State feedback control for a class of stochastic high-order feedforward nonlinear systems with time-varying delays: a homogeneous domination approach

Abstract

In this paper, the problem of global state feedback stabilization for a class of stochastic high-order feedforward nonlinear systems with different power orders and multiple time delays is investigated. A distinct property of the system to be investigated is that the control coefficients are not restricted to 1. By adding one power integrator technique and homogeneous domination approach, a state feedback controller design is recursively proposed, which ensures the global asymptotical stability in probability of the closed-loop system. Finally, a simulation example is given to illustrate the effectiveness of our results obtained in this paper.

Keywords

Stochastic high-order feedforward nonlinear systems state feedback backstepping homogeneous domination time-delay

Introduction

It is well known that stochastic systems have received great attention since stochastic modelling has come to play an important role in many branches of science and engineering applications. In order to analyse this type of system, the authors in Khas’minskii (1980), Kushner (1967), Mao (2007) and Yin et al. (2011) presented the basic stability theory of stochastic systems. Ever since backstepping designs were extended to stochastic systems, there has been constant progress on the problem of global stabilization (Deng and Krstic, 1997a, 1997b, 1999; Deng et al., 2001) that has been further developed by recent work (Liu et al., 2008b; Yu and Xie, 2010).

On the other hand, global stabilization of triangular structural stochastic nonlinear systems has been fully investigated over recent decades. For lower-triangular systems (namely feedback systems), the stochastic asymptotic stabilization problem has been studied using the backstepping approach (see, e.g., Chai and Qian, 2013; Fan et al., 2012; Hou et al., 2013; Li et al., 2011; Wu et al., 2006, 2007, 2009; Yu et al., 2010; Zhai, 2013, and the references therein).

The other class of triangular structural nonlinear systems are upper-triangular systems, which are also called feedforward systems and have been fully used to model many physical devices, such as the ball-beam with a friction term (Sepulchre et al., 1997) and the cart-pendulum system (Mazenc and Bowong, 2003). Many important stabilization results have been proposed for feedforward nonlinear systems. By using a homogeneous domination approach, Qian and Li (2006) addressed the problem of global output-feedback stabilization for a class of upper-triangular systems with higher-order nonlinearities. The case of lower-order nonlinearities was considered by Ding et al. (2010). When uncertain nonlinearities appeared in a system model, a logic-based adaptive stabilizer for feedforward nonlinear systems was proposed by Ye (1999). A universal global stabilizer design was further developed by Ye (2003) for feedforward systems without a priori knowledge of system nonlinearities. However, in these results, time delays and stochastic perturbation effects had not been taken into account.

Since time delays will inevitably occur in many mechanical, physical and biological systems and their existence may lead to poor performance, even instability of systems, many researchers have paid more attention to studying stochastic nonlinear time-delay systems over recent decades. Various results concerning stochastic lower-triangular nonlinear systems with time delays have been reported by Chen et al. (2010, 2011), Fu et al. (2005) and Xie and Liu (2012). For deterministic upper-triangular systems, a state feedback controller for a class of input-delayed systems satisfying the linear growth condition was designed by Zhang et al. (2011a) based on a dynamic low-gain control method. When the nonlinear functions were quadratic and higher order in states, the adaptive stabilization problem for feedforward nonlinear systems with time delays was considered by Ye (2011) by taking a nested saturation feedback. For high-order feedforward systems, Zhang et al. (2010) was the initial attempt to solve the problem of global stabilization by introducing a transformation of coordinates. Furthermore, the problem of global strong stabilization in the sense of Kurzweil was investigated by Zhang et al. (2011b) for a family of high-order feedforward nonlinear time-delay systems. However, to the best of our knowledge, little work has been done on the feedback stabilization for stochastic feedforward systems (Liu and Xie, 2013, Zhao and Xie, 2013). It should be pointed out that the control coefficients are required to be 1 and the power orders be equal to $p$ or 1 in Liu and Xie (2013) and Zhao and Xie (2013). Nevertheless, these conditions may not be satisfied from both practical and theoretical points of view. Naturally, one may ask a challenging and interesting problem: Is it possible to relax the restrictions on the control coefficients and the growth condition? If these restrictions are relaxed, how does one design a state feedback controller for stochastic high-order feedforward systems with multiple time-varying delays?

Inspired by the aforementioned discussion, we deal with the problem of global state feedback stabilization for a class of stochastic high-order feedforward nonlinear systems with multiple time-varying delays and time-varying control coefficients, which is more general than that in Liu and Xie (2013) and Zhao and Xie (2013) and more complex to analyse the value range of the system’s homogeneous degree $τ$ and design a stabilizing state feedback controller. To cope up with these difficulties, we firstly design a state feedback controller for the nominal system using the adding one power integrator technique. Then, we propose a scaled controller that guarantees global asymptotic stability in probability of the closed-loop system.

The outline of this paper is as follows. Sections 2 and 3 offer some preliminary results and problem formulation, respectively. The state feedback controller is designed and analysed in Section 4. In Section 5, a simulation example is presented to show the effectiveness of the state feedback controller. This paper is concluded in Section 6.

Notations

The following standard notations are used throughout this paper. $R^{n}$ denotes the real n-dimensional space. $| x | = {(\sum_{i = 1}^{n} x_{i}^{2})}^{1 / 2}$ for any $x \in R^{n}$ . For any matrix $Q \in R^{n} \times R^{m}$ , $| Q |$ denotes the Frobenius norm $| Q | = {(\sum_{i, j} q_{ij}^{2})}^{\frac{1}{2}}$ and $| Q |_{\infty} = max_{1 \leq i \leq n} {(\sum_{j = 1}^{m} q_{ij}^{2})}^{\frac{1}{2}}$ . For the sake of simplicity, sometimes function $x (t)$ is denoted by $x$ , $x_{i}$ represents the ith element of vector $x$ and ${\bar{x}}_{i} = {(x_{1}, \dots, x_{i})}^{T}$ . $C^{i}$ denotes the set of all functions with continuous ith partial derivative. $K$ denotes the set of all functions: $R^{+} \to R^{+}$ , which are continuous, strictly increasing and vanishing at zero; $K_{\infty}$ denotes the set of all functions that are of class $K$ and unbounded.

Preliminary results

Consider the following stochastic time-delay system:

\begin{matrix} dx (t) = f (x (t), x (t - σ (t))) dt + g (x (t), x (t - σ (t {)))}^{T} d ω, \end{matrix}

(1)

with initial condition ${x (s) : - σ_{0} \leq s \leq 0} = ξ \in C_{F_{0}}^{b} \times ([- σ_{0}, 0], R^{n})$ , where $σ (t) : R^{+} \to [0, σ_{0}]$ is a Borel measurable function; $x (t) \in R^{n}$ denotes the state vector and $x (t - σ (t))$ is the state vector with the time delay; $ω$ is an m-dimensional standard Wiener process defined on the complete probability space $(Ω, F, {F_{t}}_{t \geq 0}, P)$ with $Ω$ being a sample space, $F$ being a $σ$ -field, ${F_{t}}_{t \geq 0}$ being a filtration and $P$ being a probability measure; $f (\cdot) \in R^{n}$ , $g (\cdot) \in R^{m \times n}$ are locally Lipschitz functions and satisfy $f (0, 0) = 0$ , $g (0, 0) = 0$ .

In the following, we borrow some definitions and lemmas that play an important role to stabilize and analyse the stochastic time-delay systems for later development in this paper.

Definition 1 (Liu et al., 2008a). Define the infinitesimal generator $L$ of function $V (x) \in C^{2}$ along system (1) as follows:

\begin{matrix} L V (x) = \frac{\partial V}{\partial x} f (x, x (t - σ (t))) \\ + \frac{1}{2} Tr {g (x, x (t - σ (t))) \frac{\partial^{2} V}{\partial x^{2}} g (x, x (t - σ (t {)))}^{T}}, \end{matrix}

(2)

in which $\frac{1}{2} Tr {g (x, x (t - σ (t))) \frac{\partial^{2} V}{\partial x^{2}} g (x, x (t - σ (t {)))}^{T}}$ is called the Hessian term of $L$ .

Definition 2 (Liu et al., 2008). The equilibrium $x = 0$ of system (1) is globally asymptotically stable (GAS) in probability if for any $ϵ > 0$ , there exists a function $γ (\cdot, \cdot) \in K L$ such that $P {| x | \leq γ (∥ ζ ∥, t)} \geq 1 - ϵ$ for any $t \geq 0$ , $ζ \in C_{F_{0}}^{b} \times ([- σ_{0}, 0], R^{n}) \ {0}$ , where $∥ ζ ∥ = su p_{θ \in [- σ_{0}, 0]} | ζ (θ) |$ .

Definition 3 (Qian and Li, 2006). For fixed coordinates $(x_{1}, \dots, x_{n})$ and real numbers $r_{i} > 0, i = 1, . . ., n$ .

The dilation $Δ_{ε} (x)$ is defined by $Δ_{ε} (x) = (ε^{r_{1}} x_{1}, \dots, ε^{r_{n}} x_{n})$ , $\forall ε > 0$ , with $r_{i}$ being called the weights of the coordinates. For simplicity of notation, we define dilation weight $Δ = (r_{1}, . . ., r_{n})$ .

A function $V \in C (R^{n}, R)$ is said to be homogeneous of degree $τ$ if there is a real number $τ \in R$ such that $V (Δ_{ε} (x)) = ε^{τ} V (x_{1}, \dots, x_{n})$ , for $\forall x \in R^{n} \ {0},$ $ε > 0$ .

A vector field $f \in C (R^{n}, R^{n})$ is said to be homogeneous of degree $τ$ if there is a real number $τ \in R$ such that $f_{i} (Δ_{ε} (x)) = ε^{τ + r_{i}} f_{i} (x_{1}, \dots, x_{n})$ , for $\forall x \in R^{n} \ {0},$ $ε > 0,$ $i = 1, . . ., n$ .

A homogeneous $p$ -norm is defined as $∥ x ∥_{Δ, p} = {(\sum_{i = 1}^{n} | x_{i} |^{p / r_{i}})}^{1 / p}$ , $\forall x \in R^{n}$ , for a constant $p \geq 1$ . For the simplicity, we choose $p = 2$ and write $∥ x ∥_{Δ}$ for $∥ x ∥_{Δ, 2}$ in this paper.

Lemma 1 (Liu et al., 2008a). Consider the stochastic time-delay system (1), if there exists a function $V (x) \in C^{2} ([- σ_{0}, \infty] \times R^{n})$ and class $K_{\infty}$ functions $α_{1}, α_{2}$ satisfying the following inequalities:

\begin{matrix} α_{1} (| x (t) |) \leq V (x) \leq α_{2} (su p_{(- σ_{0} \leq s \leq 0)} | x (t + s) |), \\ L V (x) \leq - W (x (t)), \end{matrix}

where $W (x (t))$ is continuous and positive definite, then

there exists a unique solution on $[- σ_{0}, \infty)$ ;

the equilibrium $x = 0$ of the system (1) is GAS in probability, and $P {li m_{t \to \infty} | x (t) | = 0} = 1$ .

Lemma 2 (Qian and Li, 2006). Given a dilation weight $Δ = (r_{1}, . . ., r_{n})$ , suppose $V_{1} (x)$ and $V_{2} (x)$ are homogeneous functions of degree $τ_{1}$ and $τ_{2}$ , respectively. Then $V_{1} (x) \cdot V_{2} (x)$ is also homogeneous with respect to the same dilation weight $Δ$ . Moreover, the homogeneous degree of $V_{1} (x) \cdot V_{2} (x)$ is $τ_{1} + τ_{2}$ .

Lemma 3 (Qian and Li, 2006). Suppose $V : R^{n} \to R$ is a homogeneous function of degree $τ$ with respect to the dilation weight $Δ$ . Then the following holds.

$\frac{\partial V}{\partial x_{i}}$ is homogeneous of degree $τ - r_{i}$ with $r_{i}$ being the homogeneous weight of $x_{i}$ ;

There is a constant c such that $V (x) \leq c ∥ x ∥_{Δ}^{τ}$ . Moreover, if $V (x)$ is positive definite, then

\underline{c} ∥ x ∥_{Δ}^{τ} \leq V (x), for a constant \underline{c} > 0 .

Lemma 4 (Qian and Li, 2006). For any $x \in R, y \in R$ , and $p \geq 1$ , the following inequalities hold:

| x + y |^{p} \leq 2^{p - 1} | x^{p} + y^{p} |,

(3)

{(| x | + | y |)}^{\frac{1}{p}} \leq | x |^{\frac{1}{p}} + | y |^{\frac{1}{p}} .

(4)

Moreover, when $p \in R_{odd}^{\geq 1}$

| x - y |^{p} \leq 2^{p - 1} | x^{p} - y^{p} |,

(5)

\begin{matrix} | x^{p} - y^{p} | & \leq p | x - y | | x^{p - 1} + y^{p - 1} | \\ \leq c | x - y | | {(x - y)}^{p - 1} + y^{p - 1} |, \end{matrix}

(6)

where $c > 0$ is a constant.

Lemma 5 (Zhai, 2013). Let $x$ and $y$ be any real numbers and $p \in R_{odd}^{\geq 1}$ , then

\begin{matrix} - (x - y) (x^{p} - y^{p}) \leq - \frac{1}{2^{p - 1}} {(x - y)}^{p + 1} . \end{matrix}

(7)

Lemma 6 (Zhai, 2013). Let $c$ and $d$ be two positive real numbers. For any positive number $γ > 0$ , the following inequality holds:

| x |^{c} | y |^{d} \leq \frac{c}{c + d} γ | x |^{c + d} + \frac{d}{c + d} γ^{- \frac{c}{d}} | y |^{c + d} .

(8)

Problem formulation

In this paper, we consider the following stochastic high-order feedforward nonlinear system with time-varying delays in the form

\begin{matrix} d z_{1} = d_{1} (t) z_{2}^{p_{1}} dt + f_{1} (z, z (t - σ (t))) dt + g_{1}^{T} (z, z (t - σ (t))) d ω, \\ d z_{2} = d_{2} (t) z_{3}^{p_{2}} dt + f_{2} (z, z (t - σ (t))) dt + g_{2}^{T} (z, z (t - σ (t))) d ω, \\ ⋮ \\ d z_{n - 1} = d_{n - 1} (t) z_{n}^{p_{n - 1}} dt + f_{n - 1} (z, z (t - σ (t))) dt + g_{n - 1}^{T} (z, z (t - σ (t))) d ω, \\ d z_{n} = d_{n} (t) υ^{p_{n}} dt, \end{matrix}

(9)

where $z = {[z_{1}, \dots, z_{n}]}^{T} \in R^{n}$ , $υ \in R$ denote the system state and input, respectively; $z (t - σ (t)) = [z_{1} (t - σ_{1} (t)), \dots, z_{n} (t - σ_{n} (t {))]}^{T}$ is the state vector with time delays; $ω$ is an m-dimensional standard Wiener process. $σ_{i} (t) : R^{+} \to [0, σ_{i 0}]$ for $i = 1, \dots, n$ are time-varying delays. For $i = 1, \dots, n$ , $d_{i} (t)$ are uncertain control coefficients with known sign, and $p_{i} \in R_{odd}^{\geq 3} \overset{Δ}{=} {p \in R : p \geq 3 and p is a ratio of odd integers} .$ The drift functions $f_{i} : R^{n} \times R^{n} \to R$ and the diffusion functions $g_{i} : R^{n} \times R^{n} \to R^{m}$ are locally Lipschitz in $(z, z (t - σ (t)))$ and vanish at the origin.

In order to obtain the main results of this paper, the following assumptions are needed.

Assumption 1 For $i = 1, \dots, n - 1$ , there are constants $a_{i} > 0, b_{i} > 0$ , such that

\begin{matrix} \begin{matrix} | f_{i} (z, z (t - σ (t))) | & \leq a_{i} \sum_{j = i + 2}^{n + 1} (| z_{j} (t) |^{\frac{r_{i} + τ}{r_{j}}} + | z_{j} (t - σ_{j} (t)) |^{\frac{r_{i} + τ}{r_{j}}}), \\ | g_{i} (z, z (t - σ (t))) | & \leq b_{i} \sum_{j = i + 2}^{n + 1} (| z_{j} (t) |^{\frac{2 r_{i} + τ}{2 r_{j}}} + | z_{j} (t - σ_{j} (t)) |^{\frac{2 r_{i} + τ}{2 r_{j}}}), \end{matrix} \end{matrix}

where $p_{i} \in R_{odd}^{\geq 3}$ , $τ \in [d_{0}, \infty)$ , $r_{1} = 1, r_{i + 1} = \frac{r_{i} + τ}{p_{i}}$ , $d_{1} = \frac{2 - \frac{1}{p_{1} \dots p_{n - 1}}}{1 + \sum_{s = 1}^{n - 1} \frac{1}{p_{s} \dots p_{n - 1}}}$ , $d_{i + 1} = \frac{\frac{2}{p_{1} \dots p_{i}} - \frac{1}{p_{1} \dots p_{n - 1}}}{1 + \sum_{s = 1}^{n - 1} \frac{1}{p_{s} \dots p_{n - 1}} - \sum_{s = 1}^{i} \frac{2}{p_{s} \dots p_{i}}}$ , and $d_{0} = max_{j = 1, \dots, n} {d_{j}}$ , $z_{n} (t) = z_{n} (t - σ_{n} (t)) = 0$ .

Assumption 2 For $j = 1, \dots, n$ , the time-varying delay $σ_{j} (t)$ satisfies ${\overset{\cdot}{σ}}_{j} (t) \leq σ_{j} < 1$ for a constant $σ$ .

Assumption 3 Without loss of generality, assume that there are positive real numbers $λ_{j}$ and $ν_{j}$ such that $0 < λ_{j} \leq d_{j} (t) \leq ν_{i}$ for $j = 1, \dots, n$ .

In this paper, we aim to constructively design a homogeneous state feedback controller $υ (t) = υ (z_{1} (t), \dots, z_{n} (t))$ for system (9) under Assumptions 1–3, such that the closed-loop system is GAS in probability.

Remark 1 Obviously, Assumption 1 indicates that the system (9) is indeed a high-order feedforward delayed nonlinear system (see, e.g., Ye, 1999, 2003, 2011; Zhang et al., 2011a, 2011b) for more details. Assumption 1 is more general than that in the existing results. In Ye (1999, 2003, 2011), $p_{i} = d_{i} (t) = 1, σ_{i} (t) = 0 / constant$ . In Zhang et al. (2011b), $p_{i} \geq 1, d_{i} (t) = 1, σ_{i} (t) \equiv constant$ . Furthermore, the nonlinear growth orders in Qian and Li (2006) and Zhang et al. (2010, 2011a, 2011b) are just fixed points.

Remark 2 For stochastic feedforward systems, Zhao and Xie (2013) tentatively solved the state feedback control for a class of stochastic high-order feedforward systems with $p_{i} = p$ . Then, the stabilization problem was dealt by Liu and Xie (2013) for the stochastic feedforward nonlinear time-delay systems with $p_{i} = 1$ and a linear growth condition $| f_{i} | \leq a_{i} \sum_{j = i + 2}^{n + 1} (| z_{j} (t) | + | z_{j} (t - σ_{j} (t)) |)$ . However, in this paper, the restrictions on $p_{i} = p$ and control coefficients are greatly relaxed and the nonlinear growth orders are the interval instead of fixed points. How to determine the value range of the systems homogeneous degree $τ$ and design a state feedback stabilizer is one of the difficulties in this paper.

Controller design and stability analysis

In this section, a state feedback controller is explicitly constructed for the nonlinear system (9). The adding one power integrator technique and homogeneous domination approach are used for state feedback stabilization. The design procedure can be divided into two steps: (i) we first design a state feedback controller for the nominal system without the drift and diffusion functions using the adding one power integrator technique; (ii) then, by introducing a gain, a scaled controller is proposed to guarantee global asymptotic stability in probability of the closed-loop system.

For simplicity, we assume $τ = \frac{m}{n}$ in this paper, with $m$ being an even integer and $n$ being an odd integer. Based on this assumption, it is obvious that $r_{i}$ is a ratio of odd integers. Furthermore, it should be emphasized that the obtained results in this paper can be extended for general $r_{i}$ by a similar technique in Qian and Li (2006).

Homogeneous state feedback control of the nominal nonlinear system

In this section, we firstly introduce a key lemma, which avoids the zero-division problem and serves as a basis in the following design procedure.

Lemma 7 If $p_{i} \in R_{odd}^{\geq 3}$ , $τ \in [d_{0}, + \infty)$ are guaranteed, then we can find $l_{0}, μ_{0} \in R_{odd}^{+}$ , such that

max_{1 \leq i \leq n} {2 r_{i}, \frac{r_{i} + τ}{l_{0}}} \leq μ_{0} \leq r_{n} + τ

holds.

Proof. Based on $r_{1} = 1, r_{i + 1} = \frac{r_{i} + τ}{p_{i}}$ , it yields

\begin{array}{l} \begin{array}{l} r_{i} & = \frac{1}{p_{1} \dots p_{i - 1}} + (\sum_{s = 1}^{i - 1} \frac{1}{p_{s} \dots p_{i - 1}}) τ, i = 2, \dots, n . \end{array} \end{array}

(10)

At first, by virtue of the apology in Li et al. (2011), we will prove that there exists $l_{0} \in R_{odd}^{+}$ such that $max_{1 \leq i \leq n} {2 r_{i}, \frac{r_{i} + τ}{l_{0}}} \leq r_{n} + τ$ holds for any $p_{i} \in R_{odd}^{\geq 3}$ , $τ \in [d_{0}, + \infty)$ .

$(a_{1})$ : When $i = 1$ , $τ \geq d_{1}$ , from (10) it deduces that

\begin{array}{l} \begin{array}{l} r_{n} + τ & = \frac{1}{p_{1} \dots p_{n - 1}} + (1 + \sum_{s = 1}^{n - 1} \frac{1}{p_{s} \dots p_{n - 1}}) τ \geq 2 r_{1} . \end{array} \end{array}

(11)

$(a_{2})$ : When $i = 2, \dots, n$ , $τ \geq d_{i}$ , on the one hand, one gets

\begin{matrix} r_{n} + τ & = \frac{1}{p_{1} \dots p_{n - 1}} + (1 + \sum_{s = 1}^{n - 1} \frac{1}{p_{s} \dots p_{n - 1}}) τ \\ = (1 + \sum_{s = 1}^{n - 1} \frac{1}{p_{s} \dots p_{n - 1}} - \sum_{s = 1}^{i - 1} \frac{2}{p_{s} \dots p_{i - 1}}) τ \\ + \sum_{s = 1}^{i - 1} \frac{2}{p_{s} \dots p_{i - 1}} τ + \frac{1}{p_{1} \dots p_{n - 1}} . \end{matrix}

(12)

On the other hand, due to $p_{i} \in R_{odd}^{\geq 3}$ , we obtain

\sum_{s = 1}^{i - 1} \frac{1}{p_{s} \dots p_{i - 1}} \leq \sum_{s = 1}^{i - 1} {(\frac{1}{3})}^{i - s} \leq \sum_{s = 1}^{\infty} {(\frac{1}{3})}^{s} = \frac{1}{2} \leq \frac{1}{2} + \frac{1}{2} \sum_{s = 1}^{n - 1} \frac{1}{p_{s} \dots p_{n - 1}} .

(13)

Substituting (13) into (12) renders

\begin{array}{l} \begin{array}{l} r_{n} + τ & \geq (1 + \sum_{s = 1}^{n - 1} \frac{1}{p_{s} \dots p_{n - 1}} - \sum_{s = 1}^{i - 1} \frac{2}{p_{s} \dots p_{i - 1}}) d_{i} + \sum_{s = 1}^{i - 1} \frac{2}{p_{s} \dots p_{i - 1}} τ + \frac{1}{p_{1} \dots p_{n - 1}} & \geq 2 r_{i} . \end{array} \end{array}

(14)

$(a_{3})$ : Under an appropriate choice of $l_{0} \in R_{odd}^{+}$ , $\frac{r_{i} + τ}{l_{0}} \leq μ_{0} \leq r_{n} + τ$ can be satisfied for any $1 \leq i \leq n$ .

Combining $(a_{1})$ , $(a_{2})$ with $(a_{3})$ leads to $max_{1 \leq i \leq n} {2 r_{i}, \frac{r_{i} + τ}{l_{0}}} \leq r_{n} + τ$ . According to the denseness of the real number, it is obvious that there is a $μ_{0} \in R_{odd}^{+}$ such that Lemma 7 holds.

To begin with, we embark on a homogeneous state feedback controller design for the nominal chain of power integrators:

\begin{matrix} d x_{1} & = d_{1} (t) x_{2}^{p_{1}} dt, \\ d x_{2} & = d_{2} (t) x_{3}^{p_{2}} dt, \\ ⋮ \\ d x_{n} & = d_{n} (t) u^{p_{n}} dt . \end{matrix}

(15)

Initial step: Let $η_{1} = x_{1}^{\frac{μ_{0}}{r_{1}}}$ , $x_{1}^{*} = 0$ , $q_{1} = 4 l_{0} μ_{0} - r_{1} - τ$ and consider the Lyapunov function $V_{1} = \int_{x_{1}^{*}}^{x_{1}} {(s^{\frac{μ_{0}}{r_{1}}} - x_{1}^{* \frac{μ_{0}}{r_{1}}})}^{\frac{q_{1}}{μ_{0}}} ds$ . In view of (15), the infinitesimal generator of $V_{1}$ satisfies

\begin{array}{l} \begin{array}{l} ℒ V_{1} (x_{1}) & = d_{1} (t) x_{1}^{\frac{q_{1}}{r_{1}}} x_{2}^{p_{1}} = d_{1} (t) η_{1}^{\frac{q_{1}}{μ_{0}}} x_{2}^{p_{1}} . \end{array} \end{array}

Choosing the first virtual controller

x_{2}^{*} = - {(\frac{π_{11}}{λ_{1}})}^{\frac{1}{p_{1}}} η_{1}^{\frac{r_{1} + τ}{p_{1} μ_{0}}} \overset{Δ}{=} - β_{1}^{\frac{r_{2}}{μ_{0}}} η_{1}^{\frac{r_{2}}{μ_{0}}}

leads to

\begin{array}{l} \begin{array}{l} ℒ V_{1} (x_{1}) & \leq - π_{11} η_{1}^{4 l_{0}} + d_{1} (t) η_{1}^{\frac{q_{1}}{μ_{0}}} (x_{2}^{p_{1}} - x_{2}^{* p_{1}}) . \end{array} \end{array}

Inductive step: Suppose at step $i - 1$ one has chosen a positive definite and proper Lyapunov function $V_{i - 1} ({\bar{x}}_{i - 1}) : R^{i - 1} \to R$ , which is $C^{2}$ , and a set of virtual controllers $x_{1}^{*}, \dots, x_{i}^{*}$ defined by

\begin{matrix} \begin{matrix} x_{1}^{*} & = 0, & η_{1} = x_{1}^{\frac{μ_{0}}{r_{1}}}, \\ x_{2}^{*} & = - β_{1}^{\frac{r_{2}}{μ_{0}}} η_{1}^{\frac{r_{2}}{μ_{0}}}, & η_{2} = x_{2}^{\frac{μ_{0}}{r_{2}}} - {x_{2}^{*}}^{\frac{μ_{0}}{r_{2}}}, \\ ⋮ & ⋮ \\ x_{i}^{*} & = - β_{i - 1}^{\frac{r_{i}}{μ_{0}}} η_{i - 1}^{\frac{r_{i}}{μ_{0}}}, & η_{i} = x_{i}^{\frac{μ_{0}}{r_{i}}} - {x_{2}^{*}}^{\frac{μ_{0}}{r_{i}}}, \end{matrix} \end{matrix}

(16)

such that

\begin{array}{l} \begin{array}{l} ℒ V_{i - 1} ({\bar{x}}_{i - 1}) & \leq - \sum_{j = 1}^{i - 1} π_{i - 1, j} η_{j}^{4 l_{0}} + d_{i - 1} (t) η_{i - 1}^{\frac{q_{i - 1}}{μ_{0}}} (x_{i}^{p_{i - 1}} - {x_{2}^{*}}^{p_{i - 1}}), \end{array} \end{array}

(17)

where $β_{1}, \dots, β_{i - 1}$ , $π_{i - 1, 1}, \dots, π_{i - 1, i - 1}$ are positive constants, and $q_{i - 1} = 4 l_{0} μ_{0} - τ - r_{i - 1}$ .

In the sequel, we will prove that (17) still holds for the ith Lyapunov function:

\begin{matrix} V_{i} ({\bar{x}}_{i}) & = V_{i - 1} ({\bar{x}}_{i - 1}) + Ψ_{i} ({\bar{x}}_{i}) \\ = V_{i - 1} ({\bar{x}}_{i - 1}) + \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}}} ds, q_{i} = 4 l_{0} μ_{0} - τ - r_{i} . \end{matrix}

From (17), one obtains

\begin{matrix} L V_{i} \leq & - \sum_{j = 1}^{i - 1} π_{i - 1, j} η_{j}^{4 l_{0}} + d_{i} (t) η_{i}^{\frac{q_{i}}{μ_{0}}} x_{i + 1}^{p_{i}} + d_{i - 1} (t) η_{i - 1}^{\frac{q_{i - 1}}{μ_{0}}} (x_{i}^{p_{i - 1}} - x_{i}^{* p_{i - 1}}) \\ - \frac{q_{i}}{μ_{0}} \sum_{j = 1}^{i - 1} \frac{\partial {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}} d_{j} (t) x_{j + 1}^{p_{j}} \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}} - 1} ds . \end{matrix}

(18)

Next, we focus on the last two terms in the right-hand side of (18).

When $\frac{r_{i} p_{i - 1}}{μ_{0}} \leq 1$ , Lemma 4 renders

\begin{matrix} | x_{i}^{p_{i - 1}} - x_{i}^{* p_{i - 1}} | & = | {(x_{i}^{\frac{μ_{0}}{r_{i}}})}^{\frac{r_{i} p_{i - 1}}{μ_{0}}} - {(x_{i}^{* \frac{μ_{0}}{r_{i}}})}^{\frac{r_{i} p_{i - 1}}{μ_{0}}} | \\ \leq 2^{1 - \frac{r_{i} p_{i - 1}}{μ_{0}}} | (x_{i}^{\frac{μ_{0}}{r_{i}}}) - (x_{i}^{* \frac{μ_{0}}{r_{i}}}) |^{\frac{r_{i} p_{i - 1}}{μ_{0}}} \\ {= 2}^{1 - \frac{r_{i} p_{i - 1}}{μ_{0}}} | η_{i} |^{\frac{r_{i} p_{i - 1}}{μ_{0}}} . \end{matrix}

(19)

When $\frac{r_{i} p_{i - 1}}{μ_{0}} \geq 1$ , Lemma 4 yields

\begin{matrix} | x_{i}^{p_{i - 1}} - {x_{i}^{*}}^{p_{i - 1}} | & \leq C | x_{i}^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}} | (| x_{i}^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}} |^{\frac{r_{i} p_{i - 1}}{μ_{0}} - 1} + | {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}} |^{\frac{r_{i} p_{i - 1}}{μ_{0}} - 1}) \\ = C | η_{i} | (| η_{i} |^{\frac{r_{i} p_{i - 1}}{μ_{0}} - 1} + | β_{i - 1} η_{i - 1} |^{\frac{r_{i} p_{i - 1}}{μ_{0}} - 1}) \\ \leq k_{i 1} | η_{i - 1} |^{\frac{r_{i} p_{i - 1}}{μ_{0}}} + {\bar{k}}_{i 1} | η_{i} |^{\frac{r_{i} p_{i - 1}}{μ_{0}}}, \end{matrix}

(20)

where $C, k_{i 1}, {\bar{k}}_{i 1}$ are positive constants.

On the basis of (19) and (20), one has

\begin{matrix} | d_{i - 1} (t) η_{i - 1}^{\frac{q_{i - 1}}{μ_{0}}} (x_{i}^{p_{i - 1}} - {x_{i}^{*}}^{p_{i - 1}}) | \\ \leq ν_{i - 1} | η_{i - 1} |^{\frac{q_{i - 1}}{μ_{0}}} (k_{i 1} | η_{i - 1} |^{\frac{r_{i} p_{i - 1}}{μ_{0}}} + {\tilde{k}}_{i 1} | η_{i} |^{\frac{r_{i} p_{i - 1}}{μ_{0}}}) \\ \leq δ_{i, i - 1, 1} η_{i - 1}^{4 l_{0}} + b_{i 1} η_{i}^{4 l_{0}} \end{matrix}

(21)

with ${\tilde{k}}_{i 1} = max {2^{1 - \frac{r_{i} p_{i - 1}}{μ_{0}}}, {\bar{k}}_{i 1}}, δ_{i, i - 1, 1} \geq 0, b_{i 1} \geq 0$ .

With the help of (16), Lemmas 4 and 6, one gets

\begin{matrix} | - \frac{q_{i}}{μ_{0}} \sum_{j = 1}^{i - 1} \frac{\partial {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}} d_{j} (t) x_{j + 1}^{p_{j}} \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}} - 1} ds | \\ \leq \sum_{j = 1}^{i - 1} M (ν_{j}) | x_{j} |^{\frac{μ_{0}}{r_{j}} - 1} | x_{j + 1} |^{p_{j}} | η_{i} |^{\frac{q_{i}}{μ_{0}} - 1} | x_{i} - x_{i}^{*} | \\ \leq \sum_{j = 1}^{i - 1} \bar{M} (ν_{j}) | x_{j} |^{\frac{μ_{0}}{r_{j}} - 1} | x_{j + 1} |^{p_{j}} | η_{i} |^{\frac{4 l_{0} μ_{0} - τ}{μ_{0}} - 1} \\ \leq \sum_{j = 1}^{i - 1} \bar{M} (ν_{j}) | η_{j} - β_{j - 1} η_{j - 1} |^{1 - \frac{r_{j}}{μ_{0}}} | η_{j + 1} - β_{j} η_{j} |^{\frac{r_{j + 1} p_{j}}{μ_{0}}} | η_{i} |^{\frac{4 l_{0} μ_{0} - τ}{μ_{0}} - 1} \\ \leq \sum_{j = 1}^{i - 1} \hat{M} (ν_{j}) (| η_{j} |^{1 - \frac{r_{j}}{μ_{0}}} + | η_{j - 1} |^{1 - \frac{r_{j}}{μ_{0}}}) (| η_{j + 1} |^{\frac{r_{j + 1} p_{j}}{μ_{0}}} + | η_{j} |^{\frac{r_{j + 1} p_{j}}{μ_{0}}}) | η_{i} |^{\frac{4 l_{0} μ_{0} - τ}{μ_{0}} - 1} \\ \leq \sum_{j = 1}^{i - 1} δ_{i, j, 2} η_{l}^{4 l_{0}} + b_{i 2} η_{i}^{4 l_{0}}, \end{matrix}

(22)

where $δ_{i, j, 2}, j = 1, \dots, i - 1$ and $b_{i 2}, M (ν_{j}), \bar{M} (ν_{j}), \hat{M} (ν_{j})$ are positive constants.

Substituting (21) and (22) into (18) produces

\begin{matrix} L V_{i} \leq & - \sum_{j = 1}^{i - 1} π_{i - 1, j} η_{j}^{4 l_{0}} + d_{i} (t) η_{i}^{\frac{q_{i}}{μ_{0}}} x_{i + 1}^{p_{i}} + δ_{i, i - 1, 1} η_{i - 1}^{4 l_{0}} + b_{i 1} η_{i}^{4 l_{0}} \\ + \sum_{j = 1}^{i - 1} δ_{i, j, 2} η_{j}^{4 l_{0}} + b_{i 2} η_{i}^{4 l_{0}} . \end{matrix}

(23)

Clearly, with the choice of the virtual controller and the parameters satisfying

\begin{array}{l} \begin{array}{l} x_{i + 1}^{*} & = - {(\frac{π_{i i} + b_{i 1} + b_{i 2}}{λ_{i}})}^{\frac{1}{p_{i}}} η_{i}^{\frac{r_{i + 1}}{μ_{0}}} ≜ - β_{i}^{\frac{r_{i + 1}}{μ_{0}}} η_{i}^{\frac{r_{i + 1}}{μ_{0}}}, \\ π_{i j} & = {\begin{array}{r} π_{i - 1, j} - δ_{i, j, 2} > 0, & j = 1, \dots, i - 2, \\ π_{i - 1, j} - δ_{i, j, 1} - δ_{i, j, 2} > 0, & j = i - 1, \\ π_{i, j} > 0, & j = i, \end{array} \end{array} \end{array}

(24)

(23) becomes

\begin{matrix} \begin{matrix} L V_{i} \leq - \sum_{j = 1}^{i} π_{i, j} η_{j}^{4 l_{0}} + d_{i} (t) η_{i}^{\frac{q_{i}}{μ_{0}}} (x_{i + 1}^{p_{i}} - x_{i + 1}^{* p_{i}}) . \end{matrix} \end{matrix}

(25)

Therefore, in the last step, according to (23), we can find a Lyapunov function $V_{n} (x_{1}, \dots, x_{n})$ and a set of positive constants $π_{n - 1, j}, δ_{n, n - 1, 1}, δ_{n, j, 2}, b_{n 1}, b_{n 2}, j = 1, \dots, n - 1$ such that

\begin{matrix} \begin{matrix} L V_{n} \leq & - \sum_{j = 1}^{n - 1} π_{n - 1, j} η_{j}^{4 l_{0}} + d_{n} (t) η_{n}^{\frac{q_{n}}{μ_{0}}} u^{p_{n}} + δ_{n, n - 1, 1} η_{n - 1}^{4 l_{0}} + b_{n 1} η_{n}^{4 l_{0}} + \sum_{j = 1}^{n - 1} δ_{n, j, 2} η_{j}^{4 l_{0}} + b_{n 2} η_{n}^{4 l_{0}} . \end{matrix} \end{matrix}

(26)

With the above relations in mind, we have

\begin{matrix} L V_{n} \leq & - \sum_{j = 1}^{n} π_{n, j} η_{j}^{4 l_{0}} + d_{n} (t) η_{n}^{\frac{q_{n}}{μ_{0}}} (u^{p_{n}} - u^{* p_{n}}) \\ \leq - \sum_{j = 1}^{n} π_{n, j} η_{j}^{4 l_{0}}, \end{matrix}

(27)

by choosing

\begin{matrix} u = u^{*} & = - {(\frac{π_{nn} + b_{n 1} + b_{n 2}}{λ_{n}})}^{\frac{1}{p_{n}}} η_{n}^{\frac{r_{n + 1}}{μ_{0}}} \overset{Δ}{=} - β_{n}^{\frac{r_{n + 1}}{μ_{0}}} η_{n}^{\frac{r_{n + 1}}{μ_{0}}} \\ = - {({\bar{β}}_{n} x_{n}^{\frac{μ_{0}}{r_{n}}} + {\bar{β}}_{n - 1} x_{n - 1}^{\frac{μ_{0}}{r_{n - 1}}} + \dots + {\bar{β}}_{1} x_{1}^{\frac{μ_{0}}{r_{1}}})}^{\frac{r_{n + 1}}{μ_{0}}}, \\ {\bar{β}}_{j} = β_{n} & \dots β_{j}, j = 1, \dots, n . \end{matrix}

(28)

Remark 3 It is well known that only a positive definite, proper and $C^{1}$ Lyapunov function is needed for the controller design and analysis in the deterministic work. However, for stochastic systems, because of the appearance of the Hessian term, the Lyapunov function

\begin{array}{l} \begin{array}{l} V_{i} ({\bar{x}}_{i}) & = V_{i - 1} ({\bar{x}}_{i - 1}) + Ψ_{i} ({\bar{x}}_{i}) = V_{i - 1} ({\bar{x}}_{i - 1}) + \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}}} d s \end{array} \end{array}

must be positive definite, proper and at least $C^{2}$ .

Now, we firstly prove that $V_{i}$ is positive definite and proper by two cases.

(1): When $x_{i}^{*} \geq x_{i}$ , on the basis of Lemma 4 and $\frac{μ_{0}}{r_{i}} \in R_{odd}^{\geq 2}$ , one has

\begin{matrix} Ψ_{i} ({\bar{x}}_{i}) & = \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}}} ds \\ = \int_{x_{i}}^{x_{i}^{*}} {(x_{i}^{* \frac{μ_{0}}{r_{i}}} - s^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}}} ds \\ \geq {(\frac{1}{2^{\frac{μ_{0}}{r_{i}} - 1}})}^{\frac{q_{i}}{μ_{0}}} \int_{x_{i}}^{x_{i}^{*}} {(x_{i}^{*} - s)}^{\frac{q_{i}}{r_{i}}} ds \\ = \frac{2^{\frac{q_{i}}{μ_{0}}} r_{i}}{2^{\frac{q_{i}}{r_{i}}} (q_{i} + r_{i})} {(x_{i}^{*} - x_{i})}^{\frac{4 l_{0} μ_{0} - τ}{r_{i}}} . \end{matrix}

(29)

(2): When $x_{i}^{*} \leq x_{i}$ , we can obtain the same result in (29) with a similar process.

Therefore, ( $1$ ) and ( $2$ ) imply that $Ψ_{i} ({\bar{x}}_{i}), V_{i} ({\bar{x}}_{i})$ are positive definite and proper functions.

In the following, we prove that $V_{i} ({\bar{x}}_{i})$ is $C^{2}$ . For $j, k = 1, \dots, i - 1$ , a direct calculation gives

\begin{matrix} \frac{\partial Ψ_{i}}{\partial x_{i}} = η_{i}^{\frac{q_{i}}{μ_{0}}}, \frac{\partial^{2} Ψ_{i}}{\partial x_{i}^{2}} = \frac{q_{i}}{r_{i}} η_{i}^{\frac{q_{i}}{μ_{0}} - 1} x_{i}^{\frac{μ_{0}}{r_{i}} - 1}, \\ \frac{\partial^{2} Ψ_{i}}{\partial x_{i} \partial x_{j}} = \frac{\partial^{2} Ψ_{i}}{\partial x_{j} \partial x_{i}} = \frac{q_{i}}{μ_{0}} η_{i}^{\frac{q_{i}}{μ_{0}} - 1} (- \frac{\partial {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}}), \\ \frac{\partial Ψ_{i}}{\partial x_{j}} = - \frac{q_{i}}{μ_{0}} \frac{\partial {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}} \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}} - 1} ds, \\ \frac{\partial^{2} Ψ_{i}}{\partial x_{j}^{2}} = - \frac{q_{i}}{μ_{0}} \frac{\partial^{2} {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}^{2}} \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}} - 1} ds \\ + \frac{q_{i} (q_{i} - μ_{0})}{μ_{0}^{2}} {(\frac{\partial {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}})}^{2} \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}} - 2} ds, \\ \frac{\partial^{2} Ψ_{i}}{\partial x_{j} \partial x_{k}} = \frac{\partial^{2} Ψ_{i}}{\partial x_{k} \partial x_{j}} = \frac{q_{i} (q_{i} - μ_{0})}{μ_{0}^{2}} (\frac{\partial {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}}) (\frac{\partial {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{k}}) \int_{x_{i}^{*}}^{x_{i}} {(s^{\frac{μ_{0}}{r_{i}}} - {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}})}^{\frac{q_{i}}{μ_{0}} - 2} ds, \\ \frac{\partial {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}} = - \frac{μ_{0}}{r_{j}} β_{i - 1} \dots β_{j} x_{j}^{\frac{μ_{0}}{r_{j}} - 1}, \frac{\partial^{2} {x_{i}^{*}}^{\frac{μ_{0}}{r_{i}}}}{\partial x_{j}^{2}} = - \frac{μ_{0} (μ_{0} - r_{j})}{r_{j}} β_{i - 1} \dots β_{j} x_{j}^{\frac{μ_{0}}{r_{j}} - 2} . \end{matrix}

(30)

Based on Lemma 7, one has

\begin{matrix} \begin{matrix} \frac{q_{i}}{μ_{0}} - 2 \geq 0, \frac{μ_{0}}{r_{j}} - 2 \geq 0, \end{matrix} \end{matrix}

(31)

from which we conclude that $V_{i} ({\bar{x}}_{i})$ is $C^{2}$ .

By denoting $X = {(x_{1}, \dots, x_{n})}^{T}$ , the closed-loop system (15) and (28) can be rewritten in the following compact form:

\begin{matrix} \begin{matrix} dX & = E (X) dt = {[x_{2}^{p_{1}}, \dots, x_{n}^{p_{n - 1}}, u^{p_{n}} (x)]}^{T} dt, \end{matrix} \end{matrix}

(32)

which is homogeneous of degree $τ$ with respect to $Δ = (\underset{for x_{1}, \dots, x_{n}}{\underset{︸}{r_{1}, . . ., r_{n}}})$ . Following a similar method to that of Xie and Liu (2012), it can be shown that

\begin{matrix} \begin{matrix} V (X) & = \sum_{j = 1}^{n} \int_{x_{j}^{*}}^{x_{j}} {(s^{\frac{μ_{0}}{r_{j}}} - x_{i}^{* \frac{μ_{0}}{r_{j}}})}^{\frac{q_{j}}{μ_{0}}} ds \end{matrix} \end{matrix}

(33)

is homogeneous of degree $4 l_{0} μ_{0} - τ$ about $Δ$ .

Stability analysis

To state the main result in this paper, we first introduce the following coordinate transformation:

x_{1} = z_{1}, x_{i} = \frac{z_{i}}{L^{s_{i}}}, u^{p_{n}} = \frac{v^{p_{n}}}{L^{s_{n} + 1}},

(34)

where $s_{1} = 0, s_{i} = \frac{s_{i - 1} + 1}{p_{i - 1}}, i = 2, \dots, n$ , $0 < L < 1$ is a constant to be determined later.

It follows from (34) that the system (9) can be transformed into

\begin{matrix} d x_{1} & = {Lx}_{2}^{p_{1}} dt + f_{1} (x, x (t - σ (t))) + g_{1}^{T} (x, x (t - σ (t))) d ω, \\ d x_{2} & = {Lx}_{3}^{p_{2}} dt + \frac{f_{2} (x, x (t - σ (t)))}{L^{s_{2}}} + \frac{g_{2}^{T} (x, x (t - σ (t)))}{L^{s_{2}}} d ω, \\ ⋮ \\ d x_{n} & = L u^{p_{n}} + \frac{f_{n} (x, x (t - σ (t)))}{L^{s_{n}}} + \frac{g_{n}^{T} (x, x (t - σ (t)))}{L^{s_{n}}} d ω . \end{matrix}

(35)

The closed-loop system (35) and (28) can be expressed as

\begin{matrix} \begin{matrix} dX & = LE (X) dt + Fdt + G^{T} d ω, \end{matrix} \end{matrix}

(36)

where $F = {(f_{1}, \frac{f_{2}}{L^{s_{2}}}, \dots, \frac{f_{n}}{L^{s_{n}}})}^{T}$ , $G = (g_{1}, \frac{g_{2}}{L^{s_{2}}}, \dots, \frac{g_{n}}{L^{s_{n}}})$ . Based on the above discussion, we are ready to prove the main result of this paper.

Theorem 1 Under Assumptions 1–3, there is a state feedback controller $v^{p_{n}} = u^{p_{n}} L^{s_{n} + 1}$ such that the closed-loop system consisting of (9), (28) and (34) has a global unique solution and the equilibrium $z = 0$ is GAS in probability.

Proof. The proof of Theorem 1 is divided into the following four steps. Steps 1–3 ensure that the closed-loop system (35) and (28) has a global unique solution and the equilibrium $x = 0$ is GAS in probability. With the help of (34), the main results of this paper are obtained in Step 4.

Step 1: Firstly, we prove that the locally Lipschitz condition is satisfied for the closed-loop system (35) and (28). From (28), for $j = 1, \dots, n$ , one obtains

\begin{matrix} \begin{matrix} \frac{\partial u^{p_{n}} (x)}{\partial x_{j}} & = - \frac{r_{n} + τ}{r_{j}} {\bar{β}}_{j} x_{j}^{\frac{μ_{0}}{r_{j}} - 1} {({\bar{β}}_{n} x_{n}^{\frac{μ_{0}}{r_{n}}} + {\bar{β}}_{n - 1} x_{n - 1}^{\frac{μ_{0}}{r_{n - 1}}} + \dots + {\bar{β}}_{1} x_{1}^{\frac{μ_{0}}{r_{1}}})}^{\frac{r_{n} + τ}{μ_{0}} - 1} \end{matrix} \end{matrix}

(37)

By Lemma 7, it results in $\frac{μ_{0}}{r_{j}} - 1 \geq 0$ , $\frac{r_{n} + τ}{μ_{0}} - 1 \geq 0$ , which imply that $u^{p_{n}} (x)$ is $C^{1}$ . Notice that $f_{i}, g_{i}$ are assumed to be locally Lipschitz for $i = 1, \dots, n - 1$ . From the above discussion, it obviously renders that the closed-loop system satisfies the locally Lipschitz condition.

Step 2: Construct a Lyapunov–Krasovskii functional

\begin{array}{l} \begin{array}{l} W (X) & = V (X) + \sum_{j = 1}^{n} \frac{L^{1 + ε_{0}} (a_{j 0} + b_{j 0})}{1 - σ_{j}} \int_{t - σ_{j} (t)}^{t} ∥ X (s) ∥_{Δ}^{4 l_{0} μ_{0}} d s, \end{array} \end{array}

(38)

which is positive definite, proper and $C^{2}$ on $X$ , where $ε_{0}, a_{j 0}, b_{j 0}$ are positive parameters to be determined later. According to Lemma 4.3 in Khalil (2002), there are two $K_{\infty}$ functions $α_{1}$ and $α_{21}$ such that

\begin{matrix} \begin{matrix} α_{1} (| X (t) |) \leq V (X) \leq α_{21} (| X (t) |) . \end{matrix} \end{matrix}

(39)

Then, with the notation in Xie and Liu (2012), it is easy to verify that

\begin{matrix} \begin{matrix} \sum_{j = 1}^{n} \frac{L^{1 + ε_{0}} (a_{j 0} + b_{j 0})}{1 - σ_{j}} \int_{t - σ_{j} (t)}^{t} ∥ X (s) ∥_{Δ}^{4 l_{0} μ_{0}} ds \leq α_{22} (sup_{- σ_{0} \leq s \leq 0} | X (t + s) |), \end{matrix} \end{matrix}

(40)

where $α_{22}$ is a $K_{\infty}$ function and $σ_{0} = max_{1 \leq j \leq n} {σ_{j 0}}$ . Based on $α_{21} (| X (t) |) \leq α_{21} (sup_{- σ_{0} \leq s \leq 0} | X (t + s) |)$ , one obtains

\begin{matrix} \begin{matrix} α_{1} (| X (t) |) \leq W (X) \leq α_{2} (sup_{- σ_{0} \leq s \leq 0} | X (t + s) |), \end{matrix} \end{matrix}

(41)

where $α_{2} = α_{21} + α_{22}$ .

Step 3: Since $V (X)$ and $E (X)$ are homogenous of degree $4 l_{0} μ_{0} - τ$ and $τ$ , respectively, by Lemmas 2 and 3, one has

\begin{matrix} \begin{matrix} \frac{\partial W (X)}{\partial X} LE (X) & = \frac{\partial V (X)}{\partial X} LE (X) \\ \leq - L γ_{1} ∥ X (s) ∥_{Δ}^{4 l_{0} μ_{0}}, \end{matrix} \end{matrix}

(42)

where $γ_{1} > 0$ is a constant.

Combining Assumption 1 and (34) produces

\begin{matrix} \begin{matrix} | \frac{f_{i}}{L^{s_{i}}} | & \leq a_{i} \sum_{j = i + 2}^{n + 1} (\frac{| L^{s_{j}} x_{j} (t) |^{\frac{r_{i} + τ}{r_{j}}} + | L^{s_{j}} x_{j} (t - σ_{j} (t)) |^{\frac{r_{i} + τ}{r_{j}}}}{L^{s_{i}}}), \end{matrix} \end{matrix}

(43)

and the power of $L$ in (43) is

\begin{matrix} s_{j} \frac{r_{i} + τ}{r_{j}} - s_{i} = 1 + \frac{s_{j} (r_{i} + τ) - (s_{i} + 1) r_{j}}{r_{j}} \\ = 1 + \frac{p_{i}}{r_{j}} (s_{j} r_{i + 1} - s_{i + 1} r_{j}) . \end{matrix}

(44)

Owing to $s_{1} = 0, s_{i} = \frac{s_{i - 1} + 1}{p_{i - 1}}$ , one has

\begin{matrix} \begin{matrix} s_{i} & = \sum_{s = 1}^{i - 1} \frac{1}{p_{s} \dots p_{i - 1}}, & i = 2, \dots, n . \end{matrix} \end{matrix}

(45)

For $j \geq i + 2$ , combining (10) and (45) gives

\begin{matrix} s_{j} r_{i + 1} - s_{i + 1} r_{j} & = s_{j} (\frac{1}{p_{1} \dots p_{i}} + s_{i + 1} τ) - s_{i + 1} (\frac{1}{p_{1} \dots p_{j - 1}} + s_{j} τ) \\ = \frac{s_{j}}{p_{1} \dots p_{i}} - \frac{s_{i + 1}}{p_{1} \dots p_{j - 1}} > 0 . \end{matrix}

(46)

In summary, there exists a positive constant

ε_{1} = min {\frac{p_{i}}{r_{j}} (s_{j} r_{i + 1} - s_{i + 1} r_{j}) : 1 \leq i \leq n - 1, i + 2 \leq j \leq n},

such that

\begin{matrix} | \frac{f_{i}}{L^{s_{i}}} | & \leq a_{i} L^{1 + ε_{1}} \sum_{j = i + 2}^{n + 1} (| x_{j} (t) |^{\frac{r_{i} + τ}{r_{j}}} + | x_{j} (t - σ_{j} (t)) |^{\frac{r_{i} + τ}{r_{j}}}) \\ \leq {\hat{a}}_{i} L^{1 + ε_{1}} (∥ X (t) ∥_{Δ}^{r_{i} + τ} + \sum_{j = i + 2}^{n} ∥ X (t - σ_{j} (t)) ∥_{Δ}^{r_{i} + τ}) \end{matrix}

(47)

with ${\hat{a}}_{i} > 0$ .

According to Lemma 3, we deduce that

\begin{matrix} | \frac{\partial V (X)}{\partial X} F | & \leq \sum_{i = 1}^{n - 1} | \frac{\partial V (X)}{\partial x_{i}} | | \frac{f_{i}}{L^{s_{i}}} | \\ \leq \sum_{i = 1}^{n - 1} {\hat{a}}_{i} L^{1 + ε_{1}} ∥ X (t) ∥_{Δ}^{4 l_{0} μ_{0} - r_{i} - τ} (∥ X (t) ∥_{Δ}^{r_{i} + τ} \\ + \sum_{j = i + 2}^{n} ∥ X (t - σ_{j} (t)) ∥_{Δ}^{r_{i} + τ}) \\ \leq L^{1 + ε_{1}} (a_{0} ∥ X (t) ∥_{Δ}^{4 l_{0} μ_{0}} + \sum_{i = 1}^{n} a_{i 0} ∥ X (t - σ_{i} (t)) ∥_{Δ}^{4 l_{0} μ_{0}}), \end{matrix}

(48)

where $a_{0}, a_{i 0}$ are positive constants.

Similar to (44)–(46), there is a positive constant

ε_{2} = min {\frac{s_{j} (2 r_{i} + τ) - (2 s_{i} + 1) r_{j}}{2 r_{j}} : 1 \leq i \leq n - 1, i + 2 \leq j \leq n},

such that

\begin{matrix} | \frac{g_{i}}{L^{s_{i}}} | & \leq b_{i} L^{\frac{1}{2} + ε_{2}} \sum_{j = i + 2}^{n + 1} (| x_{j} (t) |^{\frac{2 r_{i} + τ}{2 r_{j}}} + | x_{j} (t - σ_{j} (t)) |^{\frac{2 r_{i} + τ}{2 r_{j}}}) \\ \leq {\hat{b}}_{i} L^{\frac{1}{2} + ε_{2}} (∥ X (t) ∥_{Δ}^{r_{i} + \frac{τ}{2}} + \sum_{j = i + 2}^{n} ∥ X (t - σ_{j} (t)) ∥_{Δ}^{r_{i} + \frac{τ}{2}}) \end{matrix}

(49)

with ${\hat{b}}_{i} > 0$ .

By virtue of Lemma 4 and (49), one obtains

\begin{matrix} \frac{1}{2} Tr {G \frac{\partial^{2} V (X)}{\partial X^{2}} G^{T}} \leq \frac{1}{2} m \sqrt{m} \sum_{k, l = 1}^{n - 1} | \frac{\partial^{2} V (X)}{\partial x_{k} \partial x_{l}} | | \frac{g_{k}}{L^{s_{k}}} | | \frac{g_{l}}{L^{s_{l}}} | \\ \leq \tilde{b} L^{1 + 2 ε_{2}} \sum_{k, l = 1}^{n - 1} ∥ X (t) ∥^{4 l_{0} μ_{0} - τ - r_{k} - r_{l}} (∥ X (t) ∥_{Δ}^{r_{k} + \frac{τ}{2}} \\ + \sum_{j = k + 2}^{n} ∥ X (t - σ_{j} (t)) ∥_{Δ}^{r_{k} + \frac{τ}{2}}) (∥ X (t) ∥_{Δ}^{r_{l} + \frac{τ}{2}} + \sum_{j = l + 2}^{n} ∥ X (t - σ_{j} (t)) ∥_{Δ}^{r_{l} + \frac{τ}{2}}) \\ \leq L^{1 + 2 ε_{2}} (b_{0} ∥ X (t) ∥_{Δ}^{4 l_{0} μ_{0}} + \sum_{i = 1}^{n} b_{i 0} ∥ X (t - σ_{i} (t)) ∥_{Δ}^{4 l_{0} μ_{0}}), \end{matrix}

(50)

where $\tilde{b}, b_{0}, b_{i 0}$ are positive constants.

By choosing $ε_{0} = min {ε_{1}, 2 ε_{2}}$ and Definition 1, Assumption 2, (42), (48) and (50), one can obtain

\begin{matrix} L W (X) \leq \frac{\partial W (X)}{\partial X} LE (X) + \frac{\partial V (X)}{\partial X} F + \frac{1}{2} Tr {G \frac{\partial^{2} V (X)}{\partial X^{2}} G^{T}} \\ + \sum_{i = 1}^{n} L^{1 + ε_{0}} (a_{i 0} + b_{i 0}) (\frac{1}{1 - σ_{i}} ∥ X (t) ∥_{Δ}^{4 l_{0} μ_{0}} - ∥ X (t - σ_{i} (t)) ∥_{Δ}^{4 l_{0} μ_{0}}) \\ \leq - L (γ_{1} - L^{ε_{0}} (a_{0} + b_{0} + \sum_{i = 1}^{n} \frac{a_{i 0} + b_{i 0}}{1 - σ_{i}})) ∥ X (t) ∥_{Δ}^{4 l_{0} μ_{0}} . \end{matrix}

(51)

Apparently, by choosing $0 < L < min {{(\frac{γ_{1}}{a_{0} + b_{0} + \sum_{i = 1}^{n} \frac{a_{i 0} + b_{i 0}}{1 - σ_{i}}})}^{\frac{1}{ε_{0}}}, 1}$ , there is a constant $γ^{*}$ , such that $L W (X) \leq - γ^{*} ∥ X (t) ∥_{Δ}^{4 l_{0} μ_{0}}$ .

From Steps 1–3 and Lemma 1, one obtains that the closed-loop system (35) and (28) has a global unique solution and the equilibrium $x = 0$ is GAS in probability.

Step 4: Because of the equivalence in (34), then there is a global unique solution for the closed-loop system consisting of (9), (28) and $v^{p_{n}} = u^{p_{n}} L^{s_{n} + 1}$ and the equilibrium $z = 0$ is GAS in probability.

Remark 4 By adding a power integrator method and homogeneous domination technique, a state feedback stabilizer is constructed for a class of stochastic high-order feedforward nonlinear systems. To deal with the time-delay terms in this paper, we choose an appropriate Lyapunov–Krasovskii functional, which is not an easy work.

A simulation example

In this section, we provide a simulation example to show the effectiveness of our results.

Example 1 Consider the following stochastic high-order feedforward time-delay system:

\begin{matrix} d z_{1} = & d_{1} (t) z_{2}^{3} dt + \frac{1}{10} z_{3}^{5} (t - σ_{3} (t)) dt + \frac{1}{20} z_{3}^{\frac{10}{3}} (t - σ_{3} (t)) d ω, \\ d z_{2} = & d_{2} (t) x_{3}^{5} dt, \\ d z_{3} = & d_{3} (t) ν^{3} dt, \end{matrix}

where $σ_{3} (t) = \frac{1}{2} (\sin (t) + 1), d_{1} (t) = 2 | \sin (t) | + 1, d_{2} (t) = 5 co s^{2} (t) + 3, d_{3} (t) = 5$ .

Based on the definition $d_{i}$ , we obtain $d_{1} = \frac{29}{19}, d_{2} = 1, d_{3} = \frac{1}{11},$ then $τ \in [\frac{29}{19}, \infty) .$ In this example, by choosing $τ = 2$ , then $r_{1} = 1, r_{2} = 1, r_{3} = \frac{3}{5}$ . From Lemma 7, one has $l_{0} = 3, μ_{0} = \frac{11}{5}$ . It is easy to prove that Assumption 1 is satisfied with $a_{1} = \frac{1}{10}, b_{1} = \frac{1}{20}$ . Assumption 2 also holds with $\overset{\cdot}{σ} (t) \leq 0.5 < 1$ . Furthermore, the control coefficients satisfy $0 < 1 \leq d_{1} (t) \leq 3, 0 < 3 \leq d_{2} (t) \leq 8, d_{3} (t) \equiv 5$ .

Controller design: for the nominal chain of power integrators

\begin{matrix} d x_{1} & = d_{1} (t) x_{2}^{3} dt, \\ d x_{2} & = d_{2} (t) x_{3}^{5} dt, \\ d x_{n} & = d_{3} (t) u^{3} dt . \end{matrix}

(52)

Following the above design procedure, one obtains the controller

\begin{array}{l} \begin{array}{l} u & = - {({\bar{β}}_{3} x_{3}^{\frac{11}{3}} + {\bar{β}}_{2} x_{2}^{\frac{11}{5}} + {\bar{β}}_{1} x_{1}^{\frac{11}{5}})}^{\frac{13}{33}} . \end{array} \end{array}

(53)

Introducing the following coordinate transformation:

x_{1} = z_{1}, x_{2} = \frac{z_{2}}{L^{s_{2}}}, x_{3} = \frac{z_{3}}{L^{s_{3}}}, u^{3} = \frac{v^{3}}{L^{s_{3} + 1}},

(54)

where $s_{1} = 0, s_{2} = \frac{1}{3}, s_{3} = \frac{4}{15}$ , $0 < L < 1$ is a constant to be determined later and substituting (54) into (53) yields the actual controller:

\begin{array}{l} \begin{array}{l} ν & = - L^{\frac{19}{15}} {(L^{\frac{44}{45}} {\bar{β}}_{3} z_{3}^{\frac{11}{3}} + L^{\frac{11}{15}} {\bar{β}}_{2} z_{2}^{\frac{11}{5}} + {\bar{β}}_{1} z_{1}^{\frac{11}{5}})}^{\frac{13}{33}}, \end{array} \end{array}

(55)

from which one has

L W (X) \leq - γ^{*} ∥ X (t) ∥_{Δ}^{4 l_{0} μ_{0}},

(56)

where $X = {(x_{1}, x_{2}, x_{3})}^{T}, γ^{*} > 0$ , $W (X)$ is similar to that in (38). From Lemma 1 and (54), it is deduced that the closed-loop system has a global unique solution and the equilibrium $z = 0$ is GAS in probability.

In the simulation, we choose the gain $L = 0.001, {\bar{β}}_{3} = 20, {\bar{β}}_{2} = 120, {\bar{β}}_{1} = 1300$ and initial values $z_{1} (0) = 500, z_{2} (0) = - 20, z_{3} (0) = 80 .$ Figure 1 demonstrates the effectiveness of the control scheme. To explain more the global stability of the simulation example regardless of initial condition, Figures 2 and 3 give the response of the system states with $z_{1} (0) = - 500, z_{2} (0) = 200, z_{3} (0) = - 80$ and $z_{1} (0) = - 0.5, z_{2} (0) = 0.2, z_{3} (0) = - 8.0$ , respectively. Based on Theorem 1 and the choices of initial values, Figures 1 –3 effectively demonstrate that the equilibrium $z = 0$ is GAS in probability.

Conclusions

For a class of stochastic high-order feedforward nonlinear systems with different power orders, time-varying control coefficients and multiple time-varying delays, a state stabilizer has been recursively proposed on the basis of the adding one power integrator technique and the homogeneous domination approach. By constructing an appropriate Lyapunov–Krasovskii functional, GAS in probability of the equilibrium of the closed-loop system has been proved. Finally, the proposed method is demonstrated with an example.

Footnotes

Funding

This work was supported by the NSFC (grant numbers 61174137, 61203048, 61374086, 61374153), the 333 Project (grant number BRA2011143), the Qing Lan Project and the Program for Changjiang Scholars and Innovative Research Team in University.

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