Further results on adaptive state feedback stabilization for a class of stochastic feedforward nonlinear systems with time delays

Abstract

This paper further discusses the state feedback stabilization for stochastic high-order feedforward nonlinear systems with input time delay. By constructing the appropriate Lyapunov–Krasovskii functional, and using the variable transformation technique and the homogeneous domination idea, a state feedback controller is designed to ensure that the closed-loop system will be globally asymptotically stable in probability. Finally, an example is given to verify the effectiveness of the obtained analytical results.

Keywords

Stochastic feedforward systems Lyapunov–Krasovskii functional input time delay backstepping

Introduction

Feedforward systems (also called upper-triangular systems), which can be used in many engineering practical models, such as the cart–pendulum (Mazenc and Bowong, 2003) and planar vertical takeoff and landing aircraft systems (Teel, 1996), have been the subject of increasing attention over the past few decades (Krstic, 2010; Ding et al., 2011; Du et al., 2013; Qian and Li, 2006; Ye, 2011; Zhang et al., 2006, 2011a,b).

It is well known that time delay is widely encountered in practical engineering systems (Niculescu, 2001). The existence of time delay may cause instability or bad dynamic performance of the system (Kolmanovskii and Myshkis, 1992). Hence, the control design of time delay feedforward systems has received much attention, with many good results being produced in recent years. Bresh-Pietri and Krstic (2010) studied feedforward linear systems with time delay and designed an adaptive controller. Ye (2011) studied the adaptive stabilization of feedforward nonlinear systems with time delay. Zhang et al. (2006) considered the global asymptotic stabilization of feedforward nonlinear systems with a delay in the input. Based on the Lyapunov–Krasovskii theorem and the Lyapunov–Razumikhin theorem, Zhang et al. (2011a) designed a delay-independent feedback controller for a class of feedforward nonlinear time delay systems. Zhang et al. (2011b) investigated the problem of global stabilization for a class of high-order feedforward systems with time delay. More general feedforward nonlinear systems with time delay have been studied, using different design and analysis methods, in recent years.

However, all of the aforementioned results are limited to deterministic systems. Since stochastic noises and uncertain signals widely exist in practical engineering, in recent years, based on the backstepping method, many results have been obtained for stochastic feedback (also called lower-triangular) nonlinear systems (Bai et al., 2017; Deng et al., 2001; Krstic and Deng, 1998; Liu and Xie, 2012; Liu et al., 2008; Qin, 2013; Wang et al., 2017; Xie and Duan, 2010; Yin et al., 2017a,b). In contrast with stochastic feedback systems, stochastic feedforward nonlinear systems have been developed slowly, owing to their inherent structure and a lack of effective design methods. However, many practical physical devices can be described using feedforward systems. Thus, the study of feedforward systems with stochastic disturbances is a very important and significant topic. Stochastic feedforward nonlinear systems have received much attention and there has been remarkable development on the subject in recent years. Liu et al. (2016b) and Zhao and Xie (2013) handled the state feedback of stochastic high-order feedforward nonlinear systems using a homogeneous domination method. However, Liu et al. (2016b) and Zhao and Xie (2013) did not take time delay into account. Xie and Liu (2013) considered state feedback of stochastic high-order nonlinear systems with time-varying delay using the homogeneous domination concept but they only studied stochastic feedforward nonlinear systems with time delay existing in system states and did not consider the time delay in the control input. Then Liu et al. (2016a) and Xie and Zhao (2014) studied stochastic feedforward nonlinear systems with time delay in control input, but their work was limited to stochastic low-order feedforward nonlinear systems.

Motivated by the aforementioned discussions, we will further study the problem of state feedback stabilization for a class of stochastic high-order feedforward nonlinear systems with time delay appearing in the control input. The main contributions are:

By choosing an appropriate Lyapunov–Krasovskii functional and constructing a variable transformation, the negative effects brought by input time delay is successfully compensated.

Based on backstepping technique and analysis, the state feedback controller is explicitly constructed, such that the closed-loop system is globally asymptotically stable in probability.

This paper is organized as follows. The next section provides mathematical preliminaries. In the following two sections, we design the new controller and give the main results, respectively. Then following a simulation example, before conclusions are drawn.

Preliminary results

Next, we introduce notations, as well as definitions and lemmas that are to be used throughout the paper. $R_{+}$ denotes the set of all nonnegative real numbers and $R^{n}$ denotes the real $n$ -dimensional space. $R^{*} = {\tilde{q} \in R_{+} : \tilde{q} \geq 1$ is a ratio of odd integers}, $R_{odd}^{+} = {\tilde{q} \in R_{+} : \tilde{q}$ is a ratio of odd integers}. For a given vector or matrix $X$ , $X^{T}$ denotes its transpose, $tr {X}$ denotes its trace when $X$ is square, and $| X |$ is the Euclidean norm of a vector $X$ . $C_{F_{0}}^{b} ([- d, 0]; R^{n})$ denotes the family of all $F_{0}$ -measurable bounded $C ([- d, 0]; R^{n}$ -valued random variables $ξ = {ξ (θ) : - d \leq θ \leq 0} .$ $C^{2, 1} (R^{n} \times [- d, \infty); R_{+})$ denotes the family of all nonnegative functions $V (x, t)$ on $R^{n} \times [- d, \infty)$ that are $C^{2}$ in $x$ and $C^{1}$ in $t$ ; $C^{2, 1}$ denotes the family of all functions that are $C^{2}$ in the first argument and $C^{1}$ in the second argument. $K$ denotes the set of all functions $R_{+} \to R_{+}$ that are continuous, strictly increasing and vanishing at zero; $K_{\infty}$ denotes the set of all functions that are of class $K$ and unbounded. For simplicity, the smooth function $ϱ (\cdot)$ is sometimes denoted $ϱ$ .

Consider the following stochastic nonlinear system

d x (t) = f (t, x (t), x (t - d (t))) d t + g (t, x (t), x (t - d (t))) d ω, \forall t \geq 0

(1)

with initial data ${x (θ) : - d \leq θ \leq 0} = ξ \in C_{F_{0}}^{b} ([- d, 0]; R^{n})$ , where $d (t) : R_{+} \to [0, d]$ is a Borel measurable function; $ω$ is an $r$ -dimensional independent standard Wiener process defined on a complete probability space $(Ω, F, P)$ , with $Ω$ being a sample space, $F$ being a filtration and $P$ being a measure. $x \in R^{n}$ is the state of system (1); the Borel measurable functions: $f : R_{+} \times R^{n} \times R^{n} \to R^{n}$ and $g : R_{+} \times R^{n} \times R^{n} \to R^{n \times r}$ are locally Lipschitz with $f (t, 0, 0) \equiv 0$ and $g (t, 0, 0) \equiv 0$ .

Definition 1. (Deng et al., 2001). Given any $V (x, t) \in C^{2, 1}$ , for a stochastic nonlinear system (1), the differential operator $L$ is defined as

LV (x, t) = \frac{\partial V}{\partial t} + \frac{\partial V}{\partial x} f + \frac{1}{2} tr (g^{T} \frac{\partial^{2} V}{\partial x^{2}} g)

Definition 2. (Bacciotti and Rosier, 2005). For fixed coordinates $(x_{1}, \dots, x_{n})^{T} \in R^{n}$ and real numbers $r_{i} > 0, i = 1, 2, \dots, n$ , we have the following definitions:

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

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

A vector field $f \in C (R^{n}, R)$ is said to be homogeneous of degree $τ$ if there is a real number $τ > 0$ , $τ \geq - min {r_{i}, i = 1, 2, \dots, n}$ , such that $f_{i} (Δ_{ε} (x)) = ε^{τ + r_{i}} f_{i} (x)$ for any $x \in R^{n} \ {0}, ε > 0, i = 1, 2, \dots, n$ .

A homogeneous $ϱ -$ norm is defined as

∥ x ∥_{Δ, ϱ} = {(\sum_{i = 1}^{n} | x_{i} |^{\frac{ϱ}{r_{i}}})}^{\frac{1}{ϱ}}

for any $x \in R^{n}$ , where $ϱ \leq 1$ is a constant. If there is no special instruction, we write $∥ x ∥_{Δ}$ for $∥ x ∥_{Δ, 2}$ .

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

Lemma 2. (Qian and Li, 2006). Suppose that $V : R^{n} \to R$ is a homogeneous function of degree $τ$ with respect to the dilation weight $Δ$ , then:

$\partial V / \partial x_{i}$ 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 $V (x) \geq \underline{c} ∥ x ∥_{Δ}^{τ}$ . where $\underline{c}$ is a positive constant.

Lemma 3. (Xie and Duan, 2010). Let $x, y$ be real variables; then for any numbers $m > 0$ , $n > 0$ and continuous function

a (\cdot) \geq 0

a (\cdot) x^{m} y^{n} \leq b | x |^{m + n} + \frac{n}{m + n} {(\frac{m + n}{m})}^{- \frac{m}{n}} a^{\frac{m + n}{n}} (\cdot) b^{- \frac{m}{n}} | y |^{m + n}

holds, where $b > 0$ is a real number.

Lemma 4. (Liu et al., 2008). For equation (1), if there exists a function $V (x, t) \in C^{2, 1} (R^{n} \times [- d, \infty), R_{+})$ , two class $K_{\infty}$ functions $α_{1} (\cdot)$ and $α_{2} (\cdot)$ and a class $K$ function $α_{3},$ such that

α_{1} (| x (t) |) \leq V (x, t) \leq α_{2} (sup_{- d \leq s \leq 0} | x (t + s) |), LV (x, t) \leq - α_{3} (| x (t) |),

then there exists a unique solution on $[- d, \infty)$ for equation (1), such that the equilibrium $x (t) = 0$ is globally asymptotically stable in probability and $P {lim_{t \to \infty} | x (t) | = 0} = 1$ .

Lemma 5. (Liu and Xie, 2011). For any $x, y \in R$ , where $p \geq 1$ is a constant, then

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

Moreover, if $p \in R^{*}$ , then

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

Lemma 6. (Liu and Xie, 2011). Let $p \in R^{*}$ , $x$ and $y$ be real-valued function. For a constant $c > 0$ , one has $| x^{p} - y^{p} | \leq c | x - y | | (x - y)^{p - 1} + y^{p - 1} |$ .

Controller design

Problem statement

In the paper, we consider a class of stochastic high-order feedforward systems with input time delay as follows

\begin{matrix} d x_{i} = x_{i + 1}^{p_{i}} d t + f_{i} (t, {\bar{x}}_{i + 2}, u (t), u (t - d)) d t \\ + φ_{i}^{T} (t, {\bar{x}}_{i + 1}, u (t), u (t - d)) d ω, \\ i = 1, \dots, n - 1 \\ d x_{n} = u (t - d) d t \end{matrix}

(2)

where $x = [x_{1}, \dots, x_{n}]^{T} \in R^{n}$ and $u (t) \in R$ are the system states and control input, respectively. ${\bar{x}}_{i} = (x_{i}, x_{i + 1}, \dots, x_{n})^{T}$ , $(i = 2, \dots, n)$ , $p_{i} \in R_{odd}^{> 2} = {q \in R_{+} : q > 2$ and $q$ is a ratio of odd integers } is the high order of the system. $ω \in R^{r}$ is an $r$ -dimensional standard Wiener process defined on the complete probability space $(Ω, F, P)$ , with $Ω$ being a sample space, $F$ being a filtration and $P$ being a measure. The drift and diffusion terms $f_{i} (\cdot), φ_{i} (\cdot)$ are assumed to be locally Lipschitz and smooth, vanishing at the origin $(x_{1}, x_{2}, \dots, x_{i}) = (0, 0, \dots, 0)$ .

The purpose of this study is to design the state feedback controller for equation (2). The proposed controller can guarantee that the closed-loop system is globally asymptotically stable. To design the controller for equation (2), the following assumption is needed.

Assumption 1. For $i = 1, 2, \dots, n - 1$ , there are positive constants $c_{1}$ , $c_{2}$ and $δ \in (d_{m}, + \infty)$ , such that

f_{i} (t, {\bar{x}}_{i + 2}, u (t), u (t - d)) \leq c_{1} (\sum_{j = i + 2}^{n} | x_{j} |^{\frac{r_{i} + δ}{r_{j}}} + | u (t {) |}^{\frac{r_{i} + δ}{r_{n + 1}}} + | u (t - d {) |}^{\frac{r_{i} + δ}{r_{n + 1}}})

φ_{i} (t, {\bar{x}}_{i + 1}, u (t), u (t - d)) \leq c_{2} (\sum_{j = i + 1}^{n} | x_{j} |^{\frac{r_{i} + δ}{2 r_{j}}} + | u (t {) |}^{\frac{r_{i} + δ}{2 r_{n + 1}}} + | u (t - d {) |}^{\frac{r_{i} + δ}{2 r_{n + 1}}})

where

r_{1} = p_{n} = 1

r_{i + 1} = \frac{r_{i} + δ}{p_{i}} > 0, i = 1, 2, \dots, n

d_{m} = max_{1 \leq i \leq n} {d_{i}}

d_{1} = \frac{2 - \frac{1}{Π_{i = 1}^{n - 1} p_{i}}}{1 + \frac{1}{\sum_{s = 1}^{n - 1} p_{s} \dots p_{n - 1}}}

d_{i} = \frac{\frac{2}{Π_{k = 1}^{i - 1} p_{k}} - \frac{1}{Π_{i = 1}^{n - 1} p_{i}}}{1 + \frac{1}{\sum_{s = 1}^{n - 1} p_{s} \dots p_{n - 1}} - \frac{2}{\sum_{s = 1}^{i - 1} p_{s} \dots p_{i - 1}}}, i = 2, 3, \dots, n

Remark 1. When $p_{i} = 1$ , $i = 1, 2, \dots, n - 1$ and $d = 0$ , equation (2) reduces to the well-known form, for which the feedback control problem has been well developed in recent years (Ding et al., 2011; Du et al., 2013; Mazenc and Bowong, 2003; Qian and Li, 2006; Teel, 1996).

Remark 2. Obviously, equation (2), satisfying Assumption 1, is the stochastic feedforward nonlinear system. Assumption 1 is similar to Assumption A1 in Liu and Xie (2011) and is a frequently used condition.

Lemma 7. For any $p_{i} \in R_{odd}^{> 2}$ and $δ \in (d_{m}, + \infty)$ , there exist parameters $l_{1} > 1$ and $u \in R_{odd}^{+}$ , such that

r_{n} + δ \geq u > max_{1 \leq i \leq n} {2 r_{i}, \frac{(r_{i} + δ)}{l}}

Using a similar method to that of Liu and Xie (2011), we can prove Lemma 7; thus we omit the proof here.

State feedback controller design

Introducing a set of coordinate transformations

\begin{matrix} z_{i} = \frac{x_{i}}{N^{k_{i}}}, i = 1, 2, \dots, n - 1 \\ z_{n} = \frac{{\tilde{x}}_{n}}{N^{k_{n}}} \end{matrix}

(3)

and a set of variable transformations

\begin{matrix} {\tilde{x}}_{n} = x_{n} + \int_{t - d}^{t} u (s) d s \\ v = \frac{u}{N^{k_{n + 1}}} \end{matrix}

(4)

where $0 < N < 1$ is a gain to be determined; $k_{1} = 0$ , $k_{i + 1} = (k_{i} + 1) / (p_{i}), i = 1, 2, \dots, n .$

Using equations (3) and (4), equation (2) can be transformed into

\begin{matrix} d z_{i} = {Nz}_{i + 1}^{p_{i}} d t + {\bar{f}}_{i} (t, {\bar{z}}_{i + 2}, v (t), v (t - d)) \\ d t + {\bar{ψ}}_{i}^{T} (t, {\bar{z}}_{i + 1}, v (t), v (t - d)) d ω, i = 1, \dots, n - 2 \\ d z_{n - 1} = N {z_{n}}^{p_{n - 1}} + {\bar{f}}_{n - 1} (t, {\bar{z}}_{n}, v (t), v (t - d)) d t \\ + {\bar{ψ}}_{n - 1}^{T} (t, {\bar{z}}_{n}, v (t), v (t - d)) d ω \\ d z_{n} = Nv d t \end{matrix}

(5)

where

{\bar{f}}_{i} (t, {\bar{z}}_{i + 2}, v, v (t - d)) = \frac{f_{i} (t, {\bar{x}}_{i + 2}, u (t), u (t - d))}{N^{k_{i}}}, i = 1, 2, \dots, n - 2

{\bar{f}}_{n - 1} (t, {\bar{z}}_{n}, v, v (t - d)) = \frac{f_{n - 1} (t, u, u (t - d)) + \sum_{k = 0}^{p_{n - 1}} C_{p_{n - 1}}^{k} {(N^{k_{n}} z_{n})}^{k} {(- \int_{t - d}^{t} u (s) d s)}^{p_{n - 1} - k}}{N^{k_{n - 1}}}

{\bar{ψ}}_{i} (t, {\bar{z}}_{i + 1}, v, v (t - d)) = \frac{φ_{i} (t, {\bar{x}}_{i + 1}, u (t), u (t - d))}{N^{k_{i}}}, i = 1, 2, \dots, n - 1

Next, we will design the state feedback controller for equation (5).

Step 1. Let

η_{1} = z_{1}^{\frac{u}{r_{1}}}

and construct the first Lyapunov function

V_{1} (z_{1}) = \int_{z_{1}^{*}}^{z_{1}} {(s^{\frac{u}{r_{1}}} - {z_{1}^{*}}^{\frac{u}{r_{1}}})}^{\frac{q_{1}}{u}} d s

where $z_{1}^{*} = 0$ , $q_{1} = 4 l_{1} u - δ - r_{1}$ . By Definition 1 and equation (5), one has

L V_{1} = N η_{1}^{\frac{q_{1}}{u}} z_{2}^{p_{1}} + F_{1} + Ψ_{1}

where $F_{1} = \partial V_{1} {\bar{f}}_{1} / \partial z_{1}$ and

Ψ_{1} = \frac{1}{2} tr {{\bar{ψ}}_{1} \frac{\partial^{2} V_{1}}{\partial z_{1}^{2}} {\bar{ψ}}_{1}^{T}}

The virtual controller $z_{2}^{*} = - {β_{1}}^{\frac{r_{2}}{u}} {η_{1}}^{\frac{r_{2}}{u}}$ , $β_{1} = {c_{11}}^{\frac{u}{r_{2} p_{1}}} > 0$ leads to

L V_{1} \leq - c_{11} N η_{1}^{4 l_{1}} + N η_{1}^{\frac{q_{1}}{u}} (z_{2}^{p_{1}} - z_{2}^{* p_{1}}) + F_{1} + Ψ_{1}

(6)

Step $i$ $(i = 2, 3, \dots, n)$ . With the help of Lemma 7, we can find the following property.

Lemma 8. Suppose that at step $i - 1$ , there exists a $C^{2}$ , positive definite and proper Lyapunov function

V_{i - 1} ({\bar{z}}_{i}) = \sum_{k = 1}^{i - 1} \int_{z_{k}^{*}}^{z_{k}} (s^{\frac{u}{r_{k}}} - {z_{k}^{*}}^{\frac{u}{r_{k}}}) d s

and a series of virtual controllers $z_{1}^{*}, z_{2}^{*}, \dots, z_{i}^{*}$ , defined by

\begin{matrix} z_{1}^{*} = 0, η_{1} = z_{1}^{\frac{u}{r_{1}}}, z_{k}^{*} = - β_{k - 1}^{\frac{r_{k}}{u}} η_{k - 1}^{\frac{r_{k}}{u}}, η_{k} = z_{k}^{\frac{u}{r_{k}}} - z_{k}^{* \frac{u}{r_{k}}}, k = 2, \dots, i \end{matrix}

(7)

such that

\begin{matrix} L V_{i - 1} \leq - \sum_{j = 1}^{i - 1} N c_{i - 1 j} η_{j}^{4 l_{1}} + N η_{i - 1}^{\frac{q_{i - 1}}{u}} (z_{i}^{p_{i - 1}} - z_{i}^{* p_{i - 1}}) + F_{i - 1} + Ψ_{i - 1} \end{matrix}

(8)

where $β_{j - 1}$ and $c_{i - 1 j}, j = 1, 2, \dots, i - 1$ are positive constants and

q_{i - 1} = 4 l_{1} u - δ - r_{i - 1}

F_{i - 1} = \sum_{j = 1}^{i - 1} \frac{\partial V_{i - 1}}{\partial z_{j}} {\bar{f}}_{j}

Ψ_{i - 1} = \sum_{k, m = 1}^{i - 1} \frac{1}{2} tr {{\bar{ψ}}_{k} \frac{\partial^{2} V_{i - 1}}{\partial z_{k} \partial z_{m}} {\bar{ψ}}_{m}^{T}}

Then, the $i$ th Lyapunov function

V_{i} ({\bar{z}}_{i}) = V_{i - 1} ({\bar{z}}_{i - 1}) + W_{i} ({\bar{z}}_{i})

W_{i} ({\bar{z}}_{i}) = \int_{z_{i}^{*}}^{z_{i}} {(s^{\frac{u}{r_{i}}} - {z_{k}^{*}}^{\frac{u}{r_{i}}})}^{\frac{q_{i}}{u}} d s, q_{i - 1} = 4 l_{1} u - δ - r_{i}

(9)

is $C^{2}$ , positive definite and proper, and there exists a virtual control law

z_{i}^{*} = - β_{i - 1}^{\frac{r_{i}}{u}} η_{i - 1}^{\frac{r_{i}}{u}}

such that

L V_{i} \leq - N \sum_{j = 1}^{i} c_{ij} η_{j}^{4 l_{1}} + N η_{i}^{\frac{q_{i}}{u}} (z_{i + 1}^{p_{i}} - z_{i + 1}^{* p_{i}}) + F_{i} + Ψ_{i}

(10)

where

F_{i} = F_{i - 1} + \sum_{j = 1}^{i} \frac{\partial W_{i}}{\partial z_{j}} {\bar{f}}_{j} = \sum_{j = 1}^{i} \frac{\partial V_{i}}{\partial z_{j}} {\bar{f}}_{j}

\begin{matrix} Ψ_{i} = Ψ_{i - 1} + \sum_{j, k = 1}^{i} \frac{1}{2} tr {{\bar{ψ}}_{j} \frac{\partial^{2} W_{i}}{\partial z_{j} \partial z_{k}} {\bar{ψ}}_{k}^{T}} \\ = \sum_{j, k = 1}^{i} \frac{1}{2} tr {{\bar{ψ}}_{j} \frac{\partial^{2} V_{i}}{\partial z_{j} \partial z_{k}} {\bar{ψ}}_{k}^{T}} \end{matrix}

Proof. Using a similar method to that of Liu and Xie (2011), we can prove Lemma 8; thus we omit the proof here.

Finally, when $i = n$ , choosing $V_{n} ({\bar{z}}_{n}) = V_{n - 1} ({\bar{z}}_{n - 1}) + W_{n} (z_{n - 1})$ and the control law

v = z_{n + 1}^{*} = - (β_{n} η_{n})^{\frac{r_{n + 1}}{u}} = - {({\bar{β}}_{n} z_{n}^{\frac{u}{r_{n}}} + {\bar{β}}_{n - 1} z_{n - 1}^{\frac{u}{r_{n - 1}}} + \dots + {\bar{β}}_{1} z_{1}^{\frac{u}{r_{1}}})}^{\frac{r_{n + 1}}{u}}

(11)

such that

L V_{n} \leq - N \sum_{j = 1}^{n} c_{nj} η_{j}^{4 l_{1}} + F_{n} + Ψ_{n}

(12)

where

F_{n} = \sum_{j = 1}^{n - 1} \frac{\partial V_{n}}{\partial z_{j}} {\bar{f}}_{j}

Ψ_{n} = \sum_{m, n = 1}^{n - 1} \frac{1}{2} {{\bar{ψ}}_{m} \frac{\partial^{2} V_{n}}{\partial z_{m} \partial z_{n}} {\bar{ψ}}_{n}^{T}}

{\bar{β}}_{i} = β_{n} β_{n - 1} \dots β_{i}, c_{ni}, i = 1, 2, \dots, n

are positive constants. Equations (5) and (11) can be rewritten in the following compact form

d z = N E (z) d t + F (t, z, v (t), v (t - d)) d t + Ψ^{T} (t, z, v (t), v (t - d)) d ω

(13)

where $z = (z_{1}, \dots, z_{n})^{T}$ , $E (z) = (z_{2}^{p_{2}}, \dots, z_{n}^{p_{n - 1}}, v)^{T}$ , $F (t, z, v, v (t - d)) = ({\bar{f}}_{1}, \dots, {\bar{f}}_{n - 1}, 0)^{T}$ and $Ψ (t, z, v, v (t - d)) = ({\bar{ψ}}_{1}, \dots, {\bar{ψ}}_{n - 1}, 0)$ . Introducing the dilation weight

Δ = \underset{for (z_{1}, z_{2}, \dots, z_{n})}{\underset{︸}{(r_{1}, r_{2}, \dots r_{n})}}

by equations (7) and (9), one obtains

\begin{matrix} V_{n} (Δ_{ε} (z_{1}, \dots, z_{n})) = \sum_{i = 1}^{n} \int_{ε^{r_{i}} z_{i}^{*}}^{ε^{r_{i}} z_{i}} {(s^{\frac{u}{r_{i}}} - ε^{u} {z_{i}^{*}}^{\frac{u}{r_{i}}})}^{\frac{q_{i}}{u}} d s \\ = \sum_{i = 1}^{n} \int_{z_{i}^{*}}^{z_{i}} {(ε^{u} θ^{\frac{u}{r_{i}}} - ε^{u} {z_{i}^{*}}^{\frac{u}{r_{i}}})}^{\frac{q_{i}}{u}} ε^{r_{i}} d θ \\ = ε^{4 l_{1} u - δ} \sum_{i = 1}^{n} \int_{z_{i}^{*}}^{z_{i}} {(θ^{\frac{u}{r_{i}}} - {z_{i}^{*}}^{\frac{u}{r_{i}}})}^{\frac{q_{i}}{u}} d θ \\ = ε^{4 l_{1} u - δ} V_{n} (z_{1}, \dots, z_{n}) \end{matrix}

(14)

where $s = ε^{r_{i}} θ$ , from which, with Definition 2, we know that $V_{n} (z_{1}, \dots, z_{n})$ is homogeneous of degree $4 l_{1} u - δ$ .

Stability analysis

Next,we give the main result of this paper.

Theorem 1. If Assumption 1 holds for the stochastic feedforward system (equation 2), under the state feedback controller $u = N^{k_{n + 1}} v$ and equation (11), then:

The closed-loop system has a unique solution on $[- d, \infty)$ .

The equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability.

Proof. We prove Theorem 1 using the following steps.

Step 1. We first prove that $v (z)$ in equation (11) is $C^{1}$ . From equation (11), one gets

\frac{\partial v}{\partial z_{i}} = - \frac{r_{n + 1}}{r_{i}} {\bar{β}}_{i} z_{i}^{\frac{u - r_{i}}{r_{i}}} {({\bar{β}}_{n} z_{n}^{\frac{u}{r_{n}}} + {\bar{β}}_{n - 1} z_{n - 1}^{\frac{u}{r_{n - 1}}} + \dots + {\bar{β}}_{1} z_{1} z_{1}^{\frac{u}{r_{1}}})}^{\frac{r_{n} + δ - u}{u}}, i = 1, 2, \dots, n

(15)

With the help of Lemma 7, we can easily know that $\partial v / \partial z_{i}$ is continuous; then $v (z)$ is $C^{1}$ .

Since $f_{i} (\cdot)$ and $ψ (\cdot)$ are assumed to be locally Lipschitz, the system consisting of equations (5) and (11) satisfies the locally Lipschitz condition.

Step 2. Using Lemma 3 and equation (12), there exists a positive constant $c_{01}$ , such that

\frac{\partial V_{n}}{\partial z} NE (z) \leq - c_{01} N | | z | |_{Δ}^{4 l_{1} u}

(16)

by Assumption 1, Lemma 5, the definitions of $k_{i}$ and $r_{i}$ , equations (3), (4) and (11) and $0 < N < 1$ , for $i = 1, 2, \dots, n - 2$ , one gets

\begin{array}{l} | {\bar{f}}_{i} | \leq \frac{c_{1}}{N^{k_{i}}} (\sum_{j = i + 2}^{n} | x_{j} |^{\frac{r_{i} + δ}{r_{j}}} + | u (t) |^{\frac{r_{i} + δ}{r_{n + 1}}} + | u (t - d) |^{\frac{r_{i} + δ}{r_{n + 1}}}) \leq c_{1} N^{p_{i + 2} k_{i + 2} - k_{i}} \\ (\sum_{j = i + 2}^{n} | z_{j} |^{\frac{r_{i} + δ}{r_{j}}} + | v (t) |^{\frac{r_{i} + δ}{r_{n + 1}}} + | v (t - d) |^{\frac{r_{i} + δ}{r_{n + 1}}} + {(| z_{n} - \int_{t - d}^{t} v (s) d s |)}^{\frac{r_{i} + δ}{r_{n}}}) \\ \leq {\tilde{c}}_{1} N^{1 + r_{i 1}} (\sum_{j = i + 2}^{n} | z_{j} |^{\frac{r_{i} + δ}{r_{j}}} + | v (t) | \frac{r_{i} + δ}{r_{n + 1}} + | v (t - d) |^{\frac{r_{i} + δ}{r_{n + 1}}} + {(\int_{t - d}^{t} | v (s) | d s)}^{\frac{r_{i} + δ}{r_{n}}}) \\ \leq b_{1} N^{1 + r_{i 1}} (∥ z ∥_{Δ}^{r_{i} + δ} + ∥ z (t - d) ∥_{Δ}^{r_{i} + δ} + \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{r_{i} + δ} d s) \end{array}

(17)

where ${\tilde{c}}_{1}, b_{1}$ and

r_{i 1} = \frac{p_{i}}{1 + \sum_{k = 1}^{i + 1} \frac{p_{1} p_{2} \dots p_{i + 1}}{p_{s} \dots p_{i + 1}} d}

are positive constants.

For $i = n - 1$ , by the definitions of ${\bar{f}}_{n - 1}$ , $k_{n - 1}$ , $k_{n}$ , $k_{n + 1}$ and $r_{n}$ and by Assumption 1, Lemma 1, Lemma 5, equations (3), (4) and (11) and $0 < N < 1$ , one has

\begin{matrix} | {\bar{f}}_{n - 1} | \leq \frac{c_{1}}{N^{k_{n - 1}}} (| u (t {) |}^{\frac{r_{n - 1} + δ}{r_{n + 1}}} + | u (t - d {) |}^{\frac{r_{n - 1} + δ}{r_{n + 1}}}) + \sum_{k = 0}^{p_{n - 1} - 1} C_{p_{n - 1}}^{k} {(N^{k_{n}} | z_{n} |)}^{k} {(| \int_{t - d}^{t} u (s) | d s)}^{p_{n - 1} - k} \\ \leq {\tilde{a}}_{1} N^{k_{n + 1} - k_{n - 1}} (| u (t {) |}^{\frac{r_{n - 1} + δ}{r_{n + 1}}} + | u (t - d {) |}^{\frac{r_{n - 1} + δ}{r_{n + 1}}} + \sum_{k = 0}^{p_{n - 1} - 1} C_{p_{n - 1}}^{k} (γ | z_{n} |^{p_{n - 1}} + \frac{p_{n - 1} - k}{p_{n - 1}} γ^{- \frac{k}{p_{n - 1} - k}} {(\int_{t - d}^{t} | v (s) | d s)}^{p_{n - 1}})) \\ \leq b_{2} N^{1 + \frac{1 + (1 - p_{n - 1}) k_{n - 1}}{k_{n - 1}}} (∥ z ∥_{Δ}^{r_{n - 1} + δ} + ∥ z (t - d) ∥_{Δ}^{r_{n - 1} + δ} + \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{r_{n - 1} + δ} d s) \end{matrix}

(18)

where ${\tilde{a}}_{1}, b_{2}$ and $γ$ are positive constants.

Combining equations (17) and (18) for $i = 1, 2, \dots, n - 1$ , one gets

| {\bar{f}}_{i} | \leq b_{01} N^{1 + δ_{i 1}} (∥ z ∥_{Δ}^{r_{i} + δ} + ∥ z (t - d) ∥_{Δ}^{r_{i} + δ} + \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{r_{i} + δ} d s)

(19)

where

b_{01} = max (b_{1}, b_{2}), δ_{i 1} = min_{1 \leq i \leq n - 2} {r_{i 1}, \frac{1 + (1 - p_{n - 1}) k_{n - 1}}{k_{n - 1}}}

From Lemma 2, Lemma 3, Lemma 5 and equation (17), one has

\begin{matrix} \frac{\partial V_{n}}{\partial z} F (t, z, v (t), v (t - d)) = \sum_{i = 1}^{n - 1} \frac{\partial V_{n}}{\partial z_{i}} {\bar{f}}_{i} (t, {\bar{z}}_{i + 2}, v (t), v (t - d)) \\ \leq b_{01} N^{1 + δ_{i 1}} \sum_{i = 1}^{n - 1} ∥ z (t) ∥_{Δ}^{q_{i}} (∥ z ∥_{Δ}^{r_{i} + δ} + ∥ z (t - d) ∥_{Δ}^{r_{i} + δ} + \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{r_{i} + δ} d s) \\ \leq N^{1 + δ_{1}} ({\tilde{b}}_{01} ∥ z ∥_{Δ}^{4 l_{1} u} + {\bar{b}}_{01} ∥ z (t - d) ∥_{Δ}^{4 l_{1} u} + {\hat{b}}_{01} \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{4 l_{1} u} d s) \end{matrix}

(20)

where $δ_{1} = min_{1 \leq i \leq n - 1} {δ_{i 1}}$ , ${\tilde{b}}_{01}, {\bar{b}}_{01}$ and ${\hat{b}}_{01}$ are positive constants.

Like equation (17), there exist positive constants ${\tilde{a}}_{2}, {\hat{b}}_{2}$ and

r_{i 2} = \frac{p_{i} + \frac{1}{2}}{1 + \sum_{k = 1}^{i + 1} \frac{p_{1} p_{2} \dots p_{i + 1}}{p_{s} \dots p_{i + 1}} d}

such that

\begin{matrix} | {\bar{g}}_{i} | \leq {\tilde{a}}_{2} N^{\frac{1}{2} + r_{i 2}} (\sum_{j = i + 2}^{n} | z_{j} |^{\frac{r_{i} + δ}{r_{j}}} + | v (t {) |}^{\frac{r_{i} + δ}{r_{n + 1}}} + | v (t - d {) |}^{\frac{r_{i} + δ}{r_{n + 1}}} + {(\int_{t - d}^{t} | v (s) | d s)}^{\frac{r_{i} + δ}{r_{n}}}) \\ \leq {\hat{b}}_{2} N^{\frac{1}{2} + r_{i 2}} (∥ z ∥_{Δ}^{r_{i} + \frac{δ}{2}} + ∥ z (t - d) ∥_{Δ}^{r_{i} + \frac{δ}{2}} + \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{r_{i} + \frac{δ}{2}} d s) \end{matrix}

(21)

From Lemma 2, Lemma 3, Lemma 5 and equation (21), one has

\begin{matrix} \frac{1}{2} tr {Ψ \frac{\partial^{2} V_{n}}{\partial z^{2}} Ψ^{T}} \leq b \sum_{i, j = 1}^{n - 1} \frac{\partial^{2} V_{n}}{\partial z_{i} \partial z_{j}} | {\bar{g}}_{i} | | {\bar{g}}_{j} | \\ \leq b_{02} N^{1 + δ_{i 2}} \sum_{i = 1}^{n - 1} ∥ z (t) ∥_{Δ}^{q_{i}} (∥ z ∥_{Δ}^{r_{i} + δ} + ∥ z (t - d) ∥_{Δ}^{r_{i} + δ} + \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{r_{i} + δ} d s) \\ \leq N^{1 + δ_{2}} ({\tilde{b}}_{02} ∥ z ∥_{Δ}^{4 l_{1} u} + {\bar{b}}_{02} ∥ z (t - d) ∥_{Δ}^{4 l_{1} u} + {\hat{b}}_{02} \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{4 l_{1} u} d s) \end{matrix}

(22)

where $δ_{2} = min_{1 \leq i, j \leq n - 1} (r_{i 2} + r_{j 2})$ , $b, b_{02}, {\tilde{b}}_{02}, {\bar{b}}_{02}$ and ${\hat{b}}_{02}$ are positive constants.

Step 3. Considering the following Lyapunov–Krasovskii functional

V (z (t)) = V_{n} (z (t)) + N^{1 + δ} {({\bar{b}}_{01} + {\bar{b}}_{02}) \int_{t - d}^{t} ∥ z (s) ∥_{Δ}^{4 l_{1} u} d s + ({\hat{b}}_{01} + {\hat{b}}_{02}) \int_{- d}^{0} \int_{t + σ}^{t} ∥ z ∥_{Δ}^{4 l_{1} u} d s d σ}

(23)

where $0 < δ = min (δ_{1}, δ_{2}) \leq 1$ . Using Definition 1 and equations (13), (16), (20), (22) and (23), one can obtain

\begin{matrix} LV \leq \frac{\partial V_{n}}{\partial z} NE (z) + \frac{\partial V_{n}}{\partial z} F + \frac{1}{2} tr {Ψ \frac{\partial^{2} V_{n}}{\partial z^{2}} Ψ^{T}} \\ + N^{1 + δ} ({\bar{b}}_{01} + {\bar{b}}_{02}) (∥ z ∥_{Δ}^{4 l_{1} u} - ∥ z (t - d) ∥_{Δ}^{4 l_{1} u}) \\ + N^{1 + δ} ({\hat{b}}_{01} + {\hat{b}}_{02}) d ∥ z ∥_{Δ}^{4 l_{1} u} - N^{1 + δ} ({\hat{b}}_{01} + {\hat{b}}_{02}) \\ \int_{t - d}^{t} ∥ z ∥_{Δ}^{4 l_{1} u} d s \leq - N c_{01} ∥ z ∥_{Δ}^{4 l_{1} u} + N^{1 + δ} ({\bar{b}}_{01} + {\bar{b}}_{02}) \\ ∥ z ∥_{Δ}^{4 l_{1} u} + N^{1 + δ} ({\hat{b}}_{01} + {\hat{b}}_{02}) d ∥ z ∥_{Δ}^{4 l_{1} u} \\ = - N (c_{01} - N^{δ} ({\bar{b}}_{01} + {\bar{b}}_{02} + {\hat{b}}_{01} d + {\hat{b}}_{02} d)) ∥ z ∥_{Δ}^{4 l_{1} u} \end{matrix}

(24)

Since $c_{01}$ is a positive constant independent of ${\bar{b}}_{01}, {\bar{b}}_{02}, {\hat{b}}_{01}$ and ${\hat{b}}_{02}$ , we can choose

0 < N < N^{*} = : min {1, {(\frac{c_{01}}{{\bar{b}}_{01} + {\bar{b}}_{02} + {\hat{b}}_{01} d + {\hat{b}}_{02} d})}^{\frac{1}{δ}}}

(25)

and equation (24) becomes $LV (z (t)) \leq - c_{0} ∥ z ∥_{Δ}^{4 l_{1} u}$ , where $c_{0}$ is a positive constant. From that and Lemma 4.3 in Khalil (2002), there exists a class $K_{\infty}$ function $α_{3}$ , such that

LV (z (t)) \leq α_{3} (| z (t) |)

(26)

From this analysis and Lemma 4, the system consisting of equations (5) and (11) has almost surely a unique solution on $[- d, \infty)$ and $z (t) = 0$ is globally asymptotically stable in probability. Since equations (3) and (4) are equivalent transformations, the closed-loop system consisting of equation (2), $u = N^{k_{n + 1}} v$ and equation (11) has the same properties as equations (5) and (11). Thus, Theorem 1 holds.

Simulation example

In this section, we give a simulation to validate the design procedure of the control scheme and use the simulation results to demonstrate the effectiveness of the designed adaptive controller.

We consider the following system

\begin{matrix} d x_{1} = x_{2}^{3} d t + \frac{1}{5} x_{3}^{\frac{55}{46}} (t - 0.2) d ω \\ d x_{2} = x_{3}^{\frac{5}{3}} d t \\ d x_{3} = u (t - 0.2) d t \end{matrix}

(27)

where $p_{1} = 3, p_{2} = 5 / 3, p_{3} = 1$ and $τ = 0.2$ . Obviously, Assumption 1 is satisfied with $c_{1} = 0$ , and $c_{2} = 1 / 5$ . And by Assumption 1, we have $d_{1} = 3 / 2, d_{2} = 7 / 8, d_{3} = 1 / 4$ , then $δ \in (3 / 2, \infty)$ . Following the design procedure in the ‘Controller design’ section, the state feedback controller is designed as

\begin{array}{l} u (t) = \\ - N^{\frac{9}{5}} {β_{3} N^{- \frac{196}{115}} {(x_{3} + \int_{t - 0.2}^{t} u (s) d s)}^{\frac{49}{23}} + β_{3} β_{2} N^{- \frac{49}{40}} x_{2}^{\frac{147}{40}} + β_{3} β_{2} β_{1} x_{1}^{\frac{49}{15}}}^{\frac{48}{49}} \end{array}

(28)

In the simulation, $N = 0.02,$ $β_{1} = 10$ , $β_{2} = 20$ and $β_{3} = 5$ , with the initial values $x_{1} (0) = 2, x_{2} (0) = - 0.5, x_{3} (0) = - 0.5$ . Figure 1 demonstrates the effectiveness of the control scheme.

Remark 3. For equation (27), because it is a high-order system and the control includes a time delay, the controller design method in Xie and Zhao (2014) cannot be directly applied. Thus, we redesign the state feedback controller and choose the low gain $N$ in this note. Based on the controller (equation 28), the system (equation 27) can be obtained to produce stabilization. From Figure 1, one can see that system states converge to zero at about 20 s, while the control input converges to the origin at about 10 s. This may result from the state delay and input delay. However, from the results, one verifies that, through the control design in this paper, the closed-loop system can be finally stable, as desired.

Conclusions

This paper further studies the state feedback stabilization for stochastic high-order feedforward nonlinear systems with input time delay. With the help of the homogeneous domination method, the backstepping technique and the choice of an appropriate Lyapunov–Krasovskii functional, the state feedback controller is constructed to deal with the effect of the nonlinear terms. The designed controller can be ensure that the closed-loop system is globally stable in probability. An avenue for further work is to investigate the problem of designing the output-feedback control for equation (2).

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 61573172), the Shandong Province Natural Science Foundation of China (grant number ZR2016AL05) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (grant number KYCX17_0364).

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