State-Feedback Stabilization for a Class of Stochastic Feedforward Nonlinear Time-Delay Systems

We investigate the state-feedback stabilization problem for a class of stochastic feedforward nonlinear time-delay systems. By using thehomogeneousdominationapproachandchoosinganappropriateLyapunov-Krasovskiifunctional,thedelay-independentstate-feedbackcontrollerisexplicitlyconstructedsuchthattheclosed-loopsystemisgloballyasymptoticallystableinprobability.Asimulationexampleisprovidedtodemonstratetheeffectivenessoftheproposeddesignmethod.

Feedforward (also called upper-triangular) system is another important class of nonlinear systems.Firstly, from a theoretical point of view, since they are not feedback linearizable and maybe not stabilized by applying the conventional backstepping method, the stabilization problem of these systems is more difficult than that of lower-triangular systems.Secondly, many physical devices, such as the cart-pendulum system in [20] and the ball-beam system with a friction term in [21], can be described by equations with the feedforward structure.In the recent papers, the stabilization problems for feedforward nonlinear (or time-delay) systems have achieved remarkable development; see, for example, [22][23][24][25][26][27][28][29] and the references therein.
However, all these above-mentioned results are limited to deterministic systems.There are fewer results on stochastic feedforward nonlinear systems until now, due to the special characteristics of this system.To the best of the authors' knowledge, [30] is the only paper to consider this kind of stochastic feedforward nonlinear systems, but the assumptions on the nonlinearities are restrictive.
The purpose of this paper is to further weaken the assumptions on the drift and diffusion terms of system (1) and solve the state-feedback stabilization problem.By using the homogeneous domination approach in [26] and choosing an appropriate Lyapunov-Krasovskii functional, a delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is globally asymptotically stable in probability.
The paper is organized as follows.Section 2 provides some preliminary results.The design and analysis of statefeedback controller are given in Sections 3 and 4, respectively, following a simulation example in Section 5. Section 6 concludes this paper.

Preliminary Results
The following notations, definitions, and lemmas are to be used throughout the paper.
+ denotes the set of all nonnegative real numbers, and   denotes the real -dimensional space.For a given vector or matrix ,   denotes its transpose, Tr{} denotes its trace when  is square, and || is the Euclidean norm of a vector .  1 denotes the family of all functions which are C 2 in the first argument and C 1 in the second argument.K denotes the set of all functions  + →  + , which are continuous, strictly increasing, and vanishing at zero; K ∞ denotes the set of all functions which are of class K and unbounded; KL is the set of all functions (, ):  + ×  + →  + , which are of K for each fixed  and decrease to zero as  → ∞ for each fixed .
Lemma 4 (see [6]).For system (2), if there exist a function then there exists a unique solution on [−, ∞) for (2), the equilibrium () = 0 is GAS in probability, and Lemma 5 (see [26]).Given a dilation weight Δ = ( 1 , . . .,   ), suppose that  1 () and  2 () are homogeneous functions of degrees  1 and  2 , respectively.Then  1 () 2 () is also homogeneous with respect to the same dilation weight Δ.Moreover, the homogeneous degree of Lemma 6 (see [26]).Suppose that  :   →  is a homogeneous function of degree  with respect to the dilation weight Δ; then (i) /  is homogeneous of degree  −   with   being the homogeneous weight of where  is a positive constant.

Design of State-Feedback Controller
The objective of this paper is to design a state-feedback controller for system (1) such that the equilibrium of the closedloop system is globally asymptotically stable in probability.

State-Feedback Controller Design.
We construct a statefeedback controller for system (7).
Proof.See the Appendix.

Stability Analysis
We state the main result in this paper.
Theorem 12.If Assumptions 8 and 9 hold for the stochastic feedforward nonlinear time-delay system (1), under the statefeedback controller  =   V and (13), then (i) the closed-loop system has a unique solution on [−, ∞); (ii) the equilibrium at the origin of the closed-loop system is GAS in probability.
Proof.We prove Theorem 12 by four steps.
Remark 13.In this paper, the homogeneous domination idea is generalized to stochastic feedforward nonlinear time-delay systems (1).The underlying philosophy of this approach is that the state-feedback controller is first constructed for system (7) without considering the drift and diffusion terms, and then a low gain  in (6) (whose the value range is (28)) is introduced to state-feedback controller to dominate the drift and diffusion terms.Remark 14. Due to the special upper-triangular structure and the appearance of time-varying delay, there is no efficient method to solve the stabilization problem of system (1).By combining the homogeneous domination approach with stochastic nonlinear time-delay system criterion, the statefeedback stabilization of system (1) was perfectly solved in this paper.
Remark 15.One of the main obstacles in the stability analysis is how to deal with the effect of time-varying delay.In this paper, by constructing an appropriate Lyapunov-Krasovskii functional (17), this problem was effectively solved.
Remark 16.It is worth pointing out that the rigorous proof of Theorem 12 is not an easy job.

A Concluding Remark
By using the homogeneous domination approach, this paper further studied the state-feedback stabilization problem for a class of stochastic feedforward nonlinear time-delay systems (1).The delay-independent state-feedback controller is explicitly constructed such that the closed-loop system is globally asymptotically stable in probability.
There still exist some problems to be investigated.One is to consider the output-feedback control of switched stochastic system (1) by using average dwell time method in [32].Another is to find a practical example (similar to [33][34][35]) for system (1).The last is to generalize the networked control systems (such as [36][37][38][39][40][41]) to stochastic feedforward networked systems.

Figure 1 :
Figure 1: (a) The response of the closed-loop system (30) and (b) the response of the controller (37).