in [28]-[31]. Nevertheless, the aforementioned research results in [27]-[31] are limited to the delay-free system form. Time delay occurs frequently in a variety of actual systems,such as network control system, chemical process, aircraft control system, and so on. The existence of time delay usually gives rise to the degradation of system performance, even the instability. Therefore, there has been a sustainable, deep interest in the stability analysis and control synthesis of time-delay systems (e.g., see in [32]-[47]) in the past decades. In recent years, it is inspiring that there have reported some new works on network-based non2

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linear time-delay systems with including signal quantization effects. In [48], Lyapunov-Krasowskii functionals are employed to develop quantized control laws for a class of nonlinear continuous-time systems with constant input delay. Moreover, [49] is concerned with an adaptive quantized tracking problem for switched nonstrict-feedback nonlinear systems in the presence of discrete and distributed time-varying delays. Taking into account the effect of stochastic disturbance and sector-bounded input quantization, the adaptive quantized tracking control issue was solved for a class of switched stochastic strict-feedback nonlinear systems with unknown time-varying delays in [50]. It is worthy to point out that the preceding results in [48]-[50] are available based on input quantization only and state-feedback control. Besides, for a class of discrete nonlinear stochastic time-delay systems where the measured output and the input signals are quantized respectively by two logarithmic quantizers, the quantized H∞ control problem is studied in [51]. Unfortunately, to the best of our knowledge, no more attention is paid to the output feedback adaptive tracking control design for stochastic nonlinear time-delay systems in the presence of input and output quantization. Inspired by the above observations, this paper will be concerned with exploring the problem of output feedback adaptive neural tracking control for a more general class of stochastic nonlinear time-delay systems with nonstrictfeedback structure and quantized input and output. A state observer is designed to estimate the unmeasured state variables. The proposed adaptive neural output feedback controller can guarantee that all signals in the closed-loop system are 4-moment (or 2-moment) SGUUB and the tracking error is convergent to a neighborhood of the origin. In contrast to the existing results, the main contributions of this paper are highlighted as follows: (1) It is the first time to address the output feedback adaptive quantized tracking control problem for a more general class of stochastic nonstrictfeedback nonlinear systems with time-varying delays, which can cover many popular classes of stochastic nonlinear time-delay systems (e.g., see in [35], [38]-[43]). Moreover, an observer-based adaptive neural quantized control scheme is proposed to acquire the satisfactory tracking performance for such systems with less conservative assumption. (2) Sector-bounded quantized input and output are included in the networkbased stochastic nonlinear time-delay system. Some special techniques are presented to deal with the sector-bounded quantization errors. (3) The whole control scheme contains only one adaptive parameter to be updated, which facilitates its online computation and practical implementa3

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tion. The rest of the paper is structured as follows. Some preliminary knowledge and problem statement are introduced in Section 2. Output feedback adaptive neural control design and stability analysis of the closed-loop system are addressed in Section 3. Two illustrative examples are provided in Section 4. Finally, Section 5 draws a conclusion of this paper. 2. Preliminary knowledge and problem statements

Section 2.1 gives some preliminaries, and system model and control objective are formulated in Section 2.2.

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2.1. Preliminaries Consider the following stochastic systems.

dx = f (x)dt + g T (x)dω,

(1)

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where x ∈ Rn denotes the state, ω is r-dimensional standard Wiener process, and f (·) : Rn → Rn and g(·) : Rn → Rn are vector-value or matrix-value functions with appropriate dimensions. Definition 1 [52]. For any given V (x) ∈ C 2 , infinitesimal generator L is defined as ∂V 1 ∂ 2V ∂V + f + T r{g 2 g T }, (2) LV (x) = ∂t ∂x 2 ∂x where T r(A) is the trace of A. 2 Remark 1. In general, the term 21 T r{g ∂∂xV2 g T } is called the Itˆo correction 2 term, and the second-order differential ∂∂xV2 in this term will make the design of controller much more complicated than that of the deterministic systems. Lemma 1 [28]. For the stochastic system (1), if there exists a C 2 function V (x, t), two positive constants a and b, class-K∞ functions α1 and α2 such that  α1 (kxk) ≤ V (x, t) ≤ α2 (kxk) LV ≤ −aV (x, t) + b, where ∀x ∈ Rn and ∀t > 0. Then, there exists an unique strong solution of system (1) for each x0 ∈ Rn satisfying b E [V (x)] ≤ V (x0 )e−at + , a 4

∀t > t0 .

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Furthermore, the states of the system (1) is bounded in probability. Lemma 2 (Young’s Inequality)[52]. For ∀ (x, y) ∈ R2 , the following inequality is true. 1 εp p |x| + q |y|q , xy ≤ p qε where ε > 0, p > 1, q > 1 and (p − 1)(q − 1) = 1. Lemma 3 [53]. For any η ∈ R and ε > 0, the following inequality holds. η 0 ≤ |η| − η tanh( ) ≤ 0.2785ε. ε

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Lemma 4 [37]. Consider the dynamic system of the form ˆ˙ = −λθ(t) ˆ + τ χ(t), θ(t)

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where λ and τ are positive constants and χ(t) is a positive function. For ˆ 0 ) ≥ 0, then θ(t) ˆ ≥ 0 ∀t > t0 and any given bounded initial condition θ(t holds. In this paper, Radial basis function (RBF) neural network is exploited to approximate any unknown continuous nonlinear function. For an unknown continuous function f (Z) : Rn → R over a compact set Z ∈ ΩZ ⊂ Rn , T there exists a RBF NN W ∗ Φ(Z) with arbitrary accuracy satisfy T

f (Z) = W ∗ Φ(Z) + δ, ∀Z ∈ ΩZ .

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where W ∗ is the ideal constant weight vector and is defined as W ∗ := arg min { sup f (Z) − W T Φ(Z) },

(3)

ˆ ∈Rn Z∈ΩZ W

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where W = [w1 , . . . , wl ]T ∈ Rl is denoted as the weight vector, and the kernel vector is defined as Φ(Z) = [s1 (Z), s2 (Z), . . . sl (Z)]T with active func2 ik tion si (Z) being chosen as the Gaussian function si (Z) = exp[ −kZ−µ ], i = ηi T 1, 2, . . . , l, where µi = [µi1 , . . . , µin ] is the center of the receptive field, ηi is the width of the Gaussian function, and l is the number of NN nodes. Assumption 1[27, 39]. For ∀Z ∈ ΩN N , there exists an ideal constant weight vector W ∗ such that kW ∗ k∞ ≤ wmax and |δ| ≤ δmax with bounds wmax > 0 and δmax > 0.

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From (3), it is easy to obtain W ∗T Φ(Z) + δ ≤ W ∗T Φ(Z) + |δ|

where β(Z) =

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|sm (Z)|wmax + δmax ≤ θβ(Z),

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s2m (Z) + 1) and θ = max{δmax , wmax }.

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Remark 2. In the light of (4), an upper bound of the NN parameterization can be obtained. Hence, for a Gaussian RBF NN, only one positive parameter θ instead of the NN weight vector W ∗T needs to be updated during the subsequent control design.

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2.2. System model and control objective Consider a class of stochastic nonlinear time-delay systems as follows  T   dxi = (xi+1 + fi (x, x(t − di (t))))dt + gi (x, x(t − di (t)))dω,   i = 1, . . . , n − 1  dxn = (q1 (u) + fn (x, x(t − dn (t))))dt + gnT (x, x(t − dn (t)))dω,   y = x1 ,    x(t) = φ(t), −d ≤ t ≤ 0,

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(5) where xi ∈ R (i = 1, . . . , n), u ∈ R, y ∈ R are system state, control input and output, respectively; and x := [x1 , . . . , xn ]T ; fi : Rn × Rn → R and gi : Rn × Rn → Rr are unknown smooth functions, and satisfy fi (0, 0) = 0 and giT (0, 0) = 0 for i = 1, 2, . . . , n; di (t) : R+ → [0, d] are uncertain timevarying delays which satisfy d˙i (t) ≤ di < 1, i = 1, 2, . . . , n; the initial function φ(t) is smooth and bounded. ω is defined as in (1). This paper assumes that the states of the system (5) are unknown and only the output y is available for measurement. The sector-bounded quantizer is a quantizer with its quantization error satisfying the following condition [6] |q(u) − u| ≤ δ|u| + (1 − δ)d,

where 0 < δ < 1 and d are known parameters of quantizer. Remark 3. As [7] has pointed out, most popular quantizers including hysteresis quantizer [4], [7] and logarithmic quantizer [3], [7] can be brought 6

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into a special case of the sector-bounded quantizer. The detailed models of logarithmic quantizer and hysteresis quantizer can be referred to [3], [4] and [7]. For system (5), there are two quantizers, input quantizer q1 (·) and output quantizer q2 (·), to be utilized. Furthermore, the input quantization error satisfies the following sector-bounded condition |q1 (u) − u| ≤ δ1 |u| + (1 − δ1 )d1 ,

and the output quantization error also satisfies the sector-bounded condition as follows

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|q2 (y) − y| ≤ δ2 |y| + (1 − δ2 )d2 ,

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where δ1 and d1 are quantization parameters of input quantizer; δ2 and d2 are quantization parameters of output quantizer. The control objective of this paper is to design an output feedback adaptive NN controller for system (5) such that the system output y can track a reference signal yd while all closed-loop error signals are 4-moment (or 2-moment) SGUUB. Using Lemma 2.1 in [54], the following inequalities can be derived. (6)

|gi (x, x(t − di (t)))| ≤ Gi1 (x) + Gi2 (x(t − di (t))),

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|fi (x, x(t − di (t)))| ≤ Fi1 (x) + Fi2 (x(t − di (t))),

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where Fij (·) and Gij (·) are nonnegative smooth functions satisfying Fij (0) = 0 and Gij (0) = 0 with i = 1, . . . , n and j = 1, 2. Remark 4. It is worth pointing out some following statements on system (5). (i) The unknown system functions fi (·) and gi (·) are evidently related to the whole state variables x1 , . . . , xn , thus system (5) can be regard as a nonlinear system in a nonstrict-feedback form, which can be exploited to represent many real applications, such as the mechanical movement system in [41], Brusselator model [22, 43] and the one-link manipulator model in [17, 26, 35]. (ii) In the existing results (e.g., see in [35] and [38]-[43]), some restrictive assumptions are imposed on the system functions. However, with the aid of the above processing (6) and (7), the time-delay terms are effectively

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separated from the unknown nonlinear function. And then the conservative assumption on the system functions fi and gi can be removed. (iii) Unlike the results about input and output quantization in ([13]-[15]), this paper concentrates on studying the tracking control problem for networkbased nonstrict-feedback nonlinear system with stochastic disturbance and unknown time-varying delays. As a result, the designed control scheme will be applied in much more applications. Based on the above discussion, it is significant to explore an effective output feedback adaptive quantized control scheme for such network-based systems. For the ease of the control design, the following common assumption on the reference signal yd is introduced. Assumption 2. The desired trajectory yd (t) and its n-th order derivatives are continuous and bounded. Specifically, it is assumed there exist a positive constant M to satisfy |yd (t)| ≤ M . In order to reconstruct the unknown states, the observer can be designed as  xˆ˙ i = xˆi+1 + ki (q2 (y) − xˆ1 ), i = 1, · · · , n − 1 (8) xˆ˙ n = q1 (u) + kn (q2 (y) − xˆ1 ),

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where xˆi denotes the estimation of the states xi , and ki s are constant design parameters to be specified. The initial condition is given by xˆi (0) = xˆi0 . Define the observer error as x˜ = x − xˆ with xˆ = [ˆ x1 , . . . , xˆn ]T . Combining (5) and (8), the observer error dynamics of x˜ is rewritten as the following compact form

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d˜ x = (A˜ x + F − K(q2 (y) − y))dt + Gdω,

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where F = [f1 , . . . , fn ]T , G = [g1T , . . . , gnT ]T , K = [k1 , . . . , kn ]T , and   −k1 In−1   A =  ... . −kn 0 · · · 0

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It is necessary to point out that positive constants k1 , . . . , kn need to be properly chosen such that A is asymptotically stable. Then there must exist symmetric positive definite matrices P and Q to satisfy AT P + P A = −Q.

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3. Main results

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z1 = xˆ1 − yd , zi = xˆi − αi−1 − yd

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To begin the backstepping design procedure, an unknown positive constant is firstly defined as ϑ = max{θi , i = 0, 1, . . . , n} with θi being given later. ϑˆ represents the estimate of ϑ, and the estimate error is expressed as ϑ˜ = ϑˆ − ϑ. The adaptive backstepping control design is based on the following coordinate transformation: i = 2, . . . , n,

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where αi−1 is the virtual control law to be specified below. Additionally, it is requisite to point out that the backstepping design procedure for stochastic nonlinear time-delay systems consisting of (9) and (8) will include n + 1 design steps. At step i(i = 0, 1, . . . , n), an unknown function Ψi (·) will be approximated by employing a RBF NN Wi∗T Si (Zi ) with Z0 = x, Z1 = (i) (i) ˆ T , Zi = [x ¯ˆi , y¯(i) , ϑ] ˆ T ,x ¯ˆi = [ˆ [ˆ x1 , ϑ] x1 , . . . , xˆi ]T , and y¯d = [yd , y˙ d , . . . , yd ]T for d i = 2, . . . , n. For stochastic nonlinear time-delay system (5), a feasible adaptive neural control scheme is constructed as follows: 3

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ˆ i (Zi ) tanh( zi βi (Zi ) ), αi = −ci zi − ϑβ ai

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n X z 3 βi (Zi ) ˙ ˆ ˆ ϑ = λ( zi3 βi (Zi ) tanh( i ) − γ ϑ), a i i=1

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ˆ n (Zn ) tanh( zn βn (Zn ) ), u = −cn zn − ϑβ an

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(13)

s2mi (Zi ) + 1); ci , ai , λ and γ are positive design

constants for i = 1, · · · n. Moreover, the initial condition of ϑˆ is set to meet ˆ ≥ 0 , which can ensure that ϑ(t) ˆ ≥ 0 holds for all t > 0. ϑ(0) Remark 5. A common weakness in adaptive NN control or adaptive fuzzy control is that the number of parameters to be updated will greatly increase as the number of nodes in the neural network or the number of fuzzy 9

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rules increases, which gives rise to a dramatical increase of online learning time (e.g., see in [17],[22], [23] and [43]). Therefore, for the sake of alleviating the online computing burden, the maximum value of the NN parameters instead of the NN parameters is estimated such that the overall control strategy consisting of (11)-(13) contains only one adaptive parameter. The following n + 1 steps depict the whole process of adaptive outputfeedback control design. xT P x˜)2 . Along the trajectory of (9), one has Step 0. Define V0 = 2 (˜

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LV0 = (˜ xT P x˜)˜ xT (AT P + P A)˜ x + 2(˜ xT P x˜)˜ xT P (F − K(q2 (y) − y)) +2T r{GT (2P x˜x˜T P + x˜T P x˜P )G}. (14)

Then, by using Young’s inequality, one can obtain the following inequalities. 8 3 43 µd  xk4 + 4 e−µd kF k4 σ0 e 3 kP k 3 k˜ 2 2σ0 4 8 µd 3 3 xk 4 ≤ σ0 e 3 kP k 3 k˜ 2 n 4n X 4 4 + 4 µd (F (x) + Fj2 (x(t − dj (t)))), σ0 e j=1 j1

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2(˜ xT P x˜)˜ xT P F ≤

2(˜ xT P x˜)˜ xT P K(δ2 |y| + (1 − δ2 )d2 ) 2kP k2 kKkk˜ xk3 (δ2 |y| + η2 ) 2kP k2 kKkk˜ xk3 (δ2 (|yd | + |z1 | + |˜ x1 |) + η2 ) 2 3 2kP k kKkk˜ xk δ2 (|yd | + |z1 | + k˜ xk) 2 3 +2kP k kKkk˜ xk η 2 4 4 4 8 3 (σ13 + σ23 + σ33 )kP k 3 k˜ xk4 ≤ 2  +2δ2 kP k2 kKkk˜ xk4 + 4 kKk4 δ24 z14 2σ1   4 4 4 + 4 kKk δ2 M + 4 kKk4 η24 , (16) 2σ2 2σ3 √ 2T r{GT (2P x˜x˜T P + x˜T P x˜P )G} ≤ 3n nσ4 eµd kP k4 k˜ xk 4 √ n 24n2 n X 4 (G (x) + G4j2 (x(t − dj (t)))), (17) + σ4 eµd j=1 j1

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2(˜ xT P x˜)˜ xT P K(q2 (y) − y) ≤ ≤ ≤ ≤

(15)

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where , µ and σi (i = 0, 1, 2, 3, 4) are positive design parameters, and η2 = (1 − δ2 )d2 . In order to overcome the difficulty arising from the time-delay terms, a continuous function Ψ0 (x) to be determined is utilized. According to the RBF NN approximation property, Ψ0 (x) can be approximated by a RBF NN W0 ∗T S0 (x) on the compact set Ωx , such that Ψ0 (x) = W0∗T S0 (x)+δ0 (x), ∀x ∈ Ωx , where W0 ∗T S0 (x) represents the ’ideal’ neural network approximation of Ψ0 (x) andsδ0 (x) denotes the approximation error. From (4), and noting that l0 P s2m (x) + 1) ≤ l0 + 1 and θ00 = max{w0 max , δ0 max } β0 (x) = (l0 + 1)( m=1

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with w0 max and δ0 max being the bound of kW0∗ k∞ and |δ0 (x)| respectively (wi max and δi max are similarly defined as w0 max and δ0 max at step i later), we have Ψ0 (x) ≤ θ00 β0 ≤ (l0 + 1)θ00 := θ0 ≤ ϑ.

(18)

From (18), substituting (14)-(17) into (13) yields

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n  4 4n X 4 4 4 ≤ Λk˜ xk + 4 δ2 kKk z1 − [Ψ0 (x) − 4 µd F (x) 2σ1 σ0 e j=1 j1 √ n n 4n X 4 24n2 n X 4 − G (x)] + 4 µd F (x(t − dj (t)))) σ2 eµd j=1 j1 σ0 e j=1 j2 √ n 24n2 n X 4 G (x(t − dj (t))) + $0 , + σ2 eµd j=1 j2 4

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4 3 8 4 µd P where Λ = −λmin (P )λmin (Q) + 23 (σ03 e 3 + σi3 )kP k 3 +2δ2 kP k2 kKk i=1 √ 4 4  4 + 3n nσ4 kP k and $0 = ϑ + 2σ4 δ2 kKk M 4 + 2σ 4 δ24 kKk4 η24 . 2 3 Step 1. The dynamic system of z1 is given by

dz1 = (ˆ x2 + k1 (q2 (y) − y) + k1 x˜1 − y˙ d )dt.

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Choose the Lyapunov function candidate V1 = V0 + 41 z14 + 12 λ−1 ϑ˜2 . From (2), one has ˆ˙ LV1 = LV0 + z13 (z2 + α1 + k1 (q2 (y) − y) + k1 x˜1 ) + λ−1 ϑ˜ϑ. 11

(21)

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By using the Young’s inequality, one has 3 43 4 1 z13 z2 ≤ ρ11 z1 + 4 z24 , 4 4ρ11

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z13 k1 (q2 (y) − y) ≤ |z1 |3 k1 (δ2 |y| + (1 − δ2 )d2 ) ≤ |z1 |3 k1 (δ2 (|z1 | + |yd | + |˜ x1 |) + η2 ) 4 4 4 3 3 1 3 ≤ k1 δ2 z14 + (ρ12 + ρ13 )(k1 δ2 ) 3 z14 + 4 M 4 4 4ρ12 4 4 3 1 1 3 xk4 + ρ14 k13 z14 + 4 η24 , + 4 k˜ 4ρ13 4 4ρ14 3 43 43 4 1 k1 z1 + 4 k˜ z13 k1 x˜1 ≤ ρ15 xk 4 , 4 4ρ15

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where ρ1j (j = 1, 2, 3, 4, 5) are positive design constants. Lump some functions as a new function Ψ1 (Z1 ), i.e.,

4 4 3 43 3 34 3 ρ11 z1 + k1 δz1 + (ρ12 + ρ13 )(k1 δ) 3 z1 4 4 3 34 43 3 43 43  + ρ14 k1 z1 + ρ15 k1 z1 + 4 δ 4 kKk4 z1 . 4 4 2σ1

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A RBF NN W1∗T S1 (z1 ) is used to compensate for the unknown function Ψ1 (Z1 ) on the compact set ΩZ1 . Similar to (4), we have Ψ1 (Z1 ) = W1∗T S1 (z1 ) + δ1 ≤ W1∗T S1 (z1 ) + |δ1 |

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where θ1 = max{w1 max , δ1 max }. Thus, using Lemma 3 gives z 3 β1 (Z1 ) z13 Ψ1 (Z1 ) ≤ z13 θ1 β1 (Z1 ) ≤ z13 ϑβ1 (Z1 ) tanh 1 + 0.2785a1 ϑ, a1

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1 1 1 + 4 ]k˜ xk4 − c1 z14 + 4 z24 + $1 4 4ρ13 4ρ15 4ρ11 12

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˜ ϑˆ˙ − λz 3 β1 (Z1 ) tanh z1 β1 (Z1 ) ) +λ−1 ϑ( 1 a1 n X 4n + 4 µd η 4 F 4 (x(t − dj (t)))) σ0 e j=1 j j2 √ n n 24n2 n X 4 4n X 4 + F (x) G (x(t − dj (t))) − [Ψ0 (x) − 4 µd σ4 eµd j=1 j2 σ0 e j=1 j1 √ n 24n2 n X 4 − G (x)], (27) σ2 eµd j=1 j1

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where $1 = $0 + 0.2785a1 ϑ + 4ρ14 M 4 + 4ρ14 η24 . 12 14 Step i (2 ≤ i ≤ n − 1). The dynamic of zi can be obtained as follows (i)

dzi = (ˆ xi+1 + ki (q2 (y) − y) + ki x˜1 − Lαi−1 − yd )dt, where Lαi−1 =

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Consider the Lyapunov function candidate Vi = Vi−1 + 14 zi4 . Similar to the inequalities (22)-(24), we have

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1 4 3 4 zi3 zi+1 ≤ ρi13 zi4 + 4 zi+1 , 4 4ρi1

AC

CE

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zi3 ki (q2 (y) − y) ≤ ki |zi |3 (δ2 (|z1 | + |yd | + |˜ x1 |) + η2 ) 3 z1 z ≤ ki δ2 z1 zi3 tanh( i ) + 0.2785ki εi δ2 εi 4 4 3 43 1 + (ρi2 + ρi33 )(ki δ2 ) 3 zi4 + 4 M 4 4 4ρi2 4 4 1 3 1 + 4 k˜ xk4 + ρi43 ki3 zi4 + 4 η24 , 4ρi3 4 4ρi4

(29)

3 4 4 1 zi3 ki x˜i ≤ ρi53 ki3 zi4 + 4 k˜ xk 4 , 4 4ρi5

where ρij (j = 1, 2, 3, 4, 5) and εi are positive design constants.

13

(30)

(31)

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Define an unknown function as 4 4 3 4 3 4 4 3 43 z1 z 3 ρi1 zi + ki δ2 z1 tanh( i ) + (ρi23 + ρi33 )(ki δ2 ) 3 zi + ρi43 ki3 zi 4 εi 4 4 i i X ∂αi−1 X ∂αi−1 (j) 1 3 4 4 xˆ˙ j − + ρi53 ki3 zi + 4 zi − y − ∆i (Zi ) (j−1) d 4 4ρi−1,1 ∂ xˆj j=2 j=2 ∂yd

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Ψi (Zi ) =

with continuous function ∆i (Zi ) being specified later. A RBF NN Wi∗T Si (Zi ) is used to compensate for the unknown function Ψi (Zi ) on the compact set ΩZi . Namely, Ψi (Zi ) = Wi∗T Si (Zi ) + δi , ∀Zi ∈ ΩZi . Thus, exploiting a similar way to (25) and (26) gives

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z 3 βi (Zi ) zi3 Ψi (Zi ) ≤ zi3 θi βi (Zi ) ≤ zi3 ϑβi (Zi ) tanh i + 0.2785ai ϑ. ai

(32)

where θi = max{wi max , δi max }. Combining (29)-(32) and (11), we have i X

i

X 1 1 4 1 4 + )]k˜ x k − cj zj4 + 4 zi+1 + $i 4 4 4ρ 4ρ 4ρ j3 j5 i1 j=1 j=1 √ n n X 4n 24n2 n X 4 4 −[Ψ0 (x) − 4 µd F (x) − G (x)] σ0 e j=1 j1 σ2 eµd j=1 j1 +

i X

(zj3 ∆j (Zj ) − 0.2785κj − zj3

PT

j=2

−λ

M

(

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LVi ≤ [Λ +

i X j=1

zj3 βj (Zj ) tanh

∂αj−1 ˆ˙ ˜ ϑˆ˙ ϑ) + λ−1 ϑ( ˆ ∂ϑ

n zj3 βj (Zj ) 4n X 4 F (x(t − dj (t)))) ) + 4 µd aj σ0 e j=1 j2

CE

√ n 24n2 n X 4 + G (x(t − dj (t))), σ2 eµd j=1 j2

(33)

AC

where $i = $i−1 + 0.2785(ai ϑ + κi ) + 4ρ14 M 4 + 0.2785ki εi δ2 + 4ρ14 η24 with κi i2 i4 being a positive design constant. Step n. The dynamic of zn can be obtained as follows (n)

dzn = (q1 (u) + kn (q2 (y) − y) + kn x˜1 − Lαn−1 − yd )dt, where Lαn−1 is defined similarly as Lαi at the step i. 14

(34)

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Choose the Lyapunov function candidate as Vn = Vn−1 + 14 zn4 . Then, using the property of the sector-bounded quantization, the following inequality can be derived

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zn3 q1 (u) − zn3 u ≤ |zn3 ||q1 (u) − u| ≤ |zn3 |(δ1 |u| + (1 − δ1 )d1 ).

Taking the actual control law (12) into consideration and applying Lemmas 2 and 3, we can obtain

PT

ED

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zn3 q1 (u) ≤ zn3 u + |zn3 |(δ1 |u| + (1 − δ1 )d1 ) 3 ˆ n (Zn ) tanh( zn βn (Zn ) )) ≤ zn3 (−cn zn − ϑβ an 3 ˆ n (Zn )| tanh( zn βn (Zn ) )|) + |z 3 |(1 − δ1 )d1 +δ1 |zn3 |(cn |zn | + ϑβ n an 3 ˆ n (Zn ) tanh( zn βn (Zn ) )) ≤ zn3 (−cn zn − ϑβ an 3 ˆ n (Zn )| tanh( zn βn (Zn ) )| + |z 3 |η1 +δ1 cn |zn4 | + δ1 |zn3 |ϑβ n an 3 ˆ n (Zn ) tanh( zn βn (Zn ) )) ≤ zn3 (−cn zn − ϑβ an 3 ˆ n (Zn ) tanh( zn βn (Zn ) ) +δ1 cn zn4 + δ1 zn3 ϑβ an 3 z +η1 zn3 tanh n + 0.2785σn η1 . (35) σn Similar to (30) and (31), we have

AC

CE

zn3 kn (q2 (y) − y) ≤ kn |zn |3 (δ2 (|z1 | + |yd | + |˜ x1 |) + η2 ) 3 z1 z ≤ kn δ2 z1 zn3 tanh( n ) + 0.2785kn εn δ2 εn 4 4 4 3 3 1 3 + (ρn2 + ρn3 )(kn δ2 ) 3 zn4 + 4 M 4 4 4ρn2 4 4 1 3 1 3 + 4 k˜ xk4 + ρn4 kn3 zn4 + 4 η24 , 4ρn3 4 4ρn4 3 43 43 4 1 zn3 kn x˜n ≤ ρn5 kn zn + 4 k˜ xk 4 , 4 4ρn5 15

(36)

(37)

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where ρnj (j = 2, 3, 4, 5), εn are positive design constants and η1 = (1 − δ1 )d1 . A continuous function is define as zn3 βn (Zn ) z3 3 43 43 kn zn ) + η1 zn3 tanh n + ρn4 an σn 4 4 4 z1 zn 3 43 3 34 43 3 +kn δ2 z1 tanh( ) + (ρn2 + ρn3 )(kn δ2 ) 3 zn + ρn5 kn zn εn 4 4 n n X X ∂αn−1 ˙ ∂αn−1 (j) 1 (n) zn − xˆj − + 4 yd − yd − ∆n (Zn ), (j−1) 4ρn−1,1 ∂ xˆj j=2 j=2 ∂yd

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ˆ n (Zn ) tanh( Ψn (Zn ) = δ1 cn zn + δ1 ϑβ

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where the function ∆n (Zn ) will be specified later. Obviously, Ψn (Zn ) can be approximated by using a RBF NN Wn∗T Sn (Zn ) on the compact set ΩZn . Similar to (32), it is easy to obtain that z 3 βn zn3 Ψn (Zn ) ≤ zn3 θn βn (Zn ) ≤ zn3 ϑβn tanh( n ) + 0.2785ϑan , an

(38)

n

X 1 1 xk 4 − cj zj4 + $n ≤ [Λ + ( 4 + 4 )]k˜ 4ρ 4ρ j3 j5 j=1 j=1 √ n n 4n X 4 24n2 n X 4 −[Ψ0 (x) − 4 µd F (x) − G (x)] σ0 e j=1 j1 σ2 eµd j=1 j1

ED

LVn

n X

M

where θn = max{wn max , δn max }. Combining (12) and (33)-(38) gives

PT

n X

CE

+

AC

λ

j=2 n X j=1

(zj3 ∆j (Zj ) − 0.2785κj − zj3

zj3 βj (Zj ) tanh

∂αj−1 ˆ˙ ˜ ϑˆ˙ − ϑ) + λ−1 ϑ( ∂ ϑˆ

n zj3 βj (Zj ) 4n X 4 ) + 4 µd F (x(t − τj (t)))) aj σ0 e j=1 j2

√ n 24n2 n X 4 + G (x(t − τj (t))), σ2 eµd j=1 j2

(39)

where $n = $n−1 + 0.2785(an ϑ + κn ) + 4ρ14 M 4 + 0.2785kn εn δ2 + 4ρ14 η24 + n2 n4 0.2785σn η1 with κn being a positive design constant. At this stage, the following theorem is presented to summarize the main result of this paper. 16

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n X j=1

Z

1 1 − dj

eµs Γj (x(s))ds,

t−dj (t)

2 √n

where Γj (x(s)) = 4n F 4 (x(s)) + 24nσ2 σ04 j2 time-delay terms. For 2 ≤ i ≤ n, specify ∆i (Zi ) as

(40)

G4j2 (x(s)) will be used to cancel the

z 3 Θi ∂αi−1 ˆ ϑ − Θi tanh( i ) + κi ∂ ϑˆ

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∆i (Zi ) : = −λγ

t

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Vd = e

−µt

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Theorem 1. Consider the uncertain stochastic nonlinear time-delay systems (5) with quantized input and output, the whole control scheme including the observer (8), the virtue control signals (11), the actual control law (12) and the adaptive law (13) are used. Then, for any bounded initial condition, all signals in the closed-loop system are 4-moment (or 2-moment) SGUUB. In particular, the system output y can satisfactorily follow the given reference signal yd by adjusting properly the design parameters ci , ai , γ and λ. Proof. Firstly, the Lyapunov-Krasovskii functional is chosen as V = Vn + Vd with Vd being defined as follows:

(41)

ED

i−1 X zj3 βj (Zj ) ∂αi−1 3 λ zj βj (Zj ) tanh( ) ˆ a i ∂ ϑ j=1

with Θi := λβi (Zi )

i P 3 ∂αj−1 zj ∂ ϑˆ , then we have

CE

PT

j=2

n X j=2

(zj3 ∆j (Zj ) − 0.2785κj − zj3

∂αj−1 ˆ˙ ϑ) ≤ 0. ∂ ϑˆ

(42)

AC

Moreover, designate Ψ0 (x) in (39) as √ n n 4n X 4 24n2 n X 4 Ψ0 (x) := 4 µd F (x) + G (x) σ0 e j=1 j1 σ2 eµd j=1 j1 +

n X j=1

√ 1 4n 4 24n2 n 4 [ F (x) + Gj2 (x)]. 1 − dj σ04 j2 σ2 17

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Given c0 > 0, choose proper parameters σi (i = 0, 1, 2), , ρj3 , and ρj5 , so that the following inequality is true n X

(

j=1

1 1 + 4 ) ≤ −c0 . 4 4ρj3 4ρj5

Then, combining (13) and (39)-(44) yields LV (t) ≤ −aV (t) + b,

(44)

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Λ+

(45)

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where a = min{2c0 (λ2max (P ))−1 , 4c1 , 4c2 , . . . , 4cn , γλ} and b = $n + γ2 ϑ2 . From the definition of V and Lemma 1, the close-loop error signals x˜i , zi (i = 1, . . . , n) and ϑ˜ are 4-moment (or 2-moment) SGUUB. Furthermore, using a similar argument to [33] and [36] gives

b b (46) 0 ≤ E[V (t)] ≤ (V (0) − )e−at + , a a where E[·] indicates an expectation operator. In the light of (46) and the definition of V , the following inequality is easily derived

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n X 4b E[ zj4 ] ≤ 4E[V (t)] ≤ , a j=1

ED

which means that

E[|z1 |] ≤

PT

Similarly, we also obtain

CE

E[λmin (P )˜ x21 ]

4

4b , a

2

E[|˜ x1 |] ≤

s

T

1 λmin (P )

(47)

t → ∞.

≤ E[λmin (P )k˜ xk ] ≤ E[˜ x P x˜] ≤

which implies that

AC

r

t → ∞,

r

2b , a

r

(48)

2b , a

t → ∞,

t → ∞.

(49)

(50)

By combining (48) and (50), the following inequality holds E[|y − yd |] ≤ E[|y − xˆ1 |] + E[|ˆ x1 − yd |] s r r 1 2b 4 4b ≤ + := ζ, λmin (P ) a a 18

t → ∞.

(51)

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System Eq.(5)

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From (51), the design parameters ki , ci , ai , λ and γ should be chosen appropriately in order to acquire theoretically a smaller bound ζ, i.e., a smaller tracking error. Remark 6. According to (46), the bounds of error signals zi , x˜i and ϑ˜ are greatly dependent on the values of a and b. Furthermore, it is follow from (51) that the bound ζ of tracking error |y − yd | is dependent on the main design parameters ki , ai , ci , λ and γ. Therefore, if these design parameters are properly selected so that the bound ζ is as small as possible, then the smaller tracking error will be theoretically obtained. In order to achieve this, some suggestions are given as follows: (i) Increasing ci and λ causes a to increase, which helps to diminish ζ; (ii) Decreasing ai and γ helps to decrease b, which also can result in a smaller bound ζ. y

Output quantizer

q2(y)

Observer Eq. (8)

Reference signal yd

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[ xˆ1, xˆ2 ,!, xˆn ]T Controller

ED

q1(u)

Step 1

D1

Input quantizer

Step i (i=2, ..., n)

AC

CE

PT

Di

Step n

[ z1, z2 ,!, zn ]T

u

Adaptive law Eq. (13)

Fig.1. The block diagram description of closed-loop system.

Remark 7. Based on the preceding design procedure and stability analysis, a block diagram of the whole control system is shown in Fig.1, and the basic steps of control design are summed up as follows: 19

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Step 1). Select a set of constant parameters k1 , k2 , . . . , kn , such that matrix A is strictly Hurwitz. Furthermore, choose a proper symmetric positive definite matrix Q and solve the equation AT P + P A = −Q to obtain a symmetric positive definite matrix P . Step 2). Select appropriate initial conditions, ie, xˆ(0) = [ˆ x1 (0), . . . , xˆn (0)]T , ˆ ≥ 0, and suitable design parameters ci > 0, ai > 0, λ > 0, and y(0), and ϑ(0) γ > 0 for i = 1, 2, . . . , n, then calculate successively the intermediate variables zi (i = 1, 2, . . . , n) and the virtual control functions αi (i = 1, 2, . . . , n−1) according to (10) and (11). Step 3). Compute the actual control signal u by (12) and the quantized input signal q1 (u) by some sector-bounded quantization scheme, respectively. Next, construct the observer dynamics by using the quantized output signal q2 (y) and the equation (8), and update the adaptive parameter ϑˆ by (13). Then return to Step 2). 4. Simulation examples

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In this section, two examples are given to illustrate the effectiveness and feasibility of the proposed control scheme. Example 1. Consider the following stochastic nonlinear time-delay system:  x31 (t) 2   dx1 = (x2 + x1 (t − d1 (t)) + 1+x21 (t) )dt + x1 (t)x1 (t − d1 (t))dω, x22 (t−d2 (t)) 2 2 (52) dx = (q (u) + x x + 0.2x (t − d (t)))dt + dω, 2 1 1 2 2 2 1+x22 (t)   y = x1 ,

AC

CE

where d1 (t) = 1 + 0.5 cos(t) and d2 (t) = 1 + 0.5 sin(t). The objective is to design an adaptive neural controller such that the output signal y tracks the desired reference signal yd = 0.2 sin(2t)+0.1 cos(t). Based on the proposed control approach in Section 3, the observer is designed as  xˆ˙ 1 = xˆ2 + k1 (q2 (y) − xˆ1 ), xˆ˙ 2 = q1 (u) + k2 (q2 (y) − xˆ1 ).

The virtual control law, the actual control law, and the adaptive laws are respectively designated as 3

ˆ 1 (Z1 ) tanh( z1 β1 (Z1 ) ) α1 = −c1 z1 − ϑβ a1 20

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3

ˆ 2 (Z2 ) tanh( z2 β2 (Z2 ) ) u = −c2 z2 − ϑβ a2

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2 X z 3 βi (Zi ) ˙ ˆ zi3 βi (Zi ) tanh( i ) − γ ϑ), ϑˆ = λ( a i i=1

3

0.2

0

2 1.5

PT

-0.2

y q2 (y) x ˆ1 yd

1

-0.4 14

0.5

14.5

15

15.5

16

16.5

17

17.5

18

0

CE

y , y2 (y) , x ˆ1 and yd

2.5

ED

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ˆ and Z2 = [ˆ ˆ T . The hysteresis quantizer where Z1 = [ˆ x1 , ϑ] x1 , xˆ2 , yd , y˙ d , y¨d , ϑ] described in [7] is utilized to quantize the input and the output. In particular, the parameters of input quantizer are chosen as δ1 = 0.5 and d1 = 0.05, and the parameters of output quantizer are set by δ2 = 0.2 and d2 = 0.02. Besides, the other design parameters are chosen as follows: k1 = 20, k2 = 20, c1 = 10, c2 = 20, λ = 0.005, γ = 100 and a1 = a2 = 8. In the simulation, RBF NN W1∗T S1 (Z1 ) contains 9 nodes with centers evenly spaced in [-5,5]×[-5,5]; W2∗T S2 (Z2 ) contains 729 nodes with centers evenly spaced in [-5,5]×[-5,5]×[-5,5]×[-5,5]×[-5,5]×[0,5]; and all the widths are chosen as 1. The initial conditions are given by [x1 (s), x2 (s)]T = ˆ [0.1, −0.1]T for s ∈ [−d, 0], [ˆ x1 (0), xˆ2 (0)]T = [0.1, 0.1]T and ϑ(0) = 0.1. Figs.2-5 give the simulation results. Fig.2 illustrates the satisfactory tracking performance and the output quantization. State x2 and its estimation ˆ xˆ2 are depicted in Fig.3. Fig.4 gives the trajectory of adaptive parameter ϑ. Fig.5 shows the input signals u and q1 (u).

-0.5

0

2

4

6

8

10

12

14

16

time(s)

AC

-1

Fig.2. Output y, quantized output q2 (y), state estimation x ˆ1 , and reference signal yd .

21

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2

x2 x ˆ2

1.5

0.5 0 -0.5 -1 -1.5 -2

0

2

4

6

8

10

12

time(s)

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x2 and x ˆ2

1

14

16

18

16

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Fig.3. State x2 and its estimation x ˆ2 .

0.1

0.06 0.04 0.02

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Adaptive Parameter

0.08

0

0

2

4

6

8

ED

-0.02

10

12

14

time(s)

PT

ˆ Fig.4. Adaptive parameter ϑ.

30

10

u q1 (u)

CE

25

u and q1 (u)

AC

20 15 10

5

5

0

-5 14

14.5

15

15.5

16

16.5

17

17.5

18

0 -5 -10

0

2

4

6

8

10

12

time(s)

Fig.5. Input signals: u and q1 (u).

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14

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Example 2. A mechanical movement system [41] is given to illustrate the applicability of the proposed output feedback control design. Involving the quantization scheme, the equations of motion for the networked control system can be described as follows  ¨ g k 3  θ = l sin θ + ms2 l (x − l sin(t − δt) ) cos t, (53)  q1 (u) k ks 3 x¨ = − m1 x − m1 (x − l sin θ(t − δt) ) + m1 .

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Let x1 = θ˙ and x2 = x. ˙ Taking stochastic disturbance into consideration and setting the same parameters as the ones in [41], then (53) can be expressed in the following stochastic nonlinear time-delay system form 
1 3 x1 (t − d(t)) sin x2 (t) dt + 14 x31 dω,  dx1 = x2 + 10 (54)
 1 dx2 = q1 (u) + 10 (x31 + x32 ) dt + 41 x31 (t − d(t))dω,

AC

CE

PT

ED

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where d(t) = 31 (1 + sin(t)) denotes the time-varying delay. The input and output signals are also quantized by the hysteresis quantizer presented in [7] with δ1 = 0.4, d1 = 0.04, δ2 = 0.4, and d2 = 0.02. In order to track the reference signal which is designated as yd = 0.5 sin(0.5t) + 0.5 sin(1.5t), the whole control scheme consisting of the observer (8), the virtual control laws (10), the actual control law (11) and the adaptive law (12) are utilized. Moreover, the design parameters are given by k1 = 10, k2 = 10, c1 = 18, c2 = 22, λ = 0.001, γ = 100 and a1 = a2 = 10. The RBF NN W1∗T S1 (Z1 ) contains 9 nodes with centers spaced in [-5,5]×[-5,5]; W2∗T S2 (Z2 ) contains 729 nodes with centers evenly spaced in [-5,5]×[-5,5]×[-5,5]×[-5,5]×[-5,5]×[0,5]; and all the widths are chosen as 1. This simulation is run with the initial conditions [x1 (s), x2 (s)]T = [0.1, 0.1]T for s ∈ [−d, 0], [ˆ x1 (0), xˆ2 (0)]T = [0.1, 0.1]T and ˆ ϑ(0) = 0.5. Simulation results are depicted in Figs.6-9. The output y, the quantized output q2 (y), the state estimation xˆ1 and the reference signal are shown in Fig. 6, which indicates the output y can follow the reference signal yd . State x2 and its estimation xˆ2 are shown in Fig.7. The boundedness of adaptive parameter ϑˆ is demonstrated in Fig. 8. Fig. 9 illustrates the actual input u and the quantized input q1 (u).

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4 1

y q2 (y) x ˆ1 yd

0.8 0.6 0.4

2

0.2 0 -0.2 14

1

14.5

15

15.5

16

16.5

17

0

-1

0

2

4

6

8

10

12

time(s)

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y , y2 (y) , x ˆ1 and yd

3

17.5

14

18

16

18

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Fig.6. Output y, quantized output q2 (y) , state estimation x ˆ1 , and reference signal yd .

1.5 1

x2 and x ˆ2

0.5 0

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-0.5 -1

0

2

4

6

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-1.5

x2 x ˆ2

8

10

12

14

16

18

14

16

18

time(s)

PT

Fig.7. State x2 and its estimation x ˆ2 .

CE

0.5

Adaptive Parameter

AC

0.4 0.3 0.2 0.1

0 -0.1

0

2

4

6

8

10 time(s)

ˆ Fig.8. Adaptive parameter ϑ.

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40

10

30

0

20

-10

u q1 (u)

14

10

14.5

15

15.5

16

16.5

17

0 -10 -20

0

2

4

6

8

10 time(s)

12

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u and q1 (u)

50

17.5

14

18

16

18

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Fig.9. Actual input u and quantized input q1 (u).

Remark 8. In the preceding simulation studies, the design parameters are set by utilizing a trial-and-error method based on the proposed theoretic results. How to attain the optimal control performance by choosing the optimal parameters is still an open problem.

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5. Conclusions

AC

CE

PT

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This paper has developed an adaptive output feedback tracking control scheme for a class of stochastic nonlinear time-delay systems in which both the input and the output are quantized by the sector-bounded quantizers. Based on the quantized input and output signals, an effective observer is developed to estimate the unknown states. Besides, some special techniques and the Lyapunov-Krasovskii functional are used to deal with the sector-bounded quantization error and the unknown time-varying delays, respectively. An adaptive NN control scheme is presented to attain the satisfactory tracking performance. Both theoretical analysis and simulation studies confirm the feasibility and effectiveness of the proposed design methodology. In the future work, we will probe into the output feedback quantized control design for switched stochastic nonlinear time-delay systems. Acknowledgment The authors would like to appreciate the editors and reviewers for their valuable comments and kind help. This work is partially supported by the Chinese National Natural Science Foundation under Grant 71271132 and 25

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61673296, and Shanghai Pujiang Program under Grant 15PJC049,and Fundamental Research Funds for the Central Universities under Grant 222201714055 and 222201717006.

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References

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[2] X.M. Zhang, Q.L. Han, X.H. Yu, Survey on Recent Advances in Networked Control Systems IEEE Trans. Industrial Informatics, 12(5)(2016) 1740-1752. [3] N. Elia, S. K. Mitter, Stabilization of linear systems with limited information, IEEE Trans. Autom. Control 46(9)(2009) 1384-1440.

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[4] T. Hayakawa, H. Ishii, K. Tsumura, Adaptive quantized control for nonlinear uncertain systems, Syst. Control Lett. 58(9)(2009) 625-632.

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[5] J. Zhou, C.Y. Wen, G.H. Yang, Adaptive backstepping stabilization of nonlinear uncertain systems with quantized input signal, IEEE Trans. Autom. Control 59(2)(2014) 460-464.

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[6] L.T. Xing, C.Y. Wen, H.Y. Su, Z. Liu, L. Wang, A new adaptive control scheme for uncertain nonlinear systems with quantized input signal, J. Franklin Institute 352(12)(2015) 5599-5610.

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[7] L.T. Xing, C.Y. Wen, H.Y. Su, Robust control for a class of uncertain nonlinear systems with input quantization, Int. J. Robust Nonlinear Control 26(8)(2016) 1585-1596.

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Yekai Yang received the B.S. degree in measurement and control technology and instrumentation from Wuhan Institute of Technology, Wuhan. China. in 2016. He is currently pursuing the M.S. degree in control theory and control engineering with the East China University of Science and Technology, Shanghai. China. His current research interests include nonlinear control, stochastic systems and quantized control.

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Zhaoxu Yu received the M.S. degree in Applied Mathematics from Tongji University, Shanghai, China, and the Ph.D. degree in Control Science and Engineering from Shanghai Jiaotong University, Shanghai, China, in 2001 and 2004, respectively. He is currently an associate professor with the Department of Automation in East China University of Science and Technology. From September 2015 to February 2016, he was a Visiting Scholar in the Department of Electrical and Computer Engineering, University of Florida, USA. His research interest includes nonlinear control, adaptive control and stochastic system.

Li, Shugang is a professor in the School of management at Shanghai University, Shanghai, China. He received his Ph.D. degree in control science and engineering from Shanghai Jiao Tong University in 2004. 32

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His current research areas of interest are information system and information management, data mining, soft computing and artificial intelligence.

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Jitao Sun received the B.Sc. degree in Mathematics from the Nanjing University, China, in 1983, and the Ph.D. degree in Control Theory and Control Engineering from the South China University of Technology, China, in 2002, respectively. He was with Anhui University of Technology from July 1983 to September 1997. From September 1997 to April 2000, he was with Shanghai Tiedao University. In April 2000, he joined the Department of Mathematics, Tongji University, Shanghai, China. From March 2004 to June 2004, he was a Senior Research Assistant in the Centre for Chaos Control and Synchronization, City University of Hong Kong, China. From February 2005 to May 2005, he was a Research Fellow in the Department of Applied Mathematics, City University of Hong Kong, China. From July 2005 to September 2005, he was a Visiting Professor in the Faculty of Informatics and Communication, Central Queensland University, Australia. From February 2006 to October 2006, August 2007 to October 2007, and April 2008 to June 2008, he was a Research Fellow in the Department of Electrical & Computer Engineering, National University of Singapore, Singapore, respectively. From November 2009 to May 2010, he was a Visiting Scholar in the Department of Mathematics, College of William and Mary, USA. From November 2013 to May 2014, he was a Visiting Scholar in Johns Hopkins University, USA. From September 2015 to February 2016, he was a Visiting Scholar in the Department of Electrical Engineering and Computer Science, University of Michigan, USA. He is currently a Professor at the Tongji University, China, and a Distinguished Visiting Professor at University of Johannesburg, South Africa. Prior to this, he was a Professor at Anhui University of Technology and Shanghai Tiedao University from 1995 to 2000, respectively. He is the author or coauthor of more than 200 journals papers. His recent research interests include impulsive control, time delay systems, hybrid systems, and systems biology. Prof. Sun is the Member of Technical Committee on Nonlinear Circuits and Systems, Part of the IEEE Circuits 33

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and Systems Society, and reviewer of Mathematical Reviews on AMS.

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