Adaptive tracking control of stochastic nonlinear systems with unknown time-varying powers

Abstract

This paper investigates the adaptive output tracking problem for a class of high-order stochastic nonlinear systems with unknown time-varying powers and nonlinear parameterized uncertainties. By using the parameter separation technique and adding a power integrator design method, an adaptive controller with upper and lower bounds of the unknown time-varying power is successfully designed to guarantee that all the states of the closed-loop system are bounded in probability and the output tracking error can be regulated into a small neighborhood of the origin in probability. Finally, a simulation example is provided to illustrate the effectiveness of the designed controllers.

Keywords

High-order stochastic nonlinear systems adaptive output tracking control unknown time-varying powers

Introduction

Consider the following stochastic nonlinear systems described by

\begin{matrix} d x_{i} = ([x_{i + 1}]^{p_{i} (t)} + f_{i} ({\bar{x}}_{i}, θ)) d t + g_{i}^{T} ({\bar{x}}_{i}, θ) d ω, i = 1, \dots, n - 1, \\ ⋮ \\ d x_{n} = ([u]^{p_{n} (t)} + f_{n} ({\bar{x}}_{n}, θ)) d t + g_{n}^{T} ({\bar{x}}_{n}, θ) d ω, \\ y = x_{1}, \end{matrix}

(1)

where ${\bar{x}}_{i} = (x_{1}, \dots, x_{i})^{T} \in R^{i}$ , $u \in R$ and $y \in R$ are the measurable state, the control input and output of system, respectively. $θ \in R^{m}$ is an unknown constant vector. $ω$ is an $r$ -dimensional standard Wiener process defined on a probability space $(Ω, F, P)$ , with $Ω$ being a sample space, $F$ being a filtration, and $P$ being a probability measure. The time-varying power $p_{i} (t) : R^{+} \to R^{+}$ is a continuous unknown function satisfying $1 \leq \underline{p} \leq p_{i} (t) \leq \bar{p}$ with two known constants $\underline{p}$ and $\bar{p}$ , and the power sign function $[\cdot]^{τ} = sign (\cdot) | \cdot |^{τ}$ for a constant $τ > 0$ is adopted during our design process. The unknown functions $f_{i}$ and $g_{i}$ are smooth with $f_{i} (0, θ) = 0$ , $g_{i} (0, θ) = 0$ , $i = 1, \dots, n$ .

During the past years, motivated by Kushner (1967), Deng and Krstic (1997), Deng et al. (2001) and Pan and Basar (1999), the adaptive control of stochastic nonlinear system has received much attention. When $p_{i} (t) = 1$ , system (1) reduces to the well-known normal form. The controller design and stability analysis of such systems has been investigated in many works. Specifically, Ji and Xi (2006) and Wu et al. (2010) solved the problem of adaptive output tracking control for stochastic nonlinear systems with Wiener noise and Markovian switching, respectively. Cui et al. (2013) investigated the problem of adaptive output tracking for a class of stochastic Lagrangian control systems. When $p_{i} (t) > 1$ , system (1) is called stochastic high-order nonlinear system. For such systems, Xie and Tian (2009), Li et al. (2011) and Li and Zhang (2012) investigated the problem of adaptive state-feedback stabilization. Xie and Duan (2010) and Li and Wu (2013) solved the output tracking problem by using the backstepping method and the Dynkin formula, respectively. Xue et al. (2018) studied the adaptive output tracking control problem of stochastic systems with time-delay. It is worth to mention that the previous results on stochastic nonlinear system are all based on an assumption that the power $p_{i} (t)$ must be a known constant. In practice, the power $p_{i} (t)$ in system (1) maybe unknown and time-varying. For example, the power for boiler-turbine units in Liu et al. (2015) is not fixed because it is usually identified from the operational data obtained from the power plant. Another example is the underactuated mechanical system considered in Rui et al. (1997) in which the power is unknown time-varying because of the potential performance deterioration of hardening spring.

For deterministic systems with unknown powers, Su et al. (2017) solved the problem of global stabilization, Chen et al. (2017) investigated the output feedback stabilization, Man and Liu (2019) proposed a global adaptive output tracking controller to ensure output tracking objective. For high-order stochastic nonlinear system with time-varying powers, there is currently only one result on state-feedback stabilization and inverse optimal control in Li et al. (2020b). Moreover, the actual model is inevitably affected by white noises and parameter uncertainty. Therefore, one may ask the following interesting questions: when the system model has both unknown time-varying power and unknown parameter, can we solve the adaptive output tracking problem for high-order stochastic nonlinear systems?

To the best of our knowledge, there are no results regarding adaptive output tracking control for high-order stochastic nonlinear system with unknown time-varying powers and parameters until now. In this paper, we will solve this problem, and study the adaptive output tracking problem for high-order stochastic system with both unknown time-varying power and unknown parameters. In comparison with the existing results, the contributions of this paper are as follows:

Compared with the results on deterministic systems (Chen et al., 2017; Man and Liu, 2019; Su et al., 2017), a new design procedure is developed for stochastic high-order nonlinear system with unknown time-varying power. The design scheme can guarantee that the output tracking errors can be regulated into a small neighborhood of the origin while all the signals of the closed-loop system are bounded.

The system model in this paper is more general than that in Xie and Tian (2009); Li et al. (2011); Li and Zhang (2012); Xue et al. (2018); Xie and Duan (2010); and Li and Wu (2013), in which the power is required to be a known constant. The power considered in this paper is time-varying, unknown and maybe nondifferentiable, which makes the controller design and stability analysis method developed for time-invariant stochastic high-order systems invalid. Thus, new design tools and stability analysis techniques should be developed.

Nonlinear parameterized uncertainties are considered in the high-order stochastic system, which brings new difficulties in designing adaptive controller and stability analysis. By using the parameter separation technique in Lin and Qian (2002) and Lin and Pongvuthithum (2003), and backstepping method in Krstic and Deng (1998), an adaptive controller is designed recursively without any assumption on the unknown parameters.

The remainder of this paper is organized as follows. The second section offers some preliminary results. The adaptive output tracking controller is designed and analyzed in the third section. After that, in the fourth section, a simulation example is presented to show the effectiveness of the adaptive output tracking controller. Finally, the paper is concluded in the fifth section.

Perliminary results and useful lemmas

Consider the following time-varying stochastic nonlinear system

d x = f (x, θ) d t + g^{T} (x, θ) d ω, x (0) = x_{0},

(2)

where $x \in R^{n}$ is the state, $θ \in R^{m}$ m is an unknown constant vector, and $ω$ is an r-dimensional standard Wiener process defined on a probability space $(Ω, F, P)$ . The Borel measurable functions $f : R^{n} \times R^{m} \to R^{n}$ and $g : R^{n} \times R^{m} \to R^{n \times r}$ are locally Lipschitzin $x$ .

For any given $V (x) \in C^{2}$ , associated with stochastic system (2), the differential operator $L$ is defined as $L V (x) = \frac{\partial V}{\partial x} f (x) + \frac{1}{2} Tr {g (x) \frac{\partial^{2} V}{\partial x^{2}} g^{T} (x)}$ , where $C^{2}$ indicates the existence and continuity of the second derivative function, also called second-order continuously differentiable.

Definition 1: (Krstic and Deng, 1998) The solution process ${x (t), t \geq 0} = 0$ of stochastic system (2) is said to be bounded in probability if

lim_{c \to \infty} sup_{0 \leq t \leq + \infty} P {| x (t) | > c} = 0 .

Lemma 1: (Krstic and Deng, 1998) Let $V \in C^{2, 1} (R^{n} \times R_{+}; R_{+})$ and $τ_{1}$ , $τ_{2}$ be bounded stopping times such that $0 \leq τ_{1} \leq τ_{2}$ a.s. If $V (t, x)$ and $L V (x, t)$ are bounded on $t \in [τ_{1}, τ_{2}]$ a.s., then

E (V (x, τ_{2}) - V (x, τ_{1})) = E {\int_{τ_{1}}^{τ_{2}} L V (x, t) dt} .

Lemma 2: (Chen et al., 2017) Let $b$ be a real-valued function of $t \in R^{+}$ satisfying $b (t) \geq 1$ . For any $x, y \in R$ , the following inequality holds

| [x]^{b (t)} - [y]^{b (t)} | \leq b (t) (2^{b (t) - 2} + 2) (| x - y |^{b (t)} + | x - y | | y |^{b (t) - 1}),

where $| y |^{b (t) - 1} : = 0$ if $y = 0$ and $b (t) = 1$ .

Lemma 3: (Chen et al., 2017) Let $a,$ $b$ be positive real-valued functions of $t \in R^{+}$ and $γ$ be positive real-valued function of $x, y \in R$ . For any $x, y \in R$ , the following inequality holds

| x |^{a (t)} | y |^{b (t)} \leq \frac{a (t)}{a (t) + b (t)} γ (\cdot) | x |^{a (t) + b (t)} + \frac{b (t)}{a (t) + b (t)} γ^{\frac{- a (t)}{b (t)}} (\cdot) | y |^{a (t) + b (t)} .

Lemma 4: (Man and Liu, 2019) If $b (t) \geq 1$ is a continuous function, then for $\forall x_{i} \in R$ , $i = 1, \dots, n$ , the following inequality holds

\sum_{i = 1}^{n} | x_{i} |^{b (t)} \leq {(\sum_{i = 1}^{n} | x_{i} |)}^{b (t)} \leq n^{b (t) - 1} \sum_{i = 1}^{n} | x_{i} |^{b (t)} .

Lemma 5: (Man and Liu, 2019) If $p (t)$ is a continuous function and satisfies $1 < \underline{p} < p (t) < \bar{p}$ , then for $\forall x_{i} \in R$ , the following inequality holds

| x |^{p (t)} \leq x^{\underline{p}} + x^{\bar{p}} .

Lemma 6: (Lin and Qian, 2002) For any real-valued continuous function $f (x, y)$ , where $x \in R^{m}$ , $y \in R^{n}$ , there are smooth scalar-value functions $a (x) \geq 0$ , $b (y) \geq 0$ , $c (x) \geq 1$ and $d (y) \geq 1$ , the following inequality holds

| f (x, y) | \leq a (x) + b (y), | f (x, y) | \leq c (x) d (y) .

Controller design and stability analysis of system

Controller design

In this subsection, an adaptive output tracking controller for system (1) is proposed,

\overset{\cdot}{\hat{Θ}} = Ψ (x, y_{r}, \hat{Θ}), u = u (x, y_{r}, \hat{Θ}),

where $\hat{Θ}$ is the estimated parameter.

The following assumptions are made on system (1).

Assumption 1: For the unknown functions $f_{i} ({\bar{x}}_{i}, θ)$ and $g_{i} ({\bar{x}}_{i}, θ)$ , $i = 1, \dots, n$ , there exist nonnegative smooth functions ${\bar{f}}_{i} ({\bar{x}}_{i})$ , ${\bar{g}}_{i} ({\bar{x}}_{i})$ , ${\bar{b}}_{i 1} (θ)$ , ${\bar{b}}_{i 2} (θ)$ such that

| f_{i} ({\bar{x}}_{i}, θ) | \leq {\bar{f}}_{i} ({\bar{x}}_{i}) {\bar{b}}_{i 1} (θ), | g_{i} ({\bar{x}}_{i}, θ) | \leq {\bar{g}}_{i} ({\bar{x}}_{i}) {\bar{b}}_{i 2} (θ) .

Assumption 2: The power $p_{i} (t)$ of system (1) satisfies $1 \leq \underline{p} \leq p_{i} (t)$ $\leq \bar{p}$ , $i = 1, \dots, n$ .

Assumption 3: There exists a known constant $M \geq 0$ such that for $\forall t \in [t, + \infty)$

| y_{r} | (t) \leq M, | {\overset{\cdot}{y}}_{r} (t) | \leq M .

Remark 1: Assumption 1 indicates that the drift term $f_{i} ({\bar{x}}_{i}, θ)$ and diffusion term $g_{i} ({\bar{x}}_{i}, θ)$ are in general form, $i = 1, \dots, n$ . Compared with deterministic systems (Chen et al., 2017; Su et al., 2017), the nonlinear functions have unknown parameters in this paper. Assumption 2 shows that the unknown power belongs to an interval, and the restriction that $p_{i} (t)$ can only take odd constant is relaxed, which makes the range of unknown order larger. Assumption 3 is a common assumption frequently used in tracking control.

Defining

Θ = \max {{({1 + b}_{i 1}^{2} (θ))}^{\frac{\bar{p} + 3}{6}}, {({1 + b}_{i 2}^{2} (θ))}^{\frac{\bar{p} + 3}{4}}} .

Remark 2: By estimating the unknown constant $Θ$ instead of the unknown parameter vector $θ \in R^{m}$ , the complexity of the adaptive controller is reduced.

Now, we start the recursive step. Firstly, we introduce the coordinate transformation

z_{1} = x_{1} - y_{r}, z_{i} = x_{i} - α_{i}, i = 2, \dots, n,

(3)

where $α_{2}, \dots, α_{n}$ are virtual smooth controllers to be determined in each step.

Step 1: By (1) and (3), we get

d z_{1} = [x_{2}]^{p_{1} (t)} d t + f_{1} d t - {\overset{\cdot}{y}}_{r} dt + g_{1}^{T} d ω .

(4)

Choosing $V_{1} = \frac{1}{4} z_{1}^{4} + \frac{1}{2 γ} {\tilde{Θ}}^{2}$ , where $\tilde{Θ} = Θ - \hat{Θ}$ is the parameter estimation error, $γ > 0$ is a positive constant.

From (4), we get

L V_{1} = z_{1}^{3} ({[x_{2}]}^{p_{1} (t)} + f_{1} - {\overset{\cdot}{y}}_{r}) + \frac{3}{2} z_{1}^{2} | g_{1} |^{2} - \frac{1}{γ} \tilde{Θ} \overset{\cdot}{\hat{Θ}} .

(5)

By Assumptions 1 and 3 with Lemma 6, we get

f_{1} - {\overset{\cdot}{y}}_{r} \leq γ_{11} {(1 + b_{11}^{2} (θ))}^{\frac{1}{2}}, {| g_{1} |}^{2} \leq γ_{12} b_{12} (θ),

(6)

where $γ_{11} \geq {\bar{f}}_{1}^{2} + M, γ_{12} \geq {\bar{g}}_{1}^{2}$ are nonnegative smooth functions independent of $p_{1} (t)$ , and ${\bar{b}}_{11} (θ) \leq b_{11} (θ), {\bar{b}}_{12} (θ) \leq b_{12} (θ) .$

For any real numbers $δ_{11}$ , $δ_{12} > 0$ > 0, by Lemma 3 and (6), we get

\begin{matrix} z_{1}^{3} (f_{1} - {\overset{\cdot}{y}}_{r}) \leq {| z_{1} |}^{3} γ_{11} {(1 + b_{11}^{2} (θ))}^{\frac{1}{2}} \leq δ_{11} + {| z_{1} |}^{\bar{p} + 3} \\ ρ_{11} {(1 + b_{11}^{2} (θ))}^{\frac{\bar{p} + 3}{6}}, \frac{3}{2} z_{1}^{2} {| g_{1} |}^{2} \leq \frac{3}{2} z_{1}^{2} γ_{12} b_{12} (θ) \leq δ_{12} + {| z_{1} |}^{\bar{p} + 3} \\ ρ_{12} {(1 + b_{12}^{2} (θ))}^{\frac{\bar{p} + 3}{4}}, \end{matrix}

(7)

where

ρ_{11} = \frac{3}{\bar{p} + 3} {(1 + γ_{11})}^{\frac{\bar{p} + 3}{6}} {(\frac{δ_{11} (\bar{p} + 3)}{\bar{p}})}^{- \frac{\bar{p}}{3}}, ρ_{12} = \frac{2}{\bar{p} + 3} {(\frac{3}{2} {(1 + γ_{12}^{2})}^{\frac{1}{2}})}^{\frac{\bar{p} + 3}{6}} {(\frac{δ_{12} (\bar{p} + 3)}{\bar{p} + 1})}^{- \frac{\bar{p} + 1}{2}},

where $ρ_{11}$ and $ρ_{12}$ are nonnegative smooth functions independent of $p_{1} (t)$ .

By (7) together with (5), we get

\begin{matrix} L V_{1} \leq z_{1}^{3} [x_{2}]^{p_{1} (t)} + Θ | z_{1} |^{\bar{p} + 3} - \frac{1}{γ} \tilde{Θ} \overset{\cdot}{\hat{Θ}} + δ_{1} \\ = z_{1}^{3} ([x_{2}]^{p_{1} (t)} - [α_{2}]^{p_{1} (t)}) + z_{1}^{3} [α_{2}]^{p_{1} (t)} + δ_{1} + \hat{Θ} | z_{1} |^{\bar{p} + 3} \\ ρ_{1} + \tilde{Θ} (| z_{1} |^{\bar{p} + 3} ρ_{1} - \frac{1}{γ} \overset{\cdot}{\hat{Θ}}), \end{matrix}

(8)

where $ρ_{1} = ρ_{11} + ρ_{12}$ are nonnegative smooth functions independent of $p_{1} (t)$ .

Thus, if we take

α_{2} = - (z_{1} + {[z_{1}]}^{\bar{p}}) β_{1}^{\frac{1}{\underline{p}}} = - (z_{1} + {[z_{1}]}^{\bar{p}}) {(c_{1} + 1 + \sqrt{1 + {\hat{Θ}}^{2}} ρ_{1})}^{\frac{1}{\underline{p}}},

(9)

then we get

z_{1}^{3} [α_{2}]^{p_{1} (t)} = - z_{1}^{3} {[z_{1} + {[z_{1}]}^{\bar{p}}]}^{p_{1} (t)} {(c_{1} + 1 + \sqrt{1 + {\hat{Θ}}^{2}} ρ_{1})}^{\frac{p_{1} (t)}{\underline{p}}} .

From Lemma 4, we get

z_{1}^{3} {[z_{1} + {[z_{1}]}^{\bar{p}}]}^{p_{1} (t)} = | z_{1} |^{3} {(| z_{1} | + {| z_{1} |}^{\bar{p}})}^{p_{1} (t)} \geq ({| z_{1} |}^{p_{1} (t) + 3} + {| z_{1} |}^{\bar{p} p_{1} (t) + 3}),

such that

\begin{matrix} z_{1}^{3} [α_{2}]^{p_{1} (t)} \leq - | z_{1} |^{3} {(| z_{1} | + {| z_{1} |}^{\bar{p}})}^{p_{1} (t)} (c_{1} + 1 + \sqrt{1 + {\hat{Θ}}^{2}} ρ_{1}) \leq - ({| z_{1} |}^{p_{1} (t) + 3} + {| z_{1} |}^{\bar{p} p_{1} (t) + 3}) (c_{1} + 1 + \sqrt{1 + {\hat{Θ}}^{2}} ρ_{1}), \end{matrix}

(10)

where $c_{1} \geq 1$ is a design parameter and $β_{1} = (c_{1} + 1 + \sqrt{1 + {\hat{Θ}}^{2}} ρ_{1}) > 1$ is a smooth function independent of $p_{1} (t)$ .

By Lemma 5, we get

{| z_{1} |}^{\bar{p} + 3} \leq {| z_{1} |}^{p_{1} (t) + 3} + {| z_{1} |}^{\bar{p} p_{1} (t) + 3} .

(11)

Substituting (10) into (8) with (11) yields

\begin{matrix} L V_{1} \leq - (c_{1} + 1) ({| z_{1} |}^{p_{1} (t) + 3} + {| z_{1} |}^{\bar{p} p_{1} (t) + 3}) + z_{1}^{3} ({[x_{2}]}^{p_{1} (t)} - {[α_{2}]}^{p_{1} (t)}) + \frac{1}{γ} \tilde{Θ} (τ_{1} - \overset{\cdot}{\hat{Θ}}) + σ \tilde{Θ} \hat{Θ} + δ_{1} . \end{matrix}

(12)

where $δ_{1} = δ_{11} + δ_{12}$ and $σ$ are design parameters, and $τ_{1} = γ {| z_{1} |}^{\bar{p} + 3} ρ_{1} - γ σ \hat{Θ}$ .

Step 2: According to $It \hat{o}' s$ differentiation rule, using (1) and (3), we get

d z_{2} = ({[x_{3}]}^{p (t)} + F_{2} - \frac{\partial α_{2}}{\partial \hat{Θ}} \overset{\cdot}{\hat{Θ}}) d t + G_{2}^{T} d ω,

(13)

where

F_{2} = f_{2} - \frac{\partial α_{2}}{\partial x_{1}} ({[x_{2}]}^{p_{1} (t)} + f_{1}) - \frac{\partial α_{2}}{\partial y_{r}} {\overset{\cdot}{y}}_{r} - \frac{1}{2} \frac{\partial^{2} α_{2}}{\partial x_{1}^{2}} g_{1}^{T} g_{1}, G_{2}^{T} = g_{2}^{T} - \frac{\partial α_{2}}{\partial x_{1}} g_{1}^{T} .

(14)

Choosing $V_{2} = V_{1} + \frac{1}{4} z_{2}^{4}$ with (12)–(14), we get

\begin{matrix} L V_{2} \leq - (c_{1} + 1) ({| z_{1} |}^{p_{1} (t) + 3} + {| z_{1} |}^{\bar{p} p_{1} (t) + 3}) + z_{1}^{3} ({[x_{2}]}^{p_{1} (t)} - {[α_{2}]}^{p_{1} (t)}) + \frac{1}{γ} \tilde{Θ} (τ_{1} - \overset{\cdot}{\hat{Θ}}) \\ + σ \tilde{Θ} \hat{Θ} + z_{2}^{3} ({[x_{3}]}^{p_{2} (t)} + F_{2} + \frac{\partial α_{2}}{\partial \hat{Θ}} \overset{\cdot}{\hat{Θ}}) + \frac{3}{2} z_{2}^{2} {| G_{2} |}^{2} + δ_{1} . \end{matrix}

(15)

By Assumptions 1 and 3 and Lemma 5, we obtain

\begin{matrix} F_{2} = f_{2} - \frac{\partial α_{2}}{\partial x_{1}} ({[x_{2}]}^{p_{1} (t)} + f_{1}) - \frac{\partial α_{2}}{\partial y_{r}} {\overset{\cdot}{y}}_{r} - \frac{1}{2} \frac{\partial^{2} α_{2}}{\partial x_{1}^{2}} g_{1}^{T} g_{1} \\ \leq | f_{2} | + | \frac{\partial α_{2}}{\partial x_{1}} | ({[x_{2}]}^{\bar{p}} + {[x_{2}]}^{\underline{p}} + | f_{1} |) + M | \frac{\partial α_{2}}{\partial y_{r}} | + | \frac{1}{2} \frac{\partial^{2} α_{2}}{\partial x_{1}^{2}} | g_{1}^{2} \\ \leq {\bar{f}}_{2} {\bar{b}}_{21} (θ) + | \frac{\partial α_{2}}{\partial x_{1}} | ({[x_{2}]}^{\bar{p}} + {[x_{2}]}^{\underline{p}} + {\bar{f}}_{1} {\bar{b}}_{11} (θ)) + M | \frac{\partial α_{2}}{\partial y_{r}} | + | \frac{1}{2} \frac{\partial^{2} α_{2}}{\partial x_{1}^{2}} | {\bar{g}}_{1}^{2} {\bar{b}}_{12} (θ) \\ \leq ({\bar{f}}_{2} + | \frac{\partial α_{2}}{\partial x_{1}} | {\bar{f}}_{1} + | \frac{1}{2} \frac{\partial^{2} α_{2}}{\partial x_{1}^{2}} | {\bar{g}}_{1}^{2}) b_{21} (θ) + | \frac{\partial α_{2}}{\partial x_{1}} | ({[x_{2}]}^{\bar{p}} + {[x_{2}]}^{\underline{p}}) + M | \frac{\partial α_{2}}{\partial y_{r}} | \\ \leq ({\bar{f}}_{2} + | \frac{\partial α_{2}}{\partial x_{1}} | {\bar{f}}_{1} + | \frac{1}{2} \frac{\partial^{2} α_{2}}{\partial x_{1}^{2}} | {\bar{g}}_{1}^{2} + | \frac{\partial α_{2}}{\partial x_{1}} | ({[x_{2}]}^{\bar{p}} + {[x_{2}]}^{\underline{p}}) + M | \frac{\partial α_{2}}{\partial y_{r}} |) {(1 + b_{21}^{2} (θ))}^{\frac{1}{2}} \\ \leq γ_{21} {(1 + b_{21}^{2} (θ))}^{\frac{1}{2}}, \\ G_{2} = | g_{2}^{T} - \frac{\partial α_{2}}{\partial x_{1}} g_{1}^{T} | | g_{2}^{T} - \frac{\partial α_{2}}{\partial x_{1}} g_{1}^{T} | \leq {(| g_{2} | - | \frac{\partial α_{2}}{\partial x_{1}} | | g_{1} |)}^{2} \\ \leq {(g_{2} b_{22}^{\frac{1}{2}} (θ) + | \frac{\partial α_{2}}{\partial x_{1}} | g_{1} b_{12}^{\frac{1}{2}} (θ))}^{2} \leq {({\bar{g}}_{2} + | \frac{\partial α_{2}}{\partial x_{1}} | {\bar{g}}_{1})}^{2} b_{22} (θ) \\ \leq γ_{22} b_{22} (θ), \end{matrix}

(16)

where

\begin{matrix} γ_{21} \geq {\bar{f}}_{2} + | \frac{\partial α_{2}}{\partial x_{1}} | ({[x_{2}]}^{\bar{p}} + {[x_{2}]}^{\underline{p}} + {\bar{f}}_{1}) + M | \frac{\partial α_{2}}{\partial y_{r}} | + | \frac{1}{2} \frac{\partial^{2} α_{2}}{\partial x_{1}^{2}} | {\bar{g}}_{1}^{2}, \\ γ_{22} \geq {({\bar{g}}_{2} + | \frac{\partial α_{2}}{\partial x_{1}} | {\bar{g}}_{1})}^{2}, \\ b_{21} (θ) \geq max {{\bar{b}}_{11} (θ), {\bar{b}}_{12} (θ), {\bar{b}}_{21} (θ)}, \\ b_{22} (θ) \geq max {{\bar{b}}_{12} (θ), {\bar{b}}_{22} (θ)}, \end{matrix}

and $γ_{21}$ , $γ_{22}$ re nonnegative smooth functions independent of $p_{1} (t), p_{2} (t)$ .

For any real numbers $δ_{21}$ , $δ_{22} > 0$ , by Lemma 3 and (16), we obtain

z_{1}^{3} F_{2} \leq δ_{21} + {| z_{2} |}^{\bar{p} + 3} ρ_{21} {(1 + b_{21}^{2} (θ))}^{\frac{\bar{p} + 3}{6}}, \frac{3}{2} z_{2}^{2} {| G_{2} |}^{2} \leq δ_{22} + {| z_{2} |}^{\bar{p} + 3} ρ_{22} {(1 + b_{22}^{2} (θ))}^{\frac{\bar{p} + 3}{4}},

(17)

where $ρ_{21}$ and $ρ_{22}$ are nonnegative smooth functions independent of $p_{1} (t)$ , $p_{2} (t)$ .

By Lemma 2, (3) and (9), we get

\begin{matrix} z_{1}^{3} ([x_{2}]^{p_{1} (t)} - [α_{2}]^{p_{1} (t)}) \\ \leq \bar{p} (2^{\bar{p} - 2} + 2) (| z_{1} |^{3} | z_{2} |^{p_{1} (t)} + | z_{1} |^{p_{1} (t) + 2} | z_{2} | {(1 + | z_{1} |^{\bar{p} - 1})}^{\bar{p} - 1} β_{1}^{\bar{p} - 1}) \\ \leq \bar{p} (2^{\bar{p} - 2} + 2) (| z_{1} |^{3} (| z_{2} |^{\bar{p}} + | z_{2} |) + (| z_{1} |^{\bar{p} + 2} + | z_{1} |^{3}) | z_{2} | {(1 + | z_{1} |^{\bar{p} - 1})}^{\bar{p} - 1} β_{1}^{\bar{p} - 1}) \\ \leq \bar{p} (2^{\bar{p} - 2} + 2) | z_{1} |^{3} | z_{2} |^{\bar{p}} + \bar{p} (2^{\bar{p} - 2} + 2) (| z_{1} |^{3} + (| z_{1} |^{\bar{p} + 2} + | z_{1} |^{3}) {(1 + | z_{1} |^{\bar{p} - 1})}^{\bar{p} - 1} β_{1}^{\bar{p} - 1}) | z_{2} | . \end{matrix}

(18)

From Lemma 3, we have

\begin{matrix} \bar{p} (2^{\bar{p} - 2} + 2) | z_{1} |^{3} | z_{2} |^{\bar{p}} \leq | z_{1} |^{\bar{p} + 3} + | ξ_{2} |^{\bar{p} + 3} φ_{21}, \\ \bar{p} (2^{\bar{p} - 2} + 2) (| z_{1} |^{3} + (| z_{1} |^{\bar{p} + 2} + | z_{1} |^{3}) {(1 + | z_{1} |^{\bar{p} - 1})}^{\bar{p} - 1} β_{1}^{\bar{p} - 1}) | z_{2} | \leq δ_{23} + | z_{2} |^{\bar{p} + 3} φ_{22}, \end{matrix}

(19)

where

\begin{matrix} φ_{21} = \frac{\bar{p}}{\bar{p} + 3} {(\frac{\bar{p} + 3}{3})}^{- \frac{3}{\bar{p}}} {(\bar{p} (2^{\bar{p} - 2} + 2))}^{\frac{\bar{p} + 3}{\bar{p}}}, \\ φ_{22} = \frac{1}{\bar{p} + 3} {(\frac{δ_{23} (\bar{p} + 3)}{\bar{p} + 2})}^{- (\bar{p} + 2)} {(\bar{p} (2^{\bar{p} - 2} + 2) ({| z_{1} |}^{3} + ({| z_{1} |}^{\bar{p} + 2} + {| z_{1} |}^{3}) {(1 + {| z_{1} |}^{\bar{p} - 1}}^{)^{\bar{p} - 1}} β_{1}^{\bar{p} - 1}))}^{\bar{p} + 3} . \end{matrix}

Substituting (19) into (18) yields

z_{1}^{3} ([x_{2}]^{p_{1} (t)} - [α_{2}]^{p_{1} (t)}) \leq δ_{23} + | z_{1} |^{\bar{p} + 3} + | z_{2} |^{\bar{p} + 3} φ_{20},

(20)

where $δ_{23}$ is a design parameter and $φ_{20} = φ_{21} + φ_{22} \geq 0$ is a smooth function independent of $p_{1} (t)$ , $p_{2} (t)$ .

Substituting (17) and (20) into (15) yields

\begin{matrix} L V_{2} \leq - c_{1} ({| z_{1} |}^{p_{1} (t) + 3} + {| z_{1} |}^{\bar{p} p_{1} (t) + 3}) + z_{1}^{3} [α_{3}]^{p_{2} (t)} + z_{1}^{3} ({[x_{3}]}^{p_{2} (t)} - {[α_{3}]}^{p_{2} (t)}) + {| z_{2} |}^{\bar{p} + 3} φ_{20} + δ_{1} \\ + \hat{Θ} {| z_{2} |}^{\bar{p} + 3} ρ_{2} + (\frac{1}{γ} \tilde{Θ} + z_{2}^{3} \frac{\partial α_{2}}{\partial \hat{Θ}}) (τ_{2} - \overset{\cdot}{\hat{Θ}}) - z_{2}^{3} \frac{\partial α_{2}}{\partial \hat{Θ}} τ_{2} + σ \tilde{Θ} \hat{Θ} + δ_{21} + δ_{22} + δ_{23}, \end{matrix}

(21)

where $ρ_{2} = ρ_{21} + ρ_{22}$ is a smooth function independent of $p_{1} (t)$ and $p_{2} (t)$ , $τ_{2} = τ_{1} + γ {| z_{2} |}^{\bar{p} + 3} ρ_{2}$ .

From Lemma 3, we have

- z_{2}^{3} \frac{\partial α_{2}}{\partial \hat{Θ}} τ_{2} \leq γ {| z_{2} |}^{3} ({| z_{1} |}^{\bar{p} + 3} ρ_{1} + {| z_{2} |}^{\bar{p} + 3} ρ_{2} + σ | \hat{Θ} |) \leq δ_{24} + δ_{25} + {| z_{2} |}^{\bar{p} + 3} H_{2},

(22)

where

\begin{matrix} H_{2} = \frac{3}{\bar{p} + 3} {(\frac{\bar{p}}{δ_{24} (\bar{p} + 3)})}^{\frac{\bar{p}}{3}} {(1 + {(γ {| z_{1} |}^{\bar{p} + 3} \frac{\partial α_{2}}{\partial \hat{Θ}} ρ_{1})}^{2})}^{\frac{\bar{p} + 3}{6}} + {(1 + {(γ z_{1}^{3} \frac{\partial α_{2}}{\partial \hat{Θ}} ρ_{2})}^{2})}^{\frac{1}{2}} \\ + \frac{3}{\bar{p} + 3} {(\frac{\bar{p}}{δ_{25} (\bar{p} + 3)})}^{\frac{\bar{p}}{3}} {(1 + {(γ σ \hat{Θ} \frac{\partial α_{2}}{\partial \hat{Θ}})}^{2})}^{\frac{\bar{p} + 3}{6}}, \end{matrix}

which is independent of $p_{1} (t)$ and $p_{2} (t)$ .

Substituting (22) into (21) yields

\begin{matrix} L V_{2} \leq - c_{1} ({| z_{1} |}^{p_{1} (t) + 3} + {| z_{1} |}^{\bar{p} p_{1} (t) + 3}) + z_{1}^{3} [α_{3}]^{p_{2} (t)} + z_{1}^{3} ({[x_{3}]}^{p_{2} (t)} - {[α_{3}]}^{p_{2} (t)}) + {| z_{2} |}^{\bar{p} + 3} φ_{20} \\ + {| z_{2} |}^{\bar{p} + 3} H_{2} + \hat{Θ} {| z_{2} |}^{\bar{p} + 3} ρ_{2} + (\frac{1}{γ} \tilde{Θ} + z_{2}^{3} \frac{\partial α_{2}}{\partial \hat{Θ}}) (τ_{2} - \overset{\cdot}{\hat{Θ}}) + σ \tilde{Θ} \hat{Θ} + δ_{2}, \end{matrix}

(23)

where $δ_{2} = δ_{1} + δ_{21} + δ_{22} + δ_{23} + δ_{24} + δ_{25}$ is a design parameter.

Thus, if we take

α_{3} = - (z_{2} + {[z_{2}]}^{\bar{p}}) β_{2}^{\frac{1}{\underline{p}}} = - (z_{2} + {[z_{2}]}^{\bar{p}}) {(c_{2} + 1 + ρ_{2} \sqrt{1 + {\hat{Θ}}^{2}} + φ_{20} + H_{2})}^{\frac{1}{\underline{p}}},

then we get

z_{2}^{3} [α_{3}]^{p_{2} (t)} = - ({| z_{2} |}^{p_{2} (t) + 3} + {| z_{2} |}^{\bar{p} p_{2} (t) + 3}) (c_{2} + 1 + ρ_{2} \sqrt{1 + {\hat{Θ}}^{2}} + φ_{20} + H_{2}),

(24)

where $c_{2} \geq 1$ is a design parameter and $β_{2} = (c_{2} + 1 + ρ_{2} \sqrt{1 + {\hat{Θ}}^{2}} + φ_{20} + H_{2}) \geq 1$ is smooth function independent of $p_{1} (t), p_{2} (t)$ .

Substituting (24) into (23) yields $\begin{matrix} L V_{2} \leq - c_{1} ({| z_{1} |}^{p_{1} (t) + 3} + {| z_{1} |}^{\bar{p} p_{1} (t) + 3}) - (c_{2} + 1) ({| z_{2} |}^{p_{2} (t) + 3} + {| z_{2} |}^{\bar{p} p_{2} (t) + 3}) + z_{1}^{3} ({[x_{3}]}^{p_{2} (t)} - {[α_{3}]}^{p_{2} (t)}) \\ + (\frac{1}{γ} \tilde{Θ} + z_{2}^{3} \frac{\partial α_{2}}{\partial \hat{Θ}}) (τ_{2} - \overset{\cdot}{\hat{Θ}}) + σ \tilde{Θ} \hat{Θ} + δ_{2} . \end{matrix}$

Deductive step

Assume that at step $i - 1$ , there exists a smooth virtual controller

α_{i} = - (z_{i - 1} + {[z_{i - 1}]}^{\bar{p}}) β_{i - 1}^{\frac{1}{\underline{p}}},

(25)

such that

\begin{matrix} L V_{i - 1} \leq - \sum_{j = 1}^{i - 2} c_{j} ({| z_{j} |}^{p_{j} (t) + 3} + {| z_{j} |}^{\bar{p} p_{j} (t) + 3}) - (c_{i - 1} + 1) ({| z_{i - 1} |}^{p_{i - 1} (t) + 3} + {| z_{i - 1} |}^{\bar{p} p_{i - 1} (t) + 3}) + σ \tilde{Θ} \hat{Θ} \\ + z_{i - 1}^{3} ({[x_{i - 1}]}^{p_{i - 1} (t)} - {[α_{i - 1}]}^{p_{i - 1} (t)}) + (\frac{1}{γ} \tilde{Θ} + \sum_{j = 2}^{i - 1} z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}}) (τ_{i - 1} - \overset{\cdot}{\hat{Θ}}) + δ_{i - 1}, \end{matrix}

(26)

where $c_{2} \geq 1$ and $δ_{i - 1}$ are design parameters.

Then, we will prove that (26) still holds for the $i$ th Lyapunov function

V_{i} = V_{i - 1} + \frac{1}{4} z_{i}^{4} .

(27)

By (1) and (3), we have

d z_{i} = ({[x_{i + 1}]}^{p_{i} (t)} + F_{i} - \frac{\partial α_{i}}{\partial \hat{Θ}} \overset{\cdot}{\hat{Θ}}) dt + G_{i}^{T} dw,

(28)

where

F_{i} = f_{i} - \sum_{j = 1}^{i - 1} \frac{\partial α_{i}}{\partial x_{j}} ({[x_{j + 1}]}^{p_{j} (t)} + f_{j}) - \frac{\partial α_{i}}{\partial y_{r}} {\overset{\cdot}{y}}_{r} - \frac{1}{2} \sum_{j, l = 1}^{i - 1} \frac{\partial^{2} α_{i}}{\partial x_{j} x_{l}} g_{j} g_{l}^{T}, G_{i}^{T} = g_{i}^{T} - \sum_{j = 1}^{i - 1} \frac{\partial α_{i}}{\partial x_{j}} g_{j}^{T} .

By (26)–(28), we get

\begin{matrix} L V_{i} \leq - \sum_{j = 1}^{i - 2} c_{j} ({| z_{j} |}^{p_{j} (t) + 3} + {| z_{j} |}^{\bar{p} p_{j} (t) + 3}) - (c_{i - 1} + 1) ({| z_{i - 1} |}^{p_{i - 1} (t) + 3} + {| z_{i - 1} |}^{\bar{p} p_{i - 1} (t) + 3}) \\ + z_{i - 1}^{3} ({[x_{i}]}^{p_{i - 1} (t)} - {[α_{i}]}^{p_{i - 1} (t)}) + (\frac{1}{γ} \tilde{Θ} + \sum_{j = 2}^{i - 1} z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}}) (τ_{i - 1} - \overset{\cdot}{\hat{Θ}}) \\ + z_{i}^{3} ({[x_{i + 1}]}^{p_{i} (t)} + F_{i} - \frac{\partial α_{i}}{\partial \hat{Θ}} \overset{\cdot}{\hat{Θ}}) + \frac{3}{2} z_{i}^{2} {| G_{i} |}^{2} + σ \tilde{Θ} \hat{Θ} + δ_{i - 1} . \end{matrix}

(29)

Similar to (16), we obtain

z_{i}^{3} F_{i} \leq γ_{i 1} {(1 + b_{i 1}^{2} (θ))}^{\frac{1}{2}}, G_{i}^{T} \leq γ_{i 2} b_{i 2} (θ),

(30)

where $γ_{i 1}$ and $γ_{i 2}$ are nonnegative smooth functions independent of $p_{1} (t), \dots, p_{i} (t)$ .

By Lemma 3 and (30), we get

z_{i}^{3} F_{i} \leq δ_{i 1} + {| z_{i} |}^{\bar{p} + 3} ρ_{i 1} {(1 + b_{i 1}^{2} (θ))}^{\frac{\bar{p} + 3}{6}}, \frac{3}{2} z_{i}^{2} {| G_{i} |}^{2} \leq δ_{i 2} + {| z_{i} |}^{\bar{p} + 3} ρ_{i 2} {(1 + b_{i 2}^{2} (θ))}^{\frac{\bar{p} + 3}{4}},

(31)

where $δ_{i 1}, δ_{i 2} > 0$ are $ρ_{i 1}$ , $ρ_{i 2}$ are smooth functions independent of $p_{1} (t), \dots, p_{i} (t)$ .

By Lemma 2, Lemma 3, (3) and (25), we get

z_{i - 1}^{3} ({[x_{i}]}^{p_{i - 1} (t)} - {[α_{i}]}^{p_{i - 1} (t)}) \leq δ_{i 3} + {| z_{i - 1} |}^{\bar{p} + 3} + {| z_{i} |}^{\bar{p} + 3} φ_{i 0},

(32)

where $δ_{i 3}$ is a design parameter and $φ_{i 0} > 0$ is a smooth functions independent of $p_{1} (t), \dots, p_{i} (t)$ .

Substituting (31) and (32) into (29) yields

\begin{matrix} L V_{i} \leq - \sum_{j = 1}^{i - 1} c_{j} ({| z_{j} |}^{p_{j} (t) + 3} + {| z_{j} |}^{\bar{p} p_{j} (t) + 3}) + z_{i}^{3} {[x_{i + 1}]}^{p_{i} (t)} + {| z_{i} |}^{\bar{p} + 3} φ_{i 0} \\ + (\frac{1}{γ} \tilde{Θ} + \sum_{j = 2}^{i} z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}}) (τ_{i} - \overset{\cdot}{\hat{Θ}}) - z_{i}^{3} \frac{\partial α_{i}}{\partial \hat{Θ}} τ_{i} + \hat{Θ} {| z_{i} |}^{\bar{p} + 3} ρ_{i} \\ - \sum_{j = 2}^{i - 1} γ z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}} ρ_{i} {| z_{i} |}^{\bar{p} + 3} + σ \tilde{Θ} \hat{Θ} + δ_{i - 1} + δ_{i 1} + δ_{i 2} + δ_{i 3} . \end{matrix}

(33)

where $ρ_{i} = ρ_{i 1} + ρ_{i 2}$ is smooth functions independent of $p_{1} (t), \dots, p_{i} (t)$ and $τ_{i} = τ_{i - 1} + γ {| z_{i} |}^{\bar{p} + 3} ρ_{i} .$

From Lemma 3, we have

- z_{i}^{3} \frac{\partial α_{i}}{\partial \hat{Θ}} τ_{i} \leq γ | \frac{\partial α_{i}}{\partial \hat{Θ}} | | z_{i}^{3} | (\sum_{j = 1}^{i} {| z_{j} |}^{\bar{p} + 3} ρ_{j} + σ | \hat{Θ} |) \leq \sum_{j = 1}^{i - 1} δ_{i 4 j} + δ_{i 5} + {| z_{j} |}^{\bar{p} + 3} H_{i 1},

- \sum_{j = 1}^{i - 1} γ z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}} ρ_{i} {| z_{i} |}^{\bar{p} + 3} \leq {| z_{j} |}^{\bar{p} + 3} H_{i 2},

where

\begin{matrix} H_{i 1} = \sum_{j = 1}^{i - 1} \frac{3}{\bar{p} + 3} {(\frac{\bar{p}}{δ_{i 4 j} (\bar{p} + 3)})}^{\frac{\bar{p}}{3}} {(1 + {(γ {| z_{j} |}^{\bar{p} + 3} \frac{\partial α_{i}}{\partial \hat{Θ}} ρ_{j})}^{2})}^{\frac{\bar{p} + 3}{6}} + {(1 + {(γ z_{i}^{3} \frac{\partial α_{i}}{\partial \hat{Θ}} ρ_{i})}^{2})}^{\frac{1}{2}} \\ + \frac{3}{\bar{p} + 3} {(\frac{\bar{p}}{δ_{i 5} (\bar{p} + 3)})}^{\frac{\bar{p}}{3}} {(1 + {(γ σ \hat{Θ} \frac{\partial α_{i}}{\partial \hat{Θ}})}^{2})}^{\frac{\bar{p} + 3}{6}}, \\ H_{i 2} = \sum_{j = 2}^{i - 1} {(1 + {(γ z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}} ρ_{i})}^{2})}^{\frac{1}{2}} . \end{matrix}

we get

- z_{i}^{3} \frac{\partial α_{i}}{\partial \hat{Θ}} τ_{i} - \sum_{j = 1}^{i - 1} γ z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}} ρ_{i} {| z_{i} |}^{\bar{p} + 3} \leq δ_{i 4} + δ_{i 5} + {| z_{j} |}^{\bar{p} + 3} H_{i},

(34)

where $δ_{i 4} = \sum_{j = 1}^{i - 1} δ_{i 4 j}$ is a design parameter and $H_{i} = H_{i 1} + H_{i 2} > 0$ is a smooth function independent of $p_{1} (t), \dots, p_{i} (t)$ .

Substituting (34) into (33) yields

\begin{matrix} L V_{i} \leq - \sum_{j = 1}^{i - 1} c_{j} ({| z_{j} |}^{p_{j} (t) + 3} + {| z_{j} |}^{\bar{p} p_{j} (t) + 3}) + z_{i}^{3} ({[x_{i + 1}]}^{p_{i} (t)} - {[α_{i + 1}]}^{p_{i} (t)}) + {| z_{i} |}^{\bar{p} + 3} φ_{i 0} \\ + (\frac{1}{γ} \tilde{Θ} + \sum_{j = 2}^{i} z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}}) (τ_{i} - \overset{\cdot}{\hat{Θ}}) + \hat{Θ} {| z_{i} |}^{\bar{p} + 3} ρ_{i} + {| z_{i} |}^{\bar{p} + 3} H_{i} + δ_{i}, \end{matrix}

(35)

where $δ_{i} = δ_{i - 1} + δ_{i 1} + δ_{i 2} + δ_{i 3} + δ_{i 4} + δ_{i 5}$ and $c_{j} \geq 1$ is a design parameter, $j = 1, \dots, i - 1 .$

Thus, if we take

α_{i + 1} = - (z_{i} + {[z_{i}]}^{\bar{p}}) β_{i}^{\frac{1}{\underline{p}}} = - (z_{i} + {[z_{i}]}^{\bar{p}}) {(c_{i} + 1 + ρ_{i} \sqrt{1 + {\hat{Θ}}^{2}} + φ_{i 0} + H_{i})}^{\frac{1}{\underline{p}}},

then by (35) we get

\begin{matrix} L V_{i} \leq - \sum_{j = 1}^{i - 1} c_{j} ({| z_{j} |}^{p_{j} (t) + 3} + {| z_{j} |}^{\bar{p} p_{j} (t) + 3}) \\ + (c_{i} + 1) ({| z_{i} |}^{p_{i} (t) + 3} + {| z_{i} |}^{\bar{p} p_{i} (t) + 3}) + σ \tilde{Θ} \hat{Θ} \\ + z_{i} ({[x_{i + 1}]}^{p_{i} (t)} - {[α_{i + 1}]}^{p_{i} (t)}) + (\frac{1}{γ} \tilde{Θ} + \sum_{j = 2}^{i} z_{j}^{3} \frac{\partial α_{j}}{\partial \hat{Θ}}) \\ (τ_{i} - \overset{\cdot}{\hat{Θ}}) + δ_{i} . \end{matrix}

Step n: Choosing

V_{n} = \sum_{i = 1}^{n} \frac{1}{4} z_{i}^{4} + \frac{1}{2 γ} {\tilde{Θ}}^{2} .

(36)

By (36), similar to (35), if we take

\begin{matrix} u = - (z_{n} + {[z_{n}]}^{\bar{p}}) {(c_{n} + 1 + ρ_{n} \sqrt{1 + {\hat{Θ}}^{2}} + φ_{n 0} + H_{n})}^{\frac{1}{\underline{p}}} \\ \overset{\cdot}{\hat{Θ}} = τ_{n} = \sum_{i = 1}^{n} γ {| z_{i} |}^{\bar{p} + 3} ρ_{i} - γ σ \hat{Θ}, \end{matrix}

(37)

then we get

\begin{matrix} L V_{n} \leq - \sum_{j = 1}^{n} c_{j} ({| z_{j} |}^{p_{j} (t) + 3} + {| z_{j} |}^{\bar{p} p_{j} (t) + 3}) + + σ \tilde{Θ} \hat{Θ} + δ_{n} \\ \leq - \sum_{j = 1}^{n} c_{j} {| z_{j} |}^{\bar{p} + 3} + σ \tilde{Θ} \hat{Θ} + δ_{n} . \end{matrix}

(38)

From Lemma 3, we have

σ \tilde{Θ} \hat{Θ} = σ \tilde{Θ} Θ - σ {\tilde{Θ}}^{2} \leq - \frac{σ}{2} {\tilde{Θ}}^{2} + \frac{σ}{2} Θ^{2} .

(39)

Substituting (39) into (38) yields

L V_{n} \leq - \sum_{j = 1}^{n} c_{j} {| z_{j} |}^{\bar{p} + 3} - \frac{σ}{2} {\tilde{Θ}}^{2} + \frac{σ}{2} Θ^{2} + δ_{n},

(40)

where $c_{n} \geq 1$ and $δ_{n} = \sum_{i = 1}^{n} (δ_{i 1} + δ_{i 2} + δ_{i 3} + δ_{i 4} + δ_{i 5})$ are design parameters, $δ_{13} = δ_{14} = δ_{15}$ $= 0$ .

Remark 3: A time-varying controller seems impossible since $p_{i} (t)$ is indifferentiable. Thus, we should turn to designing a time-invariant controller. Noting system (1) is essentially time-varying, how to design a controller independent of time with the effect of noise is nontrivial. In the design process, the function $α_{2}, \dots, u$ are independent of $p_{1} (t), \dots, p_{n} (t)$ by using $\bar{p}$ and $\underline{p}$ .

Remark 4: What should be emphasized is that all the smooth functions $φ_{i 0}$ , $p_{j}$ , $H_{j}$ , $i = 1, \dots, n$ , $j = 2, \dots, n$ do exist. Taking $ρ_{11}$ for example, one can choose $ρ_{11} = \frac{3}{\bar{p} + 3} {(1 + γ_{11}^{2})}^{\frac{\bar{p} + 3}{6}} {(\frac{δ_{11} (\bar{p} + 3)}{\bar{p}})}^{- \frac{\bar{p}}{3}}$ .Obviously, due to $1 + γ_{11}^{2} \geq 1$ , the problem of division by zero for any order derivative of the item ${(1 + γ_{11}^{2})}^{\frac{\bar{p} + 3}{6}}$ can always be avoided, and $ρ_{11}$ is smooth with respect to $z_{1}$ .

Stability analysis

The following theorem gives the stability analysis of the closed-loop system.

Theorem 1: If Assumptions 1–3 hold for the high-order stochastic nonlinear system (1), then under the adaptive controller (37) one has the following:

The closed-loop system composed of (1) and (37) has a unique solution on $[0, + \infty)$ ;

The output tracking error $E {| y (t) - y_{r} (t) |}^{4}$ can be made arbitrarily small by appropriately choosing the design parameters.

All the states of the closed-loop system are bounded in probability.

Proof: Due to $f_{i}$ and $g_{i}$ being smooth functions and the function $f (y) = [y]^{p_{i} (t)}$ with unknown parameters also satisfying Lemma 1 in Li et al. (2020b), we can get that the closed-loop system satisfying the locally Lipschitz condition.

When $p_{i} (t) > 1$ . By Lemma 3 and any positive constant $ε < c_{j}$ , we have

ε^{\frac{1}{\bar{p} + 3}} z_{j}^{4} \leq c_{j} {| z_{j} |}^{\bar{p} + 3} + \frac{\bar{p} - 1}{\bar{p} + 3} {(\frac{c_{j} (\bar{p} + 3)}{4})}^{- \frac{4}{\bar{p} - 1}} ε^{\frac{1}{\bar{p} - 1}} .

A direct conclusion yields

c_{j} {| z_{j} |}^{\bar{p} + 3} \geq ε^{\frac{1}{\bar{p} + 3}} z_{j}^{4} - \frac{\bar{p} - 1}{\bar{p} + 3} {(\frac{4 \sqrt[4]{ε}}{c_{j} (\bar{p} + 3)})}^{\frac{4}{\bar{p} - 1}} .

(41)

Defining $V = V_{n}$ . Substituting (41) into (40) yields

\begin{matrix} L V \leq - \sum_{j = 1}^{n} (ε^{\frac{1}{\bar{p} + 3}} z_{j}^{4} - {\bar{b}}_{j}) - \frac{σ}{2} {\tilde{Θ}}^{2} + \frac{σ}{2} Θ^{2} + δ_{n} \\ \leq - (\sum_{j = 1}^{n} \frac{1}{4} ε^{\frac{1}{\bar{p} + 3}} z_{j}^{4} + γ σ \frac{1}{2 γ} {\tilde{Θ}}^{2}) + \sum_{j = 1}^{n} {\bar{b}}_{j} + \frac{σ}{2} Θ^{2} + δ_{n} \\ \leq - aV + b, \end{matrix}

(42)

where $a = min {ε^{\frac{1}{\bar{p} + 3}}, γ σ}, b = \sum_{j = 1}^{n} {\bar{b}}_{j} + \frac{σ}{2} Θ^{2} + δ_{n}, {\bar{b}}_{j} = \frac{\bar{p} - 1}{\bar{p} + 3} {(\frac{4 \sqrt[4]{ε}}{c_{j} (\bar{p} + 3)})}^{\frac{4}{\bar{p} - 1}} .$

When $p_{i} (t) = 1 .$ Defining $V = V_{n} .$ by (40), we have

\begin{matrix} L V \leq - (\sum_{j = 1}^{n} \frac{c_{j}}{4} z_{j}^{4} + γ σ \frac{1}{2 γ} {\tilde{Θ}}^{2}) + \frac{σ}{2} Θ^{2} + δ_{n} \\ \leq - aV + b, \end{matrix}

(43)

where $a = min {c_{j}, γ σ}, b = \frac{σ}{2} Θ^{2} + δ_{n}, σ > 0$ is the small design constant.

Noting that the local Lipschitz condition of the closed-loop system holds, and by (42), (43) and Theorem 4.1 in Khasminskii (1980), the conclusions (i) hold.

For $p_{i} (t) > 1$ . Let

h (t) = {(z_{1} (t), \dots z_{n} (t), \tilde{Θ} (t))}^{T}, η_{l} = inf {t : t > t_{0}, | h | \geq l}, \forall l \geq 0,

(44)

and $t_{l} = min {η_{l}, t}$ for all $t \geq t_{0}$ . Since $| h |$ is bounded in the interval $[t_{0}, t_{l}]$ a.s., $V (h)$ is bounded on $[t_{0}, t_{l}]$ a.s. From (42), it can be obtained that $L V$ is also bounded on $[t_{0}, t_{l}]$ a.s.

By Lemma 1 we have

\begin{matrix} E e^{at} V (h (t_{l})) \leq E \int_{t_{0}}^{t_{l}} e^{as} L V (h (s)) ds + e^{a t_{0}} EV (h (t_{0})) \\ + aE \int_{t_{0}}^{t_{l}} e^{as} V (h (s)) ds . \end{matrix}

(45)

According to $t_{l} = min {η_{l}, t}$ for all $lim_{l \to \infty} η_{l} = \infty$ , we can get

lim_{l \to \infty} t_{l} = t .

From the Fatou lemma in Mao and Yuan (2006), we get

E e^{at} V (h (t)) = E lim_{l \to \infty} inf e^{a t_{l}} V (h (t_{l})) \leq lim_{l \to \infty} inf E e^{a t_{l}} V (h (t_{l})) .

(46)

By taking the limit on the left and right sides of (45) with (46), we have

\begin{matrix} E e^{at} V (h (t)) \leq \int_{t_{0}}^{t} e^{as} L V (h (s)) ds + e^{a t_{0}} EV (h (t_{0})) \\ + aE \int_{t_{0}}^{t} e^{as} V (h (s)) ds, \end{matrix}

(47)

which together with (42) implies

EV (h (t)) \leq e^{- a (t - t_{0})} EV (h (t_{0})) + \frac{b}{a} (1 - e^{a (t - t_{0})}) .

(48)

From (36) and the defining of $V = V_{n}$ , we get

z_{1}^{4} \leq V = \sum_{i = 1}^{n} 4 z_{i}^{4} + \frac{1}{2 γ} {\tilde{Θ}}^{2} .

By (47) and $h (t) = {(z_{1} (t), \dots, z_{n} (t), \tilde{Θ} (t))}^{T}$ , we get

E {| z_{1} |}^{4} \leq 4 EV (h (t)) \leq 4 e^{a (t - t_{0})} EV (h (t_{0})) + \frac{4 b}{a} (1 - e^{a (t - t_{0})}) .

Moreover, the tracking error $z_{1} = y (t) - y_{r} (t)$ satisfies

lim_{t \to \infty} E {| y (t) - y_{r} (t) |}^{4} \leq lim_{t \to \infty} E {| z_{1} |}^{4} \leq lim_{t \to \infty} 4 EV (h (t)) \leq \frac{4 b}{a} .

(49)

By opening four roots, we get

lim_{t \to \infty} E | y (t) - y_{r} (t) | \leq lim_{t \to \infty} E | z_{1} | \leq lim_{t \to \infty} {(4 EV (h (t)))}^{\frac{1}{4}} \leq {(\frac{4 b}{a})}^{\frac{1}{4}} .

(50)

From (42), we have

\frac{b}{a} = \sum_{j = 1}^{n} \frac{\bar{p} - 1}{(\bar{p} + 3) a} {(\frac{4 \sqrt[4]{ε}}{c_{j} (\bar{p} + 3)})}^{\frac{4}{\bar{p} - 1}} + \frac{σ}{2 a} Θ^{2} + \frac{δ_{n}}{a} .

(51)

For $p_{i} (t) = 1 .$ Similar to (44)–(49), we have

lim_{t \to \infty} E | z_{1} | \leq {(\frac{4 b}{a})}^{\frac{1}{4}},

(52)

where

\frac{b}{a} = \frac{σ}{2 a} Θ^{2} + \frac{δ_{n}}{a} .

(53)

Although $Θ$ is unknown, we can choose a sufficiently large and $σ, δ_{n}$ sufficiently small in (51) and (53). Then, from (49) and (52), we can conclude that the output tracking error $E | y (t) - y_{r} (t) |$ can be tuned to be arbitrarily small.

Next, we prove conclusion (iii) holds.

From (48) we have

EV (h (t)) \leq EV (h (t_{0})) + \frac{b}{a} .

(54)

Note that

EV (h) \geq \int_{| h | > a} V (h) P (d ω) \geq inf_{| h | > a} V (h) P (| h | > a) .

(55)

Then, by (54) and (55) we have

P (| h | > a) \geq \frac{EV (h (t_{0})) + \frac{b}{a}}{inf_{| h | > a} V (h)} .

which together with the definition of $V (t)$ gives

lim_{a \to \infty} sup_{t > t_{0}} P (| h | > a) \geq lim_{a \to \infty} sup_{t > t_{0}} \frac{EV (h (t_{0})) + \frac{b}{a}}{inf_{| h | > a} V (h)} = 0 .

(56)

By (56), $h (t)$ is bounded in probability, which also indicates that $z_{1} (t), \dots, z_{n} (t)$ , $\tilde{Θ} (t)$ are bounded in probability. From $z_{1} = x_{1} - y_{r}$ and the boundedness of $y_{r}$ , we can conclude that $x_{1}$ is bounded in probability. From $z_{2} = x_{2} - α_{2}$ , we can conclude that $x_{2}$ is bounded in probability. Similarly, we can prove that $x_{j}, j = 3, \dots, n$ are bounded in probability. Therefore, all the states of the closed-loop system are bounded in probability. The proof is thus completed.

A simulation example

Consider the following system

\begin{matrix} d x_{1} = ({[x_{2}]}^{\frac{8}{7} + \frac{1}{7} \sin t} + \frac{1}{4} θ x_{1}^{\frac{7}{5}}) d t + \frac{1}{5} θ x_{1} \sin^{2} x_{1} d ω, \\ d x_{2} = [u]^{\frac{6}{5} + \frac{1}{5} \sin t} d t + θ x_{1} \sin^{2} x_{2} d ω, \\ y = x_{1}, \end{matrix}

(57)

where $\bar{p} = \frac{7}{5}$ , $\underline{p} = 1$ , $θ$ is an unknown parameter. The output tracking signal is chosen as $y_{r} (t) = \sin (\frac{1}{5} t)$ , $θ = \frac{1}{5}$ . Clearly, system (57) satisfies Assumptions 1–3. Defining $Θ = {(1 + θ^{2})}^{\frac{11}{10}}$ , by following the design procedure proposed in the third section, we can derive the adaptive controller as follows

\begin{matrix} u = - (z_{2} + {[z_{2}]}^{\frac{7}{5}}) (3 + ρ_{2} \sqrt{1 + {\hat{Θ}}^{2}} + H_{2} + φ_{20}), \\ \overset{\cdot}{\hat{Θ}} = z_{1}^{\frac{22}{5}} ρ_{1} + z_{2}^{\frac{22}{5}} ρ_{2} - γ σ \hat{Θ}, \end{matrix}

(58)

where

\begin{matrix} z_{1} = - \sin (\frac{1}{5} t), \\ z_{2} = x_{2} + z_{1} β_{1}, \\ ρ_{1} = \frac{3}{2} (2 + \frac{1}{4} {x_{1}}^{\frac{7}{5}}) + \frac{7}{2} {(1 + {x_{1}}^{6})}^{\frac{11}{5}}, \\ ρ_{2} = \frac{29}{25} {(1 + γ_{21}^{2})}^{\frac{7}{10}} + \frac{7}{5} {(1 + γ_{22}^{2})}^{\frac{11}{10}}, \\ β_{1} = 3 + \sqrt{1 + {\hat{Θ}}^{2}} ρ_{1}, \\ γ_{21} = | \frac{\partial α_{2}}{\partial x_{1}} | (| x_{2} | + {| x_{2} |}^{\frac{7}{5}} + \frac{1}{4} {| x_{1} |}^{\frac{7}{5}}) + (1 + \frac{7}{5} {z_{1}}^{\frac{2}{5}}) \\ β_{1} + \frac{1}{2} | \frac{\partial^{2} α_{2}}{\partial {x_{1}}^{2}} | {x_{1}}^{6}, \\ γ_{22} = {({| x_{1} |}^{3} + {| x_{2} |}^{3} + | \frac{\partial α_{2}}{\partial x_{1}} | {| x_{1} |}^{3})}^{2}, \\ H_{2} = \frac{3}{2} {(1 + 10 {({| z_{1} |}^{\frac{27}{5}} + {| z_{1} |}^{\frac{29}{5}})}^{2} \frac{2 {\hat{Θ}}^{2}}{1 + {\hat{Θ}}^{2}} ρ_{1}^{2})}^{\frac{7}{10}} \\ + {(1 + 10 {({| z_{1} |}^{4} + {| z_{1} |}^{\frac{22}{5}})}^{2} \frac{2 {\hat{Θ}}^{2}}{1 + {\hat{Θ}}^{2}} ρ_{1}^{2} ρ_{2}^{2})}^{\frac{1}{2}} \\ + 2 {(1 + {(| z_{1} | + {| z_{1} |}^{\frac{7}{5}})}^{2} \frac{{\hat{Θ}}^{4}}{1 + {\hat{Θ}}^{2}} ρ_{1}^{2})}^{\frac{7}{10}}, \\ φ_{20} = 8 + 10 {({| z_{1} |}^{3} + ({| z_{1} |}^{\frac{17}{5}} + {| z_{1} |}^{3}) {(1 + {| z_{1} |}^{\frac{2}{5}})}^{\frac{2}{5}} β_{1}^{\frac{2}{5}})}^{\frac{22}{5}} . \end{matrix}

(59)

In the practical simulation, we choose $c_{1} = c_{2} = 2,$ $δ_{11} = δ_{24} = δ_{25} = 0.01,$ $δ_{12} = δ_{23} = 0.04,$ $δ_{21} = δ_{22} = 0.1,$ $γ = 10,$ $σ = 0.1,$ and the initial values $x_{1} (0) = 0.23$ , $x_{2} (0) = - 1.92$ , $\hat{Θ} (0) = 1$ . Figure 1 gives the responses of (57)–(59), from which, the effectiveness of the proposed algorithm is demonstrated.

Conclusions

In this paper, the adaptive output tracking control is studied for high-order stochastic nonlinear systems with unknown time-varying powers and parameters. The designed adaptive output tracking controller can ensure that the output tracking error can be tuned arbitrarily small while all the states of the closed-loop system remain bounded in probability.

There are some related problems to be investigated; for example, how to generalize the results in this paper to more general systems (Li and Krstic 2021a, 2021b, 2021c; Li et al., 2020a, 2020c, 2021).

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 is supported by National Natural Science Foundation of China (No. 61973150), the Young Taishan Scholars Program of Shandong Province of China (No. tsqn20161043), Shandong Provincial Natural Science Foundation for Distinguished Young Scholars (No. ZR2019JQ22), Shandong Province Higher Educational Excellent Youth Innovation Team (No. 2019KJN017), and Foundation of Shandong Educational Committee (No. J17KA051).

ORCID iD

Wuquan Li

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