Adaptive NN State-Feedback Control for Stochastic High-Order Nonlinear Systems with Time-Varying Control Direction and Delays

Nussbaum-type gain function and neural network (NN) approximation approaches are extended to investigate the adaptive statefeedback stabilization problem for a class of stochastic high-order nonlinear time-delay systems. The distinct features of this paper are listed as follows. Firstly, the power order condition is completely removed; the restrictions on system nonlinearities and time-varying control direction are greatly weakened. Then, based on Lyapunov-Krasovskii function and dynamic surface control technique, an adaptive NN controller is constructed to render the closed-loop system semiglobally uniformly ultimately bounded (SGUUB). Finally, a simulation example is shown to demonstrate the effectiveness of the proposed control scheme.


Introduction
During the past decades, the control of stochastic nonlinear systems has been an interesting field and received fruitful development based on the stochastic stability theory in [1][2][3] and other references.However, the existences of nonlinearities and time delays cause the instability of system performance and give rise to much greater design difficulty in the stabilization procedure.To handle nonlinearities, neural network (NN), especially radial basis function neural network (RBF NN), has been successfully extended to stochastic nonlinear systems due to its inherent approximation capacity; see [4][5][6][7][8] and the references therein.To deal with time delays, there are often two ways, namely, Lyapunov-Krasovskii function and Lyapunov-Razumikhin technique.Recently, together with NN and backstepping, Lyapunov-Krasovskii theory was used in [9][10][11][12][13] and Lyapunov-Razumikhin approach was utilized in [14][15][16][17] to stabilize stochastic nonlinear systems with time delays, respectively.However, to the best of our knowledge, for stochastic high-order nonlinear time-delay systems, there are still a few results.
This paper focuses on solving this problem.The main contributions are listed as follows.(i) Compared with the existing results [29,30,[32][33][34], we largely weaken the restrictions on the drift and diffusion terms by extending NN approximation approach to system (1).Furthermore, to handle unknown control direction, we utilize Nussbaumtype gain function approach to allow the sign and the bounds of () to be unknown.(ii) By introducing dynamic surface control (DSC), the problem of "explosion of complexity" generalized by the repeated differentiation of virtual controllers is avoided, which greatly simplifies the control algorithm.(iii) An adaptive state-feedback controller is constructed to ensure the closed-loop system to be semiglobally uniformly ultimately bounded (SGUUB) by using backstepping technique and choosing Lyapunov-Krasovskii function skillfully.
The remainder of this paper is organized as follows.Section 2 begins with mathematical preliminaries.The design process and analysis of adaptive state-feedback controller are given in Sections 3 and 4, respectively.In Section 5, a simulation example is shown.Section 6 summarizes the paper.Finally, the necessary proof is provided in the Appendix.

Mathematical Preliminaries
The following notations, definitions, and lemmas are to be used.
Notations.R + denotes the set of all the nonnegative real numbers; R  denotes the -dimensional Euclidean space.C  denotes the family of all the functions with continuous th partial derivations; − ≤  ≤ 0}.  denotes the transpose of a given vector or matrix ; Tr{} denotes its trace when  is square.K denotes the set of all functions R + → R + , which are continuous, strictly increasing, and vanishing at zero; K ∞ denotes the set of all functions which are of class K and unbounded.To simplify the procedure, we sometimes denote () by  for any variable ().
To facilitate the control design, the following lemmas are applied.
Lemma 5 (see [31]).Let  and  be real variables; then, for any real numbers , ,  > 0 and continuous function Lemma 6 (see [31]).For ,  ∈ R, where  ≥ 1 is a constant, the following inequalities hold: In the sequel, radial basis function neural network (RBF NN) is to be applied to estimate the unknown nonlinear functions.By choosing sufficiently large node number, for any continuous function () over a compact set   ⊂ R  , there is a RBF NN  *  () such that, for an ideal level of accuracy where () is the approximation error and () = [

Design of State-Feedback Controller
The objective of this paper is to design an adaptive NN statefeedback controller for system (1) under weaker conditions such that the closed-loop system is SGUUB.To achieve the above objective, we need the following assumptions.
Remark 10.We emphasize two points.(i) To the best of our knowledge, only [33] consider the unknown control directions for stochastic high-order time-delay systems.However, in [33], the unknown control directions are of known signs and are bounded by positive constants.We allow the sign of () to be unknown and remove the bounds limitations in Assumption 8. (ii) Motivated by [14,15] for stochastic time-delay systems, the restrictions on   and   are greatly relaxed in Assumption 9 compared with the existing results in [29,30,[32][33][34].
To simplify the design process, define where  1 , . . .,   are the number of RBF NN nodes and  * 1 , . . .,  *  are the ideal constant weight vectors.Before the design procedure, introduce the following coordinate transformation: where  2 , . . .,   are the outputs of the first-order filter with virtual control laws  2 , . . .,   being inputs.Now, we give the backstepping design procedure by utilizing the technique of DSC and RBF NN approximation approach.
Proposition 11.For the th Lyapunov function there exists a virtual control law in the following form: such that where   ,   , and  1 are positive design constants, Proof.See the Appendix.
Hence, at step , we choose By exactly following the design procedure in Proposition 11 and introducing Nussbaum function, one can construct the adaptive control laws as thus, this leads to where   ,  0 > 0,  1 = 0 are design constants, ) ,  0 = 0, and Remark 12.During the design procedure, the drift and diffusion terms are technically handled for more general system (1) by RBF NN approximation approach compared with [29,30,[32][33][34].Furthermore, the repeated differentiation of virtual control laws is avoided and the unknown time-varying control direction is successfully handled by introducing DSC and Nussbaum-type gain function.

Stability Analysis
The main result of this paper is stated in the following theorem.

Conclusions
This paper considers the unknown time-varying control coefficient by generalizing Nussbaum function to a class of stochastic high-order time-delay systems.Under weaker conditions on the drift and diffusion terms, the adaptive statefeedback control is solved by using RBF NN approximation approach.The control scheme ensures the closed-loop system to be SGUUB.An issue to be investigated is how to further handle the output-feedback problem for system (1).