Adaptive NN output-feedback control for stochastic time-delay nonlinear systems with unknown control coefficients and perturbations ∗

Abstract. This paper addresses the problem of adaptive output-feedback control for more general class of stochastic time-varying delay nonlinear systems with unknown control coefficients and perturbations. By using Lyapunov–Krasovskii functional, backstepping and tuning function technique, a novel adaptive neural network (NN) output-feedback controller is constructed with fewer learning parameters. The designed controller guarantees that all the signals in the closed-loop system are 4-moment (or mean square) semi-globally uniformly ultimately bounded (SGUUB). Finally, a simulation example is shown to demonstrate the effectiveness of the proposed control scheme.


Introduction
In order to obtain global stability, some restrictions such as matching conditions, extended matching conditions or growth conditions are often imposed on system nonlinearities.To handle the above restrictions, neural networks (NNs) are gradually into people's vision due to their ability to adaptively compensate for nonlinear functions.In the past two decades, based on the theoretical results of stochastic stability in [6,7,11] and NNs in [14,16,23], radial basis function neural network (RBF NN) approximation approach has been successfully used for various classes of stochastic nonlinear systems, see [1,8,12,13,15,17,27] and the references therein.family of all the functions with continuous ith partial derivations; C 2,1 (R n ×[−d, ∞), R + ) denotes the family of all non-negative functions V (x, t) on R n × [−d, ∞), which are X T denotes the transpose of a given vector or matrix X, Tr{X} denotes its trace when X is square.X is the Euclidean norm of a vector X or its inducted matrix norm.λ min (•) and λ max (•) denote the smallest and largest eigenvalues of a square matrix, respectively.To simply the procedure, we sometimes denote X(t) by X for any variable X(t).
Consider the following stochastic nonlinear time-delay system: dx(t) = f t, x(t), x t − d(t) dt + g t, x(t), x t − d(t) dω ∀t 0 with initial data {x(θ): , where d(t) : R + → [0, d] is a Borel measurable function; ω is an r-dimensional standard wiener process defined on a probability space {Ω, F, P} with Ω being a sample space, F being a σ-field and P being the probability measure.f : R + ×R n ×R n → R n and g : R + ×R n ×R n → R n×r are locally Lipschitz with f (0, 0, t) ≡ 0 and g(0, 0, t) ≡ 0. For any given V (x(t), t) ∈ C 2,1 , together with stochastic system (1), the differential operator L is defined as Definition 1.Let p 1, consider stochastic nonlinear time-delay system (1), the solution {x(t), t 0} with initial condition ξ ∈ S 0 (S 0 is some compact set containing the origin) is said to be p-moment semi-globally uniformly ultimately bounded if there exists a constant d such that E{ x(t, ξ) p } d for all t T holds for some T 0.
In the sequel, radial basis function neural network (RBF NN) will be applied to estimate the unknown nonlinear functions.It has been proven in [23] that by choosing sufficiently large node number, for any unknown continuous function f (x) over a compact set S x ⊂ R q , there is a RBF NN W * T S(x) such that for an expected level of accuracy ε (0 < ε < 1), it holds where δ(x) is the approximation error, and S(x) = [s 1 (x), . . ., s N (x)] T is the known function vector with N > 1 being the RBF NN node number.For 1 i N , the basis functions s i (x) are chosen as Lemma 1 [Young's inequality].For all (x, y) ∈ R 2 , xy ε p p −1 |x| p + (qε q ) −1 |y| q holds, where ε > 0, p, q > 1, and (p − 1)(q − 1) = 1.
Lemma 2. (See [19].)For any smooth function f (x), x ∈ R n , there exists a smooth function f

Problem description
In this paper, we consider a class of stochastic nonlinear systems in the following form: where η = [η 1 , . . ., η n ] T ∈ R n is the system state vector and η(t T is the time-delay state variable, u ∈ R and y ∈ R are control input and system output, respectively, η 2 , . . ., η n are unmeasurable state variables, control coefficients h 1 , . . ., h n are unknown constants.d(t) : R + → [0, d] is time-varying delay.ω is an r-dimensional standard wiener process.For i = 1, . . ., n, φ i : R + ×R n ×R n → R and ϕ i : R + × R n × R n → R r are unknown smooth functions with φ i (t, 0, 0) = 0 and ϕ i (t, 0, 0) = 0.For i = 1, 2, . . ., n, θ i ∈ R mi are unknown constant system parameters and ψ i : R → R mi are known smooth vector-valued functions.
The control objective of the paper is to construct an adaptive NN output-feedback controller for system (4) such that all the signals in the closed-loop system are 4-moment (or mean-square) SGUUB.To realise this objective, we need the following assumptions.
Assumption 1.The nonzero control coefficients h 1 , . . ., h n are of known signs and satisfy h |h i | h (i = 1, . . ., n), where h and h are known positive constants denoting the low and upper bounds of h 1 , . . ., h n , respectively.
http://www.mii.lt/NARemark 1.For system (4) without perturbations θ T i ψ i (y), in the similar references [2,10], the functions φ i and ϕ i only appear in the forms of φ i (t, y(t), y(t − d(t))) and ϕ i (t, y(t), y(t − d(t))).In [25], the functions f i and h i may contain unmeasured state variables.However, the bounds of the time-delay functions should only contain y(t) and y(t−d(t)).[24] and [26] relaxed the bound functions to contain state variables and solved the tracking problems.For system (4) with perturbations θ T i ψ i (y), this paper handles the output-feedback control problems by allowing the bound functions φ ij (•) and ϕ ij (•) (i = 1, . . ., n, j = 1, 2) to be both unknown and contain state variables.

Output-feedback controller design
The design procedure of output-feedback controller is divided into two parts.Firstly, by introducing an equivalent linear state transformation to lump the control coefficients into one, RBF NN is used to estimate the unknown nonlinear functions and the tuning function method is utilized to avoid over-parameterization.Then, an adaptive outputfeedback controller is designed to guarantee all the signals in the closed-loop system to be 4-moment (or mean-square) SGUUB.

Backstepping controller design
The entire system can be rewritten as Introduce the coordinate transformation and define new states X 1 = (x, z 1 ) T and X i = (x 1 , x, zi ) T , where α 2 , . . ., α n are virtual control laws to be designed later and zi = [z 1 , . . ., z i ] T .In the sequel, we will design an adaptive output-feedback controller by n steps.
Proposition 1.For the Lyapunov function candidate there exists the ith virtual control law α i+1 in the following form: such that where According to the recursive steps, at step n, choosing the Lyapunov function and constructing the adaptive control law as yield where

Stability analysis
We now state the main theorem in this paper.
Remark 2. We give a further explanation on how to design parameters.By choosing larger NN nodes N , the approximation error ε in (3) can be reduced, which may improve the approximation accuracy.Furthermore, smaller ε 0 , . . ., ε n , i1 , i0 , i together with larger c n , k i (i = 1, . . ., n) will reduce a 2 /a 1 , which leads to smaller converging region.Hence, one can reduce the bounded compact sets Ω 1 , Ω 2 and Ω 3 by appropriately regulating the parameters.

Conclusions
This paper solves the adaptive NN output-feedback control problem for a class of stochastic time-delay nonlinear systems with unknown control coefficients and perturbations.By using RBF NN, tuning function approach and backstepping technique, the proposed control scheme requires fewer parameters and guarantees all the signals in the closed-loop system to be 4-moment(or mean-square) SGUUB.An important issue under investigation is how to extend the design scheme to high-order stochastic nonlinear time-delay systems.Now, we start to estimate the right-hand side terms of (52).Considering Lemma 1, (3), ( 5), ( 9) and ( 21 where c i = c i−1 − 4 /4, fi = F i + (∂ 2 α i /∂y 2 )h 2 M g 2 11 ( x ) + (9/2)z i and ∆ id = ∆ i−1,d +(h 8 M /4)(∂ 2 α i /∂y 2 ) 4 g 8 Define a nonlinear function β i (X i ) = −k i z i − fi , from (3), for any given 0 < ε i < 1, there exists W * T i S i (X i ) such that β i (X i ) = W * T i S i (X i ) + δ 1 (X i ), |δ 1 (X i )| ε i , where k i is a positive number, and X i ∈ S Xi = {X i | X i ∈ S x }.With the use of ( 21 where i > 0 is a constant.Substituting fi = −β i − k i z i and (57) into (56), and choosing the ith virtual control law as (33) yield (34).The proof is completed.
where ς is the width of the function, b i = [b i1 , . . ., b in ] T is the center of the receptive field.W * is the ideal constant weight vector with the form W * = arg min W ∈R N {sup x∈Sx |f (x)−W T S(x)|}, where arg min is the value of variable W when the objective function sup x∈Sx |f (x) − W T S(x)| is minimum with W = [w 1 , . . ., w N ] T being the weight vector.